Crystal Elasticity. Pascal Gadaud

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      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 (σ_ij) and deformation (ε_ij) by means of the factors of proportionality, which are the elastic constants (compliance tensor C_ijkl or stiffness tensor S_ijkl):

      [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 σ_ij and ε_kl are symmetric, it can be shown that C_ijkl=C_jikl=C_ijlk. Moreover, since the tensor results from the double differentiation of interatomic potential energy, it is also true that C_ijkl=C_klij. 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 ε_x, ε_y, ε_z, γ_xy, γ_xz and γ_yz 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 σ_x:

[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, СКАЧАТЬ