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BULETINUL INSTITUTULUI POLITEHNIC DIN IAI
Tomul LVIII (LXII) Fasc. 3
CONSTRUCII DE MAINI
2012 Editura POLITEHNIUM
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BULETINUL INSTITUTULUI POLITEHNIC DIN IAI PUBLISHED BY
GHEORGHE ASACHI TECHNICAL UNIVERSITY OF IAI Editorial Office:
Bd. D. Mangeron 63, 700050, Iai, ROMANIA
Tel. 40-232-278683; Fax: 40-232-237666; e-mail:
[email protected]
Editorial Board
President: Prof. dr. eng. Ion Giurma, Member of the Academy of
Agricultural Sciences and Forest, Rector of the Gheorghe Asachi
Technical University of Iai
Editor-in-Chief: Prof. dr. eng. Carmen Teodosiu, Vice-Rector of
the Gheorghe Asachi Technical University of Iai
Honorary Editors of the Bulletin: Prof. dr. eng. Alfred Braier,
Prof. dr. eng. Hugo Rosman
Prof. dr. eng. Mihail Voicu, Corresponding Member of the
Romanian Academy, President of the Gheorghe Asachi Technical
University of Iai
Editors in Chief of the MACHINE CONSTRUCTIONS Section
Prof. dr. eng. Radu Ibnescu, Assoc. prof. dr. eng. Aristotel
Popescu Honorary Editors: Prof. dr. eng. Gheorghe Nag, Prof. dr.
eng. Cezar Oprian
Associated Editor: Assoc. prof. dr. eng. Eugen Axinte
Editorial Advisory Board
Prof.dr.eng. Dorel Leon, Gheorghe Asachi Technical University of
Iai
Prof.dr.eng. Nicuor Amariei, Gheorghe Asachi Technical
University of Iai
Assoc.prof.dr.eng. Aristomenis Antoniadis, Technical University
of Crete, Greece
Prof.dr.eng. James A. Liburdy, Oregon State University,
Corvallis, Oregon, SUA
Prof.dr.eng. Virgil Atanasiu, Gheorghe Asachi Technical
University of Iai
Prof.dr.eng.dr. h.c. Peter Lorenz, Hochschule fr Technik und
Wirtschaft, Saarbrcken, Germany
Prof.dr.eng. Petru Berce, Technical University of Cluj-Napoca
Prof.dr.eng. Ion Bostan, Technical University of Chiinu,
Prof.dr.eng. Noura -Barbu Lupulescu, University Transilvania of
Braov
Prof.dr.eng. Fabio Miani, University of Udine, Italy Republic of
Moldova Prof.dr.eng. Walter Calles, Hochschule fr Technik und
Wirtschaft des Saarlandes, Saarbrcken, Germany
Prof.dr.eng. Mircea Mihailide, Gheorghe Asachi Technical
University of Iai Prof.dr.eng. Sevasti Mitsi, Aristotle University
of
Prof.dr.eng. Doru Clrau, Gheorghe Asachi Technical University of
Iai
Thessaloniki, Salonic, Greece Prof.dr.eng. Vasile Neculiasa,
Gheorghe Asachi Technical
Prof.dr.eng. Francisco Chinesta, cole Centrale de Nantes,
France
Assoc.prof.dr.eng. Conalves Coelho, University Nova of Lisbon,
Portugal
University of Iai Prof.dr.eng. Fernando Jos Neto da Silva,
University of Aveiro, Portugal Prof.dr.eng. Dumitru Olaru, Gheorghe
Asachi Technical
Prof.dr.eng. Juan Pablo Contreras Samper, University of Cadiz,
Spain
Assoc.prof.dr.eng. Mircea Cozmnc, Gheorghe Asachi
University of Iai Prof.dr.eng. Manuel San Juan Blanco,
University of Valladolid, Spain
Technical University of Iai Prof.dr.eng. Spiridon Creu, Gheorghe
Asachi Technical
Prof.dr.eng. Loredana Santo,University Tor Vergata, Rome,
Italy
University of Iai Prof.dr.eng. Gheorghe Dumitracu, Gheorghe
Asachi
Prof.dr.eng. Cristina Siligardi, University of Modena, Italy
Prof.dr.eng. Filipe Silva, University of Minho, Portugal
Technical University of Iai Prof.dr.eng. Ctlin Fetecu,
University Dunrea de Jos of
Prof.dr.eng. Laureniu Sltineanu, Gheorghe Asachi Technical
University of Iai
Galai Prof.dr.eng. Mihai Gafianu, Gheorghe Asachi Technical
Lecturer dr.eng. Birgit Kjrside Storm, Aalborg Universitet
Esbjerg, Denmark
University of Iai Prof.dr.eng. Radu Gaiginschi, Gheorghe Asachi
Technical
Prof.dr.eng. Ezio Spessa, Politecnico di Torino, Italy
Prof.dr.eng.Roberto Teti, University Federico II, Naples, Italy
University of Iai Prof.dr.eng. Francisco Javier Santos Martin,
University of
Prof.dr.eng. Alexei Toca, Technical University of Chiinu,
Republic of Moldova
Valladolid, Spain Prof. dr. Dirk Lefeber, Vrije Universiteit
Brussels, Belgium
Prof.dr.eng. Hans-Bernhard Woyand, Bergische University
Wuppertal, Germany
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B U L E T I N U L I N S T I T U T U L U I P O L I T E H N I C D
I N I A I B U L L E T I N O F T H E P O L Y T E C H N I C I N S T I
T U T E O F I A I Tomul LVIII (LXII), Fasc. 3 2012
CONSTRUCII DE MAINI
Pag.
MOUSSA KARAMA (Frana), Comportarea barelor compozite laminate la
interfeele dintre straturi (engl., rez.
rom.)..................................................
1 ISAAC NELSON (SUA), CONSTANTIN CIOCNEL (SUA) i HEIDI
FEIGENBAUM (SUA), Curbele de reorientare pentru un aliaj
Ni50Mn28.5Ga21.5 (engl., rez.
rom.)...............................................................
19 CIPRIAN ATNSOAEI, VIOREL GOAN i PAUL DORU
BRSNESCU, Compozite din cordierit armat cu oxid de
zirconiu(engl., rez.
rom.)..........................................................................................
31
RZVAN FLORIN BARZIC, ANDREEA IRINA BARZIC i DANA-ORTANSA
DOROHOI, Comportamentul de deformre n condiii de forfecare a unor
amestecuri de hidroxipropil celuloz/L-cistin (engl., rez.
rom.).....................................................................................................
41
IONU DUMITRACU, PAUL-DORU BRSNESCU i BOGDAN LEIOIU, Studiu
comparativ privind materialele compozite armatebidirecional cu fibr
de carbon, cu matrice termorigid i termoplastic(engl., rez.
rom.)..........................................................................................
47
IONU DUMITRACU, PAUL-DORU BRSNESCU i VIOREL GOAN, Influena
matricei asupra proprietilor ale materialelor compozite armate
unidirecional cu fibr de carbon i matricetermoplastic (engl., rez.
rom.)....................................................................
53
EDUARD GHEBAN, NORINA FORNA i BOGDAN LEIOIU, Considera-ii asupra
preciziei determinrii modulului de forfecare al unui acrilat de uz
stomatologic (engl., rez.
rom.)................................................................
59
EDUARD GHEBAN, NORINA FORNA I BOGDAN LEIOIU, Considera-ii asupra
ncercrilor la traciune asupra unui material de uzstomatologic
(engl., rez.
rom.).....................................................................
67
DAN ILINCIOIU i COSMIN MIHAI MIRIOIU, Influena ipotezelor
simplificatoare asupra distribuiei tensiunilor la structurile
metalice hiperstatice (engl., rez.
rom.).....................................................................
75
S U M A R
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DAN ILINCIOIU i COSMIN MIHAI MIRIOIU, O comparaie ntre cteva
soft-uri folosite pentru studiul tensiunilor mecanice n structuri
metalice (engl., rez. rom.).. 85
ADRIAN LIVIU PARASCHIV, MARIUS GABRIEL SURU i LEANDRU-GHEORGHE
BUJOREANU, Variaia proprietilor, datorit coninu-tului de nichel i a
prelucrrii termomecanice, la feroaliajele Fe-Cr-Ni-Siutilizate
pentru pregtirea unor aliaje cu memoria formei pe baz de Fe (engl.,
rez. rom.).. 93
MARIAN MARE, CRISTINA RACU, LILIANA BUHU i ADRIAN BUHU,
Proprieti mecanice ale unor materiale compozitebiodegradabile cu
matrice polimeric (engl., rez. rom.)..............................
101
LEONID TARTAKOVSKY (Israel), VLADIMIR BAIBIKOV (Israel), MARCEL
GUTMAN (Israel), DORON POPESCU (Israel), MARC VEINBLAT(Israel) i
YORAM ZVIRIN (Israel), Compararea emisiilor poluante de la
autobuzele urbane i de la automobilele de pasageri din Israel
(engl.,
rez.rom.)............................................................................................................
109
VICTOR BRATU, ALEXANDRU BOROIU i ION TABACU, Cercetri cu privire
la mbuntirea i modernizarea sistemului de transport public de
persoane n zona preurban din oraul Piteti. (I) Studiul factorilor
determinani pentru sistemul public (engl., rez.
rom.)................................ 117
CLAUDIU BUTNARU, Influena recirculrii gazelor de evacuare asupra
arderii i emisiilor ntr-un motor HCCI (engl., rez.
rom.)........................... 127
CLAUDIU BUTNARU, Influena turaiei i a consumului de combustibil
asupra gazelor de ardere (engl., rez.
rom.).................................................. 135
VASILE CAUNII, Aspecte privind strategiile de control a
compresorului aerului condiionat (engl., rez.
rom.)...........................................................
141
COSTIN DRAGOMIR, CONSTANTIN PAN, NICULAE NEGURESCU iALEXANDRU
CERNAT, Investigaii teoretice i experimentale ale motorului cu
aprindere prin scnteie supraalimentat (engl., rez. rom.).......
151
MARIUS ATANASIU, MIHAI MARIUS PRODAN i EMIL JUGUREANU, Analiza
comparativ ntre metoda produsului planar i metodageometriei plane
pentru modelul matematic al compresorului spiral(engl., rez.
rom.)..........................................................................................
159
MIHAI ADAM, GEORGE OVIDIU RU i GHEORGHE DUMITRACU, Cercetri
experimentale privind stocarea energiei termice n materiale
cuschimbare de faz: aliaj eutectic Sn-Pb (engl., rez. rom.)...
