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One-dimensional viscoelastodynamics with Signorini boundary conditions. (Viscoélastodynamique monodimensionnelle avec conditions de Signorini.) (French. Abridged English version) Zbl 1043.35119

Summary: Let \(\alpha\) be a positive number. The one-dimensional viscoelastic problem \[ u_{tt}-u_{xx}- \alpha u_{xxt}=f,\;x\in(-\infty,0],\;t\in[0,+\infty), \] with unilateral boundary conditions \[ u(0,\cdot)\geq 0,\;(u_x+\alpha u_{xt}) (0,\cdot)\geq 0,\;\bigl(u(u_x+ \alpha u_{xt})\bigr) (0,\cdot)=0, \] can be reduced to the following variational inequality: \(\lambda_1*w=g+b\), \(w\geq 0\), \(b\geq 0\), \(\langle w,b\rangle=0\). Here \(\widehat\lambda_1(\omega)\) is the causal determination of \(i\omega\sqrt{1+i\alpha\omega}\). We show that the energy losses are purely viscous; this result is a consequence of the relation \(\langle\dot w,b\rangle=0\); since a priori, \(b\) is a measure and \(\dot w\) is defined only almost everywhere, this relation is not trivial.

MSC:

35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
74D05 Linear constitutive equations for materials with memory
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