Heterogeneous Anisotropy Index and scaling in two-phase random polycrystals

Abstract

Under consideration is the finite-size scaling of the elastic properties in two-phase random polycrystals with individual grains belonging to any arbitrary crystal class. These polycrystals are generated by Voronoi Tessellations with varying grain sizes and volume fractions. Any given realization of such a microstructure sampled randomly is highly anisotropic and heterogeneous. Using extremum principles in elasticity, we introduce the notion of a ‘Heterogeneous Anisotropy Index $$\left( A^U_H\right) $$ ’ and examine its role in the scaling of elastic properties at finite mesoscales ( $$\delta $$ ). The relationship between $$A^U_H$$ and the Universal Anisotropy Index $$A^U$$ by Ranganathan and Ostoja-Starzewski (Phys Rev Lett 101(5):055504, 2008) is established for special cases. The index $$A^U_H$$ turns out to be a function of 43 variables—21 independent components for each phase and the volume fraction of either phase. The scale-dependent bounds are then obtained by setting up and solving 9250 Dirichlet and Neumann type boundary value problems consistent with the Hill–Mandel homogenization condition. Subsequently, the concept of an elastic scaling function is introduced that takes a power-law form in terms of $$A^U_H$$ and ( $$\delta $$ ). Finally, a material scaling diagram is constructed by employing the elastic scaling function which captures the convergence to the effective properties for any two-phase elastic microstructure.