Embed Size (px)
Citation preview
Fatigue design of lattice materials
and
application to stent-like structures
by
Ehsan Masoumi Khalil Abad
Department of Mechanical Engineering
McGill University, Montreal
December 2012
A thesis submitted to McGill University in partial fulfilment
of the requirements for the degree of doctor of philosophy
© Ehsan Masoumi Khalil Abad, 2012
I
To my parents
II
A lattice material is a cellular structure with a periodic arrangement of cells in either two
or three dimensions. Lattice materials are attractive candidates for potential use in a broad
range of applications, including battery electrodes, vibration insulators, ultra lightweight
sandwich panels, and biomedical implants. This thesis focuses on the design of planar
lattices for micro-architectured materials and medical devices.
Strength of a lattice material degrades under cyclic loading conditions. In this thesis a
computational method based on finite element analysis (FEM) is proposed to analyze and
design lattice materials and structures for fatigue failure. A comparison with available
experimental data contributes to the validity of the method. The effect of the unit cell’s
architecture on the fatigue resistance of lattice materials is investigated by considering
square and hexagonal shapes of unit cells.
A shape optimization methodology based on removing the stress concentration caused by
the presence of geometrical discontinuities at the inner boundaries of the lattice cell walls
is proposed to improve the fatigue resistance of planar lattice materials.
The shape optimization method adapted for the fatigue design of a lattice is applied to
design intravascular self-expandable characterized by a periodic arrangement of cells,
against fatigue failure. In particular, the aim is to improve the fatigue resistance of Nitinol
stent grafts with closed-cell, and to design a stent-like device functioning as a protection
for an endovascular oxygenator. A parametric study was carried out to assess the effect of
different geometrical parameters on the fatigue resistance and radial stiffness of the
generated Nitinol stent lattices. Novel stent-like concepts are proposed to protect and
guide the state-of-the art intravenous oxygenator developed by ALung Technologies Inc.
(Pittsburgh, PA) in partnership with the University of Pittsburgh. The validity of the
proposed concepts in protecting the oxygenator was tested in vitro. The structural
behavior of the proposed conceptual designs was studied by using FEM, and the level of
blood damage caused by catheter rotation is investigated through CFD analysis.
Preliminary numerical and experimental observations suggest that the proposed design
can put the oxygenator one step closer to the market.
Abstract
III
Keywords: Lattice materials, Fatigue failure, Shape optimization, Selef expandable stent,
Nitinol, Protective cage, Percutaneous respiratory assist device.
IV
Un matériau en treillis est une structure cellulaire avec une disposition périodique de
cellules en deux ou en trois dimensions. Ces structures sont utilisées dans plusieurs
applications, y compris les électrodes de la batterie, isolateurs de vibration,
panneaux ultra légers en sandwich et implants biomédicaux. Cette thèse met l'accent sur
la conception de réseaux plans pour des matériaux ayant une microarchitecture et pour les
dispositifs médicaux.
Dans plusieurs applications, la résistance d'un matériau en treillis se dégrade dans les
conditions de chargement cycliques. Dans cette thèse une méthode numérique basée sur
la mécanique de calcul est proposé afin d'analyser et de concevoir des matériaux et des
structures en treillis pour prévenir toute rupture causée par fatigue.
Une comparaison avec des données expérimentales contribue à la validité de la méthode.
L'effet de l'architecture d'une cellule de cette unité sur la tenue en fatigue des matériaux
en treillis est étudiée en tenant compte des formes carrées et hexagonales de cellules
unitaires.
En outre, une méthodologie d'optimisation de forme fondé sur l'élimination de la
concentration du stress causé par la présence de discontinuités géométriques aux
frontières intérieures des parois cellulaires en treillis est proposé pour améliorer la
résistance à la fatigue des matériaux en treillis planaires. Plusieurs topologies de
cellules augmentant la résistance à la fatigue sont proposées pour l'amélioration des
matériaux et des structures caractérisées par un arrangement périodique de cellules.
Cette méthode d'optimisation de forme adaptée pour la conception de fatigue d'un réseau
de cellule est appliquée à la conception intravasculaire d’endoprothèses auto-expansibles
et aussi à la conception d’un dispositif fonctionnant comme stent offrant une protection
pour un oxygénateur endovasculaire.
Une géométrie de la cellule avec une meilleure résistance à la fatigue est proposée pour
un réseau planaire pour stent.
Une étude paramétrique a été réalisée pour évaluer l'effet des différents paramètres
géométriques sur la résistance à la fatigue et la raideur radiale des réseaux générés de
stent.
Plusieurs concepts nouveau empruntent du stent sont proposées pour protéger et guider
un oxygénateur intraveineux mis au point par Technologies Inc. Alung (Pittsburgh, PA),
en partenariat avec l'Université de Pittsburgh. La validité des concepts proposés assurant
une protection de l'oxygénateur a été testée in vitro. Le comportement de la structure des
conceptions proposées conceptuels a été étudié en utilisant la méthode des éléments finis
tandis que et le niveau de dommages de sang causé par la rotation du cathéter a était
évaluer à travers une modélisation numérique et dynamique des fluides. Les observations
RÉSUMÉ
V
numériques et expérimentales suggèrent que la conception proposée mettrait
l'oxygénateur un pas de plus vers le marché.
Mots-clés: matériaux en treillis, la rupture par fatigue, optimisation de forme, stent auto-
expansible, Nitinol, cage de protection, dispositif d'assistance respiratoires par voie
percutanée.
VI
My PhD career was not just the challenge of doing serious research but was also an
opportunity for me to better shape my vision of the worlds in which I am involved. Many
people have contributed openly to this. I would like here to express my sincere gratitude
to my PhD supervisor, Professor Damiano Pasini, for his research supervision, patience,
technical guidance, encouragement and kindness during the course of my PhD. I shall
always consider him not only as a supervisor but as a teacher and a close friend of mine.
In all our meetings and in each of his emails I could feel his idealistic push toward that
which is the best.
I would like to give special thanks to Dr. Renzo Cecere for his support and guidance with
respect to the medical side of my project. I am also grateful for having the opportunity to
audit several of his open-heart surgeries, which were very useful to gain insight into the
medical and clinical requirements. Dr. Cecere was also a constant source of
encouragement and hope in all of the tough times of my PhD.
I would like also to thank Dr. Oren Steinmetz for providing me the chance to audit
several revascularization surgeries. There are many other individuals who have helped
me during my PhD. Among them, I would like to thank, Dr. Richard Reid Cooper,
Farhad Javid, Eric Lavoie, Toufic Azar, Hoang Tran, Mostafa Elsayed, Mario Iacobaccio,
Mehdi Sanjari, Ali Mosahebi and Sajad Arabnejad Khanoki for their assistance, support,
advice, and kindnesses. I would also like to express my special thanks to my family for
being always the best of friends, with their hands consistently full of support,
encouragement, and patience extended toward me.
I would like to thank the National Sciences and Engineering Research Council of Canada
(NSERC) for their financial support, the Faculty of Engineering of McGill University for
their academic contributions, and the Hydro-Quebec Research Institute in Varennes for
their help with my experimental studies.
Acknowledgements
VII
The author claims the originality of the main ideas and research results reported in this
thesis, the most significant being listed below:
The development of a computational method to design lattice materials for fatigue
resistance.
The novel cell geometry with improved fatigue resistance for self-expandable Nitinol
stent-grafts.
The design of the encasement to protect and guide the state-of-the art intravenous
oxygenator developed by ALung Technologies Inc.
The shrinking mechanism to retract and re-position the proposed protective
encasement for the intravenous oxygenator developed by ALung Technologies Inc.
Claims of originality
VIII
Refereed Journal Papers
1. Ehsan Masoumi Khalil Abad, Damiano Pasini, Renzo Cecere, “Shape optimization
of stress concentration-free lattice for self-expandable Nitinol stent-grafts”, Journal of
Biomechanics, Vol. 45, 6, pp. 1028-1035, 2012.
2. Ehsan Masoumi Khalil Abad, Sajad Arabnejad Khanoki, Damiano Pasini, “Fatigue
design of lattice materials via computational mechanics: application to lattices with
smooth transitions in cell geometry”, International Journal of Fatigue, vol 47, pp.
126-136, 2013.
Refereed Conference Papers
1. Ehsan Masoumi Khalil Abad, Damiano Pasini, Renzo Cecere, “Design optimization of a
lattice structural cage for the protection of a rotary endovascular catheter”, CSME
2010 conference, Victoria, British Columbia, Canada, June 7-9, 2010.
2. Ehsan Masoumi Khalil Abad, Sajad Arabnejad Khanoli, Damiano Pasini, “Shape
design of periodic cellular materials under cyclic loading”, ASME 2011 International
Design Engineering Technical Conferences& Computers and Information in
Engineering Conference IDETC/CIEC 2011, Washington, DC, USA, August 28-31,
2011.
3. Sajad Arabnejad Khanoki, Ehsan Masoumi Khalil Abad, Damiano Pasini, “Synthesis
of two-dimensional lattices free of stress concentrators”, 8th
European Solid
Mechanics Conference, ESMC-2012, Graz, Austria, July 9-13, 2012.
Publications and inventions arising from this thesis
IX
US and Canadian Patents
Damiano Pasini, Ehsan Masoumi Khalil Abad, “Stent devices made of a lattice with
smooth shape cells improving stent fatigue life”, United States patent application No.
13/659,398, 2012.
Damiano Pasini, Ehsan Masoumi Khalil Abad, “Stent devices made of a lattice with
smooth shape cells improving stent fatigue life”, Canada patent application No.
2,793,650, 2102.
Damiano Pasini, Ehsan Masoumi Khalil Abad, Renzo Cecere, “A Protective Cage For
An Intravenous Respiratory Catheter”, Provisional US patent No. 61706428, 2012.
X
Abstract ............................................................................................................................... II
RÉSUMÉ .......................................................................................................................... IV
Acknowledgements ........................................................................................................... VI
Claims of originality ........................................................................................................ VII
Publications and inventions arising from this thesis ...................................................... VIII
Table of Contents ............................................................................................................... X
List of Figures ................................................................................................................ XIV
List of Tables ............................................................................................................... XVIII
Chapter 1 ............................................................................................................................. 1
Introduction ......................................................................................................................... 1
1.1. Background and Motivation ................................................................................... 1
1.1.1. Stent-like lattice devices ................................................................................. 2
1.1.2. Fatigue failure ................................................................................................. 4
1.2. Open research issues ............................................................................................... 4
1.3. Objectives ............................................................................................................... 6
1.4. Thesis outline .......................................................................................................... 6
Chapter 2 ............................................................................................................................. 8
Literature review ................................................................................................................. 8
2.1. Objectives ............................................................................................................... 8
2.2. Introduction to the failure behavior of cellular materials ....................................... 8
2.2.1. Failure modes of the cell walls ....................................................................... 9
2.2.1.1. Cell wall buckling ................................................................................... 10
2.2.1.2. Cell wall fracture .................................................................................... 10
2.3. The role of the cell wall in the strength of planar lattice materials ....................... 14
2.4. Life-time strength degradation of cellular materials ............................................. 16
2.4.1. Fatigue loading.............................................................................................. 17
2.5. Effective mechanical properties ............................................................................ 20
2.5.1. Structural analysis ..................................................................................... 21
2.5.2. Micropolar theory ...................................................................................... 22
2.5.3. Homogenization by Bloch Theorem and Cauchy-Born hypothesis .......... 23
2.5.4. Asymptotic homogenization method ........................................................ 23
2.6. Failure surfaces ..................................................................................................... 26
2.7. Concluding remarks emerging from the literature ................................................ 28
Chapter 3 ........................................................................................................................... 30
A computational method for the design of lattice materials for fatigue resistance .......... 30
3.1. Objectives ............................................................................................................. 30
3.2. Terms and definitions ........................................................................................... 30
3.2.1. Characterization of materials ........................................................................ 32
Table of Contents
XI
3.2.1.1. Stress-life approach................................................................................. 32
3.2.1.2. Fatigue crack growth in a notched specimen .......................................... 35
3.3. Fatigue design of planar lattice ............................................................................. 35
3.3.1. Basics and assumptions................................................................................. 35
3.3.2. Cell geometries under investigation.............................................................. 37
3.3.3. Numerical Modeling ..................................................................................... 38
3.4. Results ................................................................................................................... 40
3.4.1. Stress distribution in the unit cell.................................................................. 40
3.4.2. Failure surfaces and experimental validation ............................................... 42
3.4.3. Modified Goodman diagrams ....................................................................... 44
3.5. Summary and contributions to knowledge ........................................................... 45
Chapter 4 ........................................................................................................................... 48
Shape optimization of lattice materials for fatigue resistance .......................................... 48
4.1. Objective ............................................................................................................... 48
4.2. Design methodology: Basics................................................................................. 48
4.2.1. Geometrical stress concentration .................................................................. 48
4.2.2. Mathematical formulation of the optimization problem ............................... 50
4.2.3. Cell geometries under investigation.............................................................. 53
4.3. Results ................................................................................................................... 54
4.4. Discussion ............................................................................................................. 57
4.5. Concluding remarks .............................................................................................. 60
Chapter 5 ........................................................................................................................... 61
Shape optimization of stress concentration-free lattice for self-expandable Nitinol stent-
grafts ................................................................................................................................. 61
5.1. Objectives ............................................................................................................. 61
5.2. Introduction to structural design of stents ............................................................. 61
5.3. Problem statement ................................................................................................. 63
5.4. Shape synthesis of lattice geometry ...................................................................... 64
5.4.1. Numerical modeling...................................................................................... 66
5.4.1.1. Finite element modeling ......................................................................... 66
5.4.1.2. Material model ........................................................................................ 67
5.4.1.3. Loading conditions ................................................................................. 68
5.5. Results ............................................................................................................... 68
5.6. Concluding remarks .......................................................................................... 73
Chapter 6 ........................................................................................................................... 75
Structural design of a protective cage for a rotating intravenous oxygenator .................. 75
6.1. Objectives ............................................................................................................. 75
6.2. Background, motivation, and problem statement ................................................. 76
6.2.1. Lung diseases, statistics, and available treatments ....................................... 76
6.2.2. Percutaneous Respiratory Assist Catheter (PRAC) with rotary bundle........ 77
6.2.3. Problem definition ........................................................................................ 78
6.2.4. Blood damage investigation of blood-contacting medical devices............... 79
XII
6.3. Conceptual design of the protective cages for the PRAC with rotary bundle ...... 80
6.3.1. Cage design I: 2008-2010 ............................................................................. 80
6.3.1.2. Shape optimization of the exterior wall .................................................. 81
6.3.2. Cage design II: 2010-2012 ............................................................................ 84
6.3.2.1. In vitro bench test of the second design of the cage in curved paths ...... 86
6.3.2.2. Shape optimization of the exterior wall .................................................. 87
6.3.3. Retraction/Removal mechanism ................................................................... 87
6.3.4. Hematologic design ...................................................................................... 88
6.3.4.1. Numerical modeling................................................................................ 89
a. Mesh generation ......................................................................................... 90
b. Turbulence modeling .................................................................................. 91
c. Boundary Conditions ...................................................................................... 94
d. Discrete Particle Modeling (DPM) ............................................................. 94
6.3.4.2. Results ..................................................................................................... 94
a. Turbulent model and Mesh-independency investigation ............................ 94
b. Blood flow features .................................................................................... 96
c. Evaluation of blood damage caused by the rotation of the oxygenator .... 100
d. Platelet activation ..................................................................................... 103
6.3.4.3. Discussion ............................................................................................. 103
Chapter 7 ......................................................................................................................... 106
Conclusions and future works ......................................................................................... 106
7.1. Conclusions ......................................................................................................... 106
7.2. Directions for future research ............................................................................. 107
7.2.1. Numerical method based on computational mechanics to design lattice
materials against fatigue failure. ............................................................................. 107
7.2.2. Geometrical design of the unit cell of lattice materials for fatigue resistance
108
7.2.3. Design of Nitinol self-expandable stent lattices against fatigue ................. 109
7.2.4. Design of protective cage for the PRAC with rotary bundle ...................... 109
Appendix A ..................................................................................................................... 111
A.1. The matrix operators of the shape optimization approach presented in chapter 4 .. 111
Appendix B ..................................................................................................................... 112
B.1. Principles of gas exchange in blood oxygenators ................................................... 112
Appendix C ..................................................................................................................... 116
Blood damage modeling ................................................................................................. 116
C.1. Blood ................................................................................................................... 116
C.2. Methods of blood damage investigation ............................................................. 116
C.3. Blood damage types ............................................................................................ 117
C.3.1. Thrombosis ...................................................................................................... 117
C. 3.2. Hemolysis: ...................................................................................................... 119
a. Threshold Value Approach ....................................................................... 122
b. Mass-Weighted Average Approach ......................................................... 122
c. Eulerian Approach .................................................................................... 123
d. Lagrangian Approach ............................................................................... 124
XIII
References ....................................................................................................................... 125
XIV
Figure 1.1. (a) Stent lattice consisting of three rows of unit cells; (b) individual row
consisting of 20 diamond-shaped unit cells; (c) sharp-corner diamond unit cell, used in
commercially available stents; (d) rounded diamond unit cell; (e) superelliptical unit cell
(Masoumi Khalil Abad and Pasini, 2013)........................................................................... 2
Figure 2.1. Schematic view of hexagonal lattice under biaxial compressive loading
Adapted from (Gibson et al. 1989). .................................................................................. 11
Figure 2.2. Failure surface of a 2D hexagonal lattice (Gibson et al. 1989). .................... 12
Figure 2.3. (a) Biaxial stress state acting on a hexagonal unit cell. (b) free body diagram
of the inclined member. (c) the model of plastic collapse indicating the place of plastic
hinges formed under bending (Adapted from Gibson et al. 1989). .................................. 12
Figure 2.4. Homogenization concept of a cellular structure (Masoumi et al. 2011). ...... 21
Figure 2.5. Periodic boundary conditions for a pair of nodes located on the opposite
surfaces, A and A , of the RVE. .................................................................................. 24
Figure 2.6. Flowchart of the asymptotic homogenization theory to obtain the effective
strength properties of a lattice material. ............................................................................ 28
Figure 3.1. (a) Schematic view of a cyclic load with constant stress amplitude; (b)
Schematic view of a stress-life curve................................................................................ 33
Figure 3.2. (a) Schematic view of the fatigue design diagram showing the effect of
allowable alternating stress versus mean stress for a given fatigue life; (b) Schematic
view of the logarithmic rate of fatigue crack growth versus logarithm of the amplitude of
stress intensity. .................................................................................................................. 34
Figure 3.3. Schematic views of: (a) G1 square unit cell; (b) G
1 hexagonal unit cell. ...... 39
Figure 3.4. Flowchart of the design methodology. For a given cell geometry, shape
synthesis is coupled with computational analysis followed by size optimization. The goal
of the first step is to generate the geometrical model of the unit cell. In the second
module, the effective strength properties of the lattice are determined through asymptotic
homogenization theory. The third step involves the cell size optimization to reduce at
minimum the maximum von Mises stress in the cell wall. ............................................... 39
Figure 3.5. Mesh sensitivity showing the independency of the results from the mesh size.
........................................................................................................................................... 40
Figure 3.6. von Mises stress distribution (MPa) in hexagonal and square unit cells made
out of Ti-6Al-4V. Lattices under fully reversed uni-axial loading defined by: G1 cell with
small arc (left); optimum G1 cell (right). .......................................................................... 41
Figure 3.7. von Mises stress distribution (MPa) in hexagonal and square unit cells made
out of Ti-6Al-4V. Lattices under fully reversed in-plane pure shear loading defined by:
G1 cell with small arc (left); optimum G
1 cell (right). ...................................................... 41
Figure 3.8. Yield and ultimate surfaces of Ti-6Al-4V square and hexagonal unit cells for
relative density of 10%. Projection of yield and ultimate surfaces of G1 square and
hexagonal cells with small and optimum arc radii in the yy xx
and xy xx
planes. 42
Figure 3.9. Effective yield strength of the square and hexagonal unit cells under uni-axial
and shear loading as a function of relative density. Yield strength for square cell under
List of Figures
XV
uni-axial (a) and shear loading respectively (b); (c) and (d) pertain to the hexagonal cell.
........................................................................................................................................... 44
Figure 3.10. Modified Goodman diagram for Ti-6Al-4V square and hexagonal unit cells
at given relative densities. G1
square under uni-axial loading (a), and shear loading
condition (b); G1 hexagon under uni-axial loading (c), and shear loading (d). ................ 46
Figure 4.1. Schematic views of: (a) G2 continuous square cell; (b) G
2 continuous
hexagonal cell; (c) Parameterization of the inner profile of a unit cell portion. ............... 50
Figure 4.2. von Mises stress (MPa) distribution in hexagonal and square unit cells made
out of Ti-6Al-4V. Lattices under fully reversed uni-axial loading, defined by: optimum
G1 cell (left) and optimum G
2 cell (right). ........................................................................ 54
Figure 4.3. von Mises stress (MPa) distribution in hexagonal and square unit cells made
out of Ti-6Al-4V. Lattices under fully reversed pure shear loading, defined by: optimum
G1 cell (left) and optimum G
2 cell (right). ........................................................................ 55
Figure 4.4. Yield and ultimate surfaces of Ti-6Al-4V square and hexagonal unit cells for
relative density of 10%. Projection of yield and ultimate surfaces of optimum G1 and G
2
square and hexagonal cells in the yyxx and xyxx planes. ..................................... 56
Figure 4.5. Effective yield strength of the square and hexagonal unit cells under uni-axial
and shear loading as a function of relative density. Yield strength for square cell under
uni-axial (a) and shear loading respectively (b); (c) and (d) pertain to the hexagonal cell.
........................................................................................................................................... 57
Figure 4.6. Modified Goodman diagram for Ti-6Al-4V square and hexagonal unit cells at
given relative densities. G1
and G2 square under uni-axial loading (a), and shear loading
condition (b); G1 and G
2 hexagon under uni-axial loading (c), and shear loading (d). .... 59
Figure 5.1. Commercially available stents developed for prescribed applications
(Masoumi Khalil Abad et al., 2012). ................................................................................ 62
Figure 5.2. Schematic view of Nitinol stress-strain curve. .............................................. 65
Figure 5.3. Schematic view of the proposed G2-continuous cell geometry: (a) the
proposed E cell geometry; (b) parameterization required for the synthesis of a G2-
continuous cell shape; (c) inner boundaries of initial design and structurally optimized E
cell. .................................................................................................................................... 66
Figure 5.4. Structurally optimum stent. (a) a straight row of lattice cells, (b) a row folded
into a cylinder. .................................................................................................................. 70
Figure 5.5. FEA results for E cell. (a) Strain distribution in the shrunk stent; (b) first
principal strain in the stent after stent deployment under 100 mm-Hg mean pressure.; (c)
von Mises stress (in MPa) distribution in the artery after stent deployment under 100 mm-
Hg mean pressure. The maximum value occurs at the interface between stent and artery
wall. ................................................................................................................................... 70
Figure 5.7. Plots of number of cells in the circumferential and radial direction, thickness
and width of cell elements versus radial force, fatigue safety factor, and metal area in
contact with artery for E cell geometry. (a-c) effect of nc for , nl = 10,t = 0.28mm, w =
0.45mm (d-f) effect of nl for t = 0.28mm, w = 0.45mm , nc = 8; (g-i) effect of t for, w =
0.45mm , nl = 10, nc = 8; (j-l) effect of w for, t = 0.28mm , nl = 10, nc = 8 for E cell
geometries. R stent is a benchmark stent design (Kleinstreuer et al., 2008b); its design
parameters are nc = 20, nl = 10, t = 0.28mm, w = 0.35mm. ............................................. 72
Figure 6.1. Rotary catheter with rigid internal shaft ........................................................ 79
XVI
Figure 6.2. Initial design of the protective cage. (a) Front and top views; (b) 3D view of
the cage and its supporting ring. ....................................................................................... 81
Figure 6.3. Experimental set-up for concept evaluation of the cage in a straight tube. ... 82
Figure 6.4. (a) 3D geometry of the proposed lattice; (b) portion of 2D lattice mesh of the
lateral surface of the cage; (c) parameterized quarter of the lattice unit cell. ................... 83
Figure 6.5. (a) von Mises strain distribution in the compressed cage design I; (b) radial
supportive force versus inner radius of the cage design I. ................................................ 84
Figure 6.6. Second design of protective cage. ................................................................. 85
Figure 6.7. Lattice cage on a curved geometry. The clips linking the series of lattice
hoops allow the cage to adjust to change in the vein geometry. ....................................... 85
Figure 6.8. Experimental set-up for concept evaluation of the cage in a curved path; (a)
test set-up and one of its adjustable pillars; (b) and (c) two curved paths of the the cage
design II that can successfully guide the rotary catheter bundle. ..................................... 86
Figure 6.9. (a) von Mises strain distribution in the outer wall of the compressed cage
design II; (b) radial supportive force versus the inner radius of the cage design II. ......... 87
Figure 6.10. Shrinking mechanism consisting of handle, guiding rings and threads to
shrink the lattice tubular cage. .......................................................................................... 88
Figure 6.11. 3D model of the blood flow in the reference, or R version (left) and cage-
supported, or C version (right) of the rotary oxygenator. ................................................. 90
Figure 6.12. Discretized model of the blood volume flowing in the R and C versions of
the rotary oxygenator; (a) 3D view of the meshed model of the R version. (b) The refined
mesh near the boundaries of the R version was obtained by using boundary adaption
method. (c) and (d) 3D and side views of the discretized model of the blood volume
flowing in C version of the rotary oxygenator meshed with tetrahedron elements. ......... 92
Figure 6.13. (a) and (b) 3D and side views of the CAD model of the blood flow in the C
version of the oxygenator obtained by dividing the blood volume into primitive and
irregular volumes; (c) 3D view of the meshed model; (d) the refined mesh near the
boundary walls of the C version obtained by using the boundary adaption method. ....... 92
Figure 6.14. Taylor vortices pattern in rendered views of the velocity distribution in the
R (up) and C (down) versions of the rotary oxygenator. .................................................. 97
Figure 6.15. Velocity distribution in r z and r planes of the (a) R and (b) C
versions of the rotary oxygenator. .................................................................................... 98
Figure 6.16. Shear stress distribution in the r z and r planes of the (a) R and (b) C
versions of the rotary oxygenator. .................................................................................... 98
Figure 6.17. Total Pressure distribution in the r z and r planes of the (a) R and (b)
C versions of the rotary oxygenator. ................................................................................. 99
Figure 6.18. Path lines of the particles released from the inlet surface(s) of the (a) R and
(b) C versions of the rotary oxygenator. ........................................................................... 99
Figure 6.19. The maximum shear stress and average residence time of R and C versions
of the rotary catheter (solid red circle) on the chart developed based on published
threshold values for hemolysis blood damage adapted from (Day et al., 2006). ............ 101
Figure 6.20. Results of particle tracking in C version of rotary catheter; (a) and (b)
variations of shear stress and velocity versus residence time of a completed particle
released from the inlet surface(s) of figure 6.14; (c) and (d) variations of shear stress and
velocity versus residence time of a completed particle released from the inlet surface(s)
of figures 6.14. ................................................................................................................ 102
XVII
Figure B. 1. (a) Commercially available membrane oxygenator with bundle of hollow
fiber membranes (bottom left), the micro-porous structure of fibers is shown in the
bottom right (b). Intravenous membrane oxygenator with pulsating balloon , or HC.
Pictures are adapted from (Federspiel and Svitek, 2004b) and (Svitek, 2006)............... 113
XVIII
Table 3.1. Material properties of bulk solid materials………………………………….40
Table 3.2. Yield and ultimate strength of G1 unit cells for square and hexagonal
lattices…………………………………………………………………………………....47
Table 3.3. Fatigue to monotonic performance ratio, /e us , for G1 lattices made of Ti-
6Al-4V and Al 6061T6 (material properties in Table 1) at given relative densities……47
Table 4.1. Yield and ultimate strength of the optimum G1 and G
2 unit cells for square and
hexagonal lattices………………………………………………………………………...58
Table 4.2. Fatigue to monotonic performance ratio, /e us , of optimum G1 and G
2
lattices made of Ti-6Al-4V and Al 6061T6 (material properties in Table 1 of chapter 3)
at given relative densities………………………………………………………………...58
Table 5.1. Nitinol material properties (Kleinstreuer et al., 2008b) .……………………68
Table 5.2. Comparison of stent performances. Reference cell, or namely R cell, from
(Kleinstreuer et al., 2008b)…...………………………………………………………….71
Table 6.1. Average shear stress in Pa on the catheter wall of the R version oxygenator
rotating at 7000 rpm for different number of elements and turbulence models…………95
Table 6.2. Numerical and experimental values of average shear stress in Pa on the
catheter wall of the R version of the oxygenator rotating at different rotational
speeds..……………………………………………………………………………...……96
Table 6.3. Results of mesh-sensitivity analysis of the R version of rotary catheter….....96
Table 6.4. Results of mesh-sensitivity analysis of the C version of rotary catheter…....96
Table 6.5. Blood damage caused by R and C versions of the rotary oxygenator spinning
at 7000 rpm……………………………………………………………………………..102
List of Tables
1
Introduction
1.1. Background and Motivation
Engineering cellular materials are hybrid materials obtained by distributing voids in a
solid medium. The voids can have a stochastic architecture (foams) or their shape and
size can follow a tailored pattern (lattice materials). In particular, lattice materials are
defined as periodic cellular materials obtained by tessellating the plane, or the space, with
a 2D or 3D unit cell along independent directions.
Besides being mechanically superior to other materials (Fleck et al., 2010; Schaedler et
al., 2011; Vigliotti and Pasini, 2012a), lattice materials have been shown to be attractive
for their multifunctional properties, e.g. thermal insulation, sound resistance and shock
energy damping; they are thus of potential use in a broad range of applications, including
battery electrodes, catalyst supports, vibration insulators, jet-engine nacelles, ultra
lightweight sandwich panels, and biomedical implants (Banerjee and Bhaskar, 2009;
Gibson et al., 2010; Kumar and McDowell, 2009; Mullen et al., 2009; Murr et al., 2010;
Schaedler et al., 2011). New advancements in manufacturing process, such as rapid
prototyping manufacturing techniques, have offered engineers and material scientists a
precise control over the microstructure of the lattice materials. Schaedler et al. (2011)
reported manufacturing of an ultralight (densities below 1 miligram/cm3) micro-lattice
material obtained by ordering the octahedral unit cells at the nano scale. This is currently
considered the lightest material in the world. Thanks to the tailored distribution of the
solid material at its microstructure, the stiffness of the micro-lattice scales with its
relative density following2. This shows a distinct contrast with the stiffness of
available ultralight foams, e.g. aerogels, that follow the law of 3. In addition, the
synthesized micro-lattice material has a recoverable range of strain of 50% and energy
Chapter 1
2
absorption similar to elastomers. The use of lattice materials is anticipated to grow further
in daily life applications through future advancements in the cost and volume of the
manufacturing processes.
Figure 1.1. (a) Stent lattice consisting of three rows of unit cells; (b) individual row consisting of
20 diamond-shaped unit cells; (c) sharp-corner diamond unit cell, used in commercially available
stents; (d) rounded diamond unit cell; (e) superelliptical unit cell (Masoumi Khalil Abad and
Pasini, 2013).
1.1.1. Stent-like lattice devices
This thesis deals with planar lattice materials and structures. A lattice structure, i.e. a
finite periodic arrangement of unit cells, can be conveniently used to design stent-like
implants, such as vascular and other stents.
Vascular stents (figure 1.1) are permanent tubular scaffolds that support blood vessels
from the inside of the lumen. They are typically used to prevent reclosure, or restenosis,
of a blood vessel following balloon angioplasty, which is a clinical procedure used to
treat several medical conditions, including peripheral artery disease, renal vascular
hypertension, carotid artery disease (leading to stroke), coronary artery disease (leading
to a heart attack), and the narrowing of large arteries and central veins (Schrader and
Beyar, 1998). Over one million vascular devices are implanted annually in patients to
treat the aforementioned cardiovascular diseases (James and Sire, 2010).
