Effective Dynamic Material Properties for Materials with Non-Convex Microstructures

Abstract

The determination of macroscopic, effective properties of microstructured materials is referred to as homogenization. Usually in homogenization, it is assumed that on the microscale inertia effects can be neglected. Here, contrary to these approaches, inertia effects are taken into account, leading to a frequency dependent microscopic behavior. Additionally to this effect, non-convex microstructures are considered.

It is assumed that the microstructure can be modeled as a beam framework in frequency domain which is exactly solved by a boundary integral formulation. Further, it is assumed that the structures to be treated are made up of identical unit cells. However, due to the inertia effects a mean value of the microscopic response calculated for several unit cells is used.

Under these micromechanic assumptions on the macroscopic scale a frequency dependent, i.e. viscoelastic and auxetic behavior is expected. Hence, an analytical homogenization is presumably not possible. Therefore, the homogenization is performed numerically formulated as an optimization process. The classical technique SQP and soft computing methods, in particular a Genetic Algorithm, are used.

The frequency dependent macroscopic material parameters are found for a frequency range from 0 up to 10³ kHz for a seven parameter model using as well fractional derivatives. The system responses on micro- and macroscale show a good agreement for the considered frequency range. Both optimization strategies are able to find adequate material parameters on the macroscale but the SQP needs, as expected, reliable starting values. On the contrary the Genetic Algorithm is more robust but much slower.