The cold-drawn, programmed shape memory polymer (SMP) fibers show excellent stress recovery property, which promotes their application as mechanical actuators in smart material systems. A full understanding of the thermomechanical-damage responses of these fibers is crucial to minimize the trial-and-error manufacturing processes of these material systems. In this work, a multiscale viscoplastic-viscodamage theory is developed to predict the cyclic mechanical responses of SMP fibers. The proposed viscoplastic theory is based on the governing relations for each of the individual microconstituents and establishes the microscale state of the stress and strain in each of the subphases. These microscale fields are then averaged through the micromechanics framework to demonstrate the macroscale constitutive mechanical behavior. The cyclic loss in the functionality of the SMP fibers is interpreted as the damage process herein, and this cyclic loss of stress recovery property is calibrated to identify the state of the damage. The continuum damage mechanics (CDM) together with a thermodynamic consistent viscodamage theory is incorporated to simulate the damage process. The developed coupled viscoplastic-viscodamage theory provides an excellent correlation between the experimental and simulation results. The cyclic loading-damage analysis in this work relies on the underlying physical facts and accounts for the microstructural changes in each of the micro constituents. The established framework provides a well-structured method to capture the cyclic responses of the SMP fibers, which is of utmost importance for designing the SMP fiber-based smart material systems.
Issue Section:
Research Papers
References
1.
Ayoub
, G.
, Zaïri
, F.
, Fréderix
, C.
, Gloaguen
, J. M.
, Naït-Abdelaziz
, M.
, Seguela
, R.
, and Lefebvre
, J. M.
, 2011
, “Effects of Crystal Content on the Mechanical Behaviour of Polyethylene Under Finite Strains: Experiments and Constitutive Modelling
,” Int. J. Plast.
, 27
, pp.
492
–511
.10.1016/j.ijplas.2010.07.0052.
Qi
, H. J.
, and Boyce
, M. C.
, 2005
, “Stress–Strain Behavior of Thermoplastic Polyurethanes
,” Mech. Mater.
, 37
(8
), pp.
817
–839
.10.1016/j.mechmat.2004.08.0013.
Qi
, H. J.
, Nguyen
, T. D.
, Castro
, F.
, Yakacki
, C. M.
, and Shandas
, R.
, 2008
, “Finite Deformation Thermo-Mechanical Behavior of Thermally Induced Shape Memory Polymers
,” J. Mech. Phys. Solids
, 56
(5
), pp.
1730
–1751
.10.1016/j.jmps.2007.12.0024.
Ayoub
, G.
, Zaïri
, F.
, Naït-Abdelaziz
, M.
, and Gloaguen
, J. M.
, 2010
, “Modelling Large Deformation Behaviour Under Loading-Unloading of Semicrystalline Polymers: Application to a High Density Polyethylene
,” Int. J. Plast.
, 26
(3
), pp.
329
–347
.10.1016/j.ijplas.2009.07.0055.
Srivastava
, V.
, Chester
, S. A.
, Ames
, N. M.
, and Anand
, L.
, 2010
, A Thermo-Mechanically-Coupled Large-Deformation Theory for Amorphous Polymers in a Temperature Range Which Spans Their Glass Transition
,” Int. J. Plast.
, 26
(8
), pp.
1138
–1182
.10.1016/j.ijplas.2010.01.0046.
Srivastava
, V.
, Chester
, S. A.
, and Anand
, L.
, 2010
, “Thermally Actuated Shape-Memory Polymers: Experiments, Theory, and Numerical Simulations
,” J. Mech. Phys. Solids
, 58
(8
), pp.
1100
–1124
.10.1016/j.jmps.2010.04.0047.
Westbrook
, K. K.
, Kao
, P. H.
, Castro
, F.
, Ding
, Y.
, and Qi
, H. J.
, 2011
, “A 3D Finite Deformation Constitutive Model for Amorphous Shape Memory Polymers: A Multi-Branch Modeling Approach for Nonequilibrium Relaxation Processes
,” Mech. Mater.
