Computing elastic moduli on 3-D X-ray computed tomography image stacks

A numerical task of current interest is to compute the effective elastic properties of a random composite material by operating on a 3D digital image of its microstructure obtained via X-ray computed tomography (CT). The 3-D image is usually sub-sampled since an X-ray CT image is typically of order 10003 voxels or larger, which is considered to be a very large finite element problem. Two main questions for the validity of any such study are then: can the sub-sample size be made sufficiently large to capture enough of the important details of the random microstructure so that the computed moduli can be thought of as accurate, and what boundary conditions should be chosen for these sub-samples? This paper contributes to the answer of both questions by studying a simulated X-ray CT cylindrical microstructure with three phases, cut from a random model system with known elastic properties. A new hybrid numerical method is introduced, which makes use of finite element solutions coupled with exact solutions for elastic moduli of square arrays of parallel cylindrical fibers. The new method allows, in principle, all of the microstructural data to be used when the X-ray CT image is in the form of a cylinder, which is often the case. Appendix A describes a similar algorithm for spherical sub-samples, which may be of use when examining the mechanical properties of particles. Cubic sub-samples are also taken from this simulated X-ray CT structure to investigate the effect of two different kinds of boundary conditions: forced periodic and fixed displacements. It is found that using forced periodic displacements on the non-geometrically periodic cubic sub-samples always gave more accurate results than using fixed displacements, although with about the same precision. The larger the cubic sub-sample, the more accurate and precise was the elastic computation, and using the complete cylindrical sample with the new method gave still more accurate and precise results. Fortran 90 programs for the analytical solutions are made available on-line, along with the parallel finite element codes used.