An Implicit Sweeping Process Approach to Quasistatic Evolution Variational Inequalities

Abstract

In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of Moreau's catching-up algorithm, we show the well-posedness (in the sense existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality. It is well known that the variational formulations of many mechanical problems with unilateral contact and friction lead to an evolution variational inequality. As an application, we reformulate the quasistatic antiplane frictional contact problem for linear elastic materials with short memory as an implicit sweeping process with velocity constraint. The link between the implicit sweeping process and the quasistatic evolution variational inequality is possible thanks to some standard tools from convex analysis and is new in the literature.

State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities

“…To conclude this introduction, we mention that the present work can be considered as an improvement and an extension of the research initiated by the authors in [13]. In fact, the technique utilized in the current paper differs from the one used in [13].…”

“…To conclude this introduction, we mention that the present work can be considered as an improvement and an extension of the research initiated by the authors in [13]. In fact, the technique utilized in the current paper differs from the one used in [13]. The proof here is based on an implicit time discretization combined with the construction of a Cauchy sequence of functions via linear interpolation that converges strongly to a unique solution of (11).…”

“…Later the case of unbounded moving set is relaxed in [12], by assuming only the semicoercivity of the involved linear operators. Continuing the idea, in [13] the authors showed the existence and uniqueness of solutions for a new variant (7), by taking into account a linear combination of the velocity and state in Hilbert spaces. As an application, the quasistatic antiplane frictional contact problem for linear elastic materials with short memory was formulated as an implicit sweeping process with velocity constraint.…”

“…In this paper, we are interested in a generalization of the variant studied in [13]tothe case of a state-dependent moving set (11) without using any compactness assumption, which is known to be an open problem. The additional initial viability condition (8)is no longer required here, which allows nonzero initial velocity.…”

Time-dependent inclusions and sweeping processes in contact mechanics

“…Most of the material presented in this section is standard. Therefore, we introduce it without proofs and restrict ourselves to mention that details on the definitions and statements below can be found in the monographs [5,8,12,13] as well as in the paper [1].…”