Local elastic moduli of simple random composites computed at different length scales

Abstract

Techniques like nanoindentation and atomic force microscopy can estimate the local elastic moduli in a region surrounding the probe used. For composites with phase regions much larger than the size of the probe, these procedures can identify the phases via their different elastic moduli but identifying phase regions that are on the same size scale as the indent is more problematic. This paper looks at three random 3D 8003 voxel composite models, each consisting of a matrix and spherical inclusions. One model has non-overlapping spheres and two models have overlapping spheres, with two and three distinct phases. The linear elastic problem is solved for each microstructure, and histograms are made of the local Young’s moduli over a number of sub-volumes (SVs), averaged over progressively larger SVs. The number and shape of histogram peaks change from N delta functions, where N is the number of elastically distinct phases, at the 1 voxel SV limit, to a single delta function located at the value of the effective global Young’s modulus, when the SV equals the unit cell volume. The phase volume fractions are also tracked for each bin in the Young’s modulus histograms, showing the phase make-up of bin in the histogram. There are clear differences seen between the non-overlapping and three-phase overlapping models and the two-phase overlapping sphere model, because of different size microstructural features, characterized by the average value of size as computed by the W(q) function. In the three-phase model, a peak that is originally all phase 3 persists at its same location, but as the size of the SVs increase, it is made up of a mixture of phases, so that it cannot be identified with a single phase even though it remains a clear peak. These results give some guidance as to what probe size might be useful in distinguishing different phases by local elastic moduli measurements, and how the length scales of the probe and the microstructure interact.