On incompressibility constraint and crack direction in soft solids

Most soft materials resist volumetric changes much more than shape distortions. This experimental observation led to the introduction of the incompressibility constraint in the constitutive description of soft materials. The incompressibility constraint provides analytical solutions for problems which, otherwise, could be solved numerically only. However, in the present work, we show that the enforcement of the incompressibility constraint in the analysis of failure of soft materials can lead to somewhat non- physical results. 

We use hyperelasticity with energy limiters to describe material failure, which starts via the violation of the condition of strong ellipticity. This mathematical condition physically means inability of material to propagate superimposed waves because cracks nucleate perpendicular to the direction of a possible wave propagation. By enforcing the incompressibility constraint we sort out longitudinal waves and, consequently, we can miss cracks perpendicular to longitudinal waves. In the present work, we show that such scenario, indeed, occurs in the problems of uniaxial tension and pure shear of Natural Rubber. We also find that the suppression of longitudinal waves via the incompressibility constraint does not affect the consideration of material failure in equibiaxial tension and the practically relevant problem of the failure of rubber bearings under combined shear and compression.