Extrema of Elastic Properties of Cubic Crystals

As a rule, a discussion of physical properties of crystals is accompanied by far from simple mathematical calculations based on algebraic expressions in tensor and matrix notations. Such an approach caused by the nature and uniform presentation of properties of crystalline materials makes practical calculations of their specific characteristics and parameters very difficult. By an example of the titanium nickelide crystals, it is shown that the anisotropy coefficient of elastic properties of the cubic syngony crystals, which is equal to the ratio of the extreme values (minimum and maximum) of the shear modulus, is close to the ratio of the extreme values of Young's modulus. Some variants of describing the elastic anisotropy of cubic crystals using a series of dimensional and dimensionless independent indicators are considered. It is shown on a concrete example that they can give significantly different results. Methods of visual interpretation of the elastic properties anisotropy using the corresponding characteristic surfaces and their cross sections are discussed. It is noted that the characteristic surface of the Young's modulus of normal elasticity is the most accessible for construction, although it is not a complete characteristic of the anisotropy of elastic properties of cubic crystals. A method is proposed for visualizing matrices of elastic constants of crystals using the MatLab application package, which provides visual information on the ratio of the values of matrix elements. Single crystals of titanium nickelide TiNi, widely used in various fields of science, technology, and medicine and often discussed in the literature, are considered as an example of calculating the extreme values and parameters of anisotropy, as well as constructing characteristic surfaces and their cross sections.