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NUMERICAL ANALYSIS OF DAMAGE INITIATION AND
DEVELOPMENT IN BENDS OF STEEL PIPELINES
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van Rector Magnificus prof. ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 6 april 2010 om 12:30 uur
door
Auke Edwin SWART
bouwkundig en civiel ingenieur
geboren te Plymouth, Groot Brittannië
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. J. Blaauwendraad
Copromotor:
Dr. A. Scarpas, BSc, MSc
Samenstelling promotiecommissie:
Rector Magnificus, Technische Universiteit Delft, voorzitter
Prof. dr. ir. J. Blaauwendraad , Technische Universiteit Delft, promotor
Dr. A. Scarpas, BSc, MSc, Technische Universiteit Delft, copromotor
Prof. dr. ir. L.J. Sluys, Technische Universiteit Delft
Prof. dr. ir. R. Boom, Technische Universiteit Delft
Prof. ir. F.S.K. Bijlaard, Technische Universiteit Delft
Prof. dr. ir. S. van der Zwaag, Technische Universiteit Delft
Ir. G. Kruisman, r+k Consulting Engineers
Copyright © 2010 by A.E. Swart
ISBN 978-90-9025242-1
Printed in The Netherlands
In loving memory of my father Taekele Swart
PROPOSITIONS
Numerical analysis of damage initiation and
development in bends of steel pipelines
1. In the testing and simulation of structures subjected to impact loads, the
influence of friction between the material and the impactor cannot be neglected. Bij experimenten en simulaties van constructies onder een impactbelasting mag de invloed van wrijving tussen materiaal en valgewicht niet worden onderschat.
2. By means of a simple adjustment, the classical Gurson criterion is very suitable for modeling the compaction of porous materials1. Het klassieke Gurson model kan door middel van een eenvoudige aanpassing zeer goed worden gebruikt voor het modelleren van compactie van poreuze materialen1.
3. The development of micro-damage in the ferritic phase of multiphase steels can be modeled efficiently using a macroscopic void model.
Bij multifase staalsoorten kan de ontwikkeling van microschade in het ferriet efficiënt worden gemodelleerd met een macroscopisch schademodel.
4. The dangers of asbestos were discovered in 1950 and asbestosis was
acknowledged as a job related disease. The fact that the government has been aware of these dangers since the 1960’s, due to the thesis of Dr. Stumphius2, but that they waited until 1993 to forbid the use of asbestos by law is reprehensible.
In 1950 waren de risico’s van asbest al bekend en werd asbestose erkend als beroepsziekte. Het feit dat de overheid reeds in de jaren 60, mede dankzij het proefschrift van dr. Stumphius2, wist van de gevaren van asbest en het gebruik desondanks pas in 1993 bij wet verbood moet worden geduid als laakbaar bestuur.
1 “3D Material Model for EPS Response Simulation”, A.E. Swart, W.T. van Bijsterveld, M.
Duskov and A. Scarpas. Paper presented at 3rd International Conferense on EPS Geofoam in
Salt Lake City. 2 “Asbest in de bedrijfsvoering”, Dissertation of Dr. Stumphius.
5. The Pavlov response to modify input parameters when instability in a process is observed is an admission of weakness.
De Pavlovreactie om bij instabiliteit in een proces aan de ingangsparameters te sleutelen is een zwaktebod.
6. From an ethical point of view, it is irresponsible for a medical professional to
take away all hope of recovery from a patient.
Het is ethisch niet verantwoord als een behandelend medicus elke hoop op genezing bij een patiënt wegneemt.
7. The transfer of knowledge from universities to companies is also the
responsibility of the companies.
De kennisoverdracht van universiteiten naar bedrijven is ook een verantwoordelijkheid van de bedrijven.
8. The legislated reduction in CO2 emissions from cars cannot be achieved only by reducing the weight of the steel body; even if it is reduced to zero.
De wettelijke eisen voor reductie van de CO2 uitstoot van auto’s zullen niet worden gehaald door alleen het lichter maken van de carrosserie; zelfs als het gewicht ervan wordt verlaagd tot nul.
9. Recognition often lies in the denial.
In de ontkenning schuilt vaak de erkenning.
10. Contrary to what the name suggests, the amount of documentation about minimalism is very large.
In tegenstelling tot wat men zou verwachten is de documentatie over het minimalisme zeer uitgebreid.
These propositions are considered opposable and defendable and as such have been approved by the supervisor, Prof. dr. ir. J. Blaauwendraad. Deze stellingen worden opponeerbaar en verdedigbaar geacht en zijn als zodanig goedgekeurd door de promotor, Prof. dr. ir. J. Blaauwendraad.
SAMENVATTING
Numerieke analyse van het ontstaan en ontwikkelen van
microschade in stalen buisbochten.
In Nederland ligt een uitgebreid netwerk van stalen buisleidingen waarmee gassen en
vloeistoffen worden getransporteerd. De bochten in de buisleidingen vormen een
belangrijk onderdeel in het ontwerp om bijvoorbeeld dijklichamen te passeren. Door
de lagere buigstijfheid van bochten zijn ze ook zeer geschikt om uitzettingen in het
netwerk, ten gevolge van bijvoorbeeld temperatuursveranderingen, op te vangen.
Bij de bochten kan een permanente vervorming in de vorm van een opgedrongen
kromming in combinatie met belastingswisselingen, zoals een variërende interne druk,
leiden tot een toename van de plastische rek. Met de plastische vervorming groeit de
microschade in het metaal in de vorm van kleine holtes. Dit proefschrift richt zich
daarom op de ontwikkeling van microschade in de bochten ten gevolge van low cycle
fatigue.
Voor het onderzoek is gebruik gemaakt van Eindige Elementen om het gedrag van de
bochten onder een cyclische belasting te simuleren. Voor het modelleren van een
pijpbocht zijn twee elementtypen geïmplementeerd, een schaalelement en een
bochtelement. Als het schaalelement wordt toegepast, moet een groot aantal
elementen worden gebruikt. Met een bochtelement kan worden volstaan met een klein
aantal elementen. Met betrekking tot het materiaalgedrag is een model voor monotone
en cyclische belasting ontwikkeld, waarbij rekening is gehouden met de onderlinge
relatie
ii
Eindige Elementen
Wat betreft het schaalelement zijn allereerst de Lagrangian, Serendipity en Heterosis
elementen geïmplementeerd in een nieuw Eindige Elementen Programma en
vergeleken met verschillende integratieschema’s. Ze behoren tot de dikke
schaalelementen met globale verplaatsing en rotatie als graden van vrijheid. Op basis
van 2 klassieke standaardtesten is het Heterosis element met gereduceerde integratie
gekozen om resultaten van een specifiek (gekromd) buiselement mee te vergelijken.
De formulering van dit bochtelement is gebaseerd op de vervorming in langsrichting
(liggertheorie) gecombineerd met vervormingen in de omtrek van het element
(ovalisatie en kromtrekking). Hierdoor ontstaat een efficiënt element met een beperkt
aantal integratiepunten. Een nadeel van het element is dat, door alle functies waarmee
de vervorming wordt beschreven, rekening moet worden gehouden met een groot
aantal vrijheidsgraden per integratiepunt. Ook dit element is geïmplementeerd in een
nieuw Eindige Elementen Programma.
Materiaalmodel
Voor het modelleren van het materiaalgedrag is allereerst gekeken naar de
schadeontwikkeling onder monotone belasting. Voor een ductiel materiaal kan dit
gedrag worden beschreven door de initiatie, groei en samengaan van kleine holtes
onder invloed van plastische vervorming. Deze fases in de schadeontwikkeling op
microniveau kunnen worden beschreven door middel van het bekende Gurson-
Tvergaard-Needleman (GTN) materiaal model.
Een formulering van het constitutieve model voor de schaal- en voor de
buiselementen is geïmplementeerd in de eerder genoemde codes. Na het vergelijken
van beide elementtypen bleken de berekende spanningen en rekken zeer goed overeen
te komen. De met het buiselement voorspelde schadeontwikkeling blijft echter
aanzienlijk achter bij voorspelde ontwikkeling uit simulaties met het schaalelement.
iii
Het materiaalgedrag van staal ten gevolge van een cyclische wisseling tussen twee
spanningsniveaus is onder te verdelen in drie fases. Een snelle toename van de
(plastische) rek gevolgd door een constante toename met uiteindelijk een fase waarin
het materiaal bezwijkt. Voor het modelleren van dit gedrag is een benadering met
twee driedimensionale criteria toegepast. In tegenstelling tot een aantal bekende
modellen uit de literatuur is gekozen voor een constant criterium binnen het
constitutief model (Gurson) waarmee de monotone respons kan worden gesimuleerd.
Uit experimenten blijkt dat dit omhullende criterium zeer geschikt is om het moment
van bezwijken te bepalen. Gedurende de belastingswisselingen groeit en krimpt ze ten
gevolge van een fictieve cyclische plastische rek. In combinatie met een
experimenteel eenvoudig te bepalen relatie voor de ontwikkeling van de cyclische
respons in de eerste twee fases is het hiermee mogelijk om het gedrag van de eerste
belastingswisseling tot materiaaldegradatie te modelleren. Dit cyclische model is een
belangrijk wetenschappelijk resultaat van dit project. Het beschikbaar komen van dit
model maakt het mogelijk simulaties uit te voeren voor willekeurige
belastinggeschiedenissen zoals die voorkomen bij netwerken van pijpleidingen. Het
algemene karakter van het degradatiemodel, ook voor niet-ductiele materialen, wordt
gedemonstreerd aan de hand van het bijzondere geval van asfaltbeton.
Edwin Swart
iv
SUMMARY
Numerical analysis of damage initiation and
development in bends of steel pipelines
Gasses and fluids are transported via an extensive infrastructure of steel pipelines. In
the design of pipeline systems the use of elbows (pipe bends) is important to cross
obstacles. The flexural rigidity of pipe bends is smaller than that of a straight pipe.
This added flexibility makes them able to sustain significant deformations and
therefore suitable to accommodate thermal expansions and absorb other externally
induced loads in the pipeline.
The pipelines can be subjected to various load combinations which cause permanent
plastic bending moments. The variation of the stresses in the longitudinal and the
radial directions may lead to the initiation and progressive development of plasticity.
In structural steels, after the onset of plasticity, progressive material damage can
initiate in the form of micro-void nucleation. Low cycle fatigue damage may occur in
bends of steel pipelines due to combined bending and pressure loads.
For this thesis Finite Element Analysis is used to simulate the response of pipeline
bends. Two element types are used for the modeling of a pipe bend, a shell element
and a tube element (pipe elbow element), respectively. Applying the shell element, we
need to use a large number of elements. In case of tube elements, just a small number
is needed. For the material behavior a constitutive model for monotonic and for cyclic
response is developed.
Finite Elements
In the first phase of the project the Lagrangian, Serendipity and Heterosis elements
are implemented in a new Finite Element Program and compared with different
v
integration schemes. These elements fall in the category of thick shell elements and
are formulated on the basis of a thorough understanding of the kinematic and
equilibrium conditions of the problems under consideration. On the basis of two
classical benchmark tests the selectively integrated Heterosis element was selected to
compare the results obtained with a tube element.
The tube element, also known as pipe elbow element, combines longitudinal (beam-
type) with cross-sectional deformation (ovalization and warping), and is also
implemented in a new Finite Element Program. The main advantage of this element is
the reduced calculation time, due to a limited amount of integration points. A
disadvantage is the large amount of degrees of freedom per integration point and the
distance between the integration points.
Constitutive model
Initially the material response when subjected to monotonic loading was modeled.
There are three stages commonly observed in ductile damage: micro void nucleation,
growth and coalescence. To predict the damage development in the material after the
onset of plasticity the well known Gurson-Tvergaard-Needleman (GTN) constitutive
model is implemented in both Finite Element Codes. In this document a plane stress
formulation in both a Cartesian and a Curvilinear coordinate system is described. The
model can efficiently be used to predict the development of micro damage leading to
cracks. The classical shell element and the tube element are compared in combination
with this material model. The calculated stress-strain response with both models is
close, but the predicted damage is significantly different.
In standard elastoplasticity the response of a material within the yield surface is
postulated to be elastic. In order to allow for some magnitude of energy dissipation for
load cycles at stress states within the yield surface the bounding surface concept,
proposed earlier by Dafalias, is utilized. By this means, during cycling, any
experimentally observed amount of cyclic energy dissipation can be assigned. In the
vi
proposed model, the Gurson yield surface for monotonic loading acts as the bounding
surface in which a loading surface moves. During the load cycles the yield surface
hardens and softens due to a fictitious cyclic plastic strain. This implies that the
monotonic stress degradation response envelop constitutes the limit of cyclic stress
response degradation. This is also observed in experiments. This is an important
scientific deliverable of this project. The availability of such a model will enable the
simulations of arbitrary loading histories typical of those imposed on pipeline
networks. The generality of the cyclic degradation model for other, non-ductile
materials is highlighted for the particular case of asphaltic concrete.
Edwin Swart
vii
CONTENTS
SAMENVATTING
SUMMARY
1 INTRODUCTION 3
1.1 General 3
1.2 Objectives and scope of this study 4
1.3 Design of steel pipelines 4
1.4 Cyclic damage 5
1.5 Thesis delineation 6
2 FINITE ELEMENT METHOD 7
2.1 Introduction 7
2.2 Definition of strain tensors 7
2.3 Constitutive framework 10
2.4 Stiffness matrix evaluation 12
2.5 Incremental analysis 13
3 SHELL FINITE ELEMENT 15
3.1 Introduction 15
3.2 Geometry 16
3.3 Element geometry interpolation 19
3.4 Nodal variables 20
3.5 Strain measures 23
3.6 Constitutive relation 25
3.7 Numerical examples 26
4 TUBE FINITE ELEMENT 31
4.1 Introduction 31
4.2 Geometry 32
viii
4.3 Element geometry interpolation 33
4.4 Strain measures 40
4.5 Constitutive relation 41
4.6 Numerical examples 41
5 GURSON CONSTITUTIVE MODEL 49
5.1 Introduction 49
5.2 Gurson material model 51
5.3 Hardening 54
5.4 Numerical implementation 56
5.5 Numerical examples 62
6 CYCLIC MODEL 73
6.1 Introduction 73
6.2 Constitutive framework 75
6.3 Parameter determination 84
6.4 Numerical implementation 86
6.5 Numerical examples 88
7 GENERALITY OF THE CYCLIC MODEL 99
7.1 Introduction 99
7.2 Numerical implementation 101
7.3 Numerical example 103
8 CONCLUSIONS 105
APPENDICES 107
NOTATION 119
ACKNOWLEDGEMENTS 123
REFERENCES 125
CURRICULUM VITAE 129
Chapter 1
INTRODUCTION
1.1 General
Gasses and fluids are transported via an extensive infrastructure of steel pipelines. In
the design of pipeline systems the use of elbows (pipe bends) is important to cross
obstacles, like the many rivers and canals in the Netherlands, as shown in Figure 1.1.
As shown by the pioneering work of Von Karman [1911], the flexural rigidity of pipe
bends is smaller than that of a straight pipe. This added flexibility makes them able to
sustain significant deformations and therefore suitable to accommodate thermal
expansions and absorb other externally induced loads in the pipeline.
Figure 1.1 Pipeline crossing a canal (Photo: ir. A.M. Gresnigt)
2 CHAPTER 1
The pipelines can be subjected to combinations of soil pressures, temperature
variations and soil settlements, which cause permanent plastic bending moments.
These bending moments cause the circular cross-section of the elbows to ovalize. In
addition, the initially plane cross section of the bend tends to deform out of its own
plane, which is also known as warping. In combination with alternating levels of
internal pressure, the variation of the stresses in the longitudinal and the radial
directions may lead to the initiation and progressive development of plasticity. In
structural steels, after the onset of plasticity, progressive material damage can initiate
in the form of micro-void nucleation. With fatigue loading, the micro-voids in the
metallic material can eventually grow and coalesce leading to cleavage cracking. Low
cycle fatigue damage may occur in bends of steel pipelines due to combined bending
and pressure loads.
Some parts of the gas pipe network exist more than 40 years. This raised the question
whether those parts will meet the safety standards and how long they can remain in
the network. Within the pipeline industry, there is a need for an investigation to the
safety of steel pipelines and their residual life. This study has been initiated and
guided by the need to develop an inelastic constitutive model capable of simulating
cyclic hardening and softening, which characterize the material behavior under
complex loading histories. The present work can therefore also be directly applied to
industrial pipe applications or offshore pipeline applications.
1.2 Objectives and scope of this study
The objective of this research is the development of a finite element model for the
analysis of pipe components under repeated (cyclic) loads. The motivation for this
problem stems from the remaining life of buried gas pipelines, subjected to repeated
loads. The behavior of bends in steel pipelines under the action of dynamic loading is
investigated analytically by the use of the finite element method. For this purpose the
formulation of a formalistic, plasticity based model describing all stages of micro-
mechanical fatigue damage in the material is developed.
INTRODUCTION 3
For simulation of the pipe bend geometry two element types were used, layered shell
elements and layered tube bend elements. The shell elements enable the efficient
solution of otherwise intractable (in terms of mesh size and execution requirements)
civil engineering structures. They are formulated on the basis of a thorough
understanding of the kinematic and equilibrium conditions of the problems under
consideration.
The tube element (pipe elbow) is based on the mechanics of the elastic response of
pipe bends and capable of simulating the whole pipe bend with just a few elements.
1.3 Design of steel pipelines
The use of tubular members has been quite extensive in several structural and
industrial applications. They are used as liquid or gas conductors in industrial
applications and in pipeline applications (onshore and offshore). Furthermore, they
are used in many structural applications because of their good mechanical properties,
their increased strength with respect to other sections, as well as for aesthetic
purposes.
In the previous four decades extensive research has been conducted in order to
investigate the ultimate capacity of tubular members. Very important contributions on
this subject have been motivated by the design of offshore platforms (composed by
tubular members) and offshore – and onshore – pipelines. The problem of determining
the ultimate strength of a tubular member under monotonically increasing structural
loads and pressure has been investigated in quite a detail. Simplified expressions
regarding the deformation limits of tubes have been proposed during the last 15 years
and they are used in design. On the other hand, there exists very limited information
regarding the response of those members in repeated or cyclic loading.
Tubular members exhibit significant cross-sectional deformation due to loading. In
general, tubular members used in typical applications have a diameter-to-thickness
ration (D/t) which ranges from 20 to 60. The lower limit corresponds to thick tubes
used mainly in offshore pipeline or high-pressure industrial pipe applications. The
4 CHAPTER 1
response develops a well-identified maximum followed by softening. Relatively thin-
walled tubes with D/t values near the upper limit are typical for onshore pipelines or
structural (offshore and onshore) applications. For this range of D/t values, the cross-
sectional deformation (sometimes referred to as ovalization) is followed by significant
inelastic behavior. The tubes may fail due to loss of load-carrying capacity or because
of local buckling, Kyriakides and Shaw [1987].
In case of repeated loading, accumulated inelastic effects may cause a premature
failure of the member, which should be taken into serious consideration in design.
Kyriakides and Shaw [1987] demonstrated experimentally that the cross section of
circular tubes subjected to cyclic bending progressively ovalizes. Even for structures
which are designed to be within the elastic limit, plastic zones may exist at
discontinuities or at the tip of cracks. So far, designers overcome this problem by
allowing only elastic deformations or a limited inelastic deformation of the member.
In that case, repeated loading was a factor only in local spots where stress
concentration occurs. These spots are mainly in the vicinity of welds, resulting in a
high-cycle fatigue problem. This problem is usually tackled through a standard S-N
fatigue procedure using an appropriate stress concentration factor. Structural design
has been traditionally based on an allowable stress design (ASD), where limited
inelastic effects are considered.
1.4 Cyclic damage
The modeling of cyclic plasticity responses is quite complex. It is well known that the
response of different metallic materials under cyclic loads can differ. When subjected
to a large number of cycles and at a constant load level, the permanent deformation d
of a metallic material with cyclic hardening response characteristics will develop as
shown in Figure 1.2. Experiments have shown that during the first few cycles the
permanent deformation d increases rapidly. After some cycles, the rate of permanent
deformation stabilizes. If the distress phenomena are to be simulated realistically, the
range of applicability of the model should extend beyond the point of stress
INTRODUCTION 5
degradation indicated by load cycle . Otherwise, the important phase of damage
localization preceding failure will not be captured by the analyses.
fN
d
N cycles
II I III
fN
Figure 1.2 Schematic of permanent deformation development
As the relation between stress amplitude and rate of stiffness degradation is not
necessarily proportional, cyclic tests of several thousands cycles would be necessary
especially at low amplitude levels. This need can be overcome by the postulate that
the monotonic stress degradation response envelop also constitutes the limit of cyclic
stress response degradation, as illustrated in Figure 1.3. This implies that all
experimentally observed monotonic response characteristics, like hardening, softening
and sensitivity to the state of stress, are inherited by the cyclic model.
Figure 1.3 Postulated cyclic response degradation model
σ
d
II III I
Monotonic response
In standard elastoplasticity the response of a material within the yield surface is
postulated to be elastic. In order to allow for some magnitude of energy dissipation for
load cycles at stress states within the yield surface the bounding surface concept,
6 CHAPTER 1
proposed earlier by Dafalias, is utilized. By this means, during cycling, any
experimentally observed amount of cyclic energy dissipation can be assigned.
The necessary monotonic three dimensional ultimate response envelopes of the
material can be determined by means of the recently developed Gurson-Tvergaard-
Needleman (GTN) porous ductile material model. This micromechanically based
material model contains the classical von Mises model and has been known to be
capable of reproducing accurately various aspects of metallic material post-yield
response. Extension of the monotonic Gurson model to the case of cyclic plasticity
constitutes one of the important scientific deliverables of this project. Availability of
such a model will enable the simulations of arbitrary loading histories typical of those
imposed on pipeline networks.
1.4 Thesis delineation
This research project is composed of a kinematic and a constitutive description of the
deformation in pipeline bends. In Chapter 2 a general description of the finite element
method is given as an introduction to the following chapters. In Chapter 3 the
formulation and implementation of three thick shell elements are discussed. This
includes two benchmark tests to determine which formulation is most suited to use in
the analysis. Having gathered information on the mechanisms in the pipeline structure
a smart tube element is implemented as shown in Chapter 4. This element enables an
efficient evaluation of the stresses and strains in straight or curved pipelines.