167
MIHAI ADAM, RZVAN FLORIN BARZIC, GEORGE OVIDIU RU iGHEORGHE
DUMITRACU, Evaluarea direct a conductivitii termice pentru
materiale utilizate n sisteme de stocare a cldurii prin transformri
de faz (engl., rez.
rom.).........................................................
173
GABRIEL ALEXANDRU, GHEORGHE DUMITRACU i FLORIN GABRIEL FLOREAN,
Modelarea curgerii ntr-un compresor centrifug transonic. (I)
Influena modelrii stratului limit al curgerii (engl.,
rez.rom.)............................................................................................................
179
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GABRIEL ALEXANDRU, GHEORGHE DUMITRACU i FLORIN GABRIEL FLOREAN,
Analiza numeric a efectului Coriolis ntr-un compresor centrifugal
transsonic (engl., rez. rom.).....................................
185
GEORGE OVIDIU RU, MIHAI ADAM, IVANCU IONEL i GHEORGHE DUMITRACU,
Instalaie experimental pentru studiul transferului decldur cu
schimbare de faz n termosifoane (engl., rez. rom.). 191
GEORGE OVIDIU RU, MIHAI ADAM i GHEORGHE DUMITRACU, Investigarea
experimental a limitelor de fierbere i inundare, pentru tuburi
termice termosifon cu diferite diametre (engl., rez. rom.)
............... 197
DRAGO S. CRUDU i VASILE MERTICARU, Contribuii teoretice la
optimizarea dinamic a mecanismului de lucru a ncrctorului
frontal(engl., rez. rom.)...................
......................................................................
205
DRAGO S. CRUDU i VASILE MERTICARU, Date experimentale de referin
pentru proiectarea mecanismului de lucru de la ncrctorulfrontal
(engl., rez. rom.)
.............................................................................
219
CRISTIAN IOSIFESCU i SORINEL TOFAN, Studiul unui turn de rcire
umed n contra-curent. (I) Studiul funcionrii (engl., rez. rom.)
.............. 233
SORINEL TOFAN i CRISTIAN IOSIFESCU, Studiul unui turn de rcire
umed n contra-curent. (II) Analiza exergetic (engl., rez.
rom.)................ 239
ADRIAN I. SIMEDRU, Comparaia teoretico-experimental la
funcionarea cavitaional a turbinelor hidraulice axiale de la
hidrocentrala Porile de Fier (engl., rez.
rom.)...................................................................................
245
BOGDAN CCIUL, VIOREL POPA i TNASE PANAIT, Studiu privind
temperatura optim a sursei calde care acioneaz un sistem prin
adsorbie solar de climatizare (engl., rez. rom.)
......................................... 253
GEORGIANA BOLAT i DANIEL SUTIMAN, Studiul termodinamic al
sistemelor binare formate din toluen+alcani la temperatura de
318.15 K(engl., rez. rom.)
.........................................................................................
259
FLORIN GABRIEL FLOREAN, IONU PORUMBEL, GABRIEL ALEXANDRU i
GHEORGHE DUMITRACU, Simularea numeric i determinri experimentale
ntr-un jet turbulent de gaze evacuate dintr-o turbin cu gaze
(engl., rez.
rom.).................................................................
265
FLORIN GABRIEL FLOREAN, ALEXANDRU GABRIEL i GHEORGHE DUMITRACU,
Determinri experimentale i simulri numerice n curgeri turbulente
izotermice (engl., rez.
rom.)........................................... 271
DANIELA TASMA, TNASE PANAIT, KRISZTINA UZUNEANU iCTLIN MOCANU,
Evaluarea exergetic a gazeificrii rezidurilor din agricultur
(engl., rez.
rom.)........................................................................
277
DANIELA TASMA, TNASE PANAIT, KRISZTINA UZUNEANU iCTLIN MOCANU,
Influena coeficientului de exces de aer asupra gazeificrii biomasei
(engl., rez. rom.).
................................................. 285
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OLIVIA GIUCA i IOAN NICOAR, Utilizarea planurilor de experimente
pentru verificarea stabilitii dimensionale a nanocristalelor
(engl.,
rez.rom.).............................................................................................................
295
MARIUS CATAN i DANIELA TARNI, Modelul virtual tridimensional al
articulaiei genunchiului uman (engl., rez.
rom.)..................................... 303
RALUCA SOFRONIA, ARJANA DAVIDESCU i GEORGE SAVII,Interaciunea
os-ferstru n cadrul unui simulator haptic (engl.,
rez.rom.).............................................................................................................
309
MARIAN STANCU, DANIEL DRAGOMIR-STANCIU i CONSTANTINLUCA,
Instalaie de cogenerare de mic putere cu gazeificarea lemnului
(engl., rez.
rom.)..........................................................................................
317
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B U L E T I N U L I N S T I T U T U L U I P O L I T E H N I C D
I N I A I B U L L E T I N O F T H E P O L Y T E C H N I C I N S T I
T U T E O F I A I Tomul LVIII (LXII), Fasc. 3 2012
MACHINE CONSTRUCTION
Pp.
MOUSSA KARAMA (France), Behaviour of Laminated Composite Beam at
the Interfaces of Layers (English, Romanian
summary)............................. 1
ISAAC NELSON (USA), CONSTANTIN CIOCNEL (USA) and HEIDI
FEIGENBAUM (USA), Reorientation Curves for a Ni50Mn28.5Ga21.5Alloy
(English, Romanian
summary)..........................................................
19
CIPRIAN ATNSOAEI, VIOREL GOAN and PAUL-DORU BRSNESCU,
Particulate Cordierite - Zirconia Composites (English, Romanian
summary)
...................................................................................
31
RZVAN FLORIN BARZIC, ANDREEA IRINA BARZIC and DANA-ORTANSA
DOROHOI, Deformation Behavior under Shear Field ofSome
Hydroxypropyl Cellulose/L-Cystine Blends (English, Romanian
summary)
........................................................................
41
IONU DUMITRACU, PAUL-DORU BRSNESCU and BOGDAN LEIOIU,
Comparative Study of Cross-ply CFRP with Thermoplastic and
Thermoset Matrix (English, Romanian
summary)............................... 47
IONU DUMITRACU, PAUL-DORU BRSNESCU and VIOREL GOAN, Matrix
Influence on the Mechanical Properties of Unidirectional CFRTP
(English, Romanian summary).............. 53
EDUARD GHEBAN, NORINA FORNA and BOGDAN LEIOIU,
SomeConsiderations on Shear Modulus Precision Determination of a
Dentistry Acrylate (English, Romanian summary)..... 59
EDUARD GHEBAN, NORINA FORNA and BOGDAN LEIOIU,Considerations on
Tension Test, when Applied on a Dentistry Material(English,
Romanian
summary)....................................................................
67
DAN ILINCIOIU and COSMIN MIHAI MIRIOIU, The Influence
ofSimplifying Assumptions Over the Stress Distribution for
HyperstaticMetallic Structures (English, Romanian summary).....
75
DAN ILINCIOIU and COSMIN MIHAI MIRIOIU, A Comparison Between
Some Software Used for Metallic Structures Stress Studies (English,
Romanian
summary)....................................................................................
85
ADRIAN LIVIU PARASCHIV, MARIUS GABRIEL SURU andLEANDRU-GHEORGHE
BUJOREANU, Properties Variation, Due toNickel Content and
Thermo-mechanical Processing, in Fe-Cr-Ni-Si Master Alloys Used for
the Preparation of Some Fe-Base Shape Memory Alloys (English,
Romanian summary)...........................
93
C O N T E N T S
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MARIAN MARE, CRISTINA RACU, LILIANA BUHU and ADRIANBUHU,
Mechanical Properties of Some Biodegradable Polymer MatrixComposite
Materials, with Natural Yarn Fabrics Reinforcement(English,
Romanian
summary)................................................................
101
LEONID TARTAKOVSKY (Israel), VLADIMIR BAIBIKOV (Israel), MARCEL
GUTMAN (Israel), DORON POPESCU (Israel), MARC VEINBLAT (Israel) and
YORAM ZVIRIN (Israel), Comparison of Pollutants Emission by Urban
Buses and Passenger Cars in Israel(English, Romanian
summary)....................................................................
109
VICTOR BRATU, ALEXANDRU BOROIU and ION TABACU, Researches
Regarding the Improvement and the Modernization of the Public
Passengers Transport in the Peripheral Area of Pitesti. (I) Study
of Determining Factors for the Public Transport System (English,
Romanian
summary)....................................................................................
117
CLAUDIU BUTNARU, Impact of Exhaust Gas Recirculation (EGR) on
Combustion and Emissions in a HCCI Engine (English, Romanian
summary).....................................................................................................
127
CLAUDIU BUTNARU, Influence of Engine Speed and Fuel Consumption
on Exhaust Emissions (English, Romanian
summary)..................................... 135
VASILE CAUNII, Aspects Regarding the Control Strategies of Air
Conditioning Compressor (English, Romanian
summary).......................... 141
COSTIN DRAGOMIR, CONSTANTIN PANA, NICULAE NEGURESCU and
ALEXANDRU CERNAT, Theoretical and Experimental Investigations of
the SI Engine Turbo Charging (English, Romanian
summary).....................................................................................................
151
MARIUS ATANASIU, MIHAI MARIUS PRODAN and EMIL JUGUREANU,
Comparative Analysis Between Planar Product Method and Plan
Geometry Method for Scroll Compressor Mathematical Model(English,
Romanian
summary)....................................................................
159
MIHAI ADAM, GEORGE OVIDIU RU and GHEORGHE DUMITRACU,
Experimental Research on Thermal Energy Storage Using Phase Change
Material: Eutectic Sn-Pb Alloy (English, Romanian summary). 167
MIHAI ADAM, RZVAN FLORIN BARZIC, GEORGE OVIDIU RU and GHEORGHE
DUMITRACU, Thermal Conductivity Direct Assessment of PCM Used in
Heat Storage Systems (English, Romanian summary)..... 173
GABRIEL ALEXANDRU, GHEORGHE DUMITRACU and FLORIN GABRIEL
FLOREAN, Modeling the Flow Through a Transonic Centrifugal
Compressor. (I) The Influence of the Wall Treatment Method(English,
Romanian
summary)........................................................................
179
GABRIEL ALEXANDRU, GHEORGHE DUMITRACU and FLORIN GABRIEL
FLOREAN, Numerical Analysis of the Coriolis Effect in aTransonic
Centrifugal Compressor (Englih, Romanian summary).............
