Tensor Random Fields in Mechanics

Prof Martin Ostoja-Starzewski (University of Illinois at Urbana-Champaign)

Thursday 16th November 14:00-15:00 Maths 311B/ZOOM (ID: 878 4424 4710)


Mechanics and physics of random media 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 talk, we review the canonical forms of general correlation structures of statistically stationary and isotropic TRFs of ranks 1,…,4 in 3d [1,2]. Besides “conventional” correlations, our approach can be used to construct TRFs with fractal and Hurst (long-range memory) characteristics. The current research extends our earlier work on scalar-valued RFs (including random processes) in vibration problems, rods and beams with random properties under random loadings, elastodynamics, wavefronts, fracture, homogenization of random media, and contact mechanics.

1. A. Malyarenko and M. Ostoja-Starzewski, Tensor-Valued Random Fields for Continuum Physics, Cambridge University Press, 2019.
2. A. Malyarenko, M. Ostoja-Starzewski, and A. Amiri-Hezaveh, Random Fields of Piezoelectricity and Piezomagnetism, Springer, 2020.

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