Abstract
In this paper, we develop unifying scaling laws describing the response of elastic polycrystals at finite mesoscales. These polycrystals are made up of individual grains belonging to any crystal class (from cubic to triclinic) and are generated by Voronoi tessellations with varying grain sizes. Rigorous scale-dependent bounds are then obtained by setting up and solving Dirichlet and Neumann boundary value problems consistent with the Hill–Mandel homogenization condition. The results generated are benchmarked with existing numerical results in special cases and the effect of grain shape on the scaling behavior is investigated. The convergence to the effective elastic properties with increasing number of grains is established by analyzing 5180 boundary value problems. This leads to the notion of an elastic scaling function which takes a power law form in terms of the universal anisotropy index and the mesoscale. Based on the scaling function, a material scaling diagram is constructed using which the convergence to the effective properties can be analyzed for any elastic microstructure.
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