There are three stages commonly observed in ductile damage: void nucleation, growth
and coalescence. Chapter 5 deals with the implementation of the well known Gurson-
Tvergaard-Needleman (GTN) constitutive model to simulate al stages in the
development of the micro-damage. This surface acts as a bounding surface in the
cyclic model as discussed in Chapter 6. With this concept we’re able to describe all
phases in the cyclic response of metals. This approach is also very interesting for
other constitutive models as shown in Chapter 7. In this chapter a non-associative
formulation of Desai is utilized.
Chapter 2
FINITE ELEMENT METHOD
2.1 Introduction
In the present chapter a short introduction is given of the finite element method. When
subjected to bending and internal or external pressures, a non-uniform displacement
field develops giving rise to a multitude of triaxial states of stress. Triaxiality has been
known to significantly influence the response of metallic materials.
For temperatures well below half the melting point, the inelastic deformation of
structural metals develops more or less independent of the strain rate. Because the
deformations in the continuum model remain quite small, the small strain formulation
is used.
2.2 Definition of strain tensors
Consider the deformation of a solid with volume , as shown in Figure 2.1. V
X
u
x
Figure 2.1 Reference and deformed configurations of a body
8 CHAPTER 2
The kinematics of a deformable body concerns the motion of the material and
coordinate system from a reference state to the final state. The coordinates of a single
material point in the reference configuration are determined on the basis of the nodal
coordinates of the element and denoted by . After loading this point moves to a
position .
X
x
If the vector of nodal coordinates is defined as: kA
[ Tk k1 k2 k3A A A=A ]
]
= NA
, (2.1)
the nodal coordinates of an element can be expressed as:
[ T1 2 NEN...=A A A A . (2.2)
The initial configuration of any point within the element can be interpolated in terms
of the corresponding nodal coordinates as:
( )1 2 3X ,X ,X=X , (2.3)
in which the matrix contains the interpolation polynomials. N
The vector ( )1 2 3x , x , x=x describes the position of that point after deformation:
= +x X u , (2.4)
where u represents the displacement of the material point.
The deformation in the immediate neighborhood of a point in the solid is
d d= ⋅x F X , (2.5)
where F is the deformation gradient at [Bathe, 1982] X
∂ ∂⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦x uFX X
+ I . (2.6)
The expressions for the undeformed and deformed configurations are now used to
calculate the strains, which are defined as an elongation per unit length.
( )2dO d d= ⋅X X
( ) ( ) ( ) ( )XCX
XFFXXFXFxxdd
dddddddo T2
⋅⋅≡⋅⋅⋅=⋅⋅⋅=⋅=
where C is the right Cauchy-Green deformation tensor.
FFC ⋅= T .
FINITE ELEMENT METHOD 9
The change in the squared lengths is
( ) ( )2 2do dO 2d d− = ⋅X E X× .
The Lagrangian-Green strain tensor is used to characterize the deformation near a
point:
E
(12
= ⋅ −TE F F )I , (2.7)
with I the second order identity matrix.
Because of small displacements, the linear strain tensor becomes
jiij
j i
uu12 X X⎛ ⎞∂∂
ε = +⎜⎜ ∂ ∂⎝ ⎠⎟⎟ . (2.8)
The kinematic relation can be written as:
, (2.9) =ε Lu
with the differential operator matrix . L
For a continuous displacement field can be interpolated by: u
=u Nd , (2.10)
in which contains the nodal displacements. d
Combining equations (2.9) and (2.10) the strain components can be expressed in
terms of the displacement vector d of the element as:
=ε Bd , (2.11)
in which is the strain-displacement transformation matrix: B
=B LN .
The actual forms of N and B are element type dependent and are presented in the
following chapters.
At the boundary of a small body it is required that either
p=u u , (2.12)
with u the displacements at the boundary and pu the prescribed displacements, or
b=σn t , (2.13)
with bt the boundary traction and n the outward normal to the surface of the body.
The Cauchy (true) stress is the force per unit area of the deformed configuration. σ
10 CHAPTER 2
2.3 Constitutive framework
The constitutive equation in the local system is
( )0=σ D ε ε− , (2.14)
in which denotes the vector of any initial/thermal strains. The fourth order tensor
is the elastic stiffness matrix. Isotropic elasticity is assumed so that
0ε
D
( ) (ijkl ij kl ik jl il jk2D K G G3
⎛ ⎞= − δ δ + δ δ + δ δ⎜ ⎟⎝ ⎠
) , (2.15)
where is the elastic bulk modulus, G is the shear modulus and K ijδ is the Kronecker
delta. In the above equation all tensor components are given with respect to a fixed
rectangular co-ordinate system. The stress can be decomposed into a deviatoric and a
hydrostatic part. The hydrostatic pressure for a three-dimensional system is defined as:
1p3
= − σI , (2.16)
with I the second order identity tensor.
The deviatoric stress tensor now becomes
. p= +s Iσ
The von Mises effective stress is defined as: 1 23q :
2⎛ ⎞= ⎜ ⎟⎝ ⎠
s s . (2.17)
The model in this project is a classic plasticity model and can be schematized as a
spring-sliding system. This serial arrangement of an elastic spring and a friction
element dates back to Prandtl [1924] and later Reuss [1930], who proposed the “Rate
Theory”. In small deformation problems, the strain rate of the matrix material can
be additively decomposed in an elastic and a plastic component:
ε&
e= +ε ε ε& & &p , (2.18)
where the plastic component pε& accounts for irreversible deformation.
In standard elasto-plasticity, the yield criterion in stress space can be written as:
( )f , 0κ =σ ,
FINITE ELEMENT METHOD 11
in which the scalar is a hardening or softening parameter which depends on the
strain history. Stress states inside this yield contour correspond to fully elastic
constitutive behavior. For metallic materials the yield surface f is assumed to be
identical to the plastic flow potential. The associated flow rule of plasticity is defined
as:
κ
p f∂= λ
∂ε
σ&& , (2.19)
with the standard Kuhn-Tucker conditions:
0λ ≥& , , . (2.20) f 0≤ f 0λ ⋅ =&
The non-negative scalar λ represents the (plastic) multiplier and is defined as: &
T
T
f
f f
∂⎛ ⎞⎜ ⎟∂⎝ ⎠λ =∂ ∂⎛ ⎞
⎜ ⎟∂ ∂⎝ ⎠
D
D
εσ
σ σ
&& . (2.21)
Using equations (2.16) and (2.17) the flow rule can also be written as:
p f p f qp q
⎛ ∂ ∂ ∂ ∂= λ +⎜ ∂ ∂ ∂ ∂⎝ ⎠
εσ σ
&&⎞⎟ . (2.22)
The stress tensor can be written as:
2p q3
= − +Iσ n , (2.23)
where the vector n defines the return direction on the deviatoric plane, Aravas [1987]
32q
=n s . (2.24)
For plane stress elements it is required that 33 33 0σ = Δσ = , whereas the corresponding
strain increment component is considered unknown. Substitution of the plane
stress hypothesis into the three-dimensional equation and eliminating determines
the reduced form of the constitutive equation used in the plate theory.
33Δε
33Δε
12 CHAPTER 2
2.4 Stiffness matrix evaluation
The governing equilibrium equations can be obtained from the principle of virtual
work. When a set of nodal virtual displacements δu is imposed it holds: Work done by Applied Forces Work done by Internal Actions= , (2.25)
or explicitly:
, (2.26) ( ) ( )T TT
V V
d dVΩ
δ δ Ω+ ρ δ = δ∫ ∫ ∫u P N u b N u g ε σ+ T dV
in which V is the volume of the element, Ω the surface area of the element, b the
force acting on the surface, ρ the mass density and represents the gravity force. In
Figure 2.2 a schematic of an element with coordinate system is given. Assuming that
nodal forces are the only external actions applied loads on the element and
substituting σ from equation
g
P
(2.14) and δ = δε B u from equation (2.11), it results:
, (2.27) T T T T
V V
dV dVδ = δ = δ∫ ∫u P u B DBdε σ
hence
(2.28)
T
V
1 1T
1 21 1
dV
d d t
.− −
⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞
= ξ ξ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=
∫
∫ ∫
P B DB d
B DB d
dΚ
1ξ
2ξ
-1 -1
1
1
Figure 2.2 Local coordinate system
FINITE ELEMENT METHOD 13
From equations (2.27)1 and (2.11) the nodal point forces in the global axes due to
local element stresses can be computed as:
. (2.29) T
V
dV= ∫R B σ
2.5 Incremental analysis
The Modified Newton-Raphson method has been widely adopted to solve a set of
non-linear equations. For each time step, the iterations are applied to achieve
equilibrium at the end of each step. Compared to the full Newton Raphson iteration,
only the system stiffness of the first iteration step, for each load step, is necessary to
be formed.
τd d
P
Δ P
1Δd
+1
τ Δτd
2Δd
+2
τ Δτd τ τ+Δ d
τP
τ+ΔτP
Figure 2.3 Modified Newton-Raphson iteration scheme
As shown in Figure 2.3, the incremental displacement at the first iteration is
1 .τΔ = Δd P−1Κ
Using equations (2.28) and (2.29) the displacement increment is determined via:
iter iter-1,τ τ+Δτ τ+ΔτΔ = −d P RΚ (2.30)
where is the system stiffness matrix at the previous load step, the
incremental displacement vector,
τΚ iterΔdτ+ΔτP the vector of external applied loads and
the vector of nodal point forces that are equivalent to the element stresses. iter-1τ+ΔτR
14 CHAPTER 2
The displacement is updated after every iteration using
. iter iter 1 iterτ+Δτ τ+Δτ
−= + Δd d d
This iterative loop is continued until the residual forces in the system are equal to
zero.
Chapter 3
SHELL FINITE ELEMENT
3.1 Introduction
Plate bending elements are developed from solid 3-D elements with the shape of the
required bending elements and a finite thickness . Following the Bernoulli
hypothesis, these elements are degenerated into plate bending elements having only
mid-surface nodal variables. The degeneration process of a solid 20-noded element to
an 8-noded degenerate curved shell element is shown in Figure 3.1. A ninth node is
added in the centre of the element.
t
Figure 3.1 Shell degeneration process
The Heterosis element, initially proposed by Hughes and Cohen [1978], constitutes a
hybrid between Serendipity and Lagrangian type shell elements. The nine-node
Lagrange element has nine shape functions for translations and rotations. The
Serendipity element has eight shape functions. The Heterosis element has a ninth
node, which admits only rotational degrees of freedom. The Serendipity type
16 CHAPTER 3
interpolation is used to approximate the displacement, while Lagrange interpolation is
used for the rotations. It has consistently performed well on numerical tests, including
cases in which the Serendipity and Lagrange elements are poor. In the third direction
the layered concept is adopted.
In the following sections of this Chapter the Heterosis shell element is formulated, but
with the discussed equations the Serendipity and Lagrangian element can also be
constructed. The three element types are implemented in a Finite Element Program, as
well as the FE Code INSAP, Scarpas [2004], and can be combined with various
nonlinear constitutive equations. To allow for transverse shear deformations, it is
assumed that the fibers initially normal to the plate middle surface remain straight but
not necessarily perpendicular to the middle surface during deformation, Reissner
[1945] and Mindlin [1951].
1X
2X
3X
Figure 3.2 Degenerated heterosis element
3.2 Geometry
The location of a point before deformation is determined by the position vector ,
defined in a Cartesian global axes system
X
{ }iX , i 1,2,3= , as shown in Figure 3.2. In
addition to this three additional coordinate systems are utilized in the formulation of
the degenerate shell element.
SHELL ELEMENT 17
3.2.1 Curvilinear coordinate system
Any point within the element can be determined via a natural coordinate system,
where two curvilinear axes and 1ξ 2ξ are defined on the mid-surface of the element
and a third linear axis along the thickness direction, as shown in Figure 3.3. All
axes span between (3ξ
)1, 1− + . Orientation of the axes is determined by the local nodal
numbering.
1X
2X
3X mid-surface
3
2 1
4 56
7
8
3ξ
1ξ
2ξ
53χ
Figure 3.3 Curvilinear Coordinate System
3.2.2 Nodal coordinate system
At each element node a local Cartesian axes system k { }ki ; i 1, 2,3χ = is set up, which
is used as a reference frame for rotations. Axis k3χ is defined to span from the bottom
surface of the element to the top one, as shown in Figure 3.3. It is not necessarily
normal to the mid-surface of the element. The magnitude of this vector is interpreted
as the shell local thickness kt .
Axis is defined as perpendicular to k1χ k3χ and parallel to the plane. The
axis can be constructed by setting the individual components of
1X X− 3
k1χ , Figure 3.4, as
follows:
k1,1 k3,3 k1,2 k1,3 k3,1, 0 ,χ = χ χ = χ = −χ . (3.1)
In case points along the axis (i.e., k3χ 2X k3,1 k3,3 0χ = χ = ) k1χ is defined as:
k1,1 k3,2 k1,2 k1,3,χ = −χ χ = −χ = 0 . (3.2)
18 CHAPTER 3
k1,3χ
2X
1X
χk3
k1χ
k3,3χk1,1χ
k3,1χ3X
Figure 3.4 Construction of Nodal Coordinate System
Axis is perpendicular to the plane defined by vectors k2χ k1χ and k3χ , Figure 3.5,
k2 k3 k1χ = χ χx . (3.3)
mid-surface 3
k3χ
k1χ
k2χ
2X
1X
3X
2ξ
1ξ
Figure 3.5 Nodal Coordinate System
3.2.3 Local coordinate system
In order to allow in subsequent sections material anisotropy, to be defined on a local
basis, a fourth Cartesian coordinate system { }i , i 1, 2,3ζ = is set up at each integration
point, as depicted in Figure 3.6. The axis 1ζ spans along vector 1ξ
v tangent to the 1ξ
axis. The vector 3ζ is defined by the cross product of vector 1ξ
v and vector 2ξ
v
tangent to the axis. The vector 2ξ 2ζ is perpendicular to axes 1ζ and 3ζ but does not
necessarily span parallel to the vector 2ξ
v . The vectors 1ξ
v and 2ξ
v are
SHELL ELEMENT 19
1
T31 2
1 1 1
XX Xξ
⎡ ⎤∂∂ ∂= ⎢ ∂ξ ∂ξ ∂ξ⎣ ⎦
v ⎥ , (3.4)
2
T31 2
2 2 2
XX Xξ
⎡ ⎤∂∂ ∂= ⎢ ∂ξ ∂ξ ∂ξ⎣ ⎦
v ⎥ . (3.5)
Thus
11 ξ=ζ v ♦, (3.6)
11x3 ξξ=ζ vv . (3.7)
The axis 2ζ is defined by the cross product:
132 xζζ=ζ . (3.8)
1ζ 2ζ
3ζ
1ξ
2ξ
Figure 3.6 Local coordinate system
Computation of the directional cosines matrix θ :
⎡ ⎤⎢= ⎢⎢ ⎥⎣ ⎦
θ ⎥⎥ , (3.9)
between the local coordinate system ( )i , i 1...3ζ = and the global ( ) is shown in Appendix A.
iX , i 1...3=
♦ the notation will be used to indicate a normalized vector
20 CHAPTER 3
3.3 Element geometry interpolation
The mid-surface is assumed to be the reference surface. The initial location of any
point within the element can therefore be interpolated on the basis of the mid-surface
nodal coordinates and the local shell thickness via: 8 8
kk k k 3 k3
k 1 k 1
tN N2= =
= + ξ∑ ∑X A χ , (3.10)
where the vector of mid-surface nodal coordinates of node in the global axes
system is defined as:
k
[ 1 2 3T
k k k kA A A=A ] . (3.11)
Only the 8 edge nodes are utilized for geometry interpolation. Following an
isoparametric formulation, the matrix of interpolation functions for an eight-node
shell element can be defined as:
N
[ ]1 2 8=N N N NK (3.12)
with
[ ]k k k kdiag N N N ; k 1...8=N =
4
4
4
4
(3.13)
and
( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )
1 1 2 1 22
2 1 2
3 1 2 1 22
4 1 2
5 1 2 1 22
6 1 2
7 1 2 1 22
8 1 2
N 1 1 1 /
N 1 1 / 2
N 1 1 1 /
N 1 1 / 2
N 1 1 1 /
N 1 1 / 2
N 1 1 1 /
N 1 1 / 2
= − ξ ⋅ − ξ ⋅ −ξ − ξ −
= − ξ ⋅ − ξ
= + ξ ⋅ − ξ ⋅ +ξ − ξ −
= + ξ ⋅ − ξ
= + ξ ⋅ + ξ ⋅ +ξ + ξ −
= − ξ ⋅ + ξ
= − ξ ⋅ + ξ ⋅ −ξ + ξ −
= − ξ ⋅ − ξ
(3.14)
SHELL ELEMENT 21
3.4 Nodal variables
At each edge node three nodal displacements and two rotations are specified: k
[ Tk k1 k2 k3 k1 k2d d d= ωd ]ω . (3.15)
1X
2X
mid-surface3
k1χ
k3χ
2ξ
1ξ 3X k2χ
k1ω
k2ω
Figure 3.7 Nodal rotations
The rotations are specified along the axes k1χ and k2χ respectively, as shown in
Figure 3.7. On the basis of a small rotations assumption, the displacements due to
either of , of any point on the local thickness vector at distance can be
approximated, Figure 3.8, as:
k1ω k2ω P 3ξ
kk1 3 k2
kk2 3 k1
t ,2t .2
δ = ξ ω
δ = ξ ω (3.16)
Displacement is directed along the axis k1δ k1χ and k2δ along the axis . Their
vector components in the global system are
k2χ
k 2
k1
i, k1 k1,i
i, k2 k2,i
d ,
i 1,...3d .
ω
ω
= δ χ
== −δ χ
(3.17)
22 CHAPTER 3
(a) (b)
k3χ
k1χ
k2χ k2χ
k1χ
k3χ
k2ωk1ω
k3
t2
ξ
P Pk1δ k2δ
Figure 3.8 Displacements of any point P due to rotations
According to the formulation of Hughes and Cohen [1978] only 2 rotational degrees
of freedom and are admitted for the 9-th element node. 91ω 92ω
Displacements interpolation
The 8 Serendipity shape functions in equation (3.14) for the displacements of the
edge nodes and the 9 Lagrangian shape functions for the rotations of all nodes
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )
2 21 1 1 2 2
2 22 1 2 2
2 23 1 1 2 2
2 24 2 1 1
2 25 1 1 2 2
2 26 1 2 2
2 27 1 1 2 2
2 28 2 1 1
2 29 1 2
N /
N 1 / 2
N /
N 1 / 2
N
N 1 / 2
N /
N 1 / 2
N 1 1
= +ξ − ξ ⋅ +ξ − ξ
= −ξ ⋅ −ξ + ξ
= +ξ + ξ ⋅ −ξ + ξ
= −ξ ⋅ +ξ + ξ
= +ξ + ξ ⋅ +ξ + ξ
= −ξ ⋅ +ξ + ξ
= −ξ + ξ ⋅ +ξ + ξ
= −ξ ⋅ −ξ + ξ
= −ξ ⋅ − ξ
4
4
/ 4
4
)
(3.18)
are utilized for displacement interpolation.
Then, the displacements of any point within the element with local coordinates
are interpolated via: ( i , i 1...3ζ =
SHELL ELEMENT 23
1 k1 k2,1 k1,18 9k1k
2 k k2 k 3 k2,2 k1,2k2k 1 k 1
3 k3 k2,3 k1,3
u dtu N d N2
u d= =
⎡ ⎤− χ χ⎡ ⎤ ⎡ ⎤ω⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= + ξ −χ χ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ω⎣ ⎦⎢ ⎥⎢ ⎥ ⎢ ⎥ − χ χ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
∑ ∑ , (3.19)
in which all terms have been defined earlier. In equation (3.19) it is worth noticing
that summation over nodal displacements spans only over the 8 edge nodes while
summation over rotations spans over all 9 element nodes.
