Abstract
For each of the 8 symmetry classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set mathsf {V} of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders 1 and 2 of such a field, and the field’s spectral expansion.
Highlights
Microstructural randomness is present in just about all solid materials
In the case of upscaling of elastic properties, on any finite scale there is an anisotropy, and this anisotropy, with mesoscale increasing, tends to zero hand-in-hand with the fluctuations. It is in the infinite mesoscale limit (i.e., representative volume element (RVE)) that material isotropy is obtained as a consequence of the statistical isotropy
In this paper we develop second-order tensor random field (TRF) models of linear hyperelastic media in each of the eight elasticity classes
Summary
Microstructural randomness is present in just about all solid materials. When dominant (macroscopic) length scales are large relative to microscales, one can safely work with deterministic homogeneous continuum models. In the case of upscaling of elastic properties (which are tensor in character), on any finite scale there is (almost surely) an anisotropy, and this anisotropy, with mesoscale increasing, tends to zero hand-in-hand with the fluctuations It is in the infinite mesoscale limit (i.e., RVE) that material isotropy is obtained as a consequence of the statistical isotropy. Where C is the 2nd-rank tensor random field This type of upscaling is sorely needed in the stochastic finite element (SFE) method, where, instead of assuming the local isotropy of the elasticity tensor for each and every material volume (and, the finite element), full triclinic-type anisotropy is needed [28]. We find the general form of field’s spectral expansion for each of the eight isotropy classes
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