**Development of Small Scale Plasticity Theories**

One of our major research topics is the modeling of high strength metals which are applied in modern means of transport, like cars, airplanes and trains. Besides being lightweight, they must have superior crash properties and thus need to be sufficiently ductile.

**Grain boundaries in a polycrystal **

We develop extended micromechanical plasticity models and carry out respective three-dimensional material simulations including a representative description of the material microstructure and – in particular – the so-called “grain boundary strengthening”-mechanism (or “Hall-Petch”-effect).

Interestingly, this strengthening method, which is important for favorable ductile properties, so far is mostly modeled in simplified one- and two-dimensional simulations.

Our models have been validated by comparison with experiments and discrete dislocation simulations. The results have been published in a series of international journal papers.

**Development of Damage and Fracture Models**

In mechanics, “damage of brittle materials” usually describes the degradation of material properties due to the formation of microscopic crack networks. These lead to a reduction of the material stiffness and thus to softening. Recently, damage and cracks have been mainly described by so-called phase-field methods of fracture.

However, many of the existing anisotropic damage models for brittle materials yield unphysical results. They do not correctly reflect the physical damage mechanism of microscopic growing crack networks. The result is an artificial stiffening effect in certain material directions.

**Crack and damage in x- and y-direction**

We formulate continuum mechanical damage models and related criteria, which assure that a model behaves physically correct in the aforementioned sense. Recently, these developments led to the formulation of an anisotropic model of damage and fracture, which does not suffer from artificial stiffening effects.

**Hashin-Shtrikman Type Reduction of Finite Element Microstructural Models**

Recent trends in science and industry indicate that phenomenological material models for structural computations are steadily replaced by multi-scale models. These allow to couple the finite element (FE) model of the structure with a model of the microstructure. Currently, there are two main developments. The first one deals with computational homogenization methods, which are accurate but slow.

**Left: tensile stress in horizontal direction (red: elastic inclusions, blue: elastoplastic matrix), right: comparison of macroscopic FE- and HSFE-stress-prediction.**

The second one is the class of analytical homogenization methods like so-called Hashin-Shtrikman schemes, which are much faster but often less accurate. Lately, we combined both approaches to a Hashin-Shtrikman type finite element method (HFSE) for nonlinear materials. This method allows for a massive reduction of the primary unknowns (or degrees of freedom, DOF) of a finite element model. The example in the figure shows the simulation of a nonlinear microstructure with round inclusions and a reduction from 26000 to only three degrees of freedom.