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Rigorous derivation of two-scale and eective damage models based on microstructure evolution

The reasons for the sudden complete failure of a material specimen stressed by external forces are often long-termed, gradual microscopic material changes. The formation of microscopic cracks or voids, for instance, are included in such material changes. In comparison to the expansion of the specimen, the number and the increasingly small size of the microscopic material changes lead to a high level of detail which makes the accurate simulation of such damage processes simply impossible. Therefore, reliance is placed on so-called effective models when performing a simulation. These effective models sufficiently mirror the macroscopic behavior of the specimen without, however, considering each microscopic material change. The dissertation at hand deals with the rigorous derivation of such effective models used to describe damage processes. For different rate-independent damage processes in linear elastic material these effective models are derived as the asymptotic limit of microscopic models. The starting point is represented by a unidirectional microstructure evolution model which is based on a family of ordered admissible microstructures. Each microstructure of that family possesses the same intrinsic length scale ε > 0. To derive an effective model, the limit passage ε → 0 is performed with the help of techniques of the two-scale convergence. For this purpose, a microstructure-regularizing term, which can be understood as a discrete gradient for piecewise constant functions, is needed to identify the limit model. The microstructure of the effective model is given pointwisely by a so-called unit cell problem which separates the microscopic scale from the macroscopic scale. Based on these homogenization results for unidirectional microstructure evolution models, an effective model for a brutal damage process is provided. Here, the microstructure consists of only two phases, namely undamaged material which comprises inclusions of damaged material with various sizes and shapes. The size of the inclusions is scaled by ε > 0 and the unidirectional microstructure evolution prevents that, for fixed ε > 0, the inclusions shrink for progressing time. According to the unit cell problem, the material of the limit model is then given as a mixture of damaged and undamaged material. In a specific material point of the limit model, that unit cell problem does not only define the mixture ratio but also the exact geometrical mixture distribution. Then, as a generalization, brutal damage processes are investigated, which allow the modeling of voids instead of inclusions of damaged material. This means that these considered microscopic defects do not contain any material. Despite the lack of stiffness on the set of all voids, the applicability of the provided homogenization theory is guaranteed by appropriate continuation operators. By virtue of the unit cell problem, the effective model describes a material free from voids. Compared to the ε-models, the missing defects in the limit model are taken into account due to the lower stiffness of the effective material. Last but not least, for an evolution process describing the growth of microscopic cracks, a macroscopic crack-free model can be derived. In correspondence to the brutal damage model for voids, the effective model describes a crack-free material. In this case, in every material point of the limit model, the unit cell problem models a cracked unit cell. This crack geometry is uniquely determined and varies from point to point.