Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity

Abstract: We perform via $\Gamma$-convergence a 2d-1d dimension reduction analysis of a single-slip elastoplastic body in large deformations. Rigid plastic and elastoplastic regimes are considered. In particular, we show that limit deformations can essentially freely bend even if subjected to the most restrictive constraints corresponding to the elastically rigid single-slip regime. The primary challenge arises in the upper bound where the differential constraints render any bending without incurring an additional energy cost particularly difficult. We overcome this obstacle with suitable non-smooth constructions and prove that a Lavrentiev phenomenon occurs if we artificially restrict our model to smooth deformations. This issue is absent if the differential constraints are appropriately softened.