The presentation deals with the multiscale modeling of materials with complex microstructures and shows three illustrative examples from different fields.

First, the multiscale finite element method (FEM) is used to model cancellous bone and thus to reduce the extent of necessary laboratory tests typical of other methods. The main focus here is placed on the generation of a suitable representative volume element (RVE) consisting of the solid bone phase and the fluid marrow. Macroscopic simulations deal with the ultrasonic tests and enable an analysis of the attenuation dependency on the bone density and excitation frequency. 

Second example is related to the simulation of the strain induced crystallization. This model introduces a triple decomposition of the deformation gradient and two specific types of internal variables: regularity of the polymer chain network and thermal flexibility. Starting with the minimum principle of the dissipation potential, it derives evolution equations for internal variables that are able to simulate the formation and the degradation of crystalline regions and to monitor the temperature change during cyclic tensile tests. 

The last example uses the multiscale finite element method to simulate the effective material behavior of calcified hydrogels. Within this framework, RVEs are generated to depict the biphasic material microstructure consisting of the organic hydrogel and anorganic calcium phosphate. The choice of the hydrogel matrix influences the type of agglomeration of the anorganic phase and thus dictate the geometry of the RVE. Most commonly, the anorganic phase appears in the form of spherical inclusions or honeycomb grids where the characteristic size of a typical unit might vary. The example also uses asymptotic homogenization technique for the study of the effective diffusivity of hydrogels.