Crystal Elasticity. Pascal Gadaud

[1.4]

      [1.5]

      [1.6]

[1.7]

      Moreover, very simple relations can be obtained as follows:

      [1.8]

      [1.9]

      [1.10]

      [1.11]

      [1.12]

      [1.13]

      Young’s modulus along x’ can then be written as:

      [1.14]

      This expression can be rewritten in order to highlight anisotropy:

[1.15]

      [1.16]

      [1.17]

      The function of anisotropy A follows directly from the anisotropy factor shared by all crystals with cubic symmetry, which was introduced by Zener (1948):

      [1.18]

      Furthermore, the transverse deformation during the same test along the direction y’ of the Cartesian coordinates l’, m’ and n’, perpendicular to x’, can also be written. They verify the following relations:

[1.19]

[1.20]

      The transverse deformation along this direction can be written as:

      [1.21]

Since the stress components are the same, the following can be written as:

      [1.22]

      Given:

      [1.23a]

Equations [1.19] and [1.20] yield:

      [1.23b]

      The direction y’ was randomly chosen. Consider a third direction z’ perpendicular to x’ and y’ of the coordinates l’’, m’’ and n’’, which verifies the following:

[1.24]

[1.25]

      Defining:

      [1.26]

      similarly yields:

      [1.27]

      The mean transverse deformation can be written as:

      [1.28]

It should be noted that only the traction direction is preserved, while the randomly chosen transverse directions disappear. Since in statistical terms (infinite medium), all the transverse directions are equiprobable, Poisson’s ratio along an arbitrary direction can be written as:

[1.29]

      A torsion test is now conducted between the directions x’ and y’ to determine the shear modulus by applying τ_x’y’:

[1.30]

The same relations between ε_ij and σ_kl (equations [1.4]–[1.7]) are valid, and the new relations that define the stresses are:

      [1.31]

      [1.32]

      [1.33]

      [1.34]

      [1.35]

      [1.36]

Inserting these relations into [1.30] and using [1.24] and [1.25] yield:

      [1.37]