Numerical analysis and simulation of an adhesive contact problem with damage and long memory

Hailing Xuan; Xiaoliang Cheng

doi:10.3934/dcdsb.2020205

Article Contents

2021, Volume 26, Issue 5: 2781-2804. Doi: 10.3934/dcdsb.2020205

This issue Previous Article Collective behaviors of a Winfree ensemble on an infinite cylinder Next Article Global and exponential attractors for a nonlinear porous elastic system with delay term

Numerical analysis and simulation of an adhesive contact problem with damage and long memory

Hailing Xuan^, and
Xiaoliang Cheng

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

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

Received: March 2020

Revised: April 2020

Early access: June 2020

Published: May 2021

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Abstract

This paper studies an adhesive contact model which also takes into account the damage and long memory term. The deformable body is composed of a viscoelastic material and the process is taken as quasistatic. The damage of the material caused by the compression or the tension is involved in the constitutive law and the damage function is modelled through a nonlinear parabolic inclusion. Meanwhile, the adhesion process is modelled by a bonding field on the contact surface while the contact is described by a nonmonotone normal compliance condition. The variational formulation of the model is governed by a coupled system which consists of a history-dependent hemivariational inequality for the displacement field, a nonlinear parabolic variational inequality for the damage field and an ordinary differential equation for the adhesion field. We first consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived. At the end of this paper, {we report some two-dimensional numerical simulation results} for the contact problem in order to provide numerical evidence of the theoretical results.

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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. the deformed configuration at t = 0.125s, t = 0.5s, t = 0.75s and t = 1s

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Figure 3. the damage field at t = 0.5s and t = 1s

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Figure 4. the adhesion field at several times

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Figure 5. the deformed configuration at several times

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Figure 6. the adhesion field at several times

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Figure 7. the deformed configuration at t = 0.5s and t = 1s

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Figure 8. the damage field at t = 0.5s and t = 1s

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Figure 9. the deformed configuration at $ \gamma_\nu = 1 $ and $ \gamma_\nu = 10 $, respectively

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