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 τx’y’:
[1.30]
The same relations between εij and σkl (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]
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
