Abstract

In many processes of engineering mechanics the various scales of observation interact with one another. For instance, in crack propagation problems the macroscopic geometry and boundary conditions influences the microscopic dissipation processes at the crack tip and vice versa. Another example concerns the dispersive propagation of waves through microstructured media, where every wave number propagates with a distinct velocity related to the characteristic size of the microstructure. In order to capture the material and structural behaviour properly, all relevant scales must be taken into account, for instance by means of multi-scale methods.

On of the methods to perform a multi-scale analysis is to assign a microscopic unit cell to every macroscopic material point. The macroscopic material behaviour is then evaluated not by means of a macroscopic constitutive relation but instead by solving a microscopic boundary value problem on the unit cell. This unit cell is often denoted a “Representative Volume Element” (RVE). In the talk, the following three issues will be addressed:

1. RVE determination The size of the RVE must be determined before it can be used in multi-scale analysis. A simple statistical procedure has been developed to quantify the RVE size, while it also allows to verify the existence of the RVE . The existence and size determination of RVEs will be addressed for elastic, hardening and softening materials.

2. Homogenisation Once the issue of RVE size determination is dealt with, its effects on the macroscopic response can be considered. To this end, so-called micro-macro transitions must be made, i.e. the microscopic RVE response must be homogenised in order to obtain the corresponding macroscopic continuum response. Two homogenisation schemes will be treated, leading to a classical continuum and a continuum enriched with higher-order strain gradients, respectively. For the latter, the relation between RVE size and the macroscopic intrinsic length scale will be treated in detail.

3. Multi-scale analysis of quasi-brittle materials Finally, a multi-scale analysis framework will be presented. Whereas elastic analysis is relatively straightforward, problems arise in case the material exhibits softening. As it turns out, the two scales cannot be considered entirely separated anymore. To overcome this, a novel multi-scale scheme will be suggested in which the competing influences of microscopic size effect and macroscopic softening are balanced by a volume coupling of macro-scale and micro-scale.