| 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).
