This chapter addresses an identification problem for a thin linear elastic plate under the Love–Kirchhoff plate hypothesis. The purpose of this chapter is to place this identification problem in the classical framework of elastic impedance tomography and present a possible solution using the minimization of the error on constitutive law. The introductory part of the study presents the mathematical inverse problem, and a short discussion of the uniqueness of the identification problem in the anisotropic case under the static test. The second part of the study presents a numerical reconstruction procedure based on the minimization of the error on the constitutive law. The inhomogeneous bending rigidities and the underlying elastic moduli of the material are identified from over specified displacement–force measurement data. The data represent a partial knowledge of the Dirichlet-to-Neumann data map. From a numerical point of view, the problem is presented as a constrained minimization problem for the error in constitutive law over kinematically and statically admissible fields. A numerical reconstruction procedure is presented and illustrated on some examples for isotropic plates.