Stochastic Modelling of Elasticity Tensor Fields
Sharana Kumar Shivanand, Bojana Rosić, Hermann G. Matthies
TL;DR
This work develops a probabilistic framework for random, fourth-order, physically symmetric and positive definite tensors, with elasticity tensors as a prime example, by embedding them in a Lie-algebra–based representation that respects spatial rotation invariances. The core idea is to decompose a Kelvin matrix into three components—spatial orientation $Q$, strain-distributor $V$, and diagonal Kelvin moduli $oldsymbol{oldsymbol{\Lambda}}$—via a product Lie group $oldsymbol{S}_p$ and its Lie algebra, enabling independent modelling of stiffness, eigenstrain, and orientation. To define meaningful averages and interpolation on the SPD tensor manifold, the authors introduce an elasticity metric $ ilde{oldsymbol{\vartheta}}_E$ on the product group, yielding an elasticity Fréchet mean that avoids the swelling effects of Euclidean means and preserves invariances. Random tensor fields are constructed by sampling from the Lie-algebra parameters, applying the exponential map to obtain $C = Q^T V oldsymbol{oldsymbol{\Lambda}} V^T Q$, and ensuring positivity and symmetry throughout, with Karhunen–Loève–type representations governing spatial variation. The framework spans all elasticity classes in 2D and 3D, detailing the group-reduced and spectral structures for each class and illustrating a one-dimensional orthotropic example; the approach provides a rigorous, geometrically grounded basis for uncertainty quantification in stochastic elasticity, with clear pathways for measurement conditioning and SPDE-based extensions.
Abstract
We present a novel framework for the probabilistic modelling of random fourth order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memoryless transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors, and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric.