Hydrogels are used in a variety of applications ranging from tissue engineering to soft robotics. They often undergo large deformation coupled with solvent diffusion, and structural integrity is important when they are used as structural components. This paper presents a thermodynamically consistent method for calculating the transient energy release rate for crack growth in hydrogels based on a modified path-independent J-integral. The transient energy release rate takes into account the effect of solvent diffusion, separating the energy lost in diffusion from the energy available to drive crack growth. Numerical simulations are performed using a nonlinear transient finite element method for center-cracked hydrogel specimens, subject to remote tension under generalized plane strain conditions. The hydrogel specimen is assumed to be either immersed in a solvent or not immersed by imposing different chemical boundary conditions. Sharp crack and rounded notch models are used for small and large far-field strains, respectively. Comparisons to linear elastic fracture mechanics (LEFM) are presented for the crack-tip fields and crack opening profiles in the instantaneous and equilibrium limits. It is found that the stress singularity at the crack tip depends on both the far-field strain and the local solvent diffusion, and the latter evolves with time and depends on the chemical boundary conditions. The transient energy release rate is predicted as a function of time for the two types of boundary conditions with distinct behaviors due to solvent diffusion. Possible scenarios of delayed fracture are discussed based on evolution of the transient energy release rate.

References

References
1.
Jagur-Grodzinski
,
J.
,
2006
, “
Polymers for Tissue Engineering, Medical Devices, and Regenerative Medicine. Concise General Review of Recent Studies
,”
Polym. Adv. Technol.
,
17
(
6
), pp.
395
418
.10.1002/pat.729
2.
Drury
,
J. L.
, and
Mooney
,
D. J.
,
2003
, “
Hydrogels for Tissue Engineering: Scaffold Design Variables and Applications
,”
Biomaterials
,
24
(
24
), pp.
4337
4351
.10.1016/S0142-9612(03)00340-5
3.
Suciu
,
A. N.
,
Iwatsubo
,
T.
,
Matsuda
,
M.
, and
Nishino
,
T.
,
2004
, “
A Study Upon Durability of the Artificial Knee Joint With PVA Hydrogel Cartilage
,”
JSME Int. J. Ser. C
,
47
(
1
), pp.
199
208
.10.1299/jsmec.47.199
4.
Luo
,
Y.
, and
Shoichet
,
M. S.
,
2004
, “
A Photolabile Hydrogel for Guided Three-Dimensional Cell Growth and Migration
,”
Nat. Mater.
,
3
(
4
), pp.
249
253
.10.1038/nmat1092
5.
Discher
,
D. E.
,
Mooney
,
D. J.
, and
Zandstra
,
P. W.
,
2009
, “
Growth Factors, Matrices, and Forces Combine and Control Stem Cells
,”
Science
,
324
(
5935
), pp.
1673
1677
.10.1126/science.1171643
6.
Qiu
,
Y.
, and
Park
,
K.
,
2001
, “
Environment-Sensitive Hydrogels for Drug Delivery
,”
Adv. Drug Delivery Rev.
,
53
(
3
), pp.
321
339
.10.1016/S0169-409X(01)00203-4
7.
Calvert
,
P.
,
2009
, “
Hydrogels for Soft Machines
,”
Adv. Mater.
,
21
(
7
), pp.
743
756
.10.1002/adma.200800534
8.
Dong
,
L.
,
Agarwal
,
A. K.
,
Beebe
,
D. J.
, and
Jiang
,
H. R.
,
2006
, “
Adaptive Liquid Microlenses Activated by Stimuli-Responsive Hydrogels
,”
Nature
,
442
(
7102
), pp.
551
554
.10.1038/nature05024
9.
Keplinger
,
C.
,
Sun
,
J.-Y.
,
Foo
,
C. C.
,
Rothemund
,
P.
,
Whitesides
,
G. M.
, and
Suo
,
Z.
,
2013
, “
Stretchable, Transparent, Ionic Conductors
,”
Science
,
341
(
6149
), pp.
984
987
.10.1126/science.1240228
10.
Kong
,
H. J.
,
Wong
,
E.
