Crystal Elasticity. Pascal Gadaud

the dynamic resonant method or Brillouin scattering.

      Due to very strong development, the mechanics of heterogeneous continuous media could transition from the elasticity of the monocrystal and its constants to the elasticity of aggregates (poly crystals). The historical mean-field models elaborated by Woldemar Voigt and Adolph Reuss were the basis for the exploration of numerous approaches. Finally, the recent numerical progresses allowed the optimization of the whole-field methods.

      Finally, the recent emergence of nanostructures required a zooming in on interatomic interactions in order to describe fine-scale elasticity or to predict the properties of massive materials using dynamic molecular methods.

      These approaches at various scales, as well as the amount of experimental and theoretical data or data resulting from simulations involving many categories of materials, make it impossible to draw an overall view that describes crystal elasticity in a simple manner.

This book proposes a phenomenological approach that addresses this problem: at the macroscopic scale, anisotropy and a dimensionless representation can be used to simplify the mathematical formulation of the classical crystal elasticity, which is reviewed in Chapter 1, and to classify crystal materials according to their anisotropy regardless of their rigidity. This approach is presented in Chapter 2 for cubic and hexagonal symmetries for which the amount of relatively reliable data is quite sufficient. In particular, although dimensionless crystal elasticity only involves the concept of anisotropy, it is still necessary to correctly describe it. At the end of the chapter, the case of sub-structures of cubic symmetry leads to a first-scale transition from the macroscopic level to the nanoscopic level, and therefore to the atomic scale. Based on the analogy of the behavior of a monocrystal at the two scales, Chapter 3 presents this transition that involves the writing of a new Hooke’s law (which we could call nano-Hooke’s law), which deals with spatial rigidity at the atomic level.

      A second-scale transition is then approached, which passes from monocrystal elasticity to polycrystal elasticity. Chapter 4 is dedicated to this subject. First, the historical cases of mean-field homogenization are approached, and then a much simplified homogenization is proposed, which allows for the phenomenological classification of these more or less various complex approaches found in the literature. It will be noted that anisotropy remains an essential parameter for comparing the elasticity of polycrystals and mechanical approaches.

      Finally, Chapter 5 is dedicated to illustrating specific cases of crystal elasticity using the data obtained using a specific high-performance vibration method. The description of this method will be followed by multiple examples of the evolution of elasticity in relation to the structural or physical aspects of functional materials.

      Robert Hooke’s illustration of the elasticity of metals (De potentia restitutiva, 1678)

The second part of this book describes a Lagrangian approach to vibrations. The aim is to propose a unique way to describe simple geometry vibrations.

      While a numerical calculation is essential for predicting the vibrations of complex structures under widely variable loadings based on the elasticity data of the materials composing these structures, a reverse approach is proposed here: based on the vibrations of structures and the simplest possible loads (plate torsion, beam and plate bending, traction–compression on a cylindrical rod), detected resonance frequencies can be used to retrieve elasticity data on the materials, whether crystalline or not. Four chapters are dedicated to this subject.

      Furthermore, a high-performance experiment must be set up, as described in the first part, and a proper formalism should be proposed to connect elasticity and vibrations. Various approaches developed in the 20th century can be found in the literature, but this part presents a unique approach that best addresses the problem. It involves writing the Lagrangian of a dynamic system and applying the Hamilton principle to minimize the energy.

      The case of massive, (multi)coated or gradient materials is presented to address the increasingly diverse current needs. Examples of experimental characterization are presented for various cases as a way to lighten and illustrate the various calculations.

      Finally, Chapter 10 focuses on the coupling between vibrations and internal macroscopic stresses, and proposes an alternative method for their analysis.

PART 1 Crystal Elasticity: Dimensionless and Multiscale Representation

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 (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