damage-mechanics

name	damage-mechanics
description	Continuum damage mechanics (CDM) — Lemaitre damage model, effective stress, damage variable D, ductile damage, fatigue damage, coupled damage-plasticity, progressive failure, XFEM, Gurson model.
metadata	{"priority":7,"promptSignals":{"phrases":["damage mechanics","continuum damage","CDM","Lemaitre damage","damage variable","ductile damage"],"minScore":3}}

Continuum Damage Mechanics (CDM) — Complete Skill

Fundamental Concepts

Damage variable D: scalar (isotropic damage) or tensor (anisotropic) representing density of micro-defects D = 0: virgin material (undamaged) D = 1: complete failure (fully damaged) D_critical = 0.2–0.5 (crack initiation, material-dependent)

Effective stress principle (Kachanov-Rabotnov): σ̃ = σ / (1 - D) [effective stress acting on undamaged cross-section] For σ_n = nominal stress and D = damage: σ̃ = σ_n / (1-D)

Effective strain equivalence (Lemaitre): ε_elastic = σ̃ / E = σ / (E(1-D)) Effective modulus: Ẽ = E(1-D) → can measure D by stiffness degradation: D = 1 - E_meas/E₀

Lemaitre Damage Model (Ductile)

Damage driving force (strain energy release rate): Y = σ_eq² / (2E(1-D)²) × [2/3(1+ν) + 3(1-2ν)(σ_H/σ_eq)²] σ_eq = von Mises equivalent stress; σ_H = hydrostatic stress; ν = Poisson's ratio R_v = 2/3(1+ν) + 3(1-2ν)(σ_H/σ_eq)² = triaxiality function

Damage evolution law: dD/dε_p = (Y/S)^s × 1/(1-D) [Lemaitre coupled model] Or simplified power law: dD/dε_p = (Y/S)^s

Material constants (Lemaitre): S = strain energy release rate threshold [MPa]; s = exponent (typically 1.0) Determine from notch tensile tests or round-robin specimens to fracture

Coupled damage-plasticity: ε̇_p = γ̇ × ∂f/∂σ_ij × 1/(1-D) [plastic flow modified by damage] Consistent with thermodynamics of irreversible processes

Kachanov-Rabotnov Creep Damage

Creep rate and damage coupled ODEs: dε/dt = A × (σ/(1-ω))^n [creep rate; ω = Kachanov damage variable] dω/dt = B × (σ/(1-ω))^χ [damage rate] ω = 0: undamaged; ω = 1: rupture

Rupture time: t_r = 1 / (B × (χ+1) × σ^χ) [for monotonic creep at constant σ; consistent with LMP]

Tertiary creep: modeled as accelerating ε̇ due to growing ω; S-curve creep behavior

Gurson-Tvergaard-Needleman (GTN) Model — Porous Plasticity

Yield function: Φ = (σ_eq/σ̄)² + 2q₁f* × cosh(3q₂σ_H / 2σ̄) - (1 + q₃f²) = 0 f = effective void volume fraction; σ̄ = matrix yield stress; q₁, q₂, q₃ = Tvergaard constants (q₁=1.5, q₂=1, q₃=q₁²)

Void growth: df_growth = (1-f) × dε_kk_plastic [volumetric strain of matrix → void expansion]

Void nucleation: df_nucl = A_N × dε_eq_p [strain-controlled nucleation] A_N = (f_N / S_N√(2π)) × exp(-0.5 × ((ε_p - ε_N)/S_N)²) Parameters: f_N = nucleation volume fraction; ε_N = mean nucleation strain; S_N = standard deviation

Coalescence (Tvergaard-Needleman): f* = f for f ≤ f_c (critical void fraction; typically 0.1–0.15) f* = f_c + (f_u* - f_c)/(f_F - f_c) × (f - f_c) for f > f_c [accelerated softening]

GTN constants for steel: q₁ = 1.5; q₂ = 1.0; f_c = 0.10–0.15; f_F = 0.20–0.25; ε_N = 0.3; S_N = 0.1; f_N = 0.04

Fatigue Damage (Chaboche)

Fatigue damage evolution (Chaboche 1988): dD/dN = [1 - (1-D)^(β+1) / (β+1)] × [ΔσD / M₀(1 - b_σ σ_m)]^β

where: D = cumulative fatigue damage; N = number of cycles; Δσ_D = stress amplitude; σ_m = mean stress M₀, β = material constants from S-N data; b_σ = mean stress sensitivity

D = 0 → N = 0 cycles; D = 1 → failure Integrating for constant amplitude: D = 1 - [1 - N/N_R]^(1/(β+1)) [N_R = cycles to failure] Nonlinear → captures loading sequence effects (unlike linear Miner's rule)

Miner's rule as special case: when β → ∞, Chaboche → Miner's rule (D = Σ n_i/N_i)

Progressive Failure in Composites

Hashin failure criteria (fiber composites): Fiber tension: (σ₁/X_T)² + (τ₁₂/S₁₂)² = 1 Fiber compression: σ₁ = -X_C Matrix tension: (σ₂/Y_T)² + (τ₁₂/S₁₂)² = 1 Matrix compression: (σ₂/2S₂₃)² + (Y_C/2S₂₃)² - 1 + (τ₁₂/S₁₂)² = 1

Progressive damage: on criterion met → degrade stiffness (ply discounting) E₁ → 0 (fiber failure); E₂, G₁₂ → 0 (matrix failure); gradual degradation via damage variable

XFEM (Extended FEM) for crack propagation: Enrichment functions near crack tip: singular fields captured without remeshing Level set method: φ(x,t) = 0 defines crack surface; ψ(x,t) = 0 defines crack front Cohesive zone: traction-separation law (bilinear); G_c = fracture toughness from test

FEA Implementation

Damage in ABAQUS: ABAQUS/Explicit: *DAMAGE INITIATION (Ductile, Shear, FLD criteria) + *DAMAGE EVOLUTION Ductile damage: η = σ_H/σ_eq (stress triaxiality); fracture strain ε_f(η) from experiment

Triaxiality-dependent fracture locus (Johnson-Cook): ε_f = [D₁ + D₂ × exp(D₃ × η)] × [1 + D₄ × ln(ε̇*)] × [1 + D₅ × T*] η = σ_H/σ_eq; D₁...D₅ = material constants from tests at varied triaxiality and strain rate

Standards

Standard	Scope
ASTM E647	Fatigue crack growth rate (da/dN basis)
ASTM E1820	J-integral fracture toughness
ISO 12135	Metallic materials — fracture toughness testing
ASTM E2472	Cohesive zone fracture calibration

Output

Provide: damage model (Lemaitre/GTN/Chaboche/Hashin), damage variable D at current state [0–1], effective stress σ̃ [MPa], effective modulus Ẽ [GPa] and damage from stiffness ratio, material constants (S, s for Lemaitre; q₁, f_c, f_F for GTN; D₁–D₅ for JC), critical damage D_c for crack initiation, void volume fraction f (GTN), fatigue damage D(N) vs. N cycles (Chaboche), triaxiality η = σ_H/σ_eq and fracture strain ε_f(η), XFEM crack tip enrichment zone size, progressive failure ply discount factors, and applicable standard (ASTM E1820, ASTM E647).