3
Vascular stents are also used to deploy and support endovascular grafts, arterial
endoprostheses, and self-expanding heart valve implants (Kleinstreuer et al., 2008a). A
stent-graft (or simply graft) consists of a tubular fabric sutured to a stent; it is commonly
inserted into a vessel, such as the aorta, to create a pressure seal that prevents blood flow
around the stent-grafts into the aneurysm. After insertion, a stent-graft provides a new,
normal-sized lumen that is conducive to blood flow.
In 2009, we exploited the structural characteristics of a planar lattice to design a self-
expandable and retractable protective cage for a state-of-the-art percutaneous respiratory
assist catheter (PRAC) with rotary bundle (Masoumi Khalil Abad et al., 2011). The
oxygenator catheter is designed to be inserted percutaneously from the femoral vein and
to be deployed into the vena cava. After the seven or ten days required for the lungs to
heal, the cage-catheter assembly should be removed from the body. It is expected that the
designed device should contribute to make the rotary oxygenator market-ready.
Stent walls may consist of a planar lattice obtained by the replication of a unit cell along
periodic directions (see Fig. 1.1). Durability is one of the critical design requirements that
a stent should possess to guarantee patient survival. During 10 to 15 years of device use
with an average heart rate of 72 heartbeats per minute, vascular stents and grafts must
withstand 400 to 600 million loading cycles. As a result of repeated cyclic loading,
fatigue fracture can occur and subsequently cause restenosis, thrombosis, perforation of
the blood vessel, or, in the case of grafts, aneurysm rupture. Durability of stent-grafts can
be further compromised as a result of fabric erosion and suture breakage, which also
leads to aneurysm rupture. Another critical mechanical function of stents is sustaining
anchorage to the blood vessel to prevent device migration. Moreover, the stent must be
able to resist collapse under the external pressure of the blood vessel while ensuring that
the blood vessel remains unscathed.
The shape and topology of the unit cell of stent lattices can be designed to improve its
fatigue resistance and to increase the compressibility of the device required for its
percutaneous deployment via a catheter. Size and thickness of a lattice unit cell are also
4
geometric variables that can be tailored to improve the mechanical performance and
fatigue fracture resistance of stents.
1.1.2. Fatigue failure
In several applications, the load applied to a lattice material or structure is far from being
static. The recurring waves on a ship’s hull, the aero-acoustic excitation of a turbine
engine, and the rhythmic forces acting on an orthopedic implant during the swing phase
of walking are all classical examples of time-dependent loads (Côté et al., 2006; Côté et
al., 2007a; Côté et al., 2007b; Fleck and Eifler, 2010; Fleck and Qiu, 2007; Fleck and
Quintana-Alonso, 2009). A cyclic load generally has a detrimental impact on the
resistance of lattice materials. Factors that govern fatigue failure are not limited to the
alternating and mean stresses, load frequency, environment conditions, and macroscopic
form of the structure, but also include the geometry of the microstructure, i.e. the
geometry in which the unit cell of the lattice is shaped. If geometric discontinuities are
embedded at either the macro or the micro scale or both, then a severe drop in fatigue
resistance is observed. For example, crack propagation in cellular materials can be
tailored by using a designed pattern of thickness distribution among different walls of
each unit cell (Lipperman et al., 2009; Simone and Gibson, 1998).
1.2. Open research issues
There are several open research issues within the scope of the fatigue design and shape
synthesis of lattice materials that need to be addressed. The following summarizes the
research topics that this thesis aims to address.
1) Computational fatigue design of lattice materials considering a realistic microscopic
stress distribution in cell walls. As the state of the art survey in the second chapter of
the present thesis illustrates, the fatigue failure of cellular materials, and in particular
lattice materials, has received less attention than their monotonic quasi-static and
dynamic properties (Banerjee and Bhaskar, 2005; Banerjee and Bhaskar, 2009;
Gibson and Ashby, 1999b; Masters and Evans, 1996; Ruzzene, 2004; Schraad and
Harlow, 2006; Wang and McDowell, 2004a, 2005). From the literature on lattice
5
materials, it appears that most of the methods for fatigue design rely on experiments
that are tailored to handle selected lattice topologies and materials; they are time
consuming and often expensive. Theoretical approaches, on the other hand, seem to
lack accuracy since they may fail to capture the real stress distribution in the lattice
cells. Developing a computational methodology that can predict the fatigue behavior
of lattice materials considering a realistic stress distribution in the cells of the lattice
is a topic of interest that has not yet been fully addressed.
2) Geometrical design of the architecture of the lattice cells for fatigue. Besides the
fatigue behavior of their constituent solid material, fatigue resistance of lattice
materials is controlled by the shape and size of their unit cells. However, to date there
is a limited number of studies aimed at improving the fatigue resistance of lattice
materials by tailoring the geometrical architecture of their unit cells. The
implementation of established structural optimization strategies for the fatigue design
of the micro-architecture of the lattice materials is an open research topic.
3) Application of the design method (point 2) developed for lattice materials to improve
the fatigue resistance of stents. Since a stent may be considered as a planar lattice
structure, its structural properties can be tailored to attain desired structural functions
and performance. The shape optimization of a planar lattice cell to improve the
durability of stents is an open research issue.
4) Design of a lattice cage for the protection of an intravenous oxygenator with rotary
fiber bundle. The rotary oxygenator developed by ALung Technologies Inc. is a very
promising alternative to current mechanical ventilation devices for treating patients
with acute and chronic respiratory diseases. Technical complications resulting from
the rotation of the catheter at high speeds inside the vena cava have recently halted
the further development of this oxygenator. The conceptual design and optimization
of a protective cage that can guide the rotation of the catheter is an open research
issue that has not been satisfactorily addressed yet.
6
1.3. Objectives
The goals of this thesis are to:
1. Develop a computational method for the fatigue design of lattice materials that can
capture accurately the stress distribution in the unit cells of the lattice.
2. Tailor a design method to optimize the unit cell geometry of a lattice for fatigue.
3. Apply the fatigue optimization method developed in point 2 to improve the fatigue
strength of self-expandable stent grafts.
4. Design and optimize the protective cage made of a lattice for PRAC with rotary
bundle.
1.4. Thesis outline
After the introduction, a literature review describes the basics of structural analysis to
model the monotonic and fatigue resistance of lattice materials. The various failure
modes of cell walls and their role in the strength of planar lattices are described, and the
roles of cell walls on the mechanical properties of lattice materials are explained. The
literature of the fatigue of cellular materials is reviewed. The chapter continues with
explaining the computational methods developed to obtain the homogenized properties of
lattice materials. It concludes with the open research areas that motivate the rest of this
thesis.
5. In chapter 3 a computational method to design planar lattice materials against fatigue
failure is introduced. The proposed method is validated with the experimental results
available in literature. Two cell geometries are selected and their fatigue behavior is
investigated using the method proposed in this thesis. It should be noted that in
chapter 3 we consider fatigue resistance of lattice materials in general and do not
target a specific application, such as stent-like devices.
7
In chapter 4 a cell shape optimization method is implemented to improve the fatigue
strength of planar lattice materials. The methodology is applied to improve the fatigue
resistance of cellular materials with hexagonal and square unit cell shapes. The results are
compared with the fatigue strength of the unit cells studied in chapter 3. Similar to
chapter 3, in chapter 4 we optimize the geometry of planar lattice materials in general and
do not consider specific application.
In chapter 5 the shape optimization method to improve the fatigue resistance of planar
lattice materials described in chapter 4 is applied to improve the durability of Nitinol self-
expandable stent-grafts. This chapter first briefly reviews the description of stent
typology and stents’ application and design challenges. Then the shape optimization
strategy described in chapter 4 is implemented to design a self-expandable Nitinol stent-
graft against fatigue failure. A parametric study is carried out to investigate the effect of
the selected geometric parameters on the fatigue resistance and radial stiffness of the
stent.
Chapter 6 presents the conceptual design and optimization of a novel protective cage for
the state-of-the-art intravenous rotary oxygenator recently developed by ALung Inc. The
in vitro tests performed to validate the proposed concepts are explained. The finite
element modeling (FEM) that investigates the structural behaviors of the cage is
described. Computational fluid dynamics (CFD) analyses that study the effect of the
lattice cage on the level of blood damage caused by the oxygenator-cage assembly are
surveyed.
The thesis ends in chapter 7 with conclusions and suggestions for future work.
8
Chapter 2
Literature review
2.1. Objectives
This chapter reviews the literature on the failure mechanisms and strength of cellular
materials. Experimental, theoretical, and numerical approaches to predict the failure of
both periodic and stochastic cellular materials are described. The role of the shape of cell
walls in the strength of planar lattices is explained. Fundamental topics on the
degradation of cellular materials are presented to demonstrate the fatigue behavior of
lattice materials. The basic concepts used to derive and obtain the failure surfaces of
cellular materials under multi-axial loading are reviewed. Alternative approaches to
determine the effective elastic and strength properties of cellular solids are also
discussed. The chapter concludes with a list of open, un-resolved issues that represent
directions for future inquiry and motivate the remainder of this thesis.
2.2. Introduction to the failure behavior of cellular materials
The definition of failure in a mechanical component varies from one design to another. In
one design the material’s yielding may be the main concern, while in another design only
avoiding plastic collapse or buckling may be required. Similar to those of composite
materials, the failure modes of a cellular component are governed by the properties of the
solid material, the cell topology and its geometrical parameters, and the macrogeometry
of the structure, as well as the loading and boundary conditions. For example, a sandwich
panel made of a cellular core under three-point bending may fail because of indentation,
fracture, buckling or wrinkling of the face sheets, de-bonding of the core and the face
sheet, or core shear failure (Triantafillou and Gibson, 1987). Various failure modes can
be illustrated on failure maps (Triantafillou and Gibson, 1987). Theoretical and
experimental approaches have been successfully used to develop failure maps for
sandwich panels with both periodic and stochastic cellular cores; the maps enable
9
designers to select material and structural variables that can prevent a given mode of
failure (Lim et al., 2004; Petras and Sutcliffe, 1999; Triantafillou and Gibson, 1987).
Besides structural failures, such as those identified above for a sandwich beam, the
microstructure of a cellular material can fail under other modes related to the material’s
microstructure. At the meso or micro scale, for example, yielding, fracture, and buckling
of the cell wall have been identified as the main failure modes of a cellular material
(Gibson et al., 1989; Lipperman et al., 2008a; Lipperman et al., 2009; Lipperman et al.,
2008b; Triantafillou and Gibson, 1990; Triantafillou et al., 1989; Zhang et al., 2008). The
results of these investigations show that the strength and failure modes of a cellular
material depends on several parameters including, but not limited to, the rate of loading
(static, dynamic, impact loadings), loading type (tensile, compressive, shear, or multi-
axial loading), manufacturing defects, and environmental conditions.
2.2.1. Failure modes of the cell walls
The effective mechanical properties and failure mechanisms of cellular materials are
governed by two main classes of parameters: first, the properties of the solid material,
such as its Young’s modulus, yield and ultimate strengths; and second, the attributes of
the voids, i.e. cell geometry and volume percentage (Gibson and Ashby, 1999a). For
example, under compressive loading a thick cell wall tends to fail because of plastic
collapse, while a thin one would buckle before fracture. On the other hand, in contrast to
a ceramic cell wall that breaks because of brittle crash under compressive loading, a
polymeric one buckles before plastic rupture. One of the main goals of this dissertation is
to study the fatigue resistance of various cell shapes of planar lattices, in general, and
effectively apply them to biomedical design, as shown in chapter 5and 6. The diversity in
the failure mechanisms of cellular materials due to the different failure mechanisms of
bulk solid materials is bracketed by considering only the failure mechanisms of metallic
cellular materials.
10
2.2.1.1. Cell wall buckling
In contrast to solid bulk materials that are normally weaker under tension, cellular
materials are more sensitive to compressive loading. Buckling of cell walls, especially at
low relative densities, is one contributing factor in this characteristic of cellular materials
(Gibson et al., 1989; Triantafillou and Gibson, 1990; Triantafillou et al., 1989). The cell
wall of a lattice at low relative densities can be compared to a column with constrained
joints, which, under microscopic compressive loads, is more prone to buckle than to
fracture from excessive plastic deformation. Lattices with both bending and stretching
dominant failure modes may fail because of cell wall buckling. Gibson et al. (1989)
studied the buckling of cell walls of a hexagonal lattice under compressive macroscopic
loading (figure 2.1). Both axial ( 1 2 ) and biaxial ( 2 1) modes of buckling were
considered (figure 2.1). The analysis was performed by postulating that buckling of a cell
wall under a critical compressive axial load is governed by the Euler formula for the
buckling of structural columns:
2 2
2
scrit
n E IP
h
(2.1)
where h is the cell member height, I is the second moment of area of the member cross-
section, and n is a constant depending on the rigidity of the cell joints. A theoretical
study on the hexagonal lattice under various macroscopic stress ratios, or 1 2/ , was
carried out to find the n values for the first two modes of buckling of cell walls. Figure
2.2 shows how cell wall buckling controls the failure of a unit cell under compression,
while the plastic collapse of the cell walls is the dominant failure mechanism under
tension.
2.2.1.2. Cell wall fracture
Schaffner et al. (2000), in a numerical study on the crack accumulation in 2D Voronoi
honeycombs under cyclic loading, showed that the fracture of only 1% of the cell walls of
a lattice can reduce the stiffness of the whole material by 15%. Nieh et al. (2000)
experimentally showed that the strength of cellular foams at low relative densities
11
changes with the morphology and orientation of the cells. Cell wall rupture in a lattice
made of either ductile, brittle metallic, or non-metallic bulk materials under various
loading conditions has been extensively studied in the recent past (Huang and Gibson,
1991; Lipperman et al., 2008a; Lipperman et al., 2007a, b, 2009; Lipperman et al., 2008b;
Maiti et al., 1984a; Maiti et al., 1984b; Ryvkin et al., 2004).
Figure 2.1. Schematic view of hexagonal lattice under biaxial compressive loading Adapted from
(Gibson et al. 1989).
In a series of fundamental studies, Gibson and Ashby (1999a) and Gibson et al. (1989)
found simple expressions describing cell wall fracture for both foam and lattice materials.
The ductile and brittle ruptures of cell walls were examined. Equilibrium equations for a
hexagonal lattice under bi-axial loading (figure 2.3 (a)) were used to find axial forces and
bending moments that act on each cell wall. The cell walls were assumed to behave as
beam elements with constant cross-section that can resist both bending moments and
axial forces. The material properties of the cell walls were taken to be the bulk solid
material, i.e. Young’s modulus of sE , the Poisson ratio of v , and a rupture modulus of
s . The maximum stress found to occur at the cell joints of each unit cell in figure 2.3(c)
is given by:
2
aeq
F Mt
bt I (2.2)
where eq , aF , and M are, respectively, the maximum stress, the axial force, and the
maximum bending moment acting on the cell wall. The cell wall fractures under
12
excessive plastic deformation when eq reaches the rupture modulus s of the cell
wall’s bulk material. This condition limits the allowable stress level to a certain
boundary, as shown in figure 2.2. It can be seen that the allowable boundary in
compression is dominated by elastic buckling rather than by plastic collapse.
Figure 2.2. Failure surface of a 2D hexagonal lattice (Gibson et al. 1989).
Figure 2.3. (a) Biaxial stress state acting on a hexagonal unit cell. (b) free body diagram of the
inclined member. (c) the model of plastic collapse indicating the place of plastic hinges formed
under bending (Adapted from Gibson et al. 1989).
The presence of defects, e.g. fractured and missing cell walls, reduces the strength of a
cellular material under tension. Thus, while the boundaries of figure 2.2 accurately
represent the strength of the cellular material under compressive loading, they
overestimate the strength of a flawed cellular material under tensile loading. The
13
macroscopic tensile strength of 2D honeycombs in the presence of macroscopic cracks
has been thoroughly investigated by many authors (Gibson and Ashby, 1999a; Gibson et
al., 1989; Lipperman et al., 2008a; Lipperman et al., 2007a, b, 2009; Lipperman et al.,
2008b; Ryvkin et al., 2004). Gibson and Ashby (1999a) and Gibson et al. (1989) assumed
that a macroscopic crack of the length of 2a is embedded in an infinite lattice, and that
the crack tip advances for one unit cell by breaking the nearest cell wall. The
macroscopic stress required to advance the crack was determined by using a fracture
mechanics method, in which the crack length is assumed to be sufficiently large to
consider the cellular material ahead of the crack tip behaving as a continuum. Thus, the
stress distribution around the crack tip of a sample loaded in fracture mode I can be
expressed as:
1r
a
r
(2.3)
where r is the stress at a distance r from the crack tip. If the crack tip is assumed to be
located at the center of the unit cell, the equilibrium of moments in the out-of-plane
direction gives the bending moment, M , acting on the nearest cell wall as:
sin
2
0
sin( )
2
h l
r
h lM b r dr
(2.4)
where h , l , , and b are the cell wall parameters as shown in figure 2.3(a). The axial
component of the stress is found, as shown in figure 2.3(b), for the plastic collapse of the
cell walls. These two components can be added to find the maximum stress in a cell wall
as expressed by equation (2.2).
The cell wall is assumed to fracture when the maximum equivalent stress exceeds the
ultimate strength of the cell wall’s solid bulk material, s . However, as shown by Fleck
and Qiu (2007) , for a crack length beyond a transition length, the tensile strength of the
periodic lattices is independent of the length of the crack. The value of the transition
crack length is defined to be a function of the shape of the unit cell as well as the loading
condition. Fleck and Qiu (2007) obtained the deformation field, and hence the stress
distribution of the material via the FEA of a 2D lattice made of a large number of unit
14
cells with a pre-existent macroscopic crack. Closed-form expressions of the stress
intensity factor were obtained as a function of the relative density for lattices with
Kagome, hexagonal, and triangular unit cells. While the approach used in this study has a
theoretically established base, a large number of unit cells were required to capture
accurately the deformation field ahead of the macroscopic crack. Thus this method is
deemed to be computationally expensive, especially when the design objective is to find
an optimum or at least a proper, material distribution within each unit cell. To overcome
this obstacle, (Lipperman et al., 2008a; Lipperman et al., 2007a, b; Ryvkin et al., 2004),
in a series of publications, employed the discrete Fourier transform to find the exact
deformation (stress) field in an infinite 2D lattice with an embedded finite crack. This
method postulates that the periodicity of the lattice, which is violated by a group of
fractured or missed cell walls at the crack edges, can be restored by using a series of self-
equilibrating forces and moments to close the crack. Given a periodic arrangement of unit
cells, the system of the nodal displacement and forces of the lattice was calculated by
using the discrete Fourier transform. The unknown values of self-equilibrating forces and
moments were then obtained by applying free traction conditions at the crack boundaries.
This method has a robust mathematical base and is claimed to save computation time.
The method has been further developed to find a material distribution within one unit cell
that improves the fracture toughness of the lattice. This cell design method is explained in
more detail in the following sections of this chapter.
2.3. The role of the cell wall in the strength of planar lattice materials
The studies reviewed in the previous section investigated the failure mechanisms of the
lattices at the microscopic level. However, it is generally assumed that the cell walls
behave as slender structural beam elements with a constant cross-sectional area. This
assumption simplifies both the mathematical formulation and the FE models. Several
authors have argued about the accuracy of this assumption. For example, (Simone and
Gibson, 1998) showed the significant effect of the distribution of material within each
unit cell on the stiffness and strength of a 2D hexagonal lattice at the macro-level. In their
study, the strength and stiffness of the unit cells with walls of variable cross-sectional
area were normalized with respect to those of the unit cells modeled with structural beam
15
elements of constant cross-sectional area. The cell walls were defined by a series of
plateau segments which were joined with an arc fillet. The radius of the fillet, was set as a
design variable that governs the amount of material confined between two plateaus. The
representative volume element was meshed with continuum planar elements that can
capture more accurately the material distribution within the unit cells. At each relative
density, two radii of curvature maximizing the stiffness and strength of the lattice were
identified. These optimum values were observed to be almost equal. This shape of the
unit cell was later used by Lin and Huang (2005) to study the creep response of
hexagonal lattices; similar conclusion on the dependency of the creep behavior on the
material distribution in the cell walls was drawn.
In another study highlighting the effect of material distribution on the mechanical
properties of cellular materials, Li et al. (2003) estimated the stiffness and strength of a
dual cell size aluminum foam by using an idealized FE model of a 2D lattice. They
identified the position of plastic hinges and showed that the stiffness and strength of the
material for a given relative density are controlled by the ratio of the void radii, r R .
The studies of Lipperman et al. (2009) and Lipperman et al. (2008b) were carried out
with the aim of improving the fracture toughness of planar lattices. In these works, the
methodology they previously introduced, as described earlier in this chapter, was used to
find the stress distribution in a unit cell of a cracked lattice. Lipperman et al. (2008b)
optimized the fracture toughness of cracked 2D planar lattices meshed with continuum
planar elements. In this study, void profiles with an elliptical shape as well as those
defined by a given mapping function, ( )f z , were considered. The stress intensity in each
lattice was plotted as a function of a series of the geometrical design parameters. It is
shown that each unit cell shape has an optimum geometrical parameter that maximizes
the normalized fracture toughness of the lattice.
In another study, Lipperman et al. (2009) proposed two different methods to optimize the
fracture toughness of hexagonal and triangular planar lattices. First, they assumed un-
symmetric RVEs that have cell walls with uniform but different cross-sectional areas.
They considered all possible directions for crack propagation in each lattice and then
16
found the optimum solution that maximizes the fracture toughness. The minimum and
maximum allowable wall thicknesses were considered as design constraints. The
optimum distribution of the cell walls for each lattice was obtained. It was found that this
method can improve the fracture toughness of the triangular and hexagonal unit cells,
respectively, by 20% and 6%. In a next step, they maximized the fracture toughness of
hexagonal and triangular lattices made of cell walls of variable thicknesses as shown in
figure 2.6(a). The cell wall profiles were considered to be polynomials of degrees 2, 3,
and 4. The polynomial coefficients that maximize the fracture toughness of each lattice
were found by formulating an optimization problem. The optimization problem was
solved by using the sequential quadratic programming method implemented with the
fminmax MATLAB function. The optimum cell wall profiles for a hexagonal lattice are
shown in figure 2.6. It was demonstrated that using an optimum thickness profile
significantly improves the fracture toughness of the hexagonal lattices up to 103.7%,
while it has a relatively negligible effect, 4.8%, on that of the triangular lattice. This work
shows that the material distribution within the cell wall plays a key role in the strength
and fracture toughness of the lattice materials. However, the optimum cell wall
geometries were obtained with polynomial functions of degrees 0 to 4. It was shown that
the fracture toughness of the lattice increases when using higher degree polynomials to
define the cell wall’s profile. But using higher degree polynomials leads to oscillation of
the cell wall thickness between its two extremes, what is called Runge’s phenomenon. As
will be described in detail in chapter 4, this limitation can be resolved by using piecewise
smooth polynomials, or splines.
2.4. Life-time strength degradation of cellular materials
It is known that the strength of a material degrades during its service life. Creep and
fatigue are two well-known phenomena that deteriorate the strength of cellular materials.
The former happens in components that work at elevated temperatures, such as heat
exchangers and fuel cell interconnects; in such cases, cellular materials fail because of
creep buckling and rupture of their cell walls (Andrews et al., 1999). Fatigue is caused by
crack propagation in cell walls undergoing cyclic loading (Liu, 2005). It has been shown
that cell wall geometry plays a significant role in the creep response and fatigue
17
resistance of cellular materials (Lin and Huang, 2005; McCullough et al., 2000). In the
present thesis, a numerical method is presented to design a cellular material against
fatigue failure. Thus, the purpose of the next section is to review the available literature
on the fatigue of cellular materials.
2.4.1. Fatigue loading
In many instances, the lattice material has to withstand cyclic loads; Fatigue is thus an
essential aspect to be considered in the design under repetitive loads. From the literature,
it appears that fatigue failure of cellular materials, e.g. foams and lattices, has received
less attention than their monotonic, quasi-static, and dynamic counterparts (Banerjee and
Bhaskar, 2005; Banerjee and Bhaskar, 2009; Gibson and Ashby, 1999b; Masters and
Evans, 1996; Ruzzene, 2004; Schraad and Harlow, 2006; Wang and McDowell, 2004a,
2005). Among the work available on the fatigue behavior of cellular materials, both
theoretical and experimental approaches have been used. The latter, however, are
predominant and stem mainly from experiments on foams (Burman, 1998; Kolluri et al.,
2008; Kulkarni et al., 2003; Kulkarni et al., 2004; McCullough et al., 2000; Noble, 1983;
Noble and Lilley, 1981; Olurin et al., 2001). The goal of these works has been generally
to determine the stress-life curves of foams under shear and axial loading conditions. For
example, Burman (1998) conducted an extensive experimental investigation on fatigue
behavior of sandwich panels made of polymeric foam cores under common loadings
including uni-axial and pure shear loading. Various experimental data and graphs, such as
stress-life curves of PVC and PMI foam cores, were extracted. The data can be used in
the design of sandwich panels made of these materials. Observation on the formation and
propagation of fatigue cracks showed that over the entire length of the shear zone, several
micro-cracks were initiated along a horizontal line at the middle of the specimen; FEM
results showed that this region had a maximum shear stress. Propagation and inter-
connections of these micro-cracks, under alternating loading, led to a horizontal macro-
crack between two load supports. Eventually, the horizontal crack kinked and extended
toward the panel faces and led to the fracture of the sandwich panel. In this study, the
effect of mean stress on the fatigue failure of foam cores was also examined by
experimentally developing Haigh diagrams. Haigh diagrams typically show safe
18
combinations of alternating and mean stress levels that lead to a desirable fatigue life. It
was shown that i) the effect of alternating strain on the fatigue life of the core is more
significant than the effect of mean stress, and ii) the reversible cyclic loadings, tension-
compression loading, lead to a significant reduction of the fatigue life. In another work,
Olurin et al. (2001) experimentally characterized the fatigue crack propagation rate in
closed-cell aluminum alloy foams, Alulight and Alporas, under the alternating loading
condition of fracture mode I. Fatigue crack propagation was observed to be controlled by
linearly elastic fracture mechanics, or LEFM. However, the high sensitivity of the crack
propagation rate to the level of alternating stress intensity showed that the conventional
damage-tolerant methodology is not appropriate for the fatigue design of foams. Motz et
al. (2005) experimentally compared the fatigue crack propagation in two types of cellular
solids, hollow sphere structures made of a stainless steel (316L) versus a closed-cell
aluminum foam. Due to stress concentration at the sphere-sphere bonding, the fatigue
strength of the former was found to drop from 50% to 65% compared to that of the latter.
McCullough et al. (1999) found that ratcheting in closed-cell Alulight foam is the
dominant cyclic deformation mode for a tension–tension and compression–compression
loads.
Studies on the fatigue of lattice materials are much fewer than those on foams. Among
important works are those of Côté et al. (2006), who experimentally characterized the
stress-life curves of sandwich panels with a lattice core. It was shown that sandwich
panels with diamond shape cells under alternating shear stress are weaker than under
axial stress. In another study, Côté et al. (2007b) extracted the S-N curves for sandwich
beams with pyramidal core. Also, collapse maps for monotonic and cyclic loadings of the
sandwich beam were developed. These maps can be used for the design of sandwich
cores to predict the failure mechanism of the lattice with given geometrical parameters
under certain cyclic/monotonic loadings. Moreover, they showed that the weakness of
brazed joints significantly reduces the fatigue strength of the sandwich core under
alternating shear loading.
In another work, Abbadi et al. (2010) experimentally studied the damage accumulation in
sandwich panels with composite honeycomb cores connected to aluminum faces. In
19
addition to extracting of the S-N curves for the studied lattice core, two non-linear
damage accumulation models were proposed and were experimentally verified for fatigue
design of composite honeycomb cores under a sequence of variable amplitudes. These
models can be used for the design of other cellular materials, but their validity should be
verified for the design of new cores. In yet another study, Côté et al. (2007a)
experimentally investigated the fatigue behavior of diamond lattice cores made of
stainless steel 316. They reported that the available experimental data for lattice cores
show the fatigue to monotonic strength ratio, ulte /*, to be around 0.3 and 0.2,
respectively, for maximum to minimum load ratio of 0.1 and 0.5 (R=0.1 and R=0.5). The
data gathered from a wide range of lattice cores suggest that the fatigue to monotonic
strength ratio is independent of core topology, relative density, and material.
As mentioned in some of the above studies, experiments show that the fatigue crack
initiates from the connecting region of one unit cell to its neighboring cell Côté et al.
(2006) and Motz et al. (2005) or to the face sheet (Côté et al., 2007b). The crack
initiation can be due to high stress concentration at these points (Motz et al., 2005). A
geometrical stress concentration can happen at a location where two geometric primitive
entities need to be connected (Neuber, 1961b; Teng et al., 2007; Waldman et al., 2001).
Strategies that aim at removing the causes of stress concentration can have a beneficial
effect on the fatigue resistance of a cellular solid. This subject is dealt with in chapter 4 of
this dissertation.
On the theoretical side, several works were carried out to model the fatigue behavior of
lattice materials subjected to both uni-axial and multi-axial loading conditions (Huang
and Liu, 2001a, b; Huang and Lin, 1996). In these studies, the fatigue of honeycomb
lattices under high cycle fatigue (lattice subjected to alternating loads with amplitudes
lower than yield strength of the material) and low cycle fatigue (lattice subjected to
alternating loadings with amplitudes higher than yield strength of the material) were
investigated. Lattices with and without pre-existing macro cracks were examined. Cell
walls were modeled with beam elements and cell-wall-bending was used to study their
fatigue. The material properties of the cell walls, including their fatigue behavior, were
taken to be the same as of their constituent bulk solid material. The maximum
20
microscopic stress in cell walls both in the absence and in the presence of a macro crack
was estimated by assuming a unit cell under uniform in-plane multi-axial loading, i.e. (
x , y , xy ). The degradation of cell wall material was studied under two possible failure
scenarios, i.e. the failure of cell walls with and without pre-existing micro cracks. Several
fatigue mechanisms were used to model these failure conditions. In the case of a cell wall
with a micro crack, Paris law relation was used to determine the rate of micro-crack
propagation in a cell under a given alternating loading condition. The fatigue of cell walls
without pre-existing micro cracks was studied by using the Basquin empirical law for
high-cycle fatigue and the Coffin–Manson empirical relation for cells under low-cycle
fatigue (Huang and Liu, 2001a). However, the underlying assumption in these models is
that the cell walls behave as a beam element. As mentioned earlier in this chapter, the use
of beam elements rather than continuum elements, such as planar elements, to model cell
walls does not allow accounting for the real stress distribution of the lattice cell. For
example, these methods do not consider the effect of stress concentration at the corners of
unit cells, a factor that is proven to reduce the fatigue life of a cell; hence these methods
can lead to unrealistic results that limit their use for fatigue design. Chapter 3 presents a
numerical method for fatigue design of cellular solids under multi-axial loading that
considers the real stress distribution within cell walls. The proposed method will be used
to study the effect of cell design on the fatigue strength of lattice materials.
2.5. Effective mechanical properties
The analysis of a lattice material is often conducted by isolating a representative volume
element (RVE) and calculating its properties, which under certain assumptions, represent
the homogenized properties of the macro material. Theoretical, numerical, and
experimental approaches can be used to homogenize the properties of a lattice, including
its stiffness matrix, Poison ratio(s), yield and ultimate strength (Andrews et al., 2001;
Chen and Huang, 1998; Christensen, 2000; Fang et al., 2004; Gibson and Ashby, 1999a;
Hassani and Hinton, 1998a; Hassani and Hinton, 1998b; Kumar and McDowell, 2004;
Masters and Evans, 1996; Vigliotti and Pasini, 2012a, b; Wang and McDowell, 2004b;
Wang et al., 2006; Warren and Byskov, 2002). Figure 2.4 shows that the periodic cellular
21
body (on the left) subjected to traction t at the traction boundaryt, a displacement
d at the displacement boundaryd
, and a body force f is replaced by a homogenized
continuum body (on the right) with the same traction boundaries as , but without
the geometrical details and voids of the local coordinate system. Since cellular materials
are typically obtained by a periodic tessellation of a unit cell, the theoretical and
numerical studies are usually performed on their representative volume element (RVE).
From the analysis of the RVE, the equivalent properties of the entire lattice can be
obtained.
