Extreme values of Young’s modulus of tetragonal crystals

Abstract

The extreme values of Young’s modulus for six- and seven-constant tetragonal crystals are found using the necessary and sufficient conditions for the extrema of the functions of two variables. Theoretical and numerical analyzes of stationary and extreme values are performed based on experimental data on elastic constants collected in the Landolt–Börnstein reference book. Five such Young’s moduli are formed in the case of six-constant and seven-constant tetragonal crystals. It is found how the stationary and extremal values of Young’s modulus depend on three anisotropy coefficients that disappear in the limit of an isotropic material. Simple analytical dependences of some stationary and extreme values of Young’s moduli of six-constant crystals are obtained. In the case of stationary values of Young’s modulus of seven-constant tetragonal crystals, the coefficients included in the sufficient conditions for the extremum of the function of two variables are estimated numerically. Tetragonal crystals (Hg2I2, Hg2Br2, Hg2Cl2, TeO2, (NH2)2CO, LiY0.5Tb0.5F4 and C(CH2OH)4) with a large difference between the maximum and minimum values of Young’s modulus are revealed. It is shown that six-constant tetragonal crystals may have a greater difference between global extrema than seven-constant tetragonal crystals. It is found that six-constant tetragonal crystals with a large difference between the global extrema of Young’s modulus have a negative Poisson’s ratio. In the case of seven-constant tetragonal crystals, such relationship has not been identified. A classification scheme based on the dependence of three stationary values of Young’s modulus on two dimensionless parameters is proposed.