This paper presents a micromechanically based constitutive model within the framework of the continuum mechanics to characterize the inelastic elastomeric and swelling behaviors of double network (DN) hydrogels, such as the stress-softening, necking instability, hardening, and stretch-induced anisotropy. The strain-energy density function of the material is decomposed into two independent contributions from the tight and brittle first network and the soft and loose second network, each of which is obtained by integrating the strain energy of one-dimensional (1D) polymer chains in each direction of a unit sphere. The damage process is derived from the irreversible breakages of sacrificial chains in the first network and characterized by the directional stretch-dependent evolution laws for the equivalent modulus and the locking stretch in the non-Gauss statistical model of a single polymer chain. The constitutive model with the optimized-material evolution law predicts stress–stretch curves in a good agreement with the experimental results during loading, unloading, and reloading paths for both ionic and covalent DN hydrogels. The deformation-induced anisotropy is investigated and demonstrated by the constitutive model for the free swelling of damaged specimen. The constitutive model is embedded into the finite-element (FE) procedure and proved to be efficient to model the necking and neck propagation in the plane-strain uniaxial elongation. Based on the procedure, the effects of imperfection and boundary conditions on the loading path and the material evolution during different stages of deformation are investigated.

References

1.
Dong
,
L.
,
Agarwal
,
A. K.
,
Beebe
,
D. J.
, and
Jiang
,
H.
,
2006
, “
Adaptive Liquid Microlenses Activated by Stimuli-Responsive Hydrogels
,”
Nature
,
442
(
7102
), pp.
551
554
.
2.
Oh
,
J. K.
,
Drumright
,
R.
,
Siegwart
,
D. J.
, and
Matyjaszewski
,
K.
,
2008
, “
The Development of Microgels/Nanogels for Drug Delivery Applications
,”
Prog. Polym. Sci.
,
33
(
4
), pp.
448
477
.
3.
Sidorenko
,
A.
,
Krupenkin
,
T.
, and
Aizenberg
,
J.
,
2008
, “
Controlled Switching of the Wetting Behavior of Biomimetic Surfaces With Hydrogel-Supported Nanostructures
,”
J. Mater. Chem.
,
18
(
32
), pp.
3841
3846
.
4.
Calvert
,
P.
,
2009
, “
Hydrogels for Soft Machines
,”
Adv. Mater.
,
21
(
7
), pp.
743
756
.
5.
Zhao
,
X.
,
2014
, “
Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretchy Networks
,”
Soft Matter
,
10
(
5
), pp.
672
687
.
6.
Gong
,
J. P.
,
2010
, “
Why Are Double Network Hydrogels So Tough?
,”
Soft Matter
,
6
(
12
), pp.
2583
2590
.
7.
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
.
8.
Nakajima
,
T.
,
Kurokawa
,
T.
,
Ahmed
,
S.
,
Wu
,
W.
, and
Gong
,
J. P.
,
2013
, “
Characterization of Internal Fracture Process of Double Network Hydrogels Under Uniaxial Elongation
,”
Soft Matter
,
9
(
6
), pp.
1955
1966
.
9.
Sun
,
J.-Y.
,
Zhao
,
X.
,
Illeperuma
,
W. R. K.
,
Chaudhuri
,
O.
,
Oh
,
K. H.
,
Mooney
,
D. J.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2012
, “
Highly Stretchable and Tough Hydrogels
,”
Nature
,
489
(
7414
), pp.
133
136
.
10.
Ahmed
,
S.
,
Nakajima
,
T.
,
Kurokawa
,
T.
,
Anamul Haque
,
M.
, and
Gong
,
J. P.
,
2014
, “
Brittle-Ductile Transition of Double Network Hydrogels: Mechanical Balance of Two Networks as the Key Factor
,”
Polymer
,
55
(
3
), pp.
914
923
.
11.
Mullins
,
L.
,
1948
, “
Effect of Stretching on The Properties of Rubber
,”
Rubber Chem. Technol.
,
21
(
2
), pp.
281
300
.
12.
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
.
13.
Wang
,
X.
, and
Hong
,
W.
,
2011
, “
Pseudo-Elasticity of a Double Network Gel
,”
Soft Matter
,
7
(
18
), pp.
8576
8581
.
14.
Diani
,
J.
,
Fayolle
,
B.
, and
Gilormini
,
P.
,
2009
, “
A Review on the Mullins Effect
,”
Eur. Polym. J.
,
45
(
3
), pp.
601
612
.
15.
Holzapfel
,
G. A.
,
Gasser
,
T. C.
, and
Ogden
,
R. W.
,
2000
, “
A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models
,”
J. Elast.
,
61
(
1–3
), pp.
1
48
.
16.
Dargazany
,
R.
,
Khiêm
,
V. N.
, and
Itskov
,
M.
,
2014
, “
A Generalized Network Decomposition Model for the Quasi-Static Inelastic Behavior of Filled Elastomers
,”
Int. J. Plast.
,
63
, pp.
94
109
.
17.
Dargazany
,
R.
, and
Itskov
,
M.
,
2009
, “
A Network Evolution Model for the Anisotropic Mullins Effect in Carbon Black Filled Rubbers
,”
Int. J. Solids Struct.
,
46
(
16
), pp.
2967
2977
.
18.
