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Author: Hanke, Hauke ×

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    Derivation of an effective damage model with evolving micro-structure
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hanke, Hauke; Knees, Dorothee
    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 shape but of different time-dependent sizes. Scaling the micro-structure of this microscopic model by a parameter " > 0 the limit passage " ! 0 is preformed via two-scale convergence techniques. Therefore, a regularization approach for piecewise constant functions is introduced to guarantee enough regularity for identifying the limit model. In the limit model the material tensor depends on a damage variable z : [0, T ] ! W1,p( ) taking values between 0 and 1 such that, in contrast to the microscopic model, some kind of intermediate damage for a material point x 2 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
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    Homogenization of elliptic systems with non-periodic, state dependent coefficients
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hanke, Hauke; Knees, Dorothee
    In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.
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    Homogenization in gradient plasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Hanke, Hauke
    This paper yields a two-scale homogenization result for a rate-independent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by e > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor e? (? = 0) . We derive different limiting models for e ? 0 in dependence of &gamma ;. For ? > 1 the limiting model is the two-scale model derived in [MielkeTimofte07], where no gradient term was present. For ? = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for ? ? [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part