Figure 2.4. Homogenization concept of a cellular structure (Masoumi et al. 2011).
Structural analysis, micro-polar elasticity, homogenization based on the Cauchy–Born
hypothesis, and the asymptotic homogenization are four well-established methodologies
used to homogenize lattice materials. Assumptions, benefits, and limitations of each
method for estimating the homogenized properties of lattice materials are briefly
described below.
2.5.1. Structural analysis
In the classical structural analysis, the lattice cell walls are considered relatively slender
to allow the use of simple beam theory. Pure bending, axial, and shear loading or their
combinations are possible in-plane loading conditions acting on a lattice unit cell. The
deformation of the beam elements of a unit cell under a given loading condition can be
22
first obtained either theoretically or numerically. Both axial and in-plane shear loadings
are considered. The stress and strain values can be readily extracted from the nodal
deformation and used to calculate the homogenized mechanical properties of interest.
Wang and McDowell (2004a) implemented “structural analysis” technique to derive the
effective elastic stiffness and initial yield strength of metal honeycombs at a given
relative density under in-plane compression, shear, and diagonal compression. The details
of this method can be found in the work of (Wang and McDowell, 2004a). Although
structural analysis allows finding the overall mechanical properties of a cellular material,
it cannot accurately capture the stress distribution within cell walls of complex geometry,
such as cell walls with smooth variable thickness.
2.5.2. Micropolar theory
The classical continuum theory assumes that the kinematic state of a material medium is
only a function of the displacement field, and its and derivatives, of material points. As
an alternative, Eringen (1966) and Eringen (1999) proposed micropolar theory.
Its characteristic features are the existence of couple stresses and asymmetric shear
stresses, and the independence of microrotation at the joints from the displacement field.
The strain measures of micropolar continua are the asymmetric strain tensor and the
gradient of rotations. In other words, two neighboring points of material interact with
each other by transferring both moment and force vectors. Micropolar theory can be used
effectively to capture the rotation and displacement fields of the joints of cell members of
a lattice material. The kinematic field (rotation and displacement fields) of a material
point on the cell wall is described by the macroscopic displacement of the joints and a
microscopic rotation associated with the joint rotation. In other words, general
deformation of a typical cell wall of the lattice is characterized by cell wall stretching
(due to force vector) and bending (due to moment vector). Micropolar theory is not able
to capture accurately the stress distribution within cell walls of complicated geometry.
The mathematical details of this method are beyond the scope of the present thesis and
can be found in the literature (Eringen, 1966; Eringen, 1999).
23
2.5.3. Homogenization by Bloch Theorem and Cauchy-Born hypothesis
The Cauchy-Born hypothesis (Bhattacharya, 2003; Bom and Huang, 1954; Maugin,
1992; Pitteri and Zanzotto, 2003) states that the infinitesimal displacement field of a
periodic joint is equal to the deformation obtained by a macroscopically-homogeneous
strain field plus the periodic displacement field of the joint (Elsayed and Pasini, 2010a).
This method has been effectively used to homogenize the properties of lattice materials
through formulating the microscopic nodal deformations of the lattice in terms of the
material macroscopic strain field (Elsayed and Pasini, 2010b; Hutchinson, 2004). Like
the other methods described above, this technique is also unable to accurately capture the
stress distribution within the cell walls. The mathematical formulation of this method can
be found in the literature (Elsayed and Pasini, 2010a).
2.5.4. Asymptotic homogenization method
Asymptotic homogenization is an effective method that can be used not only to calculate
the effective properties of a cellular material but also its strength (Kalamkarov et al.,
2009). Asymptotic homogenization allows obtaining the macroscopic stress field that
lead to the microscopic yield, or fracture, as well as the endurance limit of the lattice.
Asymptotic homogenization is applicable to unit cells with two axes of symmetry and its
underlying assumption is that each field quantity depends on two different scales: one on
the macroscopic level x, and the other on the microscopic level, y=x/ε, where ε is a
magnification factor that scales the dimensions of a unit cell to those of the material at the
macroscale. In addition, the field quantities, such as displacement, stress, and strain are
considered periodic at the microscale and assumed to vary smoothly at the macroscopic
level (Hassani and Hinton, 1998b; Hollister and Kikuchi, 1992).
24
Figure 2.5. Periodic boundary conditions for a pair of nodes located on the opposite surfaces,
A and A , of the RVE.
Under the assumption of small deformation, the standard weak form of the equilibrium
equations for a cellular body with the pertinent geometrical details of voids and cell
wall is (Hollister and Kikuchi, 1992)
0 1 *ε ( ) ε ( ) E ε( ) ε ( ) t v t
T Tv v u u d d (2.5)
where E is the local elasticity tensor that depends on the position within the RVE,
0ε ( )v and 1ε ( )v are the virtual macroscopic and microscopic strains, respectively,
ε( )u is the average or macroscopic strain, *ε ( )u is the fluctuating strain varying
periodically at the microscale level, and { }t is the traction at the traction boundary t .
The virtual displacement v may be chosen to vary only on the microscopic level and
be constant on the macroscopic level. Based on this assumption, the microscopic
equilibrium equation can be obtained as:
1 *ε ( ) E ε( ) ε ( ) 0T
v u u d (2.6)
Taking the integral over the RVE volume (VRVE), equation (2.6) may be rewritten as
1 * 1ε ( ) E ε ( ) ε ( ) E ε( ) RVE RVE
T T
RVE RVEV V
v u dV v u dV (2.7)
1
1
1
w
v
u
A A
12
12
12
ww
vv
uu
25
The above equation represents a local problem defined on the RVE. For a given applied
macroscopic strain, the material can be characterized if the fluctuating strain, *ε ( )u , is
known. The periodicity of the strain field is ensured by imposing periodic boundary
conditions on the RVE edges (figure 2.6); the nodal displacements on the opposite edges
are set to be equal (Hassani, 1996; Hollister and Kikuchi, 1992). Equation (2.7) can be
discretized and solved for nonlinear material properties via finite element analysis as
described in the literature (Guedes and Kikuchi, 1990; Hassani and Hinton, 1998a;
Hollister and Kikuchi, 1992; Jansson, 1992). For this purpose, equation (2.7) can be
simplified to obtain a relation between the microscopic displacement field D and the
force vector f as
K D f
(2.8)
where K is the global stiffness matrix defined as
1
K km
e
e
, tk B E Be
Te e
YdY
(2.9 a,b)
m
1e
)( being the finite element assembly operator, m the number of elements, B the
strain-displacement matrix, tE the tangent modulus of the elastic–plastic bulk material,
and eY the element volume. The force vector f in equation (2.8) is expressed as
1
f fm
e
e
, tf B E ε( )
e
e e
Yu dY
(2.10)
It is interesting to note that ε( )u in (2.10) describes the initial strain field applied to
each element of the unit cell. As a result, the force vector f is a function not only of the
applied strain but also of the properties of the solid material. If the plastic deformation is
required to be considered, equation (2.8) can be considered as a system of nonlinear
equations that can be solved with the Newton-Raphson scheme. As shown in figure 2.6,
the computational analysis is accomplished by imposing increments of macroscopic
26
strain, as described in plastic theory (Bathe, 1996; Cook et al.). While the macroscopic
strain ε( )u is increased monotonically, the force vector increment is computed and
replaced in (2.8) to evaluate the displacement field. During the procedure (Fig. 2.6), the
effective material coefficients are kept constant until the stress reaches the yield strength
of the base material. The mechanical properties of the yielding elements are replaced,
iteratively, with the tangent moduli of the elastic–plastic bulk material (Bathe, 1996;
Cook et al.). Once the displacement field is obtained, the fluctuating strain *ε ( )u is
determined by the product of the strain-displacement matrix,
B , and the nodal
displacement vector. The increment of the microscopic strain is then determined through
the equation:
*ε( ) ε( ) ε ( )u u u (2.11)
The stress field in the unit cell is updated with respect to the elastic strain, and the stress
average is computed over the unit cell to attain the effective macroscopic strength as:
1σ( )
RVERVE
VRVE
u dVV (2.12)
The macroscopic stress, Eq. (2.12), represents either the yield y
or the ultimate ult
strength of the unit cell when the stress level at any local point of the microstructure
reaches respectively the yield or ultimate strength of the solid material. The whole
procedure, which includes the generation of the model and its mesh, as well as the
nonlinear plasticity analysis, can be implemented into available commercial FEM codes,
where the von Mises yield criterion with the associated flow rule has been considered for
plasticity analysis.
2.6. Failure surfaces
As previously described, the mechanical properties of a lattice material, e.g. fatigue
resistance, can be expressed in terms of homogenized properties under different loading
conditions. Often expensive and time-consuming to obtain, the experimental
homogenized values are generally recorded for a limited number of simple loading
27
conditions and stress states. As an alternative, numerical approaches can be effectively
used to find the homogenized mechanical properties of a lattice material under various
loading condition. Several theories have been introduced to derive the yield/ failure of a
material in a multi-axial stress state from the uni-axial yield values obtained via
experiments. For example, von Mises, Tresca and Tsai-Wu’s (Tsai and Wu, 1971) are
well-known failure criteria in the design of isotropic and composite materials. In the
realm of cellular materials, several experimental, theoretical, or numerical studies were
conducted to find or derive failure surfaces of these materials (Deshpande and Fleck,
2001; Gibson et al., 1989; Puso and Govindjee, 1995). On the experimental side, several
phenomenological yield functions for cellular materials have been proposed (Deshpande
and Fleck, 2001; Wang and Pan, 2006). On the theoretical side, however, the behavior of
a cell wall under a given loading condition, such as its nodal deformation, moment, force,
strain, and stress values, was obtained theoretically and was used to find the critical
macroscopic loads. For example, Gibson et al. (1989) derived yield and ultimate failure
surfaces for honeycomb lattices based on various failure modes of cell struts under a
given loading condition (recall Fig. 2.3). In their study, the elastic buckling, plastic yield,
and brittle fracture modes of failure were modeled. Based on this work, Puso and
Govindjee (1995) developed a constitutive relation for failure surfaces of foams. These
surfaces were coded in FEA to predict the failure of components made of foams. Again,
the above studies assume that cell walls flex like a thin beam, and thus beam elements
can be used to find the critical loading for the failure of a cell wall strut.
Despite their simplicity, the studies described above do not consider the real stress
distribution within the cell walls. Consequently, in this thesis we aim to find failure
surfaces of lattice materials based on accurate stress distribution at the microscopic level.
28
Figure 2.6. Flowchart of the asymptotic homogenization theory steps used to obtain the effective
strength properties of a lattice material.
2.7. Concluding remarks emerging from the literature
From the literature on the strength of lattice materials under cyclic loading, a number of
challenges, which have not been fully addressed yet, emerge. Among these, the thesis
aims to address the following issues:
Fatigue design of cellular materials, in general and not limited to special
application such as stent-like devices, through a realistic stress distribution of cell
walls with tapered cross-section. In many instances, a cellular material undergoes
29
cyclic loading, which requires a design against fatigue failure. Available
numerical studies on the fatigue design of lattice materials aim to find the number
of cycles to failure for a lattice under a given loading. It is generally assumed that
the cell walls are structural beam elements with a constant cross-sectional area. It
has been shown that these methods cannot accurately predict the stress
distribution within a unit cell. This challenge is addressed in chapter 3, where a
fatigue design method is described to account for the real stress distribution
within the lattice’s RVE.
Design optimization of lattice materials for fatigue strength. Several studies have
been performed to find a proper material distribution within RVE that improves
either their strength, or fracture toughness, or creep response. However, no study
can be found that aims to optimize the unit cell geometry to improve fatigue
resistance. Chapter 4 examines how fatigue resistance of a lattice can be improved
via the design of a smooth variable thickness profile of the cell elements.
Exploitation of lattice properties to optimize the design of two biomedical devices for
fatigue. Biomedical implants and scaffolds generally operate under specific physiological
environment and dynamic loading conditions. The method employed in chapter 4 for the
fatigue design of a lattice is applied to design an intravascular self-expandable stents for
fatigue in chapter 5, as well in chapter 6 to design a stent-like device functioning as a
protection for an endovascular oxygenator.
30
Chapter 3
A computational method for the design of lattice
materials for fatigue resistance
3.1. Objectives
This chapter presents a numerical method to evaluate the fatigue resistance of planar
lattices. It is proposed that asymptotic homogenization be used to model the real stress
distribution in the cell walls of a lattice. The fatigue of two hexagonal and square unit
cells is investigated by constructing their modified Goodman diagrams. The effect of
material distribution on fatigue resistance is investigated by using a 2D model that is
meshed by continuum plane elements, instead of structural beam elements. The failure
surfaces are obtained to be used in evaluating the monotonic and fatigue strengths under
multiaxial in-plane loading.
This chapter is organized as follow: first, the common terminology and classic theory of
fatigue design of mechanical components are reviewed (Reifsnider, 1991; Stephens et al.,
2000; Suresh, 1998). The second section describes the practical results of the proposed
numerical methodology for the design of lattice materials against fatigue failure. Finally
the main contributions of the presented study to knowledge are summarized.
3.2. Terms and definitions
The ASTM defines material fatigue as follows (Liu, 2005):
“The process of progressive localized permanent structural change occurring in a
material subjected to conditions which produce fluctuating stresses and strains at some
point or points and which may culminate in cracks or complete fracture after a sufficient
number of fluctuations.”
31
Although this definition highlights the progressive nature of fatigue crack growth, often
the final stage of fatigue failure occurs suddenly. This behavior can lead to catastrophic
and even fatal consequences. Thus, several fatigue design methodologies and criteria
have been developed to design fatigue-resistant components. Infinite-life, safe-life, fail-
safe, and damage-tolerant criteria are well-established methods introduced to predict
fatigue fracture and degradation. For a given application, the choice of a design
methodology greatly depends on the design objectives and constraints of the component
at hand.
In infinite-life design methodology, the components are designed to have stress levels
below the material fatigue threshold, after applying a safety factor. This methodology is
of interest for the fatigue design of components whose regular inspection is difficult or
expensive and/or whose overall weight is not the main design objective or constraint.
Safe-life fatigue design methodology is often used in aeronautic applications where a
lightweight and reliable component is desired. Through the use of experimental,
theoretical or numerical approaches, a “safe life” is assigned to a component or structure
to indicate when the component should be replaced or inspected after its expiration date.
High maintenance cost is a disadvantage of this method.
Fail-safe design methodology states that after failure of one or more individual
components of a large structure, the other parts should maintain the integrity of the whole
until the defective part or parts are detected and repaired. Regular and often costly
inspection by appropriate techniques and instruments is required to detect probable
cracks. This procedure increases the maintenance costs.
Damage-tolerant design methodology postulates that there are pre-existing cracks or
defects in the component. The component will fracture when the size of the crack(s)
reaches a critical length estimated by fracture mechanics. The instantaneous lengths of
the cracks must be assessed by precise and periodic measurements of their length. This
method offers a high level of safety and reliability, but it requires advanced and costly
inspections; hence, its application is limited to the design of components involving high
safety concerns, such as those used in the aerospace and nuclear power industries.
32
3.2.1. Characterization of materials
The application of all the above-mentioned design methodologies requires knowledge of
specific material properties, such as fatigue endurance limit and yield or ultimate strength
of the material. The fatigue properties of the materials are generally obtained from
traditional approaches, such as the stress (strain)-life method, or more recent techniques
based on fracture mechanics. These are experimental methods and are briefly explained
in the following subsections.
3.2.1.1. Stress-life approach
The stress (strain)-life method is an empirical approach introduced by Wholer in 1860 to
find the number of cycles that a smooth test specimen can resist under alternating stresses
(strains) of constant amplitude before fatigue fracture. The applied cyclic load is a time-
periodic load that alternates between minimum,minS , and maximum,
maxS , stress levels
(figure 3.1(a)). The alternating stress, altS , and mean stress, meanS , can be easily
calculated as follows:
max min
2alt
S SS (3.1)
max min
2mean
S SS (3.2)
The experimental data are plotted in a logarithmic or semi-logarithmic scale and called
stress-life or S-N curves. Figure 3.1(b) shows a schematic view of a stress-life curve.
Depending on the application and behavior of the material at hand, the stress values in an
S-N curve can be replaced with strain or stress intensity values. Typically at high cycles,
an S-N curve exhibits a stress plateau, which is referred to as its fatigue endurance limit
or, simply, endurance limit. For stress levels below the endurance limit, the material
seems to have infinite life and thus is of interest for infinite-life design methodology.
The fatigue life of a material under a given alternating loading condition may change as a
function of the maximum to minimum load ratio, R, defined as:
33
min
max
SR
S (3.3)
R is also a representation of the loading sign in a uni-axial loading condition; for example
a negative R represents a compression-tension loading condition, while a positive one
shows either a tension-tension or a compression-compression loading condition. Usually,
S-N curves include information for various R ratios.
Figure 3.1. (a) Schematic view of a cyclic load with constant stress amplitude; (b) Schematic
view of a stress-life curve.
Mean stress level is another loading parameter that may adversely affect the fatigue
resistance of a mechanical component. In contrast to the load ratio, R, the effect of mean
stress cannot be explicitly plotted on the S-N curves. For this purpose, other methods,
such as the constant-life diagrams (figure 3.2(a)) can be constructed by curve fitting of
experimental data for various combinations of stress amplitudes and mean stresses.
Goodman, Soderberg, Gerber, and modified Goodman are the most common models. The
first three are expressed as follows:
Goodman:
1mean alt
u f
(3.4)
Soderberg:
1mean alt
y f
(3.5)
Gerber:
S
log( )N
Endurance Limit
N OR
log(
)S
OR
34
2( ) 1mean alt
u f
(3.6)
where f is the allowable alternating stress for a given life under a fully reversible
loading ( 1R ), and u
and y are respectively the monotonic ultimate and yield
strengths of the material. For the infinite-life design methodology, f is equal to the
endurance limit of the material. Figure 3.2(a) shows the schematic representation of the
above curve-fitting formulas as well as the modified Goodman diagram. The latter is
constructed from the intersection of i) the line connecting the material ultimate and
fatigue strengths for a given life with ii) a 45 degree inclined line originating from the
monotonic yield strength of the material.
Figure 3.2. (a) Schematic view of the fatigue design diagram showing the effect of allowable
alternating stress versus mean stress for a given fatigue life; (b) Schematic view of the
logarithmic rate of fatigue crack growth versus logarithm of the amplitude of stress intensity.
The selection of one of the above models is strongly dependent on the failure mode of the
material at hand as well as the design objectives and constraints. In this chapter, modified
Goodman diagrams are developed for lattice materials under high cycle fatigue
conditions (Nicholas and Zuiker, 1989). To use this model, the most critical issue to bear
in mind is the degree of initial induced damage in the material, i.e. the material of the
samples used for the experiments must be free of damage (Nicholas and Zuiker, 1989).
As shown in figure 3.2(a), this model requires three material properties: yield strength,
ultimate strength, and fatigue strength for a given number of cycles to failure, or the
endurance limit for materials with infinite life.
e
y u
Goodman
Soderberg
Modified Goodman
45
Gerber
log( )K
I
log(
)da
dN
m
II III
Toward
higher R-ratio
cKthK
35
3.2.1.2. Fatigue crack growth in a notched specimen
This experimental method uses fracture mechanics to find the rate of crack propagation in
a material under a given loading. Various pre-cracked samples need to be tested to
characterize the fatigue behavior of the material based on the rate of fatigue crack
propagation, da dN , versus stress-intensity amplitude level, K (figure 3.2(b)). The
testing method and the geometry of various standard pre-cracked specimens can be found
in the ASTM standards (In, 1995).
The stress-intensity versus crack propagation rate, K - da dN , of several materials can
be divided into three phases based on the rates of crack propagation, which are the
initiation, stable crack growth, and unstable crack growth phases (figure 3.2(b)). During
the initiation phase, the crack grows with a non-continuous failure process and average
rate below10-6mm/cycle . The second phase, namely the Paris region, is characterized by
a linear relationship in logarithmic scale between the crack propagation rate and the
alternating stress-intensity amplitude. Thus, the equation governing this region can be
formulated using Paris’ law as follows:
( )mdaC K
dN (3.7)
where m and C are material constants, which represent respectively the slope of the curve
and the intersection point between the curve extension and 1K MPa m . When the
crack propagation rate reaches the unstable growth phase, the crack propagates at higher
rates because of a high stress intensity level at the tip of the crack, and eventually the
material collapses.
3.3. Fatigue design of planar lattice
3.3.1. Basics and assumptions
As described in the previous chapter, most of the work on the fatigue design of lattice
materials is based on experiments that are often time consuming, may be expensive, and
are generally focused on a specific lattice topology and material. The theoretical
36
approaches, on the other hand, seem to lack accuracy. The crucial issue is to accurately
model the real stress distribution in the lattice cells, a condition that is essential for
developing a reliable model. Thus, the development of a fatigue design methodology
based on a numerical approach that can assess with accuracy the real stress distribution in
the lattice cell wall is of interest.
In this chapter, the fatigue performance of planar lattices is studied via modified
Goodman fatigue diagrams. As mentioned earlier, the endurance limit, the yield and
ultimate strength of the unit cell are the properties required to construct the modified
Goodman diagram for any type of cell topology. Here, asymptotic homogenization theory
is used to determine the real stress distribution and the required monotonic strengths.
Since for a high cycle fatigue failure, as it is considered in this work, the stress level is
lower than the yield of the bulk material, the linear elasticity assumption holds. Hence,
the fatigue strength of the unit cell can be obtained through the product of the unit cell
yield strength with the ratio of the endurance limit to yield strength of the bulk material
as:
se
=sy
ses
sys
(3.8)
where ys and es are respectively the yield strength and the endurance limit of the
bulk material, and se
is the endurance limit of the unit cell. These properties are
required to construct the modified Goodman diagram (figure 3.2(a)) for any type of cell
topology, as shown in the next section. It should be noted that it is assumed that the
reference lattice material is free of defects, i.e. no type of damage has occurred in the
lattice cells. It is also assumed that the material has only one level of hierarchy, which
means that a cell wall’s material behaves as a continuum with properties comparable to
those of its bulk material. This assumption limits the applicability of this method to a
minimum length-scale that depends on the material’s microstructure, slip system, grain
size distribution (Dancygier et al., 1996; Li and Guiu, 1995; Lütjering et al., 2000). If the
effects of nanostructure defects, grain, and size of the material have to be taken into
37
account, then two levels of hierarchy need to be considered, as described by (Zienkiewicz
and Taylor, 2005).
In this chapter, the failure surfaces of selected lattice cells, hexagonal and square, will be
examined; and the results will be used to generate modified Goodman fatigue diagrams.
A comparison with the experimental results available in the literature will be provided to
validate the method. It should be noted that we examined lattice cells with length
normalized to unity. In this and subsequent chapters, we do not consider any specific
application.
3.3.2. Cell geometries under investigation
The topology and geometry of the unit cell has a significant effect on the properties of a
lattice material, including its static and fatigue performance (Banerjee and Bhattacharyya,
2010; Harders et al., 2005; Lipperman et al., 2009). To capture the effect of cell design,
as shown in figure 3.3, we examine regular square and hexagonal unit cells whose inner
boundaries are rounded by an arc of constant radius. As shown in the flowchart of figure
3.4, the methodology to obtain the fatigue properties of the optimum cell geometry
consists of three integrated parts that combine notions of design optimization and
asymptotic homogenization theory. First, the shape of unit cell is generated for given
geometrical parameters; then its fatigue properties are computed to generate the modified
Goodman diagram at given relative densities; the last step involves the minimization of
the von Mises stress in the cell wall. For square and hexagonal lattices, we study cell
shapes with the following characteristics:
1. G1 cells with small arc. These cells represent a conventional lattice with regular cell
geometry. The fillets at the cell joints are specified by an arc with radius equal to 1%
of the RVE length, which is representative of a sharp fillet resulting from a given
manufacturing process. The choice of the small-arc fillet is to obtain a realistic
material distribution in a cell member Simone and Gibson (1998), approximately
similar to that measured through experiments (Côté et al., 2007a; Côté et al., 2007b).
These cells are named here G1, where G represents “geometry” and the superscript
shows the degree of continuity of the geometry. As seen in figure 3.3, G1 cells have
38
continuous tangent along their inner boundary profile, but the curvature at the
blending points between the straight and arc geometric primitives is discontinuous.
2. G1 cells with optimum fillet radius. The G
1 cells with sharp corners are here
optimized to obtain a value of the fillet radius that reduces the effect of stress
concentration. A classical optimization problem is formulated with the objective of
minimizing the maximum value of the microscopic von Mises stress in the lattice
under a given loading. The fillet radius and the thickness at the middle of the struts
(figure 3.3) are considered as interdependent design variables. A design constraint is
set on the relative density. The problem is solved by implementing a conjugate
gradient method. For given relative density and design variables a unique solution is
found.
3.3.3. Numerical Modeling
The 2D cell geometries have been automatically obtained by means of Matlab
(MathWorks, Natick, Massachusetts), which have been coupled with ANSYS
(Canonsburg, Pennsylvania, U.S.A) to build, mesh, and solve the 2D model of the lattice
material. Assuming in-plane loading conditions, a 2D eight-node element type, Plane 82,
with plane strain formulation was used because of its capacity to model curved
boundaries with high accuracy. Different combinations of axial and shear loadings were
considered to obtain the data required to plot the failure surfaces.
The effect of the material properties on the normalized fatigue to monotonic strength
ratio, /e us , was studied by using aluminum and titanium alloys as bulk solid
materials. Aluminum was selected because of its broad application for lightweight foams,
while titanium is of interest because of its wide range of applications in the aerospace,
automobile, sports equipment and biomedical industries. Table 3.1 lists the material
properties used in this study (Case et al., 1999; Defense, 1966; Ducheyne et al., 1987;
Nicholas, 1981). A bilinear elasto-plastic constitutive law was considered to obtain the
macroscopic ultimate stress of the cellular material. The periodicity of the strain field was
assured by imposing periodic boundary conditions on the RVE edges (Hassani, 1996). A
39
mesh sensitivity test was performed to ensure the independency of the results from the
mesh size, as reported in figure 3.5.
Figure 3.3. Schematic views of: (a) G1 square unit cell; (b) G
1 hexagonal unit cell.
Figure 3.4. Flowchart of the design methodology. For a given cell geometry, shape synthesis is
coupled with computational analysis followed by size optimization. The goal of the first step is to
generate the geometrical model of the unit cell. In the second module, the effective strength
properties of the lattice are determined through asymptotic homogenization theory. The third step
involves the cell size optimization to reduce at minimum the maximum von Mises stress in the
cell wall.
rt
rt
Blending points Blending points
40
Table 3.1. Material properties of bulk solid materials.
Young’s
Modulus (
sE )
Poisson’s
ratio ( ) Yield
strength
Ultimate
tensile
strength
Fatigue
endurance
limit
Elongation to
fracture (%)
Ti-6Al-4V (Case
et al., 1999;
Ducheyne et al.,
1987)
GPa 110 3.0 901 MPa 984 MPa 486 MPa 8.9
Aluminum 6061
(Defense, 1966;
Nicholas, 1981) GPa 69 3.0 276 MPa 310 MPa 158 MPa 12
Figure 3.5. Mesh sensitivity showing the independency of the results from the mesh size.
3.4. Results
3.4.1. Stress distribution in the unit cell
Figures 3.6 and 3.7 show the stress distribution in the titanium square and hexagonal unit
cells with G1 cell topologies under fully reversed uni-axial and shear loading conditions.
The stress level changes significantly at the blending points of the unit cells, and the
curvature of their inner profile changes discontinuously. It is well known that a
discontinuity in the curvature of the struts causes stress concentration by locally
perturbing the stress flow. Thus, the removal of a geometrical stress concentration
10000 20000 30000 40000 500000
0.015
0.03
0.045
0.06
0.075
Hexagonal
Square
xx
ys
Number of Elements
41
improves the monotonic and fatigue strengths of a lattice material. An approach to
improve the fatigue resistance of cellular materials will be discussed in the next chapter.
Figure 3.6. von Mises stress distribution (MPa) in hexagonal and square unit cells made out of
Ti-6Al-4V. Lattices under fully reversed uni-axial loading defined by: G1 cell with small arc
(left); optimum G1 cell (right).
Figure 3.7. von Mises stress distribution (MPa) in hexagonal and square unit cells made out of
Ti-6Al-4V. Lattices under fully reversed in-plane pure shear loading defined by: G1 cell with
small arc (left); optimum G1 cell (right).
The results of FE modeling, figures 3.6 and 3.7, indicate the pivotal role of the unit cell
geometry on the yield, ultimate, and fatigue strengths of the lattices under investigation.
For each cell topology, a high local stress concentration occurs at the blending points of
G1 cells with a small arc, while the stress distribution in optimum G
1 cells is more
uniform. For example, optimum square and hexagonal titanium G1 cells have,
respectively, 32 % ( 35%) and 23% ( 32%) higher yield strength in comparison with G1
42
cells with sharp edges for uni-axial (shear) loadings. These results suggest that the fatigue
strength of G1 cells can be substantially improved by selecting proper design variables
that reduce stress concentration within the walls of the unit cell.
Figure 3.8. Yield and ultimate surfaces of Ti-6Al-4V square and hexagonal unit cells for relative
density of 10%. Projection of yield and ultimate surfaces of G1 square and hexagonal cells with
small and optimum arc radii in the yy xx
and xy xx
planes.
3.4.2. Failure surfaces and experimental validation
Figure 3.8 shows the calculated yield (solid lines) and ultimate (open marks) surfaces for
G1cells with small and optimum fillet radiuses. Here, the fillet radius of the unit cells was
selected to optimize the fatigue performance of the lattice under uni-axial and in-plane
-0.06 -0.03 0 0.03 0.06
-0.06
-0.03
0
0.03
0.06
-0.06 -0.03 0 0.03 0.06
-0.002
-0.001
0
0.001
0.002
-0.04 -0.02 0 0.02 0.04
-0.04
-0.02
0
0.02
0.04
-0.01 0 0.01
-0.01
0
0.01
(a) (b)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
(c) (d)yy ys
*x
*xy
-0.006 -0.003 0 0.003 0.006
-0.006
-0.003
0
0.003
0.006
xx ys
yy ys
xx ys xx ys
xx ys
xy ys
xy ys
43
pure shear loading conditions. The results show that optimum G1 cells have lower stress
concentration by effectively distributing the materials within the RVE to reach lower
maximum stress, especially for hexagonal unit cells (figures 3.6 and 3.7). Thus,
compared to G1 cells with a small arc, the failure surface of the optimum G
1 cells covers
a wider range of loads before failure. Figure 3.9 shows the effect of the relative density
on the effective yield strength of the G1 lattices under uni-axial and in-plane pure shear
loadings. As mentioned previously, the values are extracted from the intersection of
yield/ultimate surfaces with the xx
and xy
axes. Figures 3.9(c) and (d) show that at
low relative densities, 10%, the optimum G1 cells with square and hexagonal shapes have
respectively 32.8%1 (36.16%) and 37.7% (32.5%) higher yield strength than G
1 cells
with a small arc under uni-axial (in-plane pure shear) loading. The gained improvement
increases at higher relative densities. Table 3.2 shows the yield/ultimate stresses of the G1
cells for different relative densities. Table 3.3 illustrates the fatigue to monotonic
performance, e us
, of the G1 lattices made of titanium and aluminum 6061T6 alloy
(Defense, 1966; Nicholas, 1981). It can be seen that for lattices with a small arc, the
estimated numerical results are close to the experimental values (0.3 and 0.2 respectively
for R=0.1 and R=0.5) reported by (Côté et al., 2007a). Furthermore, in agreement with
their findings (Côté et al., 2007a), the fatigue to ultimate monotonic stress ratio is
independent of the topology of the unit cell, its relative density, and its material. It should
be noted that in this study, we used a bilinear plasticity model to consider the material
non-linearity of the cell walls of the lattices. For lattices with different microstructure of
the bulk material and described by other plasticity lawsmodel, the above analysis should
be repeated to ensure their validity. These results confirm the experimental data obtained
by Coté et al. and validate the numerical approach presented here for the fatigue design of
lattice materials. This method can be further extended to consider the stress distribution
in the unit cells close to the material boundaries and to model the effect of the grain size
distribution within unit cells that possess very thin cell walls (Zienkiewicz and Taylor,
1 The percentage is calculated as follows:
44
2005). Summarizing a finding of the work presented in this chapter, table 3.3 also shows
that the fatigue to ultimate monotonic stress ratio of a lattice material is a function of the
cell design and varies significantly with the arc radius of the fillet. It also demonstrates
that the proposed numerical method can be used to study the fatigue performance of
newly designed lattice cells.
