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  • 1.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, US.
    A Random Field Formulation of Hooke’s Law in All Elasticity Classes2017In: Journal of elasticity, ISSN 0374-3535, E-ISSN 1573-2681, Vol. 127, no 2, p. 269-302Article in journal (Refereed)
    Abstract [en]

    For each of the 8 symmetry classes of elastic materials, we consider a homogeneousrandom field taking values in the fixed point set V of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders 1 and 2 of such a field, and the field’s spectral expansion.

  • 2.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Fractal planetary rings: energy inequalities and random field model2017In: International Journal of Modern Physics B, ISSN 0217-9792, Vol. 31, no 30, article id 1750236Article in journal (Refereed)
    Abstract [en]

    This study is motivated by a recent observation, based on photographs from the Cassini mission, that Saturn’s rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of such a fractal structure of planetary rings is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking planetary rings’ spatial structure as being statistically stationary in time and statistically isotropic in space, but statistically nonstationary in space. An answer to the first challenge is given through an energy analysis of circular rings having a self-generated, noninteger-dimensional mass distribution [V. E. Tarasov, Int. J. Mod Phys. B 19, 4103 (2005)]. The second issue is approached by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square F^2 of the radial cross-section F, where F is a fat fractal.

  • 3.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign.
    Tensor Random Fields in Continuum Mechanics2018In: Encyclopedia of Continuum Mechanics / [ed] Altenbach, Holm and Öchsner, Andreas, Berlin, Heidelberg: Springer Berlin/Heidelberg, 2018, p. 1-9Chapter in book (Refereed)
  • 4.
    Malyarenko, Anatoliy
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Tensor-Valued Random Fields in Continuum Physics2016In: Materials with internal structure: Multiscale and Multifield Modeling and Simulation / [ed] P. Trovalusci, Berlin/Heidelberg: Springer Science+Business Media B.V., 2016, p. 75-88Chapter in book (Refereed)
    Abstract [en]

    This article reports progress on homogeneous isotropic tensor random fields (TRFs) for continuum mechanics. The basic thrust is on determinin most general representations of the correlation functions as well as their spectral expansions. Once this is accomplished, the second step is finding the restrictionsdictated 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.

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