Abstract

In this paper, we review constitutive models for soft materials. We specifically focus on physically based models accounting for hyperelasticity, visco-hyperelasticity, and damage phenomena. For completeness, we include the thermodynamically based viscohyperelastic and damage models as well as the so-called mixed models. The models are put in the frame of statistical mechanics and thermodynamics. Based on the available experimental data, we provide a quantitative comparison of the hyperelastic models. This information can be used as guidance in the selection of suitable constitutive models. Next, we consider visco-hyperelasticity in the frame of the thermodynamic theory and molecular chain dynamics. We provide a concise summary of the viscohyperelastic models including specific strain energy density function, the evolution laws of internal variables, and applicable conditions. Finally, we review the models accounting for damage phenomenon in soft materials. Various proposed damage criteria are summarized and discussed in connection with the physical interpretations that can be drawn from physically based damage models. The discussed mechanisms include the breakage of polymer chains, debonding between polymer chains and fillers, disentanglement, and so on.

Issue Section:
Review Article
Keywords:
constitutive modeling of soft materials, mechanical properties of soft materials
Topics:
Chain, Damage, Deformation, Stress, Constitutive equations

References

1.
Lee
,
J. H.
,
Chung
,
Y. S.
, and
Rodrigue
,
H.
,
2019
, “
Long Shape Memory Alloy Tendon-Based Soft Robotic Actuators and Implementation as a Soft Gripper
,”
Sci. Rep.
,
9
(
1
), p.
11251
. 10.1038/s41598-019-47794-1
2.
Schaffner
,
M.
,
Faber
,
J. A.
,
Pianegonda
,
L.
,
Ruhs
,
P. A.
,
Coulter
,
F.
, and
Studart
,
A. R.
,
2018
, “
3D Printing of Robotic Soft Actuators With Programmable Bioinspired Architectures
,”
Nat. Commun.
,
9
(
1
), p.
878
. 10.1038/s41467-018-03216-w
3.
Tyagi
,
M.
,
Pan
,
J.
, and
Jager
,
E. W. H.
,
2019
, “
Novel Fabrication of Soft Microactuators With Morphological Computing Using Soft Lithography
,”
Microsyst. Nanoeng.
,
5
(
1
), p.
44
. 10.1038/s41378-019-0092-z
4.
Rogers
,
J. A.
,
2013
, “
A Clear Advance in Soft Actuators
,”
Science
,
341
(
6149
), pp.
968
–
969
. 10.1126/science.1243314
5.
Qin
,
L.
,
Cao
,
J.
,
Tang
,
Y.
, and
Zhu
,
J.
,
2018
, “
Soft Freestanding Planar Artificial Muscle Based on Dielectric Elastomer Actuator
,”
ASME J. Appl. Mech.
,
85
(
5
), p.
5
. 10.1115/1.4039289
6.
Rus
,
D.
, and
Tolley
,
M. T.
,
2015
, “
Design, Fabrication and Control of Soft Robots
,”
Nature
,
521
(
7553
), pp.
467
–
475
. 10.1038/nature14543
7.
Li
,
T.
,
Zou
,
Z.
,
Mao
,
G.
,
Yang
,
X.
,
Liang
,
Y.
,
Li
,
C.
,
Qu
,
S.
,
Suo
,
Z.
, and
Yang
,
W.
,
2019
, “
Agile and Resilient Insect-Scale Robot
,”
Soft Rob.
,
6
(
1
), pp.
133
–
141
. 10.1089/soro.2018.0053
8.
Gu
,
G.
,
Zou
,
J.
,
Zhao
,
R.
,
Zhao
,
X.
, and
Zhu
,
X.
,
2018
, “
Soft Wall-Climbing Robots
,”
Sci Rob.
,
3
(
25
), p.
2874
. 10.1126/scirobotics.aat2874
9.
Hu
,
W.
,
Lum
,
G. Z.
,
Mastrangeli
,
M.
, and
Sitti
,
M.
,
2018
, “
Small-Scale Soft-Bodied Robot With Multimodal Locomotion
,”
Nature
,
554
(
7690
), pp.
81
–
85
. 10.1038/nature25443
10.
Rafsanjani
,
A.
,
Zhang
,
Y. R.
,
Liu
,
B. Y.
,
Rubinstein
,
S. M.
, and
Bertoldi
,
K.
,
2018
, “
Kirigami Skins Make a Simple Soft Actuator Crawl
,”
Sci Rob.
,
3
(
15
), p.
eaar7555
. 10.1126/scirobotics.aar7555
11.
Wang
,
Y.
,
Loh
,
L. Y. W.
,
Gupta
,
U.
,
Foo
,
C. C.
, and
Zhu
,
J.
,
2020
, “
Bio-Inspired Soft Swim Bladders of Large Volume Change Using Dual Dielectric Elastomer Membranes
,”
ASME J. Appl. Mech.
,
87
(
4
), p.
041007
. 10.1115/1.4045901
12.
Suzuki
,
Y.
,
Tanaka
,
T.
,
Kaneko
,
S.
,
Moromugi
,
S.
, and
Feng
,
M.
