Surface influence on the tensor of elasticity moduli

Abstract

The consideration of a change of independent components of the tensors of elasticity moduli in a vicinity of flat surfaces for all 32 bulk crystallographic classes is carried out. The calculations are performed for the fourth rank tensors of elasticity moduli and compliances on the basis of the symmetry theory with respect to the transformation of tensor components under a symmetry group. The increase in the number of nonzero components is obtained due to the symmetry lowering in a vicinity of the surface. In particular, for the cubic symmetry, the moduli c₃₃, c₁₃, and c₆₆ appear in addition to the known three nontrivial moduli (c₁₁, c₁₂, c₄₄) for surfaces of the (100) type and new nonzero components c₁₄, c₂₂=-c₁₄, c₅₆=-c₁₄ are obtained for those of the (111) type. Comparative analysis of elastic moduli in bulk and in a vicinity of the surface has shown that the Cauchy relations are not valid in a vicinity of the surface because of the inversion symmetry absence. Analysis of the experimental results and those of ab initio calculations for nanostructured materials allows us to conclude that the surface influence is essential for sizes up to 100 nm.