Figure 3.9. Effective yield strength of the square and hexagonal unit cells under uni-axial and
shear loading as a function of relative density. Yield strength for square cell under uni-axial (a)
and shear loading respectively (b); (c) and (d) pertain to the hexagonal cell.
3.4.3. Modified Goodman diagrams
Figure 3.10 shows the modified Goodman diagrams of G1 cells at different relative
densities; the plots are obtained by using the values of endurance limit and ultimate
strength listed in table 3.2. The intersection of the diagram with the horizontal axis is the
0 0.1 0.2 0.3 0.40
0.01
0.02
0.03
0 0.1 0.2 0.3 0.40
0.02
0.04
0.06
0 0.1 0.2 0.3 0.40
0.005
0.01
0.015
0.02
0 0.1 0.2 0.3 0.40
0.04
0.08
0.12
0.16
xx
ys
xx
ys
xy
ys
(a) (b)
(c) (d)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
xy
ys
45
ratio of the yield strength to the relative density of the unit cell. The corresponding value
on the vertical axis is the alternative macroscopic stress, which, as described in section
2.1. , generates a stress level equal to the fatigue endurance limit of the solid material at
the microstructural level.
As expected, the modified Goodman diagrams of the optimum G1 cells cover a wider
range of applied alternating/mean stresses in comparison to G1 cells with a small arc.
It should be noted that figure 3.10 shows only the modified Goodman diagram of lattices
under uni-axial or pure shear loadings. In practice, however, a mechanical component
should resist complex load combinations, which produce multi-axial macroscopic stress
states in each point. Thus for fatigue design, the modified Goodman diagrams should be
obtained at critical regions subjected to a multi-axial stress state. Figure 3.8 illustrates the
failure surfaces, which are given here for this purpose. For example, for a prescribed
multi-axial stress state, such as 12 5.0 and 0 , the distance between the origin and
intersection points with the ultimate/yield surfaces can be chosen as yield and ultimate
stresses to derive the corresponding modified Goodman diagram. Such a methodology
can be of interest to find the optimum unit cells in a component made of a graded lattice
material.
3.5. Summary and contributions to knowledge
A methodology based on finite element modeling has been presented to design
planar lattice materials for fatigue resistance. Asymptotic homogenization has
been used to determine the macroscopic yield strength, ultimate strength, and
endurance limit, each required to generate the modified Goodman diagrams of the
lattice. A comparison with the experimental data available in the literature reveals
a good agreement of the results.
It has been shown that the geometric design of the unit cell plays a pivotal role in
the fatigue strength of the lattice material. Using an optimum radius for rounding
the unit cell corner of G1 unit cells can substantially increase the fatigue resistance
of square and hexagonal lattices.
46
Failure surfaces of hexagonal and square G1 cells have been obtained. These
results can be used to construct modified Goodman diagrams for cells under
multi-axial loading.
Figure 3.10. Modified Goodman diagram for Ti-6Al-4V square and hexagonal unit cells at given
relative densities. G1
square under uni-axial loading (a), and shear loading condition (b); G1
hexagon under uni-axial loading (c), and shear loading (d).
0 0.004 0.008 0.0120
0.004
0.008
0.012
0 0.05 0.1 0.150
0.025
0.05
0.075
0.1
0.125
0.15
0 0.02 0.04 0.060
0.01
0.02
0.03
0.04
0.05
0.06
0 0.01 0.02 0.030
0.005
0.01
0.015
0.02
0.025
0.03
0 50 100 150 200 250 3000
50
100
150
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
0 50 100 150 200 250 3000
50
100
150
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
0 50 100 150 200 250 3000
50
100
150
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
mean
ys
(a) (b)
(c) (d)
alt
ys
mean
ys
alt
ys
alt
ys
alt
ys
mean
ys
mean
ys
47
Table 3.2. Yield and ultimate strength of G1 unit cells for square and hexagonal lattices.
%10 %10 %20 %20 %30 %30
uni-axial tension shear uni-axial tension shear uni-axial tension shear
y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa)
Square
unit cell
G1 unit cell with small arc
25.3 49.2 0.519 1.29 46.2 112.4 1.48 3.7 65.4 159.3 3.29 8.5
Optimum G1
unit cell 37.65
(0.16)1 48.17
0.813
(0.16) 1.5
73
(0.25) 98.4
3.89
(0.265) 7.1
127
(0.35) 138.7 11.3 19.8
Hexagon
al unit
cell
G1 unit cell with small arc
2.29 5.7 1.1 2.56 7.7 21.8 3.78 11.23 17.45 53.18 8 25.6
Optimum G1
unit cell 3.68
(0.2) 6.52
1.63
(0.2) 2.9
20.6
(0.5) 40.5
9.63
(0.5) 16
54.2
(0.5) 110.4
25.8
(0.5) 44.7
1- The values of optimum fillet radius.
Table 3.3. Fatigue to monotonic performance ratio, /e us , for G1 lattices made of Ti-6Al-4V and Al 6061T6 (material properties in Table
3.1) at given relative densities.
%10 %20 %30
e
us
( 1.0R ) e
us
( 5.0R ) e
us
( 1.0R ) e
us
( 5.0R ) e
us
( 1.0R ) e
us
( 5.0R )
AL(1)
Ti(2)
AL Ti AL Ti AL Ti AL Ti AL Ti
Square
unit cell G1cell with small arc 0.35 0.34 0.19 0.2 0.3 0.34 0.16 0.2 0.26 0.335 0.14 0.19
Optimum G1 unit cell 0.44 0.43 0.26 0.27 0.44 0.44 0.265 0.27 0.445 0.45 0.267 0.29
Hexagonal
unit cell G
1cell with small arc 0.36 0.36 0.2 0.21 0.31 0.3 0.17 0.17 0.28 0.28 0.15 0.156
Optimum G1 unit cell 0.45 0.446 0.27 0.282 0.46 0.467 0.278 0.3 0.47 0.45 0.29 0.288
(1) AL: Aluminum (2) Ti: Titanium
48
Shape optimization of lattice materials for fatigue
resistance
4.1. Objective
It was mentioned in chapter 3 that the geometrical stress concentration at the blending
points of discontinuous curvature reduces the fatigue resistance of a lattice material. In
this chapter, we propose to improve the monotonic and fatigue strengths of planar lattices
by synthesizing the cell boundary profile with curves that are continuous in their
curvature, i.e. G2-continous curves. Furthermore, in order to avoid high bending moments
caused by curved cell members, the cell walls are designed to be as straight as possible,
i.e. with the smallest possible curvature.
This chapter is organized as follows. First the shape optimization strategy for improving
the fatigue strength of cellular material is described. Then this methodology is applied to
improve the fatigue resistance of cellular materials with hexagonal and square shapes of
unit cells. The results are compared with those of optimum G1-cells presented in the
previous chapter. Concluding remarks are the last part of this chapter.
4.2. Design methodology: Basics
4.2.1. Geometrical stress concentration
It is well known that stress concentration can reduce the fatigue resistance of a
mechanical part. Stress concentration can occur in the presence of abrupt changes in
curvature (Dunn et al., 1997; Neuber, 1961a; Pedersen, 2007; Pilkey, 2007; Williams,
1952). Notches, circular fillets, and grooves are common examples of stress
concentrators. Their role is to perturb locally the stress flow because of a curvature
discontinuity in the geometry of a structural element (Neuber, 1961a). Their detrimental
Chapter 4
49
impact has been studied in the literature (Dunn et al., 1997; Neuber, 1961a; Pedersen,
2007; Pilkey, 2007; Williams, 1952) , starting from the seminal work of Neuber, who
first developed a theory of notch stresses with reference to the form and the material of an
element. Neuber showed that the stress concentration factor increases by reducing the
radius of the curvature of the boundary profile of a structure (Neuber, 1961a) . In addition
Neuber (1961c) showed that under pure shear loading condition the elastic stress
concentration, tK , is equal to the multiplication of the notch stress concentration factor,
K , and the strain concentration factor , K . Later studies performed by Topper et al.
(1969) and Walker (1970) showed that this relationship is valid also for cyclic stress
states. More recently, optimization strategies were proposed to reduce the effect of stress
concentration on the strength of mechanical components under monotonic and cyclic
loadings. It has been shown that by reducing the curvature of a fillet, the stress flow
might be smoothed and stress values decrease. For example, Desrochers (2008) looked at
how the shape profile of an element can be optimized to reduce its stress regime under
static condition. Waldman et al. (2001) studied the fatigue of shaft shoulders under
tension and bending loading; he showed that an optimal free-form shape fillet can
provide 23% higher fatigue life than a circular-shape fillet.
It was shown in chapter 3 that stress concentration reduces the ultimate and fatigue
endurance of cellular materials. The reason for local stress concentration is that although
the joints of the unit cells were filleted at the blending points, but the change of curvature
at each blending point was discontinuous. To remove the occurrence of geometry
discontinuity in a lattice, it is proposed here to synthesize the unit cell of the lattice with
curves that are continuous in their curvature i.e. G2-continuous curves (Teng et al., 2008).
Through the formulation of a structural optimization problem explained in the next
section, we first impose that each cell members be G2- continuous at the blending points
with adjacent elements; second, to reduce the high bending stresses caused by curved cell
members, we impose that each cell member be as straight as possible, i.e. with the
smallest possible curvature.
50
Figures 4.1(a) and (b) show the unit cell of the lattice consisting of G2-continous curves.
The next sections describe how the geometry of the lattice cells can be optimized to
improve fatigue life.
4.2.2. Mathematical formulation of the optimization problem
The design method is proposed to find smooth lattice cell topologies based on the
synthesis of structural members with G2-continuous curves that have minimum root mean
square, or rms, value of the curvature (Teng et al., 2008).
Figure 4.1. Schematic views of: (a) G2 continuous square cell; (b) G
2 continuous hexagonal cell;
(c) Parameterization of the inner profile of a unit cell portion.
The shape synthesis of the lattice strut is stated as follows: under given end conditions,
find a boundary-curve Γ that connects two given end points A and B of the cell strut as
smoothly as possible and with a G2-continuous curve. By parametrizing of the cell strut
boundary-curve Γ as a function of the arc-length s along the strut, we can formulate the
optimization problem as
21( )
B
A
J dsL
min
( )s (4.1)
where J is the rms value of the curvature of the boundary-curve of a cell member, L is
the member length, A and B are its end-points, and ds is the arc-length along the
member, starting from 0 at point A, as shown in figure 4.1 (c). The member boundary-
curve is subjected to four constraints at each end-point. Two constraints define the end-
point coordinates, while the other two set the tangent and curvature of the curve at these
points.
1t 2t1t 2t
(b)(a) (c)
y
x
B
A ds kP
51
Equation 4.1 can be treated as a problem of mathematical programming by means of non-
parametric cubic splines (Spath, 1995). Hence, each boundary curve is discretized by n+2
supporting points n+1k 0{P } that are defined by k k kP (ρ ,θ ) in a polar coordinate system. As
shown in figure 4.1(c), kP is a generic point of the curve; 0P =A and n+1P = B , where
A AA(ρ ,θ ) , and B BB(ρ ,θ )are two end-points of the boundary-curve of each cell element.
Moreover, if we assume that the discrete points are located at constant tangential
intervals, the tangential increment will be:
1
B A
n (4.2)
A cubic spline, ( ) , between two consecutive supporting points kP and 1kP can be
defined as:
3 2 2( ) ( ) ( ) ( )k k k k k k kA B C D (4.3)
The radial coordinates, the first and second derivatives of the cubic splines at the kth
supporting point, , and , respectively, are represented by the following three
vectors:
T0 1 n n+1
T0 1 n n+1
T0 1 n n+1
=[ρ ,ρ ,....ρ ,ρ ]
=[ρ ,ρ ,....ρ ,ρ ]
=[ρ ,ρ ,....ρ ,ρ ]
ρ
ρ
ρ
(4.4)
Imposing the G2-continuity condition results in the following linear relationships between
and and between and :
=6Aρ Cρ and =Pρ Qρ (4.5)
where A , C , P , and Q are defined in appendix A. Furthermore, 0 Aρ =ρ and n+1 Bρ =ρ are
known from the given boundary condition representing the cell parameters. Now, if x is
the vector of the design variables, defined as
T1 n=[ρ ,....ρ ]x (4.6)
The discretized shape optimization problem can be written as
52
1
1z( )
n2
k kw κn
x (4.7)
where wk is the weighting coefficient of point kth
defined at each supporting point; it
represents the contribution of each point to the objective function. Furthermore, the
curvature at each point kP is given by:
kk
=rk
2 + 2( ¢rk)2 - r
k¢¢rk
(rk
2 + ( ¢rk)2)3/2
(4.8)
Discretizing the objective function of equation(4.7) and applying the constraints at the
end points of the boundary curve, allow solving the problem with mathematical
programming. The required number of supporting points depends on the geometric
boundary conditions. After performing a sensitivity analysis, the figure of 100 supporting
points has been selected for the boundary curves.
To solve the optimization problem, we used a sequential quadratic programming
algorithm employing orthogonal decomposition algorithm. The details of this method can
be found in the work of (Teng and Angeles, 2001). Furthermore for comparison purposes,
we tried the fminmax subroutine of MATLAB, which uses a gradient based approach,
and found the results were in agreement.
The last note on the optimization strategy described above is the effect of the weighting
coefficients in Eq. (4.7). A weighting coefficient represents the contribution of each point
to the objective function, i.e. the weighted curvature of the optimum curve. For example,
allocating a weighting coefficient equal to unity for each supporting point along the fillet
results in a geometrically optimum shape that does not consider the material properties
and the imposed loading condition. The attribute and stress-strain curve of the material
can be taken into account by defining an outer loop that uses FEA to iteratively define the
weighting coefficient, wk, of equation (4.7) as a function of the stress (or strain) regime.
The weight coefficients are therefore not uniform along the cell strut boundary-curve and
they are defined as (Javid et al. , 2010):
53
k
k
T
w = (4.9)
where k and
T are, respectively, the rms value of the von Mises stress at the thk
supporting point of the profile curve, and the rms value of the stress over the whole cell
element of cell wall and are defined as:
2
1
1 m
T iim
(4.10)
2
1
1 k
k kiik
, m
μ =k 50 (4.11)
where m is the total number of nodes in the FE model, i is the von Mises stress at ith
node and ki is the von Mises stress of the k nodes (2% of the total nodes of FE model),
which are relatively closer to the kth
supporting point. The structural optimization
algorithm is set to end when the reduction in the maximum stress value is smaller than
2%. It should be noted that for pseudo elastic materials, which exhibit a stress-strain
plateau, it is more appropriate to use strain values to define the weighting factors. This
strategy will be adopted in the next chapter where the planar lattice under investigation is
made out of Nitinol. In this chapter we apply the above optimization strategy to design a
planar lattice with hexagonal and square unit cells against fatigue failure.
4.2.3. Cell geometries under investigation
In this section, we apply the methodology described above to the synthesis of lattices
with square and hexagonal cells free of stress concentration. The inner profiles of these
unit cells are synthesized with G2-continuous curves with minimum curvature;
throughout this chapter we call them optimum G2 cells. Figure 4.1(a-b) shows the
parametric view of the synthesized square and hexagonal cells. Besides curvature
minimization, a step of size optimization follows the computational analysis of the yield
strength (recall figure 3.4). Here, the wall size in the middle and corner of a cell (Figure
4.1) is optimized with the goal of minimizing the maximum von Mises stress. The
optimum values of these variables are found through a conjugate gradient optimization
54
method, where the density is set as a constraint. This method is selected because of its
fast convergence that reduces the computational time.
In this chapter we apply the fatigue design methodology presented in chapter 3 to study
the fatigue resistance of the optimum G2 cells. To study the effectiveness of the above
method in optimizing the fatigue resistance of a lattice material, we compare here the
fatigue resistance of G1 unit cell, as presented in chapter 3, with that of optimum G
2 cells.
To this end, the numerical details described in section 3.3.3 apply here.
4.3. Results
Figures 4.2 and 4.3 show the stress distribution in the titanium square and hexagonal unit
cells with G1 and G
2 cell shapes under fully reversed uni-axial and shear loading
conditions. Stress concentration can be seen at the blending points of the G1 cells, while
the stress in the G2 cells is distributed more uniformly along the cell struts. In addition, it
can be seen that under uni-axial loading there is no stress concentration at the joints of the
G2 square cell.
Figure 4.2. von Mises stress (MPa) distribution in hexagonal and square unit cells made out of
Ti-6Al-4V. Lattices under fully reversed uni-axial loading, defined by: optimum G1 cell (left) and
optimum G2 cell (right).
55
Figure 4.4 shows the yield (solid marks) and ultimate (open marks) surfaces for both G1
and G2 cells, calculated at relative density of 10%. Here, the geometric parameters of the
unit cells were selected to optimize the fatigue performance of the lattice under uni-axial
and shear loading conditions. Figure 4.4 shows that G2 cells have higher yield and
ultimate strength than those of G1 cells.
Figure 4.3. von Mises stress (MPa) distribution in hexagonal and square unit cells made out of
Ti-6Al-4V. Lattices under fully reversed pure shear loading, defined by: optimum G1 cell (left)
and optimum G2 cell (right).
Figure 4.5 shows the effective yield strength of optimum G1 and G
2 unit cells under uni-
axial and shear loadings as a function of the relative density. As mentioned previously,
the values are extracted from the intersection of yield/ultimate surfaces with the xx and
xy axes. Figures 4.5(a) and (b) show that although both optimum G1 and G
2 square
lattices have comparable yield strengths under axial loading condition, the stress
concentration at the joints of the G1 cell reduces its yield strength under shear loading.
Figures 4.5 (c) and (d) show that at low relative densities, the hexagonal G2 cell has up to
50% higher yield strength in comparison to the optimum hexagonal G1 cell; for higher
relative densities, on the other hand, the two unit cells have comparable yield strength.
Table 4.1 lists the yield/ultimate stresses of the optimum G1 and G
2 cells for different
relative densities. Table 4.2 reports the fatigue to monotonic performance ratio, /e us ,
56
of the G1 and G
2 lattices made of titanium and aluminum 6061T6 alloy (Defense, 1966;
Nicholas, 1981). Figure 4.6 shows the modified Goodman diagrams of the G1 and G
2
cells at given relative densities. These diagrams are obtained by using the values listed in
table 4.1.
Figure 4.4. Yield and ultimate surfaces of Ti-6Al-4V square and hexagonal unit cells for relative
density of 10%. Projection of yield and ultimate surfaces of optimum G1 and G
2 square and
hexagonal cells in the yyxx and xyxx planes.
-0.06 -0.03 0 0.03 0.06
-0.06
-0.03
0
0.03
0.06
-0.06 -0.03 0 0.03 0.06
-0.002
-0.001
0
0.001
0.002
-0.04 -0.02 0 0.02 0.04
-0.04
-0.02
0
0.02
0.04
-0.01 -0.005 0 0.005 0.01
-0.01
-0.005
0
0.005
0.01
(a) (b)
(c) (d)yy ys
*xy
xx ys
yy ys
xx ys xx ys
xx ys
xy ys
xy ys
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
57
Figure 4.5. Effective yield strength of the square and hexagonal unit cells under uni-axial and
shear loading as a function of relative density. Yield strength for square cell under uni-axial (a)
and shear loading respectively (b); (c) and (d) pertain to the hexagonal cell.
4.4. Discussion
The plots in figures 4.2 and 4.3 show the pivotal role of cell geometry on yield, ultimate
and fatigue strengths of the studied lattice materials. For each case, the local stress
concentration observed in G1 cells is removed by using unit cells with smooth G
2 corners.
A comparison between the maximum stress (figures 4.2 and 4.3) and failure surfaces
(figure 4.4) of the optimum G1 and G
2 cells shows the advantage of optimizing the cell
geometry to improve the fatigue resistance of the lattice geometries under investigation.
The results show that G2 cells, especially the hexagonal cell, can distribute the materials
more efficiently within the RVE so as to yield a lower maximum stress without stress
concentration (recall Figs. 4.2 and 4.3). Thus, in comparison with G1 cells, the failure
surfaces of G2 cells are wider, showing a higher fatigue resistance.
0 0.1 0.2 0.3 0.40
0.01
0.02
0.03
0 0.1 0.2 0.3 0.40
0.005
0.01
0.015
0.02
0 0.1 0.2 0.3 0.40
0.02
0.04
0.06
0 0.1 0.2 0.3 0.40
0.04
0.08
0.12
0.16 (a) (b)
(c) (d)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
-40 -20 0 20 40
-40
-20
0
20
40
Opt arc (Yield)
Opt arc (Ult)
G2 (Yield)
G2 (Ult)
Small arc (Yield)
Small arc (Ult)
xx
ys
xx
ys
xy
ys
xy
ys
58
Table 4.1. Yield and ultimate strength of the optimum G1 and G
2 unit cells for square and hexagonal lattices.
%10 %10 %20 %20 %30 %30
Uni-axial tension shear Uni-axial tension shear Uni-axial tension shear
y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u
(MPa) y
(MPa) u (MPa)
Square unit
cell
Optimum
G1 unit cell
37.65 48.17 0.813 1.5 73 98.4 3.89 7.1 127 138.7 11.3 19.8
Optimum
G2 unit cell
38.97 43.39 1.26 1.98 77.34 93.4 6.11 9.9 131.7 145.8 15.5 25.3
Hexagonal
unit cell
Optimum
G1 unit cell
3.68 6.52 1.63 2.9 20.6 40.5 9.63 16 54.2 110.4 25.8 44.7
Optimum
G2 unit cell
5.48 8.96 2.45 4.25 22.4 44.6 10.4 17.3 56.3 113.86 27.2 47.3
Table 4.2. Fatigue to monotonic performance ratio, /e us , of optimum G1 and G
2 lattices made of Ti-6Al-4V and Al 6061T6 (material
properties in Table 4.1 of chapter 3) at given relative densities.
%10 %20 %30
e
us
( 1.0R ) e
us
( 5.0R ) e
us
( 1.0R ) e
us
( 5.0R ) e
us
( 1.0R ) e
us
( 5.0R )
Al Ti Al Ti Al Ti Al Ti Al Ti Al Ti
Square
unit cell
Optimum G1 unit cell 0.44 0.43 0.26 0.27 0.44 0.44 0.265 0.27 0.445 0.45 0.267 0.29
Optimum G2 unit cell 0.48 0.487 0.28 0.318 0.475 0.476 0.28 0.31 0.47 0.474 0.28 0.31
hexagonal
unit cell
Optimum G1 unit cell 0.45 0.446 0.27 0.282 0.46 0.467 0.278 0.3 0.47 0.45 0.29 0.288
Optimum G2 unit cell 0.455 0.45 0.27 0.288 0.46 0.467 0.28 0.3 0.475 0.45 0.29 0.288
59
Figure 4.6. Modified Goodman diagram for Ti-6Al-4V square and hexagonal unit cells at given
relative densities. G1
and G2 square under uni-axial loading (a), and shear loading condition (b);
G1 and G
2 hexagon under uni-axial loading (c), and shear loading (d).
By comparing the failure surfaces and modified Goodman diagrams of G2 and G
1 cells
(figures 4.4 and 4.6), we observe that G2 cells have a higher fatigue resistance for
bending dominated lattices. On the other hand, for the square cell, which is stretching
dominated under uni-axial loading, G2 cells have fatigue resistance comparable to
optimum G1 cells. In this lattice the stress regime of cell struts parallel to the loading
direction is fairly uniform and is mainly controlled by the thickness of the cell members.
0 0.05 0.1 0.150
0.05
0.1
0.15
0 0.005 0.01 0.0150
0.005
0.01
0.015
0 0.01 0.02 0.030
0.01
0.02
0.03
0 0.02 0.04 0.060
0.02
0.04
0.06
mean
ys
(a) (b)
(c) (d)
alt
ys
mean
ys
alt
ys
alt
ys
alt
ys
mean
ys
mean
ys
0 50 100 150 200 250 3000
50
100
150
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
0 50 100 150 200 250 3000
50
100
150
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
0 50 100 150 200 250 3000
50
100
150
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
G2 cell
Small arc
Opt arc
60
Thus, the material distribution around the RVE joints and the curvature discontinuity of
the RVE profile has a minor effect on the stress regime. On the other hand, for a bending
dominated lattice, the bending moment is maximum at the cell corners. This suggests that
the distribution of material in this region governs the maximum stress and the fatigue
resistance.
4.5. Concluding remarks
A shape optimization strategy has been described to improve the fatigue
resistance of lattice materials by reducing the effect of geometrical stress
concentration. The inner profile of these optimized unit cells are synthesized with
G2-continuous curves of minimum weighted curvature. Failure surfaces and
modified Goodman diagrams of the optimized G2 cells were extracted. The results
show that for bending dominated lattices, the optimum G2 cells have higher
fatigue resistance than optimum G1 cells. Thus it is suggested that for a bending
dominated lattices, if manufacturability is not a concern, optimum G2 cell lattices
should be preferred over unit cells with arc-rounded joints.
As a practical engineering application, the following chapters will describe how
the shape optimization formulation described above can be integrated in the
design of self-expandable Nitinol stent grafts and a novel state-of-the-art
protective cage for rotary intravenous respiratory assist catheter.
61
Chapter 5
Shape optimization of stress concentration-free lattice
for self-expandable Nitinol stent-grafts
5.1. Objectives
The method to reduce stress concentration described in chapter 4 is applied here to
improve the fatigue resistance of Nitinol self-expandable stent-grafts. These stents grafts
are here considered as made of periodic closed-cells. The shape of the unit cell is
optimized to obtain smooth profiles of the unit cell and then used to generate a planar
array for a stent-graft. Design optimization is systematically applied to minimize the
curvature and to reduce the bending stresses of the elements defining the stent cells.
The chapter is organized as follows. First, a description of stent typology, their
application and design challenges are briefly reviewed. Then, the shape optimization
strategy described in chapter 4 is implemented to design a self-expandable Nitinol stent-
graft against fatigue failure. In section 4, a parametric study is carried out to study the
effect of the selected geometric parameters on the fatigue resistance and radial stiffness
of the stent. Concluding remarks are given at the end of the chapter.
5.2. Introduction to structural design of stents
Intravascular stents are primarily used to open and scaffold tubular passages or lumens
such as blood vessels, biliary ducts and the esophagus (Duerig et al., 1999). They may
consist of expandable lattice meshes that can deploy and hold endovascular grafts,
arterial endoprosthesis and self-expanding heart valve implants. Figures 5.1 (a-e) show
recent commercial applications of stent devices, which are designed to deploy into the
body by minimally invasive percutaneous intervention (Kleinstreuer et al., 2008b; Rose
et al., 2001; Vergnat et al., 2009; Webb, 2008). Based on the deployment mechanisms,
stents can be loosely classified into balloon expanding (BE) structures and self expanding
(SE) structures. BE structures, which are manufactured from a tube with a radius smaller
62
than the radius of the target vessel, are deployed using a retractable inflating balloon. The
plastically deformed structure preserves its deployed shape after the balloon deflation and
retraction. In contrast, SE stents are manufactured from tubes with a diameter larger than
the diameter of the target vessel. For delivery and insertion purposes, the stent must be
compressed elastically into a smaller diameter delivery catheter, which is then inserted
percutaneously into the body. Upon reaching the desired position, the stent is deployed to
its original shape by removing the casing catheter.
Figure 5.1. Commercially available stents developed for prescribed applications (Masoumi
Khalil Abad et al., 2012).
Depending on the stent application, its structure should address multiple functional
requirements and often conflicting objectives. For example, bare metal stents used for
opening the occluded arteries, as shown in figures 5.1. (d-e), are required to provide a
combination of high radial force and axial flexibility in order to keep the artery open,
prevent stent migration, conform to the curved blood vessels, and flex during the body
movement (Cheng et al., 2006). Stents used as prostheses of aortic valve (Figure 5. (a-b))
are required to provide high radial strength to exclude the calcified leaflets and to avoid
recoil. They must also fix the stent in the ascending aorta and provide longitudinal
stability (Grube et al., 2006). In endovascular repair for abdominal aortic aneurisms
(AAAs), the structure of a stent-graft (Figure 5. (c)) should provide sufficiently high radial
63
force to prevent graft migration and blood leakage into the aneurysm cavity (Kleinstreuer
and Li, 2006; Kleinstreuer et al., 2008b).
Since 1990, an ever increasing demand for endovascular stents has led to significant
advancements in the field of analysis, modeling and design of stent structures. Restenosis
rate, vessel patency, cloth formation, stent migrations, stent collapse, stent positioning,
and stent expansion behavior are common concerns that have attracted the attention of
several researchers (Bedoya et al., 2006; Chua et al., 2002; Duerig et al., 1999;
Flueckiger et al., 1994; Kleinstreuer et al., 2008b; Lally et al., 2005; Lim et al., 2008;
Martin and Boyle, 2010; Migliavacca et al., 2002; Petrini et al., 2004; Timmins et al.,
2007; Wang and Masood, 2006). These studies have shown that besides mechanical and
biological factors, the geometry and typology of a stent is a crucial aspect that governs
the device function and its performance. A parametric FE study on an actual balloon
expandable (BE) stent demonstrated the considerable influence of the metal surface area
in contact with artery on the stent radial stiffness and its radial expansion behavior
(Migliavacca et al., 2002). In addition, the impact of the typology on stent dogboning and
foreshortening, and the level of induced wall stress in the atherosclerotic arteries has been
examined for a number of commercially available BE stent designs.(Lally et al., 2005;
Lim et al., 2008). Bedoya et al. (2006) performed first a parametric study on their
proposed generic BE stent. Then they optimized the performance of the stent design by
minimizing the normalized weighted sum of the wall stress, lumen gain, and cyclic
deflection of the artery wall (Timmins et al., 2007). A work by Early and Kelly (2010)
investigated the wall stress induced in the stented femoral and coronary arteries after
deploying two typologically different SE stents and one BE stent. Their conclusion is that
SE stents generally induce lower stress level in the vessel wall than BE stents do.
5.3. Problem statement
Shapes, size as well as the thickness and width of a lattice cell are geometric variables
that can be tailored to improve the mechanical performance of a stent structures, such as
its fatigue life, axial flexibility and radial stiffness. Design optimization can be used to
find the values of these variables that best optimize one or more performance metrics of a
stent device. So far, however, the synthesis of a stent through systematic design
64
optimization has received minor attention. For example, figure 5.1(f) shows the structural
geometry of a recent stent consisting of a 2D lattice of closed-cells (Zhi et al., 2008). At
the blending points between the arcs and the linear segments of each cell, the curvature
has a discontinuity that acts as a stress concentrator (Neuber, 1967). In ten years life, a
stent can undergo nearly four hundred millions of cycles, mainly because of pulsating
blood pressure, and body movement. Such a cyclic load drastically amplifies the effect of
stress concentration that eventually reduces the fatigue life of the stent. The need to
reduce the level of stress concentration in a modular structure motivates this chapter. Due
to the existence of several stent applications, each entailing the fulfillment of specific
requirements, the focus of this chapter is on stent grafts used for treating abdominal aortic
aneurism. The success of these stent grafts is often undermined by stent fatigue, graft
migration, and blood leakage into the aneurysm cavity (Kleinstreuer and Li, 2006; Li and
Kleinstreuer, 2005). Three strategies can be adopted to reduce these risks in stent grafts
made of metallic materials: i) stiffen the stent in the radial direction to reduce
endovascular leakage and device migration; ii) reduce the level of the alternating strain
generated by a pulsating blood pressure to increase the stent’s fatigue life; iii) to use
biodegradable stents.
5.4. Shape synthesis of lattice geometry
We employ in this chapter the design strategy presented in the previous chapter to
synthesize a planar lattice free of stress-concentration. In contrast to the weighting
coefficients chosen in equ. (4.7), wk in this chapter is determined as a function of the
strain regime of the material. We consider strain, rather than stress because the plateau
region of the Nitinol stress-strain curve (figure 5.2), is much more sensitive to strain
changes. This region corresponds to the stress induced phase transformation from the
austenite to the martensite state and has a strong impact on the fatigue life of the Nitinol.
