Research Papers

Effect of Solvent Diffusion on Crack-Tip Fields and Driving Force for Fracture of Hydrogels

[+] Author and Article Information
Nikolaos Bouklas, Chad M. Landis, Rui Huang

Department of Aerospace Engineering
and Engineering Mechanics,
University of Texas,
Austin, TX 78712

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 21, 2015; final manuscript received May 6, 2015; published online June 9, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(8), 081007 (Aug 01, 2015) (16 pages) Paper No: JAM-15-1153; doi: 10.1115/1.4030587 History: Received March 21, 2015; Revised May 06, 2015; Online June 09, 2015

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.

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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. [CrossRef]
Drury, J. L., and Mooney, D. J., 2003, “Hydrogels for Tissue Engineering: Scaffold Design Variables and Applications,” Biomaterials, 24(24), pp. 4337–4351. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
Qiu, Y., and Park, K., 2001, “Environment-Sensitive Hydrogels for Drug Delivery,” Adv. Drug Delivery Rev., 53(3), pp. 321–339. [CrossRef]
Calvert, P., 2009, “Hydrogels for Soft Machines,” Adv. Mater., 21(7), pp. 743–756. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Baumberger, T., Caroli, C., and Martina, D., 2006, “Solvent Control of Crack Dynamics in a Reversible Hydrogel,” Nat. Mater., 5(7), pp. 552–555. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
Tanaka, Y., Fukao, K., and Miyamoto, Y., 2000, “Fracture Energy of Gels,” Eur. Phys. J. E, 3(4), pp. 395–401. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Gong, J. P., 2010, “Why are Double Network Hydrogels so Tough?,” Soft Matter, 6(12), pp. 2583–2590. [CrossRef]
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. [CrossRef] [PubMed]
Zhao, X., 2014, “Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretchy Networks,” Soft Matter, 10(5), pp. 672–687. [CrossRef] [PubMed]
Hu, Y., and Suo, Z., 2012, “Viscoelasticity and Poroelasticity in Elastomeric Gels,” Acta Mech. Solida Sin., 25(5), pp. 441–457. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
Knauss, W. G., 1973, “The Mechanics of Polymer Fracture,” ASME Appl. Mech. Rev., 26, pp. 1–17.
Schapery, R. A., 1975, “A Theory of Crack Initiation and Growth in Viscoelastic Media,” Int. J. Fract., 11(1), pp. 141–159. [CrossRef]
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. [CrossRef]
Schapery, R. A., 1975, “A Theory of Crack Initiation and Growth in Viscoelastic Media,” Int. J. Fract., 11(4), pp. 549–562. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Wang, X., and Hong, W., 2012, “Delayed Fracture in Gels,” Soft Matter, 8(31), pp. 8171–8178. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
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. [CrossRef]
Kishimoto, K., Aoki, S., and Sakata, M., 1980, “On the Path Independent Integral-J,” Eng. Fract. Mech., 13(4), pp. 841–850. [CrossRef]
Chien, N., and Herrmann, G., 1996, “Conservation Laws for Thermo or Poroelasticity,” ASME J. Appl. Mech., 63(2), pp. 331–336. [CrossRef]
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. [CrossRef]
Gao, Y. F., and Zhou, M., 2013, “Coupled Mechano-Diffusional Driving Forces for Fracture in Electrode Materials,” J. Power Sources, 230, pp. 176–193. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Dolbow, J., Fried, E., and Ji, H., 2004, “Chemically Induced Swelling of Hydrogels,” J. Mech. Phys. Solids, 52(1), pp. 51–84. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Prigogine, I., 1968, Introduction to Thermodynamics of Irreversible Processes, Wiley, New York.
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. [CrossRef]
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. [CrossRef]
Rice, J. R., 1968, Mathematical Analysis in the Mechanics Fracture. Fracture: An Advanced Treatise, Vol. 2, Academic, New York, pp. 191–311.
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. [CrossRef]
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. [CrossRef]
Wan, J., 2002, “Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow,” Ph.D. dissertation, Stanford University, Stanford, CA.
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. [CrossRef]
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. [CrossRef]
Bouklas, N., and Huang, R., 2012, “Swelling Kinetics of Polymer Gels: Comparison of Linear and Nonlinear Theories,” Soft Matter, 8(31), pp. 8194–8203. [CrossRef]
Broberg, K. B., 1999, Cracks and Fracture, Academic, San Diego, CA.
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. [CrossRef]
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.
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. [CrossRef] [PubMed]
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. [CrossRef]
Geubelle, P. H., 1995, “Finite Deformation Effects in Homogeneous and Interfacial Fracture,” Int. J. Solids Struct., 32(6–7), pp. 1003–1016. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]


Grahic Jump Location
Fig. 2

A simply connected region A2 enclosed by a closed contour C=C4-C1+C2+C3 around a crack tip

Grahic Jump Location
Fig. 1

Schematics of (a) a sharp crack and (b) a rounded notch model, both in the reference configuration

Grahic Jump Location
Fig. 3

A hydrogel specimen with a center crack, subject to remote tension. A two-dimensional finite element mesh for one quarter of the specimen is shown.

