Elastic constants Cauchy relations

For most distortions, both overlap interactions and Madelung terms contribute to the elastic constants. However, for the distortion associated with C44 in the rocksalt structure, only the Madelung term enters. (There are no changes in nearest-neighbor distance to first order in the strain.) Thus the experimental values can be compared with the Madelung term, and by virtue of the Cauchy relation, c, 2 should take the same value. That contribution has been calculated by Kellerman (1940) and is Values of this expression arc given in Table... 

Models that include ionic distortions are much more accurate than those based on rigid ions [26].In particular, the many-hody effects introduced by the distortions are necessary to account for the ohserved violations of the Cauchy relations among elastic constants [27, 28]. Since the form of lT(r) is model dependent, we defer its discussion until the following section. 

In the case of FCC transition and noble metals one has to introduce non central Interactions. In fact for central forces should hold the Cauchy relation between the elastic constants c... 

When it is assumed, as is usually done, that the stiffiiess and compliance tensors are additionally symmetric with respect to their diagonals, the total number of independent components is reduced from 36 to 21 (so-called Green elasticity or hyperelasticity, in contrast to the so-called Cauchy elasticity, where this is not the case). Thus in the most general case of well-defined anisotropy (triclinic monociystals) the (6 x 6) stiffness or compliance matrices or, alternatively, the fourth-order stiffness or compliance tensors, have 36 elastic constants or coefficients, respectively, 21 of which can be assumed to be independent, cf. [Pabst Gregorova 2003a]. For reasons of convenience we confine ourselves to the stiffiiess matrices in the sequel. It is understood, however, that completely analogous relations and symmetry considerations are valid in the case of the compliance matrices. 

The elastic constants are closely related to the potential energies between atoms. When the interatomic force is a central force, Cauchy s relationships hold true for the elastic stiffness constants (Love, 1944) ... 

In the case of cubic crystals, the Cauchy relations require that C " 44 so that there are only two independent elastic constants, namely, C 3ind... 

The experimental values of the elastic constants for most alkali halides satisfy the Cauchy relations moderately well (Table 3.2). However, from Table 3.2 it is seen that this is not the case for metals. In an alkali metal such as sodium, it is true that the ions are at centers of inversion symmetry and that the screened Coulomb interaction between the ions is a central force however, the crystal is not in equilibrium under the action... 

Current theoretical research on ionic crystals includes treatment of noncentral or three-body forces and their effects on shear forces (e.g., deviations from the Cauchy relation for elastic constants). Such calculations are of secondary interest from a chemical and structural point of view. 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

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