Global injectivity in second-gradient Nonlinear Elasticity and its approximation with penalty terms

We present a new penalty term approximating the Ciarlet-Ne\v{c}as 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-Ne\v{c}as condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, self-interpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.