Based on a linear poroelastic formulation, we present an asymptotic analysis of the transient crack-tip fields for stationary cracks in polymer gels under plane-strain conditions. A center crack model is studied in detail, comparing numerical results by a finite element method to the asymptotic analysis. The time evolution of the crack-tip parameters is determined as a result of solvent diffusion coupled with elastic deformation of the gel. The short-time and long-time limits are obtained for the stress intensity factor and the crack-tip energy release rate under different chemo-mechanical boundary conditions (immersed versus not-immersed, displacement versus load controlled). It is found that, under displacement-controlled loading, the crack-tip energy release rate increases monotonically over time for the not-immersed case, but for the immersed case, it increases first and then decreases, with a long-time limit lower than the short-time limit. Under load control, the energy release rate increases over time for both immersed and not-immersed cases, with different short-time limits but the same long-time limit. These results suggest that onset of crack growth may be delayed until the crack-tip energy release rate reaches a critical value if the applied displacement or traction is subcritical but greater than a threshold.

References

References
1.
Drury
,
J. L.
, and
Mooney
,
D. J.
,
2003
, “
Hydrogels for Tissue Engineering: Scaffold Design Variables and Applications
,”
Biomaterials
,
24
(
24
), pp.
4337
4351
.
2.
Langer
,
R.
,
2006
, “
Biomaterials for Drug Delivery and Tissue Engineering
,”
MRS Bull.
,
31
(
6
), pp.
477
485
.
3.
Peppas
,
N. A.
,
Hilt
,
J. Z.
,
Khademhosseini
,
A.
, and
Langer
,
R.
,
2006
, “
Hydrogels in Biology and Medicine: From Molecular Principles to Bionanotechnology
,”
Adv. Mater.
,
18
(
11
), pp.
1345
1360
.
4.
Calvert
,
P.
,
2009
, “
Hydrogels for Soft Machines
,”
Adv. Mater.
,
21
(
7
), pp.
743
756
.
5.
Suo
,
Z.
,
2012
, “
Mechanics of Stretchable Electronics and Soft Machines
,”
MRS Bull.
,
37
(
3
), pp.
218
225
.
6.
Yuk
,
H.
,
Lin
,
S. T.
,
Ma
,
C.
,
Takaffoli
,
M.
,
Fang
,
N. X.
, and
Zhao
,
X.
,
2017
, “
Hydraulic Hydrogel Actuators and Robots Optically and Sonically Camouflaged in Water
,”
Nat. Commun.
,
8
, p.
14230
.
7.
Yang
,
C.
, and
Suo
,
Z.
,
2018
, “
Hydrogel Ionotronics
,”
Nat. Rev. Mater.
,
3
(
6
), pp.
125
142
.
8.
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
.
9.
Sun
,
J. Y.
,
Zhao
,
X. H.
,
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.
Zhao
,
X.
,
2014
, “
Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretchy Networks
,”
Soft Matter
,
10
(
5
), pp.
672
687
.
11.
Long
,
R.
, and
Hui
,
C. Y.
,
2016
, “
Fracture Toughness of Hydrogels: Measurement and Interpretation
,”
Soft Matter
,
12
(
39
), pp.
8069
8086
.
12.
Creton
,
C.
,
2017
, “
50th Anniversary Perspective: Networks and Gels: Soft but Dynamic and Tough
,”
Macromolecules
,
50
(
21
), pp.
8297
8316
.
13.
Lefranc
,
M.
, and
Bouchaud
,
E.
,
2014
, “
Mode I Fracture of a Biopolymer Gel: Rate-Dependent Dissipation and Large Deformations Disentangled
,”
Extreme Mech. Lett.
,
1
, pp.
97
103
.
14.
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 Hydrocolloid
,
46
, pp.
180
190
.
15.
Baumberger
,
T.
,
Caroli
,
C.
, and
Martina
,
D.
,
2006
, “
Solvent Control of Crack Dynamics in a Reversible Hydrogel
,”
Nat. Mater.
,
5
(
7
), pp.
552
555
.
16.
Noselli
,
G.
,
Lucantonio
,
A.
,
McMeeking
,
R. M.
, and
DeSimone
,
A.
,
2016
, “
Poroelastic Toughening in Polymer Gels: A Theoretical and Numerical Study
,”
J. Mech. Phys. Solids
,
94
, pp.
33
46
.
17.
Yu
,
Y.
,
Landis
,
C. M.
, and
Huang
,
R.
,
2018
, “
Steady-State Crack Growth in Polymer Gels: A Linear Poroelastic Analysis
,”
J. Mech. Phys. Solids
,
118
, pp.
15
39
.
18.
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
.
19.
Skrzeszewska
,
P. J.
,
Sprakel
,
J.
,
de Wolf
,
F. A.
,
Fokkink
,
R.
,
Stuart
,
M. A. C.
, and
van der Gucht
,
J.
,
2010
, “
Fracture and Self-Healing in a Well-Defined Self-Assembled Polymer Network
,”
Macromolecules
,
43
(
7
), pp.
3542
3548
.
20.
Tang
,
J.
,
Li
,
J.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2017
, “
Fatigue Fracture of Hydrogels
,”
Extreme Mech. Lett.