,
2005
, “
Soft Sensor Suits as Man-Machine Interface for Wearable Power Amplifier
,”
IEEE Sys. Man. Cybern.
,
2
, pp.
1680
–
1685
.
13.
Liu
,
J.
,
Mao
,
G.
,
Huang
,
X.
,
Zou
,
Z.
, and
Qu
,
S.
,
2015
, “
Enhanced Compressive Sensing of Dielectric Elastomer Sensor Using a Novel Structure
,”
ASME J. Appl. Mech.
,
82
(
10
), p.
101004
. 10.1115/1.4030889
14.
Kell
,
A. J.
,
Paquet
,
C.
,
Mozenson
,
O.
,
Djavani-Tabrizi
,
I.
,
Deore
,
B.
,
Liu
,
X.
,
Lopinski
,
G. P.
,
James
,
R.
,
Hettak
,
K.
,
Shaker
,
J.
,
Momciu
,
A.
,
Ferrigno
,
J.
,
Ferrand
,
O.
,
Hu
,
J. X.
,
Lafreniere
,
S.
, and
Malenfant
,
P. R. L.
,
2017
, “
Versatile Molecular Silver Ink Platform for Printed Flexible Electronics
,”
ACS Appl. Mater Interfaces
,
9
(
20
), pp.
17226
–
17237
. 10.1021/acsami.7b02573
15.
Gates
,
B. D.
,
2009
, “
Flexible Electronics
,”
Science
,
323
(
5921
), pp.
1566
–
1567
. 10.1126/science.1171230
16.
Edwards
,
S. F.
, and
Vilgis
,
T.
,
1986
, “
The Effect of Entanglements in Rubber Elasticity
,”
Polymer
,
27
(
4
), pp.
483
–
492
. 10.1016/0032-3861(86)90231-4
17.
Kaliske
,
M.
, and
Heinrich
,
G.
,
1999
, “
An Extended Tube-Model for Rubber Elasticity: Statistical-Mechanical Theory and Finite Element Implementation
,”
Rubber Chem. Technol.
,
72
(
4
), pp.
602
–
632
. 10.5254/1.3538822
18.
Meissner
,
B.
, and
Matějka
,
L.
,
2003
, “
A Langevin-Elasticity-Theory-Based Constitutive Equation for Rubberlike Networks and Its Comparison With Biaxial Stress–Strain Data. Part I
,”
Polymer
,
44
(
16
), pp.
4599
–
4610
. 10.1016/S0032-3861(03)00411-7
19.
Xiang
,
Y.
,
Zhong
,
D.
,
Wang
,
P.
,
Mao
,
G.
,
Yu
,
H.
, and
Qu
,
S.
,
2018
, “
A General Constitutive Model of Soft Elastomers
,”
J. Mech. Phys. Solids
,
117
, pp.
110
–
122
. 10.1016/j.jmps.2018.04.016
20.
Davidson
,
J. D.
, and
Goulbourne
,
N. C.
,
2013
, “
A Nonaffine Network Model for Elastomers Undergoing Finite Deformations
,”
J. Mech. Phys. Solids
,
61
(
8
), pp.
1784
–
1797
. 10.1016/j.jmps.2013.03.009
21.
Bergstrom
,
J. S.
, and
Boyce
,
M. C.
,
1998
, “
Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers
,”
J. Mech. Phys. Solids
,
46
(
5
), pp.
931
–
954
. 10.1016/S0022-5096(97)00075-6
22.
Tang
,
S.
,
Steven Greene
,
M.
, and
Liu
,
W. K.
,
2012
, “
Two-Scale Mechanism-Based Theory of Nonlinear Viscoelasticity
,”
J. Mech. Phys. Solids
,
60
(
2
), pp.
199
–
226
. 10.1016/j.jmps.2011.11.003
23.
Li
,
Y.
,
Tang
,
S.
,
Kröger
,
M.
, and
Liu
,
W. K.
,
2016
, “
Molecular Simulation Guided Constitutive Modeling on Finite Strain Viscoelasticity of Elastomers
,”
J. Mech. Phys. Solids
,
88
, pp.
204
–
226
. 10.1016/j.jmps.2015.12.007
24.
Xiang
,
Y.
,
Zhong
,
D.
,
Wang
,
P.
,
Yin
,
T.
,
Zhou
,
H.
,
Yu
,
H.
,
Baliga
,
C.
,
Qu
,
S.
, and
Yang
,
W.
,
2019
, “
A Physically Based Visco-Hyperelastic Constitutive Model for Soft Materials
,”
J. Mech. Phys. Solids
,
128
, pp.
208
–
218
. 10.1016/j.jmps.2019.04.010
25.
Miehe
,
C.
,
Göktepe
,
S.
, and
Lulei
,
F.
,
2004
, “
A Micro-Macro Approach to Rubber-Like Materials—Part I: The Non-affine Micro-Sphere Model of Rubber Elasticity
,”
J. Mech. Phys. Solids
,
52
(
11
), pp.
2617
–
2660
. 10.1016/j.jmps.2004.03.011
26.
Marckmann
,
G.
,
Verron
,
E.
,
Gornet
,
L.