, and
Mooney
,
D. J.
,
2003
, “
Independent Control of Rigidity and Toughness of Polymeric Hydrogels
,”
Macromolecules
,
36
(
12
), pp.
4582
4588
.10.1021/ma034137w
11.
Henderson
,
K. J.
,
Zhou
,
T. C.
,
Otim
,
K. J.
, and
Shull
,
K. R.
,
2010
, “
Ionically Cross-Linked Triblock Copolymer Hydrogels With High Strength
,”
Macromolecules
,
43
(
14
), pp.
6193
6201
.10.1021/ma100963m
12.
Baumberger
,
T.
,
Caroli
,
C.
, and
Martina
,
D.
,
2006
, “
Fracture of a Biopolymer Gel as a Viscoplastic Disentanglement Process
,”
Eur. Phys. J. E
,
21
(
1
), pp.
81
89
.10.1140/epje/i2006-10048-6
13.
Baumberger
,
T.
,
Caroli
,
C.
, and
Martina
,
D.
,
2006
, “
Solvent Control of Crack Dynamics in a Reversible Hydrogel
,”
Nat. Mater.
,
5
(
7
), pp.
552
555
.10.1038/nmat1666
14.
Baumberger
,
T.
, and
Ronsin
,
O.
,
2010
, “
Cooperative Effect of Stress and Ion Displacement on the Dynamics of Cross-Link Unzipping and Rupture of Alginate Gels
,”
Biomacromolecules
,
11
(
6
), pp.
1571
1578
.10.1021/bm1002015
15.
Tanaka
,
Y.
,
Fukao
,
K.
, and
Miyamoto
,
Y.
,
2000
, “
Fracture Energy of Gels
,”
Eur. Phys. J. E
,
3
(
4
), pp.
395
401
.10.1007/s101890070010
16.
Simha
,
N. K.
,
Carlson
,
C. S.
, and
Lewis
,
J. L.
,
2003
, “
Evaluation of Fracture Toughness of Cartilage by Micropenetration
,”
J. Mater. Sci.: Mater. Med.
,
15
(
5
), pp.
631
639
.10.1023/B:JMSM.0000026104.30607.c7
17.
Gamonpilas
,
C.
,
Charalambides
,
M. N.
, and
Williams
,
J. G.
,
2009
, “
Determination of Large Deformation and Fracture Behavior of Starch Gels From Conventional and Wire Cutting Experiments
,”
J. Mater. Sci.
,
44
(
18
), pp.
4976
4986
.10.1007/s10853-009-3760-9
18.
Forte
,
A. E.
,
D'Amico
,
F.
,
Charalambides
,
M. N.
,
Dini
,
D.
, and
Williams
,
J. G.
,
2015
, “
Modelling and Experimental Characterisation of the Rate Dependent Fracture Properties of Gelatine Gels
,”
Food Hydrocolloids
,
46
, pp.
180
190
.10.1016/j.foodhyd.2014.12.028
19.
Kwon
,
H. J.
,
Rogalsky
,
A. D.
, and
Kim
,
D.-W.
,
2011
, “
On the Measurement of Fracture Toughness of Soft Biogel
,”
Polym. Eng. Sci.
,
51
(
6
), pp.
1078
1086
.10.1002/pen.21923
20.
Gong
,
J. P.
,
2010
, “
Why are Double Network Hydrogels so Tough?
,”
Soft Matter
,
6
(
12
), pp.
2583
2590
.10.1039/b924290b
21.
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
.10.1038/nature11409
22.
Zhao
,
X.
,
2014
, “
Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretchy Networks
,”
Soft Matter
,
10
(
5
), pp.
672
687
.10.1039/C3SM52272E
23.
Hu
,
Y.
, and
Suo
,
Z.
,
2012
, “
Viscoelasticity and Poroelasticity in Elastomeric Gels
,”
Acta Mech. Solida Sin.
,
25
(
5
), pp.
441
457
.10.1016/S0894-9166(12)60039-1
24.
Zhao
,
X.
,
Huebsch
,
N. D.
,
Mooney
,
D. J.
, and
Suo
,
Z.
