One-dimensional viscoelastodynamics with Signorini boundary conditions

Let alpha be a positive number. The one-dimensional viscoelastic problem u(tt) - u(xx) - alphau(xxt) = f, x is an element of (-infinity, 0), t is an element of [0, +infinity), with unilateral boundary conditions u(0, (.)) greater than or equal to 0, (u(x) + alphau(xt)) (0, (.)) greater than or equal to 0, (u(u(x) + alphau(xt))) (0, (.)) = 0, can be reduced to the following variational inequality: lambda(l) * w = g + b, w greater than or equal to 0, b greater than or equal to 0, <w, b> = 0. Here (λ) over cap (l)(omega) is the causal determination of iomegaroot1 + ialphaomega. We show that the energy losses are purely viscous; this result is a consequence of the relation <(w) over dot,b> = 0 since a priori. b is a measure and (w) over dot is defined only almost everywhere, this relation is not trivial. (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.