3.5 Strain measures
At each integration point, the components of strain are defined with respect to the
local coordinate system in the reference configuration. In accordance with the
assumption of zero stresses along the shell thickness direction, the five significant
strain components are
1
21
2
11 2
2 332
1 3
31
1
2
2 1
3 2
3 1
u
u
u u
uu
uu
ζ
ζζ
ζζ ζ
ζ ζ
ζ ζζζ
ζ ζ
ζζ
∂⎡ ⎤⎢ ⎥∂ζ⎢ ⎥⎢ ⎥∂ε⎡ ⎤ ⎢ ⎥
⎢ ⎥ ∂ζ⎢ ⎥ε⎢ ⎥ ⎢ ⎥∂ ∂⎢ ⎥ ⎢γ = +⎢ ⎥ ⎢2 ⎥⎥∂ζ ∂ζ⎢ ⎥γ ⎢ ⎥
∂⎢ ⎥ ∂⎢ ⎥+⎢ ⎥γ ⎢ ⎥⎣ ⎦ ∂ζ ∂ζ⎢ ⎥∂∂⎢ ⎥
+⎢ ⎥∂ζ ∂ζ⎣ ⎦
, (3.20)
in which the notation is utilized. The strains are transferred from the global coordinate system by means of the directional cosines matrix
ij ij2γ = ε
θ , as determined in § 3.2.3:
[ ] [ ]
31 2
31 2
31 2
31 2
1 1 1 1 1 1
T 31 2
2 2 2 2 2 2
31 2
3 3 33 3 3
uu u uu uX X X
uu u uu uX X X
u uu u u uX X X
ζζ ζ
ζζ ζ
ζζ ζ
∂∂ ∂⎡ ⎤ ⎡ ⎤∂∂ ∂⎢ ⎥ ⎢ ⎥∂ζ ∂ζ ∂ζ ∂ ∂ ∂⎢ ⎥ ⎢ ⎥⎢ ⎥∂∂ ∂ ⎢ ⎥∂∂ ∂⎢ ⎥ = ⎢∂ζ ∂ζ ∂ζ ∂ ∂ ∂⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂∂ ∂ ∂ ∂⎢ ⎥ ⎢ ⎥
∂ ∂ ∂⎢ ⎥∂ζ ∂ζ ∂ζ ⎣ ⎦⎣ ⎦
⎥θ θ , (3.21)
where
24 CHAPTER 3
3 31 2 1 2
1 1 1 1 1 1
131 2 1 2c
2 2 2 2 2 2
3 31 2 1 2
3 3 3 3 3 3
u uu u u uX X X
uu u u uX X X
u uu u u uX X X
−
⎡ ⎤ ⎡∂ ∂∂ ∂ ∂ ∂⎢ ⎥ ⎢∂ ∂ ∂ ∂ξ ∂ξ ∂ξ⎢ ⎥ ⎢⎢ ⎥ ⎢∂∂ ∂ ∂ ∂
=⎢ ⎥ ⎢∂ ∂ ∂ ∂ξ ∂ξ ∂ξ⎢ ⎥ ⎢⎢ ⎥ ⎢∂ ∂∂ ∂ ∂ ∂⎢ ⎥ ⎢∂ ∂ ∂ ∂ξ ∂ξ ∂ξ⎣ ⎦ ⎣
J 3u
⎤⎥⎥⎥∂⎥⎥⎥⎥⎦
, (3.22)
in which is the coordinate Jacobian matrix. cJ
1 2
1 1 1
1 2c
2 2 2
1 2
3 3 3
X X
X X
X X
⎡ ⎤∂ ∂ ∂⎢ ⎥∂ξ ∂ξ ∂ξ⎢ ⎥⎢ ⎥∂ ∂ ∂
= ⎢ ∂ξ ∂ξ ∂ξ⎢ ⎥⎢ ⎥∂ ∂ ∂⎢ ⎥∂ξ ∂ξ ∂ξ⎣ ⎦
J
3
3
3
X
X
X
⎥ . (3.23)
On the basis of equation (3.10) the individual terms of are computed as: cJ
1 2
8 8J k k k
kJ 3 k3,Jk 1 k 1,
8J k
k k3,J3 k 1
X N t NA ,2
J 1,...3
X t N .2
= =ξ=ξ ξ
=
∂⎛ ⎞ ∂ ∂= + ξ χ⎜ ⎟∂ξ ∂ξ ∂ξ⎝ ⎠
=
⎛ ⎞∂= χ⎜ ⎟∂ξ⎝ ⎠
∑ ∑
∑
(3.24)
cJ is evaluated at every integration point of the element. Once 1c−J is known
1kc
u − ⎛ ⎞∂ ∂⎛ ⎞ =⎜ ⎟ ⎜∂⎝ ⎠ ⎝ ⎠J
X ξku⎟∂
. (3.25)
Similarly
1 2
9 9k1J k k k
kJ 3 k2,J k1,Jk2k 1 k 1,
9k1J k
k k2,J k1,Jk23 k 1
u N t Nd ,2
J 1,...3
u t N .2
= =ξ=ξ ξ
=
ω⎡ ⎤∂⎛ ⎞ ∂ ∂ ⎡ ⎤= + ξ −χ χ⎜ ⎟ ⎢ ⎥⎣ ⎦ ω∂ξ ∂ξ ∂ξ⎝ ⎠ ⎣ ⎦
=
ω⎛ ⎞ ⎡ ⎤∂ ⎡ ⎤= −χ χ⎜ ⎟ ⎢ ⎥⎣ ⎦ ω∂ξ ⎣ ⎦⎝ ⎠
∑ ∑
∑
(3.26)
By means of the above, the strain components can be expressed in terms of the
displacement vector d of the shell element as:
SHELL ELEMENT 25
[
1
2
1 2
2 3
1 3
1 2 9...
ζ
ζ
ζ ζ
ζ ζ
ζ ζ
ε⎡ ⎤⎢ ⎥ε⎢ ⎥
⎢ ⎥γ =⎢ ⎥⎢ ⎥γ⎢ ⎥⎢ ⎥γ⎣ ⎦
B B B d] . (3.27)
The individual elements of are given by equation d (3.15),
1
2
9
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
dd
d
dM
, (3.28)
while sub-matrices iB are [5x5] matrices which terms are computed on the basis of
equations (3.21) to (3.26).
3.6 Constitutive relation
The five stress components in the local system ( )i ,1 1...3=ζ are
1
2
1 2
2 3
1 3
ζ
ζ
ζ ζζ ζ
ζ ζ
ζ ζ
⎡ ⎤σ⎢ ⎥⎢ ⎥σ⎢ ⎥⎢ ⎥σ = = ετ⎢ ⎥⎢ ⎥τ⎢ ⎥⎢ ⎥τ⎣ ⎦
D . (3.29)
The elasticity matrix for the case of isotropic plane stress is determined from
equation (2.15) and can be expressed as:
D
( )
( )
2
1 0 0 01 0 0 0
10 0 0 0E 2c 11
0 0 0 02
c 10 0 0 0
2
ν⎡ ⎤⎢ ⎥ν⎢ ⎥⎢ ⎥− ν⎢ ⎥⎢ ⎥=
− ν− ν ⎢ ⎥⎢ ⎥⎢ ⎥
− ν⎢ ⎥⎢ ⎥⎣ ⎦
D , (3.30)
26 CHAPTER 3
where is a correction factor for the transverse shear strains. For a homogeneous
shell material
c
c 5 6= .
3.7 Numerical examples
In order to test the robustness, accuracy and efficiency of the shell element, a number
of numerical tests are presented for a set of representative shell problems. The
numerical results of the Heterosis element are presented in comparison with the nine-
noded Lagrangian and the eight-noded Serendipity element to demonstrate the
influence of the Heterosis ninth node on the behaviour of the element.
It is well known that displacement based Mindlin-Reissner plate/shell elements often
exhibit shear locking when elements become thin. In the following paragraphs the
results are shown for a pinched cylinder and the Scordelis-Lo roof with respect to
existing analytical solutions. These well-known benchmark test examples are prone to
induce locking.
During the analysis a number of integration schemes were compared. Both uniform
integration (U) and selective integration (S) are considered. The number of Gaussian
points is 2 and 3, respectively (U2, U3, S2). The use of a uniform 2-by-2 Gauss-
Legendre integration with respect to the 1ξ and 2ξ axes results in an element that is
less stiff than the element with only 3-by-3 integration for both shear and bending.
The analysis with the Heterosis element is also performed with selective reduced
integration whereby the virtual work associated with the shearing stress components
is under-integrated to avoid locking. The Heterosis element with selective integration
has no problems with zero-energy-modes and shear locking, Hughes and Cohen
[1978]. Five Gauss points are used through the thickness of the element.
SHELL ELEMENT 27
3.7.1 Pinched cylinder
A cylinder supported by rigid diaphragms at the end edges is loaded with two
opposite concentrated loads, P. The geometrical and material properties of the
cylinder are depicted in Figure 3.9. Due to its symmetry, only one eighth of the
cylinder is discretized.
E=3x106 N/mm2
ν = 0.3
P = 1.0 N
r = 300.0 mm
t = 3.0 mm
L/2 = 300.0 mm
P
P L
r
Figure 3.9 Schematic of pinched cylinder
A part of the deformed cylinder, compared with the undeformed structure (dotted
line), is shown in Figure 3.10.
P4
Rigid diaphragm
(ux = uz = 0)
Symmetry
Sym
m.
Sym
met
ry
Undeformed structure
Figure 3.10 Pinched cylinder; displacement under load
28 CHAPTER 3
The small slenderness ratio (t/R = 1/100) of the cylindrical shell is chosen to
demonstrate the capability of the Heterosis (9H) element to overcome shear and
membrane locking phenomena. In Table 3.1 the displacement under the applied load,
normalized with respect to the analytical solution computed by Lindberg et al. [1969]
(wref = 1.8245x10-5) is compared to solutions obtained with nine-node Lagrangian
(9L) and eight-node Serendipity (8S) elements.
Table 3.1 Pinched cylinder; comparison of the displacement (the displacement is normalized with respect to the analytical solution)
Mesh a 9L-U3 9L-U2 8S-U3 8S-U2 9H-U3 9H-U2 9H-S2 4 x 4 0.16 1.01 0.15 0.92 0.15 0.97 0.84
8 x 8 0.57 1.04 0.55 1.01 0.55 1.02 0.86
12 x 12 0.83 1.05 0.81 1.02 0.81 1.03 0.96
16 x 16 0.93 1.05 0.92 1.03 0.92 1.03 1.01 a Octant cylinder
For a considerable range of finite elements, this example is associated with poor mesh
convergence. For the element types used here, the difference between the elements is
small. The influence of the used integration scheme, however, is large. All elements
with uniform 3-by-3 integration (U3) are too stiff compared to elements with uniform
2-by-2 integration (U2). When elements with “selective reduced integration” (S2) are
used, more elements are required, to get close to the analytical solution.
SHELL ELEMENT 29
3.7.2 Scordelis-Lo roof
The Scordelis-Lo roof has also achieved the status of a de facto standard test,
appearing numerous times in the literature. A cylindrical roof, supported by rigid
diaphragms at the curved edges, is loaded by its own weight , as illustrated in Figure
3.11. In Table 3.2 the computed displacement at the middle of one of the free edges,
point A, normalized with respect to the reference solution computed by MacNeal and
Harder [1985] (w
p
ref = 0.3024), is also compared to solutions obtained with nine-node
Lagrangian (9L) and eight-node Serendipity (8S) elements.
E = 4.32x108 N/mm2
ν = 0.0
p = 90.0 N/mm2
r = 250.0 mm
t = 0.25 mm
L = 50.0 mm
φ = 40°
r
A
L
φ
Figure 3.11 Schematic of Scordelis-Lo roof
A plot of the deformed structure is shown in Figure 3.12.
Rigid diaphragm
Rigid diaphragm
A
Figure 3.12 Deformed Scordelis-Lo roof
30 CHAPTER 3
Due to its symmetry, only a quarter of the model is studied. For meshes with 64 or
more elements, the results obtained with the Lagrangian and the Serendipity elements
are very close to the results obtained with the Heterosis elements. For a mesh with 4 x
4 elements the Heterosis en Lagrangian elements with uniform 2-by-2 integration
(U2) perform best.
Table 3.2 Scordelis-Lo roof; comparison of the displacement (the displacement is normalized with respect to the analytical solution)
Mesh a 9L-U3 9L-U2 8S-U3 8S-U2 9H-U3 9H-U2 9H-S2 4 x 4 0.831 1.031 0.558 0.737 0.830 1.031 0.842
8 x 8 1.008 1.027 1.008 1.027 1.008 1.027 1.012
12 x 12 1.021 1.027 1.021 1.027 1.021 1.027 1.023
16 x 16 1.023 1.027 1.023 1.027 1.023 1.027 1.026 a Quarter surface
3.7.2 Evaluation of numerical examples
In general it is known that the Heterosis element performs better than the Lagrangian
and the Serendipity element. In the examples shown here, the Serendipity element
performs less, and the performance of the nine-node Lagrangian and nine-node
Heterosis element is very close. In this study the Heterosis element with both selective
reduced integration (S2) and uniform 2-by-2 integration (U2) are chosen in the
following chapters.
Chapter 4
TUBE FINITE ELEMENT
4.1 Introduction
In principle, finite element shell models can be employed to obtain very accurate
solutions for the nonlinear analysis of piping structures. To reduce the cost of
analysis, various different formulations of efficient tube bend elements have been
developed. Von Karman [1911] analyzed “elbows” subjected to a constant in-plane
bending moment and showed that the cross-section deforms to an oval. In the
analysis, the longitudinal and circumferential strains due to ovalisation of the cross
section are superimposed on curved beam theory displacements. Vigness [1943] later
showed that out-of-plane flexibility factors were identical to the in-plane values. Clark
and Reissner [1951] proposed equations for the bending of a toroidal shell segment
and, derived from an asymptotic solution, introduced the flexibility and stress factors.
Among others, Rodabough and George [1957] extended the work by Von Karman and
used the potential energy approach to investigate the effects of internal pressure for
the case of in-plane bending under a closing moment. They formulated the pressure
reduction effect on the flexibility and stress intensification factors. With zero pressure
their results reduce to von Karman’s.
Bathe and Almeida [1980, 1982] proposed an efficient formulation for a tube bend
element with axial, torsional, and bending displacements and the Von Karman
ovalization deformations. The main characteristic of the tube element is the
combination of longitudinal (beam-type) with cross-sectional deformation
(ovalization). Based on this concept, Karamanos and Tassoulas [1996] developed a
32 CHAPTER 4
nonlinear three-node tube element, capable of describing accurately in-plane and out-
of-plane deformation. This element has been used successfully for predicting the
ultimate capacity of inelastic tubes under the combined action of thrust, moment and
pressure. The isoparametric beam finite element concept is used to describe
longitudinal deformation, with three nodes defined along the tube axis, as shown in
Figure 4.1. Geometry and displacements are interpolated using quadratic polynomials.
node 1
tube axis
node 3
node 2
X2
X3
X1
Figure 4.1 Tube elbow element
4.2 Geometry
The location of a point before deformation is determined by the position vector ,
defined in a Cartesian global axes system
X
{ }iX , i 1, 2,3= , as shown in Figure 4.1. The
tube element is assumed to be symmetric with respect to the plane.
Regarding a beam rotation about the axis, each node possesses three degrees of
freedom (two translational and one rotational), which define its position and
orientation. In addition to this two additional coordinate systems are utilized in the
formulation of the tube element.
1X X− 3
2X
4.2.1 Curvilinear coordinate system
At each integration point a local system is introduced through the use of coordinates
in the hoop, longitudinal and along the thickness direction (denoted as , and
respectively), as presented in Figure 4.2. Due to symmetry, only half of the tube is
analyzed
iξ 1ξ 2ξ
3ξ
( )12 2−π ≤ ξ ≤ π 2. The ξ axis spans between ( )0, 1+ , where the axis
spans between ( ) .
3ξ
1, 1− +
TUBE ELEMENT 33
4.2.2 Nodal coordinate system
At each element node a local Cartesian axes system k { }ki ;i 1, 2,3χ = is defined, as
shown in Figure 4.2. This system is used as a reference frame for the cross-sectional
deformation parameters.
node 1 node 3
node 2ϕ
t
k3χ
k2χ k1χ1ξ
3ξ2ξ
2X
3X 1X
R
Figure 4.2 Coordinate systems tube finite element
4.3 Element geometry interpolation
The geometry and the displacement field of the tube element are interpolated from
Fourier terms along the circumferential direction (ovalization) and shape functions
along the longitudinal direction (beam-type).
4.3.1 Initial element geometry
The element thickness is assumed to be constant and a reference line is chosen
within the cross-section. The initial location of any point within the element can
therefore be interpolated on the basis of the node coordinates, the reference line and
the thickness via:
t
( ) ( )3 3 3
k k 2 k 1 k 2 3 k 1 k 2k 1 k 1 k 1
tN ( ) N ( ) N ( )2= = =
= ξ + ξ ξ + ξ ξ∑ ∑ ∑X A r n ξ , (4.1)
where represents the corresponding Lagrangian quadratic interpolation
functions:
k 2N ( )ξ
34 CHAPTER 4
( )(
21 2
22 2
23 2
1N21N2
N 1 .
= ξ −ξ
= ξ + ξ
= −ξ
)2
2
]
(4.2)
The position vector of node in the global axes system is defined as: k
[ Tk k1 k2 k3A A A=A . (4.3)
The position vector of the reference line with respect to the cross-section
corresponding to node k can be expressed as:
k 1 k,1 k,1 k,2 k,2 k,3 k,3( ) x x xξ = χ + χ + χr , (4.4)
where, in the original configuration,
k,1 1 1
k,2 1 1
k,3 1
x ( ) r cos
x ( ) r sin
x ( ) 0,
ξ = ξ
ξ = ξ
ξ =
(4.5)
with the radius of the undeformed reference line. r
The “in-plane” outward normal of the reference line, as shown in Figure 4.3, is
represented by:
( ) ( )k 1 k,1 1 k,1 k,2 1 k,2( ) n nξ = ξ χ + ξ χn , (4.6)
where
( ) k,2k,1 1
1
dx1nr d
⎛ξ = −⎜ ξ⎝ ⎠
⎞⎟ (4.7)
( ) k,1k,2 1
1
dx1nr d
⎛ξ = ⎜ ξ⎝ ⎠
⎞⎟ . (4.8)
TUBE ELEMENT 35
thickness
( )1ξn
r1ξ
k,1χ
k,2χ
undeformed reference line
k
Figure 4.3 Cross-section original configuration.
4.3.2 Updated element geometry
For the purposes of the present study, bending is applied about the axis (i.e.
is the plane of bending). The deformed tube axis is defined by:
2X
1X X− 2
( )3
c 2 k k 2k 1
N ( )=
ξ = ξ∑x x , (4.9)
where is the position vector of node . To describe cross-sectional deformation,
element thickness is assumed to be constant and a reference line is chosen within the
cross-section. Both in-plane (ovalization) and out-of-plane (warping) cross-sectional
deformations are considered. For in-plane deformation of the tube element, fibers
initially normal to the reference line are assumed to remain normal to the reference
line.
kx k
Following the formulation by Brush and Almroth [1975], the position vector of the
reference line at the current configuration can be expressed in terms of the radial and
tangential displacements. The updated components of ( )k 1ξr at the deformed cross-
section, as depicted in Figure 4.4, are
[ ][ ]
k,1 1 1 1 1 1
k,2 1 1 1 1 1
k,3 1 1
x ( ) r w( ) cos v( )sin
x ( ) r w( ) sin v( )cos
x ( ) ( ).
ξ = + ξ ξ − ξ ξ
ξ = + ξ ξ + ξ ξ
ξ = ψ ξ
(4.10)
36 CHAPTER 4
In the above expressions , ( )1w ξ ( )1v ξ and ( )1ψ ξ are displacements of the reference
line in the radial, tangential and out-of-plane (axial) direction, respectively.
thickness
( )1w ξ
( )1v ξ
( )1ξn
nu
r
( )1ξr
k,1χ
k,2χ deformed reference line
undeformed reference line
k
Figure 4.4 Cross-sectional deformation
The material fibers normal to the reference line may rotate in the out-of-plane direction by angle , as illustrated in Figure 4.5. ( )1γ ξ
k,3χ
k,1χ
k,2χ
( )1u ξ
non-warped reference line
warped reference line
( )1ξn
( )1γ ξ
k
( )1ξm
Figure 4.5 Out-of-plane displacement and rotation of the cross section
TUBE ELEMENT 37
The displacement due to the rotation of any point on the local thickness vector at
distance can be approximated as: 3ξ
( ) ( )3
3 1 kk 1
t N2=
⎡ ⎤δ = ξ γ ξ ξ⎢ ⎥⎣ ⎦∑ 2 (4.11)
Displacement is directed along the axis δ ( )1ξm . In case of small displacements the
vector can be taken equal to ( )1ξm k,3χ . The vector components in the global system
are
( ) ( )3
3 1 k,3 kk 1
td2=
⎡ ⎤= ξ γ ξ χ ξ⎢ ⎥⎣ ⎦∑ 2N . (4.12)
The deformation functions ( )1w ξ , ( )1v ξ , ( )1ψ ξ and ( )1γ ξ are discretized as
follows:
1 0 1 1 n 1 nn 2,4,6,... n 3,5,7,....
w( ) a a sin a cos n a sin n= =
ξ = + ξ + ξ + ξ1∑ ∑ (4.13)
1 1 1 n 1 nn 2,4,6,... n 3,5,7,....
v( ) a cos b sin n b cos n= =
ξ = − ξ + ξ + ξ∑ ∑ 1
1
(4.14)
1 n 1 nn 2,4,6,... n 3,5,7,....
( ) c cos n c sin n= =
ψ ξ = ξ + ξ∑ ∑ (4.15)
1 0 1 1 n 1 nn 2,4,6,... n 3,5,7,....
( ) sin cos n sin n= =
γ ξ = γ + γ ξ + γ ξ + γ ξ∑ ∑ 1 (4.16)
Coefficients na , nb refer to in-plane cross-sectional deformation (“ovalization”
parameters) and refer to out-of-plane cross-sectional deformation (“warping”
parameters). With the geometry and displacement functions given in equations
nc , nγ
(4.1),
(4.4), (4.10) and (4.12), the position vector of an arbitrary point at the deformed
configuration is
( ) ( ) ( ) ( )3
k k 1 3 k 1 3 1 k,3 k 2k 1
t t N2 2=
⎡= + ξ + ξ ξ + ξ γ ξ χ ξ⎢⎣ ⎦∑x x r n ⎤
⎥ , (4.17)
where the first two terms within the brackets denote the deformed reference line and
the latter two the deformations “through the thickness”.
38 CHAPTER 4
Displacements interpolation
As shown in equation 2.4, the displacement components of a material point in the tube
can be determined by subtracting the coordinates of the point before deformation from
the coordinates of that point after deformation:
= −u x X .
The difference between the configuration in the deformed position and the original
configuration can be determined by differentiation of equation (4.17): 3
k k,1 3 k,1 k,1k 1
k,2 3 k,2 k,2 k,2 3 k,2 k,2
k,3 3 k,3 k,3 3 k,3 k 2
td d dx dn2
t tdx dn x n d2 2t tdx d x d N ( ).2 2
=
⎡ ⎛ ⎞= + + ξ χ +⎜ ⎟⎢ ⎝ ⎠⎣⎛ ⎞ ⎛ ⎞+ ξ χ + + ξ χ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎤⎛ ⎞ ⎛ ⎞
+
+ ξ γ χ + + ξ γ χ ξ⎜ ⎟ ⎜ ⎟ ⎥⎝ ⎠ ⎝ ⎠ ⎦
∑x x
(4.18)
The displacement of a material point within the tube element can be obtained by
rewriting of equation (4.18): 3
k k,1 3 k,1 k,1k 1
k,2 3 k,2 k,2 k,2 3 k,2 k,2
k,3 3 k,3 k,3 3 k,3 k 2
tx n2
t tx n x n2 2t tx x2 2
=
⎡ ⎛ ⎞= + Δ + ξ Δ χ +⎜ ⎟⎢ ⎝ ⎠⎣⎛ ⎞ ⎛ ⎞Δ + ξ Δ χ + + ξ Δχ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎤⎛ ⎞ ⎛ ⎞Δ + ξ Δγ χ + + ξ γ Δχ ξ⎜ ⎟ ⎜ ⎟ ⎥⎝ ⎠ ⎝ ⎠ ⎦
∑u d
N ( ),
ξ
(4.19)
with
( )
k,1 1 1 1 1 1
k,2 1 1 1 1 1
k,3 1 1
x ( ) w( )cos v( )sin
x ( ) w( )sin v( ) cos
x ( ) .
Δ ξ = ξ ξ − ξ ξ
Δ ξ = ξ ξ + ξ
Δ ξ = ψ ξ
(4.20)
Note that , , , , k,1xΔ k,2xΔ k,3xΔ k,1nΔ k,2nΔ and kΔγ are linear functions of na , nb ,
and . In Figure 4.6 the position and orientation of every node are shown, which
are defined through:
nc nγ
k k,1 1 k,3d X d X= +d 3 (4.21)
and
TUBE ELEMENT 39
k,2 k,2 k,3
k,3 k,2 k,2.
