Determining the Relaxation Kernel in Nonlinear One-Dimensional Viscoelasticity

Abstract

We consider a viscoelastic string whose mechanical behavior is governed by a non-linear stress-strain relationship. This constitutive law is characterized by a time-dependent relaxation kernel $k$ which is assumed to be unknown. The resulting motion equation is then associated with initial and Dirichlet boundary conditions. We show that the traction measurement at one end allows to identify $k$. More precisely, we prove an existence and uniqueness result on a small time interval. Also, we show how the solution continuously depends on the data.