The weight coefficients are therefore defined as:
k
k
T
εw =
ε (5.1)
65
where kε and
Tε are, respectively, the rms values of the von Mises strain at the thk
supporting point of the profile curve, and the rms value of the strain over the whole cell
element of the stent and are defined as:
2
1
1 m
T iim
(5.2)
2
1
1 k
k kiik
, m
μ =k 50 (5.3)
where m is the total number of nodes in the FE model, i is the von Mises strain at the ith
node and ki is the von Mises strain of the k nodes (2% of the total nodes of FE model),
which are relatively closer to the kth
supporting point. The structural optimization
algorithm is set to end when the reduction in the maximum strain value is smaller than
0.1%.
Figure 5.2. Schematic view of Nitinol stress-strain curve.
Figure 5.3 shows the unit cell of the lattice stent consisting of G2-continous curves.
Because of its elliptical shape of the proposed unit cell and for the sake of brevity,
throughout this chapter we will recall this cell as E cell. The unit cell is repeated in a
planar sheet to form the lattice, which is then folded into a cylindrical surface. The lattice
cylinder is described by nc cells in the circumferential direction and nl distinct cell rows
in the longitudinal direction. The tube thickness and strut width are respectively t and w,
ASf
ASs
SAs
L
SAf
66
and we assume that the stent has a total length of 100mm and a non-shrunk diameter of
30mm (Kleinstreuer et al., 2008b).
Figure 5.3. Schematic view of the proposed G2-continuous cell geometry: (a) the proposed E cell
geometry; (b) parameterization required for the synthesis of a G2-continuous cell shape; (c) inner
boundaries of initial design and structurally optimized E cell.
5.4.1. Numerical modeling
5.4.1.1. Finite element modeling
The stent geometry is synthesized through a MATLAB subroutine, which is coupled to
ANSYS to build, mesh, and solve the 3D model of the stent. Here, only the stent rows in
contact with the aneurism neck are examined due to their importance for stent-graft
migration and fatigue life (Kleinstreuer et al., 2008b). Usually, a stent consists of a set of
separate rows that are sutured on the graft fabric. Between rows, there is a gap in the
axial direction to allow a relative movement of the stent rows and to increase the axial
flexibility of the stent. In the sealing section located at the two distal rows of the stent-
graft, the stent does not gain its original size and the graft material is not in tension. Since
the stiffness of the graft material is very low, the effect of the connectivity of the rows in
the sealing section can be neglected (Kleinstreuer et al. 2008). Because of symmetry in
both geometry and loading, only ¼ of one cell is modeled. Symmetric boundary
conditions are applied at the planes of symmetry. To mesh the stent elements of the
lattice cell, a 3D eight-node element type, SOLID 185, is selected. The arterial wall is
modeled as a cylinder and meshed by a twenty-node element type, SOLID 95. A mesh
sensitivity test is also performed to ensure the independency of the results from the mesh
size.
67
5.4.1.2. Material model
During the insertion process, SE stents should withstand large elastic deformations; thus
the mechanical properties of the material out of which they are made should be selected
to accommodate large strains. In these instances, Nitinol, as a bio-compatible material
with ability to withstand severe deformation without plastic deformation is an ideal
candidate (Duerig et al., 1999). Figure 5.2 shows the schematic view of the stress-strain
curve of Nitinol at a given temperature. The large range of the recoverable strain and the
existence of the stress plateau in the stress-strain curve, namely pseudo-elasticity
behaviour of the Nitinol, are governed by stress-induced phase transformations of the
material under mechanical loading. At low stress levels, the stress varies linearly with
respect to the strain. Under a stress increase, the material, initially in the austenite phase,
undergoes a stress-induced martensite transformation. By this transformation, the
material undergoes large strains. During unloading, through a reverse transformation
from the martensite to the austenite phase, the induced strains are fully recovered and the
material returns to the original austenitic stress–strain state.
Over the past two decades, the area of constitutive modeling of shape memory alloys,
such as Nitinol, has been the topic of research efforts, with significant advancements
(Auricchio, 1995). To characterize the pseudo-elastic response of shape memory alloys, a
class of constitutive models have been developed based on a selected hardening function
that models the stress-strain response during the stress–induced martensite
transformation. Such constitutive models introduce a linearized stress–strain relation as
seen in figure 5.2 with, , , , , and , as material constants. Here we use
the constitutive model by Auricchio (1995) to model the super-elastic properties of
Nitinol. Table 5.1 shows the material properties of Nitinol used in this study.
Aneurism artery wall is assumed to be an isotropic material with a nearly incompressible
material with a Young’s modulus of 1.2MPa and a Poisson’s ratio of 0.495. The structure
of the artery neck is assumed as a cylinder with internal diameter of 22 mm and wall
thickness of 1.5 mm (Kleinstreuer et al., 2008b).
68
Table 5.1 Nitinol material properties (Kleinstreuer et al., 2008b).
EMartensite
(GPa)
AusteniteE
(GPa) ν
ASsσ
(MPa)
ASfσ
(MPa)
SAsσ
(MPa)
SAfσ
(MPa)
Lε (MPa)
47.8 51.7 0.3 600 670 288 254 6.3%
5.4.1.3. Loading conditions
a. Shrinking loading
For delivery purposes, the stent-graft assembly with outer diameter of 30mm must be first
shrunk to fit into the 24F delivery sheath and then, when deployed, must regain its
original shape. We model the shrinking manoeuvre by applying a radial displacement to a
rigid movable surface, which is in frictionless contact with the strut’s outer surface. The
graft material is assumed to have a negligible effect on the overall behavior of the stent in
the sealing section; thus the graft is not considered for modeling.
b. Sealing loading
The stent should be anchored to the neck artery of the abdominal aortic aneurism (AAA)
after its release from the deployment system. The anchoring force should be sufficiently
high to prevent the stent-graft migration. In this , the stent deployment is modelled
in two steps. First, the stent is shrunk to a diameter close to the artery interior wall by
using a rigid contact surface. Second, the stent expanded to reach an equilibrium radius in
contact with the artery wall by gently removing the contact surface of the rigid body. The
diastolic and systolic blood pressures are modeled as constant pressures applied to the
inner surface of the artery wall.
5.5. Results
Figure 5.3(c) shows the results of minimizing the curvature of the inner boundary-profile
for the E lattice cell. Figure 5.4 shows the views of the structurally optimized stents.
Figure 5.5(a) illustrates the von Mises strain distribution in the shrunk stent. It can be
seen that the maximum strain level is below the 12% allowable threshold strain limit of
Nitinol (Kleinstreuer et al., 2008b). It shows that the stent can shrank without fracture.
However, as explained later in the discussion, the deployment constraint imposes a
maximum on the allowable number of cells in the circumferential direction. The
distribution of the first principal strain in the deployed stents is shown in figure 5.5(b).
69
Table 5.2 shows the performance of the proposed design in comparison with the available
reference stent investigated by (Kleinstreuer et al., 2008b). We will refer to this stent as R
stent. It should be noted that the requirement used for comparison in table 5.2 is the area
of the R stent in contact with artery; this area is assumed to be equal to the area of the E
stent. For a given surface area requirement, we select as design variable the strut width
and we fix as design parameters: 1) the number of cells in the longitudinal direction so as
the stents have equal share of pressure on the artery wall at each row; 2) the stent
thickness, as its effect on the blood flow and hemodynamic properties may be significant.
Table 5.2 shows that the proposed E stent has 69.1% higher fatigue safety factor2 and
82.4% larger radial supportive force per unit of stent area. Figure 5.5 (c) shows the von
Mises stress distribution induced in the artery wall after graft deployment. The stress
level in the artery wall is below 0.67MPa, the elastic limit of the artery (Raghavan et al.,
1996). However, compared to the R stent, the level of von Mises stress induced in the
artery wall exhibits a 32.4% increase. This stress level might reduce over time but it
should be below the allowable elastic limit of the artery wall after stent insertion. Figure
5.6 illustrates the radial supportive force as a function of the outer diameter for the E
stent in comparison with the R stent for a prescribed stent area and tube thickness. For a
2mm constant radial displacement, the proposed E cell design provides 165% increase in
the supportive radial force.
2 where Pelton, A.R.,
Gong, X.Y., Duerig, T., Year Fatigue testing of diamond-shaped specimens. and
70
(a)
(b)
Figure 5.4. Structurally optimum stent. (a) a straight row of lattice cells, (b) a row folded into a
cylinder.
Figure 5.5. FEA results for E cell. (a) Strain distribution in the shrunk stent; (b) first principal
strain in the stent after stent deployment under 100 mm-Hg mean pressure.; (c) von Mises stress
(in MPa) distribution in the artery after stent deployment under 100 mm-Hg mean pressure. The
maximum value occurs at the interface between stent and artery wall.
ANSYS 12.1
SEQV.249434.26072.272005.283291.294575.305859.317145.328429.339715.351124
ANSYS 12.1
EPS1.519E-04.443E-03
.002008
.002399
.002791
.003182
.003574
.001617
.001255
.834E-03
MAX
.297E-03 .010499 .020744 .030991 .041235 .051482 .061727 .071974 .082219 .092465
MAX
(a)
(b) (c)
71
Figure 5.6. Radial supportive force versus stent outer diameter of E-stents compared to R cell
stent for a given area in contact with the artery wall. The design parameters for E-stent are nc = 8,
nl = 10, t = 0.28mm, w = 0.45mm, while those for R stent are nc = 20, nl =10, t = 0.28mm, w =
0.35mm (Kleinstreuer et al., 2008b).
The results of the parametric study show that to obtain a shrinkable stent an upper limit is
required on the number of cells in the circumferential direction. For example, figure 5.7
(a-c) show that for a stent with nl = 10, t = 0.28mm, w = 0.45mm, only values of nc less
than 10 enable the stent to be shrunk without fracture.
The impact of the number of cells in the circumferential direction, nc, is illustrated in
figure 5.7 (a-c). Whereas the supportive radial force of the stent is not affected, the stent
area shows a rapid linear increase. The stent fatigue safety factor, on the other hand,
0
5
10
15
20
25
5 10 15 20 25 30
Ra
dia
l s
up
po
rtiv
e f
orc
e (
N)
Stent outer diameter (mm)
E Cell
R Cell
Radial force at
100 mmHg (N)
Fatigue safety
factor
Wall stress
(MPa)
Maximum shrunk
strain (%)
E cell 3.1 3.4 0.351 9.42
R cell 1.7 2.01 0.265 8.86
Table 5.2 Comparison of stent performances. Reference cell, or R cell, from (Kleinstreuer et al.,
2008b).
72
Figure 5.7. Plots of number of cells in the circumferential and radial direction, thickness and width of cell elements versus
radial force, fatigue safety factor, and metal area in contact with artery for E cell geometry. (a-c) effect of nc for , nl = 10,t =
0.28mm, w = 0.45mm (d-f) effect of nl for t = 0.28mm, w = 0.45mm , nc = 8; (g-i) effect of t for, w = 0.45mm , nl = 10, nc =
8; (j-l) effect of w for, t = 0.28mm , nl = 10, nc = 8 for E cell geometries. R stent is a benchmark stent design (Kleinstreuer et
al., 2008b); its design parameters are nc = 20, nl = 10, t = 0.28mm, w = 0.35mm.
1.5
1.9
2.3
2.7
3.1
3.5
3.9
5 7 9 11 13 15 17 19 21
Ra
dia
l fo
rce
(N
)
Number of cells in circumferential direction
E Cell
Ref [2]
Deployability
Constraint
1.5
2
2.5
3
3.5
4
4.5
5
5.5
5 7 9 11 13 15 17 19 21
Fa
tig
ue s
afe
ty f
acto
rNumber of cells in circumferential direction
E Cell
Ref [2]
Deployability
Constraint
1000
1200
1400
1600
1800
5 7 9 11 13 15 17 19 21
Me
tal a
rea
(m
m^
2)
Number of cells in circumferential direction
E Cell
Ref [2]Deployability
Constraint
1.5
1.9
2.3
2.7
3.1
3.5
3.9
7 9 11 13
Ra
dia
l fo
rce (
N)
Number of cells in longitudinal direction
E Cell
Ref [2]
1.5
2.5
3.5
4.5
5.5
7 9 11 13
Fa
tig
ue s
afe
ty f
ac
tor
Number of cells in longitudinal direction
E Cell
Ref [2]
1000
1400
1800
7 9 11 13
Me
tal are
a (
mm
^2)
Number of cells in longitudinal direction
E Cell
Ref [2]
1.5
1.9
2.3
2.7
3.1
3.5
3.9
0.2 0.25 0.3 0.35 0.4 0.45
Rad
ial
forc
e (
N)
Strut thickness (mm)
E Cell
R Cell
1.5
2.5
3.5
4.5
5.5
0.2 0.25 0.3 0.35 0.4 0.45
Fati
gu
e s
afe
ty f
acto
r
Strut thickness (mm)
E Cell
R Cell
1000
1200
1400
1600
1800
0.2 0.25 0.3 0.35 0.4 0.45
Meta
l are
a(m
m^
2)
Strut thickness (mm)
E Cell
R Cell
1.5
1.9
2.3
2.7
3.1
3.5
3.9
0.25 0.35 0.45 0.55
Rad
ial
forc
e (
N)
Strut width (mm)
E Cell
R Cell
1.5
2.5
3.5
4.5
5.5
0.25 0.35 0.45 0.55
Fati
gu
e s
afe
ty f
acto
r
Strut width (mm)
E Cell
R Cell
1000
1200
1400
1600
1800
0.25 0.35 0.45 0.55
Meta
l are
a (
mm
^2)
Strut width (mm)
E Cell
Ref [2]
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
73
decreases if nc does. Therefore, higher values of nc should be chosen while respecting the
deployment constraint as shown in figure 5.7 (a). It is noteworthy, also, that reducing nc
might increase the stress level in the artery wall.
Figures 5.7 (d-f) illustrate the influence of the number of cells, nl, in the longitudinal
direction on the stent performance. By increasing nl for a given arterial length, the share
of each row in supporting the arterial radial load decreases that reduces the level of radial
supportive force as shown in figure 5.7 (d). In addition, the stiffness of the stent increases
by shortening the length of each cell row. This outcome improves the stent fatigue safety
factor by reducing the level of alternating strain.
Figures 5.7 (g) and (j) show that thickening the strut and width is beneficial for both stent
radial stiffness and radial supportive force. Besides these gains, a stiffer stent would be
also more resistant to the deformation imposed by a pulsatile pressure, thereby reducing
the alternating strain experienced by its members. This is observed in figures 5.7 (h) and
(k), where the fatigue safety factor increases linearly with w and t. On the other hand,
Figure 5.7 (i) shows that the stent area is not affected by any change of the stent thickness
as opposed to the trend observed by varying nc, nl, w in figures 5.7 (c), (f), and (l).
The result of figure 5.7 (g), however, should be taken with a caution. A thicker strut will
cause a higher contact stress in the artery wall. Furthermore, blood flow in proximity with
the artery wall and stent struts will affect the selection of the strut thickness. These issues
should be determined through multi-disciplinary analysis and optimization involving both
computational fluid dynamics and structural analysis.
5.6. Concluding remarks
This chapter has presented a design methodology based on shape optimization to
improve the fatigue safety factor and to increase the radial supportive force of Nitinol
self-expandable stent-grafts with closed-cell geometry. To increase stent fatigue life,
we have proposed to synthesize the shape of each cell of the lattice with elements of
continuous curvature. Minimizing their curvature has reduced the bending moments
caused by curved cell members. A novel cell geometry has been synthesized, and its
radial supportive force and fatigue safety factor have been studied through a FEA
parametric analysis. Compared to recent stent design, the results have shown an
74
improvement of the stent anchoring performance and a reduction of the risk of fatigue
failure.
As shown by the results of the parametric study, stent radial supportive force, fatigue
failure safety factor, and stress level in the artery wall often have conflicting
outcomes. Furthermore, other parameters such as crimping and dogboning are
important properties of a stent that should be studied. An improvement of one will
penalize the other. It is, thus, necessary to formulate the shape synthesis of the lattice
cell within a multi-objective optimization framework (Messac et al., 2003), which
would be capable of providing trade-off solutions among the conflicting objectives.
In this chapter, stent design for fatigue life has been tackled by minimizing the
occurrence of stress concentration due to geometric discontinuity. This method can be
complemented by integrating a fracture mechanics approach, which is based on the
design guidelines for fatigue design of Nitinol devices (Robertson and Ritchie, 2007;
Robertson and Ritchie, 2008; Stankiewicz et al., 2007). In such a structural design
approach, various fatigue parameters, including the effect of loading ratio, R, and
phase transition in the tip of fatigue crack are considered.
The methodology proposed in this chapter can be applied to synthesize the geometry
of other types of stents, e.g. superficial femoral artery stents, to meet prescribed
design objectives imposed by the specific application.
75
Chapter 6
Structural design of a protective cage for a rotating
intravenous oxygenator
6.1. Objectives
The percutaneous respiratory assist catheter (PRAC) with rotary bundle is a promising
alternative to current mechanical ventilators for respiratory assist of patients with acute
respiratory failure. The device, developed by ALung Technologies Inc. (Pittsburgh, PA)
in partnership with the University of Pittsburgh, is a rotating intravenous oxygenator,
which is inserted through a peripheral vein into the vena cava. PRAC is expected to
drastically reduce the patient’s recovery period (from 3 weeks to 7 days), associated risks
of infection, and iatrogenic injuries. One remaining obstacle to achieve the proper
functioning of the oxygenator is the design of an encasement to protect the vena cava
from injury during the rotation at high speeds. In this chapter, we apply the method for
the synthesis of a planar lattice free of stress concentration to design a protective
encasement for PRAC with rotary bundle.
This chapter is organized as follows: first the background, motivations, and design
objectives of the protective cage are described. Since the cage-supported catheter works
in contact with the blood, it should be designed to have minimum level of blood damage.
Hence, a review of the principles required to design blood-contacting medical device is
given in appendix C at the end of this thesis while the fundamentals for gas oxygenation
are reported in appendix B. In section 6.3, we propose two conceptual designs for the
protective cage. The detailed structural and blood flow analyses of the proposed solutions
are given. The chapter ends with concluding remarks.
76
6.2. Background, motivation, and problem statement
6.2.1. Lung diseases, statistics, and available treatments
Acute and chronic lung diseases present a major healthcare problem in Canada and the
United States, affecting an estimated 10% of the population (Public Health Agency of
Canada, 2007; Svitek and Federspiel, 2004). Over 3 million Canadians cope with one of
five serious respiratory diseases – asthma, chronic obstructive pulmonary disease or
COPD, lung cancer, tuberculosis, and cystic fibrosis (Public Health Agency of Canada,
2007). Since many of these conditions are linked to an aging population, their prevalence
is predicted to increase. The economic effects of these diseases, not including lung
cancer, are tremendous: nearly $5.7 billion of direct costs and $6.7 billion of indirect
costs are incurred annually, representing 6.5% of the total healthcare cost.
Among the respiratory diseases mentioned above, chronic obstructive pulmonary disease
(COPD) is a prevalent lung disease with a high mortality rate3. Acute exacerbations of
COPD occur because of pollution, bacterial and viral infections, and changes in
environmental temperature (Petty, 2002). Patients who are in need of medical treatment
for COPD often suffer from three acute exacerbations per year (Niederman, 1998).
Fortunately, acute exacerbations may be reversible if the lungs can be temporarily
supported and allowed to heal (Davidson, 2002; Hirvela, 2000; Sethi and Siegel, 2000).
The required temporary supplemental oxygenation and carbon dioxide removal is
conventionally done by one of two methods: (1) mechanical ventilation, which forces air
into the lungs at a prescribed pressure or volume; or (2) extracorporeal membrane
oxygenation (ECMO), whereby carbon dioxide and oxygen are exchanged through
artificial membranes in an external blood circuit. Mechanical ventilation and ECMO
systems, which are often employed complementarily, are mechanically complex and
3 COPD affects 4%-10% of adults Halbert, R., Isonaka, S., George, D., Iqbal, A., 2003. Interpreting COPD
Prevalence Estimates*. Chest 123, 1684-1692.and is reported as a cause of almost 662,000 hospitalizations
in US that led to 105,000 deaths there during the year 1998 Mannino, D.M., Homa, D.M., Akinbami, L.J.,
Ford, E.S., Redd, S.C., 2002. Chronic obstructive pulmonary disease surveillance-United States, 1971-
2000. Respiratory care 47, 1184-1199..
77
bulky extracorporeal devices. Their use can lead to post-operative infections, which have
an alarmingly high mortality rate (40%). Moreover, they require intensive surgical
procedures, involving patient sedation and intubation, and are prone to iatrogenic (i.e.
treatment-induced) injuries, such as volutrauma or barotrauma (Mihelc et al., 2009).
6.2.2. Percutaneous Respiratory Assist Catheter (PRAC) with rotary bundle
Intravenous oxygenators are a promising alternative to extracorporeal systems. They
consist of a respiratory catheter that is inserted into the vena cava through a peripheral
vein. This technique is a minimally-invasive approach that uses the heart as a pump and
the body as a heat exchanger. The abandonment of large external devices and the
simplicity of the surgical procedure reduce the rate of infection and iatrogenic injury,
reduces the treatment costs and number of personnel involved (Federspiel and Svitek,
2004a). The principles of active gas exchange in blood oxygenators, including
intravenous rotary oxygenators, are briefly described in Appendix B for the reader not
familiar with the topic.
Despite the challenges imposed by stringent anatomical space constraints, in 2006 ALung
Technologies Inc. (Pittsburgh, PA) in partnership with the University of Pittsburgh
introduced the latest version of an intravascular oxygenator. This device (figure 6.1)
consists of an oxygenating bundle driven by an extracorporeal electrical motor at a speed
exceeding 7000 RPM (Hattler et al., 2006a). Figure 6.1 shows the rotary catheter, its
schematic embodiment, and its placement site within the vena cava. The rotary
oxygenator, which is inserted percutaneously and has an effective insertional diameter of
8.33 mm, consists of a 20 cm-long fiber bundle incorporating 525 micro-porous hollow
fiber membranes (HFMs) of 300 m outer diameter for a total surface area of 0.1 m2. The
fiber bundle is rolled around a stainless steel support rod with an approx. 2 mm outer
diameter that is connected to the extracorporeal gas sweep sources via a 52 cm long tube
(Eash et al., 2007c; Hattler et al., 2006b). Fiber rotation causes “active mixing” of the
blood with respect to the fiber bundle and a concomitant increase in the gas exchange
efficiency (Eash et al., 2007b). The rotary catheter is designed to provide 40-60% of the
body’s resting metabolic needs for a short period of time, thereby alleviating the burden
placed on a body that is convalescing after surgery (Eash et al., 2007a).
78
Early in vivo attempts on swine failed because the rotating catheter fibers damaged the
wall of the tortuous inferior vena cava (IVC) into which it was inserted. Aiming at
protecting the vena wall from direct shear with the catheter, several protective cages were
devised, developed, and implemented by the Pittsburgh team during in vivo tests on a
calf. Although the cages protected the vena wall successfully, the catheter bundle was
damaged from shearing against the cage wall. The animal eventually died because of
severe blood clotting induced by the spoiled catheter bundle. In response to these
unsuccessful attempts, a research collaboration was formed between McGill University,
QC, Canada, and the McGowan Institute for Regenerative Medicine, University of
Pittsburgh, PA, to overcome the challenges met during the in vivo tests and to design a
novel cage that would ultimately make the rotary oxygenator function.
6.2.3. Problem definition
The cage-supported catheter is to be percutaneously inserted into the body and must
therefore fit into a 30 Fr (10 mm) sheath. When the appropriate position in the approx. 23
mm diameter IVC is reached, the cage should expand against the vessel wall to permit
catheter rotation while providing a sufficiently large unobstructed lumen for blood flow.
After 7-10 days of oxygenation required for healing the lungs, the cage is to be shrunk to
the catheter’s diameter to permit removal of the device from the body. Thus, the proposed
design must address the following structural requirements:
1. Protect both the IVC wall and the oxygenating bundle while guiding the rotating
catheter;
2. Be able to open and resist the geometrical restrictions imposed by the curved tortuous
vena cava of the animal in which the catheter must be tested;
3. Be sufficiently stiff to withstand both distributed and point forces applied by the
curved vein;
4. Be deployable and retractable for insertion and retraction purposes.
Besides these functional requirements, the cage-supported rotary catheter is a medical
device in contact with blood; therefore, its unavoidable level of blood damage should be
investigated and minimized. Furthermore, as a medical device working in the body, the
79
protective cage should be manufactured from biocompatible materials and coatings. We
note here that the specifications and the development of the detailed material design of
the cage, which includes its coating, finishing, and its manufacturing process are beyond
the scope of this thesis; it will be the topic of further investigations in the future.
Figure 6.1. Rotary catheter with rigid internal shaft
6.2.4. Blood damage investigation of blood-contacting medical devices
Before being approved for clinical practice, a blood-contacting medical device must pass
through extensive design processes that include conceptual, embodiment and detailed
design. Each process should be supported by theoretical and numerical simulations, in
vitro experiments, and in vivo observations. From the mechanical engineering point of
view, the design process includes both the structural and hematologic aspects of the
device. While the desired objectives and the defined constraints of the structural design
differ from one medical device to another, hematologic design focuses on minimizing the
blood damage caused by the implantation of the device. The exact mechanisms leading to
blood damage are complex and not yet well understood (see Appendix C); however, it is
generally accepted that the level of blood shear stress and the existence of the regions of
flow stagnation and recirculation are among the flow-related factors that affect the level
of blood damage (Medvitz, 2008). These adverse blood flow features are often observed
within small passages, journals, steps, or crevices in the flow path that requires their
design for a lower level of blood damage (Zhao, 2008). Thrombosis and hemolysis are
well-known categories of blood damage that are observed after the implantation of blood-
contacting devices. The former refers to clot nucleation and accumulation in the vascular
80
system, while the latter refers to premature damage or rupture of RBCs (Zhao, 2008).
Both types of blood damage, especially thrombosis, are complex phenomena that besides
physiological and biological parameters are affected by certain blood flow characteristics,
including shear stress. High shear stress values have been shown to be responsible for
hemolysis and platelet activation, while platelet deposition is often observed in regions of
low shear stress and flow stagnation (Chua et al., 2005; Medvitz, 2008). In vitro
experiments, in vivo observations and computational methods can be used to assess the
level of blood damage caused by a blood-contacting medical device. Appendix C at the
end of the thesis describes the methods that are currently used to estimate the hemolysis
and thrombosis blood damage.
6.3. Conceptual design of the protective cages for the PRAC with
rotary bundle
During the past three years, we have explored several conceptual designs for a protective
cage that meets the functional requirements listed in section 6.2.3. The bulk of these
concepts are periodic structures that can be thought as planar lattices folded into
cylindrical surfaces, inspired by Nitinol stents. These concepts have primarily been tested
in vitro and structurally optimized to comply with the deployability requirement for
percutaneous insertion of the cage. The rest of this chapter describes the design details of
two of these developed concepts.
6.3.1. Cage design I: 2008-2010
In 2008 exploratory concepts of the cage were tentatively proposed by a group of
undergraduate students at McGill University. Among these concepts, in 2009, a design
embodiment shown in figure 6.2 was identified for preliminary testing. The design
consists of a stent-like exterior tube made of Nitinol that is connected to the catheter
through four bearing-like inner rings, as shown in figure 6.2(a), each containing four
compliant arms that allow the cage to expand and contract. While the lateral surface of the
cage straightens slightly and opens the tortuous vena cava, the inner rings guide the
catheter to avoid direct shearing of the bundle on the cage wall.
81
6.3.1.1. In vitro testing of proof-of-concept prototypes
The ability of the lattice cage to protect the catheter bundle in the human (straight) vena
cava has been simulated and tested in vitro. The test set-up (Fig. 6.3) consists of an
aluminum fixture, a straight and transparent plastic tube connected to a water faucet, and
an electrical motor for driving the rotary oxygenator catheter. Three different cage
prototypes made of titanium, as shown in figure 6.3(c), each with a prescribed number of
inner rings, were manufactured with the Electron Beam Melting facility located at the
Hydro Quebec research laboratory in Varennes (Montreal). In vitro tests revealed that a
minimum of four support rings (recall figure 6.2) are necessary to hold and guide the
catheter.
Figure 6.2. Initial design of the protective cage. (a) Front and top views; (b) 3D view of the cage
and its supporting ring.
6.3.1.2. Shape optimization of the exterior wall
During insertion and retraction maneuvers, the deformation of the exterior tube of the cage
should remain below the 12% allowable strain limit of Nitinol (Kleinstreuer et al., 2008a).
Preliminary FE analyses showed that the strain level in the shrunk cage exceeds 19%;
thus, the cage fails to provide the range of deformation required for the shrinking
maneuver. The FE contours of the strain distribution in the struts of the cage show that the
82
maximum strain happens at the points with discontinuous curvature. To overcome this
challenge, we apply the method to reduce stress concentration in lattice structures
described in chapter 4 to obtain an exterior lattice of smooth unit cells. Similar to the
shape design of the stents explained in chapter 5, the exterior wall of the cage is defined
by a finite periodic lattice structure obtained by folding a 2D lattice sheet into a cylindrical
surface (figure 6.4).
Figure 6.3. Experimental set-up for concept evaluation of the cage in a straight tube.
As shown in figure 6.4 (c), the strut’s geometry can be described by nc cells in the
circumferential direction, nl cell rows in the longitudinal direction, the widths of the cell-
to-cell joints, 1 2 3, , ,t t t and 4t , and the strut thickness, 5t . The cage length is 240mm and
has a non-shrunk diameter of 25mm. We use here the same material properties and
element types as those described in the previous chapter for FE simulation of the Nitinol
stent-grafts. Figure 6.5 (a) shows the von Mises strain and stress distributions in the
shrunk cage defined by 12, 4l cn n 1 3 2 4, 0.6 , 1.9 ,t t mm t t mm and 5 0.6t mm . It
can be seen that the maximum strain in the shrunk cage is 6.4%, which is far below the
allowable 12% strain limit of the Nitinol. The force-displacement curve of the designed
cage (figure 6.5 (b)) shows that for a radial contraction of 1mm, the cage applies 7N on
the vena wall. This radial force is enough to open up the torturous vena cava and to
provide a safe lumen for the catheter rotation (Federspiel, 2010). An oversized cage could
83
be manufactured to compensate for the un-deployed radius of the lattice after its insertion
in the vena cava. It is noteworthy that any of the above design scenarios requires further
experimental and numerical investigations to ensure the safe deployment of the protective
cage in the vena wall. This task is beyond the scope of the present dissertation and will be
considered as future studies. The above experimental and numerical observations show
that the structural design of the cage meets the functional requirements listed in section
6.2.3. However, this structural design poses some challenges. First, because of the
integrated rows of its exterior wall, the cage is axially stiff and does not fully comply
with the curved vena cava of the animal subjects; the axially stiff cage might excessively
straighten and damage the vena wall. Also, even for human subjects (with a straight vena
cava) safe insertion of the axially stiff cage through the curved vascular paths is a
challenging task for surgeons. Second, the four metallic compliant arms (figure 6.2)
increase the diameter of the shrunk cage, which generally should be as small as possible
for percutaneous insertion procedures. These two challenges were addressed by
introducing a second modified version of the cage, which is described in the following
section.
Figure 6.4. (a) 3D geometry of the proposed lattice; (b) portion of 2D lattice mesh of the lateral
surface of the cage; (c) parameterized quarter of the lattice unit cell.
84
Figure 6.5. (a) von Mises strain distribution in the compressed cage design I; (b) radial
supportive force versus inner radius of the cage design I.