185
GEORGE OVIDIU RU, MIHAI ADAM, IVANCU IONEL and GHEORGHE
DUMITRACU, Experimental Setup for Testing the Phase Change Heat
Transfers in Heat Pipes (English, Romanian
summary).....................................................................................................
191
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GEORGE OVIDIU RU, MIHAI ADAM and GHEORGHE
DUMITRACU, Experimental Investigation of Boiling and Flooding
Limits, using Two-Phase Closed Thermosyphons with Different
Diameters (English, Romanian
summary)................................................... 197
DRAGO S. CRUDU and VASILE MERTICARU, Theoretical Contributions
to Dynamic Optimization of the Frontal Loader Working Mechanism
(English, Romanian
summary)............................................... 205
DRAGO S. CRUDU and VASILE MERTICARU, Experimental Data Reference
to Design Working Mechanism for Frontal Loader (English, Romanian
summary)....................................................................................
219
CRISTIAN IOSIFESCU and SORINEL TOFAN, Study of a Counterflow Wet
Cooling Tower. (I) Simulation Model for Performance
Analysis(English, Romanian
summary)....................................................................
233
SORINEL TOFAN and CRISTIAN IOSIFESCU, Study of a Counterflow Wet
Cooling Tower. (II) Exergy Analysis (English, Romanian
summary)........ 239
ADRIAN I. SIMEDRU, Theoretical and Experimental Comparison of
the Axial Hydraulic Turbines Working with Cavitation at Iron Gates
I Hydro-Power Plant (English, Romanian
summary).................................... 245
BOGDAN CACIULA, VIOREL POPA, TANASE PANAIT, Study Concerning
Optimum Heat Source Temperature for a Solar Adsorption System
(English, Romanian
summary).......................................................
253
GEORGIANA BOLAT and DANIEL SUTIMAN, Thermodinamic Study of
Toluene + an Alkane at 318.15 K (English, Romanian
summary).............. 259
FLORIN GABRIEL FLOREAN, IONUT PORUMBEL, GABRIELALEXANDRU and
GHEORGHE DUMITRACU, Numerical Simulation and Experimental
Measurements in a Gas Turbine Turbulent Exhaust Jet (English,
Romanian summary).................................................
265
FLORIN GABRIEL FLOREAN, ALEXANDRU GABRIEL and GHEORGHE
DUMITRACU, Experimental Measurements and Numerical Simulations in
Isothermal Turbulent Flows (English, Romanian
summary)....................................................................................
271
DANIELA TASMA, TNASE PANAIT, KRISZTINA UZUNEANU andCTLIN MOCANU,
Exergetic Evaluation of Agricultural Residues Gasification
(English, Romanian
summary)................................................ 277
DANIELA TASMA , TNASE PANAIT, KRISZTINA UZUNEANU and CTLIN
MOCANU, Influence of Excess Air Ratio on Biomass Gasification
(English, Romanian
summary)................................................ 285
OLIVIA GIUCA and IOAN NICOAR, Experimental Plans Used for
Verification of Dimensional Stability Nanocrystals (English,
Romanian
summary).....................................................................................................
295
MARIUS CATAN and DANIELA TARNI, The Three-Dimensional Virtual
Model of the Human Knee Joint (English, Romanian
summary).....................................................................................................
303
RALUCA SOFRONIA, ARJANA DAVIDESCU and GEORGE SAVII,Physics-Based
Bone Sawing Interaction for a Haptic Simulator (English, Romanian
summary)....................................................................................
309
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MARIAN STANCU, DANIEL DRAGOMIR-STANCIU and CONSTANTIN
LUCA, Small Cogeneration System with Wood Gasifier (Englih,
Romanian
summary)....................................................................................
317
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BULETINUL INSTITUTULUI POLITEHNIC DIN IAI Publicat de
Universitatea Tehnic Gheorghe Asachi din Iai, Tomul LVIII
(LXII), Fasc. 3, 2012
Secia CONSTRUCII DE MAINI
BEHAVIOUR OF LAMINATED COMPOSITE BEAM AT THE INTERFACES OF
LAYERS
BY
MOUSSA KARAMA
Universit de Toulouse, France
Received: April 2012 Accepted for publication: June 2012
Abstract. One of the current problems connected with
multiplayer
composite structures concerns the analysis of the distribution
of the stresses around peculiarities (free edge and loaded edge)
and at the interfaces of each layer. This work presents a new shear
stress function in the form of the exponential function, to predict
the mechanical behaviour of multi-layered laminated composite
structures. As a case study, the mechanical behaviour of laminated
composite beam (90/0/0/90) is examined. The results are compared
with the model Sinus and 2D finite element method studied. Results
show that this new model is more precise than older ones as
compared to the results by the finite element analysis (Abaqus). To
introduce continuity on the interfaces of each layer, the
kinematics defined by Ossadzow is used with new exponential model.
The equilibrium equations and natural boundary conditions are
derived by the principle of virtual power.
Key words: composite materials, shear function, laminated
structure, interface, refined model, finite element.
1. Introduction
One of the major challenges in computational structural
mechanics is the development of the advanced models and numerical
techniques in order to provide efficient tools exhibiting good
interior and edge solutions. In this paper we are introducing an
exponential function as a shear stress function; the
e-mail: [email protected]
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2 Moussa Karama
exponential functions are found to be very much richer than
trigonometric sine and cosine functions in their development
series. According to the definition of the transverse shear stress
function, the existing laminated composite beam is divided into two
broad categories; firstly, the global approximation models and
secondly the discrete layer approximation models. The equivalent
single-layer laminate theories are those in which a heterogeneous
laminated plate is treated as a statically equivalent, single layer
having a complex constitutive behaviour, reducing the 3-D continuum
problem to 2-D problem.
The equivalent single layer models are:
i) the Kirchoff (1850) Love (1934) theory (or classical theory)
in which deformation due to transverse shear is neglected, implies
that the normal to the mid plane remains straight and normal at
mid-surface after deformation. This theory can be used for thin
beams.
ii) the Reissner (1945)-Mindlin (1951) theory (or first order
theory). That the first order deformation theory extends the
kinematics of the classical laminated plate theory by including a
gross transverse shear deformation in its kinematic assumption, the
transverse shear strain remain constant with respect to the
thickness coordinate, implies that the normal to the mid plane
remains straight but not normal to mid-surface after deformation
due to shear effect. The first order theory requires shear
correction factors, which are difficult to determine for arbitrary
laminated composite plate.
iii) and the higher order models are based on the hypothesis of
non-linear stress variation through thickness (Reddy, 1984),
(Touratier, 1991). These models are able to represent the section
warping in the deformed configuration.
These theories do not satisfy the continuity conditions of
transverse shear
stress at layer interfaces. Although the discrete layer
approximation theories are accurate, but they are rather complex in
solving problems because the order of their governing equations
depends on the number of layers.
Di Sciuva (1987, 1998) and then Touratier (1991, 1992) proposed
simplified discrete layer model with only five variational unknowns
(two membrane displacements, a transverse displacement and two
rotations), allowing the section to be represented wrapping in the
deformed configuration for the Touratier (1992) model.
Nevertheless, in these two cases the compatibility conditions, both
layer interfaces and the boundaries, cannot be satisfied. From
Touratiers work, Beakou (1991) and Idlbi (1995) proposed,
respectively, shell and plate models which satisfy both the stress
continuity at interfaces and the zero stress conditions at the
boundaries.
Finally, He (1994) introduced the Heaviside step function which
enables automatic satisfaction of the displacement continuity at
interfaces between different layers. The new discrete layer model
comes from the work of Di Sciuva (1993), He (1994) and Ossadzow et
al. (1995), the displacement field is
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 3
(1)
01 1 3 1 1 3 ,1 1 1 3 1 1
2
3 1 1
( , , ) ( , ) ( , ) ( ) ( , ),0,
( , ) ( , ),
U x x t u x t x w x t h x x tU
U x t w x t
= + = = with transverse shear function
1
( ) ( ) ( )3 31 3 3 1 3 3 3 3
1
( )( ) ( ) ( ) ( )2 2
Nm m
m
x f xh x g x x x H x x
=
m = + + + ,
where, H(x3 x3(m)), the Heaviside Step function is defined
as:
( )3 3( )
3 3 ( )3 3
1 for ,( )
0 for ,
mm
m
x xH x x
x x
= >
where is the diameter of the and the closed domain && is
set such that { }edge /2/ .z h= = = && From the beginning
our objective was so clear, to find out a transverse shear stress
function f(z), which gives the mechanical behaviour of the
composite laminated structures as much close as possible to the
exact 3D solution (Pagano, 1970) or finite element analysis in 2D
(stress, strain plane), and better representation of the transverse
shear stress in the thickness of the laminated structure. Since
different higher order polynomial and trigonometric function
already has been tried which are as follow:
-
4 Moussa Karama
Ambartsumain (1958), where
2 2
( ) ,2 4 3z h zf z =
Kaczkowski & Panc (1975) and Reissner (1975) where
2
25 4( ) 1 ,4 3
zf z zh
=
Levinson (1980), Muthy (1981) and Reddy (1984) where
2
24( ) 1 ,3
zf z zh
=
and finally Touratier (1991), where
( ) sin ,h zf z h
=
So, we took a start with an exponential function, because an
exponential function has all even and odd power in its expansion
unlike Sine function, which have only odd power. So an exponential
function is much richer than a Sine function. If we take a look on
the expansions of different transverse shear stress functions
as
Reddy (1984), where
2 3
2 24( ) 1 1.33 ,3
z zf z z zh h
= = (3)
Touratier (1991), where
3 5 7
2 4 6( ) sin 1.645 0.812 0.191 0.0261h z z z zf z z
h h h h= = + +9
8zh
, (4)
Present model
2
3 5 72( / )
2 4 6( ) 2 2 1.333 0.666 .z h z z z zf z ze z
h h h h= = + +
9
8 (5)
As it is clear from expansions of the transverse shear stress
functions, that the coefficient of successive terms in Sine
functions are decreasing more rapidly than present exponential
function which are the main responsible to gives different
mechanical behaviour of laminated structures. For the transverse
shear stress behaviour, it is very important that the first
derivative of the transverse shear stress function must give a
parabolic response in the thickness direction of the laminate and
satisfy the boundary conditions.
