On the Identification of Nonlinear Constitutive Laws from Indentation Tests

This paper addresses the identification of the parameters of a nonlinear constitutive law from indentation tests. The case of a standard generalized material without work hardening is extensively treated using the adjoint state method. This provides a general framework to perform optimization involving contact conditions and nonlinear material behaviour. A numerical identification is presented and proves the accuracy and the robustness of the method. INTRODUCTION The indentation test consists in pressing a punch on a material sample. It was initialy used to evaluate the hardness of metals and is now being considered as an efficient non destructive method for determining material mechanical characteristics (Taljat et al.,1998). The constitutive law should be identified from the knowledge of the indentation curve, representing the load applied on the punch versus the penetration depth. The mechanical interpretation of the indentation curve is not as straightforward as for the classical traction curve. This implies that the use of the indentation test for material characterization depends on the reliability of the identification. Most of the identification strategies are based on semiempirical formulas dedicated to a given constitutive behaviour : elasticity, perfect plasticity (Johnson,1985), power laws (Jayaraman et al., 1998), : : : Only a few studies present this problem from a general point of view, i.e. defining the identification as the minimization of a cost functional (Bui, 1994). The identification methods are generally based on simple trial & error techniques (Hasanov & Seyidmamedov , 1995). This is partly due to the mathematical complexity of the contact description, appearing independently of the constitutive behaviour of the material. A first attempt to solve the problem from a general point of view has been presented in the case of linear elasticity (Constantinescu & Tardieu, 1995). The contact conditions have been regularized by penalization, the problem was therefore described by variational equalities, and not variational inequalities as before. This enables the application of classical optimal control (Lions,1968) techniques, in particular the adjoint state method. The identification problem has been solved afterwards by the minimization of cost functional using a gradient descent method. The goal of this paper is to extend this method to the identification of the parameters of a standard generalized constitutive law without work hardening. The method presented in this paper is not based on the regularization of the contact conditions as in (Constantinescu & Tardieu, 1995), instead Lagrange multipliers are used. The gradient of the cost functional is computed from the solution of a direct and an adjoint problem. The accuracy and robustness of the method are illustrated through a numerical example for a Maxwell viscoelastic constitutive law. 1 Copyright  1999 by ASME THE DIRECT PROBLEM (P ) Let us consider an axisymmetric body, with its section occupying in its reference configuration an open subset Ω R2 with smooth boundary Γ (see Figure 1). The boundary is partioned in three parts Γ = ΓD[ΓF [ΓC : the part ΓD where displacements are imposed, the free surface ΓF , and the surface ΓC where contact might occur. n and t denote the normal and tangent vector to the boundary Γ. The axisymmetric hypothesis is taken in order to simplify the presentation and the computational burden and does not restrict the generality of the method. The problem will be treated within the theory of small strains and rotations. The validity of this hypothesis will be discussed later. Let us denote respectively by u, ε and σ the vector field of displacements and the tensor fields of small strains and stresses. The problem considered in the sequel is the indentation of the body Ω by a rigid punch whose profile is characterized by the gap g. The contact is considered without friction. An indentation experiment is driven either by the vertical displacement U or by the force F applied to the punch. The force F can be expressed as integral of the contact pressure: