Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

Victor A. Kovtunenko,Nyurgun P. Lazarev

doi:10.3390/math10020250

Victor A. Kovtunenko, Nyurgun P. Lazarev

Open Access

https://doi.org/10.3390/math10020250

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Abstract

A new type of non-classical 2D contact problem formulated over non-convex admissible sets is proposed. Specifically, we suppose that a composite body in its undeformed state touches a wedge-shaped rigid obstacle at a single contact point. Composite bodies under investigation consist of an elastic matrix and a rigid inclusion. In this case, the displacements on the set, corresponding to a rigid inclusion, have a predetermined structure that describes possible parallel shifts and rotations of the inclusion. The rigid inclusion is located on the external boundary and has the form of a wedge. The presence of the rigid inclusion imposes a new type of non-penetration condition for certain geometrical configurations of the obstacle and the body near the contact point. The sharp-shaped edges of the obstacle effect such sets of admissible displacements that may be non-convex. For the case of a thin rigid inclusion, which is described by a curve and a volume (bulk) rigid inclusion specified in a subdomain, the energy minimization problems are formulated. The solvability of the corresponding boundary value problems is proved, based on analysis of auxiliary minimization problems formulated over convex sets. Qualitative properties of the auxiliary variational problems are revealed; in particular, we have found their equivalent differential formulations. As the most important result of this study, we provide justification for a new type of mathematical model for 2D contact problems for reinforced composite bodies.

Highlights

There are various studies related to problems describing contact of elastic bodies with rigid or elastic obstacles, see for example [1,2,3,4,5,6,7,8,9,10]
In contrast to [16], we propose a class of non-linear contact problems, where non-penetration conditions can be written for a single point located on the sharp edge
The obtained results justify the new class of point-contact problems

Summary

Introduction

There are various studies related to problems describing contact of elastic bodies with rigid or elastic obstacles, see for example [1,2,3,4,5,6,7,8,9,10]. The differential formulation of (12) can be obtained without additional regularity assumptions on the solution U2 In this case, the expressions for the boundary conditions take the form of duality relations in the space of distributions. We note that the values σν(U2ω)−, στ(U2ω)− can be non-zero on (∂ω \ γ)−, despite of σν(U2ω)+ = 0, στ(U2ω)+ = 0 on (∂ω \ γ)+ due to εij(U2ω) = 0 in the rigid inclusion ω, i, j = 1, 2 This case arises when the jumps of functions σij(U2ω), i, j = 1, 2, are not equal to zero on (∂ω \ γ)− provided that U2ω ∈ H(Ω), but U2ω ∈/ H2(Ω)

The Case of Two-Hinged Thin Rigid Inclusions

Discussion

Methods