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 5
2. Governing Equations
From the virtual power principle, the equations of motion and
the natural boundary conditions can be obtained. The calculations
are made in small perturbations. According to the principle of
virtual power
(6) * * *( ) ( ) ( ) .a i eP P P= +
But the virtual power of the acceleration quantities is
* *( ) d ;T
aP U U
= && (7)
we suppose
(8)
/2 /2
3 ' 3/2 /2
/2 /22
' 3 3 1 3 3/2 /2
/2 /221 3 3 ' 3 1 3 3
/2 /2
d , d ,
d , ( )d ,
( )d , ( )d .
h h
w uwh h
h h
w uh h
h h
wh h
I x I x x
I x x I h x x
I h x x I x h x x
= =
= =
= =
3
So, the Eq. (6) becomes (see Appendix A for the mathematical
detail)
( )* ( ) * ( ) * ( ) * ( )( ) 1 1 10
dL
u o w waP u w x
= + + + *w , (9) with
( )1 ' ,1 1
( )' 1,1 ' ,11 ' 1,1
( )1 ' ,1 1
( )' 1 ' ,1 ' 1
,,
,
.
u ow uw u
w ouw w w w
ou w
w ouw w w
I u I w II u I w I w I
I u I w I
I u I w I
= + + = + = + + = + +
&&&& &&&&&& &&
&&
&&&& &&&&&&
&&
Now the virtual power of internal work is
*
*( ) : d ,
T
iP D
= (10) but
11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
, ,D D D
D D D D D D D
= =
-
6 Moussa Karama
so, in two dimension 11 11 13 13: 2D D D = + .
*.
(11) Resulting stresses N , M and P are defined as
(12)
/2 /2
11 11 3 11 3 11 3/2 /2
/2 /2
11 1 3 11 3 13 1,3 3 13 3/2 /2
d , d ,
( ) d , ( ) d ,
h h
h h
h h
h h
N x M x x
P h x x P h x x
= =
= =
so Eq. (10) becomes (see Appendix B for the mathematical
detail), * * * * * * *
( ) 11,1 1 11,11 11,1 13 1 1 11 1 11,1 11 ,1 11 10
( ( ) )dL
o oiP N u M w P P x N u M w M w P = + + + (13)
Now the virtual power of external loading is
(14) * * *( )
dT TeP U f U F
= + d
2
but 1 1
* * *1 3 2
3 3
0 , , ,Tf F
U U U f f F Ff F
= = =
with * * * *1 1 3 ,1 1 3 1*2
* *3
( ) ,
0,
.
oU u x w h x
U
U w
= +==
We define
/2 /2
33/2 /2
/2 /2
3 3 3 3/2 /2
/2 /2
1 3 3 1 3 3/2 /2
d ,d ,
d , d ,
( ) d , ( ) d ,
h h
i i iih h
h h
ii ih h
h h
ii iih h
F xn f x N
m x f x M x F xi
p h x f x P h x F x
= =
= =
= =
(15)
so Eq. (14) becomes (see Appendix C for the mathematical
detail),
* * * * * * *1 11 3 1,1 1 3 1( ) 1 1 1 1 ,1 11
0
( ( ) )d ( )L
o oeP n u n m w p x N u N m w M w P
* = + + + + + (16)
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 7
Now, by the Eqs. (5), (8), (13) and (16), governing equations
and natural boundary conditions for , *1
**1 ,, wuo
( )111,1
( )3 1,111,11
( )11,1 13 1
,
( )
.
u
w
N n
M n m
P P p
= + = + + = +
, (17)
And natural boundary conditions for: : *1,*1
**o1 w,,w,u
111( )
3 111,1
111
111
0 ,
( )
0 ,
0 .
w
N N
,M N m
P P
M M
= + = + = +=
(18)
The three-dimensional orthotropic constitutive law is
(19)
11 12 1311 11
12 22 2322 22
13 23 3333 33
4423 23
5513 13
6612 12
0 0 00 0 00 0 0
.0 0 0 0 0 20 0 0 0 0 20 0 0 0 0 2
C C C C C C C C C
C C
C
=
The dimension x2 is supposed unitary, and the effects of the 33
are neglected, so orthotropic law (19), becomes
, (20) '
11 1111
13 1355
020
C C
= with
*
11 1,1 1,1 3 ,11 1 1,1
31 1,3 12
' 11 33 1311
33
,
2 ,
.
oU u x w h
h
C C CCC
= = +=
=
Now, the generalized constitutive law
-
8 Moussa Karama
*11 1,111 11
11 ,1111 11
11 1,1
13 1
00 .0
0 0 0
oN uA B KM wB D TP K T SP Y
=
%%%% %
%
(21)
So, the governing Eqs. (17) become
( ) *111 1,11 11 ,111 1,11
( ) *3 1,111 1,111 11 ,1111 1,111
( ) *1,11 ,111 1,11 1 1
,
,
.
u o
w o
o
A u B w K n
B u D w T n m
Ku Tw S Y p
= + + = + + + = + +
%%
%% % % (22)
And the natural boundary conditions (18) become
*11 1,1 11 ,11 1,1 1
( ) *11 1,11 11 ,111 1,11 3 1
*11,1 ,11 1,1
*11 1,1 11 ,11 1,1 1
0 ,
,
0 ,
0 .
o
w o
o
o
A u B w K N
B u D w T N m
Ku Tw S P
B u D w T M
= + = + + = + += +
%
%
%% %
%
(23)
Continuity coefficients : To find out the value of the
continuity coefficients, the conditions of the
continuity of the transverse shear stress between each
interfaces of the layers were used (Fig. 1)
. (24) ( ) ( ) ( 1) ( )13 3 3 13 3 3( ) (m m m x x x x+= = =
)m
1)
Interface of layer (1) and layer (2)
(1) (1) (2) (13 3 3 13 3 3( ) ( ) x x x x= = =
(1)1 (1) (1) ( 2 ) (3) 355 1 3 1 1 1
(1)2 (1) (1) ( 2 ) (3) (1)355 1 3 1 1 1 1
'( )1( ) '( ) ( )2 2
'( )1( ) '( ) ( ) .2 2
f xQ x g x
f xQ x g x
+ + + + = = + + + +
+
.
(25)
Interface of layer (2) and layer (3)
(2) (2) (3) (2)13 3 3 13 3 3( ) ( ) x x x x= = =
Since, Q55 of the second and third layer are equal (Fig. 1),
so,
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 9
( 2 )2 ( 2 ) (1) ( 2 ) (3) (1)355 1 3 1 1 1 1
( 2 )3 ( 2 ) (1) ( 2 ) (3) (1)355 1 3 1 1 1 1 1
'( )1( ) '( ) ( )2 2
'( )1( ) '( ) ( ) .2 2
f xQ x g x
f xQ x g x
+ + + + + = = + + + + +
( 2 )+
(26)
Now by Eqs. (25) and (26),
(27) (1) (1) (2)1 1 1
(2)1
,
0.
= +=
This shows that if the mechanical characteristics of the two
consecutive layers are the same (Fig. 1), the coefficient of the
continuity will be zero ((2) = 0). Interface of layer (3) and layer
(4)
, (3) (3) (4) (3)13 3 3 13 3 3( ) ( ) x x x x= = =
(3)3 (3) (1) (2 ) (3) (1) ( 2)355 1 3 1 1 1 1 1
(3)4 (3) (1) ( 2) (3) (1) (2 ) (3)355 1 3 1 1 1 1 1 1
'( )1( ) '( ) ( )2 2
'( )1( ) '( ) ( ) ,2 2
f xQ x g x
f xQ x g x
+ + + + + + = = + + + + + + +
(28)
we have,
(1) (3)3 3
(1) (3)3 3
'( / 4) '( / 4),
'( / 4) '( / 4).
f x h f x h
g x h g x h
= = == = =
So, by Eqs (26), (27) and (28), gives
(3)
1 (3 ) (1) ( 2 ) (3 ) 355 1 3 1 1 1
(3)4 (3 ) (1) ( 2 ) (3 ) (1) (3 )355 1 3 1 1 1 1 1
'( )1( ) '( ) ( )2 2
'( )1( ) '( ) ( ) ,2 2
f xQ x g x
f xQ x g x
+ + + + = = + + + + +
+
(29) (1) (3)
1 1
(1) (3)1 1
0
= +=
So, by Eq (27) and (29), Eq (25) becomes
-
10 Moussa Karama
(30) 1 (1) 2 (1) (1)55 3 55 3 1( '( )) ( '( ) ),Q g x Q g x =
+
1 2 (1) 2 1 (1)
(1) (3)55 55 3 55 55 31 12 2
55 55
( ) '( ) ( ) '( and .Q Q g x Q Q g x Q Q
= = ) (31)
3. Finite Element Analysis
Since no exact 3D solution exists for the considered case study,
so ABAQUS (finite element analysis software) is used to show the
efficiency of the present model. In this paper finite element
results are taken as a reference for the comparison of different
models of laminated composite structures, done by Abou Harb (1998).
The 3D approximation of the behaviour is carried out by element
type CPS8 (quadrilateral element of eight nodes, 16 dof per
element). To validate the finite element results, firstly it is
necessary to find out the convergence of laminate meshing. So, for
the given problem, in static and dynamic, the convergence found to
be at 1680 elements included 24 element in thickness.
4. Some Evaluations of the Present Model
The static bending analysis is studied, so the virtual power of
acceleration quantities is cancelled. Three different bending
analyses have been developed for three different specific boundary
conditions. For the simply supported conditions, the unknown
variables are deduced directly by the equation of motions. For
clamped conditions, kinematical boundary conditions are used and,
finally, in a free edge case, natural boundary conditions are
employed. The beam studied has a length of L = 6.35 m, a unitary
width, and a thickness h = =2.794 m in the thick case and h =
0.2794 m in the thin case. The beam possesses four layers of the
same thickness at 90/0/0/90. The material used for the four layers
is boron epoxy. The mechanical properties of the 0 layer are as
follows (Pagano, 1970): E11 = 241.5 GPa, E22 = E33 = 18.89 GPa, G23
= 3.45 GPa,
G12 = G13 = 5.18 GPa, 23 = 0.25, 12 = 13 = 0.24, and the
density, = 2015 kg/m3. The continuity coefficients from the Eqs.
(27), (29) and (31)
1(1) = - 1(3) = 0.2210501411, 1(2) = 0.
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 11
Fig. 1 Description of the beam.
Problem 1: bending of a simply supported beam under
distributed
sinusoidal load.
