Toupin–Mindlin first strain gradient theory revisited for cubic crystals of hexoctahedral class: Analytical expression of the material parameters in terms of the atomic force constants and evaluation via ab initio DFT

Abstract

Capture of the discrete nature of crystalline solids for the purpose of the determination of their mechanical behavior with high precision is of interest. To achieve this objective, two fundamental contributing factors are on top of the list: (1) formulation in the mathematical framework of an appropriate higher order continuum theory rather than using classical treatment, and (2) incorporation of the true anisotropy of the media. The present work revisits Toupin–Mindlin first strain gradient theory for media with general anisotropy, and then specialize it to cubic crystals of hexoctahedral class. This formulation in addition to 3 classical material constants encountered in classical theory of elasticity, gives rise to 11 additional material parameters peculiar to first strain gradient theory. To date, there is no experimental method in the literature for the measurement of these parameters. A methodology incorporating lattice dynamics is proposed, by which all the material parameters including the classic ones are analytically expressed in terms of the atomic force constants. Subsequently, the analytical expressions for the nonzero components of the 4th and 6th order elastic moduli tensors as well as 6 characteristic lengths are derived. Finally, with the aid of ab initio calculations all the material properties in Toupin–Mindlin first strain gradient theory are numerically obtained with high precision. In this work the transformation matrices of cubic crystals of diploidal class which also falls under centrosymmetric point groups are discussed as well.