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Analytical applications and effective properties of a second gradient isotropic elastic material model

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Abstract

Recently, the works by Toupin, Mindlin, Sokolowski and Germain have been developed following two research streams. In the first one, higher-order gradient continuum models were developed based on the Cauchy tetrahedron argument (see, e.g., dell’Isola and Seppecher in Comptes Rendus de l Academie de Sciences 17 Serie IIb: Mecanique, Physique, Chimie, Astronomie 321:303–308, 1995, Meccanica 32:33–52 1997, Zeitschrift fr Angewandte Mathematik und Physik 63(6):1119–1141, 2012). In the second one, the structure of higher-order gradient models is developed with a view to the applications. In particular in the model of linear isotropic solids proposed by Dell’Isola, Sciarra and Vidoli (DSV), the main constitutive equation is obtained for the case of second gradient models. This model introduces in addition to the two well-known Lame’s elastic constants five constitutive constants. The practical applications of this model remain in its infancy since the issue of determining the new moduli it introduces is not yet completely addressed. Also, analytical solutions of simple boundary value problems that can be helpful to grasp some of the physical foundations of this model are missing. This paper aims to address these two issues by providing the analytical solutions for two model problems, a spherical shell subjected to axisymmetric loading conditions and the circular bending of a beam in plane strain, both the beam and the shell obeying the DSV second gradient isotropic elastic model. The solution of the circular bending of a beam has served to grasp some of the physical soundness of the model. A framework based on homogenization under inhomogeneous boundary conditions is also suggested to determine the unknown constitutive constants, which are provided in the particular case of elastic porous heterogeneous materials.

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  1. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, 39762, USA

    Koffi Enakoutsa

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Enakoutsa, K. Analytical applications and effective properties of a second gradient isotropic elastic material model. Z. Angew. Math. Phys. 66, 1277–1293 (2015). https://doi.org/10.1007/s00033-014-0453-2

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  • DOI: https://doi.org/10.1007/s00033-014-0453-2

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