Crystal Elasticity. Pascal Gadaud

      Similar to transverse deformations, the following relation is obtained along z’ that is orthogonal to x’ and y’:

      [1.38]

Calculating the half-sum (where directions are equally probable):

[1.39]

      1.1.2. Hexagonal symmetry

For this symmetry, there are now five independent elastic constants (S₁₁, S₁₂, S₁₃, S₃₃ and S₄₄). This symmetry is dealt with exactly the same as the cubic symmetry. For a traction test in the direction x’, relations [1.2] and [1.3] remain unchanged. However, Hooke’s law should be rewritten using this symmetry:

      [1.40]

      [1.41]

      [1.42]

      [1.43]

      [1.44]

[1.45]

      Young’s modulus along the direction x’ can therefore be written (by considering ε_x to be identical to equation [1.3]) as follows:

      [1.46]

      Moreover, with relation [1.2], the following expression can be obtained:

      [1.47]

It should be noted that S₁₂ is no longer present in the writing of Young’s modulus, but above all, l and m disappear; consequently, the modulus only depends on n, which is the sine of the angle between x’ and z. This reflects a well-known property of the elasticity of hexagonal symmetry, namely its transverse anisotropy (conventionally defined in the plane xy; this transverse anisotropy is in fact implicit in equation [1.45]).

      In order to study Poisson’s ratio, the same approach is taken for cubic symmetry. First, the transverse deformation along the direction y’ can be written as:

      [1.48]

      The expression in z’ is the same if the index ‘ is replaced by ‘‘.

      [1.49]

      And the resulting expression of Poisson’s ratio depends only on the angle between x’ and z:

      [1.50]

      Finally, the shear module during the same torsion test defined for cubic symmetry is written as.

      [1.51]

      Moreover, using the same approach yields:

      [1.52]

The expressions are somewhat complex for this symmetry, especially since while the transverse isotropy is quite visible, anisotropy is well hidden; this point will be revisited in the next chapter.

      The cases of quadratic and orthorhombic symmetries are beyond the scope of this chapter. As it will be noted in what follows, the amount of reliable data on these symmetries is insufficient.

1.2. Theory and experimental precautions

The first problem arises due to the fact that, when applying the formalism to experimental tests, a mathematical indeterminacy occurs. This is illustrated here by traction tests performed on monocrystals with cubic symmetry. Taking the first sample along the direction <100>, equation [1.15] yields:

[1.53]

      Taking the second sample in the direction of type <111> yields:

[1.54]

Since three constants need to be determined, measuring the module along the third direction brings no information, as it yields a linear combination of [1.53] and [1.54]. Therefore, another type of measurement should be considered. For example, if a torsion test is conducted along the direction <100>, equation [1.39] yields:

      [1.55]

      The indeterminacy is then lifted; S₁₁ and S₄₄ are determined by the first and third tests. S₁₂ can be deduced from the second test:

      [1.56]

The measurement precision required to correctly estimate the elastic constants can be easily imagined. The same problems are applicable to the hexagonal symmetry, while a priori two measurement directions are sufficient, but three different types of experiments are necessary.

      The second difficulty, certainly less known, arises from the strict application of the generalized Hooke’s law. As already noted, for Poisson’s ratio and the shear modulus, the fact of reasoning in an infinite medium (or a sphere) renders all the directions perpendicular to the stress direction equiprobable and simplifies the formalism. However, this formalism cannot be applied for the measurement of a sample of finite dimensions, unless it exclusively involves cylindrical rods.

Once again, consider the case of cubic symmetry and analyze Poisson’s ratio along the direction of type <100>. According to equation [1.29], the statistical value is:

      [1.57]