The directional behaviour of Young's modulus, shear modulus, and Poisson's ratio are expressed for a number of crystallographic planes for cubic materials. Their behaviour as a function of direction on a particular plane is shown to be a simple sum of sines and cosines of twice and sometimes of four times the angle. A general expression gives the coefficients for the sines and cosines and a table gives the coefficients for 19 planes on the standard triangle of the stereographic projection. Superposed on the stereographic projection are shown polar plots of the shear modulus Gprime12 (1prime direction normal to the plane) for silicon. Examples are also given showing G12prime for copper for some planes of zones [1bar above 00], [1bar above 10] and [2bar above 10].
Several methods are advanced to represent the directional dependence of elastic coefficients for cubic materials which are then illustrated using silicon, copper, and molybdenum. The extensively displayed coefficient is the shear stiffness. Also presented are some rarely shown coefficients such as C1123′ and C1312′ whose on-axes values are zero. Polar plots are made of the shear stiffness coefficient C1313′ for silicon, for some planes that have a common zone axis. A method particularly suitable for illustrating anisotropy in transverse coefficients is presented in which a family of polar plots is superimposed on the standard triangle.
By the use of Eulerian angles, expressions are given for the directionality of the sets of eighty-one compliance and stiffness constants in cubic materials. These expressions are trigonometric series in nθ (n=1, 2, 4), where θ is the angle of rotation in a given plane. Coefficients of the series are listed for planes in the standard triangle and a procedure is presented to extend the analysis to all planes of the same form.
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