Computational Physics Group

Karel Matous










Third-order Thermo-mechanical Properties for Packs of Platonic Solids Using Statistical Micromechanics

A. Gillman1, G. Amadio2, K. Matous1 and T.L. Jackson3

1Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN, 46556, USA.

2Department of Aerospace Engineering
University of Illinois
Urbana, IL, 61801, USA.

3Department of Aerospace and Mechanical Engineering
University of Florida
Gainesville, FL, 32611, USA.


        Obtaining an accurate higher order statistical description of heterogeneous materials and utilizing this information to predict effective material behavior with high fidelity has remained an outstanding problem for many years. In a recent letter, Gillman and Matous (Gillman and Matous, Physics Letters A, 378(41), 2014) accurately evaluated the three-point microstructural parameter that arises in third-order theories and predicted with high accuracy the effective thermal conductivity of highly packed material systems. Expanding this work here, we predict for the first time effective thermo-mechanical properties of granular Platonic solids packs using third-order statistical micromechanics. Systems of impenetrable and penetrable spheres are considered to verify adaptive methods for computing n−point probability functions directly from threedimensional microstructures, and excellent agreement is shown with simulation. Moreover, a significant shape effect is discovered for the effective thermal conductivity of highly packed composites, while a moderate shape effect is exhibited for the elastic constants.



        In this work, we predict with high accuracy thermal conductivity and elastic constants of isotropic packs of Platonic Solids (crystalline materials). Verification studies are conducted for systems of overlapping and hard monodisperse spheres, and numerical approaches are found very accurate. Good agreement is shown between the third-order models and finite element simulations for rigid particles in a deformable matrix, and the three-point approximation using the well-resolved microstructural parameters ζp and ηp is improved. For the first time, three-point approximations of the thermal-mechanical properties are computed for isotropic systems of Platonic solids at various volume fractions. A significant particle shape effect is predicted for thermal conductivity, whereas the effective elastic moduli are less sensitive to the microstructural configuration. Based on our statistical framework, a large class of materials with arbitrary inclusion shapes can now be easily studied. Moreover, image-based modeling, using micro-computed tomography (micro-CT) for example, can now be successfully employed for real material systems.

Authors’ contributions

        AG developed the numerical integration/interpolation and statistical sampling methods, conducted numerical experiments and performed subsequent analysis of results. GA and TJ developed the packing algorithm for Platonic solids. KM conceived and coordinated the studies, and contributed to analysis of results. AG and KM drafted the manuscript. All authors approved manuscript for publication.

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© 2015 Notre  Dame  and Dr. Karel Matous