tailieunhanh - On a nonlinear inverse problem in viscoelasticity

We consider an inverse problem for determining an inhomogeneity in a viscoelastic body of the Zener type, using Cauchy boundary data, under cyclic loads at low frequency. We show that the inverse problem reduces to the one for the Helmholtz equation and to the same nonlinear Calderon equation given for the harmonic case. A method of solution is proposed which consists in two steps : solution of a source inverse problem, then solution of a linear Volterra integral equation. | Vietnam Journal of Mechanics, VAST, Vol. 31, No. 3 &4 (2009), pp. 211 – 219 ON A NONLINEAR INVERSE PROBLEM IN VISCOELASTICITY . Bui1 and S. Chaillat2 1 Ecole Polytechnique/LMS, Palaiseau and Electricite de France/RD/LaMSID, Clamart, France 2 Georgia Institute of Technology/College of Computing, USA Abstract. We consider an inverse problem for determining an inhomogeneity in a viscoelastic body of the Zener type, using Cauchy boundary data, under cyclic loads at low frequency. We show that the inverse problem reduces to the one for the Helmholtz equation and to the same nonlinear Calderon equation given for the harmonic case. A method of solution is proposed which consists in two steps : solution of a source inverse problem, then solution of a linear Volterra integral equation. 1. INTRODUCTION Inverse problems for defect and crack identification in elasticity and viscoelasticity have many applications in medecine and the mechanics of materials. In medecine, tomography techniques using mechanical loads such as antiplane shear loading on life tissue, considered as a viscoelastic medium, have been worked out for Kelvin-Voigt’s viscoelasticity (Catheline et al [12], Muller et al [18]) and for pure elasticity. In the elastic case, solutions to crack inverse problems in 2D and 3D are already known, see Andrieux and Ben Abda [4], Andrieux et al [5], Bui et al [8]. It is thus important to know if viscoelastic inverse problems can be studied by using classical correspondence between viscoelasticity and elasticity. Such a correspondence allows for applications for mechanical tomography in viscoelastic media. We shall consider a more general viscoelastic constitutive equation of the Zener type and consider either dynamic antiplane shear loading or scalar acoustic problems in viscous materials. It has been established in Chaillat and Bui [13] that, for low frequency of the load, the correspondence between viscoelasticity and elasticity still exists. Therefore, the inverse