In late June, Professor Martin Ostoja-Starzewski gave a general lecture at the 14th International Symposium on Continuum Models and Discrete Systems in Paris, France.
The CMDS symposia brings together scientists from different backgrounds, working on continuum theories and discrete mechanical and thermodynamical systems in the fields of mathematics, theoretical and applied mechanics, physics, materials science, and engineering. The spirit of CMDS meetings is to stimulate extensive and active interdisciplinary research – culminating in an impactful exchange of ideas, methods and results.
Ostoja-Starzewski’s lecture began with the observation that deterministic models of continuum mechanics may fail for various reasons, especially in multiscale problems. From the standpoint of random microstructures, probabilistic models needed as input to stochastic partial differential equations (SPDE) and stochastic finite elements (SFE) naturally involve tensor-valued random fields (TRF) with generally anisotropic realizations and non-trivial correlation functions or variograms.
However, the commonly employed SPDE and SFE models either employ one scalar-valued RF to represent a tensorial material property (conductivity, elasticity…) or simply postulate a TRF to have white-noise correlations. Working in the setting of wide-sense homogeneous and isotropic, mean-square continuous TRFs, the lecture reviewed explicit representations of most general correlation functions of TRFs of ranks 1 through 4 in 2d or 3d [1,2]. Employing polyadic forms, such TRFs can readily be simulated from rapidly generated scalar random fields, to provide inputs to stochastic boundary value problems. Experiments can be used to determine/calibrate the correlation functions of TRF.
Besides “conventional” correlation structures, this strategy can be used to generate TRFs with fractal and Hurst characteristics, i.e., with multi-scale randomness and free of the restriction to self-similarity. 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, elastodynamics, wavefronts, fracture, buckling, homogenization of random media, statistical turbulence, and contact mechanics.
The lecture was based on the following published research:
- 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.