, 43
(12
), pp.
853
–869
.10.1016/j.mechmat.2011.09.0048.
Yakacki
, C. M.
, Nguyen
, T. D.
, Likos
, R.
, Lamell
, R.
, Guigou
, D.
, and Gall
, K.
, 2011
, “Impact of Shape-Memory Programming on Mechanically-Driven Recovery in Polymers
,” Polymer
, 52
(21
), pp.
4947
–4954
.10.1016/j.polymer.2011.08.0279.
Li
, G.
, and John
, M.
, 2008
, A Self-Healing Smart Syntactic Foam Under Multiple Impacts
,” Compos. Sci. Technol.
, 68
(15–16
), pp.
3337
–3343
.10.1016/j.compscitech.2008.09.00910.
Li
, G.
, and Nettles
, D.
, 2010
, “Thermomechanical Characterization of a Shape Memory Polymer Based Self-Repairing Syntactic Foam
,” Polymer
, 51
(3
), pp.
755
–762
.10.1016/j.polymer.2009.12.00211.
Li
, G.
, and Shojaei
, A.
, 2012, “A Viscoplastic Theory of Shape Memory Polymer Fibres With Application to Self-Healing Materials
,” Proc. R. Soc. London, Ser. A
, (in press)
.12.
Li
, G.
, and Uppu
, N.
, 2010
, “Shape Memory Polymer Based Self-Healing Syntactic Foam: 3-D Confined Thermomechanical Characterization
,” Compos. Sci. Technol.
, 70
(9
), pp.
1419
–1427
.10.1016/j.compscitech.2010.04.02613.
Voyiadjis
, G. Z.
, Shojaei
, A.
, and Li
, G.
, 2011
, “A Thermodynamic Consistent Damage and Healing Model for Self Healing Materials
,” Int. J. Plast.
, 27
(7
), pp.
1025
–1044
.10.1016/j.ijplas.2010.11.00214.
Voyiadjis
, G. Z.
, Shojaei
, A.
, and Li
, G.
, 2012
, “A Generalized Coupled Viscoplastic–Viscodamage–Viscohealing Theory for Glassy Polymers
,” Int. J. Plast.
, 28
(1
), pp.
21
–45
.10.1016/j.ijplas.2011.05.01215.
Voyiadjis
, G. Z.
, Shojaei
, A.
, Li
, G.
, and Kattan
, P.
, 2012
, “Continuum Damage-Healing Mechanics With Introduction to New Healing Variables
,” Int. J. Damage Mech.
, 21
(3
), pp.
391
–414
.10.1177/105678951039706916.
Voyiadjis
, G. Z.
, Shojaei
, A.
, Li
, G.
, and Kattan
, P. I.
, 2012
, “A Theory of Anisotropic Healing and Damage Mechanics of Materials
,” Proc. R. Soc. London, Ser. A
, 468
(2137
), pp.
163
–183
.10.1098/rspa.2011.032617.
Xu
, W.
, and Li
, G.
, 2010
, “Constitutive Modeling of Shape Memory Polymer Based Self-Healing Syntactic Foam
,” Int. J. Solids Struct.
, 47
(9
), pp.
1306
–1316
.10.1016/j.ijsolstr.2010.01.01518.
White
, S. R.
, Sottos
, N. R.
, Geubelle
, P. H.
, Moore
, J. S.
, Kessler
, M. R.
, Sriram
, S. R.
, Brown
, E. N.
, and Viswanathan
, S.
, 2001
, “Autonomic Healing of Polymer Composites
,” Nature
, 409
(6822
), pp.
794
–797
.10.1038/3505723219.
Toohey
, K. S.
, Sottos
, N. R.
, Lewis
, J. A.
, Moore
, J. S.
, and White
, S. R.
, 2007
, “Self-Healing Materials With Microvascular Networks
,” Nature Mater.
