Lattice Theory of Second- and Third-Order Elastic Constants of Aluminum, Copper, and Nickel

Abstract

Lattice theory has been used to obtain the expressions for the second- and third-order elastic constants for face-centered-cubic lattices in terms of the second- and third-order coupling parameters, considering the general interaction between the nearest-neighbor atoms. The number of constants have been reduced by expressing the third-order coupling parameters in terms of the coupling parameters of second order and second-order elastic constants. The general expressions of the elastic constants have been evaluated for Al, Cu, and Ni in a special case of central forces. The force constants involved in the expressions have been determined by representing the central interaction between the pair of atoms by the Morse potential function. It is found that the Cauchy relations for second-order elastic contants C12=C44 and for third-order elastic constants C112=C166 and C123=C456=C144 are satisfied in the case of central forces. It is also seen that the values of C111 for all the three metals are the largest and C112 is approximately half as that of C111 and almost all the values of C123 are negative and small compared with the other third-order elastic constants. The pressure derivatives of second-order elastic constants, the anisotropy factor, and the Debye temperatures have also been calculated for these metals. The values obtained are in good agreement with the experimental values available in the literature.