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Frictional viscoelastic contact problem

Let be $G$ a viscoelastic body in $\R^3$ The smooth boundary $\Gamma $ is divided into three disjoint open parts : $\Gamma =\overline {\Gamma }_u\cup\overline {\Gamma }_f\cup\overline {\Gamma }_c$. The body is subjected to volume forces of density $F_1$. On $\Gamma _u$ we have zero displacements and on $\Gamma _f$ are prescribed tractions. On $\Gamma _c$ the body comes in contact with the foundation. Denote by $u$ the displacement field, by $\sigma =(\sigma _{ij})$ is the stress tensor, by $\varepsilon =(\varepsilon _{ij}=1/2(u_{i,j}+u_{j,i})$ the strain tensor. Further $\rho$ is the mass density and ${\bf A}=(a_{ijkl})$ and ${\bf B}=(b_{ijkl})$ are the elasticity and viscoelasticity tensors, which are symmetric and elliptic.
 

In a strong formulation the problem is the following: Find $u=u(t,x)$ such that $u(0,x)=u_0, \dot{u}(0,x)=\dot{u}_0$ and for all $t\in (0,T)$

\begin{eqnarray*}div \sigma + F_1&=&\rho \ddot{u}\quad\mbox{in} \quad G\\\sigma &=&{\bf A}\varepsilon (u)+B\varepsilon (\dot{u})\end{eqnarray*}
Boundary conditions are:
\begin{eqnarray*}&&u=0\quad\mbox{on}\quad\Gamma _u\\&&\sigma \cdot n=F_2\quad\mbox{on}\quad\Gamma _f\end{eqnarray*}
$n$ is the outward normal unit vector. Note that $ u=u_Nn+u_T$ and $\sigma \cdot n=\sigma _Nn+\sigma _T$. The index $N$ or $T$ stands for normal or tangential components. We have $\sigma _N=\sum_{i,j=1}^3(\sigma _{ij})n_in_j$ and $u_N=\sum_{i=1}^3u_in_i$. Now we produce a normal compliance law for the contact with the Coulomb friction law on $(0,T)\times \Gamma _c$. This normal compliance problem was introduced by Martins JAC and Oden JT in 1987. We assume
\begin{eqnarray*}&&\sigma _N=-C_N(u_N-q)_+\\&&\vert\sigma _T\vert\le C_T(u_N-q)_+\\\end{eqnarray*}
where from $\vert\sigma _T\vert< C_T(u_N-q)_+$ follows $\dot{u}_T=0$ and $\vert\sigma _T\vert=C_T(u_N-q)_+$ then there exists a constant $\lambda \ge0$ with $\dot{u}_T=-\lambda \sigma _T$. Here $q$ represents the initial gap mostly assumed to be zero.
 

A last reference: J-M. Ricaud, E. Pratt (2001) Analysis of time discretization for an implicit variational inequality modelling dynamic contact problems with friction, MMAS, 24, ???-???.


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Nächste Seite:Navier-Stokes equations (NSE) forAufwärts:Models of Mathematical PhysicsVorherige Seite:Relativistic Vlasov-Poisson equation (RVM)
collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg