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  • 1.
    Karimi, Pouyan
    et al.
    University of Illinois at Urbana-Champaigne, USA.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Zhang, Xian
    University of Illinois at Urbana-Champaigne, USA.
    RVE problem: mathematical aspects and related stochastic mechanics2020In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 146, article id 103169Article in journal (Refereed)
    Abstract [en]

    The paper examines (i) formulation of field problems of mechanics accounting for a random material microstructure and (ii) solution of associated boundary value problems. The adopted approach involves upscaling of constitutive properties according to the Hill--Mandel condition, as the only method yielding hierarchies of scale-dependent bounds and their statistics for a wide range of (non)linear elastic and inelastic, coupled-field, and even electromagnetic problems requiring (a) weakly homogeneous random fields and (b) corresponding variational principles. The upscaling leads to statistically homogeneous and isotropic mesoscale tensor random fields (TRFs) of constitutive\ properties, whose realizations are, in general, everywhere anisotropic. A summary of most general admissible correlation tensors for TRFs of ranks 1, \dots, 4 is given. A method of solving boundary value problems based on the TRF input is discussed in terms of torsion of a randomly structured rod. Given that many random materials encountered in nature (e.g., in biological and geological structures) are fractal and possess long-range correlations, we also outline a method for simulating such materials, accompanied by an application to wave propagation.

  • 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, 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.

  • 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, 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.

  • 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.
    Polyadic random fields2022In: Zeitschrift für Angewandte Mathematik und Physik, ISSN 0044-2275, E-ISSN 1420-9039, Vol. 73, p. 1-21, article id 204Article in journal (Refereed)
    Abstract [en]

    The paper considers mean-square continuous, wide-sense homogeneous, and isotropic random fields taking values in a linear space of polyadics. We find a set of such fields whose values are symmetric and positive-definite dyadics, and outline a strategy for their simulation.

  • 5.
    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 Mechanics2020In: Encyclopedia of Continuum Mechanics / [ed] Altenbach, Holm and Öchsner, Andreas, Berlin, Heidelberg: Springer Berlin/Heidelberg, 2020, p. 2433-2441Chapter in book (Refereed)
  • 6.
    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 for Continuum Physics2018Book (Refereed)
    Abstract [en]

    Many areas of continuum physics pose a challenge to physicists. What are the most general, admissible statistically homogeneous and isotropic tensor-valued random fields (TRFs)? Previously, only the TRFs of rank 0 were completely described. This book assembles a complete description of such fields in terms of one- and two-point correlation functions for tensors of ranks 1 through 4. Working from the standpoint of invariance of physical laws with respect to the choice of a coordinate system, spatial domain representations, as well as their wavenumber domain counterparts are rigorously given in full detail. The book also discusses, an introduction to a range of continuum theories requiring TRFs, an introduction to mathematical theories necessary for the description of homogeneous and isotropic TRFs, and a range of applications including a strategy for simulation of TRFs, ergodic TRFs, scaling laws of stochastic constitutive responses, and applications to stochastic partial differential equations. It is invaluable for mathematicians looking to solve problems of continuum physics, and for physicists aiming to enrich their knowledge of the relevant mathematical tools.

  • 7.
    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.

  • 8.
    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.
    Towards stochastic continuum damage mechanics2020In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 184, p. 202-210Article in journal (Refereed)
    Abstract [en]

    In classical continuum damage mechanics, the distribution of cracks over differently oriented planes is an even deterministic function defined on the unit sphere. The coefficients of its Fourier expansion are completely symmetric and completely traceless tensors of even rank, the so-called fabric or damage tensors. We propose a stochastic generalisation of the above described mathematical model, where damage tensors are mean-square continuous wide-sense homogeneous and isotropic random fields.

  • 9.
    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.
    Amiri-Hezaveh, Amirhossein
    University of Illinois at Urbana-Champaign.
    Random fields of Piezoelectricity and piezomagnetism: Correlation structures2020 (ed. 1)Book (Refereed)
    Abstract [en]

    Random fields are a necessity when formulating stochastic continuum theories. In this book, a theory of random piezoelectric and piezomagnetic materials is developed. First, elements of the continuum mechanics of electromagnetic solids are presented. Then the relevant linear governing equations are introduced, written in terms of either a displacement approach or a stress approach, along with linear variational principles. On this basis, a statistical description of second-order (statistically) homogeneous and isotropic rank-3 tensor-valued random fields is given. With a group-theoretic foundation, correlation functions and their spectral counterparts are obtained in terms of stochastic integrals with respect to certain random measures for the fields that belong to orthotropic, tetragonal, and cubic crystal systems. The target audience will primarily comprise researchers and graduate students in theoretical mechanics, statistical physics, and probability.

  • 10.
    Zhang, Xian
    et al.
    University of Illinois at Urbana-Champaign Urbana, Champaign, IL, 61801, USA.
    Malyarenko, Anatoliy
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Porcu, Emilio
    Khalifa University at Abu Dhabi, United Arab Emirates.
    Ostoja-Starzewski, Martin
    University of Illinois at Urbana-Champaign, USA.
    Elastodynamic problem on tensor random fields with fractal and Hurst effects2022In: Meccanica (Milano. Print), ISSN 0025-6455, E-ISSN 1572-9648, Vol. 57, no 4, p. 957-970Article in journal (Refereed)
    Abstract [en]

    This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for $$\beta < 0.5$$) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.

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