Stochastic modelling of symmetric positive-definite material tensors

Spatial symmetries and invariances play an important role in the description of materials. When modelling material properties, it is important to be able to respect such invariances. Here we discuss how to model and generate random ensembles of tensors where one wants to be able to prescribe certain classes of spatial symmetries and invariances for the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly 'higher' spatial invariance class. Our special interest is in the class of physically symmetric and positive definite tensors, as they appear often in the description of materials. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear vector space of physically symmetric tensors, it may be advantageous to widen the notion of mean to the so-called Fr\'echet mean, which is based on distance measures between positive definite tensors other than the usual Euclidean one. For the sake of simplicity, as well as to expose the main idea as clearly as possible, we limit ourselves here to second order tensors. It is shown how the random ensemble can be modelled and generated, with fine control of the spatial symmetry or invariance of the whole ensemble, as well as its Fr\'echet mean, independently in its scaling and directional aspects. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is explored. It is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest, such as temperature distribution and heat flux.