, 6
(8
), pp.
581
–585
.10.1038/nmat193420.
Kirkby
, E. L.
, Michaud
, V. J.
, Månson
, J. A. E.
, Sottos
, N. R.
, and White
, S. R.
, 2009
, “Performance of Self-Healing Epoxy With Microencapsulated Healing Agent and Shape Memory Alloy Wires
,” Polymer
, 50
(23
), pp.
5533
–5538
.10.1016/j.polymer.2009.05.01421.
Zako
, M.
, and Takano
, N.
, 1999
, “Intelligent Material Systems Using Epoxy Particles to Repair Microcracks and Delamination Damage in GFRP
,” J. Intell. Mater. Syst. Struct.
, 10
(10
), pp.
836
–841
.22.
Nji
, J.
, and Li
, G.
, 2010
, “A Biomimic Shape Memory Polymer Based Self-Healing Particulate Composite
,” Polymer
, 51
(25
), pp.
6021
–6029
.10.1016/j.polymer.2010.10.02123.
Nji
, J.
, and Li
, G.
, 2010
, “A Self-Healing 3D Woven Fabric Reinforced Shape Memory Polymer Composite for Impact Mitigation
,” Smart Mater. Struct.
, 19
(3
), pp.
1
–9
.10.1088/0964-1726/19/3/03500724.
Meng
, Q. H.
, and Hu
, J. L.
, 2008
, “The Influence of Heat Treatment on Properties of Shape Memory Fibers: I. Crystallinity, Hydrogen Bonding and Shape Memory Effect
,” J. Appl. Polym. Sci.
, 109
, pp.
2616
–2623
.10.1002/app.2836325.
Clough
, S. B.
, and Schneider
, N. S.
, 1968
, “Structural Studies on Urethane Elastomers
,” J. Macromol. Sci., Phys.
, 2
(4
), pp.
553
–566
.10.1080/0022234680821245826.
Clough
, S. B.
, Schneider
, N. S.
, and King
, A. O.
, 1968
, “Small-Angle X-Ray Scattering From Polyurethane Elastomers
,” J. Macromol. Sci., Phys.
, 2
(4
), pp.
641
–648
.10.1080/0022234680821246327.
Ferguson
, J.
, Hourston
, D. J.
, Meredith
, R.
, and Patsavoudis
, D.
, 1972
, “Mechanical Relaxations in a Series of Polyurethanes With Varying Hard to Soft Segment Ratio
,” Eur. Polym. J.
, 8
(3
), pp.
369
–383
.10.1016/0014-3057(72)90102-428.
Ferguson
, J.
, and Patsavoudis
, D.
, 1972
, “Chemical Structure-Physical Property Relationships in Polyurethane Elastomeric Fibres; Property Variations in Polymers Containing High Hard Segment Concentrations
,” Eur. Polym. J.
, 8
(3
), pp.
385
–396
.10.1016/0014-3057(72)90103-629.
Chaboche
, J. L.
, 2008
, “A Review of Some Plasticity and Viscoplasticity Constitutive Theories
,” Int. J. Plast.
, 24
(10
), pp.
1642
–1693
.10.1016/j.ijplas.2008.03.00930.
Chaboche
, J. L.
, Kruch
, S.
, Maire
, J. F.
, and Pottier
, T.
, 2001
, “Towards a Micromechanics Based Inelastic and Damage Modeling of Composites
,” Int. J. Plast.
, 17
(4
), pp.
411
–439
.10.1016/S0749-6419(00)00056-531.
Eshelby
, J. D.
, 1957
, “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,” Proc. R. Soc. London, Ser. A
, 241
(1226
), pp.
376
–396
.10.1098/rspa.1957.013332.
Fotiu
, P. A.
, and Nemat-Nasser
, S.
, 1996
, “Overall Properties of Elastic-Viscoplastic Periodic Composites
,” Int. J. Plast.
, 12
(2
), pp.
