Tensor Random Fields

Mechanics and physics of random media strongly suggest that stochastic PDEs and stochastic finite element methods require mesoscale tensor-valued random fields (TRFs) of constitutive laws with locally anisotropic fluctuations. Such models are also useful when there is interest in fields of dependent quantities (velocity, strain, stress...) that need to be constrained by the balance laws (of mass, momentum...); examples are irrotational and solenoidal TRFs. In this article, we review the canonical forms of general correlation structures of second-order, mean-square continuous, wide-sense homogeneous and isotropic TRFs of ranks 1, ... , 4 in 3d. Besides “conventional” covariances, this approach can be used to construct TRFs with fractal and Hurst (long-range memory) characteristics. The current research extends the earlier work on scalar-valued RFs (including random processes) in vibration problems, rods and beams with random properties under random loadings, as well as elastodynamics, wavefronts, fracture, homogenization of random media, and contact mechanics.