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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.
##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 49J40 Variational inequalities