Ab initio calculations of characteristic lengths of crystalline materials in first strain gradient elasticity

•Additional constants in first strain gradient elasticity.•Calculates the characteristic lengths of the crystalline materials.•Obtains the characteristic lengths with consideration of the electronic structure.•Gives elastic tensors in terms of the total energy obtained from ab initio DFT.•Relates the sixth order elastic tensors and the additional constants via ab initio.

Incorporation of the first gradient of strain, in addition to the strain itself, into the strain energy density of an elastic solid leads to Mindlin’s first strain gradient theory, which is useful for examination of size effect as well as other mechanical phenomena at the nano-scale. For isotropic elastic solids, the first strain gradient theory, in addition to the two independent Lamé constants, gives rise to five new material constants which in turn reduce to two material parameters, ℓ1ℓ1 and ℓ2ℓ2 with dimension of length. The evaluation of these parameters, however, has posed serious challenges, both experimentally and theoretically. In this work ab initio method is used to compute the characteristic lengths for several fcc and bcc metal crystals. It will be seen that the elements of the Hessian matrix, obtained by taking the second derivatives of the total energy with respect to the atomic positions, are linked to the strain gradient material constants.