6.3.2. Cage design II: 2010-2012
Figure 6.6 shows the second version of the cage. As with the first version, the safe lumen
for rotation of the catheter is provided by deploying a stent-like exterior wall made of
Nitinol. In this version, however, the adaptability of the cage to the curved geometries is
achieved thorough inter-hoop clips. The flexible exterior wall of the tubular cage (figures
6.6 and 6.7) contains clipping hoops that connect different rows. The clips provide the
cage with sufficient axial flexibility to gently conform to an animal’s tortuous IVC and to
pass through the human vein paths during percutaneous insertion. The insertion size of
the cage is reduced by replacing the spiral wires of the first version with flexible radial
threads shown in figure 6.6. Made of stiff biocompatible hyper-elastic materials, the
strings are radially located to control the catheter’s movement in the radial direction
within an allowable tolerance. The hyper-elastic properties of the radial threads help to
reduce the risk of rupture, which could occur as a result of unexpectedly high vibration of
the catheter. As in the first version of the cage, the internal rings (figure 6.6) are
biocompatible strands in contact with the catheter bundles that maintain the position of
the catheter while it is rotating. It should be noted that the internal rings and the radial
threads are potential sites for excessive blood damage. Thus, experimental in vitro and in
vivo investigations are required to ensure the conceptual feasibility of these structural
features from blood flow point of view.
(a) (b)
85
Figure 6.6. Second design of protective cage.
Figure 6.7. Lattice cage on a curved geometry. The clips linking the series of lattice hoops allow
the cage to adjust to change in the vein geometry.
86
6.3.2.1. In vitro bench test of the second design of the cage in curved paths
We designed an in vitro set-up to qualitatively test the ability of the lattice cage in
protecting the rotary oxygenating bundle in the curved animal vena cava. The test set-up
(Fig. 6.8) consists of an aluminum fixture that allows simulating various curved paths and
an electrical motor for driving the rotary catheter. The vertical pillars shown in figure 6.8
(a) can be positioned in x, y, and z directions to generate various curved paths. Each pillar
is drilled with a 1in drill bit, which represents the outer wall of the cage. The catheter is
guided via three threads located 120 apart from each other, as shown in figure 6.8 (a).
Various shapes of the curves that mimic the wall of tortuous vena cava are selected and
tested as shown in figure 6.8. In contrast to the in vitro test described in section 6.3.1.1,
here we used a rotary catheter with a flexible inner shaft that gently conforms to the
desired curved paths, as shown in figures 6.8 (b) and (c). The in vitro observations
revealed that a cage with six inner rings can safely guide the catheter during its rotation
along curved paths like those shown in figures 6.8 (b) and (c).
Figure 6.8. Experimental set-up for concept evaluation of the cage in a curved path; (a) test set-
up and one of its adjustable pillars; (b) and (c) two curved paths of the the cage design II that can
successfully guide the rotary catheter bundle.
87
6.3.2.2. Shape optimization of the exterior wall
Here we used the optimized shape of the unit cell, the design method, and the
computational procedure explained in the previous chapter to generate Nitinol stents free
of geometrical stress concentration. The strain distribution in the shrunk cage for 8cn ,
10ln , 0.28t mm , and 0.28w mm is shown in figure 6.9 (a). It can be seen that the
maximum strain of the shrunk cage is 10.3%, which is below the allowable strain limit of
the Nitinol. The force-displacement curve of the synthesized cage (figure 6.9 (b)) shows
that the cage can provide up to 10 N for 1 mm radial contraction, which is around twice
the required 4-5 N radial force for opening up the torturous vena cava.
Figure 6.9. (a) von Mises strain distribution in the outer wall of the compressed cage design II;
(b) radial supportive force versus the inner radius of the cage design II.
6.3.3. Retraction/Removal mechanism
After 7-10 days of oxygenation required for lungs to heal, the cage-supported catheter is
removed from the body by a retraction mechanism shown in figure 6.10. The shrinking
mechanism functions like a belt around the outer wall of the cage. It includes wire mesh,
guiding rings, and an exterior handle. The cage contracts the catheter by means of a thin
mesh wrapping (red in figure 6.10) around the tubular cage. The mesh consists of
circumferential hooks wrapped around each row of the lattice cage and longitudinal wires
connected through guiding rings to a handle outside the body. One end of each
longitudinal thread is connected to the cage and passes through the guiding rings until it
reaches the handle ring outside the body. The cage will collapse by tightening the wire
(a) (b)
88
mesh through pressing the back of the fixed handle against the skin and pulling the
movable end in the opposite direction. This mechanism has been tested successfully on a
working rubber prototype.
Figure 6.10. Shrinking mechanism consisting of handle, guiding rings and threads to shrink the
lattice tubular cage.
6.3.4. Hematologic design
From a structural point of view, the second version of the cage meets the functional
requirements explained in section 6.2.3. For a blood-contacting medical device, the next
step is to investigate the amount of blood damage caused by the device during its
operation. In this section, we estimate the hemolysis level caused by implantation of the
second version of the protective cage by using four well-established numerical methods:
1- maximum shear stress approach; 2) mass-weighted shear stress average approach 3)
Eulerian approach, and 4) Lagrangian approach. The platelet activation state (PAS) will
also be estimated to obtain a rough approximation of the thrombosis induced by the cage-
supported rotary catheter. As mentioned earlier, the mathematical foundation of these
methods is explained in Appendix C. A comparison will be made between the numerical
89
results of blood damage caused by the cage-supported catheter (C version) with that of
the catheter without a cage, which is used as a reference model and will be recalled as R
version throughout this chapter. In this section the focus is on studying the blood damage
caused by a cage with the following parameters 8cn , 10ln , 0.28t mm , and
0.28w mm .
6.3.4.1. Numerical modeling
ANSYS-FLUENT v.13, an unstructured-mesh finite-volume based commercial CFD
package, was used to solve the incompressible transient and steady Navier-Stokes
equations. The 3D CAD models of the blood volume around the R and C versions of the
rotary oxygenator were generated by using SolidWorks 2012 (figure 6.11). Inlet and
outlet volumes shown in figure 6.11 take into account the effect of upstream and
downstream flows in the vena cava after insertion of the catheter. In these models, the
vena cava is considered as a cylinder with inner diameter of 22 mm and total length of 24
cm. It should be noted that these two distal volumes are modeled as annular regions
because the connecting tube and the stationary tip of the catheter partially block the vena
cava. It is noteworthy that this dissertation does not consider the effect of physiological
blood vessels that join to the vena cava in the oxygenation site, e.g. renal blood vessels.
The detailed effect of these joining physiological pathways should be investigated
through experimental in vivo studies.
90
Figure 6.11. 3D model of the blood flow in the reference, or R version (left) and cage-supported,
or C version (right) of the rotary oxygenator.
a. Mesh generation
We used the Mesh component of ANSYS Workbench v.13 to mesh the 3D models of the
R and versions of the rotary oxygenator (figure 6.11). The volume of the blood flow in
the R version is a primitive cylindrical geometry that can be easily meshed with
hexahedral elements, as shown in figure 6.12 (a). The mesh was then refined in the CFD
package to achieve a flow field independent of grid size, as shown in figure 6.12(b).
Because of the irregular and highly complex geometry of the cage struts, the blood
volume flowing between the vena wall and the C version of the oxygenator was meshed
by tetrahedral elements. Two approaches were used to mesh this volume. First, only
tetrahedron cells (free mesh) were used to discretize the blood control volume (figure
6.12(c) and (d)). This method is a simple and fast approach to mesh complex geometries,
but it usually produces a high number of cells and gives limited control to the analyst to
decide on the quality and size of the elements in region of interest. As can be seen in
figure 6.12 (d), despite the high number of generated cells (3,433,752 tetrahedron cells)
the mesh size around the catheter wall, which is the boundary with the maximum gradient
of shear strain rate, was too coarse to yield valid CFD results. Furthermore, because of
the high number of initially generated cells, using a sufficient level of mesh refinement
near the catheter wall was beyond our available computational resources at the time. To
resolve this challenge, the blood volume was divided into a region of primitive
cylindrical geometries that are connected with the irregular region of the exterior wall of
the cage, as shown in figures 6.13 (a) and (b). This technique requires more elaboration at
the pre-processing level, but it substantially reduces the number of cells required for grid-
independent CFD results. In this case, the primitive geometries were meshed with
hexahedron elements. The irregular region of the blood volume, which is shown in brown
in figures 6.13 (a) and (b), was meshed by tetrahedron elements.
For a turbulent flow, boundary walls are a major source of turbulent eddies that affect the
whole flow. Therefore, the fluid mesh around the boundary walls has to be refined until
mesh-independent results are attained. There are several tools to refine or coarsen the
91
initial mesh. Here, we used the boundary adaption, */y y adaption, and gradient adaption
methods (FLUENT, 2009) to refine the mesh size near the boundary walls. For example,
figure 6.12 (b) and 6.13 (d) shows the axial cross section of the R and C versions of the
catheter after using the boundary adaption method.
b. Turbulence modeling
The blood flows around the rotary catheter with axial Reynolds number of
504ax axRe V D and Taylor number 1/2( ( ) / ) ( ) 11200i o i o i iTa R R R R R R .
The experimental Taylor-Reynolds map described in (Kaye and Elgar, 1956) predicts that
the rotation of the catheter causes the blood flow to be turbulent in the vena cava. The in-
coherent and time-dependent vortices observed during in vitro testing of the rotary
catheter validated this prediction (Budilarto et al., 2009).
There are several levels of turbulence modeling in CFD, and each of them is designed to
solve a certain range of problems. Reynolds Average Navier-Stokes (RANS), Large Eddy
Simulation (LES), Detached Eddy Simulation (DES), and Direct Numerical Simulation
(DNS) are examples of these established turbulent models. Among these methods, RANS
is the simplest level of modeling of a turbulent flow, while DNS is the most detailed but
computationally expensive one.
Most of the RANS models have been developed based on the experimental results of the
steady fluid flow streaming with a high Reynolds number over simple geometries, such
as a flat plate.
Some RANS models are modified to simulate low Reynolds number flows. The
applicability of RANS models is questionable for simulating separated and unsteady
flows. Several researchers have used RANS models to design and simulate blood-
contacting medical devices. A few studies (Kim et al., 1992; Yano et al., 2003) used the
k RANS model, but most of the researchers have used the k RANS model, e.g.
(Apel et al., 2001; Bluestein et al., 2000; Gu and Smith, 2005; Yin et al., 2004). In two
separate studies, (Apel et al., 2001; Song et al., 2003b) showed that in general the k
is a better model than k for simulating blood flow in heart pumps. They showed that
92
Figure 6.12. Discretized model of the blood volume flowing in the R and C versions of the rotary
oxygenator; (a) 3D view of the meshed model of the R version. (b) The refined mesh near the
boundaries of the R version was obtained by using boundary adaption method. (c) and (d) 3D and
side views of the discretized model of the blood volume flowing in C version of the rotary
oxygenator meshed with tetrahedron elements.
Figure 6.13. (a) and (b) 3D and side views of the CAD model of the blood flow in the C version
of the oxygenator obtained by dividing the blood volume into primitive and irregular volumes; (c)
3D view of the meshed model; (d) the refined mesh near the boundary walls of the C version
obtained by using the boundary adaption method.
the k model has better accuracy in near-wall regions (low Reynolds number) while
the k model better simulates the blood flow far from the wall. (Myagmar, 2011) used
the shear-stress transport (SST) formulation of the k model to simulate the blood
93
flow in the RIT LEV-VAD heart pump. The SST k approach uses the k model
for regions close to the boundary wall and k model for regions far from the wall
(FLUENT, 2009). (Myagmar, 2011) argued that in comparison with standard k
model, the blood damage calculated based on the results of SST k model is in more
agreement with the experimental findings.
The LES and DES are two turbulence models that directly resolve the geometrically
dependent large eddies that contain most of the turbulence energy. The size and range of
the resolved eddies depend on the selected time step and grid spacing, which dictate the
required computational resources and time. Medvitz (2008) used the LES model to
simulate the pulsatile and transient blood flow in a positive displacement left ventricular
assist device (LVAD).
The DNS model directly solves the Navier-Stocks equations by resolving all turbulent
fluctuations occurring at the smallest dissipative scale of turbulence, or Kolmogorov
scales. This method requires a very fine grid size and should be solved for sufficiently
small time steps; hence, it requires substantial computational power that limits its
applicability to simple geometries.
Choosing the turbulent model that accurately simulates the blood flow around the rotary
oxygenator is a challenging task. To select the appropriate turbulent model for our
problem, we tried three models to solve the blood flow around the R version of the
oxygenator, rotating at 7000 rpm in a straight vena cava geometry. These models were
steady SST k , transient SST k , and transient LES. The results of these
simulations were then compared with those experimental observations reported by
(Budilarto et al., 2009). The rest of the CFD analyses presented in this chapter are
performed by using the turbulent model in more agreement with the experimental data. A
converged solution was achieved once the scaled residuals fell below 1e-4, or when the
velocity field did not significantly change.
94
c. Boundary Conditions
Blood flow with constant axial velocity of 0.117m/s ( 2 2, ( ) 4axial out inV Q A A d d )4
enters, through the inlet wall(s), into the control volume(s) of figures 6.11 (a) and 6.13
(a). A pressure-outlet condition was set at the outlet surface(s). All of the exterior walls
of the model as well as the interior boundaries of the inlet and outlet volumes were
considered as stationary rigid walls. The rotation of the catheter was modeled by
imposing a moving wall boundary condition with an absolute rotational speed along the
longitudinal axis of the catheter. The interface boundaries between different volumes
were defined as the interface boundary condition.
d. Discrete Particle Modeling (DPM)
The blood’s constituents, such as platelets and RBCs, were modeled as discrete particles
travelling along the blood flow. It has been shown that a sufficient number of particles
should be tracked to obtain valid hemolysis and PAS values (Chan et al., 2002). Here, we
tracked discrete particles that were released from the inlet surface shown in figures 6.11
and 6.13. The particles were assumed to have the same density and initial velocity as
those of the inlet blood flow. A sufficiently large number of time steps (50000) were
chosen to ensure the completion of the particles. Because of the large size of strain
history data files (near 861 Mb) we used the pathline data to calculate the hemolysis and
PAS values in MATLAB.
6.3.4.2. Results
This section presents the results of CFD analyses performed to solve the blood flow in
the R and C versions of the rotary oxygenator.
a. Turbulent model and Mesh-independency investigation
4 Here, axialV is the axial velocity, Q is the flow rate, A is the cross-sectional area, ind and
outd are, respectively, the outer diameter of the rotary catheter and the inner diameter of
the vena cava.
95
Table 6.1 shows the average shear stress on the catheter obtained by solving the blood
flow around the R version for a different number of elements and turbulence models i.e.
steady SST k , transient SST k , and transient LES. Experimental observation
predicted an average shear stress of 17 Pa on the catheter wall rotating at 7000 rpm. It can
be seen that in comparison to steady and transient SST k models, the results of the
transient LES model are in better agreement with experimental observations. Therefore,
we chose to use the LES model for the remainder of our analyses. It should be noted that
LES model is a computationally expensive approach that requires a sufficiently fine mesh
to capture the small eddies in fluid flow. Furthermore, this method is designed for
transient problems and requires a sufficient number of time steps to obtain statistically
constant fluid flow parameters. Table 6.2 shows the average shear stress on the catheter
wall of the R version of the oxygenator at different rotational speeds obtained by LES
method. The good agreement between these results and the experimental observations
contribute to the validity of our CFD analyses.
Tables 6.3 and 6.4 illustrate the results of the mesh-sensitivity analyses for the R and C
versions of the rotary oxygenator. We used two parameters to ensure that the value of the
predicted blood damage is independent of CFD mesh. These two parameters are: i) the
average shear stress on the catheter wall and ii) the modified index of hemolysis (MIH)
obtained based on the Eulerian method for Heuser’s empirical blood damage model (
equation C.8 and C.9 of Appendix C). The former was selected as a measure of the stress
values around the boundary of high shear stress gradient, i.e. the catheter wall, while the
latter parameter is an indicator of the average of shear stress in the entire volume of the
blood flow.
Table 6.1. Average shear stress in Pa on the catheter wall of the R version oxygenator rotating at
7000 rpm for different number of elements and turbulence models.
Number
of nodes
Number
of cells Steady State k Transient k Transient LES
136080 119520 2.5 2.5 2.9
459840 378000 5.3 5.4 6.0
1514760 1253520 11.5 11.7 16.7
1773651 1444342 11.7 12.0 16.4
2765520 1953772 11.6 12.1 16.5
96
Table 6.2. Numerical and experimental values of the average shear stress (in Pa) on the catheter
wall of the R version of the oxygenator rotating at different rotational speeds.
2000 rpm 4000 rpm 7000 rpm
CFD analyses (LES
turbulence model) 6.9 9.8 16.5
Experimental results
(Budilarto et al., 2009) 6 10.5 17
Table 6.3. Results of mesh-sensitivity analysis
on the R version of the rotary catheter.
Table 6.4. Results of mesh-sensitivity
analysis on the C version of the rotary
catheter.
Number
of cells
Average shear
stress on
catheter wall
(Pa)
MIH (Heuser
blood damage)
Number
of cells
Average shear
stress on the
catheter wall
(Pa)
MIH (Heuser
blood
damage)
119520 9.6 1.3E-01 4386293 3.4 2.4E-02
378000 17.7 2.5E-01 4500953 17.1 2.3E-01
1253520 29.7 4.2E-01 4959593 19.3 2.2E-01
1444342 29.7 4.20E-01 5101084 19.6 2.3E-01
1953772 29.7 4.2E-01 5461920 19.9 2.3E-01
b. Blood flow features
Figure 6.14 shows the Taylor vortices in the rendered view of the velocity field of the
blood flow in the R and C versions of the rotary oxygenator. Figure 6.15 shows the vector
and contour plots of the velocity (m/s) distribution in the r z and r planes of the R
and C versions of the rotary oxygenator where the time-dependent and in-coherent Taylor
vortices can be clearly seen. These results are in agreement with the experimentally
observed pattern by (Budilarto et al., 2009). Figure 6.15 shows that the velocity field of
the blood is dominated by the catheter’s rotation. Also it can be seen that the Taylor
vortices have a negligible effect on the flow pattern in the r plane. The contours of
the shear stress distribution in the r z and r planes of the R and C versions of the
oxygenator are shown in figure 6.16. As predicted, the maximum shear stress occurs at
the surface of the rotary catheter. The shear stress drops significantly in moving away
from the catheter wall towards the vena wall. Figure 6.16 (b) shows that the shear stress
locally increases at the vicinity of the exterior wall of the cage. This may adversely
97
trigger hemolysis and platelet activation mechanisms. Figure 6.17 shows the pressure
contours in the r z and r planes of the R and C versions of the rotary catheter (A-A
and B-B planes in figures 6.11 and 6.13). As expected, because of centrifugal forces, the
pressure increases from the catheter wall towards the vena wall.
Figure 6.18 shows the pathline of 30 particles injected from the inlet surface(s) of figures
6.11 and 6.13. The results showed that only a portion of the particles (35%) escaped from
the outlet surface(s) while the rest of them were trapped in the Taylor vortices. To capture
the shear stress history in the entire volume of the blood flow, we injected another group
of particles from the mid-plane of the rotary catheter, as shown in figure 6.18 (b).
Figure 6.14. Taylor vortices pattern in rendered views of the velocity distribution in the R (up)
and C (down) versions of the rotary oxygenator.
98
Figure 6.15. Velocity distribution in r z and r planes of the (a) R and (b) C versions of
the rotary oxygenator.
Figure 6.16. Shear stress distribution in the r z and r planes of the (a) R and (b) C
versions of the rotary oxygenator.
99
Figure 6.17. Total Pressure distribution in the r z and r planes of the (a) R and (b) C
versions of the rotary oxygenator.
Figure 6.18. Path lines of the particles released from the inlet surface(s) of the (a) R and (b) C
versions of the rotary oxygenator.
100
c. Evaluation of blood damage caused by the rotation of the oxygenator
For the two versions of the rotary oxygenators described in this section, we used the data
from the flow field solution obtained from the above described CFD analyses to calculate
the values of PAS and hemolysis blood damage by MATLAB.
i) Hemolysis
Method 1: Maximum shear stress
CFD analyses predict that the maximum values of shear stress in the catheter walls of the
R and C versions of the oxygenator are, respectively, 22.6 Pa and 23.42 Pa. These values
are obtained based on the Bludszuweit 1D scalar shear stress (Eq. C.4 of Appendix C). It
can be seen that the C version of the rotary oxygenator caused only 4.5% higher
maximum shear stress compared to the R version. Continuity of the input and output flow
gives the average exposure time of the blood to the rotary catheter is:
0.20 0.117 1.71secexposure axialt L V m m s . It is noteworthy to mention that the
average exposure time is the average time of several flow pathlines streamed in the
annular region between catheter wall and IVC. Thus, there exist some pathlines that have
higher exposure time because of being entrapped in recirculation zones. Figure 6.19
shows the threshold of shear stress for hemolysis damage generated by the experimental
observations of several researchers gathered by (Day et al., 2006). It can be seen that the
maximum shear stress approach predicts that neither R nor C versions of the catheter
causes a noticeable hemolysis damage. Nevertheless, on the basis of the Giersiepen 1%
and Heuser 1% blood damage lines, the maximum shear stress approach predicts 1%
hemolysis for the R and C versions of the oxygenator rotating at 7000 rpm.
Method 2: Mass-Weighted Average Approach
CFD results show that the maximum shear stress in the R and C versions of the catheter is
much below the critical 200 Pa (figure 6.19). Thus, the mass-weighted average approach
predicts no noticeable hemolysis to occur in both versions of the oxygenator.
Method 3: Eulerian Approach
101
Table 6.4 shows the MIH blood damage indices obtained based on the Eulerian method
for both the Giersiepen and Heuser RBC damage models (refer to equations C.8 and C.9
of Appendix C). Table 6.4 indicates that the cage contributes to 42.1% of the total
damage to RBCs of the C version of the oxygenator.
Method 4: Lagrangian Approach
The Lagrangian approach tracks the accumulation of blood damage as the blood particles,
either RBCs or platelets, move along their trajectories within the medical device. The
shear stress,
Figure 6.19. The maximum shear stress and average residence time of R and C versions of the
rotary catheter (solid red circle) on the chart developed based on published threshold values for
hemolysis blood damage adapted from (Day et al., 2006).
velocity, exposure time and coordinates of each particle resulted from the CFD analyses
were used in MATLAB to find the average blood damage indices. Figures 6.20 (a) and
(b) show the variation of shear stress and velocity versus residence time obtained by
tracking one completed particle released from the inlet surface of the C version of the
rotary oxygenator (figure 6.16). Figures 6.20 (c) and (d) show the variation of shear stress
and velocity fields versus residence time for all the particles released from the inlet
surface of the C version of the rotary oxygenator. Table 6.4 shows the blood damage
indices obtained based on the Lagrangian method. To obtain these variables we first
102
calculated the blood damage indices separately for all the particles (equations C.8 and
C.9). The average of the blood indices for all released particles was taken as the blood
damage index.
It can be seen that this approach estimates that the use of the protective cage increases the
hemolysis indices by 24.9% for the oxygenator rotating at 7000 rpm.
Table 6.5. Blood damage caused by R and C versions of the rotary oxygenator spinning at 7000
rpm.
Number
of cells
MIH
(Eulerian,
Heuser)
MIH
(Eulerian,
Gierespien)
MIH
(Lagrangian
, Heuser)
MIH
(Lagrangian,
Gieresiepen)
PAS
R
version 0.15 0.10 0.20 0.18 1.01
C
version 0.24 0.16 0.38 0.24 1.52
Figure 6.20. Results of particle tracking in C version of rotary catheter; (a) and (b) variations of
shear stress and velocity versus residence time of a completed particle released from the inlet
surface(s) of figure 6.14; (c) and (d) variations of shear stress and velocity versus residence time
of a completed particle released from the inlet surface(s) of figures 6.14.
103
d. Platelet activation
Table 6.5 shows the platelet activation state (PAS) caused by the R and C versions of the
rotary oxygenator at 7000 rpm. These results were obtained by tracking all the injected
particles released from the inlet surfaces of figures 6.11 and 6.13. The results show that
the deployment of the stent-like wall of the protective cage increases the PAS by 85.3%.
The numerically predicted PAS in both the R and C versions of the rotary catheter are
below the PAS values calculated for blood pump valves.
6.3.4.3. Discussion
Figure 6.14 shows that the flow pattern around the rotary oxygenator spinning at 7000
rpm needs 3-4 cm to be fully developed. The surface of the rotary bundle in contact with
this undeveloped blood flow pattern probably has lower gas exchange efficiency
compared to the rest of the oxygenating surface of the catheter. The total gas exchange
efficiency of the catheter can be improved by increasing the blood mixing in this portion
of the catheter surface. However, this numerically predicted result should be validated by
further by in vitro experiments.
Tables 6.4 show that the blood damage indices for the R and C versions of the oxygenator
rotating at 7000 rpm are below the estimated values for existing heart pumps and valves
(Myagmar, 2011). This is in agreement with results of primary ex vivo experiments
performed by (Eash et al., 2007c). Table 6.4 also shows that the cage contributes to
85.3% of the total platelet activation state (PAS) and 42.1% hemolysis damage at 7000
rpm. Thus, adding the protective cage should not cause an un-acceptable level of blood
damage. It should be noted that due to the highly complex nature of the blood damage
mechanisms the results should be taken with caution and must be validated by in vitro
and in vivo experiments. For example, Jesty et al. (2003) showed that assuming a linear
relationship between platelet activation and shear stress value (equations C.2 and C.4)
leads to a simple and easy way to assess the platelet activation level, but this strategy is
not necessarily an accurate one. However, considering the lack of available accurate
numerical method, the values listed in table 6.4 can be used - as a first step- for a
comparative study of the effect of deploying the protective cage around the whirling
blood flow.
104
The estimated hemolysis levels obtained with the four methods to estimate hemolysis are
very different from each other. However, it can be seen that the damage levels predicted
by the Lagrangian and Eulerian approaches are in the same order of magnitude. The MIH
hemolysis indices predicted by the Eulerian and Lagrangian approaches (table 6.4) for
both the R and C versions of the rotary oxygenator are less than the 5.59 hemolysis
indices calculated for the 16G needle cannulas (Garon and Farinas, 2004).
It should be noted that in this chapter we have considered only one geometrical design of
the second version of the cage while both the structural and the hematologic performance
of the protective cage depends on the selected set of design parameters. The best set of
geometrical parameters can be obtained by performing a multi-disciplinary optimization
that takes into account both solid and fluid phases. However, this task is computationally
expensive and should be pursued after successful results from in vitro and in vivo
experiments. These studies are beyond the scope of the present thesis but will be
considered in future studies.
6.4. Concluding remarks
The intravenous rotary oxygenator is a promising alternative for mechanical
ventilators currently in use for treating patients suffering from respiratory diseases.
The rotation of the catheter within the inferior vena cava poses several challenges,
including the means to protect both the body and the catheter bundle from direct
shearing on each other as well percutaneous insertion, and biocompatibility concerns.
In this chapter we proposed and studied two conceptual designs that may contribute
to eventually solve these challenges. The capability of the cage in guiding the catheter
along both straight and curved paths has been proved by in vitro experiments.
The cage and its attached rotary oxygenator should be removed from the body after
about seven days of oxygenation. Here, we have proposed a shrinking mechanism
that is able to shrink and remove the apparatus. This mechanism was tested on a
flexible prototype made of a rubber lattice.
The structural behavior of the proposed designs was studied by performing FE
simulations. It was shown that the cage can deploy and provide sufficient radial force
to open up the tortuous vena cava.
105
The contribution of the cage structure to the level of blood damage caused by the
operation of the rotary oxygenator was studied by performing CFD analyses. It was
shown that the level of the blood damage caused by the cage-supported catheter is
below the predicted values for available heart pumps currently in use for medical
practices. However, these results need to be validated through in vitro and in vivo
experiments. This can be the subject of future studies. It should be mentioned that the
results of the CFD analyses could be further refined by applying additional levels of
CFD mesh refinement, but this would require more powerful computational
resources. Even if this computation were done, the author does not predict a
significant or decisive change in the current results, i.e. the contribution of the cage
structure to the total level of blood damage.
In this chapter we studied the structural and hematologic behavior of the proposed
cage by considering only one set of the design parameters that define the geometry of
the cage. The selected design set was not necessarily the optimal one. Thus, there is a
need for future studies to obtain the optimal set. This is a computationally expensive
and time-consuming process that should only be performed after ensuring the overall
efficacy of the cage performance through further in vitro and in vivo experiments.
106
Chapter 7
Conclusions and future works
7.1. Conclusions
A numerical method based on asymptotic homogenization theory has been presented
to design lattice materials for fatigue failure. For a given multi-axial cyclic loading,
failure surfaces of metallic hexagonal and square lattices have been determined along
with their Modified Goodman diagrams to assess the effect of mean and alternating
stresses on the fatigue strength. A good agreement has been found between the results
of the developed computational method and the experimental data available in
literature.
In chapters 3 and 4, it has been shown that the geometric design of the unit cell plays a
pivotal role in the fatigue strength of lattice structures. As a case study, the fatigue
design methodology has been applied to investigate the effect of cell shape on the
fatigue/monotonic strength of lattices with hexagonal and square cells. Failure surfaces
of the unit cells with their respective material distribution within the RVE and with no
geometric stress concentration were obtained. The results have shown that for bending
dominated lattices, cell geometries with continuous and minimum curvature have
superior fatigue performance than cells with shape boundaries defined by line and arc
primitives. In addition in bending dominated lattice materials, unit cells defined by G2
curves with minimum curvature should be preferred to cells with arc-rounded joints,
especially if manufacturability is not a constraint.
A novel unit cell for generating planar lattices in a cylindrical coordinate system has
been introduced to improve the durability of Nitinol self-expandable stent-grafts with
closed-cell geometry. The geometry of the lattice unit cells has been tailored by
applying the shape optimization methodology described in chapter 4 to synthesize unit
cells free of geometrical stress concentration. The radial supportive force and fatigue
safety factor of the generated stent lattice have been studied through a FEA parametric
107
analysis. Compared to recent stent design, the results have shown an improvement of
the stent anchoring performance and a reduction in the risk of fatigue failure.
Two conceptual designs have been proposed and tested to guide Alung Technology
Inc.’s intravenous oxygenator with rotary bundle inside the vena cava. Preliminary in
vitro tests have validated the ability of the proposed concepts in guiding the rotation of
the catheter in straight and curved vascular lumens. A shrinking mechanism has been
proposed and successfully tested on a rubber prototype to retract the cage-catheter
assembly after its operation.
The deployability of the exterior wall of the cage after its percutaneous insertion
inside the vena cava has been studied by performing FEM analyses. The radial force-
inner diameter curve of the exterior wall of the cage has been evaluated. The results
have revealed that the cage is capable of providing sufficient radial force for opening
the torturous vena cava.
The level of blood damage caused by cage-catheter assembly has been studied
numerically by performing CFD analyses. The results have shown that the level of
blood damage is below the numerically predicted values for available heart pumps
and artificial valves. A comparison with the blood damage caused by rotation of the
reference oxygenating catheter with and without the cage has revealed that the cage
causes a 42.1% increase in hemolysis level and 85.3% increase in platelet activation
state (PAS).
7.2. Directions for future research
Each original research solves some existing problems but presents more questions to be
answered in the future. The following are the author’s suggestions for future work:
7.2.1. Numerical method based on computational mechanics to design lattice
materials against fatigue failure.
Lattice materials are cellular periodic materials of at least one level of hierarchy.
Their overall macroscopic fatigue resistance and behavior are a function of the fatigue
resistance and behavior of each constituent level. In this thesis we limited our focus to
metallic lattices made of one layer of hierarchy, i.e. we assumed a perfectly uniform
108
distribution of the solid metallic material within the walls of the unit cells. The
fatigue design method presented here can be further extended to address following
challenges i) model the fatigue behavior of lattices made of nonmetallic bulk
materials, such as plastics and elastomers; ii) consider the effect of the microstructure
of the solid material, e.g. its grain size and shape, and the existence of probable void
defects; and iii) consider lattices made of more than one level of hierarchy, such as
the ultralight lattice presented in (Schaedler et al., 2011). The starting point of a
future research aimed at solving the first challenge may be the detailed investigation
of the fatigue behavior of the constituent solid material and its implementation to the
fatigue methodology proposed in this thesis. The author suggests adapting a method
similar to the one presented by Zienkiewicz and Taylor (2005) to model two length
scales in the lattice microstructure to tackle the second and third challenges.