,
Chagnon
,
G.
,
Charrier
,
P.
, and
Fort
,
P.
,
2002
, “
A Theory of Network Alteration for the Mullins Effect
,”
J. Mech. Phys. Solids
,
50
(
9
), pp.
2011
–
2028
. 10.1016/S0022-5096(01)00136-3
27.
Zhong
,
D.
,
Xiang
,
Y.
,
Yin
,
T.
,
Yu
,
H.
,
Qu
,
S.
, and
Yang
,
W.
,
2019
, “
A Physically-Based Damage Model for Soft Elastomeric Materials With Anisotropic Mullins Effect
,”
Int. J. Solids Struct.
,
176-177
, pp.
121
–
134
. 10.1016/j.ijsolstr.2019.05.018
28.
Zhao
,
X.
,
2012
, “
A Theory for Large Deformation and Damage of Interpenetrating Polymer Networks
,”
J. Mech. Phys. Solids
,
60
(
2
), pp.
319
–
332
. 10.1016/j.jmps.2011.10.005
29.
Lavoie
,
S. R.
,
Millereau
,
P.
,
Creton
,
C.
,
Long
,
R.
, and
Tang
,
T.
,
2019
, “
A Continuum Model for Progressive Damage in Tough Multinetwork Elastomers
,”
J. Mech. Phys. Solids
,
125
, pp.
523
–
549
. 10.1016/j.jmps.2019.01.001
30.
Kothari
,
K.
,
Hu
,
Y.
,
Gupta
,
S.
, and
Elbanna
,
A.
,
2018
, “
Mechanical Response of Two-Dimensional Polymer Networks: Role of Topology, Rate Dependence, and Damage Accumulation
,”
ASME J. Appl. Mech.
,
85
(
3
), p.
031008
. 10.1115/1.4038883
31.
Boyce
,
M. C.
, and
Arruda
,
E. M.
,
2000
, “
Constitutive Models of Rubber Elasticity: A Review
,”
Rubber Chem. Technol.
,
73
(
3
), pp.
504
–
523
. 10.5254/1.3547602
32.
Marckmann
,
G.
, and
Verron
,
E.
,
2006
, “
Comparison of Hyperelastic Models for Rubber-Like Materials
,”
Rubber Chem. Technol.
,
79
(
5
), pp.
835
–
858
. 10.5254/1.3547969
33.
Drapaca
,
C. S.
,
Sivaloganathan
,
S.
, and
Tenti
,
G.
,
2007
, “
Nonlinear Constitutive Laws in Viscoelasticity
,”
Math. Mech. Solids
,
12
(
5
), pp.
475
–
501
. 10.1177/1081286506062450
34.
Wineman
,
A.
,
2009
, “
Nonlinear Viscoelastic Solids—A Review
,”
Math. Mech. Solids
,
14
(
3
), pp.
300
–
366
. 10.1177/1081286509103660
35.
Banks
,
H. T.
,
Hu
,
S. H.
, and
Kenz
,
Z. R.
,
2011
, “
A Brief Review of Elasticity and Viscoelasticity for Solids
,”
Adv. Appl. Math. Mech.
,
3
(
1
), pp.
1
–
51
. 10.4208/aamm.10-m1030
36.
Diani
,
J.
,
Fayolle
,
B.
, and
Gilormini
,
P.
,
2009
, “
A Review on the Mullins Effect
,”
Eur. Polym. J.
,
45
(
3
), pp.
601
–
612
. 10.1016/j.eurpolymj.2008.11.017
37.
Holzapfel
,
G. A.
,
2002
, “
Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science
,”
Meccanica
,
37
(
4
), pp.
489
–
490
. 10.1023/A:1020843529530
38.
Edwards
,
S. F.
,
1986
,
The Theory of Polymer Dynamics
,
Oxford University Press
,
Oxford
.
39.
Freed
,
K. F.
,
1972
, “
Functional Integrals and Polymer Statistics
,”
Adv. Chem. Phys.
,
22
(
1
), pp.
1
–
128
.
40.
Edwards
,
S.
, and
Freed
,
K.
,
1969
, “
The Entropy of a Confined Polymer. I
,”
J. Phys. A: General Phys.
,
2
(
2
), pp.
145
–
150
. 10.1088/0305-4470/2/2/001
41.
Cho
,
K. S.
,
2016
,
Viscoelasticity of Polymers: Theory and Numerical Algorithms
,
Springer Series in Materials Science
,
Berlin
.
42.
Treloar
,
L.
,
1943
, “
The Elasticity of a Network of Long-Chain Molecules. I
,”
Trans. Faraday Soc.
,
39
, pp.
36
–
41
. 10.1039/tf9433900036
43.
James
,
H. M.
, and
Guth
,
E.
,
1943
, “
Theory of the Elastic Properties of Rubber
,”
J. Chem. Phys.
,
11
(
10
), pp.
455
–
481
. 10.1063/1.1723785
44.
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)90013-6
45.
Ball
,
R. C.
,
Doi
,
M.
,
Edwards
,
S. F.
, and
Warner
,
M.