Δχ = ω χ
Δχ = −ω χ (4.22)
2X
3X
1X1
2
3
4
3,1d 3,3d
3,2ω
Figure 4.6 Nodal Displacements and Rotations
Depending on the number of ovalization and/or warping parameters used, a typical
nodal point in the tube element can have from 3 to n degrees of freedom. At each
node k the displacement vector is specified as: kU
k,1
k,3
k,2
k,1
k,n
k,2
k
k,n
k,2
k,n
k,0
k,n
dd
a:
ab:
bc:
c
:
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ω⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥γ⎢ ⎥⎢ ⎥⎢ ⎥γ⎢⎣
U
⎥⎦
]
.
By means of the above, can be written as: U
[ T1 2 3=U U U U .
40 CHAPTER 4
4.4 Strain measures
The stress and strain tensors are described in terms of their components with respect
to a curvilinear coordinate system along 1ξ , 2ξ and 3ξ . The partial derivatives of the
position vector allows for the definition of the convective coordinate system, defined
by the covariant basis vector in the form:
ii
∂=∂ξ
Xg ,
Because of small strains, this system is set up with respect to the reference
configuration. The covariant base vectors g1, g2, g3 are obtained by appropriate
differentiation of equation (4.1) with respect to the coordinates 1ξ , 2ξ and : 3ξ
( )
( ) ( )( ) ( )
( ) ( )
3k,1 x,1 k,2 x,2
1 3 k,1 3 k,2 k 21 1 1 1 1k 1
3k 2
2 b k,1 3 x,1 k,1 k,2 3 x,2 k,22 2k 1
3
3 x,1 k,1 x,2 k,2 k 23 k 1
x n x nN
Nx n x n
n n N .
=
=
=
⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞∂= = + ξ χ + + ξ χ ξ⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ξ ∂ξ ∂ξ ∂ξ ∂ξ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦
∂ ξ⎡ ⎤∂= = + + ξ χ + + ξ χ⎢ ⎥∂ξ ∂ξ⎣ ⎦∂ ⎡ ⎤= = χ + χ ξ⎣ ⎦∂ξ
∑
∑
∑
Xg
Xg x
Xg
Note that 1g and 2g define the shell laminas and 3g runs through the thickness. With
the base vectors the contravariant (reciprocal) base vectors can be defined from the
following relation: b b
a a⋅ = δg g ,
where baδ is the well-known Kronecker delta.
The strain tensor is written as:
( k lkl= ε ⊗ )g gε , (4.23)
where
( )kl k l l k1 u u2
ε = + and ( )k / m k
mu
∂= ⋅∂ξ
ug ,
with the covariant derivation of the incremental displacement components. k / mu
TUBE ELEMENT 41
4.5 Constitutive relation
The stress tensor
(iji= σ ⊗σ )jg g (4.24)
can be computed from
= ⋅Dσ ε ,
where, according to Green and Zerna [1968], equation (2.15) is written as:
(ijkl ij kl ik jl il jk2D K G g g G g g g g3
⎛ ⎞= − + +⎜ ⎟⎝ ⎠
) . (4.25)
Furthermore, shell theory requires that ( )⋅ ⊗σ m m is zero throughout the deformation
history, where is the unit normal vector to any lamina. It is readily shown that
is equal to
m m3 3g g . The stresses in longitudinal and circumferential direction
represent the physical components of the stress vector in the direction of the unit
vector:
1111longitudinal 11
2222circumferential 22
gg
gg
σ = σ
σ = σ
4.6 Numerical examples
The numerical results obtained with the tube elements are compared with results
obtained with the selective integrated Heterosis elements (9H-S2). For the purposes of
the present study, bending is applied about axis X2 (i.e. X1-X3 is the plane of
bending). An 5th degree expansion ( n 5≤ in equations (4.13), (4.14), (4.15) and
(4.16)) for , , and ( )1w ξ ( )1v ξ ( )1ψ ξ ( )1γ ξ is found to be adequate [Karamanos and
Tassoulas, 1993] for all cases.
Regarding the number of integration points in the circumferential direction, 19
equally spaced integration points around the half-circumference are used including the
two points on the symmetry plane. Five Gauss points are used in the radial (through
the thickness) direction. With two Gauss points the tube element is underintegrated
42 CHAPTER 4
with respect to the longitudinal coordinate 2ξ . This results in an element that is less
stiff. The presented element is implemented in a new Finite Element Program.
4.6.1 Analysis of a straight pipeline subjected to a nodal load
The straight cantilever pipe in Figure 4.7 was analyzed to demonstrate the
effectiveness in the analysis of thin structural members using one tube element. All
degrees of freedom in point A are restrained.
A
L = 4000 mm
r = 198.45 mm
t = 9.5 mm
P = 4800 N ν = 0.3 E = 2.1×105 N/mm2
L
P
2r
t
Figure 4.7 Schematic of a straight pipe
The displacement under the load P, which is applied on the end-node of the tube
element, is 2.125 mm. This is identical to the displacement calculated with the shell
elements, in which case the load is distributed over the nodes at the edge of the
elements. The calculation time when one tube element is used is only 0.14 seconds.
This is five times less expensive than a calculation with 12 shell elements (a 4x3 mesh
to model half the pipe). The result is compared with the formula for a beam with a
thin circular cross-section with both flexural and shear deformation:
( )3 2
ref2
PL rw 1 6 1 2.131 mm3EI L
⎛ ⎞= + + ν =⎜ ⎟⎜ ⎟
⎝ ⎠.
The difference between the numerical and the analytical solution is 0.3 %. For a
straight tube element the solution is very accurate.
TUBE ELEMENT 43
4.6.2 Analysis of pure bending of a straight pipeline
A straight pipe, as shown in Figure 4.8, is subjected to a constant moment M.
L = 8000 mm
r = 198.45 mm
t = 9.5 mm
M = 10000 Nmm ν = 0.3 E = 2.1×105 N/mm2
M
L
M
2r
t
Figure 4.8 Schematic of a straight pipe
The longitudinal stress at the midsurface along the circumference, obtained with one
tube element, is compared with results obtained with shell elements with selective
reduced integration (S2), as presented in Figure 4.9. Due to symmetry only a quarter
of the pipe is modeled. In this figure the results of a 10x6 mesh are shown.
-0.009
-0.006
-0.003
0
0.003
0.006
0.009
0 30 60 90 120 150 180
shell elementstube element
stre
ss
radiusangle
0.0085 N/mm2
Figure 4.9 Longitudinal stress at midsurface
The stresses calculated with the tube element are identical to the stresses calculated
with the shell elements. For comparison, the well known design formula is used:
2longitudinal,M 2
Mr M 0.0085 N mmI r t
σ = = =π
.
44 CHAPTER 4
4.6.3 Analysis of a straight pipeline subjected to internal pressure
A straight pipe, as shown in Figure 4.10, is subjected to an internal pressure.
L = 8000 mm
r = 198.45 mm
t = 9.5 mm
pint = 1.0 N/mm2 ν = 0.3 E = 2.1×105 N/mm2
pint
2r
t
L
Figure 4.10 Schematic of a straight pipe with butt plates
The structure is analyzed using one tube element and 12 shell elements (a 4x3 mesh to
model half the pipe). The tube is capped at both ends. The computed stress at the mid-
surface in the circumferential direction caused by internal pressure pint is 20.37
N/mm2. The stress in the longitudinal direction is 9.95 N/mm2. In long straight
pipelines the longitudinal strains are assumed to be zero because of the frictional
restraint of the pipe by the surrounding soil [Gresnigt, 1986]. The stresses at the mid-
surface in case of a pressure vessel [Flügge, 1993] are 2
int2
longitudinal,p
int2
circumferential,p
1p r t2 9.95 N mm
2rt1p r t2 20.38 N mm .
t
⎛ ⎞−⎜ ⎟⎝ ⎠σ = =
⎛ ⎞−⎜ ⎟⎝ ⎠σ = =
The match with the FEM result is perfect.
TUBE ELEMENT 45
4.6.4 Analysis of pure bending of a curved pipeline
Pipeline bends are a problem of great interest to many designers. As mentioned in the
introduction, they have a complex response to in-plane and out-of-plane bending
moments. When an external moment is applied to one of its ends, the cross section
tends to deform significantly both in and out of its plane. The pipe structure shown in
Figure 4.11 was analyzed using tube and shell elements. The pipeline bend is
subjected to a “closing” moment M. The radius of the pipe r is 198.45 mm. The radius
of the bend R is 609.4 mm. The structure is fixed at node A, so that the end node
cannot translate or rotate, whereas the cross-section is free to ovalize, but not to warp.
The other end is free to translate or rotate; it may ovalize but cannot warp. For the
curved part of the pipe structure 5 tube elements were used and 300 shell elements
(20x15 mesh). For the analysis in the elastic domain more elements are not needed.
L1 = 609.6 mm
L2 = 152.4 mm
RB = 609.4 mm
r = 198.45 mm
t = 9.5 mm
ν = 0.3 E = 1.66×105 N/mm2
RB
M
L1
L2
A 2r
t
Figure 4.11 Schematic of pipe structure
The results are compared with the results presented by Sobel [1977], Bathe and
Almeida [1980] and the Clark and Reissner shell theory [1951]. Only half the
circumference is analyzed due to symmetry. In the results, as shown in Figures 4.12 to
46 CHAPTER 4
4.14, the stresses calculated at the integration points are presented using the stress-
intensification factor , as proposed by Rodabaugh and George [1957]: sfi
sflongitudinal,M
i σ=σ
,
In Figure 4.12 the circumferential stress at the inside of the pipe wall is shown with
respect to the hoop direction of the cross section, where 0 degrees denotes the outside
and 180 degrees the inside of the pipe bend. In his work Sobel used the Marc
computer program to analyze the bend. Bathe and Almeida used the program
ADINAP.
-8
-6
-4
-2
0
2
4
0 15 30 45 60 75 90 105 120 135 150 165 180angle
TUBE element
MARC, n=64Clark & Reissner
ADINAPHeterosis element (U2)
(intrados)
(extrados)
sf
i
Figure 4.12 Circumferential stress at inside of the pipe wall
The results obtained with the shell element as well as the tube element are very close
to the results from theory. In Figure 4.13 the circumferential stress at the outside of
the pipe wall is shown. In this example the importance of the warping terms in the
formulation of tube element is shown.
In Figure 4.14 the longitudinal stress at the midsurface of the pipe wall is shown.
Again the results are very close, except for the heterosis element which shows a
TUBE ELEMENT 47
compressive stress at the inside of the pipe bend. The distribution of the longitudinal
stresses at the midsurface is also shown in Figure 4.15
-4
-3
-2
-1
Figure 4.13 Circumferential stress at outside of the pipe wall
Figure 4.14 Longitudinal stress at midsurface
0
1
2
3
4
5
6
0 15 30 45 60 75 90 105 120 135 150 165 180angle
7 TUBE element
TUBE without warpingsfi
MARC, n=64
Heterosis element (U2)
(intrados) (extrados)
-4
-3
-2
-1
0
1
2
3
4
0 15 30 45 60 75 90 105 120 135 150 165 180
angle
TUBE element MARC, n=32sfi
Clark & ReissnerADINAPHeterosis element (U2)
(extrados) (intrados)
48 CHAPTER 4
Tens
ile st
ress
C
ompr
essi
ve
rigid diaphragm
Figure 4.15 Longitudinal stress at midsurface
Chapter 5
GURSON CONSTITUTIVE MODEL
5.1 Introduction
Development and implementation of advanced material models is needed to improve
the predictability of material failure in Finite Element simulations. Damage is the
deterioration of materials which occurs prior to failure. In structural steels, after the
onset of plastification, progressive material damage can initiate in the form of micro-
void nucleation. These voids are first nucleated at second phase particles under the
application of external loads (Brown and Embury [1973]). Due to large plastic
deformations, the micro-voids in the material gradually grow. As shown in Figure 5.1
the deformation of the material is accompanied by the nucleation of voids that were
originally not present in the material and which have a diameter of a few microns.
(a) Undeformed specimen (b) Deformed specimen
Figure 5.1 SEM images of void formation in DP 600 steel (from: S. Celotto [2008])
50 CHAPTER 5
The ductile growth and final coalescence of micro-voids will lead to response
degradation and eventually fracture, as shown in Figure 5.2.
a. no damage b. nucleation c. expansion d. coalescence
Figure 5.2 Schematic representation of spherical void development
Constitutive models are developed to capture this evolution of damage. The existing
models were mainly developed to predict growth of cavities in a ductile matrix. Berg
[1962] has pioneered the analysis by studying the cylindrical-void growth law in a
linearly viscous material. Void nucleation and growth as the key micro-mechanism of
rupture for ductile metals was introduced by Mc Clintock [1968]. Rice and Tracey
[1969] later developed a model to evaluate the enlargement of a spherical void in an
infinite, rigid, perfectly plastic material when subjected to a remote uniform strain
field. They proposed a relation between the growth of the radius of the cavity and the
equivalent plastic strain. Since the Rice and Tracey model is based on a single void, it
does not take into account the interaction between voids, nor does it predict ultimate
failure.
Gurson formulated the plastic potential function from the analytical study of a single
isolated cavity in an elastic-perfectly-plastic material [1975]. This pressure dependent
plasticity material model contains the classical Von Mises model and is capable of
reproducing accurately various aspects of metallic material post-yield response.
According to the model, the real material consists of intact material, carrying the
stresses, and voids, which are supposed to always remain spherical. These internal
variables induce a progressive shrinkage of the yield surface until failure occurs due
to loss of stress carrying capability.
CONSTITUTIVE MODEL 51
Numerous alterations and improvements with respect to the yield function and
damage evolution, have been suggested by various authors, most notably Tvergaard
and Needleman (Tvergaard [1981, 1982], Chu and Needleman [1980], Tvergaard and
Needleman [1984]), such that it is often referred to as the Gurson-Tvergaard-
Needleman (GTN) model. Other noteworthy frameworks have been developed by
Shima and Oyane [1976]; Lemaitre [1985]; Rousselier [2001].
5.2 Gurson material model
The initiation and growth of voids within a metallic material can be elegantly
simulated by means of the Gurson material model. As compared to other models, it
has a simpler form and a fewer number of material constants. The yield function and
plastic potential in the Gurson model are expressed as:
( )2
* 212
3q pqf 2q f cosh q f2
⎡ −⎛ ⎞= + − −⎜ ⎟⎢ σσ ⎝ ⎠⎣ ⎦σ *2
3 1⎤⎥ , (5.1)
where is the effective deviatoric von Mises stress, and is the hydrostatic stress.
The model parameters q
q p
1, q2 and q3 affect the shape of the yield surface. A schematic
of equation (5.1) is shown in Figure 5.3.
2σ
3σ
1σ
Figure 5.3 Gurson yield surface in 3-D stress space.
52 CHAPTER 5
The surface is continuous and hence avoids discontinuity problems. The equivalent
tensile flow stress in the matrix material σ is a function of the equivalent plastic
strain and controls the isotropic hardening of the material response. The evolution of
damage is described by means of the non-directional parameter , which represents
the current void volume fraction. The change in void volume fraction during an
increment of deformation is partly due to the nucleation of new voids and partly due
to the growth of existing ones. As increases, the size of the yield surface decreases.
If is zero the term between brackets is also zero and equation
*f
*f*f (5.1) reduces to the
von Mises yield function.
Based upon a comparison between Gurson’s continuum model and a numerical
model, which fully accounts for the nonuniform stress field around each void and also
for the interaction between neighbouring voids, Tvergaard [1981, 1982] introduced
the material dependent parameters and . It is commonly assumed that
and . For sheet metals of various alloys is 1.5; Tvergaard [1982].
1q , 2q 3q
2q 1.= 0 12
3q q= 1q
The void growth rate is proportional to the differential change in the hydrostatic
plastic strain of the matrix material. This coincides with experiments (Dodd and
Atkins [1983]), in which no significant increase of void volume fraction due to shear
dominated stress/strain states was observed. Void nucleation occurs by debonding of
second phase particles. As proposed by Chu and Needleman [1980] the void
nucleation function is assumed to have a normal distribution and is strain related.
( )* * *
growth nucleation
* pijij
df df df
1 f d Ad
= +
= − ε δ + ε p (5.2)
where2p
N N
NN
f 1A exp2 ss 2
⎡ ⎤⎛ ⎞ε − ε⎢ ⎥= − ⎜ ⎟⎜ ⎟⎢ ⎥π ⎝ ⎠⎣ ⎦,
Pε the microscopic equivalent plastic strain, the initial volume fraction of void
nucleating particles, the mean strain for nucleation and the standard deviation.
Nf
Nε Ns
The influence of the void growth and development on the material response is shown
in Figure 5.4.
CONSTITUTIVE MODEL 53
0
100
200
300
400
0 0.1 0.2 0.3 0.4 0.5 0.6true strain
stre
ss (M
Pa)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
dam
age
f*
stress responsedamage development
Nε
*0f
Ns
Figure 5.4 Influence of void growth and development on material response.
For the analysis, a non-hardening material is used for which the following parameters,
according to Tvergaard, were adopted. The initial void volume , ,
, , ,
0.004f *0 = Nf 0.0= 4
= E 205000 MPa=N 0.3ε = Ns 0.1 0.3ν = and flow stress 400 MPaσ = .
Cyclic loading has a significant influence on the growth of the voids. Under tension
the void volume fraction increases due to growth of the existing and nucleation of new
voids, whilst under compression the existing voids will close, but new voids are
formed. This is due to fact that the equivalent plastic strain increment Pd is non-
negative. After reaching a critical void volume fraction, f , the void growth
accelerates due to interaction of voids.
ε
crit
( )
* *crit
** *u crit
crit crit critf crit
f , for f f ,f f ff f f , for f f .
f f
⎧ ≤⎪
= ⎨ −+ − >⎪ −⎩
When the final void volume fraction, , is reached, the material has lost its stress
bearing capacity and the damage variable, , then takes its ultimate value, f , which
equals
ff*f u
11 , Tvergaard and Needleman [1984]. q
54 CHAPTER 5
5.3 Material hardening
5.3.1 Isotropic hardening
The isotropic hardening rule postulates that the yield surface expands uniformly about
the origin of stress space, while the location of its center remains unchanged during
plastic flow, as shown in Figure 5.5. Isotropic hardening is defined as a function of
the equivalent plastic strain. p
y,0 isoHσ = σ + ⋅ ε (5.3)
where represents the initial flow stress in the matrix material. The size of the
flow surface is controlled by parameter , which, in the framework of this project,
is a constant.
y,0σ
isoH
1σ
2σ
Initial yield surface
Figure 5.5 Yield surface for plane-stress conditions and different yield strength.
5.3.2 Kinematic hardening
The Bauschinger effect [1881] is one of the most important phenomena of metals
under cyclic loading. This effect is characterized by a reduced yield stress upon load
reversal after plastic deformation has occurred during the initial loading. In the
mechanics literature the phenomenology of this effect are empirically described as
kinematic hardening. This hardening rule dictates the evolution of the yield surface
during a plastic loading increment by translation in stress space while the size remains
fixed, as illustrated in Figure 5.6.
The center of the yield surface , also known as the back stress, is updated via: fα
CONSTITUTIVE MODEL 55
f f dτ+Δτ τ= +α α αf . (5.4)
Prager [1956] assumed that the yield surface moves in the direction of the plastic
strain. When hardening parameter is constant the following kinematic hardening
rule is linear:
kinH
( ) pf kid H
τ+Δτ
τ+Δτ= ⋅ ⋅σ
ασ
%
%n dε , (5.5)
1σ
2σ
fτ+Δτα
fτα
Figure 5.6 Graphical representation of moving surface for plane-stress conditions
With respect to the translation of the yield function, equation (5.1) becomes:
( ) 32 2f f
2 3q pq * 2f 2q f cosh q f1 2
⎡ −⎛ ⎞= + − −⎢ ⎜ ⎟⎜ ⎟σ σ⎢ ⎥⎝ ⎠⎣ ⎦
σ%%
% *2 1⎤⎥ , (5.6)
where
f= −σ σ α% . (5.7)
The effective hydrostatic stress ( )p p= σ% % and the effective deviatoric stress
( ) ( ) 1 2f f
3q :2
= − −⎡ ⎤⎣ ⎦s a s a% ,
where indicates the deviatoric part of the back stress. fa
Using equation (5.7), the flow rule, as described in § 2.3, can be written as:
p f p f qp q
⎛ ⎞∂ ∂ ∂ ∂= λ +⎜ ∂ ∂ ∂ ∂⎝ ⎠
εσ σ%&&
% %% % ⎟%
(5.8)
The stress tensor can be written as:
2p q3
= − +Iσ % %% n% , (5.9)
where 32q
=n%%
s% . (5.10)
56 CHAPTER 5
5.4 Numerical implementation
5.4.1 Three-dimensional formulation
Aravas [1987] proposed a numerical algorithm, based on the Euler backward method,
for pressure-dependent plasticity models. First a trial state of stress is obtained,
assuming that the entire step is elastic: e τ= + ⋅ΔDσ σ ε (5.11)
Integration of equation (2.19) yields:
p f
1 f fI ,3 p q
τ+Δτ
τ+Δτ
τ+Δτ
τ+Δτ τ+Δτ
∂⎛ ⎞Δε = Δλ⎜ ⎟∂⎝ ⎠
⎛ ⎛ ⎞ ⎛ ⎞∂ ∂= Δλ − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
n
σ%
%% %
⎞ (5.12)
where
32q
τ+Δτ =n%%
s% (5.13)
The increment of plastic strain pτ+ΔτΔε can be expressed in terms of volumetric and
deviatoric components as:
pp q
13
τ+ΔτΔε = Δε + ΔεI %n , (5.14)
where
pfp τ+Δτ
⎛ ⎞∂Δε = −Δλ⎜ ⎟∂⎝ ⎠%
and qfq τ+Δτ
⎛ ⎞∂Δε = Δλ⎜ ⎟∂⎝ ⎠%
(5.15)
Elimination of gives: Δλ
p qf f 0q pτ+Δτ τ+Δτ
⎛ ⎞ ⎛ ⎞∂ ∂Δε + Δε =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠% %
(5.16)
If the yield criterion is violated, the final stress at τ + Δτ is computed through a plastic
stress correction, as shown in Figure 5.7: eτ+Δτ τ+Δτ= − ⋅ ΔεDσ σ p (5.17)
CONSTITUTIVE MODEL 57
Using equation (5.14), the term t t p+Δ⋅ ΔεD can be expressed in terms of the
hydrostatic and deviatoric plastic strain components and the elastic bulk and shear
moduli.
