Abstract We consider an initial and boundary value problem which describes the evolution of a viscoelastic body submitted to body forces and surface tractions. The viscoelastic constitutive law is assumed to be nonlinear and the process is quasistatic. We prove the existence and the uniqueness of the solution using arguments of monotone operators theory and a version of Cauchy-Lypchitz theorem. We establish the continuous dependence of the solution on the elasticity operator. Finally, we study the behavior of the solution when the viscosity operator converges to zero.