The surface and volume force components are cancelled except
13 3 3
0
d sin . h
oxn f x q qL
= = = For the simply supported boundary conditions, the Levy
solution is used, define as
1 11 1cos , sin , coso
o o ox x xu u w wL L
= = = 1
L. (32)
Now the governing Eqs (22), with P(a)* = 0 , becomes
*11 1,11 11 ,111 1,11
* 111 1,111 11 ,1111 1,111
*1,11 ,111 1,11 1
0 ,
0 s
0 .
o
oo
o
A u B w KxB u D w T qL
Ku Tw S Y
= + = + +
= +
%
%
%% % %
in , (33)
Now, by the Levy solution, the governing equations become:
2 3 211 1 11 1 1
3 4 311 1 11 1 1 1
2 3 21 1 1 1
0 cos cos cos ,
0 sin sin sin ,
0 cos cos cos cos with,
o o
o o o o
o o o o
A u x B w x K x
B u x D w x T x q xK u x T w x S x Y x L
= + = + += + =
%%
%% % %
and then in matrix form
-
12 Moussa Karama
= (34) 2 3 2
11 113 4 3
11 112 3 2
0,
0
o
o o
o
A B K u B D T w q
K T S Y
%%
%% % %
and also, the displacement (1), becomes
( )1 1 3 3 1 3 1
2
3 1
( , ) ( ) cos( ),
0,
sin( ),
o o o
o
U x x u x w h x x
U
U w x
= +==
(35)
and now by Eq. (20), stresses expression
(36) '11 1 3 11 3 1 1( , ) ( )sin( ),o o o x x C u x w h x=
+
and,
13 1 3 55 1,3 1( , ) cos( ),o x x C h x= (37)
and integration of the equilibrium equation 13,1 33,3 0 + = ,
enables us to calculate the analytical value of 33
33 55 1 3 1( ) sin( ).oC h x x = (38)
The numerical results obtained (qo = -106 Pa) using the present
model are compared with those obtained by the finite element
analysis7 and the Sine7 model by Touratier (Table 1).
Table 1 Bending of the simply supported thick beam under
distributed sinusoidal load
Model U3 (L/2)
m
U1 (0, h/2)
m
13 (L/4, 0) (Interface)
Pa
11 (L/2, -h/4+) (Interface)
Pa
33 (L/2, h/2)
Pa Present -6.370110-4 2.119610-4 -940098.0 8112840.0
-1039990.0
Error (%) 4.4 8.3 6.6 3.5 3.9 Sine7 -6.279410-4 2.018010-4
-896865.0 8158932.0 -1047274.0
Error (%) 2.9 12.7 10.8 4.1 4.6 Abaqus7 -6.100610-4 2.312510-4
-1006000.0 7835200.0 -1000900.0
For this problem, the present model present a better
mechanical
behaviour better than Sine model compared to the finite element
analysis results except the transverse deflection (U3). Percentage
error reduction is more significant in case of transverse shear
stress (13) at interfaces of layers. The efficiency of this model
is shown for (Figs. 2, 3 and 4), different stresses and
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 13
displacement plotted according to the length and thickness of
the beam, showing that the present model at every point on the
beam, is closer to the finite element results then to those of the
Sine model. Here we can see also the continuity of displacement and
transverse shear stress between layer interfaces of the present
model.
Fig. 2 Variation of the stress 11 along the direction x1 for x3
= -h/2 for
Problem 1: Present (-*-), Sine7(-), Abaqus7(- - -).
Fig. 3 Variation of the transverse shear 13 through the
thickness for x3 = 0 (Interface).
for Problem 1, Present (-*-), Sine7(-), Abaqus7(- - -).
Fig. 4 - Variation of the displacement U1 through the thickness
for x1 = L/4 for
Problem 1, Present (-*-), Sine7(-), Abaqus7(- - -).
-
14 Moussa Karama
*
d
5. Conclusions
1. Shear stress continuity of displacement and transverse shear
stresses at interfaces of the layers and the boundary conditions
for a laminated composite are exactly satisfied by this present new
multi-layered structure exponential with the help of the Heaviside
step function (Figs. 2-10). 2. For the new proposed model the
results are compared with the existing model (like Sine (1998))
model by Touratier (1991) and by the finite element method by
Abaqus (1998). Results show that the new proposed exponential model
present a better approximation than the Sine (1998) model as
compared to the finite element analysis, except for some results,
(Table 1), especially results are very much favourable on interface
of the layers. 3. The new model is also simple in so far as any
correction factor is used in opposition to the higher order models.
In the case of static analysis, (Figs. 2-4) presents the numerical
results for the bending deformation under different types of
loading and boundary conditions on a thick beam, showing that the
present model is always trying to approaches to the finite element
analysis by Abaqus (1998). As a whole we can conclude the present
exponential model is more accurate than other existing analytical
models for multi-layered structures compared to finite element
analysis.
Appendix A
(Virtual Power of the Acceleration Quantities) We have,
1 1 3 ,1 1 1 3
* * * * *1 1 3 ,1 1 1 3
, ,and
, .
o
o
U u x w h U w
U u x w h U w
= + =
= + =
&& &&&& && &&
So Eq. (6) becomes, * * *
( ) 1 1 3 3( )aP U U U U
= + && && . By the integration by parts
* * * * *( ) 1 1 3 3 1,1 1 3 1 1 3 1( ) ( ) d
oaP u u u x u w h x u x w u
= + + + && && && && &&
d
and, now
( ) ( )( ) ( )
* *( ) 1 3 ,1 1 3 1 1 3 1,1 3 ,11 1 3 1,1
* *1 3 1 3 ,1 1 3 1 1 3 1 3 ,1 1 3 1
( ) ( ( ) )
( ) ( ) d ( ) d ,
o o oa
o o
P u x w h x u w x u x w h x w
h x u x w h x x u x w h x w
= + + + ++ + +
&& && && && &&
&& && && &&
* +
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 15
/2 /2 /2* *
( ) 3 1 3 3 ,1 1 3 3 1 10 /2 /2 /2
/2 /2 /2 /22 *
3 3 1,1 3 3 ,11 3 1 3 3 1,1/2 /2 /2 /2
/
/ 2
d ) ( d ) ( ( )d )
d ) ( ) ( d ) ( ( )d )
(
L h h ho o
ah h h
h h h ho
h h h h
h
h
P x u x x w h x x u
x w dx u x x w x h x x w
= + + + + + +
+
&& &&
&& && &&
2 /2 /22 *
1 3 3 1 3 1 3 3 ,1 1 3 3 1 1/2 /2
/2 /2 /22 *
3 3 1 3 3 ,1 3 1 3 3 1/2 /2 /2
( )d ) ( ( )d ) ( ( )d ) d
( d ) ( d ) ( ( )d )
h ho
h h
h h ho
h h h
h x x u x h x x w h x x
x x u x x w x h x x w
+ + + + +
&& &&
&& &&
+
+
and now by relations (7),
( )* ( ) * ( ) * ( ) * ( )( ) 1 1 10
dL
u o w waP u w x
= + + + *w
Appendix B
(Virtual Power of the Internal Work)
By relation (11), virtual power of the internal work (10)
become
( )( )
* * * * *( ) 1,1 3 ,11 1 3 1,1 11 1,3 3 1 13
* * * * *( ) 11 1,1 3 ,11 11 1 3 11 1,1 13 1,3 3 1
1( ) 2 ( ) d ,2
( ) ( ) d .
oi
oi
P u x w h x h x
P u x w h x h x
= + + = + +
Now, by integrating each term by parts:
* * * * *( ) 11,1 1 3 11,11 1 3 11,1 1 1,3 3 13 1
* * * *11 1 3 11 ,1 3 11,1 1 3 11 1
( ) ( ) d
( ) d ,
oi
o
P u x w h x h x
u x w x w h x
= + + +
*
/2 /2 /2* * *
( ) 11,1 3 1 3 11,11 3 1 3 11,1 3 10 /2 /2 /2
/2 /2 /2* *
1,3 3 13 3 1 1 11 3 1 3 11/2 /2 /2
d d ( ) d
( ) d d - d d
L h h ho
ih h h
h h ho
h h h
P x u x x w h x x
h x x x x u x x
= + + + +
*3 ,1/2 /2
* *3 11,1 3 1 3 11 3 1
/2 /2
d ( ) d .h h
h h
w
x x w h x x
-
16 Moussa Karama
*.
Now by relations (12),
* * * * * * *( ) 11,1 1 11,11 11,1 13 1 1 11 1 11,1 11 ,1 11
1
0
( ( ) )dL
o oiP N u M w P P x N u M w M w P = + + +
Appendix C
(Virtual power of the External Loading) By relations (15),
virtual power of external loading (14), becomes:
1 1* * * * *
( ) 1 3 2 1 3 2
3 3
0 d 0e
f FP U U f U U F
f F d
= +
* ,
,
* * * * *
( ) 1 1 3 3 1 1 3 3
* * * * *( ) 1 1 1 3 ,1 1 3 1 1 3
* * * *1 1 1 3 ,1 1 3 1 1 3
/ 2 / 2 /2* *
( ) 1 3 1 3 3 3 1,1 3/ 2 / 2 /2
( )d ( )d ,
( ( ) )d
( ( ) )d ,
d d
e
oe
o
h h ho
eh h h
P f U f U F U F U
P f u f x w h x f f w
F u F x w h x F F w
P f x u f dx x f x
= + + +
= + + +
+ + +
= + +
/ 2* *1 3 1 3 1 10 /2
/2 /2 / 2 / 2 / 2* * *
1 3 1 3 3 3 1 3 1 3 1 3 1 3 1 3 ,1/2 /2 / 2 / 2 / 2
( ) d
d d d ( ) d d
L h
h
h h h h ho
h h h h h
w h x f dx x
F x u F x x f x w h x F x x F x w
+ + + + +
and now by relation (15),
* * * * * * *1 11 3 1,1 1 3 1( ) 1 1 1 1 ,1 11
0
( ( ) )d ( )L
o oeP n u n m w p x N u N m w M w P
* = + + + + + .
REFERENCES
Ambartsumian S.A., On Theory of Bending Plates. Izv. Otd. Tech.
Nauk., AN SSSR., 5, 69-77 (1958).
Beakou A., Homognisation et modlisation des coques composites
multicouches. Thesis ENSAM, Paris, 1991.
Di Sciuva M., A General Quadrilateral Multi-layered Plate
Element with Continuous Inter-laminar Stresses. Computers and
Structures, 47(1), 91-105 (1993).
Di Sciuva, M., An Improved Shear Deformation Theory for
Moderately Thick Multi-layered Anisotropic Shells and Plates. J.
