Petrov, Adrien; Schatzman, Michelle One-dimensional viscoelastodynamics with Signorini boundary conditions. (Viscoélastodynamique monodimensionnelle avec conditions de Signorini.) (French. Abridged English version) Zbl 1043.35119 C. R., Math., Acad. Sci. Paris 334, No. 11, 983-988 (2002). 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. Cited in 10 Documents MSC: 35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators 74D05 Linear constitutive equations for materials with memory Keywords:one-dimensional viscoelastic problem; unilateral boundary conditions; variational inequality; energy losses PDFBibTeX XMLCite \textit{A. Petrov} and \textit{M. Schatzman}, C. R., Math., Acad. Sci. Paris 334, No. 11, 983--988 (2002; Zbl 1043.35119) Full Text: DOI References: [1] Amerio, L., Su un problem di vincoli unilaterali per l’equazione non omogenea delle corda vibrante, Publ. I. A. C. Serie III, 109 (1976) · Zbl 0432.73062 [2] Amerio, L., On the motion of a string vibrating through a moving ring with a continuously variable diameter, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 62, 2, 134-142 (1977) · Zbl 0378.73057 [3] Crandall, M. G.; Pazy, A., Semi-groups of nonlinear contractions and dissipative sets, J. Funct. Anal., 3, 376-418 (1969) · Zbl 0182.18903 [4] Jarušek, J., Remark to dynamic contact problems for bodies with a singular memory, Comment. Math. Univ. Carolin., 39, 3, 545-550 (1998) · Zbl 0963.35137 [5] Kim, J. U., A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14, 8-9, 1011-1026 (1989) · Zbl 0704.35101 [6] Lebeau, G.; Schatzman, M., A wave problem in a half-space with a unilateral constraint at the boundary, J. Differential Equations, 53, 3, 309-361 (1984) · Zbl 0559.35043 [7] Schatzman, M., Le système différentiel \((d^2u\)/d \(t^2)+ ∂ϕ (u)\)∋\(f\) avec conditions initiales, C. R. Acad. Sci. Paris, Série A-B, 284, 11, A603-A606 (1977) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.