Extrema of elastic properties of cubic crystals

As a rule, the discussion of physical properties of crystals is accompanied by complicated mathematical calculations based on algebraic expressions in tensor and matrix notation. This approach, due to the nature and unified representation of the tensor properties of crystalline materials, makes it very difficult to calculate their specific characteristics and parameters. This report presents the final work formulas for calculating the orientation dependences of the elastic properties of cubic crystal symmetry - elastic modules and Poisson's ratio, as well as the parameters of anisotropy. Special attention is paid to the extreme - minimum and maximum - values of the parameters of elastic properties and the relationship between them. Variants of description of elastic anisotropy of cubic crystals by means of a number of independent indicators are considered. On a concrete example it is shown that they can give essentially different results. Methods of visual interpretation of anisotropy of elastic properties by means of corresponding characteristic (indicative) surfaces and their sections are discussed. It is noted that the index surface of the Young normal elasticity modulus is the most accessible for the construction, although it is not a complete characteristic of the anisotropy of elastic properties. A method of visualization of matrices of elastic constants of crystals using MATLAB application software package is proposed, which gives visual information about the ratio of the matrix elements. As an example of calculation of extreme values and parameters of anisotropy, as well as the construction of characteristic surfaces and their cross sections, the TiNi titanium nickelide single crystals, which are widely used in various fields of science, technology and medicine and are often discussed in the literature, are considered

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