The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

Abstract : We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we show also that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analysis is illustrated by examples.
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Submitted on : Tuesday, March 22, 2011 - 7:13:57 PM
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Kim Pham, Jean-Jacques Marigo, Corrado Maurini. The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. Journal of the Mechanics and Physics of Solids, Elsevier, 2011, 59 (6), pp.1163-1190. ⟨hal-00578995⟩

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