,
1981
, “
Elasticity of Entangled Networks
,”
Polymer
,
22
(
8
), pp.
1010
–
1018
. 10.1016/0032-3861(81)90284-6
46.
Doi
,
M.
, and
Edwards
,
S. F.
,
1988
,
The Theory of Polymer Dynamics
,
Oxford University Press
,
Oxford
.
47.
Treloar
,
L. R. G.
,
1944
, “
Stress-Strain Data for Vulcanised Rubber Under Various Types of Deformation
,”
Trans. Faraday Soc.
,
40
, pp.
59
–
70
. 10.1039/tf9444000059
48.
Rivlin
,
R. S.
,
1948
, “
Large Elastic Deformations of Isotropic Materials. IV. Further Developments of the General Theory
,”
Philosophical Transactions Royal Soc. A: Mathematical, Physical and Engineering Sciences
,
241
(
835
), pp.
379
–
397
. 10.1098/rsta.1948.0024
49.
Mooney
,
M.
,
1940
, “
A Theory of Large Elastic Deformation
,”
J. Appl. Phys.
,
11
(
9
), pp.
582
–
592
. 10.1063/1.1712836
50.
Ogden
,
R. W.
,
1972
, “
Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids
,”
Proc. Math. Phys. Eng. Sci.
,
326
(
1567
), pp.
565
–
584
.
51.
Yeoh
,
O. H.
,
1993
, “
Some Forms of the Strain-Energy Function for Rubber
,”
Rubber Chem. Technol.
,
66
(
5
), pp.
754
–
771
. 10.5254/1.3538343
52.
Gent
,
A. N.
,
1996
, “
A new Constitutive Relation for Rubber
,”
Rubber Chem. Technol.
,
69
(
1
), pp.
59
–
61
. 10.5254/1.3538357
53.
Reese
,
S.
, and
Govindjee
,
S.
,
1998
, “
A Theory of Finite Viscoelasticity and Numerical Aspects
,”
Int. J. Solids Struct.
,
35
(
26–27
), pp.
3455
–
3482
. 10.1016/S0020-7683(97)00217-5
54.
Muschik
,
W.
,
1990
, “
Internal Variables in Nonequilibrium Thermodynamics
,”
J. Non-Equil Thermody
,
15
(
2
), pp.
127
–
137
. 10.1515/jnet.1990.15.2.127
55.
Hong
,
W.
,
2011
, “
Modeling Viscoelastic Dielectrics
,”
J. Mech. Phys. Solids
,
59
(
3
), pp.
637
–
650
. 10.1016/j.jmps.2010.12.003
56.
Kumar
,
A.
, and
Lopez-Pamies
,
O.
,
2016
, “
On the Two-Potential Constitutive Modeling of Rubber Viscoelastic Materials
,”
Comptes Rendus Mécanique
,
344
(
2
), pp.
102
–
112
. 10.1016/j.crme.2015.11.004
57.
Silberstein
,
M. N.
, and
Boyce
,
M. C.
,
2010
, “
Constitutive Modeling of the Rate, Temperature, and Hydration Dependent Deformation Response of Nafion to Monotonic and Cyclic Loading
,”
J. Power Sources
,
195
(
17
), pp.
5692
–
5706
. 10.1016/j.jpowsour.2010.03.047
58.
Zhao
,
X. H.
,
Koh
,
S. J. A.
, and
Suo
,
Z. G.
,
2011
, “
Nonequilibrium Thermodynamics of Dielectric Elastomers
,”
Int. J. Appl. Mech.
,
3
(
2
), pp.
203
–
217
. 10.1142/S1758825111000944
59.
Amin
,
A. F. M. S.
,
Alam
,
M. S.
, and
Okui
,
Y.
,
2002
, “
An Improved Hyperelasticity Relation in Modeling Viscoelasticity Response of Natural and High Damping Rubbers in Compression: Experiments, Parameter Identification and Numerical Verification
,”
Mech. Mater
,
34
(
2
), pp.
75
–
95
. 10.1016/S0167-6636(01)00102-8
60.
Amin
,
A. F. M. S.
,
Lion
,
A.
,
Sekita
,
S.
, and
Okui
,
Y.
,
2006
, “
Nonlinear Dependence of Viscosity in Modeling the Rate-Dependent Response of Natural and High Damping Rubbers in Compression and Shear: Experimental Identification and Numerical Verification
,”
Int. J. Plasticity
,
22
(
9
), pp.
1610
–
1657
. 10.1016/j.ijplas.2005.09.005
61.
Zhang
,
J.
, and
Chen
,
H.
,
2014
, “
Electromechanical Performance of a Viscoelastic Dielectric Elastomer Balloon
,”
Int. J. Smart Nano Mater.
,
5
(
2
), pp.
76
–
85
. 10.1080/19475411.2014.893930
62.
Mao
,
G. Y.
,
Xiang
,
Y. H.
,
Huang
,
X. Q.
,
Hong
,
W.
,
Lu
,
T. Q.
, and
Qu
,
S. X.
,
2018
, “
Viscoelastic Effect on the Wrinkling of an Inflated Dielectric-Elastomer Balloon
,”
ASME J. Appl. Mech.-T.
