The problem addressed in this paper is the identification of the constitutive law of a body from the knowledge of a displacement - force curve measured during an indentation test. The inverse problem is formulated as the minimization of the distance between computed and measured forces on the displacement driven indentor, with the constitutive parameters as minimization variables. The implicit dependence of the cost functional on the minimization variables leads to a resolution using a lagrangian formulation and the adjoint state method. In spite of the non differentiable contact problem, we prove in the elastic case the existence of the adjoint problem that gives together with the direct problem an efficient evaluation of the gradient of the cost functional. The effectiveness of the method is illustrated on numerical examples for the identification of an elastic bimaterial.

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