Grahic Jump Location
Fig. 4

(a) Finite element mesh near the crack tip in the sharp crack model, with 50 quarter-point singular Taylor–Hood elements (6u3p). (b) Mesh near a rounded notch, where the radius of the notch is three orders of magnitude smaller than the crack length (rn/a = 10-3).

Grahic Jump Location
Fig. 12

Normalized J* for immersed (a) and not-immersed (b) hydrogel specimens with increasing far-field strain. The results from the rounded notch model are compared to the sharp crack model for ɛ∞ = 0.1.

Grahic Jump Location
Fig. 5

Cauchy stress distributions around a crack tip at the instantaneous ((a)–(c)) and equilibrium ((d)–(f)) limits for an immersed hydrogel specimen under Mode I loading (ɛ∞ = 10-3), with increasing distance (r) from the crack tip. Dashed lines are obtained from the LEFM solution in Eq. (4.7) with Poisson's ratio ν = 0.5 and 0.2415, respectively, for the two limits.

Grahic Jump Location
Fig. 6

(a) Chemical potential field (normalized as μ/kBT) near the crack tip as the instantaneous response (t/τ = 10-4) of the hydrogel subject to a small remote strain (ɛ∞ = 10-3). (b) Change of solvent concentration (normalized as (C-C0)Ω) near the crack tip in the equilibrium state (t/τ = 105).

Grahic Jump Location
Fig. 7

Change of solvent concentration ahead of the crack tip for (a) an immersed and (b) a not-immersed hydrogel specimens subject to a small remote strain (ɛ∞ = 10-3). The dashed line has a slope of −0.5 in the log–log scale.

Grahic Jump Location
Fig. 9

(a) Path-independent J*-integral and (b) the classical J-integral for the immersed hydrogel specimen (ɛ∞ = 10-3) at the instantaneous, transient, and equilibrium states. Here, r is the radius of the contour (C1 in Fig. 2); the annular domain (A2) is taken to be one ring of elements outside the contour for the domain integral calculations.

Grahic Jump Location
Fig. 11

Dependence of the J*-integral on the material parameters for not-immersed hydrogel specimens under a small far-field strain (ɛ∞ = 10-3). (a) and (b) show the dependence on NΩ and χ, respectively. (c) and (d) The renormalized results using the instantaneous J* in Eq. (4.12) and the poroelastic timescale, τ* = a2/D*, with the effective diffusivity in Eq. (4.14).

Grahic Jump Location
Fig. 13

(a) Crack opening profiles for an immersed hydrogel specimen under a moderately large far-field strain (ɛ∞=0.1), from the sharp crack model (thin lines) and the rounded notch model (symbols); the dashed line shows the LEFM prediction. (b) Distribution of the Cauchy stress σ22 ahead of the crack tip in the instantaneous and equilibrium limits. (c) Evolution of the chemical potential and (d) the solvent concentration ahead of the crack tip.

Grahic Jump Location
Fig. 8

Crack opening profiles for (a) immersed and (b) not-immersed hydrogel specimens subject to a small remote strain (ɛ∞ = 10-3). (c) Maximum crack opening displacement as a function of time for the two cases. Dashed lines are the LEFM solution as given in Eq. (4.11).

Grahic Jump Location
Fig. 10

J* as a function of time for the immersed and not-immersed hydrogel specimens subject to a small far-field strain (ɛ∞ = 10-3). The effect of the specimen size is shown for the immersed case with h/a = 10,20, and 100.

Grahic Jump Location
Fig. 14

Crack opening profiles and distributions of the Cauchy stress σ22 ahead of the crack tip under a large far-field strain (ɛ∞=0.5). ((a) and (b)) For an immersed hydrogel specimen and ((c) and (d)) for a not-immersed hydrogel specimen.

Grahic Jump Location
Fig. 15

Evolution of the modified J-integral for an immersed and a not-immersed hydrogel specimens under a large far-field strain (ɛ∞=0.5)



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