,
85
(
7
), p.
071003
. 10.1115/1.4039672
63.
Lubliner
,
J.
,
1985
, “
A Model of Rubber Viscoelasticity
,”
Mech. Res. Commun.
,
12
(
2
), pp.
93
–
99
. 10.1016/0093-6413(85)90075-8
64.
Rouse
,
P. E.
,
1953
, “
A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers
,”
J. Chem. Phys.
,
21
(
7
), pp.
1272
–
1280
. 10.1063/1.1699180
65.
de Gennes
,
P.-G.
,
1971
, “
Reptation of a Polymer Chain in the Presence of Fixed Obstacles
,”
J. Chem. Phys.
,
55
(
2
), pp.
572
–
579
. 10.1063/1.1675789
66.
Green
,
M. S.
, and
Tobolsky
,
A. V.
,
1946
, “
A New Approach to the Theory of Relaxing Polymeric Media
,”
J. Chem. Phys.
,
14
(
2
), pp.
80
–
92
. 10.1063/1.1724109
67.
Le Tallec
,
P.
,
Rahier
,
C.
, and
Kaiss
,
A.
,
1993
, “
Three-Dimensional Incompressible Viscoelasticity in Large Strains: Formulation and Numerical Approximation
,”
Comput. Methods Appl. Mech. Eng.
,
109
(
3
), pp.
233
–
258
. 10.1016/0045-7825(93)90080-H
68.
Ciarlet
,
P. G.
,
1988
,
Three-Dimensional Elasticity
,
Elsevier
,
New York
.
69.
Holzapfel
,
G. A.
, and
Simo
,
J. C.
,
1996
, “
A new Viscoelastic Constitutive Model for Continuous Media at Finite Thermomechanical Changes
,”
Int. J. Solids Struct.
,
33
(
20–22
), pp.
3019
–
3034
. 10.1016/0020-7683(95)00263-4
70.
Simo
,
J. C.
, and
Miehe
,
C.
,
1992
, “
Associative Coupled Thermoplasticity at Finite Strains—Formulation, Numerical-Analysis and Implementation
,”
Comput. Methods Appl. Mech. Eng.
,
98
(
1
), pp.
41
–
104
. 10.1016/0045-7825(92)90170-O
71.
Bonet
,
J.
,
2001
, “
Large Strain Viscoelastic Constitutive Models
,”
Int. J. Solids Struct.
,
38
(
17
), pp.
2953
–
2968
. 10.1016/S0020-7683(00)00215-8
72.
Perić
,
D.
,
Owen
,
D. R. J.
, and
Honnor
,
M. E.
,
1992
, “
A Model for Finite Strain Elasto-Plasticity Based on Logarithmic Strains: Computational Issues
,”
Comput. Methods Appl. Mech. Eng.
,
94
(
1
), pp.
35
–
61
. 10.1016/0045-7825(92)90156-E
73.
Vandoolaeghe
,
W. L.
, and
Terentjev
,
E. M.
,
2005
, “
Constrained Rouse Model of Rubber Viscoelasticity
,”
J. Chem. Phys.
,
123
(
3
), p.
34902
. 10.1063/1.1955445
74.
Vandoolaeghe
,
W. L.
, and
Terentjev
,
E. M.
,
2007
, “
A Rouse-Tube Model of Dynamic Rubber Viscoelasticity
,”
J. Phys. A-Math. Theor.
,
40
(
49
), pp.
14725
–
14744
. 10.1088/1751-8113/40/49/008
75.
Long
,
R.
,
Mayumi
,
K.
,
Creton
,
C.
,
Narita
,
T.
, and
Hui
,
C. Y.
,
2014
, “
Time Dependent Behavior of a Dual Cross-Link Self-Healing Gel: Theory and Experiments
,”
Macromolecules
,
47
(
20
), pp.
7243
–
7250
. 10.1021/ma501290h
76.
Miehe
,
C.
, and
Göktepe
,
S.
,
2005
, “
A Micro–Macro Approach to Rubber-Like Materials. Part II: The Micro-Sphere Model of Finite Rubber Viscoelasticity
,”
J. Mech. Phys. Solids
,
53
(
10
), pp.
2231
–
2258
. 10.1016/j.jmps.2005.04.006
77.
Linder
,
C.
,
Tkachuk
,
M.
, and
Miehe
,
C.
,
2011
, “
A Micromechanically Motivated Diffusion-Based Transient Network Model and Its Incorporation Into Finite Rubber Viscoelasticity
,”
J. Mech. Phys. Solids
,
59
(
10
), pp.
2134
–
2156
. 10.1016/j.jmps.2011.05.005
78.
Zhou
,
J.
,
Jiang
,
L.
, and
Khayat
,
R. E.
,
2018
, “
A Micro–Macro Constitutive Model for Finite-Deformation Viscoelasticity of Elastomers With Nonlinear Viscosity
,”
J. Mech. Phys. Solids
,
110
, pp.
137
–
154
. 10.1016/j.jmps.2017.09.016
79.
Mullins
,
L.
,
1948
, “
Effect of Stretching on the Properties of Rubber
,”
Rubber Chem. Technol.