163
–190
.10.1016/S0749-6419(96)00002-233.
Kruch
, S.
, and Chaboche
, J. L.
, 2011
, “Multi-Scale Analysis in Elasto-Viscoplasticity Coupled With Damage
,” Int. J. Plast.
, 27
(12
), pp.
2026
–2039
.10.1016/j.ijplas.2011.03.00734.
Nemat-Nasser
, S.
, and Hori
, M.
, 1993
, Micromechanics: Overall Properties of Heterogeneous Materials
, Elsevier
, Amsterdam
.35.
Ames
, N. M.
, Srivastava
, V.
, Chester
, S. A.
, and Anand
, L.
, 2009
, “A Thermo-mechanically Coupled Theory for Large Deformations of Amorphous Polymers. Part II: Applications
,” Int. J. Plast.
, 25
(8
), pp.
1495
–1539
.10.1016/j.ijplas.2008.11.00536.
Anand
, L.
, and Ames
, N. M.
, 2006
, “On Modeling the Micro-Indentation Response of an Amorphous Polymer
,” Int. J. Plast.
, 22
(6
), pp.
1123
–1170
.10.1016/j.ijplas.2005.07.00637.
Anand
, L.
, Ames
, N. M.
, Srivastava
, V.
, and Chester
, S. A.
, 2009
, “A Thermo-Mechanically Coupled Theory for Large Deformations of Amorphous Polymers. Part I: Formulation
,” Int. J. Plast.
, 25
(8
), pp.
1474
–1494
.10.1016/j.ijplas.2008.11.00438.
Naghdi
, P. M.
, and Trapp
, J. A.
, 1975
, “Restrictions on Constitutive Equations of Finitely Deformed Elastic-Plastic Materials
,” Q. J. Mech. Appl. Math.
, 28
(1
), pp.
25
–46
.10.1093/qjmam/28.1.2539.
Green
, A. E.
, and Naghdi
, P. M.
, 1971
, “Some Remarks on Elastic-Plastic Deformation at Finite Strain
,” Int. J. Eng. Sci.
, 9
(12
), pp.
1219
–1229
.10.1016/0020-7225(71)90086-340.
Green
, A. E.
, and Naghdi
, P. M.
, 1965
, “A General Theory of an Elastic-Plastic Continuum
,” Arch. Ration. Mech. Anal.
, 18
(4
), pp.
251
–281
.10.1007/BF0025166641.
Hansen
, N. R.
, and Schreyer
, H. L.
, 1994
, “A Thermodynamically Consistent Framework for Theories of Elastoplasticity Coupled With Damage
,” Int. J. Solids Struct.
, 31
(3
), pp.
359
–389
.10.1016/0020-7683(94)90112-042.
Kapoor
, R.
, and Nemat-Nasser
, S.
, 1998
, “Determination of Temperature Rise During High Strain Rate Deformation
,” Mech. Mater.
, 27
(1
), pp.
1
–12
.10.1016/S0167-6636(97)00036-743.
Nemat-Nasser
, S.
, Isaacs
, J. B.
, and Liu
, M.
, 1998
, “Microstructure of High-Strain, High-Strain-Rate Deformed Tantalum
,” Acta Mater.
, 46
(4
), pp.
1307
–1325
.10.1016/S1359-6454(97)00746-544.
Beheshti
, A.
, and Khonsari
, M. M.
, 2011
, “On the Prediction of Fatigue Crack Initiation in Rolling/Sliding Contacts With Provision for Loading Sequence Effect
,” Tribol. Int.
, 44
(12
), pp.
1620
–1628
.10.1016/j.triboint.2011.05.01745.
Aghdam
, A. B.
, Beheshti
, A.
, and Khonsari
, M. M.
, 2012
, “On the Fretting Crack Nucleation With Provision for Size Effect
,” Tribol. Int.
, 47
, pp.
32
–43
.10.1016/j.triboint.2011.10.00146.
Lemaitre
, J.
, and Dufailly
, J.
