Mode I crack tip stress field: σᵢⱼ = K_I/(√(2πr)) · fᵢⱼ(θ). The plastic zone radius r_p = (K_I/σ_Y)²/2π.
30
300
0.030
Linear Elastic Fracture Mechanics (Irwin 1957): Near the crack tip (r → 0), stresses are singular:
σ_yy = K_I/√(2πr) · cos(θ/2)[1 + sin(θ/2)sin(3θ/2)] + O(√r).
Critical condition: fracture when K_I ≥ K_Ic (material toughness). K_I = σ√(πa) for center crack of half-length a.
Plastic zone (Irwin): r_p ≈ (1/2π)(K_I/σ_Y)² — the region where elastic solution breaks down.
Griffith energy criterion: G_c = K_Ic²/E (plane stress). Paris law: da/dN = C(ΔK)^m governs fatigue crack growth.
Mode I = opening; Mode II = sliding; Mode III = tearing. K_I dominates for most engineering cracks.