,
21
(
2
), pp.
281
–
300
. 10.5254/1.3546914
80.
Mullins
,
L.
,
1969
, “
Softening of Rubber by Deformation
,”
Rubber Chem. Technol.
,
42
(
1
), pp.
339
–
362
. 10.5254/1.3539210
81.
Mullins
,
L.
, and
Tobin
,
N.
,
1957
, “
Theoretical Model for the Elastic Behavior of Filler-Reinforced Vulcanized Rubbers
,”
Rubber Chem. Technol.
,
30
(
2
), pp.
555
–
571
. 10.5254/1.3542705
82.
Mullins
,
L.
, and
Tobin
,
N.
,
1965
, “
Stress Softening in Rubber Vulcanizates. Part I. Use of a Strain Amplification Factor to Describe the Elastic Behavior of Filler-Reinforced Vulcanized Rubber
,”
J. Appl. Polym. Sci.
,
9
(
9
), pp.
2993
–
3009
. 10.1002/app.1965.070090906
83.
Bouasse
,
H.
, and
Carrière
,
Z.
,
1903
, “
Sur les Courbes de Traction du Caoutchouc Vulcanisé
,”
Proc. Annales de la Faculté des sciences de Toulouse: Mathématiques
,
5
(
3
), pp.
257
–
283
. 10.5802/afst.205
84.
Bueche
,
F.
,
1960
, “
Molecular Basis for the Mullins Effect
,”
J. Appl. Polym. Sci.
,
4
(
10
), pp.
107
–
114
. 10.1002/app.1960.070041017
85.
Bueche
,
F.
,
1961
, “
Mullins Effect and Rubber–Filler Interaction
,”
J. Appl. Polym. Sci.
,
5
(
15
), pp.
271
–
281
. 10.1002/app.1961.070051504
86.
Harwood
,
J.
,
Mullins
,
L.
, and
Payne
,
A.
,
1965
, “
Stress Softening in Natural Rubber Vulcanizates. Part II. Stress Softening Effects in Pure Gum and Filler Loaded Rubbers
,”
J. Appl. Polym. Sci.
,
9
(
9
), pp.
3011
–
3021
. 10.1002/app.1965.070090907
87.
Harwood
,
J.
, and
Payne
,
A.
,
1966
, “
Stress Softening in Natural Rubber Vulcanizates. Part IV. Unfilled Vulcanizates
,”
J. Appl. Polym. Sci.
,
10
(
8
), pp.
1203
–
1211
. 10.1002/app.1966.070100811
88.
Houwink
,
R.
,
1956
, “
Slipping of Molecules During the Deformation of Reinforced Rubber
,”
Rubber Chem. Technol.
,
29
(
3
), pp.
888
–
893
. 10.5254/1.3542602
89.
Kraus
,
G.
,
Childers
,
C.
, and
Rollmann
,
K.
,
1966
, “
Stress Softening in Carbon Black-Reinforced Vulcanizates. Strain Rate and Temperature Effects
,”
J. Appl. Polym. Sci.
,
10
(
2
), pp.
229
–
244
. 10.1002/app.1966.070100205
90.
Hanson
,
D. E.
,
Hawley
,
M.
,
Houlton
,
R.
,
Chitanvis
,
K.
,
Rae
,
P.
,
Orler
,
E. B.
, and
Wrobleski
,
D. A.
,
2005
, “
Stress Softening Experiments in Silica-Filled Polydimethylsiloxane Provide Insight Into a Mechanism for the Mullins Effect
,”
Polymer
,
46
(
24
), pp.
10989
–
10995
. 10.1016/j.polymer.2005.09.039
91.
Suzuki
,
N.
,
Ito
,
M.
, and
Yatsuyanagi
,
F.
,
2005
, “
Effects of Rubber/Filler Interactions on Deformation Behavior of Silica Filled SBR Systems
,”
Polymer
,
46
(
1
), pp.
193
–
201
. 10.1016/j.polymer.2004.10.066
92.
Ducrot
,
E.
,
Chen
,
Y.
,
Bulters
,
M.
,
Sijbesma
,
R. P.
, and
Creton
,
C.
,
2014
, “
Toughening Elastomers With Sacrificial Bonds and Watching Them Break
,”
Science
,
344
(
6180
), pp.
186
–
189
. 10.1126/science.1248494
93.
Clough
,
J. M.
,
Creton
,
C.
,
Craig
,
S. L.
, and
Sijbesma
,
R. P.
,
2016
, “
Covalent Bond Scission in the Mullins Effect of a Filled Elastomer: Real-Time Visualization With Mechanoluminescence
,”
Adv. Funct. Mater.
,
26
(
48
), pp.
9063
–
9074
. 10.1002/adfm.201602490
94.
Johnson
,
M.
, and
Beatty
,
M.
,
1993
, “
The Mullins Effect in Uniaxial Extension and its Influence on the Transverse Vibration of a Rubber String
,”
Continuum Mech. Thermodyn.
,
5
(
2
), pp.
83
–
115
. 10.1007/BF01141446
95.
