The stress tensor σᵢⱼ describes force per unit area on each face of an infinitesimal element. For 2D plane stress: σ = [[σ_xx, τ_xy],[τ_xy, σ_yy]]. The principal stresses are eigenvalues of this tensor:
Mohr's Circle (Otto Mohr, 1882) geometrically represents how normal and shear stresses transform under coordinate rotation. The center is at (σ_mean, 0), radius R = max shear stress. Hooke's Law: ε_xx = (σ_xx − ν·σ_yy)/E, γ_xy = τ_xy/G where G = E/2(1+ν).
The von Mises yield criterion: material yields when √(σ₁² − σ₁σ₂ + σ₂²) ≥ σ_y. This is equivalent to the second invariant of the deviatoric stress tensor reaching a critical value — a sphere in principal stress space.