, 1987
, “Damage Measurements
,” Eng. Fract. Mech.
, 28
(5–6
), pp.
643
–661
.10.1016/0013-7944(87)90059-247.
Simo
, J. C.
and Hughes
, T. J. R.
, 1997
, Computational Inelasticity
, Springer
, New York
.48.
Nemat-Nasser
, S.
, 2006
, Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials
, Cambridge University Press
, Cambridge, England
.49.
Simo
, J. C.
, and Ortiz
, M.
, 1985
, “A Unified Approach to Finite Deformation Elastoplastic Analysis Based on the Use of Hyperelastic Constitutive Equations
,” Comput. Methods Appl. Mech. Eng.
, 49
(2
), pp.
221
–245
.10.1016/0045-7825(85)90061-150.
Boyce
, M. C.
, Parks
, D. M.
, and Argon
, A. S.
, 1989
, “Plastic Flow in Oriented Glassy Polymers
,” Int. J. Plast.
, 5
(6
), pp.
593
–615
.10.1016/0749-6419(89)90003-X51.
Anand
, L.
, 1979
, “On H. Hencky's Approximate Strain-Energy Function for Moderate Deformations
,” ASME J. Appl. Mech.
, 46
(1
), pp.
78
–82
.10.1115/1.342453252.
Ping
, P.
, Wang
, W.
, Chen
, X.
, and Jing
, X.
, 2005
, “Poly(ɛ-Caprolactone) Polyurethane and Its Shape-Memory Property
,” Biomacromolecules
, 6
(2
), pp.
587
–592
.10.1021/bm049477j53.
Li
, G.
, and Xu
, W.
, 2011
, “Thermomechanical Behavior of Thermoset Shape Memory Polymer Programmed by Cold-Compression: Testing and Constitutive Modeling
,” J. Mech. Phys. Solids
, 59
(6
), pp.
1231
–1250
.10.1016/j.jmps.2011.03.00154.
Wang
, T. T.
, 1973
, “Morphology and Mechanical Properties of Crystalline Polymers. I. Transcrystalline Polyethylene
,” J. Appl. Phys.
, 44
(5
), pp.
2218
–2224
.10.1063/1.166254055.
Stevenson
, A.
, 1995
, “Effect of Crystallization on the Mechanical Properties of Elastomers Under Large Deformation
,” Current Research in the Thermo-Mechanics of Polymers in the Rubbery-Glassy Range
, ASME
, New York
.56.
Shojaei
, A.
, and Li
, G.
, 2012
, “Viscoplasticity Analysis of Semicrystalline Polymers: A Multiscale Approach Within Micromechanics Framework
,” Int. J. Plast.
, (in press).10.1016/j.ijplas.2012.09.01457.
Argon
, A. S.
, 1973
, “A Theory for the Low-Temperature Plastic Deformation of Glassy Polymers
,” Philos. Mag.
, 28
(4
), pp.
839
–865
.10.1080/1478643730822098758.
Arruda
, E. M.
, Boyce
, M. C.
, and Quintus-Bosz
, H.
, 1993
, “Effects of Initial Anisotropy on the Finite Strain Deformation Behavior of Glassy Polymers
,” Int. J. Plast.
, 9
(7
), pp.
783
–811
.10.1016/0749-6419(93)90052-R59.
Arruda
, E. M.
, and Boyce
, M. C.
, 1993
, “A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials
,” J. Mech. Phys. Solids
, 41
(2
), pp.
389
–412
.10.1016/0022-5096(93)90013-660.
Arruda
, E. M.
, and Boyce
, M. C.
, 1993
, “Evolution of Plastic Anisotropy in Amorphous Polymers During Finite Straining
,” Int. J. Plast.
, 9
(6
), pp.
697
–720
.10.1016/0749-6419(93)90034-N61.
Boyce
, M. C.
and Arruda
, E. M.
