Elasticity: Stress & Strain

Tensor Hooke's Law

The general linear elastic constitutive law:
σᵢⱼ = Cᵢⱼₖₗ εₖₗ

For isotropic materials (2D plane stress):
σₓₓ = E/(1−ν²) · (εₓₓ + ν εᵧᵧ)
σᵧᵧ = E/(1−ν²) · (εᵧᵧ + ν εₓₓ)
σₓᵧ = E/(1+ν) · εₓᵧ

Lamé parameters: λ = Eν/((1+ν)(1−2ν)), μ = E/(2(1+ν))
Bulk modulus: K = E/(3(1−2ν))
Shear modulus: G = μ

Von Mises Criterion & Elastic Energy

Von Mises stress predicts yielding:
σ_VM = √(σₓₓ²+σᵧᵧ²−σₓₓσᵧᵧ+3σₓᵧ²)

Yield occurs when σ_VM ≥ σ_yield. The criterion reflects that hydrostatic pressure alone cannot yield an isotropic material — only the deviatoric stress matters.

Elastic energy density:
W = ½ σᵢⱼ εᵢⱼ = ½ εᵢⱼ Cᵢⱼₖₗ εₖₗ

Total stored energy = ∫W dV — returned on unloading (perfectly elastic).