双边摩擦接触力学问题和第二类混合变分不等式的等价性及其数值近似

The Equality between a Bilateral Friction Contact Problem and the Second Kind Mixed Inequality and its Numerical Approximation

摩擦接触问题的数学模型是一个变分不等式,一般的变分不等式对应力,表面力及位移是利用应力-应变关系,应变-位移关系逐个进行求解,而混合变分不等形式则可同时求解应力和位移,这是混合变分不等式的优点.王烈衡[1]曾以混合变分形式为基础,利用有限元法求解无摩擦弹性力学问题.本文以弹性力学问题中的双边摩擦接触问题为背景,讨论了第二类混合变分不等形式和能量泛函的极小值问题,并对它们的等价性进行了研究,接着用有限元法求双边摩擦的弹性接触问题以及近似解的误差估计.

The mathematician model of the frictional contact problem is a variation inequality.We usually solve the solutions of stress,surface force and the displacement by the relations of stress-strain and stress-displacement,but a mixed variation inequality can obtain the solution of stress and displacement at the same time.This is an advantage of mixed variation inequality.Based on mixed variation inequality,Wang L.H.has solved the solution of no friction contact problem in elasticity by the finite element method（FEM）.In this paper,we obtain mixed variation inequality from bilateral friction contact problem,as well as the solution of the minimum energy function correspond,and give an attestation on equality between them.Then,the finite element method is used to solve the error estimates approximate of the bilateral friction contact problem.