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Finite element discretization of some variational inequalities arising in contact problems with friction. (English) Zbl 1164.74547
The subject of the paper is the numerical analysis of a contact problem with Coulomb friction. Let \(\Omega\subset \mathbb{R}^m\), \(m=2,3\) be a domain occupied by a linear elastic body, and \(\Gamma=\partial\Omega\) a Lipschitz surface, \(\Gamma=\bar\Gamma_1\cup\bar\Gamma_2\cup\bar\Gamma_3\). The displacement field \(\vec u(x,t)=(u_1,\dots,u_m)\) is governed by the equation \[ -\text{div\;}\sigma(\vec u)=\vec f,\quad x\in \Omega,\quad t\in[0,T]\tag{1} \] with boundary and initial conditions \[ \vec u=0\;\text{on}\;\Gamma_1,\quad \sigma\cdot\vec n=\vec h\;\text{on}\;\Gamma_2,\tag{2} \] \[ u_N\leq g,\quad \sigma_N(\vec u)\leq 0,\quad (u_N-g)\sigma_N(\vec u)=0 \quad\text{on}\;\Gamma_3,\tag{3} \] \[ \begin{aligned} &\| \sigma_\tau(\vec u)\|\leq \mu|\sigma_N(\vec u)|\quad\text{on}\;\Gamma_3\quad \text{such that}\\ &\text{if}\;\| \sigma_\tau(\vec u)\|< \mu|\sigma_N(\vec u)|\quad\;\text{then}\;\vec u_\tau=0,\\ &\text{if}\;\| \sigma_\tau(\vec u)\|= \mu|\sigma_N(\vec u)|\quad\;\text{then}\;\exists\;\alpha\geq 0\;\text{for which}\;\dot{\vec u}_{\tau}=-\alpha\,\sigma_{\tau}(\vec u), \end{aligned}\tag{4} \] \[ \vec u(x,0)=\vec u_0.\tag{5} \] Here \(\vec f\) is the volume forces of density, \(\vec h\) is the surface traction of density, \(g\geq 0\) is the gap between the body and the rigid foundation, \[ u_N=\vec u\cdot\vec n,\quad \vec u_\tau=\vec u -u_N\,\vec n,\quad \sigma_N=\sigma_{ij}n_in_j,\quad (\vec \sigma_\tau)_i=\sigma_{ij}n_j-\sigma_Nn_i, \] \(\sigma(\vec u)\) is the stress tensor, \(\vec n\) is the outward normal to \(\Gamma\), \(\mu\) is a given function. The condition (4) describes Coulomb friction. The problem (1) – (5) has an equivalent formulation in the form of a certain inequality. This inequality is solved by mixed finite elements. The numerical solution is studied in detail.
74S05 Finite element methods applied to problems in solid mechanics
49J40 Variational inequalities