, 1990
, “An Experimental and Analytical Investigation of the Large Strain Compressive and Tensile Response of Glassy Polymers
,” Polym. Eng. Sci.
, 30
(20
), pp.
1288
–1298
.10.1002/pen.76030200562.
Boyce
, M. C.
, Parks
, D. M.
, and Argon
, A. S.
, 1988
, “Large Inelastic Deformation of Glassy Polymers. Part II: Numerical Simulation of Hydrostatic Extrusion
,” Mech. Mater.
, 7
(1
), pp.
35
–47
.10.1016/0167-6636(88)90004-X63.
Boyce
, M. C.
, Parks
, D. M.
, and Argon
, A. S.
, 1988
, “Large Inelastic Deformation of Glassy Polymers. Part I: Rate Dependent Constitutive Model
,” Mech. Mater.
, 7
(1
), pp.
15
–33
.10.1016/0167-6636(88)90003-864.
Cohen
, A.
, 1991
, “A Pade' Approximant to the Inverse Langevin Function
,” Rheol. Acta
, 30
, pp.
270
–273
.10.1007/BF0036664065.
Parks
, D. M.
, and Ahzi
, S.
, 1990
, “Polycrystalline Plastic Deformation and Texture Evolution for Crystals Lacking Five Independent Slip Systems
,” J. Mech. Phys. Solids
, 38
(5
), pp.
701
–724
.10.1016/0022-5096(90)90029-466.
Lee
, B. J.
, Parks
, D. M.
, and Ahzi
, S.
, 1993
, “Micromechanical Modeling of Large Plastic Deformation and Texture Evolution in Semi-Crystalline Polymers
,” J. Mech. Phys. Solids
, 41
(10
), pp.
1651
–1687
.10.1016/0022-5096(93)90018-B67.
Hutchinson
, J. W.
, 1976
, “Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials
,” Proc. R. Soc. London, Ser. A
, 348
(1652
), pp.
101
–127
.10.1098/rspa.1976.002768.
Asaro
, R. J.
, 1979
, “Geometrical Effects in the Inhomogeneous Deformation of Ductile Single Crystals
,” Acta Metall.
, 27
(3
), pp.
445
–453
.10.1016/0001-6160(79)90036-169.
Asaro
, R. J.
, and Rice
, J. R.
, 1977
, “Strain Localization in Ductile Single Crystals
,” J. Mech. Phys. Solids
, 25
(5
), pp.
309
–338
.10.1016/0022-5096(77)90001-170.
Negahban
, M.
, 1997
, “Thermodynamic Modeling of the Thermomechanical Effects of Polymer Crystallization: A General Theoretical Structure
,” Int. J. Eng. Sci.
, 35
(3
), pp.
277
–298
.10.1016/S0020-7225(96)00078-X71.
Negahban
, M.
, Wineman
, A. S.
, and Ma
, R.-J.
, 1993
, “Simulation of Mechanical Response in Polymer Crystallization
,” Int. J. Eng. Sci.
, 31
(1
), pp.
93
–113
.10.1016/0020-7225(93)90068-672.
Nemat-Nasser
, S.
, 1979
, “Decomposition of Strain Measures and Their Rates in Finite Deformation Elastoplasticity
,” Int. J. Solids Struct.
, 15
(2
), pp.
155
–166
.10.1016/0020-7683(79)90019-273.
Hershey
, A. V.
, 1954
, “The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystal
,” ASME J. Appl. Mech.
, 21
, pp.
236
–241
.74.
Freed
, A. D.
, and Walker
, K. P.
, 1990
, “Steady-State and Transient Zener Parameters in Viscoplasticity: Drag Strength Versus Yield Strength
,” Appl. Mech. Rev.
, 43
(5S
), pp.
S328
–S337
.10.1115/1.312083675.
Deliktas
, B.
, Voyiadjis
, G. Z.
, and Palazotto
, A. N.
