L. O. Schwen, U. Wolfram, H.-J. Wilke, and M. Rumpf.
Determining effective elasticity parameters of microstructured
In 15th Workshop on the Finite Element Method in Biomedical
Engineering, Biomechanics and Related Fields, pages 41-62. University of
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In this paper we study homogenization of elastic materials with a periodic microstructure. Homogenization is a tool to determine effective macroscopic material laws for microstructured materials that are in a statistical sense periodic on the microscale. For the computations Composite Finite Elements (CFE), tailored to geometrically complicated shapes, are used in combination with appropriate multigrid solvers. We consider representative volume elements constituting geometrically periodic media and a suitable set of microscopic simulations on them to determine an effective elasticity tensor. For this purpose, we impose unit macroscopic deformations on the cell geometry and compute the microscopic displacements and an average stress. Using sufficiently many unit deformations, the effective (usually anisotropic) elasticity tensor of the cell can be determined. In this paper we present the algorithmic building blocks for implementing these "cell problems" using CFE, periodic boundary conditions and a multigrid solver. We apply this in case of a scalar model problem and linearized elasticity. Furthermore, we present a method to determine whether the underlying material property is orthotropic, and if so, with respect to which axes.