BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Rothschild Lecture: Tensor Random Fields in Mechanics - Martin Ost oja-Starzewski (University of Illinois at Urbana-Champaign) DTSTART:20230719T150000Z DTEND:20230719T160000Z UID:TALK202381@talks.cam.ac.uk DESCRIPTION:Deterministic models of continuum mechanics tackled by boundar y value problems may be inadequate for various reasons\, especially in mul tiscale problems. Viewed from the standpoint of random microstructures\, p robabilistic models such as stochastic partial differential equations (SPD E) and stochastic finite elements (SFE) should naturally involve tensor-va lued random fields (TRF) with generally anisotropic realizations and non-t rivial correlation functions. We discuss current work and outline some ope n problems. The setting is wide-sense homogeneous and isotropic\, second-o rder\, mean-square continuous fields on mesoscales [1]. We give explicit r epresentations of the most general correlation functions of TRFs of 1st\, 2nd\, 3rd\, and 4th ranks [2\,3].\nGoverning equations of continuum theori es (such as incompressibility or equilibrium) offer constraints on the cor relation functions of dependent fields (e.g.\, displacement\, velocity [4] \, strain\, stress&hellip\;) of various ranks in classical continua and\, similarly\, in conductivity\, electricity\, or magnetism. Analogously\, on e can establish the consequences for TRFs (of rotation\, curvature-torsion \, couple-stress&hellip\;) in stochastic micropolar theories. When there i s interest in TRFs of constitutive properties (e.g.\, conductivity\, stiff ness\, damage)\, experiments can be used to determine/calibrate the correl ation functions.\nBesides &ldquo\;conventional&rdquo\; correlation structu res\, this strategy can be generalized to TRFs with fractal and Hurst char acteristics\, 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 ano ther the Hurst exponent. The current research extends our earlier work on scalar-valued RFs (including random processes) in vibration problems\, rod s and beams with random properties under random loadings\, elastodynamics\ , wavefronts\, fracture\, homogenization of random media\, and contact mec hanics\, e.g. [7].\n \;\n\nM. 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.\nA. Malyare nko and M. Ostoja-Starzewski\, Tensor-Valued Random Fields for Continuum P hysics\, Cambridge University Press\, 2019.\nA. Malyarenko\, M. Ostoja-Sta rzewski\, and A. Amiri-Hezaveh\, Random Fields of Piezoelectricity and Pie zomagnetism\, Springer\, 2020.\nG.K. Batchelor\, The Theory of Homogeneous Turbulence\, Cambridge University Press\, 1953.\nT. Gneiting and M. Schla ther. Stochastic models that separate fractal dimension and the Hurst effe ct\, SIAM Rev. 46\, 269-282\, 2004.\nE. Porcu\, J. Mateu\, A. Zini and Pin i\, Modelling spatio-temporal data: A new variogram and covariance structu re proposal. Stat. Probab. Lett. 77\, 83-89\, 2007.\nY.S. Jetti and M. Ost oja-Starzewski\, Elastic contact of random surfaces with fractal and Hurst effects\, Proc. R. Soc. A 478\, 20220384\, 2022.\n LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR