Near a crack tip, stresses are singular: σ_ij ~ K/√(2πr) as r→0. The stress intensity factor K characterizes the strength of the singularity and is the fundamental quantity in linear elastic fracture mechanics (LEFM).
Mode I (opening), Mode II (sliding), Mode III (tearing) have separate K values. Fracture occurs when K_I ≥ K_Ic (material toughness).
The plastic zone size r_p ≈ (K_I/σ_Y)²/2π determines when LEFM breaks down. The J-integral (path-independent) generalizes K to nonlinear materials.