Advertisement

Tensor-Valued Random Fields in Continuum Physics

  • Anatoliy Malyarenko
  • Martin Ostoja-StarzewskiEmail author
Part of the Springer Tracts in Mechanical Engineering book series (STME)

Abstract

This article reports progress on homogeneous isotropic tensor random fields (TRFs) for continuum mechanics. The basic thrust is on determining most general representations of the correlation functions as well as their spectral expansions. Once this is accomplished, the second step is finding the restrictions dictated by a particular physical application. Thus, in the case of fields of material properties (like conductivity and stiffness), the restriction resides in the positive-definiteness, whereby a connection to experiments and/or computational micromechanics can be established. On the other hand, in the case of fields of dependent properties (e.g., stress, strain and displacement), restrictions are due to the respective field equations.

Keywords

Random Field Cyclic Permutation Classical Elasticity Spectral Expansion Correlation Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Batchelor GK (1953) The theory of homogeneous turbulence. In: Cambridge monographs on mechanics and applied mathematics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  2. 2.
    Lomakin VA (1964) Statistical description of the stressed state of a body under deformation. Dokl Akad Nauk SSSR 155:1274–1277MathSciNetGoogle Scholar
  3. 3.
    Lomakin VA (1965) Deformation of microscopically nonhomogeneous elastic bodies. Appl Math Mech 29(5):888–893MathSciNetGoogle Scholar
  4. 4.
    Malyarenko A (2013) Invariant random fields on spaces with a group action. In: Probability and its applications. Springer, Heidelberg. doi:10.1007/978-3-642-33406-1. http://dx.doi.org/10.1007/978-3-642-33406-1Google Scholar
  5. 5.
    Malyarenko A, Ostoja-Starzewski M (2014) The spectral expansion of the elasticity random field. In: Sivasundaram S (ed) 10th international conference on mathematical problems in engineering, aerospace, and sciences (ICNPAA 2014). AIP conference proceedings, vol 1637, Narvik, 15–18 July 2014, pp 647–655. doi:10.1063/1.4904635Google Scholar
  6. 6.
    Malyarenko A, Ostoja-Starzewski M (2014) Spectral expansions of homogeneous and isotropic tensor-valued random fields. http://arxiv.org/abs/1402.1648
  7. 7.
    Malyarenko A, Ostoja-Starzewski M (2014) Statistically isotropic tensor random fields: correlation structures. Math Mech Complex Syst 2(2):209–231zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Monin AS, Yaglom AM (2007) Statistical fluid mechanics: mechanics of turbulence, vol II. Dover, MineolaGoogle Scholar
  9. 9.
    Ostoja-Starzewski M (2008) Microstructural randomness and scaling in mechanics of materials. CRC series: modern mechanics and mathematics. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  10. 10.
    Ostoja-Starzewski M, Shen L, Malyarenko A (2015) Tensor random fields in conductivity and classical or microcontinuum theories. Math Mech Solids 20(4):418–432. doi:10.1177/1081286513498524MathSciNetCrossRefGoogle Scholar
  11. 11.
    Robertson HP (1940) The invariant theory of isotropic turbulence. Proc Camb Philos Soc 36:209–223CrossRefGoogle Scholar
  12. 12.
    Shermergor T (1971) Relations between the components of the correlation functions of an elastic field. Appl Math Mech 35(3):432–437Google Scholar
  13. 13.
    Spencer A (1971) Theory of invariants. In: Continuum physics, vol 1. Academic, New York, pp 239–353Google Scholar
  14. 14.
    Taylor G (1935) Statistical theory of turbulence. Proc R Soc A 151:421–478zbMATHCrossRefGoogle Scholar
  15. 15.
    von Kármán T, Horwath L (1938) On the statistical theory of isotropic turbulence. Proc R Soc A 164:192–215CrossRefGoogle Scholar
  16. 16.
    Yaglom AM (1957) Certain types of random fields in n-dimensional space similar to stationary stochastic processes. Teor Veroyatnost i Primenen 2:292–338zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Division of Applied MathematicsMälardalen UniversityVästeråsSweden
  2. 2.Department of Mechanical Science & Engineering, also Institute for Condensed Matter Theory and Beckon InstituteUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations