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

• The definition of optimal zeroth-order bounds is corrected assuring their positive definiteness.
• A novel anisotropy ratio is defined for complete composites.
• Application on material data bases for speeding up material design objectives.

Zeroth-order bounds of elastic properties have been discussed by Kröner (1977) and by Nadeau and Ferrari (2001). These bounds enclose the effective linear elastic properties of multiphase materials constituted of materials with arbitrary symmetry and of an arbitrary number of phases by using solely the material 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 considered stiffness is positive definite. A simple analytic example for this undesirable property is given, together 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.

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