Let α be a positive number. The one-dimensional viscoelastic problem utt−uxx−αuxxt=f, x∈(−∞,0], t∈[0,+∞), with unilateral boundary conditions u(0,·)⩾0, (ux+αuxt)(0,·)⩾0, (u(ux+αuxt))(0,·)=0, can be reduced to the following variational inequality: λ1*w=g+b, w⩾0, b⩾0, 〈w,b〉=0. Here λˆ1(ω) is the causal determination of iω√ of 1+iαω. We show that the energy losses are purely viscous; this result is a consequence of the relation 〈w˙,b〉=0; since a priori, b is a measure and w˙ is defined only almost everywhere, this relation is not trivial. To cite this article: A. Petrov, M. Schatzman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 983–988. [Copyright &y& Elsevier]