A Quasistatic Evolution Model for the Interaction Between Fracture and Damage

Abstract

To cite this version:Jean-François Babadjian. A quasistatic evolution model for the interaction between fracture and damage. Archive for Rational Mechanics and Analysis, Springer Verlag, 2011, 200 (3) JEAN-FRANÇ OIS BABADJIANAbstract. This paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture. In both cases, the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process, the minimality at each time, and the energy balance. From a mechanical point of view, it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium. The brutal character of the damage process is then replaced by a progressive one, where the original damage internal variable, i.e. the characteristic function of the damaged part, is replaced by the local volume fraction. The analysis rests on a locality property for mixtures which enables to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the G-closure set.

Young-Measure Quasi-Static Damage Evolution

“…From the one hand, this prevents us from considering linear mixtures energies of the form of (1.1) in our frame. In particular, the relaxed models from [14,16,2] seem not directly recoverable in the present setting. From the other hand, this gives us the advantage of tracing the damage variable z into the evolution.…”

“…We shall refer to Francfort & Garroni [14], Garroni & Larsen [16], and Babadjian [2] for recent contributions on damage models for brittle materials, namely assuming z ∈ {0, 1}. Besides brittleness, we have to remark that the mechanical stand of the latter papers is quite different from ours.…”

“…We consider a stored energy defined by 2). We observe that the function g is C 2 (0, 2) with g ≤ 0 and g ≤ 0 in (0, 2) and g is constant on (−∞, 0]; it is now easy to see that the Hessian matrix of W is positive definite and hence W is a convex function on [0, 2) × R (see [35,Lemma 5.1…”

“…Damage evolution is governed by the interplay of energy minimization and dissipation. In particular, damage is often very well assimilable to a quasi-static evolution process and, in this regard, the first possible choice for a dissipation mechanism from the damage state z old to updated state z new may be assumed to be This very frame for a variational theory of rate-independent damage has attracted a good deal of attention among in recent years and rigorous mathematical results are to be found, for instance, in [2,3,14,16,28,30]. The analysis of this paper moves exactly within the setting of the result by Thomas & Mielke [35] where the Authors develop an existence theory for incomplete damage by directly including a gradient term of the internal variable z into the energy.…”

“…with φ(e) = e 2 /2 (in the 1-dimensional case) in [14,16] and a more general convex function φ in [2]. As no gradient terms in the damage variable are considered, evolution via time-discretization immediately calls for quasi-convexification and the passage to the limit is performed by determining the limiting materials via its elasticity tensor by homogenization tools.…”