composites

name	composites
description	Composite materials and structures — CLT (Classical Laminate Theory), ply stiffness, laminate ABD matrix, Tsai-Wu/Tsai-Hill failure criteria, CFRP/GFRP properties, sandwich panels, interlaminar shear.
metadata	{"priority":8,"promptSignals":{"phrases":["composite","CFRP","GFRP","laminate","CLT","fiber reinforced","ply","Tsai-Wu","sandwich panel","carbon fiber"],"minScore":3}}

Composite Materials & Structures — Complete Skill

Constituent Properties

Micromechanics (Rule of Mixtures)

V_f = fiber volume fraction (typical: 0.5-0.65 for aerospace, 0.30-0.45 for commercial) V_m = 1 - V_f (matrix volume fraction)

Longitudinal modulus (fiber direction): E₁ = V_f × E_f + V_m × E_m [Rule of mixtures — exact]

Transverse modulus: E₂ = E_f × E_m / (V_m × E_f + V_f × E_m) [inverse ROM — approximate, lower bound] Better: Halpin-Tsai: E₂ = E_m(1 + ξη V_f)/(1 - η V_f), η = (E_f/E_m - 1)/(E_f/E_m + ξ), ξ = 2

Shear modulus: G₁₂ = G_f × G_m / (V_m G_f + V_f G_m) [inverse ROM] Poisson ratio: ν₁₂ = V_f ν_f + V_m ν_m [ROM] Density: ρ = V_f ρ_f + V_m ρ_m

Typical Ply Properties (Unidirectional)

Material	E₁ [GPa]	E₂ [GPa]	G₁₂ [GPa]	ν₁₂	ρ [kg/m³]	F₁t [MPa]	F₂t [MPa]
IM7/8552 CFRP	165	8.4	5.6	0.34	1570	2800	60
T300/914 CFRP	138	9.5	5.5	0.30	1580	1500	40
E-Glass/epoxy	38	8.3	4.1	0.26	2100	1062	31
S-Glass/epoxy	55	16	7.6	0.28	2000	1620	40
Kevlar 49/epoxy	76	5.5	2.3	0.34	1390	1400	12

Strengths: F₁t (longitudinal tension), F₂t (transverse tension), F₁c (longitudinal compression), F₂c (transverse compression), F₆ (in-plane shear)

Ply Stress-Strain (On-Axis, Principal Material Coordinates)

[σ₁] [Q₁₁ Q₁₂ 0 ] [ε₁] [σ₂] = [Q₁₂ Q₂₂ 0 ] [ε₂] [τ₁₂] [ 0 0 Q₆₆] [γ₁₂]

Q₁₁ = E₁/(1-ν₁₂ν₂₁), Q₂₂ = E₂/(1-ν₁₂ν₂₁) Q₁₂ = ν₁₂E₂/(1-ν₁₂ν₂₁) = ν₂₁E₁/(1-ν₁₂ν₂₁) Q₆₆ = G₁₂ ν₂₁ = ν₁₂ × E₂/E₁ [reciprocal relation]

Transformed Ply Stiffness (Off-Axis, Angle θ from x-axis)

Q̄ = T⁻¹ × Q × Tᵀ (plane stress transformation) m = cos θ, n = sin θ

Q̄₁₁ = Q₁₁m⁴ + 2(Q₁₂+2Q₆₆)m²n² + Q₂₂n⁴ Q̄₂₂ = Q₁₁n⁴ + 2(Q₁₂+2Q₆₆)m²n² + Q₂₂m⁴ Q̄₁₂ = (Q₁₁+Q₂₂-4Q₆₆)m²n² + Q₁₂(m⁴+n⁴) Q̄₆₆ = (Q₁₁+Q₂₂-2Q₁₂-2Q₆₆)m²n² + Q₆₆(m⁴+n⁴) Q̄₁₆ = (Q₁₁-Q₁₂-2Q₆₆)m³n - (Q₂₂-Q₁₂-2Q₆₆)mn³ Q̄₂₆ = (Q₁₁-Q₁₂-2Q₆₆)mn³ - (Q₂₂-Q₁₂-2Q₆₆)m³n

Classical Laminate Theory (CLT) — ABD Matrix

Laminate Stiffness Matrices

For N plies with midplane at z=0, ply k from z_{k-1} to z_k:

A_ij (extensional stiffness): A_ij = Σ_k Q̄_ij^(k) × (z_k - z_{k-1}) [N/m]

B_ij (bending-extension coupling): B_ij = ½ Σ_k Q̄_ij^(k) × (z_k² - z_{k-1}²) [N]

D_ij (bending stiffness): D_ij = ⅓ Σ_k Q̄_ij^(k) × (z_k³ - z_{k-1}³) [N·m]

