Representation of Elastic Behavior in Cubic Materials for Arbitrary Axes

Abstract

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.

What is the Young's Modulus of Silicon?

“…Expanded expressions for G in a cubic crystal are given in [9], [10], [34], and [35]. The value of G is 79.6 GPa between the [100] direction and any other direction in the (100) plane.…”

The Directional Dependence of Elastic Stiffness and Compliance Shear Coefficients and Shear Moduli in Cubic Materials

“…For example, it is apparent from Fig. 2 of [6] that for planes of the form (0kl), (hhl) and (hkk) defining the borders of the standard 001-011-111 stereographic triangle for cubic materials, the polar plots of c rr as a function of shear vector in the plane are symmetrical about [100] and [0lk], [110] and [llh], and [011] and [khh] respectively, and have 2mm point symmetry at the origin. However, for a general plane (hkl) within this standard stereographic triangle, such as (156), the two perpendicular directions about which the polar plots are symmetrical are not specified in terms of directions [uvw] lying in (hkl).…”

“…Later, in a series of papers, Turley and Sines [6][7][8] reported the results of numerical computations of the directional dependence of elastic stiffness and compliance shear coefficients and shear moduli in silicon, copper and molybdenum using the same graphical representations as Workman and Evans, with, in addition, a very powerful graphical representation using a standard stereographical triangle on which to represent c rr and G r for silicon for various (hkl). The mathematical expressions given by these authors for the shapes of c 44 and G 4 as a function of the shear direction and the normal to the shear plane are highly complex, although readily amenable to numerical computation.…”

Interatomic bonds and the tensile anisotropy of trialuminides in the elastic limit: a density functional study for Al3(Sc, Ti, V, Cr)

“…The anisotropic behaviour of the Young modulus (Y ) and Poisson ratio (ν) can be calculated using the general transformation rule for fourth-order tensors [45][46][47]. Following Cazzani and Rovati [48,49], the reciprocal value of Young modulus for uniaxial tension in the direction of the unit vector n can be expressed as…”