Numerical analysis and simulations of a frictional contact problem with damage and memory

Hailing Xuan; Xiaoliang Cheng

doi:10.3934/mcrf.2021037

Article Contents

2022, Volume 12, Issue 3: 621-639. Doi: 10.3934/mcrf.2021037

This issue Previous Article Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement Next Article Local null controllability of the penalized Boussinesq system with a reduced number of controls

Numerical analysis and simulations of a frictional contact problem with damage and memory

Hailing Xuan^, and
Xiaoliang Cheng

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

^* Corresponding author: Hailing Xuan
^* Corresponding author: Hailing Xuan

Received: October 31, 2020

Revised: February 28, 2021

Early access: July 2021

Published: July 2022

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 65M15, 74G30, 65N22; Secondary: 74M15, 47J20.

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Figure 1. Reference configuration of the two-dimensional example

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Figure 2. Results with deformed configuration(left) and damage field(right)

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Figure 3. Results with $ a = 1, \ \mu = 0 $ (left) and with $ a = 1, \ \mu = 1 $ (right)

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Figure 4. Results with deformed configuration(left) and damage field(right)

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Figure 5. numerical errors

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Table 1. 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	0.1940

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