Velocity dependent boundary conditions for the displacement in a one dimensional viscoelastic material
In this paper a boundary value problem of one dimensional viscoelastic material is analysed. The governing equation is \[ {\partial^ 2 u \over \partial t^ 2} = - \bigl( P(V) + \alpha (V) u_{tx} \bigr)_ x \tag{1} \] where \(x\) is the Lagrange mass variable, \(t\) is the time, \(u = u(x,t)\) is the displacement, \(V = V(x,t)\) is the specific volume, \(P\) is the pressure and the term \(-\alpha (V) u_{tx}\) gives the viscous stress. We have \[ V(x,t) = {\partial u \over \partial x} + V_ 0 (x), \tag{2} \] here \(V_ 0(x)\) is the initial value of the specific volume \(V\). The boundary conditions are as follows \[ - P \bigl( V(t,0) \bigr) + \alpha \bigl( V(t,0) \bigr) u_{tx} (t,0) = k_ 0 \tag{3} \] \[ - P \bigl( V(t,1) \bigr) + \alpha \bigl( V(t,1) \bigr) u_{tx} (t,1) = k_ 1. \tag{4} \] In this paper \(k_ 0\), \(k_ 1\) are depending on the time and velocity. It is assumed that (a) \(P\) satisfies a global Lipschitz condition, (b) the coefficient of viscosity \(\alpha (V)\) is bounded. Existence, regularity and uniqueness theorems for the global solutions to the boundary value problem (1), (3), (4) are given.
scientific article; zbMATH DE number 645222
scientific article