Etude de l'endommagement pendant la mise en forme à froid de tôles d'aluminium

Annexe A Homogenisation procedures for non linear heterogeneous materials Cette annexe est extraite d’un rapport du projet européen VIRFORM D3. Homogenisation procedures such as the self-consistent scheme or distribution-based models [Pon95][Bor96] for microheterogeneous materials have been originally formulated in the field of linear elasticity theory, for constant moduli given to the constitutive phases. The methods formally hold for any linear phase behaviour law, and have also been considered (since [Hil65]) as incrementally applicable to "linearizable behaviour laws". Many works concerning elastic-plastic materials have thus been modelized using a linear rate (or incremental) formulation for the phase behaviour laws, say: φ φ φ Q ' = t q ' ij ijkl kl φ φ φ(for the appropriate local conjugate stress and strain rate tensors Q ' and q ', or dQ φ φand dq increments), using the current (instantaneous) tangent (t ) phase moduli to approximate the overall effective related ones. These approaches are uncorrect when considering that the current moduli distribution in each phase, which is not uniform since strain-dependant (and the strain field is in general not uniform) can be represented by an average current value. In the mean time, theoretical considerations, not to be discussed here, have yield to consider as a "better" linearized formulation, the one using current ...