Читать книгу Crystal Elasticity - Pascal Gadaud - Страница 12
Оглавление
1
Macroscopic Elasticity: Conventional Writing
This chapter reviews the fundaments of classical crystal elasticity. It summarizes the already existing calculations that are scattered throughout the literature with very different notations. The written formalism presented here employs stiffnesses, which are less complex than compliances and better highlight crystal anisotropy. It is also important to note that the transition from theory to experimental applications requires several precautions.
1.1. Generalized Hooke’s law
The generalized Hooke’s law gives the linear relations between the components of stress (σij) and deformation (εij) by means of the factors of proportionality, which are the elastic constants (compliance tensor Cijkl or stiffness tensor Sijkl):
[1.1]
This is valid only under the hypothesis of small deformations. This tensor calculus (a fourth-order tensor having a priori 81 independent parameters) is applicable to any anisotropic crystal. Since tensors σij and εkl are symmetric, it can be shown that Cijkl=Cjikl=Cijlk. Moreover, since the tensor results from the double differentiation of interatomic potential energy, it is also true that Cijkl=Cklij. Consequently, the number of independent parameters is limited to a maximum of 21 for triclinic symmetry crystals, and this is even lower for higher degree of symmetry.
1.1.1. Cubic symmetry
Along an arbitrary direction x’ of the crystal, the Cartesian coordinates l, m and n are defined in the orthonormal reference system (x,y,z), which corresponds to the symmetry directions of the crystal of type <100>. They verify the following relation:
[1.2]
For this symmetry, there are only three independent elastic constants (S11, S12 and S44), and the deformations εx, εy, εz, γxy, γxz and γyz are classically defined with respect to the axes of the reference system. Stresses can be written in a very simple form, depending on the applied stress. Consider a traction test along x’ by applying a stress σx:
[1.3]
Using the elasticity matrix, the tensor calculus yields:
[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 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 x’ can therefore be written (by considering εx 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]
However, if all these transverse directions are considered individually, this value evolves angularly between two extrema:
[1.58]
[1.59]
According to the anisotropy, each one corresponds either to a maximum or a minimum. Therefore, for a rectangular section, for example, the weighting calculation of the directions in the formalism is needed, which additionally requires the knowledge about the orientations of the sides of the rectangle.
The problem can be solved by measuring Poisson’s ratio or the shear modulus only along the directions of type <100> or <111> in which these two quantities are anisotropic.
As far as the hexagonal symmetry is concerned, the same type of reasoning leads to conducting the measurements of these two quantities only along the direction z in which they are equally isotropic.