K
G
The updated stress state can be written as: e
p qK 2Gτ+Δτ τ+Δτ τ+Δτ τ+Δτ= − ⋅ Δε ⋅ − ⋅ Δε ⋅Iσ σ %% % n . (5.18)
Figure 5.7 Graphical representation of the backward Euler algorithm in stress space
Equation (5.9) can be written as:
2p q3
τ+Δτ τ+Δτ τ+Δτ τ+Δτ= − ⋅ + ⋅ ⋅I nσ % %%
( )f 0τ =σ%( )f 0τ+Δτ =σ%
τσ
τ+Δτσ
eσ
% , (5.19)
from which the stress correction along the hydrostatic and the deviatoric axes becomes
apparent: e
p
eq
p p K
q q 3G
τ+Δτ
τ+Δτ
= + ⋅Δε
= − ⋅Δε
% %
% % (5.20)
Equations (5.6) and (5.16), constitute a nonlinear algebraic system of pΔε and ,
which are chosen as the primary unknowns. Using
qΔε
p∂Δε and q∂Δε as the corrections,
the Newton-Raphson equations are
( )
1 1
p q p p q
2 2 q
p q
r rf fq p
r r f
∂ ∂⎡ ⎤∂ ∂⎡ ⎤⎢ ⎥∂Δε ∂Δε ∂Δε − Δε − Δε⎡ ⎤ ⎢ ⎥⎢ ⎥ ∂ ∂=⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂ ∂Δε⎢ ⎥⎣ ⎦ −⎢ ⎥⎢ ⎥ ⎣ ⎦∂Δε ∂Δε⎢ ⎥⎣ ⎦
σ% %
%
,
where r and are 1 2r
58 CHAPTER 5
( )
1 p
2
f frq p
r f
∂ ∂= Δε + Δε∂ ∂
= σ% %
%
q
The terms involved in the solution of the equations given in Appendix B. The
equations are solved for p∂Δε and q∂Δε by means of the Newton-Raphson iterative
procedure set up at local material level. The values of pΔε and qΔε are then updated:
p p p
q q
Δε → Δε + ∂Δε
Δε → Δε + ∂Δεq
During the iterative procedure, the stress is corrected along the hydrostatic and
deviatoric axes p and using equation % q% (5.20).
5.4.2 Plane stress formulation
For plane stress elements it is required that the stress perpendicular to the surface
, whereas the corresponding strain increment component 3
0ζσ =3ζ
Δε is considered
unknown. Application of the backward Euler method requires some modifications to
the method described in the previous section. To enforce the zero stress condition in
the 3ζ -direction, the strain increment is decomposed in two parts
3ζΔ = Δ + Δεε ε ψ (5.21)
where
( ) ( ) (11 1 1 22 2 2 12 1 2 2 1 13 1 3 3 1 23 2 3 3 2Δ = Δε + Δε + Δε + + Δε + + Δε +ε e e e e e e e e e e e e e e e e )
is the known part of the strain increment, and
3 3=ψ e e ,
with , being the unit vectors along the coordinate axes. i , i 1, 2,3=e
Therefore, equation (5.18) becomes
( )
( ) ( )
3
3
ep q
qp
K 2G
K 3Gq
τ+Δτζ
τζ
= + Δε − Δε − Δε
Δε= Δε + Δε − Δε −
D n
D D
σ σ ψ
σ + ψ
%% %
%%%
s (5.22)
where the left superscript τ + is omitted for the sake of simplicity in , and ,
and is the elasticity matrix, as described in equation 3.30.
Δτ p q s
D
It should be underlined that eσ% is not equal to the elastic predictor tensor . eσ%
CONSTITUTIVE MODEL 59
Using equation (5.19), the hydrostatic and deviatoric parts of the final stress state are
given by the following relationships:
3
epp p K K ζ= + Δε − Δε% % (5.23)
3
qe 32G2 qζΔε⎡ ⎤
= + Δε −⎢ ⎥⎣ ⎦
s s y s% % %%
(5.24)
where ep% and es% are the hydrostatic and deviatoric parts of eσ% , which corresponds to
Δε , and y is the deviatoric part of ψ .
Using equation (5.24) we find:
( )2
3 3 3
1 2e e 2 2
qq 3G q 6Gs 4Gζ ζ ζ= − Δε + + Δε + Δε% % , (5.25)
where
e e3q2
= ⋅s s% % e% . (5.26)
The condition of zero stress normal to the surface ( )30ζσ =% is equivalent to the
following condition
3
es pζ − =% % 0 (5.27)
and using equation (5.24), the following expression is obtained:
( ) 3 3
eq
4q 3G p s G q 03ζ ζ
⎛+ Δε − + Δε =⎜⎝ ⎠
% % % ⎞⎟ % . (5.28)
Equations (5.6), (5.16) and (5.28) constitute a nonlinear algebraic system of pΔε ,
and
qΔε
3ζΔε , which are chosen as the primary unknowns. The equations are solved by
means of the Newton-Raphson iterative method
3
p11 12 13 1
21 22 23 q 2
31 32 33 3
Y Y Y zY Y Y zY Y Y z
ζ
⎡ ⎤∂Δε⎡ ⎤ ⎢ ⎥⎢ ⎥ ∂Δε =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ∂Δε⎣ ⎦ ⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
,
where the terms and are given in Appendices B and C. During the iterative
procedure, the values of and are updated using equations
ijY iz
p% q% (5.23) and (5.25),
respectively.
60 CHAPTER 5
5.4.3 Plane stress formulation for a curvilinear coordinate system
The method presented in this paragraph is similar to the plane stress algorithm.
Because of the curvilinear coordinate system the covariant and contravariant vectors
are introduced in this application of the backward Euler method. For the tube
elements it is required that the stress perpendicular to the surface 33 0σ = , whereas the
corresponding strain increment component 33Δε is considered unknown. The strain
increment is decomposed in two parts
33 cΔ = Δ + Δεε ε ψ , (5.29)
where
( )k mkmΔ = Δε ⊗ε g g , (5.30)
and
(3 3 3k 3mc g g= ⊗ = ⊗ψ )k mg g g g . (5.31)
Equation (5.18) becomes
( )
( ) ( )
e33 c p q
q33 c p
K 2G
K 3Gq
τ+Δτ
τ
= + Δε − Δε − Δε
Δε= Δε + Δε − Δε −
D n
D D
σ σ ψ
σ + ψ
%% %
%%%
s
where also the left superscript is omitted for the sake of simplicity in p , q and
and is the elasticity matrix, as described in equation 4.25.
τ + Δτ
s D
The hydrostatic and deviatoric parts of the final stress state are now given by the
following relationships: e
p 33p p K K g= + Δε − Δε% % 33 (5.32)
qe33 c
32G2 qΔε⎡ ⎤
= + Δε −⎢ ⎥⎣ ⎦
s s y s% % %%
, (5.33)
where
pqe epq
1p3
= − σ g% % (5.34)
and
ij ij pqe e ij epq
13
= −s σ g σ g% % % (5.35)
CONSTITUTIVE MODEL 61
Equation (5.34) can also be written as:
]
e 11 12 1311 12 13
21 22 2321 22 23
31 3231 32
p p Kτ ⎡= − Δε + Δε + Δε +⎣
Δε + Δε + Δε +
Δε + Δε
g g g
g g g
g g
% %
(5.36)
Using equation (5.33) and the fact the contravariant components of cy are
km 3k 3m km 33c
1y g g g g3
= − , (5.37)
it is possible to obtain an expression for the final effective stress q : %
( )2 1 2e e33 2 2 33 33
q 33 33q 3G q 6Gs 4G g g= − Δε + + Δε + Δε% % , (5.38)
where
ij kme eij km
3q2
= e⋅g g s s% % % . (5.39)
The condition of zero stress normal to the surface ( )33 0σ =% is equivalent to the
following condition e33 33s pg− =% % 0 (5.40)
and using equation (5.33), the following expression is obtained
( ) 33 e33 33 33q 33
4q 3G pg s G g g q 03
⎛+ Δε − + Δε =⎜⎝ ⎠
% % % %⎞⎟ . (5.41)
As described in the previous paragraph, equations (5.6), (5.16) and (5.41) are solved
by means of the Newton-Raphson iterative method. During the iterative procedure,
the values of and are updated using equations p% q% (5.32) and (5.38) respectively. See
also Appendix D.
62 CHAPTER 5
5.5 Numerical examples
The Gurson material model is implemented in the Finite Element Programs for shell
and tube elements, as well as the FE Code INSAP, Scarpas [2004]. In the following
paragraphs the numerical results from analysis with the Gurson material model for a
straight pipeline and a pipeline bend are shown. The numerical results obtained in
combination with the tube elements are compared with results obtained in
combination with selective integrated Heterosis elements (S2) and underintegrated
Heterosis elements (U2). For the formulation of the tube element the use of the
warping terms is essential. As shown in chapter 4, the results with the tube elements
and the shell elements in the elastic domain are very close.
For the analysis, the following material parameters were adopted. The used values for
the Gurson parameters are commonly applied for metallic strip material. The initial
void volume and the hypothetical initial yield stress .
The Young’s modulus and the Poisson ratio . The
parameters and are 1.5, 1.0 and 2.25 respectively. The hypothetical
isotropic hardening parameter = 500 N/mm
*0f 0.004= 2
y,0 400 N / mmσ =
MPa210000E = 0.3ν =
1q , 2q 3q
ISOH 2. The volume fraction of void
nucleating particles = 0.04, the standard deviation = 0.1 and the mean strain for
nucleation = 0.3.
Nf Ns
Nε
The pipe structures are subjected to a prescribed rotation pκ . As shown in Figure
5.8a, this rotation is imposed on the end node on the axis of the tube elements. On the
shell elements this rotation is enforced via prescribed displacements on the element
nodes.
(a) (b)
Figure 5.8 Prescribed rotation with tube elements (a) and shell elements (b)
symmetry line
CONSTITUTIVE MODEL 63
5.5.1 Analysis of pure bending of a straight pipeline
A straight pipe element, as shown in Figure 5.9 is subjected to a prescribed rotation
rad. The same geometric properties are used as the example in paragraph
4.6.2. Due to symmetry only a quarter of the pipeline is modeled. The structure is
fixed at L/2, so that this node cannot translate or rotate, whereas the cross-section is
free to ovalize, but not to warp. The other end is free to translate perpendicular to the
pipe axis. The cross-section may ovalize, but cannot warp. For the analysis only one
tube element was used.
p 0.2κ =
L = 8000 mm
r = 198.45 mm
t = 9.5 mm
ν = 0.3 E = 2.1×105 N/mm2
pκ
L 2r
t
Figure 5.9 Schematic of a straight pipe
The longitudinal stresses and the micro-damage development in the pipe structure are
shown in Figures 5.10 and 5.11, respectively, where 0 degrees denotes the top and
180 degrees the bottom of the pipe structure. Both elements are used in combination
with the Gurson model.
-500
-400
-300
-200
-100
0
100
200
300
400
500
0 20 40 60 80 100 120 140 160 180angle
Lon
gitu
dina
l str
ess (
MPa
)
Heterosis element (U2)
Heterosis element (S2)
TUBE element
Figure 5.10 Longitudinal stresses at outside of straight pipe due to bending
64 CHAPTER 5
0.00398
0.004
0.00402
0.00404
0.00406
0.00408
0.0041
0.00412
0.00414
0 30 60 90 120 150 180angle
dam
age
f*Heterosis element(U2), insideHeterosis element(U2), outsideTUBE element, insideTUBE element, outside
Figure 5.11 Void volume development in straight pipe due to bending
The results obtained with the tube element are very close to the results calculated with
the Heterosis element. The damage development on the outside of the pipe wall is
slightly larger than on the inside of the pipe wall. In Figure 5.12 the micro-damage
development in the pipeline is shown. The color red corresponds to the initial void
volume and the color blue to the maximum value. *0f
Figure 5.12 Damage development in pipeline due to bending
CONSTITUTIVE MODEL 65
5.5.2 Curved pipeline, subjected to a “closing” rotation
A pipeline bend, as shown in Figure 5.13, is considered while subjected to a
monotonic prescribed rotation rad. The same geometric properties are used
as the example in paragraph 4.6.4. The radius of the pipe r is 198.45 mm. The radius
of the pipeline bend is 609.4 mm. The structure is fixed at node A, so that the end
node cannot translate or rotate, whereas the cross-section is free to ovalize, but not to
warp. Corresponding to the boundary condition as shown in Figure 5.8b, the other end
is free to translate perpendicular to the pipe axis and restrained in the other direction.
The cross-section may ovalize, but cannot warp. For the analysis 11 tube elements
were used.
p 0.2κ =
BR
Figure 5.13 Schematic of pipe structure
Only half the circumference is analyzed due to symmetry. The material parameters are
shown in paragraph 5.5. In the following graphs, the stresses and micro-damage
are shown with respect to the hoop direction of the cross section, where 0 degrees
denotes the outside and 180 degrees the inside of the pipe bend. Figures 5.14 and 5.15
show the circumferential stresses at the inside and the longitudinal stresses at the
outside of the pipe wall, respectively.
*f
L1 = 609.6 mm
L2 = 152.4 mm
RB = 609.4 mm
r = 198.45 mm
t = 9.5 mm
ν = 0.3 E = 2.1×105 N/mm2
RB
pκ
L1
L2
A 2r
t
66 CHAPTER 5
-600
-400
-200
0
200
400
600
0 30 60 90 120 150 180angle
Cir
cum
fere
ntia
l str
ess (
MPa
)
tube element
heterosis element (U2)
Figure 5.14 Circumferential stresses at inside of the pipe wall, p 0.2κ = rad
-600
-400
-200
0
200
400
600
0 30 60 90 120 150 180angle
Lon
gitu
dina
l str
ess (
MPa
)
tube element
heterosis element (U2)
Figure 5.15 Longitudinal stresses at outside of the pipe wall, p 0.2κ = rad
The circumferential and longitudinal stresses, along the circumference of the pipeline
bend, determined with the tube element in combination with the Gurson constitutive
model are very close to the stresses obtained with the Heterosis element. The response
obtained with the tube elements, however, is much smoother.
(extrados) (intrados)
(extrados) (intrados)
CONSTITUTIVE MODEL 67
In Figure 5.16 the developed damage determined with shell elements in combination
with the GTN material model is shown for p 0.2κ = rad. The onset of plasticity is at
the inside of the pipe wall due to the circumferential stress, as shown in Figure 5.16a.
Due to the longitudinal stress micro damage develops at the outside of the pipe wall as
shown in Figure 5.16b. The red color corresponds to the initial void volume and
the blue color to the maximum value.
*0f
(a) (b)
Figure 5.16 Damage at inside and outside of pipe wall with shell elements
In Figure 5.17 the development of micro-damage at the inside of the pipe wall is
shown. When tube elements in combination with the material model are used, the
maximum developed damage is less than the damage predicted with the Heterosis
shell elements, but the zone is wider. Analyses of the pipe structure with different
integration schemes do not show a significant difference, as the strains are not far in
the softening zone. The observed difference in damage development is caused by
different formulations to describe the deformation of the elements.
*f
68 CHAPTER 5
0.004
0.0043
0.0046
0.0049
0.0052
0.0055
0.0058
0.0061
0.0064
0.0067
0 30 60 90 120 150 180angle
dam
age
f*tube elementheterosis element, 3-2 integrationheterosis element, 2-2 integration
Figure 5.17 Damage development at inside of pipe wall, p 0.2κ = rad
In Figure 5.18 the deformed cross-sections of the pipe structure with shell and tube
elements are shown. The predicted deformation of the pipe bend is almost identical.
When the out-of-plane rotations are not allowed the tube element ovalizes, but the
cross-section remains symmetric.
Figure 5.18 Cross-sectional deformation at end of pipe structure, rad p 0.2κ =
(extrados) (intrados)
0
50
100
150
200
250
300
-400-300-200-1000100200300horizontal coordinates
vert
ical
coo
rdin
ates
original cross-sectioncross-section tube elementscross-section shell elements
(extrados) (intrados)
CONSTITUTIVE MODEL 69
5.5.3 Curved pipeline, geometrical influence on damage development
A pipeline bend, as shown in Figure 5.19, is considered while subjected to a
monotonic prescribed rotation rad. The angle of the pipeline bend is 45°. The
length of L
p 0.2κ =
1 is identical to L2. The influence of the radius of the pipe on the damage
development is compared for two radii, = 198.45 mm and = 125 mm. The radius
of the axis of the pipeline bend is 609.4 mm. The structure is fixed at node A, so
that the end node cannot translate or rotate, whereas the cross-section is free to
ovalize, but not to warp. The other end is free to translate in both directions. The
cross-section may ovalize, but cannot warp. For the analysis 13 tube elements were
used.
r r
BR
L1 = 609.6 mm
L2 = 609.6 mm
RB = 609.4 mm
t = 9.5 mm
ν = 0.3 E = 2.1×105 N/mm2
RB
pκ
L1
L2
A 2r
t
Figure 5.19 Schematic of pipe structure with 45° bend
As the geometry and boundary conditions are not identical to the example in § 5.5.2,
the calculation is repeated with a 90° bend angle and pipe radius = 198.45 mm to
demonstrate the influence of the pipeline bend angle on the damage development.
Only half the circumference is analyzed due to symmetry. The material parameters are
shown in paragraph 5.5. In the following graphs, the stresses and micro-damage
are shown with respect to the hoop direction of the cross section, where 0 degrees
denotes the outside and 180 degrees the inside of the pipe bend.
r
*f
70 CHAPTER 5
The circumferential stresses in the bends are shown in Figure 5.20. The location
where the stress changes from tension into compression is different for both
geometries. This results in a smaller compressive zone on the inside and tensile zone
on the outside of the pipe wall for the pipe with radius r = 198.45 mm.
-600
-400
-200
0
200
400
600
0 30 60 90 120 150 180
angle
stre
ss [M
Pa]
r=198.45_inside r=198.45_outsider=125_inside r=125_outside
Figure 5.20 Circumferential stresses for pipelines with different radius, rad p 0.2κ =
The longitudinal stresses in the bends are shown in Figure 5.21. The observed
difference in response is related to the difference in bend radius at the inside of the
pipeline bend.
-600
-400
-200
0
200
400
600
0 30 60 90 120 150 180
angle
stre
ss [M
Pa]
r=198.45_inside r=198.45_outside
r=125_inside r=125_outside
Figure 5.21 Longitudinal stresses for pipelines with different radius, rad p 0.2κ =
(extrados) (intrados)
(extrados) (intrados)
CONSTITUTIVE MODEL 71
The damage development on the inside of the pipe wall is shown in Figure 5.22. The
increase of micro voids in the pipeline bend with a 45° angle is approximately two
times larger than in the pipe structure with a 90° angle. The cross-section with the
smallest diameter shows less damage at 96° in the circumference but more damage
development on the inside of the pipe bend. This difference is related to the amount of
ovalization of the cross-sections, as shown in Figures 5.23 and 5.24. In these graphs
the nodal displacements are not taken into account.
0.004
0.0042
0.0044
0.0046
0.0048
0.005
0.0052
0.0054
0 30 60 90 120 150 180angle
dam
age
f*
r=198.45_45° bend
r=125_45° bend
r=198.45_90° bend
Figure 5.22 Damage development at inside of pipe wall, p 0.2κ = rad
0
50
100
150
200
250
300
-300 -200 -100 0 100 200 300horizontal coordinates
vert
ical
coo
rdin
ates
original cross-sectionradius = 198.45, 45° pipe bendradius = 198.45, 90° pipe bend
Figure 5.23 Cross-sectional deformation in pipeline bend, p 0.2κ = rad
(extrados) (intrados)
(extrados) (intrados)
72 CHAPTER 5
0
20
40
60
80
100
120
140
160
180
-150 -100 -50 0 50 100 150horizontal displacement
vert
ical
dis
plac
emen
t
original cross-section
radius = 125, 45° pipe bend
(extrados) (intrados)
IP 31
Figure 5.24 Cross-sectional deformation in pipeline bend, p 0.2κ = rad
The pipe with radius r = 198.45 mm shows more ovalization than the pipe with
radius = 125 mm. As mentioned before, the radius of the pipeline bend is
determined with respect to the tube axis. This implies that the radius of the bend on
the inside of the pipe is different for the compared structures.
r
Chapter 6
CYCLIC MODEL
6.1 Introduction
To simulate the cyclic response including the degradation phase a new concept is
presented in this chapter. In standard elastoplasticity, the region inside the yield
surface corresponds to fully elastic constitutive behavior. Consequently, at transition
from elastic to elastoplastic behavior, the stiffness changes abruptly from elastic to
elastoplastic. To account for a smooth transition from elasticity to plasticity a number
of constitutive models have been developed. Numerous publications (e.g. Bari and
Hassan [2000]) have shown that models in which only kinematic hardening is related
to plastic strain perform poorly in case of multiaxiality.
Dafalias and Popov [1975] developed a well-known concept in the modeling of
ratcheting response. A loading surface moves inside a fixed bounding surface in such
a way that the bounding surface is approached asymptotically, as shown in Figure
6.1a. On the basis of this Bounding Surface Concept, many researchers have proposed
improved models for simulation of ratcheting (e.g. Mróz, Norris and Zienkiewicz
[1981], Voyiadjis and Abu-Lebdeh [1994] and Montáns [2000]). The particularity of
these models lies on the fact that traditional bounding surface models impose the
consistency condition on the loading surface. The disadvantage of these models is that
they are not able to predict the response degradation after a certain maximum amount
of cycles. To predict this behaviour the model must keep track of the total accumulated
amount of damage.
74 CHAPTER 6
fixed Bounding Surface
Yield Surface ( )f σ%
Yield Surface ( )f σ%
Loading Surface ( )g σ)
1σ
3σ
2
1σ
3σ
σ 2σ
(a) (b)
Figure 6.1 Classic Bounding Surface concept (a) and proposed model (b)
In the framework of this thesis a two-surface model, based on the Bounding Surface
Concept, is proposed. The developed model is based on a new concept and imposes
the consistency condition on the bounding surface and scales the response to the
current state of stress. This means that the dimensions of the bounding surface are no
longer fixed and introduces the possibility to determine the accumulated cyclic
damage. This implies that the space in which the loading surface can move varies. This
model consists of a yield surface ( )f σ% (Gurson) which acts as a bounding surface for
a smaller surface, also known as loading surface, as illustrated in Figure 6.1b.
