# Rothschild Lecture: Tensor Random Fields in Mechanics

USM - Uncertainty quantification and stochastic modelling of materials

Deterministic models of continuum mechanics tackled by boundary value problems may be inadequate for various reasons, especially in multiscale problems. Viewed from the standpoint of random microstructures, probabilistic models such as stochastic partial differential equations (SPDE) and stochastic finite elements (SFE) should naturally involve tensor-valued random fields (TRF) with generally anisotropic realizations and non-trivial correlation functions. We discuss current work and outline some open problems. The setting is wide-sense homogeneous and isotropic, second-order, mean-square continuous fields on mesoscales [1]. We give explicit representations of the most general correlation functions of TRFs of 1st, 2nd, 3rd, and 4th ranks [2,3]. Governing equations of continuum theories (such as incompressibility or equilibrium) offer constraints on the correlation functions of dependent fields (e.g., displacement, velocity [4], strain, stress…) of various ranks in classical continua and, similarly, in conductivity, electricity, or magnetism. Analogously, one can establish the consequences for TRFs (of rotation, curvature-torsion, couple-stress…) in stochastic micropolar theories. When there is interest in TRFs of constitutive properties (e.g., conductivity, stiffness, damage), experiments can be used to determine/calibrate the correlation functions. Besides “conventional” correlation structures, this strategy can be generalized to TRFs with fractal and Hurst characteristics, i.e., with multi-scale randomness, long memory, and free of the restriction to self-similarity. Out of a large menu of correlation functions in probability theory, two models can accomplish that: Cauchy [5] and Dagum [6]. In each case, the correlation function is controlled by two independent parameters, one specifying the fractal dimension and another the Hurst exponent. 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, e.g. [7].

M. Ostoja-Starzewski, S. Kale, P. Karimi, A. Malyarenko, B. Raghavan, S.I. Ranganathan, and J. Zhang, Scaling to RVE in random media, Adv. Appl. Mech. 49, 111-211, 2016. A. Malyarenko and M. Ostoja-Starzewski, Tensor-Valued Random Fields for Continuum Physics, Cambridge University Press, 2019. A. Malyarenko, M. Ostoja-Starzewski, and A. Amiri-Hezaveh, Random Fields of Piezoelectricity and Piezomagnetism, Springer, 2020. G.K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, 1953. T. Gneiting and M. Schlather. Stochastic models that separate fractal dimension and the Hurst effect, SIAM Rev. 46, 269-282, 2004. E. Porcu, J. Mateu, A. Zini and Pini, Modelling spatio-temporal data: A new variogram and covariance structure proposal. Stat. Probab. Lett. 77, 83-89, 2007. Y.S. Jetti and M. Ostoja-Starzewski, Elastic contact of random surfaces with fractal and Hurst effects, Proc. R. Soc. A 478 , 20220384, 2022.

This talk is part of the Isaac Newton Institute Seminar Series series.