, 2009
, “Simulation of Perforation and Penetration in Metal Matrix Composite Materials Using Coupled Viscoplastic Damage Model
,” Composites, Part B
, 40
(6
), pp.
434
–442
.10.1016/j.compositesb.2009.04.01976.
Voyiadjis
, G. Z.
and Deliktas
, B.
, 2000
, “A Coupled Anisotropic Damage Model for the Inelastic Response of Composite Materials
,”. Comput. Methods Appl. Mech. Eng.
, 183
(3–4
), pp.
159
–199
.10.1016/S0045-7825(99)00218-277.
Voyiadjis
, G. Z.
, Deliktas
, B.
, and Palazotto
, A. N.
, 2009
, “Thermodynamically Consistent Coupled Viscoplastic Damage Model for Perforation and Penetration in Metal Matrix Composite Materials
,” Composites, Part B
, 40
(6
), pp.
427
–433
.10.1016/j.compositesb.2009.01.00878.
Lemaitre
, J.
, and Chaboche
, J. L.
, 1990
, Mechanics of Solid Materials
, Cambridge University Press
, Cambridge, England
.79.
Lemaitre
, J.
, 1985
, “Coupled Elasto-Plasticity and Damage Constitutive Equations
,” Comput. Methods Appl. Mech. Eng.
, 51
(1–3
), pp.
31
–49
.10.1016/0045-7825(85)90026-X80.
Echle
, R.
, and Voyiadjis
, G. Z.
, 1999
, “Simulation of Damage Evolution in a Uni-Directional Titanium Matrix Composite Subjected to High Cycle Fatigue
,” Int. J. Fatigue
, 21
(9
), pp.
909
–923
.10.1016/S0142-1123(99)00082-181.
Voyiadjis
, G. Z.
, and Echle
, R.
, 1998
, “High Cycle Fatigue Damage Evolution in Uni-Directional Metal Matrix Composites Using a Micro-Mechanical Approach
,” Mech. Mater.
, 30
(2
), pp.
91
–110
.10.1016/S0167-6636(98)00040-482.
Shojaei
, A.
, Eslami
, M.
, and Mahbadi
, H.
, 2010
, “Cyclic Loading of Beams Based on the Chaboche Model
,” Int. J. Mech. Mater. Des.
, 6
(3
), pp.
217
–228
.10.1007/s10999-010-9131-583.
Horgan
, C. O.
, Ogden
, R. W.
, and Saccomandi
, G.
, 2004
, “A Theory of Stress Softening of Elastomers Based on Finite Chain Extensibility
,” Proc. R. Soc. London, Ser. A
, 460
(2046
), pp.
1737
–1754
.10.1098/rspa.2003.124884.
Govindjee
, S.
, and Simo
, J.
, 1992
, “Transition From Micro-Mechanics to Computationally Efficient Phenomenology: Carbon Black Filled Rubbers Incorporating Mullins' Effect
,” J. Mech. Phys. Solids
, 40
(1
), pp.
213
–233
.10.1016/0022-5096(92)90324-U85.
Govindjee
, S.
, and Simo
, J.
, 1991
, A Micro-Mechanically Based Continuum Damage Model for Carbon Black-Filled Rubbers Incorporating Mullins' Effect
,” J. Mech. Phys. Solids
, 39
(1
), pp.
87
–112
.10.1016/0022-5096(91)90032-J86.
Simo
, J. C.
, Taylor
, R. L.
, and Pister
, K. S.
, 1985
, “Variational and Projection Methods for the Volume Constraint in Finite Deformation Elasto-Plasticity
,” Comput. Methods Appl. Mech. Eng.
, 51
(1–3
), pp.
177
–208
.10.1016/0045-7825(85)90033-787.
Li
, G.
, Meng
, H.
, and Hu
, J.
, 2012, “Healable Thermoset Polymer Composite Embedded With Stimuli-Responsive Fibers
,” J. R. Soc., Interface
, (submitted)
.Copyright © 2013 by ASME
You do not currently have access to this content.
