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On optimal zeroth-order bounds of linear elastic properties of multiphase materials and application in materials design

Mauricio Lobos
Thomas Böhlke
Journal / Anthology

International Journal of Solids and Structures
Year: 2016
Volume: 84
Page range: 40-48

Zeroth-order bounds of elastic properties have been discussed by Kröner (1977) and by Nadeau and Fer- rari (2001). These bounds enclose the effective linear elastic properties of multiphase materials consti- tuted of materials with arbitrary symmetry and of an arbitrary number of phases by using solely the ma- terial constants of the single materials. Nadeau and Ferrari showed that these bounds are isotropic tensors and presented an algorithm for the determination of the upper and the lower zeroth-order bound. It is shown in this paper that a problem arises for the lower bound, since the algorithm presented in Nadeau and Ferrari (2001), results in a negative compression modulus and/or shear modulus although the con- sidered stiffness is positive definite. A simple analytic example for this undesirable property is given, to- gether with a short Mathematica ®code of the algorithm. In the present work, the definition of the lower bound by Nadeau and Ferrari is modified, thereby assuring its positive definiteness. The Mathematica ®code of the corrected algorithm is also given. Furthermore, new bounds for non-diagonal components are derived, which give information of, in principle, accessible values for non-diagonal stiffness components using the zeroth-order bounds of the present work. The practical application of zeroth-order bounds for local and online material data bases of stiffness tensors is presented, in order to accelerate purposes in materials design through efficient materials screening.

*Engineering > Materials and Metallurgical Engineering