# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021037
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## Numerical analysis and simulations of a frictional contact problem with damage and memory

 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Hailing Xuan

Received  November 2020 Revised  March 2021 Early access July 2021

In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.

Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulations of a frictional contact problem with damage and memory. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021037
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##### References:
Reference configuration of the two-dimensional example
Results with deformed configuration(left) and damage field(right)
Results with $a = 1, \ \mu = 0$ (left) and with $a = 1, \ \mu = 1$ (right)
Results with deformed configuration(left) and damage field(right)
numerical errors
numerical errors
 $h+k$ 0.046875 0.09375 0.1875 0.375 $\Vert {\boldsymbol{u }}- {\boldsymbol{u }}^{hk}\Vert_V+\Vert \zeta-\zeta^{hk}\Vert_{Z_0}$ $0.324\times 10^{-1}$ $0.631\times 10^{-1}$ 0.1082 0.194
 $h+k$ 0.046875 0.09375 0.1875 0.375 $\Vert {\boldsymbol{u }}- {\boldsymbol{u }}^{hk}\Vert_V+\Vert \zeta-\zeta^{hk}\Vert_{Z_0}$ $0.324\times 10^{-1}$ $0.631\times 10^{-1}$ 0.1082 0.194
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