TY - GEN
T1 - Tensor Random Fields
AU - Ostoja-Starzewski, Martin
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - Correlation structure
KW - Hurst exponent
KW - Random fields, fractals
KW - Stochastic mechanics
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U2 - 10.1007/978-3-031-58665-1_2
DO - 10.1007/978-3-031-58665-1_2
M3 - Conference contribution
AN - SCOPUS:85206107045
SN - 9783031586644
T3 - Springer Proceedings in Mathematics and Statistics
SP - 15
EP - 27
BT - Continuum Models and Discrete Systems - CMDS-14
A2 - Willot, François
A2 - Jeulin, Dominique
A2 - Willot, François
A2 - Dirrenberger, Justin
A2 - Forest, Samuel
A2 - Cherkaev, Andrej V.
PB - Springer
T2 - 14th International Symposium on Continuum Models and Discrete Systems, CMDS 2023
Y2 - 26 June 2023 through 30 June 2023
ER -