Hereafter we may refer to this material model as two-surface model.
2σ
1σ
gα
Δσ
( )f σ%
fα
( )g σ)
2σ
1σ
τσ
τ+Δτσ
*σ
gα
Loading surface ( )g σ)
Yield surface ( )f σ%
(a) (b)
Figure 6.2 Two-surface model for plane-stress conditions
CYCLIC MODEL 75
( )g σ)The loading surface is formulated with the same shape parameters as the
bounding surface:
( ) 32g g
2 3q pq * 2g 2q f cosh q f1 2
⎡ ⎤⎛ ⎞−⎢ ⎜ ⎟= + − −
⎜ ⎟σ σ⎢ ⎥⎝ ⎠⎣ ⎦σ *2 1⎥
))) , (6.1)
where
g= −σ σ α) .
( )p p= σ) )The effective hydrostatic stress and the effective deviatoric stress
( ) ( ) 1 2g g
3q :2⎡ ⎤= − −⎣ ⎦s a s a) ,
where indicates the deviatoric part of the back stress. ga
When the state of stress is inside this inner surface the material response is elastic. The
loading surface can change in size and moves via kinematic hardening within the
bounding surface, as shown in Figure 6.2a. If the size of the loading surface remains
unchanged, the translation of the origin of this surface gα is identical to the stress
increment. gd d=α σ
6.2 Constitutive framework
Initially the bounding surface controls the response of the material as described in
chapter 5. Due to hardening the bounding surface grows and moves in the loading
direction.
As shown in Figure 6.2b, when the state of stress is on the yield surface the loading
surface does not change in size but moves with the yield surface in the direction of the
elastic stress increment via:
( ) gg
f
τ+Δτ τ+Δτ τ+Δτ τ+Δτfσ
= ασ
α σ − σ − , (6.2)
fσis the yield stress of the loading surface and where gσ the yield stress of the
bounding surface.
76 CHAPTER 6
t t+Δ σAfter reversal of the loading direction the state of stress is inside the loading
surface, until it reaches the surface. After that the surface moves with the state of stress
until it reaches the bounding surface.
Assuming, for visualization purposes, proportional loading, the state of stress * can
be defined by the intersection of the projection of the elastic stress increment onto the
outer surface, as illustrated in Figure 6.3.
σ
2σ
1σ
τσ
τ+Δτσ
( )g σ)
( )f σ%
2σ
1σ
τστ+Δτσ
*σ
gα
( )g σ)
( )f σ%eΔσ
fα
RVSσ
Figure 6.3 Two-surface model for plane-stress conditions
RVSσDuring stress reversal at any point between the state of stress at load reversal, ,
and * , the increment of stress σ Δσ is computed via
( )pcΔ = Δ −ΔDσ ε ε , (6.3)
pcΔε is evaluated via: The fictitious plastic strain increment tensor
( )( )*pc p1 ω
σΔ = − δ Δε ε , (6.4)
where is the plastic strain increment in the “projected” stress state at the
bounding surface and ω controls the size of the cycles.
( )*p
σΔε
The relative distance in stress space between the current state of stress and the
projection point at the bounding surface is described via:
* RVS
∗ τ+Δτ⎛ ⎞−δ = ⎜⎜ −⎝ ⎠
σ σσ σ
⎟⎟ . (6.5)
Initially, when , the response is elastic. Gradually the state of stress moves to the
bounding surface,
1δ =
0δ = , and the response is plastic.
CYCLIC MODEL 77
Substituting equation (6.4) into equation (6.3) yields:
( ) (*p 1 )ω
σΔ = Δ − ⋅ Δ ⋅ − δD Dσ ε ε , (6.6)
which, physically, implies that the stress increment tensor Δσ varies from at
to the stress increment postulated by plasticity theory at *
ΔD εRVSσ σ . In paragraph 6.3
computation of will be addressed. ( )*p
σΔε
6.2.1 Cyclic response development Extensive studies of the mechanical behavior of metals under uniaxial cyclic loading
histories have revealed that, under such loading conditions, metals can harden or
soften. When subjected to strain controlled cycles, as shown in Figures 6.6b and 6.7b,
the hysteresis loop tends to stabilize to one that is closed after a number of cycles. In
case of stress controlled cycles, as shown in Figures 6.6a and 6.7a, it is observed that
the induced hysteresis loops never close. The strain in the direction of the mean stress
gradually creeps, or ratchets, as a result of these cycles. These phenomena are well
known and have been reported among others by Landgraf [1970] and Hassan and
Kyriakides [1992].
Under multiaxial loading, ratcheting can occur if at least one component of stress is
prescribed in a multiaxial cyclic loading history involving some plastic deformation, as
shown in Hassan et al. [1992]. In such cases ratcheting will be in the direction of the
prescribed stress(es).
Parameter is a material parameter representing material stiffness degradation with
increasing number of cycles. Preliminary experiments have shown that the energy
dissipation of the material reduces with every cycle. The width of every cycle is
determined via equation
ω
(6.7), where ω varies from 1ω at the first cycle to at the
end of ratcheting
∞ω
( )1 1 ∞ω = ηω + −η ω . (6.7)
The parameter represents the normalized function of the development of the size of
the cycles. It depends on the equivalent fictitious plastic strain
η
pcε , and therefore
controls the degradation.
( ) pccHpc
c1 H e− ⋅εη = + ⋅ ε ⋅ , (6.8)
78 CHAPTER 6
where the parameter can be determined via curve fitting . cH
ηThe influence of on the material response is shown in Figure 6.4, whereas a
variation of ( )1 ω− δ with for various δ ω values is shown in Figure 6.5.
0
0.2
0.4
0.6
0.8
1
1.2
0 4 8 12εpc
η
H = 0.5H = 1.0H = 1.5
Hc = 0.5Hc = 1.0H = 1.5c
( )1 ω− δFigure 6.4 Proposed response development function, variation of with δ
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1δ
1−δω ω = 1.5
ω = 2ω = 5ω = 10
( )1 ω− δ with δ Figure 6.5 variation of
In many applications, structures and structural components must be designed to
withstand not only mechanical loads but also the internal or external environment.
Depending on these conditions different materials are used in the design. It is well
known that the response of these metallic materials under cyclic loads can differ.
Among others, Hassan and Kyriakides [1992] demonstrated experimentally that, in
CYCLIC MODEL 79
the course of cyclic loading, stainless steel 304 hardens while carbon steel 1020 is
seen to soften.
Cyclic softening When the material softens the loops become larger and the stress-strain curves in the
plastic range become flatter. In terms of the present model this can be simulated when
parameter is smaller than . The influence of an increasing value of ω on the
response is shown in Figures 6.6a and 6.6b.
1ω ∞ω
ε
σ
1ω ∞ω
ε
σ
(a) (b)
Figure 6.6 Cyclic softening, stress controlled load (a) and strain controlled load (b)
Cyclic hardening When the material hardens the curves become steeper, and the width of the cycles
decreases. In terms of the present model this can be simulated when parameter is
larger than .The influence of a decreasing value of
1ω
∞ω ω on the response is shown in
Figures 6.7a and 6.7b.
σ
ε ε
σ
1ω ∞ω
(a) (b)
Figure 6.7 Cyclic hardening, stress controlled load (a) and strain controlled load (b)
80 CHAPTER 6
6.2.2 Cyclic response degradation When subjected to moment-controlled cycling, degradation of the tube response
occurs after accumulation of the deformation, as shown in Figure 6.8. Corona and
Kyriakides [1991] showed that the tube buckled at approximately the same value of
curvature as a tube subjected to a monotonic load.
Figure 6.8 Response of a tube subjected to moment-controlled cycling
(From E. Corona & S. Kyriakides [1991]. Reprinted by permission.)
In this case, the tube suffered two types of degradation which lead to buckling. The
first consisted of an accumulation of curvature while the second consisted of an
accumulation of curvature with the same sense as the prescribed mean moment. This
accumulation of curvature approximately corresponds to the axial strain ratcheting of a
bar under axial loading.
Initially the width of the cycles reduces to a constant level due to cyclic hardening.
After only a few cycles, the width of the loops is constant. Approximately at the point
where the response of the monotonically loaded tube starts to degrade, the cycles
become wider until failure occurs. This type of reversal in cyclic behavior has been
observed in low cycle fatigue as well as high cycle fatigue by, amongst others, Laird
[1977].
In the proposed model the monotonic stress degradation response envelop constitutes
the limit of cyclic stress response degradation, as illustrated in Figure 6.9. When
subjected to a monotonic load, hardening and softening of the yield surface are
CYCLIC MODEL 81
pεcontrolled by the total plastic strain and the equivalent plastic strain , as shown in
chapter 5. When subjected to a cyclic load, the permanent deformation is postulated to
increase due to the fictitious plastic strain increment tensor pcΔε . The bounding
surface hardens and softens due to the increasing plastic strain. When the yield
surface grows the cycles become smaller for load-controlled cycling. This is also
observed in high cycle fatigue.
The cyclic stress-strain response and the damage development of the model are shown
in Figure 6.9. For this load-controlled example the material parameters for SS304, as
determined in paragraph 6.5.1 are used. Initially the loading surface has the same size
as the bounding surface but when the latter increases in size, due to isotropic
hardening, the loading surface can move within the bounding surface. As shown in
equation (6.5), the relative distance between the current state of stress and the state of
stress at the projection point changes when the bounding surface increases or reduces
in size. When the relative distance δ increases the width of the cycles will reduce.
-150
-100
-50
0
50
100
150
200
250
0 0.05 0.1 0.15 0.2 0.25
Total strain
Stre
ss (M
Pa)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Dam
age
deve
lopm
ent f
*
Cyclic responseMonotonic responseDamage development
Figure 6.9 Material response of SS304 subjected to load-controlled cycling
The opposite mechanism can be seen when the bounding surface reduces in size. As
shown in chapter 5, the size of the yield surface reduces due to an increasing void
volume. This affects also the width of the cycles, as they become significantly wider
when the bounding surface begins to shrink.
82 CHAPTER 6
During unloading the total plastic strain is reduced with the fictitious cyclic plastic
strain increment pcΔε , and during reloading this cyclic plastic strain is added to the
total plastic strain pε . During the loading history the maximum value of the total
plastic strain is stored as a limit value, p, limitε . When the absolute value of the
updated total plastic strain is smaller than this fixed value, the bounding surface will
not change in size. When the absolute value of the total plastic strain and the fictitious
plastic strain increment pcΔε is larger than this fixed value, the total plastic strain will
increase, as shown in Figure 6.10.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.005 0.01 0.015 0.02total strain
Equivalent total plastic strain
Equivalent cyclic plastic strain
Equivalent plastic strain at projection point
Limit value for equivalent plastic strain
Equ
ival
ent p
last
ic st
rain
uniaxial total strain Figure 6.10 (Pseudo) plastic strain development during load cycles
Following equation (5.3), the yield stress of the bounding surface is updated via:
( )( )p pc p.limitf f isoHτ+Δτ τσ = σ + ⋅ ε + Δε − ε . (6.9)
As mentioned above, the relative distance becomes larger when the bounding surface
grows. This makes the curves steeper, corresponding to a cyclic hardening response of
the model, as shown in Figure 6.11. During cyclic loading the micro damage develops
according to
( )( ) ( )( )* * * p pc p, limit p pc p,limitijij ij ijf f 1 f Aτ+Δτ τ τ τ= + − ε + Δε − ε δ + ⋅ ε + Δε − ε . (6.10)
A schematic of this equation is shown in Figure 6.12.
CYCLIC MODEL 83
-150
-100
-50
0
50
100
150
200
250
0 0.005 0.01 0.015 0.02 0.025 0.03
total strain
Stre
ss (M
Pa)
0.0034
0.0036
0.0038
0.004
0.0042
0.0044
0.0046
0.0048
0.005
Dam
age
deve
lopm
ent f
*
Cyclic responseMonotonic responseDamage development
Figure 6.11 Cyclic and monotonic response
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 0.1 0.2 0.3 0.4 0.5total strain
Dam
age
devo
pmen
t f*
Damage development(cyclic load)Damage development(monotonic load)
Figure 6.12 Damage development due to cyclic and monotonic load
84 CHAPTER 6
6.3 Parameter determination
The development of the cyclic response can be determined from experimental data
from uni-axial tests. From equations (6.7) and (6.8) the development of ω with
respect to the equivalent fictitious cyclic plastic strain can be determined via:
( ) pccHpc
c1
1 H e− ⋅ε∞
∞
ω−ω= + ⋅ ε ⋅
ω −ω. (6.11)
This relation, for different values of , is shown in Figure 6.13. cH
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5εpc
ω
H=1H=2H=4
1ω
∞ω
Figure 6.13 Parameter with respect to the fictitious cyclic plastic strain ω
The width of the cycles, and 1ω ∞ω , can easily be determined at the points where the
loading surface touches the bounding surface. By means of equations (6.4) and (6.5)
can be determined via: 1ω
( )( ) ( )( )
( ) ( ) ( )
* *
1 1
*
pc p p
C D
pc
p
1 1
2 1
τ τ+Δτω ω
σ σ
ω ωτ τ+Δτ
σ
⎛ ⎞⎡ ⎤ ⎡ ⎤Δ = −δ Δ + −δ Δ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠⎛ ⎞
Δ⎜ ⎟δ + δ = ⋅ −⎜ ⎟Δ⎜ ⎟⎝ ⎠
ε ε ε
ε
ε
/ 2
(6.12)
RVSσδ is determined via equation (6.5), where The relative distance in stress space
corresponds to the stress in point C and τ+Δτσ to point D. These points are shown in
Figure 6.14, which represents a schematic representation of the uniaxial response. C is
the point where the response during reloading becomes nonlinear. D is the point
where the stress state is close to the bounding surface.
CYCLIC MODEL 85
Cε Dε Aε
σ A D
C
B
Figure 6.14 Determination of cyclic response parameters
The plastic strain increment ( )*p
σΔε at the projection point on the bounding surface is
constant and can be calculated via equation (6.20). The fictitious cyclic plastic strain
increment pcΔε , from point C to point D, can be determined from the experimental
data. Because 0τ+Δτδ = , equation (6.12) can be written as:
( )( )
*
pc
p
1C
ln 1
2ln
σ
⎛ ⎞Δ⎜ ⎟−
⎜ ⎟Δ⎝ ⎠ω = ⋅
δ
ε
ε.
For the determination of a similar procedure can be followed. ∞ω
86 CHAPTER 6
6.4 Numerical implementation
6.4.1 Three-dimensional formulation ( )f ,κσ%The truncated Taylor expansion of about the state of stress at the projection
point * is σ
( ) ( ) ( )* * *
T* * *
f ff
f f ff fτ+Δτ τ+Δτ τ+Δτ τ+Δτ
σ σ σ
⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞≈ + − + κ − κ + − =⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂κ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠σ σ α α
σ α%* 0
t
.
(6.13)
The change of the yield condition from state * to state t +Δ must be zero. This is the
classic consistency condition of Prager, which on account of equation (6.6) becomes:
* * * *
T*
ff
f f f ff 0σ σ σ σ
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ = Δ −Δλ + Δκ+ Δ =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂κ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠D Dε α
σ σ α% %, (6.14)
in which is some measure of isotropic hardening/softening. κ
( )pκ = σ ε , then: Assuming
* *f
f f
σ σ
⎛ ⎞∂ ∂⎛ ⎞ = ⎜ ⎟⎜ ⎟∂κ ∂σ⎝ ⎠ ⎝ ⎠, (6.15)
and
( )( )
*
*
Tpf
p
pisoH .
σ
σ
∂σ⎛ ⎞Δκ = ⋅ Δε⎜ ⎟∂ε⎝ ⎠
= ⋅ Δε
(6.16)
The incremental displacement of the center of the bounding surface, equation 5.5, can
be written as:
( )*
*p
f kin*HσΔ = ⋅ ⋅Δ
σα
σ
%
%ε . (6.17)
pΔεIn the Gurson model the microscopic equivalent plastic strain is assumed to vary
according to the equivalent plastic work expression:
( )* pf1 f :− σ Δε = Δσ ε% p ,
or equivalently for the stress at projection point *σ :
CYCLIC MODEL 87
( ) ( )( )
*
*
* pp
*f
:
1 fσ
σ
ΔΔε =
− σ
σ ε%. (6.18)
As shown in equation (5.12), ( )*p
σΔε can be computed as:
( )**
p * fσ σ
∂⎛ ⎞Δ = Δλ ⎜ ⎟∂⎝ ⎠ε
σ%. (6.19)
Δλ(6.14) and solving in terms of : Substituting this in equation
( )
*
* * * **
T
*
T *
iso kin *f f
f
f f f f fH H1 f
σ
∗
∗σ σ σ σ
∂⎛ ⎞ ⋅ ⋅Δ⎜ ⎟∂⎝ ⎠Δλ =⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⋅ ⋅ − ⋅ + ⋅ ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂σ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠− σ⎝ ⎠⎣ ⎦
D
D
εσ
σ σσ σ α σσ
%
% %
% % %% σ
(6.20)
as shown in equation (2.21), but with the isotropic and kinematic hardening of the
bounding surface taken into account. The terms involved in the numerical
implementation for the three-dimensional system are
*
** * 2
2 1* 2f
3 2
q p33q f pq sinh2f 2 q
σ
⎛ ⎞−⎜ ⎟σ∂ −⎛ ⎞ ⎝ ⎠= +⎜ ⎟∂σ σ σ⎝ ⎠
%%
%
* ** *
f f p f qp qσ σσ σ
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠σ σ% %
% %% % *σσ%
* ** *f f
f f p f qp qσ σσ σ
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠α α
% %
% % *f σα.
Equation (6.6) can be written as:
( ) ( )
( )*
*
p
*
1
f 1 .
ω
σ
ω
σ
⎡ ⎤Δ = − ⋅ Δ ⋅ − δ Δ⎢ ⎥⎣ ⎦⎡ ⎤∂⎛ ⎞− ⋅Δλ ⋅ − δ Δ⎢ ⎥⎜ ⎟∂⎝ ⎠⎣ ⎦
D D
D D
σ ε ε
= εσ%
88 CHAPTER 6
6.4.2 Plane stress formulation For plane stress elements as the shell and the tube element it is required that the stress
perpendicular to the surface is zero. As shown by Crisfield [1997], the corresponding
strain increment component 33Δε, respectively 3ζ
Δε , is also considered to be zero.
The stress, strain and back stress tensors in this algorithm are identical to the tensors in
the three-dimensional system, but exist of five components.
6.5 Numerical examples In the following paragraphs the cyclic model is evaluated with a set of uniaxial cyclic
hardening and softening responses from Hassan and Kyriakides [1994]. The Stainless
Steel SS 304 is a material that hardens under cyclic loading, while Carbon Steel CS
1018 tends to soften under repeated cycles. With the parameters obtained for SS 304
the response of a curved pipeline is simulated. For this analysis underintegrated
Heterosis elements (U2) were used.
The material parameters for the cyclic model are determined from the available
experimental data. The typical parameters for the Gurson model are from chapter 5
and not modified. The initial void volume fraction = 0.004 and the parameters ,
and are 1.5, 1.0 and 2.25 respectively. The volume fraction of void nucleating
particles = 0.04, the standard deviation = 0.1 and the mean strain for nucleation
= 0.25.
*0f 1q
2q 3q
Nf Ns
Nε
No cyclic degradation, as discussed in paragraph 6.2.2, is observed in the
experiments.
CYCLIC MODEL 89
6.5.1 Cyclic hardening: SS 304
The use of the cyclic model is demonstrated at the following example where a strain
controlled experiment has been performed on stainless steel SS 304. The SS 304
material is a material that hardens under uniaxial strain-symmetric cyclic loading.
The yield stress of the material is 205.5 N/mm2, the Young’s modulus is 191700
N/mm2 and the Poison ratio is 0.33. The bounding surface model has a constant
isotropic hardening of =1400 and a constant kinematic hardening of
=1100.
ISOH
KINH
∞ω1ω = 1.30, The parameters which control the width of the cycles = 0.4 and the
degradation control parameter η = 0.4. The values of the control parameters and
( > ) are consistent with the theoretical anticipated ones as the stainless steel
SS304 is a material that hardens under cyclic loads.
1ω
∞ω ∞ω1ω
Strain controlled experiments
The strain-symmetric cyclic experiment with a strain limit of 1% has been conducted
by Kyriakides [1994] in order to obtain the hardening characteristics of the mentioned
material.
Figure 6.15a shows the experimental stress-strain response for the material. The
simulated stress-strain response is shown in Figure 6.15b. The differences between the
two responses are very small. The computed progression of the stress amplitude of
each cycle with respect to the number of cycles is shown in Figure 6.16. It seems that
the stress amplitude increases by approximately 42% during the first 25 cycles and
remains constant from that point on.
In Figure 6.17 the development of the void volume fraction is shown with respect to
the total uniaxial strain. The void volume fraction increases during reloading and
remains constant during unloading following from equation (6.10). Figure 6.18 shows
the development of the void volume with respect to the number of load cycles.
90 CHAPTER 6
(a)
-60
-40
-20
0
20
40
60
-1.0% -0.5% 0.0% 0.5% 1.0%
(b)
Figure 6.15 Cyclic hardening of SS 304 (a) Experiment (b) Numerical
0
10
20
30
40
50
60
0 5 10 15 20 25
Max
imum
stre
sses
(ksi
)M
axim
um st
ress
es (k
si)
Number of cycles (N)Number of cycles (N) Figure 6.16 Computed stress amplitude as a function of number of cycles (N)
CYCLIC MODEL 91
0
0.01
0.02
0.03
0.04
0.05
-0.01 -0.0075 -0.005 -0.0025 0 0.0025 0.005 0.0075 0.01
total strain
void volume fractionV
oid
volu
me
frac
tion
Figure 6.17 Void volume fraction as a function of total strain
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 5 10 15 20 25Number of cycles (N)
void volume fraction
Stra
ins
Voi
d vo
lum
e fr
actio
n
Figure 6.18 Void volume fraction as a function of number of cycles (N)
92 CHAPTER 6
6.5.2 Cyclic softening: CS 1018
Another example demonstrating the use of the cyclic model is that of a strain
controlled experiment performed for Carbon Steel CS 1018. The CS 1018 is cold
worked during its manufacturing and tends to soften under repeated cycles.
