Mathematical and computational models of incompressible materials subject to shear

Numerical modelling of incompressible non-linear elastic materials plays an increasing role in computational science and engineering, particularly in the high-fidelity simulation of rubber-like materials and many biological tissues. Our present study focuses on the treatment of the incompressibility constraint in finite-element discretizations for a cube subject to 'simple shear'. We demonstrate that this test problem is not easily captured in three-dimensional mathematical and computational models, with challenges related to the incompressibility constraint that are unique to each approach. Specifically, we review the mathematical model, which presupposes the simple shear deformation and requires additional assumptions to determine the response. A computational model avoids these difficulties, but gives rise to questions regarding the mismatch between continuum and discrete representations of material incompressibility. We consider three distinct finite-element formulations to enforce (near) incompressibility, with particular attention paid to the competing goals of physical fidelity and computational efficiency. We demonstrate that some of these standard approaches fail to resolve incompressibility in a pointwise manner, despite holding it in an averaged sense. Numerical results indicate that the maximum ratio of deformed to undeformed volumes grows sharply with mesh refinement; this is in contrast to the mathematical model, but occurs in a manner consistent with finite-element convergence theory.