Appl. Mech., ASME, 54, 589-596 (1987).
Gachon H., Sur le flambage des plaques: modle de calcul, modles
exprimentaux. Construction Mtallique, 4, 23-52 (1980).
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 17 He L.H., A
Linear Theory of Laminated Shells Accounting for Continuity of
Displacements and Transverse Shear Stresses at Layer Interface.
Int. J. Solids Structures, 31(5), 613-627 (1994).
Idlbi A., Comparaison de thories de plaque et estimation de la
qualit des solution dans la zone bord. Thesis, ENSAM, Paris,
1995.
Kaczkowski Z., Plates. Statistical Calculations. Arkady, Warsaw,
1968. Karama M., Abou H. et al., Bending, Buckling and Free
Vibration of Laminated
Composite with Transverse Shear Stress Continuity Model.
Composite Part B, 29B, 223-234 (1998).
Karama M., Touratier M. et al., An Evaluation of the Edge
Solution for a Higher-order Laminated Plate Theory. Composite
Structures, 25, 495-502 (1993).
Kirchhoff G.J., Uber das Gleichgewicht und die Bewegung einer
elastischen Schreitbe. Reine Angew. Math, 40, 51-58 (1850).
Levinson M., An Accurate Simple Theory of the Statics and
Dynamics of Elastic Plates. Mech. Res. Common., 7, 343-350
(1980).
Love A.E.H., A Treatise on the Mathematical Theory of
Elasticity. 4th Ed., Cambridge University Press, 1934.
Mindlin R.D., Influence of Rotary Inertia and Shear on Flexural
Motions of Isotropic Elastic Plates. J. Appl. Mech., ASME, 18,
31-38 (1951).
Murthy M.V.V., An Improved Transverse Shear Deformation Theory
for Laminated Anisotropic Plates. NASA Technical Paper, 1981.
Ossadzow C., Muller P. et al., Une thorie gnrale des coques
composites multi-couches. 2ime Colloque National en Calcul des
Structures, 1, Hermes, 1995.
Pagano N. J., Exact Solution for Rectangular Bi-directional
Composites and Sandwich Plates. J. Composite Materials, 4, 20-34
(1970).
Palardy R.F., Palazotto A.N., Buckling and Vibration of the
Composite Plates Using the Levy Method. Composite Structures, 14,
61-85 (1990).
Panc V., Theories of Elastic Plates. Academia, Prague, 1975.
Reddy J.N., A Simple High-order Theory of Laminated Composite
Plate. J. App. Mech.,
51, 745-752 (1984). Reissner E., Reflection on the Theory of
Elastic Plates. J. Appl. Mech., 38, 1453-1464
(1945). Reissner E., On Transverse Bending of Plates, Including
the effects of Transverse Shear
Deformation. Int. J. Solid Structure, 25, 495-502 (1975).
Touratier M., An Efficient Standard Plate Theory. Int. J. Eng.
Sci., 29(8), 901-916
(1991). Touratier M., A Generalization of Shear Deformation
Theories for Axisymmetric Multi-
layered Shells. Int. J. Solids Structures, 29(11), 1379-139
(1992). Widera G.E.O., Logan D.L., Refined Theories for
Non-homogeneous Aniso-tropic
Cylindrical Shells. J. Eng. Mech. Division, EM6, 1053-1090
(1980).
COMPORTAREA BARELOR COMPOZITE LAMINATE LA INTERFEELE DINTRE
STRATURI
(Rezumat)
Una din problemele actuale legate de structurile compozite
multi-strat se refer la analiza distribuiei tensiunilor n zonele cu
particulariti geometrice i la interfeele
-
18 Moussa Karama
dintre straturi. Pentru a prognoza comportamentul structurilor
compozite multistrat, n cadrul acestei lucrri am introdus o funcie
exponenial determinat n raport cu tensiunea de forfecare, avnd o
acuratee mai mare fa de funciile trigonometrice sinus i cosinus,
relativ la dezvoltarea n serii. Ca studiu de caz se prezint
comportamentul mecanic al barei compozite laminate (900/00/00/900).
Rezultatele sunt comparate cu modelul Sinus i analiza bidimensional
cu elemente finite. Datele arat c noul model propus n cadrul
acestei lucrri furnizeaz rezultate mai bune dect cele anterioare.
De asemenea, pentru definirea strii de tensiuni n compozitele
multistrat, noul model propus este mai simplu i mai precis dect
vechile modele existente.
-
BULETINUL INSTITUTULUI POLITEHNIC DIN IAI Publicat de
Universitatea Tehnic Gheorghe Asachi din Iai, Tomul LVIII
(LXII), Fasc. 3, 2012
Secia CONSTRUCII DE MAINI
REORIENTATION CURVES FOR A Ni50Mn28.5Ga21.5 ALLOY
BY
ISAAC NELSON, CONSTANTIN CIOCNEL and HEIDI FEIGENBAUM
Northern Arizona University, Department of Mechanical
Engineering
Received: April 2012 Accepted for publication: June 2012
Abstract. This paper presents experimental results for variant
reorientation curves specific to a Ni50Mn28.5Ga21.5 alloy.
Reorientation curves show the stress and magnetic field values when
reorientation begins and ends for various load conditions. The
curves in this work have been generated based on material response
under three loading conditions: constant stress variable field,
constant field variable stress, and variable stress variable field.
Reorientation curves can give insight into the material behaviour
and are key features of phenomenological models for MSMAs.
Previously, variant reorientation curves have been generated only
for the constant field variable stress loading case, and models
have been limited to predicting either constant field - variable
stress or constant stress variable field conditions, but not both.
Thus, to this point, it has not been clear if it is possible to
develop one model that can predict the magneto-mechanical behaviour
of MSMAs subject to any possible stress field loading conditions.
The experimental results in this work suggest that a set of
reorientation curves covering the whole 2D magneto-mechanical
loading spectrum is achievable, which implies that the
magneto-mechanical behaviour of MSMAs can indeed be captured with a
single model. Accordingly, the results of this work will be useful
in developing such a model.
Key words: NiMnGa alloy, variant reorientation, reorientation
curves.
1. Introduction
NiMnGa is a relatively new magnetic shape memory alloy that
exhibits the shape memory effect in the presence of a magnetic
field. The shape memory effect is facilitated by the microstructure
of the material that consists of three
Corresponding author: e-mail: [email protected]
-
20 Issac Nelson, Constantin Ciocnel and Heidi Feigenbaum
martensite variants, as shown in Fig. 1. The tetragonal
martensite variants are characterized by a magnetic easy axis that
is aligned with the short side (side c in Fig. 1) of the unit
cell.
Upon the application of a magnetic field, the variants
magnetization easy axis tends to align in the direction of the
applied field, causingup to 10% macroscopic reorientation strain
(Marioni et al., 2003). The magnitude of thereorientation strain is
contingentupon the chemistry of the alloy, the materials training,
the applied magnetic field and the compressive stress that is
experienced by the material; a compressive stress as small as 5MPa
(Karaca et al., 2005) can prevent variant reorientation under any
magnitude magnetic field.
From an applications point of view, this material is suitable
for actuation, sensing and power harvesting applications, at a
minimum. In all of these applications the stress is compressive and
applied axially, while the magnetic field is applied perpendicular
to the stress, in the transverse direction. For actuation
applications, a MSMA element should be subject to a constant axial
compressive stress and a variable transverse magnetic field. During
this type of loading, the variants will reorient to align with the
magnetic field, thus elongating the MSMA element and beginning the
actuation process. The limiting factor for actuation applications
is the blocking stress specific to the material. For sensing and
power harvesting applications, a MSMA element should be subject to
constant transverse magnetic field and variable axial compressive
stress. During this type of loading, the variants will reorient to
align with the compressive stress, and as they do so the direction
of the internal magnetization changes, which can be sensed or
harvested using a pick-up coil.
Fig 1. The three martensitic variants with magnetic easy axis
parallel to the short axis (adopted from Kiefer and Lagoudas,
2005).
2
1
2
1
For a robust design of a MSMA based device, one has to be able
to predict the magneto-mechanical response of the material to
various loading conditions. Furthermore, new applications of MSMAs
may be discovered through a better understanding and more
comprehensive modelling of the magneto-mechanical behaviour of
MSMAs.
Reorientation curves are one way to get a better understanding
of the magneto-mechanical behaviour of MSMAs, and reorientation
functions are key features of phenomenological models of MSMAs.
To illustrate what the reorientation curves are, consider the
material response to a constant field-variable load condition, as
shown in Fig. 2. In this case, at the beginning of the test,
ideallythe materialis allin variant 2 (the
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Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 21
magnitude of the field causes all variants to reorient such that
the easy axes align with the external field, forcing variants 1 and
3 into variant 2). Upon the application of the compressive stress,
after some period of pure elastic deformation, some material in
variant 2 will start to reorient into a stress preferred
orientation, i.e. into variant 1. This point is called start 2 to 1
and is identified as S(2,1) in Fig. 2. Reorientation then continues
until all material is in variant 1, i.e. the finish 2 to 1 point,
identified as F(2,1) in Fig. 2. Once full reorientation from
variant 2 to variant 1 is completed, further increase of the
compressive load causes the material to deform only elastically.
Should the compressive stress then be decreased, the material will
initially deform only elastically, however, at some point the
removal of the stress and the presence of the magnetic field will
cause some material in martensite variant 1 to begin reorientation
back into variant 2, this is called start 1 to 2 and is identified
as S(1,2) in Fig. 2. Finally, the point where reorientation from
variant 1 to 2 ends is called finish 1 to 2 and is identified as
F(1,2) in the same figure. Reorientation curves show these start
and finish points for various magnitudes of load and load paths on
a stress vs. field graph.
Fig. 2 Variant reorientation points identified on the
magneto-mechanical response of the material experiencing a variable
compressive stress under a constant magnetic field.
Prior to this work, Couch et al. (2007) performed constant
stress variable field and constant field variable stress tests;
data from the later was used to generate reorientation curves. Both
types of tests were performed on a2mm x 3mm x 16mm specimens
provided by Adaptamat Ltd., same manufacturer that supplied the
specimens used in this study. In all of their experiments, the
stress was compressive and applied axially, while the magnetic
field was applied perpendicular to the stress in the transverse
direction. To determine the reorientation curves, the stresses at
which variant reorientation
-
22 Issac Nelson, Constantin Ciocnel and Heidi Feigenbaum
occurred from the field preferred to the stress preferred
orientation and back, from the stress preferred to the field
preferred orientation were recorded for various constant field
values. The reorientation curves from Couch et al. show that the
start and finish values of stress vary in a nonlinear way with
applied magnetic field. Such curves can be used for model
development and calibration when simple constant field variable
stress loading conditions are being simulated. However, to date no
researchers have determined reorientation curves for other loading
conditions, such as constant stress variable load or variable field
variable load. As a result, the objective of work is to determine
these curves.
