Derivation of an effective damage model with evolving micro-structure
- Hanke, Hauke
- Knees, Dorothee
2010 Mathematics Subject Classification
- 74A45 74C05 74R05 74Q15 76M50
- Two-scale convergence, folding and unfolding operator, rate-independent damage evolution, Γ-convergence, irreversibility, broken Sobolev function
In this paper rate-independent damage models for elastic materials are considered. The aim is the derivation of an effective damage model by investigating the limit process of damage models with evolving micro-defects. In all presented models the damage is modeled via a unidirectional change of the material tensor. With progressing time this tensor is only allowed to decrease in the sense of quadratic forms. The magnitude of the damage is given by comparing the actual material tensor with two reference configurations, denoting completely undamaged material and maximally damaged material (no complete damage). The starting point is a microscopic model, where the underlying micro-defects, describing the distribution of either undamaged material or maximally damaged material (but nothing in between), are of a given time-dependent shape but of different sizes. Scaling the microstructure of this microscopic model by a parameter ε>0 the limit passage ε→0 is preformed via two-scale convergence techniques. Therefore, a regularization approach for piecewise constant functions is introduced to guaranty enough regularity for identifying the limit model. In the limit model the material tensor depends on a damage variable z:[0,T]→ W1,p(Ω) taking values between 0 and 1 such that, in contrast to the microscopic model, some kind of intermediate damage for a material point x∈Ω is possible. Moreover, this damage variable is connected to the material tensor via an explicit formula, namely, a unit cell formula known from classical homogenization results.