,
10
, pp.
24
31
.
21.
Petch
,
N. J.
, and
Stables
,
P.
,
1952
, “
Delayed Fracture of Metals Under Static Load
,”
Nature
,
169
(
4307
), pp.
842
843
.
22.
Pearson
,
S.
,
1956
, “
Delayed Fracture of Sintered Alumina
,”
P. Phys. Soc. B
,
69
(
12
), pp.
1293
1296
.
23.
Knauss
,
W. G.
,
1970
, “
Delayed Failure—the Griffith Problem for Linearly Viscoelastic Materials
,”
Int. J. Fract. Mech.
,
6
(
1
), pp.
7
20
.
24.
Sprakel
,
J.
,
Lindstrom
,
S. B.
,
Kodger
,
T. E.
, and
Weitz
,
D. A.
,
2011
, “
Stress Enhancement in the Delayed Yielding of Colloidal Gels
,”
Phys. Rev. Lett.
,
106
(
24
), p.
248303
.
25.
Lindstrom
,
S. B.
,
Kodger
,
T. E.
,
Sprakel
,
J.
, and
Weitz
,
D. A.
,
2012
, “
Structures, Stresses, and Fluctuations in the Delayed Failure of Colloidal Gels
,”
Soft Matter
,
8
(
13
), pp.
3657
3664
.
26.
Shahidzadeh-Bonn
,
N.
,
Vie
,
P.
,
Chateau
,
X.
,
Roux
,
J.-N.
, and
Bonn
,
D.
,
2015
, “
Delayed Fracture in Porous Media
,”
Phys. Rev. Lett.
,
95
(
17
), p.
175501
.
27.
van der Kooij
,
H. M.
,
Dussi
,
S.
,
van de Kerkhof
,
G. T.
,
Frijns
,
R. A. M.
,
van der Gucht
,
J.
, and
Sprakel
,
J.
,
2018
, “
Laser Speckle Strain Imaging Reveals the Origin of Delayed Fracture in a Soft Solid
,”
Sci. Adv.
,
4
(
5
), p.
eaar1926
28.
Wang
,
X.
, and
Hong
,
W.
,
2012
, “
Delayed Fracture in Gels
,”
Soft Matter
,
8
(
31
), pp.
8171
8178
.
29.
Bouklas
,
N.
,
Landis
,
C. M.
, and
Huang
,
R.
,
2015
, “
Effect of Solvent Diffusion on Crack-Tip Fields and Driving Force for Fracture of Hydrogels
,”
ASME J. Appl. Mech.
,
82
(
8
), p.
081007
.
30.
Atkinson
,
C.
, and
Craster
,
R. V.
,
1991
, “
Plane Strain Fracture in Poroelastic Media
,”
Proc. R. Soc. London. Ser. A
,
434
(
1892
), pp.
605
633
.
31.
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
.
32.
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
.
33.
Yang
,
C.-H.
, and
Lin
,
Y.-Y.
,
2018
, “
Time-Dependent Fracture of Mode-I Cracks in Poroviscoelastic Media
,”
Eur. J. Mech. - A/Solids
,
69
, pp.
78
87
.
34.
Böger
,
L.
,
Keip
,
M.-A.
, and
Miehe
,
C.
,
2017
, “
Minimization and Saddle-Point Principles for the Phase-Field Modeling of Fracture in Hydrogels
,”
Comput. Mater. Sci.
,
138
, pp.
474
485
.
35.
Mao
,
Y.
, and
Anand
,
L.
,
2018
, “
A Theory for Fracture of Polymeric Gels
,”
J. Mech. Phys. Solids
,
115
, pp.
30
53
.
36.
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
.
37.
Hong
,
W.
,
Liu
,
Z.
, and
Suo
,
Z.
,
2009
, “
Inhomogeneous Swelling of a Gel in Equilibrium With a Solvent and Mechanical Load
,”
Int. J. Solids Struct.
,
46
(
17
), pp.
3282
3289
.
38.
Kang
,
M. K.
, and
Huang
,
R.
,
2010
, “
A Variational Approach and Finite Element Implementation for Swelling of Polymeric Hydrogels Under Geometric Constraints
,”
ASME J. Appl. Mech.
,
77
(
6
), p.
061004
.
39.
Bouklas
,
N.
, and
Huang
,
R.
,
2012
, “
Swelling Kinetics of Polymer Gels: Comparison of Linear and Nonlinear Theories
,”
Soft Matter
,
8
(
31
), pp.
8194
8203
.
40.
Hui
,
C.-Y.
,
Lin
,
Y. Y.
,
Chuang
,
F.-C.
,
Shull
,
K. R.
, and
Lin
,
W.-C.
,
2006
, “
A Contact Mechanics Method for Characterizing the Elastic Properties and Permeability of Gels
,”
J. Polym. Sci., Part B: Polym. Phys.
,
44
(
2
), pp.
359
370
.
41.
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
.
42.
Craster
,
R. V.
, and
Atkinson
,
C.
,
1992
, “
Shear Cracks in Thermoelastic and Poroelastic Media
,”
J. Mech. Phys. Solids
,
40
(
4
), pp.
887
924
.
You do not currently have access to this content.