Webber
,
R. E.
,
Creton
,
C.
,
Brown
,
H. R.
, and
Gong
,
J. P.
,
2007
, “
Large Strain Hysteresis and Mullins Effect of Tough Double-Network Hydrogels
,”
Macromolecules
,
40
(
8
), pp.
2919
2927
.
19.
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
.
20.
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
.
21.
Zhao
,
X.
,
2012
, “
A Theory for Large Deformation and Damage of Interpenetrating Polymer Networks
,”
J. Mech. Phys. Solids
,
60
(
2
), pp.
319
332
.
22.
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
.
23.
Nakajima
,
T.
,
Furukawa
,
H.
,
Tanaka
,
Y.
,
Kurokawa
,
T.
,
Osada
,
Y.
, and
Gong
,
J. P.
,
2009
, “
True Chemical Structure of Double Network Hydrogels
,”
Macromolecules
,
42
(
6
), pp.
2184
2189
.
24.
Diani
,
J.
,
Brieu
,
M.
, and
Vacherand
,
J. M.
,
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
.
25.
Ehret
,
A. E.
,
Itskov
,
M.
, and
Schmid
,
H.
,
2010
, “
Numerical Integration on the Sphere and Its Effect on the Material Symmetry of Constitutive Equations—A Comparative Study
,”
Int. J. Numer. Methods Eng.
,
81
(
2
), pp.
189
206
.
26.
Bažant
,
P.
, and
Oh
,
B. H.
,
1986
, “
Efficient Numerical Integration on the Surface of a Sphere
,”
ZAMM—J. Appl. Math. Mech.
,
66
(
1
), pp.
37
49
.
27.
Chester
,
S. A.
, and
Anand
,
L.
,
2010
, “
A Coupled Theory of Fluid Permeation and Large Deformations for Elastomeric Materials
,”
J. Mech. Phys. Solids
,
58
(
11
), pp.
1879
1906
.
28.
Kuhn
,
W.
, and
Grün
,
F.
,
1942
, “
Beziehungen Zwischen Elastischen Konstanten und Dehnungsdoppelbrechung Hochelastischer Stoffe
,”
Kolloid-Z.
,
101
(
3
), pp.
248
271
.
29.
Merckel
,
Y.
,
Diani
,
J.
,
Brieu
,
M.
, and
Caillard
,
J.
,
2013
, “
Constitutive Modeling of the Anisotropic Behavior of Mullins Softened Filled Rubbers
,”
Mech. Mater.
,
57
, pp.
30
41
.
30.
Neale
,
K. W.
, and
Tuḡcu
,
P.
,
1985
, “
Analysis of Necking and Neck Propagation in Polymeric Materials
,”
J. Mech. Phys. Solids
,
33
(
4
), pp.
323
337
.
31.
Liu
,
Y.
,
Zhang
,
H.
,
Zhang
,
J.
, and
Zheng
,
Y.
,
2015
, “
Constitutive Modeling for Polymer Hydrogels: A New Perspective and Applications to Anisotropic Hydrogels in Free Swelling
,”
Eur. J. Mech. A: Solids
,
54
, pp.
171
186
.
32.
Orwoll
,
R. A.
, and
Arnold
,
P. A.
,
2007
, “
Polymer–Solvent Interaction Parameter χ
,”
Physical Properties of Polymers Handbook
,
J. E.
Mark
, ed.,
Springer
,
New York
, pp.
233
257
.
33.
de Souza Neto
,
E. A.
,
Perić
,
D.
,
Dutko
,
M.
, and
Owen
,
D. R. J.
,
1996
, “
Design of Simple Low Order Finite Elements for Large Strain Analysis of Nearly Incompressible Solids
,”
Int. J. Solids Struct.
,
33
(
20
), pp.
3277
3296
.
34.
Hutchinson
,
J. W.
, and
Neale
,
K. W.
,
1983
, “
Neck Propagation
,”
J. Mech. Phys. Solids
,
31
(
5
), pp.
405
426
.
35.
Zhou
,
W.
,
Li
,
X.
,
Lu
,
J.
,
Huang
,
N.
,
Chen
,
L.
,
Qi
,
Z.
,
Li
,
L.
, and
Liang
,
H.
,
2014
, “
Toughening Mystery of Natural Rubber Deciphered by Double Network Incorporating Hierarchical Structures
,”
Sci. Rep.
,
4
, p.
7502
.
36.
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
.
37.
Argun
,
A.
,
Can
,
V.
,
Altun
,
U.
, and
Okay
,
O.
,
2014
, “
Nonionic Double and Triple Network Hydrogels of High Mechanical Strength
,”
Macromolecules
,
47
(
18
), pp.
6430
6440
.
38.
Yang
,
W.
,
Furukawa
,
H.
, and
Gong
,
J. P.
,
2008
, “
Highly Extensible Double-Network Gels With Self-Assembling Anisotropic Structure
,”
Adv. Mater.
,
20
(
23
), pp.
4499
4503
.
39.
Crisfield
,
M. A.
,
1997
,
Non-Linear Finite Element Analysis of Solids and Structures
, Vol.
2
,
Wiley
,
Chichester, UK
.
40.
Wriggers
,
P.
,
2008
,
Nonlinear Finite Element Methods
,
Springer
,
Berlin
.
You do not currently have access to this content.