These types of reorientation curves will be used in future work
to help
derive a model that captures the behaviour of MSMAs subject to
general loading conditions. Similar to yield functions in
plasticity theory, phenomenological models of MSMAs, such as those
by Kiefer and Lagoudas (2005, 2009) and Waldauer et al. (2011),
rely on reorientation functions to determine whether or not
reorientation is occurring. Once the material reaches a stress
field state for reorientation to begin, the material must harden
for reorientation to continue until the finished point is reached.
Accordingly, reorientation functions play a vital role in the
modelling of MSMAs and the experimental work reported here can be
used to motivate future models.
Moreover, robust and general constitutive model should be
calibrated on
one set of data, and with that unique calibration the model to
be able to predict the magneto-mechanical response predictions
under any loading condition. However, as Waldauer et al. (2011)
showed, current material models for MSMAs, such as those by Kiefer
and Lagoudas (2005, 2009) and Waldauer et al. (2011), can only be
used for loading similar to which the model was calibrated. For
example, to simulate variable field-constant stress loading, the
model must be calibrated with variable field constant stress
experimental data. Furthermore, Waldauer et al. demonstrated that
the material parameters found will be quite different if calibrated
with variable field-constant stress data than they will be if
calibrated with variable stress- constant field data. So at this
point it is unclear if a single model that captures the
magneto-mechanical behaviour of MSMAs under any load is even
feasible.
Accordingly, the goal of this paper is to establish
experimentally a set of
reorientation curves that can be used to verify whether a single
model for all load cases is possible. And if so, the experimentally
found reorientation curves in this work can help researchers
formulate a constitutive model that can predict material response
under any loading conditions. The following sections describe the
experimental setup and the experimental data analysis procedure
used to acquire and develop the variant reorientation curves.
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Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 23
2. Experimental Setup and Procedure 2.1. Experimental Setup
The experimental setup consisted of a bi-axial 8874 Instron
hydraulic test rig equipped with a 3470 GMW 45mm dipole
electromagnet (Fig. 3). The magnetic field generated by the
electromagnet was applied normal to the direction of the applied
compressive stress and was monitored using an AC/DC 5170 Gaussmeter
with a transverse Hall probe.The tested specimen wasa single
crystal NiMnGa specimen with dimensions 3mm 3mm 20mm, provided by
AdaptaMat Ltd.
Electromagnet
MSMA specimen
Magnetic field probe
Fig. 3 Experimental setup.
a b Fig. 4 a Load profile as applied using the 8874 test rig and
a constant load
device; b Constant load device as installed on the Instron 8874
frame.
A power supply controls the current passing through the
electromagnet coils and allows for both, constant and variable
magnetic fields to be applied on
-
24 Issac Nelson, Constantin Ciocnel and Heidi Feigenbaum
the specimen. The compressive load is applied directly by the
machine, in the case of variable stress-constant field and variable
stress-variable field experiments, or using a device that applies a
constant load on the specimen, for the constant stress-variable
field case. The need for the constant load device comes from the
fact that the stiffness of the specimen changes significantly
during tests and the 8874 system cannot keep the load truly
constant on the specimen, as it continuously tries to accommodate
the change in stiffness. As a result, the applied load from the
test rig can vary significantly from the desired constant load,as
shown in Fig. 4 a. However, when the constant load device is used
there is almost no change in the applied load, as shown in Fig. 4
a.
2.2. Experimental Procedure
The constant loaddevice, shown in Fig. 4 b, is made of
acetyl-butyl styrene copolymer (ABS) and on one side it has a push
rod similar to the aluminium push rods used to perform the tests,
while on the other side it has a cylindrical bearing that
facilitates the sliding up and down, along a guiding rod, of the
device as the specimen expands or compresses upon the application
of the magnetic field. The sliding occurs with minimal friction.
The constant load device was designed such that it allows the
addition of various size weights on it to simulate different stress
levels on the tested specimen. As an example, Fig. 4.b shows the
constant load device with its own mass of 199 grams, and on top of
it another mass of 434 grams, simulating approximately 0.7MPa of
stress on the specimen. Additional weights have been manufactured
to allow application of stress in increments of 0.25MPa, between
0.25 and 1.75MPa.
Experiments have been performed in three loading modes: a)
constant field variable load, b) constant load variable field, and
c) variable load- variable field. In cases a) and c), the tests
started with the specimen in variant 2 (fully elongated), while in
case b), the tests started with the specimen in variant 1 (fully
compressed). Also, in cases a) and b), the load and field have been
varied between zero and a maximum value, respectively. In case c),
the field was varied between a maximum value and a minimum value
equal to 0.25 the magnitude of the maximum value, while the load
was varied between zero and a desired maximum value. In all cases,
the direction of the loading remained the same as shown in Fig.
3.
3. Results
Fig. 5 illustrates the magneto-mechanical response of the
material for each loading case considered. The constant field tests
were performed at fields ranging between 0.6T and 0.95T in
increments of 0.05T and with the compressive stress varying between
zero and a maximum of 6.67MPa. The constant stress tests were
performed under constant stresses applied in increments of 0.25MPa,
between 0.25MPa and 1.75MPa, and magnetic field
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Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 25
varying between 0 and 1T. The complex loading tests were
performed with the stress and field varying linearly between 0 and
2.5MPa, and between maximum values for magnetic field of 0.65T,
0.75T, 0.85T, 0.95T and a quarter of this magnitude, respectively.
These tests started with the specimen fully elongated.
The data recorded during these tests has been used to determine
the start and finish reorientation stress-field values. It is
important to note that for low stress levels (between 0.25MPa and
approximately 0.4MPa) in the constant stress tests and field
magnitudes above 0.75T only reorientation from variant 1 to 2 can
be observed experimentally. This can be explained as follows. In
the constant stress case, the test is started with the specimen
fully in variant 1 from the constant stress applied. As the
magnetic field is increased, the variants reorient into variant 2.
Then, as the magnetic field is removed, the stress is too weak to
be able to induce reorientation back to variant 1. Accordingly,
from these tests, only the first two start and finish reorientation
stress-field pairs may be obtained.
In order to produce reorientation curves, a clear criteria for
selecting the start and finish reorientation points is needed and
previous work has not yet established such criteria. In an effort
to be consistent in the selection of the stress and field
reorientation values, the authors used a 0.05 strain offset from
the elastic region of the reorientation curve, as illustrated in
Fig. 5. The start and finish reorientation values have been
collected from material response plots that exhibited full
reorientation, similar to those shown in Fig. 5. For the constant
field variable load case, Fig. 5 a, the material is initially in
variant 2 and upon the application of the increasing stress, the
variants reorient into variant 1. Therefore, the red line
intersects the material response at the points of start
reorientation 2 to 1 (S(2,1)) and at finish reorientation 1 to 2
(F(1,2)). The green line intersects the material response at the
points of finish reorientation 2 to 1 (F(2,1)) and at start
reorientation 1 to 2 (S(1,2)). For the constant stress variable
field case, the material is initially in variant 1 and upon
application of the increasing magnetic field, it reorients into
variant 2. Fig. 5 b shows the selection of the start and finish
points for reorientation for this type of loading using the same
criteria of 0.05 strain offset from the elastic region.
For these two cases, where either field or stress is constant,
the selection of the stress-field values for the beginning and end
reorientation points is fairly straightforward once the off-set
strain is defined. However, for the variable stress variable field
case, the approach in determining the start and finish
reorientation points is a little more involved. One could follow
the same procedure as done for the cases where one load is
constant, however, now there will be both field strain curves as
well as stress strain curves to from which key points may be
gathered. If the off-set strain is used for both the field strain
plots as well as the stress strain plots there is the potential to
select field and stress values that are not happening
simultaneously.
-
26 Issac Nelson, Constantin Ciocnel and Heidi Feigenbaum
Specifically, it could be seen that the strain value is
different if the key point is identified on the strain field curve
as compared to if the key point is identified on the stress strain
curve. Thus, the following procedure is proposed to identify the
stress-field reorientation values with complex loads. First, one
picks the start-finish reorientation values for the stress at the
0.05 strain offset, from the stress-strain plot (Fig. 5 c) and
then, for that specific strain value reads the corresponding
magnetic field value for reorientation from the raw experimental
data file.
Using this procedure start and finish points for reorientation
were found for various magnitudes of load for constant stress
variable field, constant field variable stress and variable field
variable stress loading. A summary of all the results is shown in
Fig. 6.
It must be noted that the experimental data plotted in Fig. 6
shows an unexpectedly large scattering of the data. This may be due
to the fact that the specimen developed several cracks between the
time the constant field tests
a b
F(2,1)
c d
Fig. 5 a Constant field variable stress; b Constant stress
variable field; c & d Variable stress variable field. The
dotted line is plotted parallel to the elastic region during the
loading unloading sequence, while the red and green continuous
lines are drawn parallel to the dotted line with an offset of
0.05strain.
S(2,1)
F(1,2)
S(1,2)
F(1,2) S(1,2)
S(2,1)
F(2,1)
0.05
S(2,1)F(2,1)
F(2,1) F(1,2)S(2,1)
S(1,2) S(1,2)
F(1,2)
-
Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 27
were performed and the time the other tests were performed (Fig.
7). These cracks developed in the specimen during several tests
performed in a dynamic regime to assess the material suitability
for power harvesting. As no other specimen was available at the
time of data collection, the cracked/chipped specimen was used to
perform the remaining tests necessary for this study. The authors
expect that once a new fully intact specimen is available it will
be used to gather similar type of data, and the scatter should be
significantly reduced.
In addition, the method used for the magnetic field measurement
may have contributed to the scattering of the data. Specifically,
the field was measured using a Hall probe whose position along the
specimen may have not been the same when measurements were made for
the various tests performed. While the field distribution between
the magnetic poles, along the specimen height, is uniform in the
absence of the MSMA specimen, as the variant reorientation occurs
within the MSMA there is a change in the direction of internal
magnetization, which can affect the external magnetic field. So if
there are variations along the length of the specimen of the
variant volume fraction, reading the field at slightly different
locations along the specimen may lead to slightly different
measured field values at key points. This would lead to scatter in
the reorientation curves as seen in Fig. 6.