,
2010
, “
Stress-Relaxation Behavior in Gels With Ionic and Covalent Crosslinks
,”
J. Appl. Phys.
,
107
(
6
), p.
063509
.10.1063/1.3343265
25.
Hu
,
Y.
,
Chen
,
X.
,
Whitesides
,
G. M.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2011
, “
Indentation of Polydimethylsiloxane Submerged in Organic Solvents
,”
J. Mater. Res.
,
26
(
6
), pp.
785
795
.10.1557/jmr.2010.35
26.
Galli
,
M.
,
Fornasiere
,
E.
,
Gugnoni
,
J.
, and
Oyen
,
M. L.
,
2011
, “
Poroviscoelastic Characterization of Particle-Reinforced Gelatin Gels Using Indentation and Homogenization
,”
J. Mech. Behav. Biomed. Mater.
,
4
(
4
), pp.
610
617
.10.1016/j.jmbbm.2011.01.009
27.
Knauss
,
W. G.
,
1973
, “
The Mechanics of Polymer Fracture
,”
ASME Appl. Mech. Rev.
,
26
, pp.
1
17
.
28.
Schapery
,
R. A.
,
1975
, “
A Theory of Crack Initiation and Growth in Viscoelastic Media
,”
Int. J. Fract.
,
11
(
1
), pp.
141
159
.10.1007/BF00034721
29.
Schapery
,
R. A.
,
1975
, “
A Theory of Crack Initiation and Growth in Viscoelastic Media II. Approximate Methods of Analysis
,”
Int. J. Fract.
,
11
(
3
), pp.
369
388
.10.1007/BF00033526
30.
Schapery
,
R. A.
,
1975
, “
A Theory of Crack Initiation and Growth in Viscoelastic Media
,”
Int. J. Fract.
,
11
(
4
), pp.
549
562
.10.1007/BF00116363
31.
Schapery
,
R. A.
,
1984
, “
Correspondence Principles and a Generalized J Integral for Large Deformation and Fracture Analysis of Viscoelastic Media
,”
Int. J. Fract.
,
25
(
3
), pp.
195
223
.10.1007/BF01140837
32.
Hui
,
C. Y.
,
Long
,
R.
, and
Ning
,
J.
,
2013
, “
Stress Relaxation Near the Tip of a Stationary Mode I Crack in a Poroelastic Solid
,”
ASME J. Appl. Mech.
,
80
(
2
), p.
021014
.10.1115/1.4007228
33.
Wang
,
X.
, and
Hong
,
W.
,
2012
, “
Delayed Fracture in Gels
,”
Soft Matter
,
8
(
31
), pp.
8171
8178
.10.1039/c2sm25553g
34.
Zhang
,
J.
,
An
,
Y.
,
Yazzie
,
K.
,
Chawla
,
N.
, and
Jiang
,
H.
,
2012
, “
Finite Element Simulation of Swelling-Induced Crack Healing in Gels
,”
Soft Matter
,
8
(
31
), pp.
8107
8112
.10.1039/c2sm25399b
35.
Rice
,
J. R.
, and
Cleary
,
M. P.
,
1976
, “
Some Basic Stress Diffusion Solutions for Fluid-Saturated Elastic Porous Media With Compressible Constituents
,”
Rev. Geophys.
,
14
(
2
), pp.
227
241
.10.1029/RG014i002p00227
36.
Ruina
,
A.
,
1978
, “
Influence of Coupled Deformation-Diffusion Effects on the Retardation of Hydraulic Fracture
,”
19th U.S. Symposium on Rock Mechanics
, Reno, NV, May 1–3.
37.
Detournay
,
E.
, and
Cheng
,
A. H.-D.
,
1991
, “
Plane Strain Analysis of a Stationary Hydraulic Fracture in a Poroelastic Medium
,”
Int. J. Solids Struct.
,
27
(
13
), pp.
1645
1662
.10.1016/0020-7683(91)90067-P
38.
Adachi
,
J. I.
, and
Detournay
,
E.
