tailieunhanh - Stability of non localized responses for damaging materials

This work is devoted to the analysis of the stability of the homogeneous states of a bar made of a brittle strain softening material submitted to a tensile loading. We distinguish two types of damage models: local damage models and gradient damage models. We show that a local damage model necessarily leads to the unstability of the homogeneous response once the first damage threshold is reached. On the contrary, in the case of a gradient damage model, viewed as a regularization of the underlying local model, the homogeneous damage states of “sufficiently small" bars are stable. | Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 307 – 318 Special Issue of the 30th Anniversary STABILITY OF NON LOCALIZED RESPONSES FOR DAMAGING MATERIALS K. Pham and . Marigo Université Paris 6, Institut Jean le Rond d’Alembert, 4 Place Jussieu 75005 Paris Abstract. This work is devoted to the analysis of the stability of the homogeneous states of a bar made of a brittle strain softening material submitted to a tensile loading. We distinguish two types of damage models: local damage models and gradient damage models. We show that a local damage model necessarily leads to the unstability of the homogeneous response once the first damage threshold is reached. On the contrary, in the case of a gradient damage model, viewed as a regularization of the underlying local model, the homogeneous damage states of “sufficiently small" bars are stable. 1. INTRODUCTION Prior to their complete rupture, many engineering materials such as concrete, rocks, wood or various composites show a strain-softening behavior when they are deformed beyond a certain limit. The theory of damage is generally used to model this behavior at a continuum level. Limiting our analysis to rate independent behaviors, we can distinguish two types of damage models: (i) the so-called local models where the only variables characterizing the state of the material point are the strain and the damage variable; (ii) the so-called non local models where additional information on the neighborhood of the material point are involved. From the theoretical viewpoint, the boundary-value problem associated with local models is mathematically ill posed (Benallal et al. 1989 [2], Lasry and Belytschko, 1988 [7]) and lead to multiple (and even an infinite number of) solutions. From the numerical viewpoint, the computations give rise to spurious mesh dependences: upon refinement of the meshsize, no convergence is observed or more precisely the deformation is localized into narrow bands whose thickness