[Paper Review] Boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials

Qiang Li, Valentin L. Popov|arXiv (Cornell University)|Dec 26, 2016

Adhesion, Friction, and Surface Interactions10 references3 citations

TL;DR

This paper presents a boundary element method (BEM) for normal contact mechanics of power-law graded elastic materials, both non-adhesive and adhesive. It extends the fundamental solution for a half-space with spatially varying modulus to calculate deformation under pressure using a rectangular grid and conjugate-gradient inversion, accelerated by FFT. The method enables accurate simulation of adhesive contact using a Pohrt–Popov detachment criterion, validated against analytical solutions with high numerical precision.

ABSTRACT

Recently proposed formulation of the Boundary Element Method for adhesive contacts has been generalized for contacts of functionally graded materials with and without adhesion. First, proceeding from the fundamental solution for single force acting on the surface of a half space with a power-law varying elastic modulus, the deformation produced by constant pressure acting on a rectangular element was calculated and the influence matrix was obtained for a rectangular grid. The inverse problem for the calculation of required stress in contact area from a known surface deformation was solved by use of conjugate-gradient technique. For the transformation between the stresses and displacements, the Fast Fourier Transformation is used which drastically reduces the computation time. For the adhesive contact of graded material, the detachment criterion based on the method of Pohrt and Popov was proposed. A number of numerical test for the problem having exact analytical solution have been carried out confirming the correctness of underlying ideas and numerical implementation.

Motivation & Objective

• To extend the boundary element method to functionally graded materials with power-law varying elastic modulus.
• To model both non-adhesive and adhesive normal contact problems in such materials.
• To develop an efficient numerical scheme for solving the inverse problem of stress reconstruction from surface displacement.
• To incorporate an adhesive contact criterion based on the Pohrt–Popov method for graded materials.
• To validate the method against exact analytical solutions for benchmark cases.

Proposed method

• Derives the fundamental solution for a point force on a half-space with power-law varying elastic modulus.
• Calculates deformation due to constant pressure on a rectangular boundary element using the fundamental solution.
• Constructs an influence matrix for a rectangular grid to relate surface tractions to displacements.
• Solves the inverse problem (displacement to stress) via the conjugate-gradient method to ensure numerical stability.
• Employs Fast Fourier Transform (FFT) to accelerate the matrix-vector multiplication in the BEM formulation.
• Introduces a detachment criterion for adhesive contact based on the Pohrt–Popov method, adapted for graded materials.

Experimental results

Research questions

• RQ1 How can the boundary element method be extended to handle normal contact in materials with power-law graded elasticity?
• RQ2 What is the most efficient numerical approach to solve the inverse contact problem in such materials?
• RQ3 How can adhesive contact be modeled in functionally graded materials using a consistent detachment criterion?
• RQ4 Can the proposed BEM formulation achieve high accuracy when compared to analytical solutions?
• RQ5 What is the computational efficiency gain from using FFT in the BEM formulation for graded materials?

Key findings

• The BEM formulation accurately reproduces analytical solutions for benchmark contact problems, confirming the correctness of the numerical implementation.
• The use of FFT reduces computation time significantly, enabling efficient solution of large-scale contact problems.
• The conjugate-gradient method effectively solves the ill-posed inverse problem of reconstructing surface stress from displacement data.
• The Pohrt–Popov-based detachment criterion successfully captures adhesive behavior in graded materials.
• The method demonstrates high numerical stability and convergence for both non-adhesive and adhesive contact scenarios.
• The influence matrix for rectangular elements is accurately computed using the fundamental solution, forming a solid basis for further simulations.