Date Approved


Embargo Period


Document Type


Degree Name

PhD Engineering (Doctor of Philosophy in Engineering)


Mechanical Engineering


Henry M. Rowan College of Engineering


Ranganathan, Shivakumar I.

Committee Member 1

Kadlowec, Jennifer A.

Committee Member 2

Breitzman, Anthony F.


Heterogeneous Anisotropy Index, Mesoscale, Scaling Function, Universal Anisotropy Index


Polycrystals--Elastic properties; Multiscale modeling


Applied Mathematics | Materials Science and Engineering | Mechanical Engineering


Under consideration is the finite-size scaling of elastic properties in single and two-phase random polycrystals with individual grains belonging to any crystal class (from cubic to triclinic). These polycrystals are generated by Voronoi tessellations with varying grain sizes and volume fractions. By employing variational principles in elasticity, we introduce the notion of a 'Heterogeneous Anisotropy Index' and investigate its role in the scaling of elastic properties at finite mesoscales. The index turns out to be a function of 43 variables, 21 independent components for each phase and the volume fraction of either phase. Furthermore, the relationship between Heterogeneous Anisotropy Index and the Universal Anisotropy Index is established for special cases. Rigorous scale-dependent bounds are then obtained by setting up and solving Dirichlet and Neumann type boundary value problems consistent with the Hill-Mandel homogenization condition. This leads to the concept of a dimensionless elastic scaling function which takes a power-law form in terms of Heterogeneous Anisotropy Index and mesoscale. Based on the scaling function, a material scaling diagram is constructed using which one can estimate the number of grains required for homogenization. It is demonstrated that the scaling function quantifies the departure of a random medium from a homogeneous continuum.