Convergence of penalty and regularization method for contact problems in locking materials with normal compliance

Abstract

We study a static contact problem involving a locking body and an obstacle considering normal compliance and frictional effects. The mechanical behavior follows an elastic constitutive law for a locking material, which is modeled by using a subdifferential approach. The weak formulation is presented as a variational inequality for the displacement field. We establish the existence and uniqueness of the weak solution under suitable regularity conditions. Next, we introduce a penalty formulation expressed as a variational identity achieved by integrating the penalization of the terms in the constitutive law on one hand and, on the other hand, by regularizing the contact condition. Under the scaling relation $\epsilon = \frac{1}{n}$ between the penalty parameters, we establish the convergence of the penalized solutions as $n \to \infty$. Finally, we analyze error estimates for the finite element discretization of the penalty method, showing that for the discretization parameter $h= \frac{1}{n^{2}}$ the error terms yield estimates comparable to the constrained problem case. Our analysis demonstrates consistent convergence behavior between the penalized and constrained formulations.