,
2008
, “
Plane Strain Propagation of a Hydraulic Fracture in a Permeable Rock
,”
Eng. Fract. Mech.
,
75
(
16
), pp.
4666
4694
.10.1016/j.engfracmech.2008.04.006
39.
Kishimoto
,
K.
,
Aoki
,
S.
, and
Sakata
,
M.
,
1980
, “
On the Path Independent Integral-J
,”
Eng. Fract. Mech.
,
13
(
4
), pp.
841
850
.10.1016/0013-7944(80)90015-6
40.
Chien
,
N.
, and
Herrmann
,
G.
,
1996
, “
Conservation Laws for Thermo or Poroelasticity
,”
ASME J. Appl. Mech.
,
63
(
2
), pp.
331
336
.10.1115/1.2788869
41.
Yang
,
F.
,
Wang
,
J.
, and
Chen
,
D.
,
2006
, “
The Energy Release Rate for Hygrothermal Coupling Elastic Materials
,”
Acta Mech. Sin.
,
22
(
1
), pp.
28
33
.10.1007/s10409-006-0087-5
42.
Gao
,
Y. F.
, and
Zhou
,
M.
,
2013
, “
Coupled Mechano-Diffusional Driving Forces for Fracture in Electrode Materials
,”
J. Power Sources
,
230
, pp.
176
193
.10.1016/j.jpowsour.2012.12.034
43.
Haftbaradaran
,
H.
, and
Qu
,
J.
,
2014
, “
A Path-Independent Integral for Fracture of Solids Under Combined Electrochemical and Mechanical Loadings
,”
J. Mech. Phys. Solids
,
71
, pp.
1
14
.10.1016/j.jmps.2014.06.007
44.
Bouklas
,
N.
,
Landis
,
C. M.
, and
Huang
,
R.
,
2015
, “
A Nonlinear, Transient Finite Element Method for Coupled Solvent Diffusion and Large Deformation of Hydrogels
,”
J. Mech. Phys. Solids
,
79
, pp.
21
43
.10.1016/j.jmps.2015.03.004
45.
Dolbow
,
J.
,
Fried
,
E.
, and
Ji
,
H.
,
2004
, “
Chemically Induced Swelling of Hydrogels
,”
J. Mech. Phys. Solids
,
52
(
1
), pp.
51
84
.10.1016/S0022-5096(03)00091-7
46.
Hong
,
W.
,
Zhao
,
X.
,
Zhou
,
J.
, and
Suo
,
Z.
,
2008
, “
A Theory of Coupled Diffusion and Large Deformation in Polymeric Gels
,”
J. Mech. Phys. Solids
,
56
(
5
), pp.
1779
1793
.10.1016/j.jmps.2007.11.010
47.
Duda
,
F. P.
,
Souza
,
A. C.
, and
Fried
,
E.
,
2010
, “
A Theory for Species Migration in a Finitely Strained Solid With Application to Polymer Network Swelling
,”
J. Mech. Phys. Solids
,
58
(
4
), pp.
515
529
.10.1016/j.jmps.2010.01.009
48.
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
.10.1016/j.jmps.2010.07.020
49.
Wang
,
X.
, and
Hong
,
W.
,
2012
, “
A Visco-Poroelastic Theory for Polymeric Gels
,”
Proc. R. Soc. London, Ser. A: Math. Phys. Eng. Sci.
,
468
(
2148
), pp.
3824
3841
.10.1098/rspa.2012.0385
50.
Prigogine
,
I.
,
1968
,
Introduction to Thermodynamics of Irreversible Processes
,
Wiley
,
New York
.
51.
Coleman
,
B. D.
, and
Noll
,
W.
,
1963
, “
The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity
,”
Arch. Ration. Mech. Anal.
,
13
(
1
), pp.
167
178
.10.1007/BF01262690
52.
Rice
,
J. R.
,
1968
, “
A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks
,”
ASME J. Appl. Mech.
,
35
(
2
), pp.
379
386
.10.1115/1.3601206
53.
Rice
,
J. R.
,
1968
,
Mathematical Analysis in the Mechanics Fracture. Fracture: An Advanced Treatise
, Vol.