The yield stress of the material is 602 N/mm2, the Young’s modulus is 196500 N/mm2
and the Poison ratio is 0.33. The experimental response shows a limited amount of
hardening during the first cycle. This is modeled with a nonlinear isotropic hardening
function for . The kinematic hardening remains constant for all the cycles
=2000.
ISOH KINH
∞ω ηThe control parameter = 0.93, 1ω = 1.5 and the degradation control parameter
= 12,0. The values of the control parameters ∞ω ∞ω1ω 1ω and ( < ) are consistent
with the theoretical anticipated ones as the carbon steel is a material that softens under
cyclic loads.
Strain controlled experiments
The strain-symmetric cyclic experiment with a strain limit of 1% has been conducted
by Kyriakides [1994] in order to obtain the softening characteristics of the mentioned
material.
In Figure 6.19 the experimental and numerical stress-strain diagrams are shown. The
differences between the two responses are small. Figure 6.20 shows the maximum
stress development according to the number of cycles. The computed analysis curve
show that the stress amplitude of the cycles decreases by approximately 17% in 15
cycles. After this number of cycles the rate of additional softening was rather small.
CYCLIC MODEL 93
(a)
-120
-80
-40
0
40
80
120
-1.0% -0.5% 0.0% 0.5% 1.0%
(b)
Figure 6.19 Cyclic softening of CS 1018 (a) Experiment (b) Numerical
0
20
40
60
80
100
120
0 2 4 6 8 10 12 14
Number of cycles (N)
Max
imum
stre
sses
(ksi
)
Figure 6.20 Computed stress amplitude as a function of number of cycles (N)
CS 1018
xc 1.0%ε =
x (%)ε
xσ
(ksi)
Max
imum
stre
sses
(ksi
)
Number of cycles (N)
94 CHAPTER 6
6.5.2 Curved pipeline, subjected to cyclic load
A pipeline bend, as shown in Figure 6.21, is considered while subjected to a cyclic
prescribed rotation rad. The same geometric properties are used as
the example in paragraph 4.6.4. The radius of the pipe is 198.45 mm. The radius of
the bend is 609.4 mm. The structure is fixed at node A, so that the end node
cannot translate or rotate, whereas the cross-section is free to ovalize, but not to warp.
Corresponding to the boundary condition as shown in Figure 5.8b, the other end is
free to translate perpendicular to the pipe axis and restrained in the other direction.
The cross-section may ovalize, but cannot warp.
p 0.0 0.04κ = ↔
r
BR
κ
Figure 6.21 Schematic of pipe structure
The results are obtained with the selective integrated Heterosis elements (S2). For the
analysis the material parameters for SS304 were used. The initial void volume
fraction = 0.004 and the parameters , and are 1.5, 1.0 and 2.25
respectively. The volume fraction of void nucleating particles = 0.04, the standard
deviation = 0.1 and the mean strain for nucleation
*0f 1q 2q 3q
Nf
Ns Nε = 0.25. The yield stress of
the material is 205.5 N/mm2, the Young’s modulus is 191700 N/mm2 and the Poison
ratio is 0.33. The bounding surface model has a constant isotropic hardening of
L1 = 609.6 mm
L2 = 152.4 mm
RB = 609.4 mm
r = 198.45 mm
t = 9.5 mm
RB
p
L1
L2
t
2r A
CYCLIC MODEL 95
ISOH =1400 and a constant kinematic hardening of =1100. The parameters
which control the width of the cycles
KINH
1ω 2ω = 1.30, = 0.4 and the degradation
control parameter η = 0.4.
Only half the circumference is analyzed due to symmetry. In Figure 6.22, the
responses after 8 load cycles are shown. The stresses are plotted with respect to the
hoop direction of the cross section, where 0 degrees denotes the outside and 180
degrees the inside of the pipe bend. The circumferential stresses on the inside and
outside of the pipe wall show a large jump near the outside of the pipe bend (cross-
sectional angle between 20 and 25 degrees). The discontinuous response is, to a
smaller degree, already observed in Figures 5.13 to 5.15. It is most likely an effect of
the element because the response on integration point level is smooth, as shown in
Figures 6.24 to 6.27.
-240
-180
-120
-60
0
60
120
180
240
0 15 30 45 60 75 90 105 120 135 150 165 180anglest
ress
[Mpa
]
circumferential stress, inside
circumferential stress, outside
longitudinal stress, outside
Figure 6.22 Stresses, at inside and outside of the pipe wall, after 8 load cycles
In Figure 6.23 the longitudinal strain is shown with respect to the circumferential
strain. Both are on the inside of the pipe wall at 97.1 degrees of the cross section. This
angle corresponds with the maximum response as shown in Figure 6.22. The figure
demonstrates that, on the inside of the pipe wall, the strains in the circumferential
direction are significantly bigger than the strains in longitudinal direction. Figures
(extrados)
(intrados)
97.1°
115.1°
96 CHAPTER 6
6.24 and 6.25 show the circumferential and longitudinal stresses on both sides of the
pipe wall at the angle of 97.1°. The response is plotted versus the circumferential
strain on the inside of the pipe wall. The angle corresponds with the location of the
integration point on the circumference of the pipe element.
-0.003
-0.0025
-0.002
-0.0015
-0.001
-0.0005
0-0.014-0.012-0.01-0.008-0.006-0.004-0.0020
circumferential strain, inside
long
itudi
nal s
trai
n, in
side
angle = 97.1 degr.
Figure 6.23 Circumferential vs. longitudinal strain at inside of the pipe wall
-300
-200
-100
0
100
200
300-0.014-0.012-0.01-0.008-0.006-0.004-0.0020
stre
ss [M
pa],
insi
de
circumferential stress, angle = 97.1 degr.longitudinal stress, angle = 97.1 degr.
Figure 6.24 Stresses at inside of the pipe wall, at 97.1° in the circumference
CYCLIC MODEL 97
-250
-200
-150
-100
-50
0
50
100
150
200
250
300-0.014-0.012-0.01-0.008-0.006-0.004-0.0020
,
stre
ss [M
pa],
outs
ide
circumferential stress, angle = 97.1 degr.longitudinal stress, angle = 97.1 degr.
Figure 6.25 Stresses at outside of the pipe wall, at 97.1° in the circumference
Figures 6.26 and 6.27 show the circumferential and longitudinal stresses on both sides
of the pipe wall at the angle of 115.1°. The response is plotted versus the
circumferential strain on the inside of the pipe wall.
-250
-200
-150
-100
-50
0
50
100
150
200-0.0025-0.002-0.0015-0.001-0.00050
stre
ss [M
pa],
insi
de
circumferential stress, angle = 115.1 degr.longitudinal stress, angle = 115.1 degr.
Figure 6.26 Stresses at inside of the pipe wall, at 115.1° in the circumference
98 CHAPTER 6
-200
-150
-100
-50
0
50
100
150-0.0025-0.002-0.0015-0.001-0.00050
stre
ss [M
pa],
outs
ide
circumferential stress, angle = 115.1 degr.longitudinal stress, angle = 115.1 degr.
Figure 6.27 Stresses at outside of the pipe wall, at 115.1° in the circumference
Hardly any hardening or softening is observed in this numerical example. As already
shown in Figure 6.22, the response depends on the location of the integration point.
The peak stresses are at the cross-sectional angle of approximately 97.1 degrees. The
response at integration point level is continuous.
Chapter 7
GENERALITY OF THE CYCLIC MODEL
7.1 Introduction
So far, the cyclic model is applied to steel grades. In fact the model is of a generality,
which makes it valuable for other materials as well. To demonstrate, here an example
is shown for a frictional asphaltic material with a non-associate flow rule. In this
application the focus will be on the response degradation of the bounding surface. The
proposed model is published in a special ASME Geotechnical publication of the
McMat conference (Swart et al., [2005]). The used notations do not match in all cases
the notations in the other chapters. The Desai yield function, as proposed by Desai in
the context of the hierarchical approach, is utilized to model the monotonic
mechanical response of asphaltic material [1980]. One attractive feature of this
particular surface is that it includes most of the currently common used plasticity
models as special cases. Like the Gurson model, the surface is continuous and hence
avoids the problems of multisurface models. The chosen form of the model yield
function is given by:
( )n m
2 1 12
a aa
J I R I Rf , 0p pp
⎡ ⎤⎛ ⎞ ⎛ ⎞+ +⎢σ α = − −α ⋅ + γ ⋅ =⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
⎥ , (7.1)
where and 1I 2J are the first and second stress invariants respectively, , and
are material parameters.
,α γ m
n ap is the atmospheric pressure. Parameter m controls the
nonlinearity of the ultimate surface. An extensive elaboration on the parameters is
shown by Scarpas [2004].
100 CHAPTER 7
In the theory of plasticity, non-associated flow rules are commonly used for plasticity
modeling of frictional material. In the hierarchical approach, the potential surface is
given by:
( )n m
2 1 1Q2
a aa
J I R I RQ ,p pp
⎡ ⎤⎛ ⎞ ⎛ ⎞+ +⎢σ α = − −α ⋅ + γ ⋅⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
⎥
)v
, (7.2)
in which . Parameter ( )(Q c 0 1α = α + κ α −α −χ 0α is the value of α at the
initiation of non-associativeness, cκ is the non-associative material parameter, vχ
controls the contribution of volumetric plastic deformation to the expansion of the
potential surface. All parameters in equations 7.1 and 7.2 are experimentally
determinable. Specific forms for the hardening/softening parameter , as well as the
numerical implementation, can be found in Liu et al. [2004].
α
The yield surface (Desai) acts as a bounding surface to the loading surface, which is
formulated with the same shape parameters as the bounding surface, Figure 7.1.
Initially its size and origin coincide with those of the surface, i.e. f
( ) (g 0g , f ,σ α = σ α ) with σ = σ−χ . The loading surface moves within f in the
direction of the stress increment via kinematic hardening.
Figure 7.1 Schematic of 2-surface model in p-q space
As shown in equation 6.4, the fictitious plastic strain increment pcΔε is evaluated via:
( )( )*Q
pc p1 ω
σΔ = − δ Δε
( ), 0gg σ α =
( )f , 0σ α =
χ
2J
1I
ε , (7.3)
where ( is the plastic strain increment at the potential surface Q . )*Q
pσ
Δε 0=
GENERALITY OF THE CYCLIC MODEL 101
The state of stress can be defined by the intersection of the projection of the
elastic stress increment onto the potential surface, as shown in Figure 7.2.
*Qσ
Figure 7.2 Projection onto the potential surface
2σ
1σ
Q 0= *Q
Q
σ
∂⎛ ⎞⎜ ⎟∂σ⎝ ⎠
7.2 Numerical implementation
7.2.1 Constitutive model
The numerical implementation of the Desai yield surface is based on the same
numerical algorithm as discussed in chapter 5. This was initially proposed by Aravas
[1987] for pressure-dependent plasticity models and is based on the backward Euler
concept.
Integration of Eq. (2.19) yields:
pij
ij
ijij
Q
s1 Q Q 33 p q 2 q
τ+Δτ
τ+Δτ
τ+Δτ τ+Δτ
⎛ ⎞∂Δε = Δλ ⋅⎜ ⎟⎜ ⎟∂σ⎝ ⎠
⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂= Δλ ⋅ − ⋅δ + ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
(7.4)
The increment of plastic strain t t p+Δ Δε can be expressed in terms of volumetric and
deviatoric components as:
ijpij p ij q
s1 33 2
τ+Δτ τ+Δτ τ+Δτ
qτ+Δτ
⎛ ⎞Δε = Δε ⋅δ + Δε ⋅⎜ ⎟
⎝ ⎠, (7.5)
where pQp τ+Δτ
⎛ ⎞∂Δε = −Δλ⎜ ⎟∂⎝ ⎠
and qQq τ+Δτ
⎛ ⎞∂Δε = Δλ⎜ ⎟∂⎝ ⎠
(7.6)
Elimination of gives: Δλ
102 CHAPTER 7
p qQ Q 0q pτ+Δτ τ+Δτ
⎛ ⎞ ⎛ ⎞∂ ∂Δε + Δε =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
(7.7)
Equations (7.7) and (7.2), constitute a nonlinear algebraic system in terms of pΔε and
, which are chosen as the primary unknowns. The equations are solved by means
of a Newton-Raphson iteration process at constitutive law level. During the iterative
procedure, the stress is corrected along the hydrostatic and the deviatoric axes
qΔε
p and
. The stresses are finally updated via: q
ep q
3K 2G2 q
τ+Δτ τ+Δτ τ+Δτ
τ+Δτ
⎛ ⎞− Δε ⋅ − Δε ⋅⎜ ⎟
⎝ ⎠
sIσ = σ . (7.8)
7.2.2 Cyclic model
The truncated Taylor expansion of ( )f ,ασ about the state of stress at the projection
point * is: σ
( ) ( )* *
T* *f ff fτ+Δτ τ+Δτ τ+Δτ
σ σ
∂ ∂⎛ ⎞ ⎛ ⎞≈ + − + α − α =⎜ ⎟ ⎜ ⎟∂ ∂κ⎝ ⎠ ⎝ ⎠σ σ
σ* 0 (7.9)
The change of the yield condition from state * to state τ+ Δτ must be zero. This is
the classic consistency condition of Prager, which on account of Eq. (6.6) becomes:
* *
Te * ef Q fF 0
σ σ
⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ = Δ −Δλ + Δα =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂α⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠C Cε
σ σ *σ
(7.10)
in which is some measure of hardening/softening. αp
p∂α
Δα = ⋅Δε∂ε
(7.11)
Substituting this in Eq. (7.10) and solving in terms of Δλ :
*
* *
Te
*T
ep
f
f Q fσ
σ σ
∂⎛ ⎞ ⋅ ⋅Δ⎜ ⎟∂σ⎝ ⎠Δλ =∂ ∂ ∂ ∂α ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ ⋅ − ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂α ∂∂ε⎝ ⎠ ⎝ ⎠ ⎝ ⎠
C
C
ε
σ σ *
Q
σσ
(7.12)
Hence, ( in Eq. (6.6) can be computed as: )*Q
pσ
Δε
( )**Q
p * Qσ
σ
∂⎛ ⎞Δ = Δλ ⎜ ⎟∂σ⎝ ⎠ε (7.13)
GENERALITY OF THE CYCLIC MODEL 103
7.3 Numerical example
Utilization of the above proposed model is illustrated in this section for the case of an
asphaltic material subjected to uniaxial cyclic compression. The actual model
parameters are evaluated on the basis of an extensive experimental investigation
carried out at the University of Nottingham. The parameters of the monotonic model
are: , and E 220 MPa= 0.35ν = ap 0.1 MPa= − . All other parameters are functions
of the strain rate and temperature, as shown by Dunhill [2002]. The parameters of the
cyclic model, as discussed in 6.2.1 are: 1 1.0ω = , 0.43∞ω = and H 2.0= .
The stress-strain response of the material with increasing number of cycles is shown
in Figure 7.3. The response of the model is compared to the experimental monotonic
response of the asphaltic material.
Figure 7.3 Stress-strain relation numerical example
This response and the material characteristics are used for the bounding surface. The
cyclic load is applied with a maximum stress of -3.55 MPa. As typically observed in
laboratory tests with constant stress, the model allows for a gradual decrease of the
amount of energy dissipated per cycle and a smooth transition to the steady state
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0-0.1-0.08-0.06-0.04-0.020
strain (%)
stre
ss M
Pa
experimentmonotoniccyclic, max 3.55 MPa
104 CHAPTER 7
response indicated as phase II in Figure 7.4. Due to softening of the bounding surface
the relative distance between the surfaces decreases. Due to this effect, the width of
the cycles increases in phase III.
Figure 7.4 Schematic of permanent deformation development
The proposed cyclic model exhibits expected phenomena for the chosen frictional
material. Needed calibration is beyond the goal of this Chapter and scope of the
thesis.
d
N cycles
II I III
fN
Chapter 8
CONCLUSIONS
The objective of this research project is the development of a finite element model for
the prediction of material degradation in pipe components under repeated (cyclic)
loads. To accomplish this goal finite element formulations of a classical shell element
and a tube element are implemented in new Finite Element Codes. To model the
cyclic material degradation a two-surface model with the well known Gurson-
Tvergaard-Needleman model acting as bounding surface is developed.
Element formulation
As shell element preference is given to the Heterosis element. In general it can be
observed that this element performs better than the Lagrangian and the Serendipity
element, but in the selected benchmark tests the differences are minor. The influence
of the used integration scheme is large. For comparison with the tube element the
uniform 2x2 and selective reduced integrated elements are used.
The tube element is designed to specifically simulate the response of pipe lines.
Therefore, it has less integration points than a pipe model of conventional shell
elements. Even though the amount of degrees of freedom per integration point is large
it is less expensive in computational time. For an accurate formulation of the tube
element, it is essential to incorporate the out-of-plane deformation, as shown in
paragraph 4.6.4.
106 CHAPTER 8
Constitutive models
The plasticity driven damage formulation as originally proposed by Gurson for
monotonic loading is an efficient and accurate model for the considered geometry.
In general, for a three dimensional problem, we need to solve for all 6 independent
stress components. Aravas [1987] presented an efficient backward Euler method in
which the system can be solved with the volumetric and deviatoric stress components.
The Gurson model consists of less material parameters than other constitutive models
which describe the nucleation, growth and coalescence of voids. Nevertheless it is
difficult to tell what physical mechanism is responsible for the observed softening.
Modelling cyclic degradation by the proposed two-surface material model, the
proposed concept of a loading surface inside a yield surface, has proven to be
successful. It facilitates modelling of materials that harden and materials that soften
during repeated loading. The model is very suitable to describe the response
degradation.
The used approach is not limited to steel grades, but can also be used for yield
surfaces of other materials as shown in chapter 7.
Appendix A
θ-MATRIX SHELL ELEMENT
As mentioned in § 3.2.3, the directional cosines matrix θ determines the relation
between the local and the global coordinate system:
11 12 13
21 22 23
31 32 33
θ θ θ⎡ ⎤⎢ ⎥= θ θ θ⎢ ⎥⎢ ⎥θ θ θ⎣ ⎦
θ .
The axis 3ζ is used for purposes of invoking the plane stress hypothesis and
perpendicular to axes 1ζ and 2ζ . The direction 3ξ
v , normal to 1ξ
v (Eq. 3.4) and 2ξ
v
(Eq. 3.5), is
( )
( )
( )2
1
1
2
2
2
1
1
3
3ξ
2
3
1
1
2
1
1
3
3
2ξ
2
2
1
3
2
3
1
2
3
1ξ
XXXXX3v
XXXXX2v
XXXXX1v
3
3
3
ξξξξξ
ξξξξξ
ξξξξξ
∂∂
∂∂
−∂∂
∂∂
=∂∂
=
∂∂
∂∂
−∂∂
∂∂
=∂∂
=
∂∂
∂∂
−∂∂
∂∂
=∂∂
=
With the tensors i
vξ the matrix is composed: V
1 1 1
1 2 311 12 13
2 2 221 22 23
1 2 331 32 33
3 3 3
1 2 3
X X X
V V VX X XV V V
V V VX X X
⎡ ⎤∂ ∂ ∂⎢ ⎥∂ξ ∂ξ ∂ξ⎢ ⎥⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥= = ⎢ ⎥⎢ ⎥ ∂ξ ∂ξ ∂ξ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥∂ ∂ ∂⎢ ⎥∂ξ ∂ξ ∂ξ⎢ ⎥⎣ ⎦
V
108 APPENDIX A
When the axes and are not tangent, vector 1ξ 2ξ 2ξv is also not tangent to . As
shown in Figure A.1, the orthonormal base vector
1ξv
2ζ does not have to be equal to
2ξv . After determination of , the following transformations are therefore
performed:
V
11 33
21
31 13
V VV 0.0V V
=
== −
If 1011V 1.0 10−< ⋅ and 10
31V 1.0 10−< ⋅ then 11 23V V= −
12 23 31 33 21
22 33 11 13 31
32 13 21 23 11
V V V V VV V V V VV V V V V
= −
= −
= −
1ξ
1ζ2ξ
3ζ
2ζ
Figure A.1 Orthonormal base vectors local coordinate system
The matrix contains the normalized vectors of V : θ
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
1311 123 3 3
i1 i1 i2 i2 i3 i3i 1 i 1 i 1
2321 223 3 3
i1 i1 i2 i2 i3 i3i 1 i 1 i 1
31 32 333 3 3
i1 i1 i2 i2 i3 i3i 1 i 1 i 1
VV V
V V V V V V
VV V
V V V V V V
V V V
V V V V V V
= = =
= = =
= = =
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⋅ ⋅⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⋅ ⋅⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⋅ ⋅⎢ ⎥⎣ ⎦
∑ ∑ ∑
∑ ∑ ∑
∑ ∑ ∑
θ
⋅
⋅
⋅
Appendix B
NEWTON EQUATIONS CONSTITUTIVE MODEL
The magnitudes of the volumetric pΔε and the deviatoric qΔε equivalent plastic
strain increments can be computed on the basis of the Newton-Raphson iterative
process. The terms involved in the solution of the equations for the three-dimensional
formulation (§ 5.4.1) are given in this appendix. The equations can be written as:
p11 12 1
q21 22 2
Y Y z
Y Y z
∂Δε⎡ ⎤ ⎡⎡ ⎤=⎢ ⎥ ⎢⎢ ⎥
∂Δε⎢ ⎥⎢ ⎥ ⎢⎣ ⎦⎣ ⎦ ⎣
⎤⎥⎥⎦
,
where the right hand terms are
( )
1 p
2
f fzq p
z f .
