Crystal Elasticity

4Macroscopic Elasticity: From Monocrystal to Polycrystal

The passage from the elastic constants of a monocrystal to the polycrystal-characteristic elasticity data involves the transition from the writing of elasticity of a continuous medium to that of a heterogeneous medium composed of randomly oriented monocrystalline grains. This chapter focuses on several historical cases of mean-field homogenization among the multitude of existing approaches by inserting the simplified relations of Chapter 2 for elasticity data sets describing the monocrystal anisotropy.

4.1. Homogenization: several historical approaches and a simplified approach

The first to be presented is the Reuss model (1929), which involves a constraint homogenization, and it is applied to a cubic symmetry. Resuming equation [1.15], a random distribution of grain orientation yields the following value:

[4.1]

For the monocrystal, Σ_lmn varies from 0 to 1/3. Inserting this value, the following well-known relation is obtained:

[4.2]

If this writing is rendered dimensionless and the relations between constants established in the second chapter and the Zener anisotropy are added, then:

[4.3]

Similarly, for the shear modulus, ...