| name | fea-fracture-mechanics |
| description | FEA fracture mechanics — singular crack-tip elements, J-integral domain method, SIF from FEA (K_I/K_II/K_III), VCCT, cohesive zone model, contour integral ABAQUS/ANSYS. |
| metadata | {"priority":7,"promptSignals":{"phrases":["FEA fracture mechanics","crack tip element","J-integral FEA","stress intensity factor FEA","VCCT","cohesive zone model","contour integral ABAQUS"],"minScore":3}} |
FEA Fracture Mechanics — Complete Skill
Crack-Tip Stress Singularity
Near crack tip, stresses follow r^(-1/2) singularity:
σ_ij = K_I / √(2πr) × f_ij(θ) [Mode I]
Standard isoparametric elements cannot capture this — they assume polynomial displacement fields
Singular Crack-Tip Elements
Quarter-Point Elements (QPE)
Concept: move mid-side nodes of 8-node serendipity quad (or 20-node hex) to quarter-point positions (0.25L from crack tip)
This produces r^(1/2) displacement variation → r^(-1/2) strain/stress singularity
2D degenerate triangle:
Collapse one side of 8-node quad; mid-side nodes at quarter-point
Results in r^(-1/2) singularity in all directions from collapsed edge
Construction:
- Ring of QPE (wedge/spider) around crack tip
- Next ring: standard elements
- Fan mesh: 8–16 elements around crack tip typical
Rules:
- QPE size r_tip ≈ a/8 to a/15 (a = crack length)
- Mesh at crack faces: compatible, no penetration (contact or duplicate nodes)
K Extraction from QPE Mesh
Displacement correlation method:
K_I = E'/(8π^0.5) × Σ[w_j × (v(r_j, π) - v(r_j, -π)) / r_j^0.5]
E' = E (plane stress); E' = E/(1-ν²) (plane strain)
v = crack-opening displacement (COD)
Extrapolate K from multiple r-values; use 2-4 points; linear extrapolation to r → 0
More precise: use 3-parameter fit: K = α₀ + α₁√r + α₂r
J-Integral Domain Method
Domain (Virtual Crack Extension) Form
J = ∫∫_A [(W δ₁ⱼ - σᵢⱼ ∂uᵢ/∂x₁) ∂q/∂xⱼ] dA
q(x) = smooth weight function: q = 1 at crack tip, q = 0 on outer contour of domain A
W = strain energy density
Advantages over contour integral:
- Domain integral is less sensitive to mesh quality
- Can use standard elements (no QPE needed) if domain is large enough
- Implemented in ABAQUS, ANSYS, MSC Nastran as "contour integral" request
Multiple contours (J_1, J_2, J_3): all should converge to same value; path-independence check
Deviation > 5% between contours → mesh too coarse or near strong singularity
Separating K_I, K_II, K_III from J
J = (K_I² + K_II²)/E' + K_III²/(2μ) [combined mode]
Interaction integral (M-integral):
K_I = E'/2 × M^(I) (auxiliary Mode I field)
Decomposes mixed-mode J into individual K contributions
Virtual Crack Closure Technique (VCCT)
Concept: energy release rate G equals work to close crack from current state
G_I = F_y^tip × v_tip / (2 × b × Δa)
F_y^tip = nodal force at crack tip node (mode I)
v_tip = crack face opening behind tip
b = element width (3D: out-of-plane thickness element behind tip)
Δa = crack advance step
Advantages: simple; no re-meshing; works with standard elements
Limitations: assumes self-similar crack growth; mode I only (unless extended)
Implementation (ABAQUS): DEBOND with VCCT or *FRACTURE CRITERION, TYPE=VCCT
ANSYS: CINT command with VCCT option; results as G_I, G_II, G_III per element
Cohesive Zone Model (CZM)
Traction-separation law:
Linear softening: T = K × δ (before damage); 0 (after)
Bilinear: T rises to T_max at δ_0, drops to zero at δ_f
Exponential (PPR, van den Bosch): physical for polymer fracture
Key parameters:
G_Ic = area under T-δ curve = ½ × T_max × δ_f [J/m²]
T_max = cohesive strength ≈ σ_ultimate of interface
δ_f = critical opening ≈ 2 × G_Ic / T_max
Mixed-mode (Benzeggagh-Kenane criterion):
G_total = G_Ic + (G_IIc - G_Ic) × (G_II/G_total)^η
Application: delamination in composites; adhesive joint failure; fatigue crack propagation
ABAQUS: COH3D8 / COH2D4 cohesive elements; or surface-based cohesion
ANSYS: INTER204/205 interface elements; CZM material model TBDATA
Extended FEM (XFEM) in Commercial Codes
ABAQUS XFEM: *ENRICHMENT, TYPE=STATIONARY/PROPAGATION
No conforming mesh to crack; crack tracked by level sets
K_I, K_II, K_III extracted from J via interaction integral
Automatic crack propagation in static (quasi-static) steps
ANSYS XFEM: SMART (Separating, Morphing, Adaptive, Remeshing Technology)
Automatic crack growth direction from maximum hoop stress criterion
Paris law for fatigue cycle-by-cycle crack growth
SIF Validation Benchmarks
Always validate against known solutions:
- Edge crack in tension: K_I = F_I × σ√(πa); F_I from Tada/Rooke-Cartwright
- Center crack: F_I = sec(πa/W)^0.5 (Isida) for W/a correction
- Compact Tension: K_I = (P/BW^0.5) × f(a/W); ASTM E399 expression
Target: < 2% difference from handbook for identical geometry
3D vs. 2D Analysis
Plane stress: thin plates, free surface condition; K plane stress ≈ K 2D
Plane strain: valid when B > 2.5(K_Ic/σ_y)² — gives lower bound K_Ic
3D: required for through-thickness variation (surface flaw, semi-elliptical crack)
K varies along crack front: K = K(s) where s = position along crack front
Fatigue Crack Growth in FEA
Paris law integration:
da/dN = C × (ΔK)^m
K from FEA at each crack increment → cycle integration → life N_f
NASGRO (AFGROW, FRANC3D): more accurate with threshold and R-ratio effects
Output
Provide: K_I, K_II, K_III [MPa√m] at key crack locations, J [kJ/m²], G_I, G_II, G_III [J/m²], path independence check (J contours), comparison to handbook solution, recommended element size at crack tip r_tip [mm], mesh type (QPE ring, XFEM, CZM), failure criterion (K_I vs. K_Ic, J vs. J_Ic), fatigue life N_f [cycles].