∂ ∂= − Δε − Δε
∂ ∂
= − σ% %
%
q
The terms in the matrix are
11 p qp p
f fYq p
⎛ ⎞ ⎛∂ ∂ ∂ ∂= Δε + Δε⎜ ⎟ ⎜∂Δε ∂ ∂Δε ∂⎝ ⎠ ⎝% %
⎞⎟⎠
If we replace fq∂∂%
with and 1w fp∂∂%
with we get 2w
*1 2 2
11 p q *2
p p p p
w w w wf fYq pf
⎛ ⎞ ⎛∂ ∂ ∂ ∂∂ ∂σ ∂ ∂σ= + Δε + Δε + +⎜ ⎟ ⎜⎜ ⎟ ⎜∂ ∂σ ∂Δε ∂Δε ∂σ ∂Δε ∂ ∂Δε∂⎝ ⎠ ⎝
%
% %
p ⎞∂⎟⎟⎠
12 p qq q
f fYq p
⎛ ⎞ ⎛∂ ∂ ∂ ∂= Δε + Δε⎜ ⎟ ⎜∂Δε ∂ ∂Δε ∂⎝ ⎠ ⎝% %
⎞⎟⎠
*1 1 2 2
12 p q *q q q
w w w wq f fYq p f
⎛ ⎞ ⎛∂ ∂ ∂ ∂∂ ∂σ ∂ ∂ ∂= Δε + + + Δε +⎜ ⎟ ⎜⎜ ⎟ ⎜∂ ∂Δε ∂σ ∂Δε ∂ ∂Δε ∂σ ∂Δε∂⎝ ⎠ ⎝
%
% % q
⎞σ⎟⎟⎠
110 APPENDIX B
*
21 *p p p
f p f f fYp f∂ ∂ ∂ ∂ ∂ ∂σ
= + +∂ ∂Δε ∂Δε ∂σ ∂Δε∂
%
%
*
22 *q q
f q f f fYq f∂ ∂ ∂ ∂ ∂ ∂σ
= + +∂ ∂Δε ∂Δε ∂σ ∂Δε∂
%
% q
where
p
p K∂=
∂Δε%
q
q 3G∂= −
∂Δε%
1 2f 2qwq∂
= =∂ σ
%
% and
* 21 2
2
q p33q q f sinhf 2wp
⎛ ⎞− −⎜ ⎟∂ σ⎝ ⎠= =∂ σ
%
%
* 22 2 1
3 2
q p33q f pq sinhf 2q 2
⎛ ⎞−⎜ ⎟∂ − σ⎝ ⎠= +∂σ σ σ
%%
%
*21 3*
q pf 32q cosh 2q f2f
∂ ⎛ ⎞= − −⎜ ⎟σ∂ ⎝ ⎠
%
1w f 2q q q
⎛ ⎞∂ ∂ ∂= =⎜ ⎟∂ ∂ ∂⎝ ⎠% % % σ
13
w f 4q
⎛ ⎞∂ ∂ ∂ −= =⎜ ⎟∂σ ∂σ ∂ σ⎝ ⎠
%
%
q
*
22 21 2 2
w qf 9 f 3q q coshp p p 2 2
⎛ ⎞∂ ∂ ∂ ⎛ ⎞= = − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ σσ ⎝ ⎠⎝ ⎠
%
% % %
p
* *
22 21 2 1 22 3
w q pf f 3 9 f 33q q sinh q q p coshp 2 2
⎛ ⎞∂ ∂ ∂ ⎛ ⎞ ⎛ ⎞= = − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂σ ∂σ ∂ σ σσ σ⎝ ⎠ ⎝ ⎠⎝ ⎠
% %
%2q p
2
21 2
2* *
q p33q q sinhw f 2
pf f
⎛ ⎞− −⎜ ⎟⎛ ⎞∂ ∂ ∂ σ⎝ ⎠= =⎜ ⎟∂ σ∂ ∂ ⎝ ⎠
%
%
The terms p
∂σ∂Δε
and q
∂σ∂Δε
are determined with respect to Eq. 5.3 and formulated as:
p p
pp p p
H∂σ ∂σ ∂Δε ∂Δε= =
∂Δε ∂Δε ∂Δε∂Δε
APPENDIX B 111
p p
pq q
H∂σ ∂σ ∂Δε ∂Δε= =
∂Δε ∂Δε ∂Δε∂Δε q
The derivations with respect to the nucleation model (Eq. 5.2) are formulated as:
2p p
N N N Np 2 2
NN NN 2
fA 1exp As 2 ss sπ
⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ε − ε ε − ε ε − ε∂ ⎢ ⎥= − − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ε ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
p
p
pp p
A A∂ ∂ ∂ε=
∂Δε ∂Δε∂ε
p
pq q
A A∂ ∂ ∂ε=
∂Δε ∂Δε∂ε
The void volume fraction can be written as:
* pp*
p
f Af
1
ττ+Δτ + Δε + Δε
=+ Δε
( )
( )
pp
* p*p pp
2p p pp
p pp
p* pp pp
2p pp
A Af Af 1
1 11
A Af A1
1 11
τ
τ
⎛ ⎞∂ ∂ΔεΔε +⎜ ⎟⎜ ⎟∂Δε ∂Δε+ Δε + Δε∂ ⎝ ⎠= − +
∂Δε + Δε + Δε+ Δε
⎛ ⎞∂ ∂Δε ∂ΔεΔε +⎜ ⎟⎜ ⎟∂Δε ∂Δε∂ε+ Δε + Δε ⎝ ⎠= − +
+ Δε + Δε+ Δε
pp
*q q
q p
A Af
1
⎛ ⎞∂ ∂ΔεΔε +⎜ ⎟⎜ ⎟∂Δε ∂Δε∂ ⎝ ⎠=
∂Δε + Δε
The increment of microscopic equivalent plastic strain can be written as
( )p qp
*
p q
1 f
τ+Δτ τ+Δτ− Δε + ΔεΔε =
− σ
% %
112 APPENDIX B
( )( ) ( )
( )( )
( )( )
( )( )
* pp q pp
2 2 ** pp p
pppp q p q
2 * 2* p
p q p Kf A11 1 f11 f
A Ap q p q H1
1 1 f1 f
τ⎧ ⎫⎧ ⎫− Δε + Δε − − Δε+ Δε + Δε⎪ ⎪⎪ ⎪− − +⎨ ⎬⎨ ⎬+ Δε − σ+ Δε⎪ ⎪⎪ ⎪− σ∂Δε ⎩ ⎭⎩ ⎭=∂∂Δε ⎧ ⎫⎧ ⎫ Δε + ⎧ ⎫− Δε + Δε − Δε + Δε⎪ ⎪⎪ ⎪⎪ ⎪ ⎪∂ε+ +⎨ ⎬⎨ ⎬ ⎨+ Δε − σ⎪ ⎪⎪ ⎪ ⎪− σ ⎩ ⎭⎩ ⎭⎪ ⎪⎩ ⎭
% % %
% % % % ⎪⎬⎪
( )( )
( )( )
( )( )
q*p
pqpp q p q
2 * 2* p
q 3G
1 f
A Ap q p q H1
1 1 f1 f
− Δε
− σ∂Δε=
∂∂Δε ⎧ ⎫⎧ ⎫ Δε + ⎧ ⎫− Δε + Δε − Δε + Δε⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪∂ε+ +⎨ ⎬⎨ ⎬ ⎨ ⎬+ Δε − σ⎪ ⎪⎪ ⎪ ⎪ ⎪− σ ⎩ ⎭⎩ ⎭⎪ ⎪⎩ ⎭
%
% % % %
After implementation of p
p
∂Δε∂Δε
and p
q
∂Δε∂Δε
follows:
( )
( )( ) ( )
( )( )
( )( )
* p*p
2p p p
p * pp p q pp2 2 **p p p
ppp q
2* p
f Af 11 1
A A p q p Kf A11 1 1 f11 f
A Ap q1
11 f
τ
τ
+ Δε + Δε∂= − +
∂Δε + Δε + Δε
∂⎛ ⎞⎛ ⎞⎧ ⎫⎧ ⎫Δε + − Δε + Δε − − Δε⎜ ⎟ + Δε + Δε⎜ ⎟⎪ ⎪⎪ ⎪∂ε − − +⎜ ⎟ ⎨ ⎬⎨ ⎬⎜ ⎟+ Δε + Δε − σ⎜ ⎟ + Δε⎪ ⎪⎪ ⎪− σ⎜ ⎟⎜ ⎟ ⎩ ⎭⎩ ⎭⎝ ⎠⎝ ⎠∂⎧⎧ ⎫ Δε +− Δε + Δε ⎪⎪ ⎪⎪ ∂ε+ ⎨ ⎬⎨ + Δε⎪ ⎪⎪− σ⎩ ⎭⎩
% % %
% % ( )( )
p q* 2
p q H
1 f
⎫ ⎧ ⎫− Δε + Δε⎪⎪ ⎪ ⎪+⎬ ⎨ ⎬− σ⎪ ⎪ ⎪⎩ ⎭⎪ ⎪⎭
% %
( )( )
( )( )
( )( )
pp q
*p*
pqpp q p q
2 * 2* p
A A q 3G1 1 f
fA Ap q p q H
11 1 f1 f
∂⎧ ⎫Δε + ⎧ ⎫− Δε⎪ ⎪⎪ ⎪⎪ ⎪∂ε⎨ ⎬⎨ ⎬+ Δε − σ⎪ ⎪⎪ ⎪⎩ ⎭∂ ⎪ ⎪⎩ ⎭=∂∂Δε ⎧ ⎫⎧ ⎫ Δε + ⎧ ⎫− Δε + Δε − Δε + Δε⎪ ⎪⎪ ⎪⎪ ⎪ ⎪∂ε+ +⎨ ⎬⎨ ⎬ ⎨+ Δε − σ⎪ ⎪⎪ ⎪ ⎪− σ ⎩ ⎭⎩ ⎭⎪ ⎪⎩ ⎭
%
% % % % ⎪⎬⎪
Appendix C
NEWTON EQUATIONS PLANE STRESS
The magnitudes of the equivalent plastic strain increments pΔε , qΔε and 3ζ
Δε can be
computed on the basis of the Newton-Raphson iterative process. The equations can be
written as:
.
3
p11 12 13 1
21 22 23 q 2
31 32 33 3
Y Y Y zY Y Y zY Y Y z
ζ
⎡ ⎤∂Δε⎡ ⎤ ⎢ ⎥⎢ ⎥ ∂Δε =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ∂Δε⎣ ⎦ ⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
The terms involved in the solution of the equations for the three-dimensional
formulation are given in appendix B. The additional terms involved in the solution of
the Newton equations for the plane stress formulation (§ 5.4.2) are given in this
appendix. The additional right hand term is
( ) 33 3
e3 q
4z q 3G p s G3
ζζ ζ
⎛ ⎞= −σ = − + Δε + + Δε⎜ ⎟⎝ ⎠
% % q%
The additional terms in the matrix are
3
3 3
13 p q
1 2p q
f fYq p
w wq pq p
ζ
ζ ζ
⎛ ⎞∂ ∂ ∂= Δε + Δε⎜ ⎟∂Δε ∂ ∂⎝ ⎠
⎛ ⎞ ⎛∂ ∂∂ ∂= Δε + Δε⎜ ⎟ ⎜⎜ ⎟ ⎜∂Δε ∂ ∂Δε ∂⎝ ⎠ ⎝
% %
% %
% %
⎞⎟⎟⎠
3 3 3
*
23 *f p f q f f fYp q f
3ζ ζ ζ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂σ= + + +∂ ∂Δε ∂ ∂Δε ∂Δε ∂σ ∂Δε∂
% %
% % ζ
( )31 qY K q 3G= + Δε%
33
e 232Y 3Gs 4Gζ
ζ= + Δε
114 APPENDIX C
( ) 33
3 3 3
e33 q
q p 4 4Y p q 3G Gq s G3 3
ζζ
q
ζ ζ ζ
∂ ∂ ∂⎛ ⎞= + + Δε − − + Δε⎜ ⎟∂Δε ∂Δε ∂Δε⎝ ⎠
% %% % %
%,
where
3
p Kζ
∂= −
∂Δε%
3
3
3
e 2
q
3Gs 4Gqq 3G
ζζ
ζ
+ Δε∂=
∂Δε + Δε%
%
3 3 3
p p
p Hζ ζ ζ
∂σ ∂σ ∂Δε ∂Δε= =
∂Δε ∂Δε ∂Δε∂Δε
3 3
p
pA A
ζ ζ
∂ ∂ ∂ε=
∂Δε ∂Δε∂ε
3 3
3
pp
*
p
A Af
1ζ ζ
ζ
⎛ ⎞∂ ∂ΔεΔε +⎜ ⎟⎜ ⎟∂Δε ∂Δε∂ ⎝ ⎠=∂Δε + Δε
( )( )
( )( )
( )
3 3
3
p q
*p
ppp q p q
2 * 2* p p
p q
1 f
Ap q p q HA1
1 1 1 f1 f
ζ ζ
ζ
⎛ ⎞∂ ∂− Δε + Δε⎜ ⎟⎜ ⎟∂Δε ∂Δε⎝ ⎠
− σ∂Δε=
∂∂Δε ⎧ ⎫⎧ ⎫ Δε ⎧ ⎫− Δε + Δε − Δε + Δε⎪ ⎪⎪ ⎪⎪ ⎪ ⎪∂ε+ + +⎨ ⎬⎨ ⎬ ⎨+ Δε + Δε − σ⎪ ⎪⎪ ⎪ ⎪− σ ⎩ ⎭⎩ ⎭⎪ ⎪⎩ ⎭
% %
% % % % ⎪⎬⎪
Appendix D
NEWTON EQUATIONS CONSTITUTIVE MODEL TUBE ELEMENT
The terms involved in the solution of the Newton equations for the plane stress
formulation with respect to a curvilinear coordinate system (§ 5.7) are given in this
appendix. These terms are similar to the terms in appendix D, but with respect to the
curvilinear coordinate system. The additional right hand term is
( ) 33 e33 33 333 33 q 33
4z q 3G pg s G g3
⎛ ⎞= −σ = − + Δε + + Δε⎜ ⎟⎝ ⎠
% % g q%
The additional terms in the matrix are
13 p q
33
1 2p q
33 33
f fYq p
w wq pq p
⎛ ⎞∂ ∂ ∂= Δε + Δε⎜ ⎟∂Δε ∂ ∂⎝ ⎠
⎛ ⎞ ⎛∂ ∂∂ ∂= Δε + Δε⎜ ⎟ ⎜∂Δε ∂ ∂Δε ∂⎝ ⎠ ⎝
% %
% %
% %
⎞⎟⎠
*
23 *33 33 33 33
f p f q f f fYp q f∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂σ
= + + +∂ ∂Δε ∂ ∂Δε ∂Δε ∂σ ∂Δε∂
% %
% %
( ) 3331 qY K q 3G g= + Δε%
e33 2 33 3332 33Y 3Gs 4G g g= + Δε
( )33 33 33 33 e33 33 3333 q 33
33 33 33
q p 4 4Y pg q 3G g Gqg g s G g g3 3
∂ ∂ ⎛ ⎞= + + Δε − − + Δε⎜ ⎟∂Δε ∂Δε ∂Δε⎝ ⎠
% %% % %
q∂%
where
33
33
p Kg∂= −
∂Δε%
e33 2 33 33
33
33 q
3Gs 4G g gqq 3G+ Δε∂
=∂Δε + Δε
%
%
116 APPENDIX D
p p
p33 33 33
H∂σ ∂σ ∂Δε ∂Δε= =
∂Δε ∂Δε ∂Δε∂Δε
p
p33 33
A A∂ ∂ ∂ε=
∂Δε ∂Δε∂ε
pp
*33 33
33 p
A Af
1
⎛ ⎞∂ ∂ΔεΔε +⎜ ⎟∂Δε ∂Δε∂ ⎝ ⎠=∂Δε + Δε
( )( )
( )( )
( )
p q33 33
*p
p33pp q p q
2 * 2* p p
p q
1 f
Ap q p q HA1
1 1 1 f1 f
⎛ ⎞∂ ∂− Δε + Δε⎜ ⎟∂Δε ∂Δε⎝ ⎠
− σ∂Δε=
∂∂Δε ⎧ ⎫⎧ ⎫ Δε ⎧ ⎫− Δε + Δε − Δε + Δε⎪ ⎪⎪ ⎪⎪ ⎪ ⎪∂ε+ + +⎨ ⎬⎨ ⎬ ⎨+ Δε + Δε − σ⎪ ⎪⎪ ⎪ ⎪− σ ⎩ ⎭⎩ ⎭⎪ ⎪⎩ ⎭
% %
% % % % ⎪⎬⎪
Appendix E
EQUATIONS CYCLIC MODEL
The terms involved in the numerical implementation for the three-dimensional system
(Eq. 6.16) are given in this appendix. For plane stress elements it is assumed that the
stress, strain and back stress perpendicular to the surface are zero.
*
** * 2
2 1* 2
3 2
q p33q f pq sinh2f 2 q
σ
⎛ ⎞−⎜ ⎟σ∂ −⎛ ⎞ ⎝ ⎠= +⎜ ⎟∂σ σ σ⎝ ⎠
%%
%
* ** *
f f p f qp qσ σσ σ
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠σ σ% %
% %% % *σσ%
* ** *
f f p f qp qσ σσ σ
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠α α% %
% % *σα
where
*
*
2f 2q σ
⎛ ⎞∂=⎜ ⎟∂
qσ⎝ ⎠
%
%
*
** 2
1 2q p33q q f sinh
2fp σ
⎛ ⎞− −⎜ ⎟σ⎛ ⎞∂ ⎝ ⎠=⎜ ⎟∂ σ⎝ ⎠
%
%
*ij
i
p 13σ
⎛ ⎞∂= − δ⎜ ⎟∂σ⎝ ⎠
%
%
*ij
i
p 13σ
⎛ ⎞∂= δ⎜ ⎟∂α⎝ ⎠
%
and
118 APPENDIX E
*
*
i*
i 1,2,3
i*
i 4,5,6
sq 32 q
sq 3q
= σ
= σ
⎛ ⎞∂=⎜ ⎟⎜ ⎟∂σ⎝ ⎠
⎛ ⎞∂=⎜ ⎟⎜ ⎟∂σ⎝ ⎠
%%
% %
%%
% %
*
*
i*
i 1,2,3
i*
i 4,5,6
sq 32 q
sq 3q
= σ
= σ
⎛ ⎞∂= −⎜ ⎟⎜ ⎟∂α⎝ ⎠
⎛ ⎞∂= −⎜ ⎟⎜ ⎟∂α⎝ ⎠
%%
%
%%
%
NOTATIONS
Lowercase roman symbols
a deviatoric part of back stress
b surface load vector
c correction factor for transverse shear strains, shell element
d nodal displacements vector
e unit vector along coordinate axis
f yield function and plastic potential *f void volume fraction, Gurson model
Nf initial void volume fraction
critf critical void volume fraction
ff final void volume fraction
uf ultimate void volume fraction
ig covariant base vector ig contravariant base vector
k element node
n outward normal to surface
o unit length in deformed configuration
p hydrostatic stress
intp internal pressure
q Von Mises effective stress
1q , material dependent parameters, Gurson model 2q , 3q
r position of the cross-sectional reference line
r radius of pipe element
s deviatoric stress tensor
Ns standard deviation in void nucleation function
bt boundary traction
120 NOTATIONS
t element thickness
kt local element thickness, shell element
u displacement of any point in the structure after deformation
pu prescribed displacements refw deformation
x location of any point in the structure after deformation
kx location of nodal point in the structure after deformation
cx location of axis in the tube structure after deformation
y deviatoric part of
z tensor with Newton-Raphson equations
Uppercase roman symbols
A nodal coordinates vector
A void nucleation function
B strain-displacement transformation matrix
C Gauchy-Green deformation tensor
D material stiffness matrix
E Lagrangian-Green strain tensor
E modulus of elasticity
F deformation gradient
G shear modulus
cH material parameter cyclic model
isoH isotropic hardening parameter
kinH kinematic hardening parameter
I second order identity tensor
I moment of inertia
cJ coordinate Jacobian matrix
K stiffness matrix
K bulk modulus
L differential operator matrix
NOTATIONS 121
L length of the pipeline
M bending moment in the pipeline
N matrix of interpolation functions
O unit length in undeformed configuration
P nodal load vector
R nodal force vector
BR pipe bend radius
V volume
X initial location of any point in the structure
Y matrix for Newton-Raphson equations
Lowercase greek symbols
fα back stress tensor of the yield surface
gα back stress tensor of the loading surface
ijδ Kronecker delta
δ relative distance between current state of stress and projection point at bounding
surface
ε strain tensor
0ε vector of any initial/thermal strains eε elastic strain tensor pε plastic strain tensor pcε (fictitious) cyclic plastic strain
Nε mean strain in void nucleation function
pε volumetric, or hydrostatic, strain
qε deviatoric strain pε equivalent plastic strain
ζ local coordinate system
η material parameter, controls development of the width of the cycles
θ directional cosines matrix
pκ prescribed rotation
122 NOTATIONS
λ plastic multiplier
ν Poisson’s ratio
ξ curvilinear coordinate system
ρ mass density
σ stress tensor
σ yield stress
yσ initial yield stress
τ time
φ angle
χ nodal coordinate system
ω material parameter, controls width of cycles
ACKNOWLEDGEMENTS
The research presented in this thesis has been carried out at the Section of Structural
Mechanics of the Faculty of Civil Engineering at Delft University of Technology.
I would like to thank Professor Johan Blaauwendraad and Tom Scarpas for the
opportunity to begin this research project, Spyros Karamanos, of the University of
Thessaly, for his support with the formulation of the tube element and Ralf Peek, of
Shell RD&T, for the actual information about pipeline design.
Most of all, I want to thank Natasja Tak and my parents for their love and support.
The news we heard the 18th of October 2001 changed our lives more than we thought
possible. For more than a year our emotions changed from moments with hope into
moments with deep despair. Even though this period was very difficult you’ve given
me the encouragement to continue this project.
From 2006 on Thijmen, Nanon and Noëlle enriched our lives with their arrival! The
three of you made it an enjoyable challenge for me to finally finish this thesis….
Edwin Swart
124
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CURRICULUM VITAE
April 28th, 1972 Born in Plymouth, United Kingdom, as Auke Edwin Swart
1984 – 1991 Atheneum, Schagen, the Netherlands
1991 – 1995 Faculty of Building Engineering, Structural Mechanics
Higher Polytechnic School (B.Sc.), Amsterdam
1995 – 1998 Faculty of Civil Engineering, Structural Mechanics
University of Technology, Delft
Distinction: Cum Laude
1995 Structural Engineer at Tebodin, Beverwijk, the Netherlands
1997 – 1998 Teaching assistant, Department of Infrastructure
1998 – 1999 Representative of Delft University of Technology at
“Amadeus”, European project for evaluation of software for
road engineering
1999 – 2004 Research assistant, Faculty of Civil Engineering
Delft University of Technology, the Netherlands
2004 – 2007 Research fellow, Centre for Pavement Engineering
University of Nottingham, United Kingdom
2007 – Researcher, CORUS RD&T, IJmuiden, the Netherlands
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