Linearized elasticity as Mosco limit of finite elasticity in the presence of cracks

The small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Γ-converge to the linearized elastic energy with a local constraint of non-interpenetration along the crack.

Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms

We present a new penalty term approximating the Ciarlet–Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher-order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet–Nečas condition. Moreover, the penalization can be chosen in such a way that for all low-energy deformations, self-interpenetration is avoided completely already at all sufficiently small finite values of the penalization parameter. We also present numerical experiments in two dimensions illustrating our theoretical results and provide own MATLAB code available for download and testing.