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Existence Results for the Time-incremental Elastic Contact Problem with Coulomb Friction in 2D

Patrick Ballard, Flaviana Iurlano

TL;DR

This work addresses the incremental Signorini-Coulomb problem in 2D linear elasticity with Coulomb friction by recasting each time step as a variational inequality driven by a nonlinear Leray–Lions operator. The authors prove existence using Brézis’s pseudomonotone framework, contingent on a key 2D compactness property of the Neumann-to-Dirichlet operator, which they establish for isotropic elasticity. In the anisotropic case, they show that a friction-bound condition $|\alpha|f<1$ (with $\alpha = \mathscr C_2/\mathscr C_1$) is necessary for solvability and provide a modified operator $A^\alpha$ that retains Leray–Lions structure to recover existence; they also demonstrate a nonexistence result for large friction in a steady-sliding scenario. Overall, the paper delivers sharp 2D existence results for the incremental Signorini-Coulomb problem, clarifies when friction can be arbitrarily large (isotropic) or must be restrained (anisotropic), and develops a rigorous framework linking elasticity NtD operators to variational inequalities with pseudomonotone operators.

Abstract

In this article, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini-Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray-Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray-Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini-Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.

Existence Results for the Time-incremental Elastic Contact Problem with Coulomb Friction in 2D

TL;DR

This work addresses the incremental Signorini-Coulomb problem in 2D linear elasticity with Coulomb friction by recasting each time step as a variational inequality driven by a nonlinear Leray–Lions operator. The authors prove existence using Brézis’s pseudomonotone framework, contingent on a key 2D compactness property of the Neumann-to-Dirichlet operator, which they establish for isotropic elasticity. In the anisotropic case, they show that a friction-bound condition (with ) is necessary for solvability and provide a modified operator that retains Leray–Lions structure to recover existence; they also demonstrate a nonexistence result for large friction in a steady-sliding scenario. Overall, the paper delivers sharp 2D existence results for the incremental Signorini-Coulomb problem, clarifies when friction can be arbitrarily large (isotropic) or must be restrained (anisotropic), and develops a rigorous framework linking elasticity NtD operators to variational inequalities with pseudomonotone operators.

Abstract

In this article, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini-Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray-Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray-Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini-Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.
Paper Structure (12 sections, 27 theorems, 146 equations)

This paper contains 12 sections, 27 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Formulation of the Signorini-Coulomb problem
  3. Historical background
  4. Strategy of proof and organization of the article
  5. The case of isotropic elasticity
  6. Data regularity and trace spaces
  7. Existence in the case of isotropic elasticity
  8. Proof of Theorem \ref{['thm:cornerstone2D']}
  9. The case of anisotropic elasticity
  10. A frictional contact problem on the half-space with no solution
  11. Existence in the case of anisotropic elasticity
  12. The Neumann-to-Dirichlet operator of the anisotropic elastic 2D half-space

Key Result

Proposition 2.1

Let $J_\Lambda:H^{1/2}(\Gamma_C;\mathbb{R}^N)\to \{\mathbf v\in H^1(\Omega,\mathbb{R}^N):\mathbf{v}_{|\Gamma_U}=\mathbf{0}\}$ be defined as Then, $\| J_\Lambda(\mathbf{u})\|_{H^1(\Omega,\mathbb{R}^N)}$ is a norm on $H^{1/2}(\Gamma_C;\mathbb{R}^N)$ which is equivalent to that of $H^{1/2}(\Gamma_C;\mathbb{R}^N)$. In an analogous way, let $J'_\Lambda:H^{-1/2}(\Gamma_C;\mathbb{R}^N)\to \{\mathbf v\in

Theorems & Definitions (39)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 29 more