Stochastic Modelling of Symmetric Positive Definite Material Tensors

TL;DR

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 is discussed.

Abstract

Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of 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. We formulate a modelling framework which not only respects these two requirements$-$positive definiteness and invariance$-$but also allows a fine control over orientation on one hand, and strength/size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. 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 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.

Figures (18)

• Figure 1: Boundary conditions of 2D femur bone
• Figure 2: Stochastic tensor $\boldsymbol\kappa_{s}(\omega_s)$ with fixed symmetry (iso-iso-scl): \ref{['Fig:isoisoLN']} log-normal pdf of identical random scaling values ${{\lambda}}^{(1)}_s(\omega_s) ={{\lambda}}^{(2)}_s(\omega_s)$, \ref{['Fig:isoisoquad']} geometric visualization of reference tensor $\widehat{\boldsymbol\kappa}^{iso}_{s}$ and realisations of random tensor $\boldsymbol\kappa_{s}(\omega_s)$; the geometries are enhanced by scaling the eigenvalues by a factor of 0.5
• Figure 3: Stochastic tensor $\boldsymbol\kappa_{s}(\omega_s)$ with varying symmetry (iso-ortho-scl): \ref{['Fig:isoorthoLN']} log-normal pdf of i.i.d. random scaling values $\{{{\lambda}}_s^{(i)}(\omega_s)\}^2_{i=1}$, \ref{['Fig:isoorthoquad']} geometric visualization of reference tensor $\widehat{\boldsymbol\kappa}^{iso}_{s}$ and realisations of random tensor $\boldsymbol\kappa_{s}(\omega_s)$; the geometries are enhanced by scaling the eigenvalues by a factor of 0.5
• Figure 4: Stochastic tensor $\boldsymbol\kappa_r(\omega_r)$ with fixed symmetry (ortho-ortho-dir): \ref{['Fig:orthoorthoVM']} von Mises pdf of random rotation angle $\phi(\omega_r)$, \ref{['Fig:orthoorthoVMunit']} realisations of circular random variable $\phi(\omega_r)$ on unit circle with resultant mean vector (solid straight line), \ref{['Fig:orthoorthorot']} geometric visualization of reference tensor $\widehat{\boldsymbol\kappa}^{ortho}_{r}$ and realisations of random tensor $\boldsymbol\kappa_r(\omega_r)$; the geometries are enhanced by scaling the eigenvalues by a factor of 0.5, and the orientations with respect to the mean are scaled by a factor of 1.5
• Figure 5: mc mean estimate of nodal temperature