Multiscale tools in Z-set

This section presents some tools about the non-linear multi-scale modeling of multi-phase materials and composites, developed in Z-set. The aim of these approaches is to describe the behavior of a structure from the local constitutive equations of the constituents.

Several Mean-Field homogenization approaches are available, to allow accurate prediction of the macroscopic stress-strain response of composite materials, which is related to the description of their complex microstructural behavior exemplified by the interaction between the constituents, such as Voigt, Reuss, Self consistent, etc… In the development of the homogenization procedures for non linear materials, we have to define both the homogenization step itself, which consists in averaging the local fields to obtain the overall ones and the often more complicated localization step, where a local problem for each sub-volume is solved in order to obtain approximations for the local field behavior. In this context, the microstructure of the material under consideration is basically taken into account by representative volume elements (RVE).

From a general point of view, the description of the macroscopic behavior of solid heterogeneous media with multi-materials is a very difficult task. In the multiscale tools in Z-set, one distinct set of constitutive equations is attributed to each sub-volume, which are treated independently. Each sub-volume then possesses its own stress/strain tensor. The macroscopic behaviour is obtained by averaging the corresponding non-uniform local behaviour law using the well-known homogenization schemes. Consequently, it is possible to mix different types of constitutive equations for each sub-volume. In principle, all mechanical Z-mat models can be used in the multi-mat model, including other multi-mat models themselves.

Multiscale tools in Z-set, are called “Multimat” for static interface and “Phase-field” for dynamic interface, with additional evolution equation related to the interface mobility.