Force-Moment to Strain-Curvature

{N} [A B] {ε°} {M} = [B D] {κ }

N = in-plane force resultant [N/m], M = moment resultant [N·m/m] ε° = midplane strains, κ = curvatures [1/m]

Inversion (solve for strains): {ε°} = [A*]{N} + [B*]{M} {κ} = [C*]{N} + [D*]{M} Where [A* B*; C* D*] = [A B; B D]⁻¹

Symmetric Laminates

[θ/-θ/0/90]_s notation: s = symmetric about midplane Symmetric → B_ij = 0 (no bending-extension coupling) Balanced: equal +θ and -θ plies → A₁₆ = A₂₆ = 0 (no shear-extension coupling) Quasi-isotropic [0/±60]_s or [0/90/±45]_s: A₁₁=A₂₂, A₁₆=A₂₆=0

Ply Stress Recovery

For given global strains {ε°, κ}: Ply k strain (at any z within ply): {ε_x,ε_y,γ_xy} = {ε°} + z{κ} Ply stress (global): {σ_x,σ_y,τ_xy} = Q̄^(k){ε} Ply stress (local): transform back to 1-2 axes using T matrix

Failure Criteria

Max Stress Criterion (Non-interactive)

Failure if ANY of: σ₁ > F₁t (or σ₁ < -F₁c), σ₂ > F₂t (or σ₂ < -F₂c), |τ₁₂| > F₆ Simple, non-conservative for off-axis loading

Tsai-Hill Criterion

(σ₁/F₁)² - σ₁σ₂/F₁² + (σ₂/F₂)² + (τ₁₂/F₆)² = 1 (failure when = 1) F₁ = F₁t if σ₁ > 0, else F₁c; F₂ = F₂t if σ₂ > 0, else F₂c

Tsai-Wu Criterion (Most comprehensive)

F₁σ₁ + F₂σ₂ + F₁₁σ₁² + F₂₂σ₂² + F₆₆τ₁₂² + 2F₁₂σ₁σ₂ = 1

F₁ = 1/F₁t - 1/F₁c, F₂ = 1/F₂t - 1/F₂c F₁₁ = 1/(F₁t×F₁c), F₂₂ = 1/(F₂t×F₂c), F₆₆ = 1/F₆² F₁₂ = -½√(F₁₁×F₂₂) [conservative estimate; from biaxial test ideally]

Strength Ratio R: RF × {σ} = failure; R = 1/Σ(F terms) [how close to failure]

Hashin Criteria (Mode-specific, most physically correct)

Fiber tension (σ₁ > 0): (σ₁/F₁t)² + (τ₁₂/F₆)² ≥ 1 Fiber compression (σ₁ < 0): σ₁/F₁c ≥ 1 Matrix tension (σ₂ > 0): (σ₂/F₂t)² + (τ₁₂/F₆)² ≥ 1 Matrix compression (σ₂ < 0): (σ₂/2F₂c)² + [(F₂c/2F₂c)²-1]σ₂/F₂c + (τ₁₂/F₆)² ≥ 1

Interlaminar Shear (Short-Beam Shear Test — ASTM D2344)

τ_ILS = 0.75 × F_max / (b × h) [short-beam, 3-point, L/h=5] Typical ILSS: CFRP 60-90 MPa; GFRP 25-35 MPa Interlaminar failure: delamination (critical for thick laminates, impact damage)

Sandwich Panels

Core: foam (Rohacell, Divinycell), honeycomb (Al, Nomex) Facesheets: thin laminates carry in-plane loads (membrane + bending)

Flexural rigidity: D_sandwich = E_f × t_f × (t_f + t_c)² / 2 [per unit width, symmetric] t_f = facesheet thickness, t_c = core thickness

Core shear: τ_core = V/(t_c × b) ≤ G_core × γ_allow Facesheet wrinkling (local buckling): σ_wr = 0.5(E_f × E_c × G_c)^(1/3) ← critical

Design Rules

• 10% rule: minimum 10% of plies in each of 4 directions (0°, ±45°, 90°)
• No more than 4 consecutive plies of same orientation
• Taper: drop no more than 1 ply per 6mm (avoid interlaminar stress concentration)
• Impact damage: barely visible impact damage (BVID) tolerance required (aerospace)
• Repair: scarf repair for strength-critical; patch/bolted for fatigue

Output

Provide: E₁, E₂, G₁₂, ν₁₂ for laminate (from CLT A matrix), A/B/D matrices, ply-by-ply stress in 1-2 frame, first-ply-failure (FPF) load from Tsai-Wu, safety reserve R, delamination risk assessment.