In this thesis, the presented method for fatigue design of lattice materials was
validated by a comparison with available experimental data in the literature. The
author believes that the application of the proposed method for design of new lattices
should first be verified and calibrated through following standard experimental
procedures. The crack in cellular material advances for one unit cell, following the
rupture of a cell wall containing an unstable micro-crack. Experimental observation
of the behavior of micro-crack propagation, such as its nucleation sites, rate, and the
path of its propagation, is an interesting topic for future research. The results of these
studies could be used for modifying or adjusting the computational method presented
in this thesis for fatigue design of lattice materials.
7.2.2. Geometrical design of the unit cell of lattice materials for fatigue
resistance
In this thesis, the significant role of the architecture of the unit cells of lattices on
their fatigue resistance has been studied by comparing the optimized and the regular
shapes of the lattice unit cell. The presented shape optimization method in this thesis
is able to improve the fatigue life of 2D planar lattices with hexagonal and square
shapes of unit cells. But its implementation to planar Kagome and auxetic lattices as
well 3D unit cells revealed several programming and conceptual complications. Other
109
optimization strategies need to be adopted or developed to improve the fatigue
resistance of the aforementioned lattices.
7.2.3. Design of Nitinol self-expandable stent lattices against fatigue
In this thesis, we considered only the shape synthesis of self-expandable stent-grafts
used for abdominal aortic aneurysms. The proposed shape optimization method can
be adapted to improve the durability of other types of Nitinol self-expandable stents,
especially for those used in superficial femoral arteries (SFA).
The results of the parametric study in chapter 5 of this thesis showed that the stent
radial supportive force, fatigue failure safety factor, and stress level in the artery wall
are often conflicting objectives. The shape synthesis of the lattice cell can be
formulated within a multi-objective optimization framework (Messac et al., 2003)
that is capable of providing trade-off solutions among the aforementioned conflicting
objective functions.
In this thesis, stent design for fatigue life was tackled by minimizing the occurrence
of stress concentration due to geometric discontinuity. This method can be
complemented by integrating a fracture mechanics approach based on the design
guidelines for fatigue design of Nitinol devices (Robertson and Ritchie, 2007;
Robertson and Ritchie, 2008; Stankiewicz et al., 2007).
7.2.4. Design of protective cage for the PRAC with rotary bundle
The results of chapter 6 show that the proposed conceptual designs for the protective
cage have the potential to protect the catheter’s rotation inside the vena cava.
However, a real-size assembly that includes the cage, rotary oxygenator, and the
devised shrinking mechanism should be manufactured to validate the structural
requirement for percutaneous insertion of the oxygenating assembly, including its
deployabilty, retractability, and axial flexibility. Furthermore, the radial stiffness of
the cage should be experimentally measured to ensure its ability in opening the
torturous vena cava. Various in vitro and in vivo experiments need to be done i) to
observe the cage’s ability to protect the catheter during its rotation inside the vena
cava, ii) to measure the level of blood damage triggered by the rotation of the
110
catheter, and iii) to validate the flow field parameters predicted by the CFD analyses.
These experimental observations will open the way for modifying the proposed
conceptual designs or for devising new concepts that can finally make the oxygenator
market-ready.
In chapter 6 of this thesis, we studied the behavior of the proposed cages for only one
set of geometrical design parameters. These selected parameters are not necessarily
the optimal ones. Thus, the performance of the cage can be further improved by using
multi-disciplinary optimization. This is a computationally expensive and time-
consuming process that should only be performed after ensuring the overall efficacy
of the cage performance through further in vitro and in vivo experiments.
111
Appendix A
A.1. The matrix operators of the shape optimization
approach presented in chapter 4
210000
141000
014100
001410
000141
000012
A
nnc
c
10000
121000
012100
001210
000121
00001
1
11
C
100000
141000
014100
001410
000141
000001
P
B
A
t
t
100000
303000
030300
003030
000303
000001
1
Q
112
Appendix B
B.1. Principles of gas exchange in blood oxygenators
Blood oxygenators are alternative devices to mechanical ventilators for treating patients
suffering from respiratory diseases. A blood oxygenator can be intracorporeal or
extracorporeal. Figure B.1 shows two oxygenators: a commercially available
extracorporeal oxygenator and an intracorporeal one that is under development. The
respiratory gas exchange function of recent oxygenators is fulfilled by microporous,
hollow fiber membranes that are potted together in a bundle configuration (Svitek, 2006).
The oxygenator bundle can be configured for intravascular (Federspiel et al., 2000),
intrathroacic (Boschetti et al., 2003), or paracorporeal (Zwischenberger et al., 2002)
placements. Figure B.1(a) shows microscopic views of the oxygenator bundle and the
wall of its fibers. A pure O2 gas sweep, which is provided by an external source, flows
through the hollow fibers while the outer surface of the fibers is exposed to the blood
flow. The micro-porous wall of the fibers is the place for gas exchange between the blood
and the flowing gas within the fiber. The gas exchange mechanism is driven by the
difference between the concentration, or partial pressure, of the respiratory gases in and
out of the hollow fibers. The oxygen diffuses from the region of higher oxygen partial
pressure (pO2), i.e. inside the hollow fibers, into the region of lower pO2, i.e. in the blood
flow. Simultaneously, carbon dioxide is removed from the blood that has a higher
concentration of CO2 and is swept away by the gas flow within the fibers that contain a
lower concentration of CO2. The exchange rate of oxygen and carbon dioxide is a
function of the geometry of the fiber bundle, surface area, partial pressure of the gas side,
blood flow rate, and hemoglobin concentration. However, because of the higher storage
capacity of blood for CO2, its rate of exchange is less dependent on the blood flow rate
113
Figure B. 1. (a) Commercially available membrane oxygenator with bundle of hollow fiber
membranes (bottom left), the micro-porous structure of fibers is shown in the bottom right (b).
Intravenous membrane oxygenator with pulsating balloon , or HC. Pictures are adapted from
(Federspiel and Svitek, 2004b) and (Svitek, 2006).
than that of O2. At the rated flow5 of blood oxygenators with micro-porous hollow fiber
membranes, the rate of CO2 removal exceeds the rate of O2 delivery (Svitek, 2006).
Intravenous oxygenators, such as the catheter fiber with rotary bundle, are a subset of
intracorporeal devices that are designed to be placed in a vein through either surgical
operation or percutaneous insertion. During the last three decades several versions of
intravenous oxygenators have been designed and developed by research or industrial
groups. However, most of these efforts were unsuccessful because of an insufficient level
of gas exchange. This challenge should be resolved by improving the gas exchange
efficiency of the fiber bundle, i.e. the gas exchange rate per unit of the fiber bundle area.
The percutaneous respiratory assist catheter (PRAC) with rotary fiber bundle is a solution
for improving the gas exchange efficiency of the fiber bundle. The catheter consists of a
5 The rated flow of an oxygenator is defined as the minimum flow rate of the blood with
75% hemoglobin saturation that can enter the oxygenator and leave it with 100%
hemoglobin saturation Svitek, R.G., 2006. Development of a Paracorporeal Respiratory
Assist Lung (PRAL). University of Pittsburgh..
(a) (b)
114
bundle of hollow fibers that is wrapped around a central rotary shaft (Fig. B.1(a)). In vitro
experiments in both water and blood have shown the promising gas exchange efficiency
of the fiber bundle oxygenator. For example, rotation of the fiber bundle caused a twofold
increase in the gas exchange efficiency of the rotary fiber bundles over HC that leaded to
a comparable level of respiratory gas exchange for a smaller insertion size (a 25 Fr
insertion size and 20cm bundle length) (Budilarto et al., 2009). The in vitro PIV flow
visualization of the rotary catheter in water exhibits the well-known Taylor vortices,
which form in the fluid flow in an annular gap between two concentric cylinders where
the inner cylinder is
rotating with respect to the fixed outer cylinder. The nature of the Taylor vortices is
characterized by the dimensionless Taylor number, Ta , as follows (Budilarto et al.,
2009):
1/2
( )i o i o i
i
R R R R RTa
R (B.1)
(Budilarto et al., 2009) investigated the flow pattern caused by the rotation of the fiber
bundle in a mock tube of 25.4mm inner diameter. The water was fed axially into the
annular gap between the fiber bundle and the mock tube. A range of the catheter’s
rotational speeds between 500-7000 rpm, which cover a range of Ta numbers between
2200-31000, was studied. The incoherent and time-dependent nature of the observed
vortices was in agreement with the predicted experimental and theoretical data in the
literature6. The experimental observations showed that the gained improvement in the
level of respiratory gas exchange of the rotary fiber bundle reached its plateau at
rotational speeds near 6000 rpm. This mass transfer characteristic of the rotary bundle is
contrary to the previously reported direct monotonic relation between the mass transfer
rate and the Taylor number of rotary oxygenators; hence, the authors concluded that the
6 The characteristic feature of the vortices formed in the annular gap between two
concentric cylinders caused by rotating the inner cylinder with respect to a fixed outer
cylinder is predicted to be ibid.: steady vortices for 40 800Ta , wavy vortices for
800 2000Ta , turbulent vortices for 2000 10000 15000Ta , and turbulent flow
for 150000Ta .
115
Taylor-like flow pattern is not the predominant parameter that increases the gas exchange
efficiency of the rotary catheter. Instead, it was hypothesized that the level and the
velocity of the fluid penetration into the layered fiber bundle, which increases by
increasing the rotational speed, are the key parameters in the higher exchange rate of
respiratory gases.
116
Appendix C
Blood damage modeling
C.1. Blood
Blood is a complex and fragile tissue that consists mainly of plasma, red blood cells,
white blood cells, and platelets. Plasma is composed largely of water (92%) and some
nutrients and metabolic waste products of cells. White blood cells are part of the immune
system, which defends the body against foreign bacteria, viruses, and other
microorganisms. Red blood cells, or RBCs, are biconcave flexible disc-shaped cells that
contain hemoglobin, a protein pigment, which transports oxygen to the tissues and
absorbs their metabolic wastes, including carbon dioxide. Platelets are ellipsoidal disks of
a diameter of 2-4 μm that prevent bleeding through hemostasis mechanisms (Zhao,
2008).
Blood density is generally assumed to be 1050 kg/m3, which is close to the density of
pure water. Blood is a shear thinning fluid and its viscosity is a function of temperature
and blood hematocrit. However, blood under high strain rates, above 100s-1
, is often
modeled as a Newtonian fluid with a constant viscosity of 0.0035 Pa.s ( or 3.5 cP) . This
assumption significantly simplifies the numerical simulations (Myagmar, 2011).
C.2. Methods of blood damage investigation
In vitro experiments, in vivo experiments, and numerical simulations are three recognized
complementary methods to evaluate the level of blood damage caused by a blood-
contacting medical devices. Each method has its own benefits and limitations. The aim of
in vitro experiments is to simulate the performance of a medical device in a physiological
environment by using a mechanical set-up, while, the purpose of in vivo tests is to
evaluate the performance of a medical device in both animal and human cases. Numerical
modeling, on the other hand, has evolved as an essential element in the design of blood-
contacting medical devices. Both finite element modeling (FFM) and computational fluid
117
dynamics (CFD) methods have received increasing attention for simulation of blood flow
in a range of biomedical devices, including heart pumps (Behbahani et al., 2009). In
comparison with in vitro and in vivo experiments, computational methods are a cheaper
approach that can be used to explore alternative designs. Furthermore, these methods give
detailed quantitative views of various mechanical parameters of interest, such as 3D
velocity and shear stress fields as well as particle residence time. All of these parameters,
which are essential in the assessment of blood damage caused by a blood-contacting
medical devices, are not easily accessible by experimental techniques. However, the
replacement of the experimental in vitro and in vivo investigations of highly complex
biological systems with numerical simulations is currently not possible; further advances
in both computational hardware and software resources are required.
C.3. Blood damage types
Living blood constituents flowing in or on a biomedical device are exposed to artificial
biomaterials; hence they experience non-physiological flow features, such as high shear
stresses, flow stagnation, and cavitation. Blood damage mechanisms are complex
phenomenon that are related to many factors including, but not limited to mechanical and
biological parameters. Thrombosis and hemolysis are two categories of blood damage
that needs to be studied for design of blood-contacting medical devices. The remaining of
this appendix briefly reviews these two types of blood damage
C.3.1. Thrombosis
Thrombosis is a multi-step blood damage process that often starts with platelet activation
and ends with aggregation and deposition of the activated platelets on those device
surfaces that are in contact with the blood flow. These clots may be washed out from the
surface and travel with the blood flow and, in turn, may block narrow vessels, leading to
fatal consequences, such as organ failure, heart attack, or stroke (Lloyd-Jones et al.,
2009). The thrombus formation is triggered by Virchow’s triad: i) change in the blood
flow features, including formation of regions of high shear stress, recirculation, or
stagnation; ii) vascular endothelium trauma; and iii) change in the blood constitution,
leading to hypercoagulability (Behbahani et al., 2009). The first element in Virchow’s
118
triad is related to the hematologic design of the device and can be reduced or, ideally,
resolved by the design of the blood flow in the device. It has been shown that even low
shear stress values, on the order of 10 Pa, can trigger the platelet activation mechanism
(Behbahani et al., 2009; Giersiepen et al., 1990). Platelet aggregation and deposition are
much more complex phenomena that have not yet been fully understood. However, it is
known that besides biological factors, flow-dependent parameters, including shear stress
and the residence, or exposure, time of platelets to this stress level govern platelet
deposition (Medvitz, 2008). Experimental studies have indicated that for a given surface
material in contact with blood, there is a critical strain rate that maximizes the platelet
deposition. For example, (Hubbell and McIntire, 1986) reported that the platelet
deposition on polyurethane surfaces is maximum for a shear strain of 500 s-1
. Below this
value, by decreasing the strain rate, the platelet deposition on the surface is reduced
because of the lower convective transport of the activated platelets to the surface. Above
this value, however, more platelets are activated, but the high level of shear stress on the
wall surface washes them out from the wall. Thus the level of platelet deposition
decreases. It is anticipated that by further increasing the shearing force applied from the
blood flow on the surfaces there should be a threshold at which the thrombus deposition
would be negligible (Medvitz, 2008). Quantitative prediction of platelet activation and
deposition, using CFD analysis, has attracted researcher attention (Balasubramanian and
Slack, 2002). The literature shows that platelet activation is a more quantitatively
expressed mechanism than platelet deposition. Several models have been proposed to
predict this phenomenon. For example, (Giersiepen et al., 1990) proposed the power-law
model shown below to predict platelet activation:
6 0.77 3.075(%) 3.31 10LDH
tLDH
(C.1)
where LDH is the platelet release of the cytoplasm enzyme during platelet activation,
is the shear stress, and t is the time of exposure to that shear stress. In another study,
(Cheng et al., 2004) developed a 3D CFD model to simulate the blood flow around a bi-
leaflet valve. They showed that wall shear stress and exposure time play a determinant
role in thrombus formation, particularly platelet activation. (Avrahami et al., 2006) used a
platelet level-of-action parameter, LOA, as a quantitative measure to estimate the risk of
119
flow induced thrombus formation in a Berlin ventricular assist device chamber with
mono-leaflet valves. The LOA represents the cumulative effect of shear stress history on
platelet activation and may be defined according to Hellum’s platelet stimulation
function, PSF, for blood flow through stenosed arteries (Boreda et al., 1995; Jesty et al.,
2003). (Avrahami et al., 2006) used the following relation to define LOA:
0.453.n
path
LOA t (C.2)
where t is the exposure, or residence, time and n is a scalar measure of the 3D stress
tensor. They used Bludszuweit’s approach, which is an analog to the von Mises criterion
for solid materials, to define scalar n of a 3D stress tensor.
2 2 2 2 2 213( )
3n xx yy zz xx yy yy zz zz xx xy yz zx (C.3)
where , ,xx yy zzare the normal components and , ,xy yz zx are the shear components of
the stress tensor. If Hellum’s criterion is adopted, the LOA values above 3.5 Pa.s are
susceptible to thrombus formation and should thus be avoided. (Yin et al., 2004) used
platelet activation state (PAS) parameter as a measure of platelet activation in mono-
leaflet and bi-leaflet valves of heart pumps.
.n
path
PAS t (C.4)
where (ASTM, 1997) t is the exposure, or residence, time and n is a scalar measure of
the 3D stress tensor obtained by Boussinesq approximation. They CFD results showed
that the studied bi-leaflet valves can produce a platelet activation state of 2 Pa.s.
C. 3.2. Hemolysis:
Red blood cells can resist high normal deformations, but they are fragile under shear
strain. Sufficiently long residence of RBCs in regions of high shear strain rate (1 ms
exposure to 400 Pa shear stress) may lead to premature rupture of RBCs, or hemolysis
(Sallam and Hwang, 1984). The reduction in the rate of oxygen and carbon dioxide
exchange following hemolysis can lead to dysfunction or failure of some body organs. It
120
can also lead to kidney saturation following rupture of RBCs and release of their toxic
constitutes, including hemoglobin, into the blood stream; each kidney can clear around 14
grams of hemoglobin per day (Olsen, 2000). The hemolysis level can be experimentally
determined by measuring the concentration of free hemoglobin in blood samples.
(ASTM, 1997) proposes the following three indices to evaluate the hemolysis caused by a
blood-contacting medical device:
Normalized index of hemolysis (NIH):
NIH( /100 ) (1 ) 100fHb V
g L Htt Q
(C.5)
NIH represents the increase in concentration of plasma-free hemoglobin ( fHb ) in grams
per 100 L of pumped blood. In this equation V is the total volume of the pumped blood, Q
is the pump flow rate, and Ht is the hematocrit fraction of the pumped blood.
Normalized milligram index of hemolysis (mgNIH):
mgNIH( /100 ) 1000 NIH (1 ) 100fHb V
mg L Htt Q
(C.6)
mgNIH is a measure of the increase in plasma-free hemoglobin in milligram per 100 liter
of pumped blood.
Modified index of hemolysis (MIH)
4 6NIH 1MIH 10 (1 ) 10
fHb VHt
Hb Hb t Q (C.7)
MIH is a unit-free index of hemolysis that indicates the increase in the concentration of
hemoglobin in the pumped blood that is normalized by the hemoglobin concentration
(Hb).
Extensive experimental efforts have been devoted to find an empirical model that
describes the level of hemolysis damage as a function of the flow field features. These
models are often obtained by measuring the hemolysis level in blood samples that flow
within the annular gap of a concentric cylinder viscometer, which is an experimental
setup for generating a uniform shear stress field. Because of the natural complications of
the hemolysis mechanism, which depends on the adopted experimental procedure and the
precision of the implemented equipments, the reported experimental results are to some
121
extent scattered. For example, (Leverett et al., 1972) reported a hemolysis threshold level
of 150 Pa for blood flow in a concentric cylinder geometry. In another study, (Sutera and
Mehrjardi, 1975) reported a critical shear stress of 250 Pa with 4 minutes of exposure
time in a turbulent shear flow. (Paul et al., 2003) investigated the hemolysis caused by a
wider range of shear stresses (0<τ<450 Pa) and exposure times ( 25 1238ms t ms ).
In these experiments they used porcine blood samples, which closely mimic human
blood. The authors did not detect an increase of hemolysis for shear stresses below
425Pa for residence times values of 620t ms. (Giersiepen et al., 1990), using
experimental data on human blood reported by (Wurzinger L. J. et al., 1986), proposed
the following empirical power-law relation to correlate the hemolysis level to shear stress
and residence time.
7 2.416 0.785( , ) 3.62 10Hb
D t tHb
(C.8)
Here D is the hemolysis damage, Hb is the hemoglobin concentration in blood, is
shear stress, and t is the exposure time to that shear stress. (Heuser and Opitz, 1980)
proposed another power-law model for predicting the hemolysis damage; it was based on
experimental data taken from a laminar flow in a Couette viscometer for a range of shear
stress of 40-700 Pa and exposure time of 3.4-600 ms:
6 1.991 0.765( , ) 1.8 10Hb
D t tHb
(C.9)
where D is the hemolysis damage, Hb is the hemoglobin concentration in blood, is the
shear stress and t is the exposure time to that shear stress.
The hemolysis damage (D) is related to the normalized indices of hemolysis as (Garon
and Farinas, 2004):
( )( ) 1 ( , ) 100
fMb tNIH t D t Hb
Mb (C.10)
6( )( ) 1 ( , ) 10
fMb tMIH t D t
Mb (C.11)
122
where Mb is the total hemoglobin and fMb is the free-hemoglobin in the pumped blood.
For small ratios of free-hemoglobin to total hemoglobin in the pumped blood,
1fMb Mb and the above equations are simplified to:
( ) ( , ) 100NIH t D t Hb (C.12)
6( ) ( , ) 10MIH t D t (C.13)
Numerically, hemolysis caused by a blood-contacting medical device is often evaluated
by solving CFD models that are designed to find the shear stress history, i.e. the shear
stress and residence time values along several flow stream lines (Behbahani et al., 2009).
The results of these simulations are incorporated with four well-established numerical
approaches to assess the hemolysis damage (D) caused by the device. These approaches
have been successfully implemented by several authors to design blood-contacting
medical devices against hemolysis. The remaining part of this appendix briefly describes
these four numerical approaches.
a. Threshold Value Approach
The threshold value approach assumes that hemolysis occurs when blood is exposed to
shear stress values above the threshold level. This approach is easy to implement and fast,
but it does not consider the decisive role of residence time on the premature rupture of
RBCs. Furthermore, based on its generated hemolysis level, the threshold value approach
only either accepts or rejects a design; while in many practical applications the regions of
critically high shear stress occupy a small portion of the whole volume of the device.
Hence, this approach is not suitable for design optimization purposes.
b. Mass-Weighted Average Approach
The mass-weighted average approach assumes that the hemolysis level caused by a
blood-contacting medical device is proportional to the mass-weight average of the
regions of critically high shear stress values. This approach is a modification of the
threshold value method. It is a fast numerical method that requires low computational
resources. Furthermore, this method gives a quantitative measure of hemolysis that can
be used for design optimization purposes. However, it neglects the significant role of
123
residence time in accumulation of RBC damage. The details of this method can be found
in the works of (Apel et al., 2001; Chua et al., 2006; Chua et al., 2007; Mitoh et al.,
2003).
c. Eulerian Approach
The Eulerian approach was proposed by (Farinas et al., 2006) as a fast 3D method to
assess the hemolysis level caused by an artificial blood-contacting medical device. This
approach implements an empirical power-law relation, e.g. equations (C.8) and (C.9), to
assess a damage index independent of residence time. Equations C.8 and C.9 are non-
linear with respect to residence time and can be represented in the following general
form:
D C t (C.14)
where , ,C are constants that are identified after regression analysis on measured
experimental data. (Farinas et al., 2006) defined the following linear damage model:
1/ 1/ /
ID D C t (C.15)
The exact derivation of the linear blood damage with respect to time yields a time
independent parabolic transport equation that represents the blood damage source
parameter, which is defined as:
1/ /[ ]I
dI D C
dt (C.16)
By assuming a time independent and incompressible blood flow, an average linear
hemolysis index can be obtained as follows:
1I
V
D IdVQ
(C.17)
where Q is the flow rate into the device and dV is the infinitesimal volume of the blood.
The average blood damage can then be readily obtained from the above defined average
linear damage index.
124
ID D (C.18)
As a major advantage, instead of performing time-consuming analysis on the stream
lines, this method computes the blood damage index by performing volume integration
over the whole stress domain. Furthermore, this approach can in theory highlight those
problematic features of the device that can cause high level of blood damage. (Zhang et
al., 2006), who performed a numerical and experimental study on a centrifugal pump,
reported a good agreement between the predicted hemolysis damage by this approach and
that of experimental investigations.
d. Lagrangian Approach
The Lagrangian approach is the most accurate but computationally expensive method for
predicting hemolysis levels caused by a blood-contacting medical device. This approach
monitors the damage accumulation for a sufficient number of RBCs along their flow
path-lines. Each RBC is often modeled by a discrete phase while the empirical damage
models, such as equations (C.8) and (C.9), are used to correlate hemolysis to the shear
stress history of each RBC. The total blood damage is taken as the average of the damage
to all of the modeled RBCs. It was shown by (Chan et al., 2002) that tracking more
particles, which comes at the expense of higher computational time, increases the
accuracy of the results. Many authors have used this method in the design of various
blood contacting device, including heart pumps (Apel et al., 2001; Arora et al., 2006;
Chan et al., 2002; Song et al., 2003a; Yano et al., 2003).
125
References
Abbadi, A., Azari, Z., Belouettar, S., Gilgert, J., Freres, P., 2010. Modelling the fatigue
behaviour of composites honeycomb materials (aluminium/aramide fibre core) using
four-point bending tests. International Journal of Fatigue 32, 1739-1747.
Andrews, E., Gibson, L., Ashby, M., 1999. The creep of cellular solids. Acta materialia
47, 2853-2863.
Andrews, E., Gioux, G., Onck, P., Gibson, L., 2001. Size effects in ductile cellular solids.
Part II: experimental results. International Journal of Mechanical Sciences 43, 701-713.
Apel, J., Paul, R., Klaus, S., Siess, T., Reul, H., 2001. Assessment of hemolysis related
quantities in a microaxial blood pump by computational fluid dynamics. Artificial Organs
25, 341-347.
Arora, D., Behr, M., Pasquali, M., 2006. Hemolysis Estimation in a Centrifugal Blood
Pump Using a Tensor‐based Measure. Artificial Organs 30, 539-547.
ASTM, 1997. Designation: F 1841-97. Standard practice for assessment of hemolysis in
continuous flow blood pumps. Annual Books of ASTM Standards, F, pp. 1288-1292.
Auricchio, F., 1995. Shape Memory Alloys: Applications, Micromechanics,
Macromodeling and Numerical Simulations. University of California at Berkely.
Avrahami, I., Rosenfeld, M., Einav, S., 2006. The hemodynamics of the Berlin pulsatile
VAD and the role of its MHV configuration. Annals of biomedical engineering 34, 1373-
1388.
Balasubramanian, V., Slack, S., 2002. The effect of fluid shear and co-adsorbed proteins
on the stability of immobilized fibrinogen and subsequent platelet interactions. Journal of
Biomaterials Science, Polymer Edition 13, 543-561.
Banerjee, S., Bhaskar, A., 2005. Free vibration of cellular structures using continuum
modes. Journal of Sound and Vibration 287, 77-100.
Banerjee, S., Bhaskar, A., 2009. The applicability of the effective medium theory to the
dynamics of cellular beams. International Journal of Mechanical Sciences 51, 598-608.
Banerjee, S., Bhattacharyya, D., 2010. Optimal design of sandwich panels made of wood
veneer hollow cores. Composites Science and Technology.
Bathe, K.J., 1996. Finite element procedures. Prentice hall Englewood Cliffs, NJ.
Bedoya, J., Meyer, C.A., Timmins, L.H., Moreno, M.R., Moore, J.J.E., 2006. Effects of
Stent Design Parameters on Normal Artery Wall Mechanics. Journal of Biomechanical
Engineering 128, 757-765.
Behbahani, M., Behr, M., Hormes, M., Steinseifer, U., Arora, D., Coronado, O., Pasquali,
M., 2009. A review of computational fluid dynamics analysis of blood pumps. European
Journal of Applied Mathematics 20, 363-397.
126
Bhattacharya, K., 2003. Microstructure of martensite: why it forms and how it gives rise
to the shape-memory effect. Oxford University Press, USA.
Bluestein, D., Rambod, E., Gharib, M., 2000. Vortex shedding as a mechanism for free
emboli formation in mechanical heart valves. Journal of biomechanical engineering 122,
125.
Bom, M., Huang, K., 1954. Dynamical theory of crystal lattices. Oxford: Clarendon
Press.
Boreda, R., Fatemi, R., Rittgers, S., 1995. Potential for platelet stimulation in critically
stenosed carotid and coronary arteries. J. Vasc. Invest 1, 26-37.
Boschetti, F., Cook, K.E., Perlman, C.E., Mockros, L.F., 2003. Blood flow pulsatility
effects upon oxygen transfer in artificial lungs. ASAIO journal 49, 678.
Budilarto, S.G., Frankowski, B.J., Hattler, B.G., Federspiel, W.J., 2009. Flow
visualization study of a novel respiratory assist catheter. Artificial Organs 33, 411-418.
Burman, M., 1998. Fatigue crack initiation and propagation in sandwich structures. PhD
theses. Department of Aeronautics, Royal Institute of Technology, Stockholm, Sweden.
Case, J., Chilver, L., Ross, C.T.F., 1999. Strength of Materials and Structures (4th
Edition). Elsevier.
Chan, W., Wong, Y., Ding, Y., Chua, L., Yu, S., 2002. Numerical investigation of the
effect of blade geometry on blood trauma in a centrifugal blood pump. Artificial Organs
26, 785-793.
Chen, J., Huang, M., 1998. Fracture analysis of cellular materials: a strain gradient
model. Journal of the Mechanics and Physics of Solids 46, 789-828.
Cheng, C.P., Wilson, N.M., Hallett, R.L., Herfkens, R.J., Taylor, C.A., 2006. In Vivo MR
Angiographic Quantification of Axial and Twisting Deformations of the Superficial
Femoral Artery Resulting from Maximum Hip and Knee Flexion. Journal of vascular and
interventional radiology : JVIR 17, 979-987.
Cheng, R., Lai, Y.G., Chandran, K.B., 2004. Three-dimensional fluid-structure
interaction simulation of bileaflet mechanical heart valve flow dynamics. Annals of
biomedical engineering 32, 1471-1483.
Christensen, R.M., 2000. Mechanics of cellular and other low-density materials.
International Journal of Solids and Structures 37, 93-104.
Chua, L.P., Song, G., Lim, T.M., Zhou, T., 2006. Numerical analysis of the inner flow
field of a biocentrifugal blood pump. Artificial Organs 30, 467-477.
Chua, L.P., Song, G., Yu, S.C.M., Lim, T.M., 2005. Computational fluid dynamics of gap
flow in a biocentrifugal blood pump. Artificial Organs 29, 620-628.
Chua, L.P., Su, B., Lim, T.M., Zhou, T., 2007. Numerical simulation of an axial blood
pump. Artificial Organs 31, 560-570.
Chua, S.N.D., Mac Donald, B.J., Hashmi, M.S.J., 2002. Finite-element simulation of
stent expansion. Journal of Materials Processing Technology 120, 335-340.
127
Cook, R., Malkus, D., Plesha, M., Witt, R., Concepts and applications of finite element
analysis. 2002. Wiley, New York.
Côté, F., Deshpande, V., Fleck, N., Evans, A., 2006. The compressive and shear
responses of corrugated and diamond lattice materials. International Journal of Solids and
Structures 43, 6220-6242.
Côté, F., Deshpande, V.S., Fleck, N.A., 2007a. Shear fatigue strength of a prismatic
diamond sandwich core. Scripta Materialia 56, 585-588.
Côté, F., Fleck, N.A., Deshpande, V.S., 2007b. Fatigue performance of sandwich beams
with a pyramidal core. International Journal of Fatigue 29, 1402-1412.
Dancygier, A., Yankelevsky, D., Nimrod, A., Benson, W., Shibata, H., Tokaji, K.,
Ogawa, T., Shiota, H., 1996. Microstructure dependence of fatigue strength and fatigue
crack propagation in titanium aluminide. International Journal of Fatigue 18, 119-125.
Davidson, A., 2002. The pulmonary physician in critical care• 11: Critical care
management of respiratory failure resulting from COPD. Thorax 57, 1079-1084.
Day, S.W., Song, X., Throckmorton, A.L., Kirk, J.B., Olsen, D.B., 2006. Comparison of
Blood Damage Predictions From Fluid Measurements and Computations To in Vitro and
in Vivo Observations in A Centrifugal Blood Pump. ASAIO journal 52, 49A.
Defense, U.S.D.o., 1966. Aluminum and aluminum alloys. US Dept. of Defense.
Deshpande, V., Fleck, N., 2001. Multi-axial yield behaviour of polymer foams. Acta
materialia 49, 1859-1866.
Desrochers, S., 2008. Optimum design of simplicial uniaxial accelerometers. McGill
University.
Ducheyne, P., Kohn, D., Smith, T., 1987. Fatigue properties of cast and heat treated Ti---
6Al---4V alloy for anatomic hip prostheses. Biomaterials 8, 223-227.
Duerig, T., Pelton, A., Stöckel, D., 1999. An overview of nitinol medical applications.