Johnson
,
M. A.
, and
Beatty
,
M. F.
,
1995
, “
The Mullins Effect in Equibiaxial Extension and Its Influence on the Inflation of a Balloon
,”
Int. J. Eng. Sci.
,
33
(
2
), pp.
223
–
245
. 10.1016/0020-7225(94)E0052-K
96.
Bergstrom
,
J. S.
, and
Boyce
,
M. C.
,
1999
, “
Mechanical Behavior of Particle Filled Elastomers
,”
Rubber Chem. Technol.
,
72
(
4
), pp.
633
–
656
. 10.5254/1.3538823
97.
Qi
,
H.
, and
Boyce
,
M.
,
2004
, “
Constitutive Model for Stretch-Induced Softening of the Stress–Stretch Behavior of Elastomeric Materials
,”
J. Mech. Phys. Solids
,
52
(
10
), pp.
2187
–
2205
. 10.1016/j.jmps.2004.04.008
98.
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-J
99.
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-U
100.
Göktepe
,
S.
, and
Miehe
,
C.
,
2005
, “
A Micro–Macro Approach to Rubber-Like Materials. Part III: The Micro-Sphere Model of Anisotropic Mullins-Type Damage
,”
J. Mech. Phys. Solids
,
53
(
10
), pp.
2259
–
2283
. 10.1016/j.jmps.2005.04.010
101.
Lion
,
A.
,
1996
, “
A Constitutive Model for Carbon Black Filled Rubber: Experimental Investigations and Mathematical Representation
,”
Continuum Mech. Thermodyn.
,
8
(
3
), pp.
153
–
169
. 10.1007/BF01181853
102.
Simo
,
J. C.
,
1987
, “
On a Fully Three-Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects
,”
Comput. Methods Appl. Mech. Eng.
,
60
(
2
), pp.
153
–
173
. 10.1016/0045-7825(87)90107-1
103.
Neto
,
E. D. S.
,
Perić
,
D.
, and
Owen
,
D.
,
1994
, “
A Phenomenological Three-Dimensional Rate-Idependent Continuum Damage Model for Highly Filled Polymers: Formulation and Computational Aspects
,”
J. Mech. Phys. Solids
,
42
(
10
), pp.
1533
–
1550
. 10.1016/0022-5096(94)90086-8
104.
Miehe
,
C.
, and
Keck
,
J.
,
2000
, “
Superimposed Finite Elastic–Viscoelastic–Plastoelastic Stress Response With Damage in Filled Rubbery Polymers. Experiments, Modelling and Algorithmic Implementation
,”
J. Mech. Phys. Solids
,
48
(
2
), pp.
323
–
365
. 10.1016/S0022-5096(99)00017-4
105.
Kaliske
,
M.
,
Nasdala
,
L.
, and
Rothert
,
H.
,
2001
, “
On Damage Modelling for Elastic and Viscoelastic Materials at Large Strain
,”
Comput. Struct.
,
79
(
22–25
), pp.
2133
–
2141
. 10.1016/S0045-7949(01)00061-X
106.
Mao
,
Y.
,
Lin
,
S.
,
Zhao
,
X.
, and
Anand
,
L.
,
2017
, “
A Large Deformation Viscoelastic Model for Double-Network Hydrogels
,”
J. Mech. Phys. Solids
,
100
, pp.
103
–
130
. 10.1016/j.jmps.2016.12.011
107.
Ogden
,
R. W.
, and
Roxburgh
,
D. G.
,
1999
, “
A Pseudo-Elastic Model for the Mullins Effect in Filled Rubber
,”
Proc. Math. Phys. Eng. Sci.
,
455
(
1988
), pp.
2861
–
2877
. 10.1098/rspa.1999.0431
108.
Dorfmann
,
A.
, and
Ogden
,
R. W.
,
2004
, “
A Constitutive Model for the Mullins Effect With Permanent Set in Particle-Reinforced Rubber
,”
Int. J. Solids Struct.
,
41
(
7
), pp.
1855
–
1878
. 10.1016/j.ijsolstr.2003.11.014
109.
Diani
,
J.
,
Brieu
,
M.
, and
Gilormini
,
P.
,
2006
, “
Observation and Modeling of the Anisotropic Visco-Hyperelastic Behavior of a Rubberlike Material
,”
Int. J. Solids Struct.
,
43
(
10
), pp.
3044
–
3056
. 10.1016/j.ijsolstr.2005.06.045
110.
Diani
,
J.
,
Brieu
,
M.
, and
Vacherand
,
J.
,
2006
, “
A Damage Directional Constitutive Model for Mullins Effect With Permanent set and Induced Anisotropy
,”
Eur. J. Mech.-A/Solids
,
25
(
3
), pp.
483
–
496
. 10.1016/j.euromechsol.2005.09.011
111.
Marckmann
,
G.
,
Chagnon
,
G.
,
Le Saux
,
M.
, and
Charrier
,
P.
,
2016
, “
Experimental Investigation and Theoretical Modelling of Induced Anisotropy During Stress-Softening of Rubber
,”
Int. J. Solids Struct.
,
97
, pp.
554
–
565
. 10.1016/j.ijsolstr.2016.06.028
112.
