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.

References

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
.
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