F(2,1)
S(2,1)
S(1,2)
F(1,2)
Fig. 6 Variant reorientation data for a
Ni50Mn28.5Ga21.5alloy.
-
28 Issac Nelson, Constantin Ciocnel and Heidi Feigenbaum
To improve the reorientation plots shown in Fig. 6, additional
experiments need to be performed to provide data particularly in
the regions below 2MPa stress or 0.6T field. Data points in these
regions are the most difficult to acquire because when a low
constant stress is combined with a medium to low field,
reorientation will occur only in one direction, from variant 1 to
2, as the stress will not be able to overcome the field to reorient
the variants back to variant 1. And similarly should for a low
constant field with medium to low compressive stress value.
chip
Fig. 7 Tested specimen exhibiting cracks and chips after being
loaded dynamically.
crack
4. Conclusions
1. The results shown in Fig. 6 demonstrate that a single model
for when reorientation begins and ends which can capture all the
three loading conditions used in this work (variable stress
constant field, variable field constant stress, and variable field
variable stress) is possible.
2. Lines have been drawn in Fig. 6 to show the approximate
reorientation curves given by the data. Despite the large scatter
of the data, these lines show that a single curve can capture
relatively well each of the key reorientation points (the start 1
to 2, start 2 to 1, finish 1 to 2 and finish 2 to 1) no matter
which type of loading the material was subject to (constant stress
variable field, constant field variable stress or variable stress
variable field).
3. While this only shows that a single model to capture the
three load cases used here could be developed, these load cases are
diverse enough to suggest that a single model that can simulate any
2D load case is possible.
4. Reorientation functions that mimic the curves seen in Fig. 6
will be important features of a phenomenological model that can
capture any loading case with a single calibration.
Acknowledgements. This material is based upon work supported by
the
National Science Foundation under Grant No. 0923517 and Grant
No. 1101108. The authors would also like to thank AdaptaMat Ltd.
for providing the MSMA samples.
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Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 29
REFERENCES
Couch R.N., Sirohi J., Chopra I., Development of a Quasi-static
Model of NiMnGa Magnetic Shape Memory Alloy. Journal of Intelligent
Material Systems and Structures, 18, 611-622 (2007).
Karaca H.E., Karaman I., Basaran B., Chumlyakov Y.I., Maier
H.J., Magnetic Field and Stress Induced Martensite Reorientation in
NiMnGa Ferromagnetic Shape Memory Alloy Single Crystals. Acta
Materialia, 54, 233245 (2006).
Kiefer B., Lagoudas D., Magnetic Field-induced Martensitic
Variant Reorienta-tion in Magnetic Shape Memory Alloys.
Philosophical Magazine, 85, 42894329 (2005).
Kiefer B., Lagoudas D., Modeling the Coupled Strain and
Magnetization Response of Magnetic Shape Memory Alloys under
Magnetomechanical Loading. Journal of Intelligent Material Systems
and Structures, 20, 143-170 (2009).
Marioni M., Bono D., Banful A., del Rosario M., Rodrigue E.,
Peterson B., Allen S., OHandley R., Pulsed Field Actuation of
Ni-Mn-Ga Ferromagnetic Shape Memory Alloy Single Crystal. Phys. IV
France, 112, 10011004 (2003).
Waldauer A.B., Feigenbaum H.P., Ciocanel C.,, Improvements to
the Kiefer and Lagoudas Model for Prediction of the
Magneto-Mechanical Behavior of Magnetic Shape Memory Alloys. ASME
2011 Conference on Smart Materials, Adaptive Structures and
Intelligent Systems (SMASIS 2011), Scottsdale, AZ, 2011.
CURBE DE REORIENTARE PENTRU UN ALIAJ Ni50Mn28.5Ga21.5
(Rezumat) Lucrarea prezint curbele de reorientare obinute
experimental pentru un aliaj
magnetic cu memoria formei. Aceste curbe de reorientare
caracterizeaz rspunsul materialului la diferite tipuri de ncrcri i
sunt necesare pentru simularea rspunsului materialului cu modele de
material fenomenologice. Rezultatele susin posibilitatea crerii
unui model fenomenologic care s foloseasc un singur set de curbe de
reorientare pentru obinerea parametrilor modelului, parametri cu
care modelul poate simula toate tipurile de ncrcri la care poate fi
supus materialul.
-
BULETINUL INSTITUTULUI POLITEHNIC DIN IAI Publicat de
Universitatea Tehnic Gheorghe Asachi din Iai, Tomul LVIII
(LXII), Fasc. 3, 2012
Secia CONSTRUCII DE MAINI
PARTICULATE CORDIERITE - ZIRCONIA COMPOSITES
BY
CIPRIAN ATNSOAEI, VIOREL GOAN and PAUL DORU BRSNESCU
Gheorghe Asachi Technical University of Iasi, Department of
Mechanical Engineering, Mechatronics and Robotics
Received: May 2012 Accepted for publication: June 2012
Abstract. The literature proposes many studies on improving
the
mechanical properties of zirconia (ZrO2) reinforced cordierite.
Strengthening mechanisms operating in the system cordierite -
zirconia are based on large difference between thermal expansion of
the two components and transformation of zirconia from tetragonal
to monoclinic state (transformation toughening). The best results
were obtained especially when transformation of zirconia in zircon
was avoided by using Yttria Stabilized Zirconia (YSZ) combined with
a high heating rate or Rate-Controlled-Sintering (RCS).
Key words: cordierita, zirconia, zircon, composite
materials.
1. Introduction
Cordierite is a ceramic with a very low thermal expansion (2
10-6/C), good dielectric properties (~4 at 1 MHz), high chemical
stability and a relatively low cost price. These qualities are
often valued in the manufacture of structural ceramic components
for use in aggressive environments (catalytic converters,
multilayer circuit boards, kiln furniture, thermal insulation
parts, diesel particulate filters, etc.). However, the poor
mechanical properties limit the usefulness of this material. It is
widely accepted that the best way to improve the mechanical
properties of cordierite ceramics is to use fibers or
Corresponding author: e-mail: [email protected]
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32 Ciprian Atnsoaei et al.
particulate reinforcements. The range of reinforcement materials
used is quite large: diamond (Hasselman et al., 1994), mullite
(Ebadzadeh & Lee, 1998), SiC (Wadsworth & Stevens, 1991),
ZrO2 (Wadsworth et al., 1990), Si3N4 (Zamir et al., 2010) etc. A
quality composite requires careful control of each stage of
processing. This is even more important in the case of cordierite
because its sintering temperature range is very narrow (about 50C).
In addition, zirconia reinforced cordierite is prone to a
particular problem: the tendency of zirconia to react with the
cordierite matrix, resulting in zircon - a compound with poor
mechanical properties.
The scientific literature gave pretty much attention to
reinforcement of cordierite with zirconia particles proposing
several variants of technological routes, each having results more
or less significant. This paper aims at highlighting the
technological details that lead to obtain substantial improvements
of mechanical properties of zirconia toughened cordierite.
2. Toughening Mechanisms in Zirconia-Cordierite Composites 2.1.
Transformation Toughening
In terms of crystalline structure, zirconia shows three states:
monoclinic (m zirconia), tetragonal (t zirconia) and cubic (c
zirconia) (Xia & Langdon, 1994). The monoclinic structure can
be kept up to 950C. The temperatures between 9502370C correspond to
tetragonal structure and beyond 2370C usually we can find only the
cubic structure. Regarding the transformation toughening mechanism,
the most important fact is zirconia transition from tetragonal to
monoclinic structure (t-m transformation), because it implies a
volume increase of about 4%. If zirconia is dispersed in a
(ceramic) matrix with a similar thermal expansion, t-m
transformation will significantly change the stress field within
the ceramic body. Therefore, transformation toughening alters the
normal propagation of macrocracks by creating stress concentrators
into the ceramic body.
In the case of zirconia toughened cordierite, transformation
toughening is not always a very important mechanism influencing the
composite performance because the thermal expansion of cordierite
is much lower than that of zirconia, This allows reinforcing
particle to have a volume increase greater than that induced by the
t-m transformation (Wadsworth et. al., 1990). Besides this, the
increase in volume due to t-m transformation is similar to volume
shrinkage after sintering, and thus the two phenomena are partially
or totally cancelling their effects. On the other hand, the
combination of several factors (the use of additives, particle
size, homogeneity, etc.) can inhibit the t-m transformation.
As one can see, when we are talking about zirconia toughened
cordierite, the contribution of transformation toughening to
mechanical performance is hard to predict. In practice, the
presence of the transformation toughening mechanism can be
evaluated by X-ray diffraction (XRD), comparing the values
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Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 3, 2012 33
measured for m-zirconia, before and after polishing the surface
studied (Jang, 1995).
2.2. Microcracking
The trajectory of a macrocrack can be deflected if it encounters
areas with different mechanical properties (obstacles, different
stress fields, etc.). The mechanism of microcracking involves the
appearance of cracks in the ceramic body. These microcracks occur
in areas with high tensile stresses which attract the microcrack
and divert its trajectory (Fig. 1).
Fig. 1 Macrocrack deflected by Fig. 2 A crack front pinned by
particles microcracks (Xia & Langdon, 1994). (Xia &
Langdon, 1994).
Increasing the length of macrocrack path can reduce its energy
below the
limit at which propagation is possible. In the
zirconia-cordierite system, microcracks are induced by the
large
difference of thermal expansion of the constituents. Of course,
despite the fact that it is not always very efficient, t-m
transformation can be another microcracks source.
2.3. Crack Pinning
Crack pinning mechanism consists in stopping the propagation of
a macrocrack when it meets a particle or a fiber which has the
tensile strength greater than the macrocrack tip stress (Fig. 2).
In the zirconia-cordierite system, the big difference between the
mechanical properties of the constituents makes the crack pinning
mechanism to be very effective. However, the efficiency of this
mechanism is closely related to the quality of cohesion between
matrix and reinforcement material.
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34 Ciprian Atnsoaei et al.
3. Effect of Starting Powders Preparation 3.1. Particle Size
Effect (Milling and Mixing Effect)
Both theoretical and practical (Jang, 1995), is well known that
densification of powders can be significantly improved by reducing
the size of the starting powders. This is also true in
zirconia-cordierite system, where milling up to 24h or more can
reduce the optimal sintering temperature below 1300C. The
mechanical properties, especially fracture toughness, hardness and
f
