5^th SEECCM2023 ECCOMAS Conference

Abstract

Uncertain material properties are often described in probabilistic terms. And although the properties of the material may be uncertain, some relations may not be uncertain at all. One possible kind of such a relation are spatial symmetries and invariances of the material. It is well known that such symmetries and invariances play an important role in the behaviour of materials, and thus one should try to respect this in the probabilistic description and modelling of material properties. The focus is here on material properties described by the class of physically symmetric fourth order tensors, of which the elasticity tensor is a prime example. We shall concentrate on spatial point symmetries or invariances under rotations, a classical subject in deterministic elasticity, a topic which has already attracted considerable attention, e.g. [5], and offer a sketch how this can be connected to stochastic geometric symmetries or invariances in the modelling of spatial tensor fields. The culmination of a long series of works on such representations of tensors of any order in the presence of spatial symmetries or invariances is given in [7], but the additional property of physical symmetry implies a possible spectral theory and decomposition to separate scale or strength from orientation, and this will be the topic here.
The characterisation of elasticity tensors through their spectral decomposition, i.e. eigenvalues, is is an old subject, already started by Lord Kelvin (W.K. Thomson) [13, 14], but little appreciated at the time and then apparently almost forgotten, to be picked up again by [3], and later, among others, by [9, 8, 11]. A historical account in modern notation may be found in [6].
An approach on the basis of the spectral decomposition was used in [10] to model random second order symmetric tensors fields. As for second order symmetric tensors there are only three classes in 3D and two classes in 2D, this is quickly reviewed to easily explain the concepts.
The elasticity tensor is not only a physically symmetric tensor, but each realisation usually has to be positive definite. Looking at random ensembles resp. probability distributions on such tensors, one wants to preserve those properties. This is achieved by constructing a mapping from a product Lie group to the positive tensors. The corresponding Lie algebra is then a direct sum of vector spaces, and the appropriate exponential map, which is easily computed in this case, guarantees positive definiteness. Additionally often one wants 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.
In formulating a modelling framework which respects these requirements, one has to explore the mathematical structure of the underlying sample space of all such tensors, in the course of which it appears expedient to introduce a new distance measures or metric between such tensors. It is shown how the proposed framework allows the random ensemble to be modelled and generated, independently in its scaling and orientational or directional aspects.
To achieve this modelling we collect for the fourth-order elasticity tensor for all eight elastic symmetry classes [4] in 3D and for the four symmetry classes [1] in 2D the invariant subspaces with appropriate Kelvin eigenvector basis and the corresponding eigenvalue — the Kelvin stiffnesses or Kelvin elastic moduli [13, 14, 6] — are introduced. In a slight deviation [8] from the traditional Voigt notation, e.g. [12], the fourth order elasticity tensor is mapped onto a 6-dimensional tensor of second order, the Kelvin notation, which will then be used for the representation. In this 6-dimensional strain-stress space one might distinguish the rotations induced by true spatial rotations from others, which are strain redistributions. The rotations induced by spatial rotations allow the well-known group theoretic reduction, e.g. [2], to the well known zero-non zero patterns resp. block diagonal forms which characterise the different elasticity classes in 2D and 3D, e.g. [1, 4]. Rotations in the 6-dimensional strain-stress space which only redistribute strains can then be used to achieve a complete diagonalisation.
On this basis a method to generate random elasticity tensors is described. The tensor is in this way modelled by one spatial vector which describes a spatial orientation through its direction and magnitude — leading to a reduction of the symmetry group representation and the well-known reduced forms of the tensor-tuples of parameters to describe the logarithms of the eigenvalues resp. Kelvin moduli, plus tuples of parameters to describe the ‘strain distributors’, which furnish an orthogonal transformation in strain space to completely diagonalise the tensor from its reduced form.

References
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1 Institute of Scientific Computing, TU Braunschweig, Germany, h.matthies@tu-bs.de
2 The Alan Turing Institute, London, United Kingdom, sshivanand@turing.ac.uk
3 Applied Mechanics and Data Analysis, University of Twente, Netherlands, b.rosic@utwente.nl