2
,
Academic
,
New York
, pp.
191
311
.
54.
Li
,
F. Z.
,
Shih
,
C. F.
, and
Needleman
,
A.
,
1985
, “
A Comparison of Methods for Calculating Energy Release Rates
,”
Eng. Fract. Mech.
,
21
(
2
), pp.
405
421
.10.1016/0013-7944(85)90029-3
55.
Murad
,
M. A.
, and
Loula
,
A. F.
,
1994
, “
On Stability and Convergence of Finite Element Approximations of Biot's Consolidation Problem
,”
Int. J. Numer. Meth. Eng.
,
37
(
4
), pp.
645
667
.10.1002/nme.1620370407
56.
Wan
,
J.
,
2002
, “
Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow
,” Ph.D. dissertation,
Stanford University
,
Stanford, CA
.
57.
Phillips
,
P. J.
, and
Wheeler
,
M. F.
,
2009
, “
Overcoming the Problem of Locking in Linear Elasticity and Poroelasticity: An Heuristic Approach
,”
Comput. Geosci.
,
13
(
1
), pp.
5
12
.10.1007/s10596-008-9114-x
58.
Taylor
,
C.
, and
Hood
,
P.
,
1973
, “
A Numerical Solution of the Navier–Stokes Equations Using the Finite Element Technique
,”
Comput. Fluids
,
1
(
1
), pp.
73
100
.10.1016/0045-7930(73)90027-3
59.
Bouklas
,
N.
, and
Huang
,
R.
,
2012
, “
Swelling Kinetics of Polymer Gels: Comparison of Linear and Nonlinear Theories
,”
Soft Matter
,
8
(
31
), pp.
8194
8203
.10.1039/c2sm25467k
60.
Broberg
,
K. B.
,
1999
,
Cracks and Fracture
,
Academic
,
San Diego, CA
.
61.
Yoon
,
J.
,
Cai
,
S.
,
Suo
,
Z.
, and
Hayward
,
R. C.
,
2010
, “
Poroelastic Swelling Kinetics of Thin Hydrogel Layers: Comparison of Theory and Experiment
,”
Soft Matter
,
6
(
23
), pp.
6004
6012
.10.1039/c0sm00434k
62.
Bouklas
,
N.
,
2014
, “
Modelling and Simulation of Hydrogels With Coupled Solvent Diffusion and Large Deformation
,” Ph.D. dissertation,
The University of Texas at Austin
,
Austin, TX
.
63.
Bonn
,
D.
,
Kellay
,
H.
,
Prochnow
,
M.
,
Ben-Djemiaa
,
K.
, and
Meunier
,
J.
,
1998
, “
Delayed Fracture of an Inhomogeneous Soft Solid
,”
Science
,
280
(
5361
), pp.
265
267
.10.1126/science.280.5361.265
64.
McMeeking
,
R. M.
,
1977
, “
Finite Deformation Analysis of Crack-Tip Opening in Elastic–Plastic Materials and Implications for Fracture
,”
J. Mech. Phys. Solids
,
25
(
5
), pp.
357
381
.10.1016/0022-5096(77)90003-5
65.
Geubelle
,
P. H.
,
1995
, “
Finite Deformation Effects in Homogeneous and Interfacial Fracture
,”
Int. J. Solids Struct.
,
32
(
6–7
), pp.
1003
1016
.10.1016/0020-7683(94)00174-U
66.
Krishnan
,
V. R.
,
Hui
,
C. Y.
, and
Long
,
R.
,
2008
, “
Finite Strain Crack Tip Fields in Soft Incompressible Elastic Solids
,”
Langmuir
,
24
(
24
), pp.
14245
14253
.10.1021/la802795e
67.
Bouchbinder
,
E.
,
Livne
,
A.
, and
Fineberg
,
J.
,
2009
, “
The 1/r Singularity in Weakly Nonlinear Fracture Mechanics
,”
J. Mech. Phys. Solids
,
57
(
9
), pp.
1568
1577
.10.1016/j.jmps.2009.05.006
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