Finite strain theory of a Mode III crack in a rate dependent gel consisting of chemical and physical cross-links

In previous works we developed a three-dimensional large deformation constitutive theory for a dual cross-link hydrogel gel with permanent and transient bonds. This theory connects the breaking and reforming kinetics of the transient bonds to the nonlinear elasticity of the gel network. We have shown that this theory agrees well with experimental data from both tension and torsion tests. Here we study the mechanics of a Mode III or anti-plane shear crack for this particular class of rate dependent solids. We first show that our constitutive model admits a non-trivial state of anti-plane shear deformation. We then establish a correspondence principle for the special case where the chains in the network obey Gaussian statistics. For this special case there is a one to one analogy between our model and standard linear viscoelasticity. We also study the asymptotic behavior of the time dependent crack tip field, and show that for a wide class of network energy density functions, the spatial singularities of these fields are identical to a hyper-elastic, cracked body with the same but undamaged networks.