Abstract
We establish spectral expansions of tensor-valued homogeneous and isotropic random fields in terms of stochastic integrals with respect to orthogonal scattered random measures previously known only for the case of tensor rank 0. The fields under consideration take values in the 3-dimensional Euclidean space \({E^3}\) and in the space \({\mathsf{S}^2(E^3)}\) of symmetric rank 2 tensors over \({E^3}\). We find a link between the theory of random fields and the theory of finite-dimensional convex compact sets. These random fields furnish stepping-stone for models of rank 1 and rank 2 tensor-valued fields in continuum physics, such as displacement, velocity, stress, strain, providing appropriate conditions (such as the governing equation or positive-definiteness) are imposed.
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We are grateful to Professor Hanspeter Kraft of Universität Basel for useful discussions of invariant theory. MO-S acknowledges the NSF support under Grants CMMI-1462749 and IIP-1362146(I/UCRC on Novel High Voltage/Temperature Materials and Structures).
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Malyarenko, A., Ostoja-Starzewski, M. Spectral expansions of homogeneous and isotropic tensor-valued random fields. Z. Angew. Math. Phys. 67, 59 (2016). https://doi.org/10.1007/s00033-016-0657-8
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DOI: https://doi.org/10.1007/s00033-016-0657-8