Crystal Elasticity

3Crystal Elasticity: From Monocrystal to Lattice

When passing from macroscopic to atomic scale elasticity, it must be considered that during a simple test, such as traction, the mechanical behavior must be the same for any monocrystal, and for the lattice cell. Moreover, the description at the nanoscopic scale involves the well-known atomic springs, which represent the interatomic cohesion forces. This type of scale transition can be found in the literature. A first study related to the cubic and hexagonal symmetries addressed this problem by using central force interactions between nearest neighbors (Chen et al. 2015), while another study adding shear and torsion springs focused on isotropic materials (Zhang et al. 2014).

This chapter first presents this approach, which allows a first description of the elasticity at the atomic scale by formulating the problem in a simple manner, which will be useful later on.

3.1. Discrete representation

First, springs are inserted between nearest neighbors, as shown in Figure 3.1 for sc symmetry (atoms in green). k₁, k₂, k₃ represent the stiffnesses per unit length between nearest neighbors in the directions <100>, <110> and <111>, respectively. For the sake of clarity, the figure shows all the k₁ representing the cubic structure, and only one of the other types. When counting them, while apparently there are, respectively, 12, 12 and 8, actually there are 3, 6 and 8, the <100> edges belonging to 4 lattices, and the <110> diagonals belonging ...