The constitutive relation (relation between stress and strain) in linear elasticity (known as the Hook’s law) is given by the proportionality between stress and strain, σ=Dε, where σ is a second-rank stress tensor, ε is a second-rank strain tensor and D is a fourth-rank tensor of elastic constants.
The simplest but most often implemented tensor of elastic constants (mainly for polycrystalline materials) is that of isotropic elasticity.
D=[D11D12D12000D12D11D12000D12D12D11000000D44000000D44000000D44],
in which
D11=E(1−ν)(1+ν)(1−2ν)D12=Eν(1+ν)(1−2ν)D44=E2(1+ν),
where E is the Young’s modulus and μ is the Poisson’s ratio. It is only these two constants sufficient for describing the isotropic behaviour.
This constitutive relation in 2D is simplified into
[σxxσyyτxy]=[C11C120C12C11000C44][εxxεyyγxy],
where the elastic constants C11, C12, and C44 are defined for
- plane strain
C11=E(1−μ)(1+μ)(1−2μ)C12=Eμ(1+μ)(1−2μ)C44=E2(1+μ).
- plane stress
C11=E1−μ2C12=Eμ1−μ2C44=E2(1+μ).
For an elastically isotropic material the following condition holds: C44=C11−C122. There is no directionality in the elastic properties. Polycrystalline materials may be considered isotropic as the sufficient amount of grains with a random (non-textured) grain orientation will average the elastic properties.