Materials Science and Engineering A 273-275, 149-160.
Dunn, M.L., Suwito, W., Cunningham, S., 1997. Stress intensities at notch singularities.
Engineering Fracture Mechanics 57, 417-430.
Early, M., Kelly, D.J., 2010. The role of vessel geometry and material properties on the
mechanics of stenting in the coronary and peripheral arteries. Proceedings of the
Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 224,
465-476.
Eash, H.J., Mihelc, K.M., Frankowski, B.J., Hattler, B.G., Federspiel, W.J., 2007a.
Evaluation of Fiber Bundle Rotation for Enhancing Gas Exchange in a Respiratory Assist
Catheter. ASAIO 53.
Eash, H.J., Mihelc, K.M., Frankowski, B.J., Hattler, B.G., Federspiel, W.J., 2007b.
Evaluation of fiber bundle rotation for enhancing gas exchange in a respiratory assist
catheter. Asaio J. 53, 368-373.
128
Eash, H.J., Mihelc, K.M., Frankowski, B.J., Hattler, B.G., Federspiel, W.J., 2007c.
Evaluation of fiber bundle rotation for enhancing gas exchange in a respiratory assist
catheter. ASAIO journal (American Society for Artificial Internal Organs: 1992) 53, 368.
Elsayed, M., Pasini, D., 2010a. Analysis of the elastostatic specific stiffness of 2D
stretching-dominated lattice materials. Mechanics of Materials.
Elsayed, M.S.A., Pasini, D., 2010b. Multiscale structural design of columns made of
regular octet-truss lattice material. International Journal of Solids and Structures 47,
1764-1774.
Eringen, A.C., 1966. Linear theory of micropolar elasticity. Journal of Mathematics and
Mechanics 15, 909-923.
Eringen, A.C., 1999. Microcontinuum Field Theories I: Foundations and Solids.
Springer, New York.
Fang, Z., Sun, W., Tzeng, J., 2004. Asymptotic homogenization and numerical
implementation to predict the effective mechanical properties for electromagnetic
composite conductor. Journal of Composite Materials 38, 1371.
Farinas, M.I., Lacasse, D., N’dri, D., 2006. Asymptotically consistent numerical
approximation of hemolysis. Journal of biomechanical engineering 128, 688.
Federspiel, P.c.w.D.B.J., 2010.
Federspiel, W.J., Hewitt, T.J., Hattler, B.G., 2000. Experimental evaluation of a model
for oxygen exchange in a pulsating intravascular artificial lung. Annals of biomedical
engineering 28, 160-167.
Federspiel, W.J., Svitek, R.G., 2004a. Lung, Artificial: Current Research and Future
Directions, Encyclopedia of Biomaterials and Biomedical Engineering. Informa
Healthcare, pp. 922-931.
Federspiel, W.J., Svitek, R.G., 2004b. Lung, artificial: current research and future
directions. Encyclopedia of Biomaterials and Biomedical Engineering. New York:
Marcel Dekker, Inc, 922-931.
Fleck, C., Eifler, D., 2010. Corrosion, fatigue and corrosion fatigue behaviour of metal
implant materials, especially titanium alloys. International Journal of Fatigue 32, 929-
935.
Fleck, N., Deshpande, V., Ashby, M., 2010. Micro-architectured materials: past, present
and future. Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Science 466, 2495.
Fleck, N., Qiu, X., 2007. The damage tolerance of elastic-brittle, two-dimensional
isotropic lattices. Journal of the Mechanics and Physics of Solids 55, 562-588.
Fleck, N., Quintana-Alonso, I., 2009. Damage Tolerance of a Sandwich Panel Containing
a Cracked Square Lattice Core. Journal of Sandwich Structures and Materials.
Flueckiger, F., Sternthal, H., Klein, G.E., Aschauer, M., Szolar, D., Kleinhappl, G., 1994.
Strength, Elasticity, and Plasticity of Expandable Metal Stents: In Vitro Studies with
Three Types of Stress. Journal of Vascular and Interventional Radiology 5, 745-750.
129
FLUENT, A., 2009. 12.0/12.1 Documentation. Users Guide Manual, Ansys Inc.
Garon, A., Farinas, M.I., 2004. Fast Three‐dimensional Numerical Hemolysis
Approximation. Artificial Organs 28, 1016-1025.
Gibson, L., Ashby, M., 1999a. Cellular solids: structure and properties. Cambridge Univ
Pr.
Gibson, L., Ashby, M., Zhang, J., Triantafillou, T., 1989. Failure surfaces for cellular
materials under multiaxial loads--I. Modelling. International Journal of Mechanical
Sciences 31, 635-663.
Gibson, L.J., Ashby, M.F., 1999b. Cellular solids: structure and properties. Cambridge
Univ Pr.
Gibson, L.J., Ashby, M.F., Harley, B.A., 2010. Cellular Materials in Nature and
Medicine. Cambridge University Press, New York.
Giersiepen, M., Wurzinger, L., Opitz, R., Reul, H., 1990. Estimation of shear stress-
related blood damage in heart valve prostheses--in vitro comparison of 25 aortic valves.
The International journal of artificial organs 13, 300.
Grube, E., Laborde, J.C., Gerckens, U., Felderhoff, T., Sauren, B., Buellesfeld, L.,
Mueller, R., Menichelli, M., Schmidt, T., Zickmann, B., Iversen, S., Stone, G.W., 2006.
Percutaneous Implantation of the CoreValve Self-Expanding Valve Prosthesis in High-
Risk Patients With Aortic Valve Disease. Circulation 114, 1616-1624.
Gu, L., Smith, W.A., 2005. Evaluation of computational models for hemolysis
estimation. ASAIO journal 51, 202.
Guedes, J., Kikuchi, N., 1990. Preprocessing and postprocessing for materials based on
the homogenization method with adaptive finite element methods. Computer methods in
applied mechanics and engineering 83, 143-198.
Halbert, R., Isonaka, S., George, D., Iqbal, A., 2003. Interpreting COPD Prevalence
Estimates*. Chest 123, 1684-1692.
Harders, H., Hupfer, K., Rosler, J., 2005. Influence of cell wall shape and density on the
mechanical behaviour of 2D foam structures. Acta materialia 53, 1335-1345.
Hassani, B., 1996. A direct method to derive the boundary conditions of the
homogenization equation for symmetric cells. Communications in numerical methods in
engineering 12, 185-196.
Hassani, B., Hinton, E., 1998a. A review of homogenization and topology opimization I I-
-analytical and numerical solution of homogenization equations. Computers & Structures
69, 719-738.
Hassani, B., Hinton, E., 1998b. A review of homogenization and topology optimization I-
-homogenization theory for media with periodic structure. Computers & Structures 69,
707-717.
Hattler, B.G., Federspiel , W.J., Frankowski, B., 2006a. Percutaneous Respiratory Assist
Catheter Incorporating a Spinning Fiber Bundle. US Patent, USA.
130
Hattler, B.G., Federspiel, W.J., Frankowski, B.J., 2006b. Percutaneous respiratory assist
catheter incorporating a spinning fiber bundle. Google Patents.
Heuser, G., Opitz, R., 1980. A Couette viscometer for short time shearing of blood.
Biorheology 17, 17.
Hirvela, E.R., 2000. Advances in the management of acute respiratory distress syndrome:
protective ventilation. Archives of Surgery 135, 126.
Hollister, S., Kikuchi, N., 1992. A comparison of homogenization and standard
mechanics analyses for periodic porous composites. Computational Mechanics 10, 73-95.
Huang, J.-S., Liu, S.-Y., 2001a. Fatigue of honeycombs under in-plane multiaxial loads.
Materials Science and Engineering: A 308, 45-52.
Huang, J.-S., Liu, S.-Y., 2001b. Fatigue of isotropic open-cell foams under multiaxial
loads. International Journal of Fatigue 23, 233-240.
Huang, J.S., Gibson, L.J., 1991. Fracture toughness of brittle foams. Acta Metallurgica et
Materialia 39, 1627-1636.
Huang, J.S., Lin, J.Y., 1996. Fatigue of cellular materials. Acta Materialia 44, 289-296.
Hubbell, J.A., McIntire, L.V., 1986. Visualization and analysis of mural thrombogenesis
on collagen, polyurethane and nylon. Biomaterials 7, 354-360, IN351, 361-363.
Hutchinson, R.G., 2004. Mechanics of lattice materials. University of Cambridge.
In, A., 1995. Annual book of ASTM standards. American Society for Testing and
Materials: West Conshohocken, PA 8, 182.
James, B.A., Sire, R.A., 2010. Fatigue-life assessment and validation techniques for
metallic vascular implants. Biomaterials 31, 181-186.
Jansson, S., 1992. Homogenized nonlinear constitutive properties and local stress
concentrations for composites with periodic internal structure. International Journal of
Solids and Structures 29, 2181-2200.
Javid, F., Angeles, J., Pasini, D., Cecere, R., Year Optimum Design of an Anchoring
System for Percutaneous Mitral Valve Repair.
Jesty, J., Yin, W., Perrotta, P., Bluestein, D., 2003. Platelet activation in a circulating
flow loop: combined effects of shear stress and exposure time. Platelets 14, 143-149.
Kalamkarov, A.L., Andrianov, I.V., Danishevs’kyy, V.V., 2009. Asymptotic
homogenization of composite materials and structures. Applied Mechanics Reviews 62,
030802.
Kaye, J., Elgar, E.C., 1956. Fluid flow and heat transfer in an annulus with an inner
rotating cylinder. Massachusetts Institute of Technology.
Kim, S., Chandran, K., Chen, C., 1992. Numerical simulation of steady flow in a two-
dimensional total artificial heart model. Journal of biomechanical engineering 114, 497.
Kleinstreuer, C., Li, Z., 2006. Analysis and computer program for rupture-risk prediction
of abdominal aortic aneurysms. BioMedical Engineering Online 5, 19.
131
Kleinstreuer, C., Li, Z., Basciano, C., Seelecke, S., Farber, M., 2008a. Computational
mechanics of Nitinol stent grafts. Journal of biomechanics 41, 2370-2378.
Kleinstreuer, C., Li, Z., Basciano, C.A., Seelecke, S., Farber, M.A., 2008b.
Computational mechanics of Nitinol stent grafts. Journal of Biomechanics 41, 2370-
2378.
Kolluri, M., Mukherjee, M., Garcia-Moreno, F., Banhart, J., Ramamurty, U., 2008.
Fatigue of a laterally constrained closed cell aluminum foam. Acta Materialia 56, 1114-
1125.
Kulkarni, N., Mahfuz, H., Jeelani, S., Carlsson, L.A., 2003. Fatigue crack growth and life
prediction of foam core sandwich composites under flexural loading. Composite
Structures 59, 499-505.
Kulkarni, N., Mahfuz, H., Jeelani, S., Carlsson, L.A., 2004. Fatigue failure mechanism
and crack growth in foam core sandwich composites under flexural loading. Journal of
reinforced plastics and composites 23, 83.
Kumar, R., McDowell, D., 2004. Generalized continuum modeling of 2-D periodic
cellular solids. International Journal of Solids and Structures 41, 7399-7422.
Kumar, R.S., McDowell, D.L., 2009. Multifunctional design of two-dimensional cellular
materials with tailored mesostructure. International Journal of Solids and Structures 46,
2871-2885.
Lally, C., Dolan, F., Prendergast, P.J., 2005. Cardiovascular stent design and vessel
stresses: a finite element analysis. Journal of Biomechanics 38, 1574-1581.
Leverett, L., Hellums, J., Alfrey, C., Lynch, E., 1972. Red blood cell damage by shear
stress. Biophysical journal 12, 257-273.
Li, J., Cheng, H., Yu, J., Han, F., 2003. Effect of dual-size cell mix on the stiffness and
strength of open-cell aluminum foams. Materials Science and Engineering: A 362, 240-
248.
Li, M., Guiu, F., 1995. Subcritical fatigue crack growth in alumina--I. Effects of grain
size, specimen size and loading mode. Acta metallurgica et materialia 43, 1859-1869.
Li, Z., Kleinstreuer, C., 2005. Blood flow and structure interactions in a stented
abdominal aortic aneurysm model. Medical Engineering & Physics 27, 369-382.
Lim, D., Cho, S.-K., Park, W.-P., Kristensson, A., Ko, J.-Y., Al-Hassani, S., Kim, H.-S.,
2008. Suggestion of Potential Stent Design Parameters to Reduce Restenosis Risk driven
by Foreshortening or Dogboning due to Non-uniform Balloon-Stent Expansion. Annals
of Biomedical Engineering 36, 1118-1129.
Lim, T.S., Lee, C.S., Lee, D.G., 2004. Failure modes of foam core sandwich beams under
static and impact loads. Journal of composite materials 38, 1639-1662.
Lin, J.Y., Huang, J.S., 2005. Creep of hexagonal honeycombs with Plateau borders.
Composite Structures 67, 477-484.
Lipperman, F., Fuchs, M., Ryvkin, M., 2008a. Stress localization and strength
optimization of frame material with periodic microstructure. Computer methods in
applied mechanics and engineering 197, 4016-4026.
132
Lipperman, F., Ryvkin, M., Fuchs, M., 2007a. Fracture toughness of two-dimensional
cellular material with periodic microstructure. International Journal of Fracture 146, 279-
290.
Lipperman, F., Ryvkin, M., Fuchs, M., 2007b. Nucleation of cracks in two-dimensional
periodic cellular materials. Computational Mechanics 39, 127-139.
Lipperman, F., Ryvkin, M., Fuchs, M., 2009. Design of crack-resistant two-dimensional
periodic cellular materials. Journal of Mechanics of Materials and Structures, 441.
Lipperman, F., Ryvkin, M., Fuchs, M.B., 2008b. Crack arresting low-density porous
materials with periodic microstructure. International Journal of Engineering Science 46,
572-584.
Liu, A.F., 2005. Mechanics and mechanisms of fracture: an introduction. ASM
International (OH).
Lloyd-Jones, D., Adams, R., Carnethon, M., De Simone, G., Ferguson, T.B., Flegal, K.,
Ford, E., Furie, K., Go, A., Greenlund, K., 2009. Heart Disease and Stroke Statistics—
2009 Update A Report From the American Heart Association Statistics Committee and
Stroke Statistics Subcommittee. Circulation 119, 480-486.
Lütjering, G., Williams, J., Gysler, A., 2000. Microstructure and mechanical properties of
titanium alloys. Microstructure and Properties of Materials 2, 1-74.
Maiti, S., Ashby, M., Gibson, L., 1984a. Fracture toughness of brittle cellular solids.
SCRIPTA METALLURG. 18, 213-218.
Maiti, S.K., Ashby, M.F., Gibson, L.J., 1984b. Fracture toughness of brittle cellular
solids. Scripta Metallurgica 18, 213-217.
Mannino, D.M., Homa, D.M., Akinbami, L.J., Ford, E.S., Redd, S.C., 2002. Chronic
obstructive pulmonary disease surveillance-United States, 1971-2000. Respiratory care
47, 1184-1199.
Martin, D., Boyle, F.J., 2010. Computational structural modelling of coronary stent
deployment: a review. Computer Methods in Biomechanics and Biomedical Engineering.
Masoumi Khalil Abad, E., Pasini, D., Cecere, R., 2012. Shape optimization of stress
concentration-free lattice for self-expandable Nitinol stent-grafts. Journal of
biomechanics.
Masoumi Khalil Abad, E., Arabnejad Khanoki, S., Pasini, D., 2013. Fatigue design of
lattice materials via computational mechanics: application to lattices with smooth
transitions in cell geometry. International Journal of Fatigue 47, 126-136.
Masters, I., Evans, K., 1996. Models for the elastic deformation of honeycombs.
Composite structures 35, 403-422.
Maugin, G., 1992. The thermomechanics of plasticity and fracture. Cambridge Univ Pr.
McCullough, K., Fleck, N., Ashby, M., 1999. Uniaxial stress-strain behaviour of
aluminium alloy foams. Acta Materialia 47, 2323-2330.
133
McCullough, K., Fleck, N., Ashby, M., 2000. The stress-life fatigue behaviour of
aluminium alloy foams. Fatigue and Fracture of Engineering Materials and Structures 23,
199-208.
Medvitz, R.B., 2008. Development and validation of a computational fluid dynamic
methodology for pulsatile blood pump design and prediction of thrombus potential.
ProQuest.
Messac, A., Ismail-Yahaya, A., Mattson, C., 2003. The Normalized Normal Constraint
Method for Generating the Pareto Frontier. Structural and Multidisciplinary Optimization
25, 86-98.
Migliavacca, F., Petrini, L., Colombo, M., Auricchio, F., Pietrabissa, R., 2002.
Mechanical behavior of coronary stents investigated through the finite element method.
Journal of Biomechanics 35, 803-811.
Mihelc, K.M., Frankowski, B.J., Lieber, S.C., Moore, N.D., Hattler, B.G., Federspiel,
W.J., 2009. Evaluation of a Respiratory Assist Catheter that Uses an Impeller Within a
Hollow Fiber Membrane Bundle. Asaio J. 55, 569-574.
Mitoh, A., Yano, T., Sekine, K., Mitamura, Y., Okamoto, E., Kim, D.W., Yozu, R.,
Kawada, S., 2003. Computational Fluid Dynamics Analysis of an Intra‐Cardiac Axial
Flow Pump. Artificial Organs 27, 34-40.
Motz, C., Friedl, O., Pippan, R., 2005. Fatigue crack propagation in cellular metals.
International Journal of Fatigue 27, 1571-1581.
Mullen, L., Stamp, R., Fox, P., Jones, E., Ngo, C., Sutcliffe, C., 2009. Selective laser
melting: A unit cell approach for the manufacture of porous, titanium, bone in-growth
constructs, suitable for orthopedic applications. II. Randomized structures. Journal of
Biomedical Materials Research Part B: Applied Biomaterials 92, 178-188.
Murr, L., Gaytan, S., Medina, F., Lopez, H., Martinez, E., Machado, B., Hernandez, D.,
Martinez, L., Lopez, M., Wicker, R., 2010. Next-generation biomedical implants using
additive manufacturing of complex, cellular and functional mesh arrays. Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
368, 1999.
Myagmar, O., 2011. Evaluation of CFD based hemolysis prediction methods.
Neuber, H., 1961a. Theory of notch stresses: Principles for Exact Calculation of Strength
with Reference to Structural Form and Material, 2nd ed. United States Atomic Energy
Commission, Washington.
Neuber, H., 1961b. Theory of notch stresses: principles for exact calculation of strength
with reference to structural form and material. USAEC Office of Technical Information.
Neuber, H., 1961c. Theory of Stress Concentration for Shear Strained Prismatic Bodies
with Arbitrary Nonlinear Stress-Strain Law. Journal of Applied Mechanics,Transactions
of the ASME 28, 554.
Neuber, H., 1967. Theory of notch stresses: Principles for Exact Calculation of Strength
with Reference to Structural Form and Material, 2nd ed. United States Atomic Energy
Commission, Washington.
134
Nicholas, T., 1981. Tensile testing of materials at high rates of strain. Experimental
Mechanics 21, 177-185.
Nicholas, T., Zuiker, J., 1989. On the use of the Goodman diagram for high cycle fatigue
design. International Journal of Fracture 80, 219-235.
Niederman, M.S., 1998. Introduction: mechanisms and management of COPD. Chest
113, 233S-234S.
Nieh, T., Higashi, K., Wadsworth, J., 2000. Effect of cell morphology on the compressive
properties of open-cell aluminum foams. Materials Science and Engineering A 283, 105-
110.
Noble, F., 1983. Acoustic emission and fatigue crack growth in rigid polyurethane foam.
Journal of Materials Science 18, 1827-1835.
Noble, F., Lilley, J., 1981. Fatigue crack growth in polyurethane foam. Journal of
Materials Science 16, 1801-1808.
Olsen, D.B., 2000. The History of Continuous‐Flow Blood Pumps. Artificial Organs 24,
401-404.
Olurin, O., McCullough, K., Fleck, N., Ashby, M., 2001. Fatigue crack propagation in
aluminium alloy foams. International Journal of Fatigue 23, 375-382.
Pasini, D., Masoumi Khalil Abad, E., 2012. Stent devices made of a lattice with smooth
shape cells improving stent fatigue life, United States patent application No. 13/659,398.
Pasini, D., Masoumi Khalil Abad, E., 2012. Stent devices made of a lattice with smooth
shape cells improving stent fatigue life, Canada patent application No. 2,793,650.
Pasini, D. Masoumi Khalil Abad, E., Cecere, R., 2012. A Protective Cage For An
Intravenous Respiratory Catheter, US patent No. 61706428.
Paul, R., Apel, J., Klaus, S., Schügner, F., Schwindke, P., Reul, H., 2003. Shear stress
related blood damage in laminar couette flow. Artificial Organs 27, 517-529.
Pedersen, P., Year Some Benchmarks for Optimized Shapes with Stress Concentration.
Pelton, A.R., Gong, X.Y., Duerig, T., Year Fatigue testing of diamond-shaped specimens.
Petras, A., Sutcliffe, M., 1999. Failure mode maps for honeycomb sandwich panels.
Composite Structures 44, 237-252.
Petrini, L., Migliavacca, F., Auricchio, F., Dubini, G., 2004. Numerical Investigation of
the Intravascular Coronary Stent Flexibility. Journal of Biomechanics 37, 495-501.
Petty, T.L., 2002. COPD in Perspective*. Chest 121, 116S-120S.
Pilkey, W.D., 2007. Frontmatter, Peterson's Stress Concentration Factors. John Wiley &
Sons, Inc., pp. i-xxxii.
Pitteri, M., Zanzotto, G., 2003. Continuum models for phase transitions and twinning in
crystals. CRC Press.
Public Health Agency of Canada, 2007. Life and Breath: Respiratory Disease in Canada
135
Puso, M., Govindjee, S., 1995. A phenomenological constitutive model for rigid
polymeric foam. ASME APPLIED MECHANICS DIVISION-PUBLICATIONS-AMD
215, 159-176.
Raghavan, M., Webster, M., Vorp, D., 1996. Ex vivo biomechanical behavior of
abdominal aortic aneurysm: Assessment using a new mathematical model. Annals of
Biomedical Engineering 24, 573-582.
Reifsnider, K., 1991. Fatigue of composite materials. Amsterdam and New York,
Elsevier(Composite Materials Series. 4.
Robertson, S.W., Ritchie, R.O., 2007. In Vitro Fatigue-crack Growth and Fracture
Toughness Behavior of Thin-walled Superelastic Nitinol Tube for Endovascular Stents:
A Basis for Defining the Effect of Crack-like Defects. Biomaterials 28, 700-709.
Robertson, S.W., Ritchie, R.O., 2008. A fracture-mechanics-based approach to fracture
control in biomedical devices manufactured from superelastic Nitinol tube. Journal of
Biomedical Materials Research Part B: Applied Biomaterials 84B, 26-33.
Rose, J.D.G., Pimpalwar, S., Jackson, R.W., 2001. A new stent-graft for transjugular
intrahepatic portosystemic shunts. Br J Radiol 74, 908-912.
Ruzzene, M., 2004. Vibration and sound radiation of sandwich beams with honeycomb
truss core. Journal of Sound and Vibration 277, 741-763.
Ryvkin, M., Fuchs, M., Lipperman, F., Kucherov, L., 2004. Fracture analysis of materials
with periodic microstructure by the representative cell method. International Journal of
Fracture 128, 215-221.
Sallam, A., Hwang, N., 1984. Human red blood cell hemolysis in a turbulent shear flow:
contribution of Reynolds shear stresses. Biorheology 21, 783.
Schaedler, T., Jacobsen, A., Torrents, A., Sorensen, A., Lian, J., Greer, J., Valdevit, L.,
Carter, W., 2011. Ultralight Metallic Microlattices. science 334, 962-965.
Schaffner, G., Guo, X.D.E., Silva, M.J., Gibson, L.J., 2000. Modelling fatigue damage
accumulation in two-dimensional Voronoi honeycombs. International Journal of
Mechanical Sciences 42, 645-656.
Schraad, M., Harlow, F., 2006. A multi-field approach to modeling the dynamic response
of cellular materials. International Journal of Mechanical Sciences 48, 85-106.
Schrader, S.C., Beyar, R., 1998. Evaluation of the compressive mechanical properties of
endoluminal metal stents. Catheter. Cardiovasc. Diagn. 44, 179-187.
Sethi, J.M., Siegel, M.D., 2000. Mechanical ventilation in chronic obstructive lung
disease. Clinics in chest medicine 21, 799-818.
Simone, A., Gibson, L., 1998. Effects of solid distribution on the stiffness and strength of
metallic foams. Acta Materialia 46, 2139-2150.
Song, X., Throckmorton, A.L., Wood, H.G., Antaki, J.F., Olsen, D.B., 2003a.
Computational fluid dynamics prediction of blood damage in a centrifugal pump.
Artificial Organs 27, 938-941.
136
Song, X., Wood, H.G., Day, S.W., Olsen, D.B., 2003b. Studies of turbulence models in a
computational fluid dynamics model of a blood pump. Artificial Organs 27, 935-937.
Spath, H., 1995. One Dimensional Spline Interpolation Algorithms. A K Peters,
Wellesley, Mass.
Stankiewicz, J., Robertson, S., Ritchie, R., 2007. Fatigue-crack growth properties of thin-
walled superelastic austenitic Nitinol tube for endovascular stents. Journal of Biomedical
Materials Research Part A 81A, 685-691.
Stephens, R.I., Fatemi, A., Stephens, R.R., Fuchs, H.O., 2000. Metal Fatigue in
Engineering. New York: John Willey & Sons, INC.
Suresh, S., 1998. Fatigue of materials. Cambridge Univ Pr.
Sutera, S., Mehrjardi, M., 1975. Deformation and fragmentation of human red blood cells
in turbulent shear flow. Biophysical journal 15, 1-10.
Svitek, R.G., 2006. Development of a Paracorporeal Respiratory Assist Lung (PRAL).
University of Pittsburgh.
Svitek, R.G., Federspiel, W.J., 2004. Lung, Artificial: Basic Principles and Current
Applications, Encyclopedia of Biomaterials and Biomedical Engineering. Informa
Healthcare, Pittsburgh, PA, pp. 910-921.
Teng, C.P., Angeles, J., 2001. A sequential-quadratic-programming algorithm using
orthogonal decomposition with Gerschgorin stabilization. Journal of mechanical design
123, 501-509.
Teng, C.P., Bai, S., Angeles, J., 2008. Shape Synthesis in Mechanical Design. Acta
Polytechnica 47, 56-62.
Timmins, L., Moreno, M., Meyer, C., Criscione, J., Rachev, A., Moore, J., 2007. Stented
artery biomechanics and device design optimization. Medical and Biological Engineering
and Computing 45, 505-513.
Topper, T., Wetzel, R., Morrow, J.D., 1969. Neubers Rule Applied to Fatigue of Notched
Specimens. J MATER 4, 200-209.
Triantafillou, T., Gibson, L., 1990. Multiaxial failure criteria for brittle foams.
International Journal of Mechanical Sciences 32, 479-496.
Triantafillou, T., Zhang, J., Shercliff, T., Gibson, L., Ashby, M., 1989. Failure surfaces
for cellular materials under multiaxial loads--II. Comparison of models with experiment.
International Journal of Mechanical Sciences 31, 665-678.
Triantafillou, T.C., Gibson, L.J., 1987. Failure mode maps for foam core sandwich
beams. Materials Science and Engineering 95, 37-53.
Tsai, S.W., Wu, E.M., 1971. A general theory of strength for anisotropic materials.
Journal of composite materials 5, 58.
Vergnat, M., Henaine, R., Kalejs, M., Bommeli, S., Ferrari, E., Obadia, J.-F., Von
Segesser, L.K., 2009. A new self-expanding aortic stent valve with annular fixation: in
vitro haemodynamic assessment. Eur J Cardiothorac Surg 35, 970-976.
137
Vigliotti, A., Pasini, D., 2012a. Linear multiscale analysis and finite element validation of
stretching and bending dominated lattice materials. Mechanics of Materials 46, 57-68.
Vigliotti, A., Pasini, D., 2012b. Stiffness and strength of tridimensional periodic lattices.
Computer methods in applied mechanics and engineering 229–232, 27-43.
Waldman, W., Heller, M., Chen, G., 2001. Optimal free-form shapes for shoulder fillets
in flat plates under tension and bending. International journal of fatigue 23, 509-523.
Walker, K., 1970. The effect of stress ratio during crack propagation and fatigue for
2024-T3 and 7075-T6 aluminum. Effects of environment and complex load history on
fatigue life, ASTM STP 462, 1-14.
Wang, A., McDowell, D., 2004a. In-plane stiffness and yield strength of periodic metal
honeycombs. Journal of Engineering Materials and Technology 126, 137.
Wang, A., McDowell, D., 2004b. In-plane stiffness and yield strength of periodic metal
honeycombs. Journal of Engineering Materials and Technology 126, 137-156.
Wang, A., McDowell, D., 2005. Yield surfaces of various periodic metal honeycombs at
intermediate relative density. International Journal of Plasticity 21, 285-320.
Wang, D.A., Pan, J., 2006. A non-quadratic yield function for polymeric foams.
International journal of plasticity 22, 434-458.
Wang, W.-X., Luo, D., Takao, Y., Kakimoto, K., 2006. New solution method for
homogenization analysis and its application to the prediction of macroscopic elastic
constants of materials with periodic microstructures. Computers & Structures 84, 991-
1001.
Wang, X., Masood, S., 2006. Investigation of expansion characteristics of coronary slot
stents using finite flement analysis, in: Wang, K., Kovacs, G., Wozny, M., Fang, M.
(Eds.), Knowledge Enterprise: Intelligent Strategies in Product Design, Manufacturing,
and Management. Springer Boston, pp. 735-742.
Warren, W., Byskov, E., 2002. Three-fold symmetry restrictions on two-dimensional
micropolar materials. European Journal of Mechanics-A/Solids 21, 779-792.
Webb, J.G., 2008. Percutaneous Aortic Valve Replacement Will Become a Common
Treatment for Aortic Valve Disease. Journal of the American College of Cardiology:
Cardiovascular Interventions 1, 122-126.
Williams, M., 1952. Stress singularities resulting from various boundary conditions in
angular corners of plates in extension. J. appl. Mech 19, 526-528.
Wurzinger L. J., Opitz R., H., E., 1986. Mechanical blood trauma: an overview.
Angeiologie 38, 81-97.
Yano, T., Sekine, K., Mitoh, A., Mitamura, Y., Okamoto, E., Kim, D.W., Nishimura, I.,
Murabayashi, S., Yozu, R., 2003. An estimation method of hemolysis within an axial
flow blood pump by computational fluid dynamics analysis. Artificial Organs 27, 920-
925.
Yin, W., Alemu, Y., Affeld, K., Jesty, J., Bluestein, D., 2004. Flow-induced platelet
activation in bileaflet and monoleaflet mechanical heart valves. Annals of biomedical
engineering 32, 1058-1066.
138
Zhang, J., Gellman, B., Koert, A., Dasse, K.A., Gilbert, R.J., Griffith, B.P., Wu, Z.J.,
2006. Computational and experimental evaluation of the fluid dynamics and
hemocompatibility of the CentriMag blood pump. Artificial Organs 30, 168-177.
Zhang, Y., Fan, H., Fang, D., 2008. Constitutive relations and failure criterion of planar
lattice composites. Composites Science and Technology 68, 3299-3304.
Zhao, R., 2008. Experimental and theoretical investigation of blood flow dynamics
associated with blood contacting devices. ProQuest.
Zhi, Y., Wang, X., Gao, Z., Liu, Y., Yue, Z., 2008. Mechanical property analysis of
Nitinol defective stent under uniaxial loading/unloading. Materialwissenschaft und
Werkstofftechnik 39, 479-485.
Zienkiewicz, O.C., Taylor, R.L., 2005. The finite element method for solid and structural
mechanics. Butterworth-Heinemann.
Zwischenberger, J.B., Wang, D., Lick, S.D., Deyo, D.J., Alpard, S.K., Chambers, S.D.,
2002. The paracorporeal artificial lung improves 5-day outcomes from lethal
smoke/burn-induced acute respiratory distress syndrome in sheep. The Annals of thoracic
surgery 74, 1011-1018.