Wang
,
X.
, and
Hong
,
W.
,
2011
, “
Pseudo-Elasticity of a Double Network gel
,”
Soft Matter
,
7
(
18
), pp.
8576
–
8581
. 10.1039/c1sm05787a
113.
Lu
,
T.
,
Wang
,
J.
,
Yang
,
R.
, and
Wang
,
T. J.
,
2016
, “
A Constitutive Model for Soft Materials Incorporating Viscoelasticity and Mullins Effect
,”
ASME. J. Appl. Mech
,
84
(
2
), p.
021010
.
114.
Wang
,
Z.
,
Tang
,
J.
,
Bai
,
R.
,
Zhang
,
W.
,
Lian
,
T.
,
Lu
,
T.
, and
Wang
,
T.
,
2018
, “
A Phenomenological Model for Shakedown of Tough Hydrogels Under Cyclic Loads
,”
ASME J. Appl. Mech.
,
85
(
9
), p.
091005
. 10.1115/1.4040330
115.
Lu
,
T.
,
Wang
,
Z.
,
Tang
,
J.
,
Zhang
,
W.
, and
Wang
,
T.
,
2020
, “
A Pseudo-Elasticity Theory to Model the Strain-Softening Behavior of Tough Hydrogels
,”
J. Mech. Phys. Solids
,
137
, p.
103832
. 10.1016/j.jmps.2019.103832
116.
Chagnon
,
G.
,
Verron
,
E.
,
Marckmann
,
G.
, and
Gornet
,
L.
,
2006
, “
Development of New Constitutive Equations for the Mullins Effect in Rubber Using the Network Alteration Theory
,”
Int. J. Solids Struct.
,
43
(
22
), pp.
6817
–
6831
. 10.1016/j.ijsolstr.2006.02.011
117.
Gong
,
J. P.
,
Katsuyama
,
Y.
,
Kurokawa
,
T.
, and
Osada
,
Y.
,
2003
, “
Double-Network Hydrogels With Extremely High Mechanical Strength
,”
Adv. Mater.
,
15
(
14
), pp.
1155
–
1158
. 10.1002/adma.200304907
118.
Zhu
,
P.
, and
Zhong
,
Z.
,
2020
, “
Modelling the Mechanical Behaviors of Double-Network Hydrogels
,”
Int. J. Solids Struct.
,
193–194
, pp.
492
–
501
. 10.1016/j.ijsolstr.2020.03.003
119.
Nakajima
,
T.
,
Kurokawa
,
T.
,
Ahmed
,
S.
,
Wu
,
W.-l.
, and
Gong
,
J. P.
,
2013
, “
Characterization of Internal Fracture Process of Double Network Hydrogels Under Uniaxial Elongation
,”
Soft Matter
,
9
(
6
), pp.
1955
–
1966
. 10.1039/C2SM27232F
120.
Lavoie
,
S. R.
,
Rong
,
L.
, and
Tian
,
T.
,
2016
, “
A Rate-Dependent Damage Model for Elastomers at Large Strain
,”
Extreme Mech. Lett.
,
8
, pp.
114
–
124
. 10.1016/j.eml.2016.05.016
121.
Vernerey
,
F. J.
,
Roberto
,
B.
,
Rong
,
L.
, and
Tong
,
S.
,
2018
, “
Statistical Damage Mechanics of Polymer Networks
,”
Macromolecules.
,
51
(
17
), pp.
6609
–
6622
.
122.
Taylor
,
C. A.
, and
Figueroa
,
C. A.
,
2009
, “
Patient-Specific Modeling of Cardiovascular Mechanics
,”
Annu. Rev. Biomed. Eng.
,
11
(
1
), pp.
109
–
134
. 10.1146/annurev.bioeng.10.061807.160521
123.
Aguado-Sierra
,
J.
,
Krishnamurthy
,
A.
,
Villongco
,
C.
,
Chuang
,
J.
,
Howard
,
E.
,
Gonzales
,
M. J.
,
Omens
,
J.
,
Krummen
,
D. E.
,
Narayan
,
S.
,
Kerckhoffs
,
R. C.
, and
McCulloch
,
A. D.
,
2011
, “
Patient-Specific Modeling of Dyssynchronous Heart Failure: a Case Study
,”
Prog. Biophys. Mol. Biol.
,
107
(
1
), pp.
147
–
155
. 10.1016/j.pbiomolbio.2011.06.014
124.
Tartakovsky
,
A. M.
,
Marrero
,
C. O.
,
Tartakovsky
,
D.
, and
Barajas-Solano
,
D.
,
2018
, “
Learning Parameters and Constitutive Relationships With Physics Informed Deep Neural Networks
,”
arXiv preprint arXiv:1808.03398
.
125.
Tipireddy
,
R.
,
Perdikaris
,
P.
,
Stinis
,
P.
, and
Tartakovsky
,
A.
,
2019
, “
A Comparative Study of Physics-Informed Neural Network Models for Learning Unknown Dynamics and Constitutive Relations
,”
arXiv preprint arXiv:1904.04058
.
Copyright © 2020 by ASME
You do not currently have access to this content.