Field statistics in linearized elastic and viscous composites and polycrystals

Abstract

This paper presents a general method to estimate the first and second moments of the stress and deformation fields in linearized elastic and viscous composites and polycrystals. The methodology can be seamlessly extended by means of the ‘linear comparison’ variational homogenization methods (Ponte Castañeda, 1991, 2016) to extract the corresponding field statistics in composites and polycrystals with nonlinear properties, including hyperelastic composites, as well as viscoplastic composites, undergoing finite deformations. Expressions are obtained for the overall response and statistics of the deformation gradient field in hyperelastic composites with linearized (incremental) response characterized by elasticity tensors exhibiting major symmetry, but not minor symmetry. These results are then specialized for composites with fully symmetric elasticity tensors, and use is made of a well-known analogy to convert these results into corresponding estimates for the overall viscosity tensor and statistics of the velocity gradient for linearly viscous composites. Subsequently, the velocity-gradient statistics are projected onto the appropriate symmetric and anti-symmetric subspaces to extract the strain-rate and spin tensor statistics, respectively. The formulation recovers known relations for the overall response and fields statistics, but also allows for the computation of hitherto unexplored quantities, such as the second moments of the spin fluctuations in viscous composites. To illustrate the results, applications are considered for various examples, including two-phase viscous composites, as well as viscous polycrystals, with special types of isotropic and anisotropic linearized response.