Towards scaling laws in random polycrystals

We propose a methodology to set up unifying scaling laws describing the response of multifarious random polycrystals. The methodology employed falls in the realm of stochastic micro-mechanics and is consistent with the Hill condition. Within this framework, we introduce the concept of a scaling function that describes “finite size scaling” of both elastic and inelastic crystalline aggregates. While the finite size is represented by the mesoscale, the scaling function depends on an appropriate measure quantifying the single crystal anisotropy. Based on the scaling function, we construct a material scaling diagram, from which one can assess the approach to a representative volume element (RVE) for many different polycrystals. We demonstrate these concepts on the scaling of the fourth-rank elasticity and the second-rank thermal conductivity tensors.