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Variational inequalities of multilayer elastic systems with interlayer friction: existence and uniqueness of solution and convergence of numerical solution

Zhizhuo Zhang, Xiaobing Nie, Jinde Cao

TL;DR

This work addresses the displacement–stress analysis of multilayer pavement systems with interlayer Coulomb friction by casting the mechanics into a variational inequality framework. The authors formulate a nonlinear, small-deformation PDE model with layerwise operators $\mathcal{A}^i$ and friction functionals $g_N^i,g_T^i$, deriving a variational inequality $P_1$ on a constraint set $\mathcal{K}$. They prove existence and uniqueness under strong monotonicity of $A$ and appropriate Lipschitz properties of $j$, via a Banach fixed-point argument requiring $M>m$, and develop finite-element discretizations with convergence results and a quantified convergence rate. The results provide theoretical justification and practical guidance for solving pavement displacement–strain problems using variational inequalities, offering a path toward more realistic, friction-inclusive pavement models and efficient FE solvers.

Abstract

Based on the mathematical-physical model of pavement mechanics, a multilayer elastic system with interlayer friction conditions is constructed. Given the complex boundary conditions, the corresponding variational inequalities of the partial differential equations are derived, so that the problem can be analyzed under the variational framework. First, the existence and uniqueness of the solution of the variational inequality is proved; then the approximation error of the numerical solution based on the finite element method is analyzed, and when the finite element space satisfies certain approximation conditions, the convergence of the numerical solution is proved; finally, in the trivial finite element space, the convergence order of the numerical solution is derived. The above conclusions provide basic theoretical support for solving the displacement-strain problem of multilayer elastic systems under the framework of variational inequalities.

Variational inequalities of multilayer elastic systems with interlayer friction: existence and uniqueness of solution and convergence of numerical solution

TL;DR

This work addresses the displacement–stress analysis of multilayer pavement systems with interlayer Coulomb friction by casting the mechanics into a variational inequality framework. The authors formulate a nonlinear, small-deformation PDE model with layerwise operators and friction functionals , deriving a variational inequality on a constraint set . They prove existence and uniqueness under strong monotonicity of and appropriate Lipschitz properties of , via a Banach fixed-point argument requiring , and develop finite-element discretizations with convergence results and a quantified convergence rate. The results provide theoretical justification and practical guidance for solving pavement displacement–strain problems using variational inequalities, offering a path toward more realistic, friction-inclusive pavement models and efficient FE solvers.

Abstract

Based on the mathematical-physical model of pavement mechanics, a multilayer elastic system with interlayer friction conditions is constructed. Given the complex boundary conditions, the corresponding variational inequalities of the partial differential equations are derived, so that the problem can be analyzed under the variational framework. First, the existence and uniqueness of the solution of the variational inequality is proved; then the approximation error of the numerical solution based on the finite element method is analyzed, and when the finite element space satisfies certain approximation conditions, the convergence of the numerical solution is proved; finally, in the trivial finite element space, the convergence order of the numerical solution is derived. The above conclusions provide basic theoretical support for solving the displacement-strain problem of multilayer elastic systems under the framework of variational inequalities.
Paper Structure (11 sections, 6 theorems, 87 equations, 1 figure)

This paper contains 11 sections, 6 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The physical model and variational formulation
  3. The physical model and PDEs
  4. Variational inequality
  5. The existence and uniqueness of solution
  6. Approximation and Convergence
  7. Discrete approximation
  8. Convergence Analysis
  9. The convergence order of the error estimate
  10. Conclusion
  11. Appendix

Key Result

Theorem 3.1

Let (3.2)-(3.3) hold and $\boldsymbol{f}\in V$. Then, if $M>m$, there exists a unique solution $\boldsymbol{u}\in \mathcal{K}$ to the problem (3.1).

Figures (1)

  • Figure 1: The physical model of multilayer contact problem

Theorems & Definitions (13)

  • Remark 1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 2
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 3 more