Composite laminae, laminates and classical lamination theory

Why it matters

Half the structural weight of a modern airliner (wing boxes, fuselage barrels, tails) is carbon-fibre composite. A composite part is built from thin unidirectional layers, each very stiff and strong along its fibres and weak across them, stacked at chosen angles. Classical lamination theory (CLT) tells you how the stacking sequence sets the laminate's stiffness and couplings, and lets you recover the stress in every ply to check it for failure.

Key ideas

Lamina (ply). A single layer of parallel fibres in a matrix, typically 0.125–0.25 mm thick. In its own axes (1 along the fibres, 2 transverse in the plane) it is orthotropic, described by four independent in-plane constants: E₁, E₂, G₁₂ and ν₁₂ (with ν₂₁ = ν₁₂·E₂/E₁).

Micromechanics (rule of mixtures). With fibre volume fraction V_f and matrix fraction V_m = 1 − V_f:

• Along the fibres the constituents strain equally (parallel model), so E₁ = E_f·V_f + E_m·V_m, and ν₁₂ follows the same rule. This is accurate.
• Across the fibres they carry equal stress (series model), so 1/E₂ = V_f/E_f + V_m/E_m and similarly for G₁₂. This is a lower bound and underestimates; semi-empirical corrections (Halpin–Tsai) are used in practice. Carbon fibres are themselves anisotropic, so their transverse modulus, not the axial one, should be used for E₂; measured ply data from the material supplier are always preferred.

Ply stress–strain law. For a thin ply in plane stress, {σ₁, σ₂, τ₁₂} = [Q]{ε₁, ε₂, γ₁₂} with the reduced stiffnesses Q₁₁, Q₂₂, Q₁₂, Q₆₆ below. For a ply at angle θ to the laminate x-axis, the transformed stiffness [Q̄] is obtained by rotating [Q]; it generally has non-zero Q̄₁₆ and Q̄₂₆, which couple shear with extension.

Laminate. Plies are bonded and assumed to strain together (Kirchhoff hypothesis): ε = ε⁰ + z·κ, where ε⁰ are mid-plane strains and κ curvatures. Integrating ply stresses through the thickness gives the force resultants N (N/m) and moment resultants M (N) in terms of the A, B and D matrices:

• [A]: in-plane (extensional) stiffness.
• [B]: coupling between extension and bending. Zero for a symmetric laminate (plies mirrored about the mid-plane).
• [D]: bending stiffness, dominated by plies far from the mid-plane.

Stacking rules.

• Symmetric (e.g. [0/±45/90]s): B = 0, so the laminate does not bend or twist when stretched, and does not warp on cooling from cure.
• Balanced (every +θ has a −θ): A₁₆ = A₂₆ = 0, so no shear–extension coupling.
• Quasi-isotropic layups ([0/±45/90]s, [0/±60]s) have isotropic A but not isotropic D.
• Aircraft practice: at least 10% of plies in each of 0°, ±45° and 90°, ±45° plies on the outer surfaces for impact and buckling, and avoid more than about four identical plies in a row.

Ply stresses and failure. From N and M, invert to get ε⁰ and κ, compute the strain at each ply's height, transform to ply axes and multiply by [Q]. Compare with ply strengths X (fibre), Y (transverse) and S (shear) using a criterion: maximum stress, maximum strain, Tsai–Hill or Tsai–Wu. The first ply to reach its criterion defines first-ply failure, often transverse cracking of the 90° plies. Strengths are material data; take them from the supplier or design allowables.

Formulas

E₁ = E_f·V_f + E_m·V_m and ν₁₂ = ν_f·V_f + ν_m·V_m

• E: moduli (Pa), V: volume fractions, subscripts f fibre and m matrix.

1/E₂ = V_f/E_f + V_m/E_m and 1/G₁₂ = V_f/G_f + V_m/G_m

• Transverse and in-plane shear moduli (Pa); series model, lower bound.

ν₂₁ = ν₁₂·E₂ / E₁

• Reciprocal relation.

Q₁₁ = E₁/(1 − ν₁₂·ν₂₁), Q₂₂ = E₂/(1 − ν₁₂·ν₂₁), Q₁₂ = ν₁₂·E₂/(1 − ν₁₂·ν₂₁), Q₆₆ = G₁₂

• Reduced stiffnesses of a ply in plane stress (Pa).

{N} = [A]{ε⁰} + [B]{κ} and {M} = [B]{ε⁰} + [D]{κ}

• N: force per unit width (N/m), M: moment per unit width (N), ε⁰: mid-plane strains, κ: curvatures (1/m).

A_ij = Σ Q̄_ij,k·(z_k − z_(k−1)), B_ij = ½·Σ Q̄_ij,k·(z_k² − z_(k−1)²), D_ij = ⅓·Σ Q̄_ij,k·(z_k³ − z_(k−1)³)

• z_k: coordinate of the top of ply k measured from the mid-plane (m).

E_x = (A₁₁·A₂₂ − A₁₂²) / (A₂₂·h)

• Effective in-plane modulus (Pa) of a symmetric, balanced laminate of thickness h (m).

(σ₁/X)² − σ₁·σ₂/X² + (σ₂/Y)² + (τ₁₂/S)² = 1

• Tsai–Hill failure criterion; X, Y, S: ply strengths (Pa) for the sign of stress present.

Worked examples

Example 1 (standard): glass/epoxy ply properties. Given: E_f = 72 GPa, ν_f = 0.22, G_f = 30 GPa; E_m = 3.5 GPa, ν_m = 0.35, G_m = 1.3 GPa; V_f = 0.6.

1. E₁ = 72 × 0.6 + 3.5 × 0.4 = 43.2 + 1.4 = 44.6 GPa.
2. 1/E₂ = 0.6/72 + 0.4/3.5 = 0.00833 + 0.11429 = 0.12262, so E₂ = 8.16 GPa.
3. ν₁₂ = 0.22 × 0.6 + 0.35 × 0.4 = 0.272.
4. 1/G₁₂ = 0.6/30 + 0.4/1.3 = 0.0200 + 0.3077 = 0.3277, so G₁₂ = 3.05 GPa. Answer: E₁ = 44.6 GPa, E₂ = 8.16 GPa, ν₁₂ = 0.272, G₁₂ = 3.05 GPa; the ply is about 5.5 times stiffer along the fibres.

Example 2 (GATE level): [0/90]s cross-ply laminate. Given: carbon/epoxy ply E₁ = 140 GPa, E₂ = 10 GPa, G₁₂ = 5 GPa, ν₁₂ = 0.3, ply thickness 0.125 mm; laminate [0/90]s (four plies, h = 0.5 mm) under N_x = 100 N/mm.

1. ν₂₁ = 0.3 × 10/140 = 0.02143; 1 − ν₁₂·ν₂₁ = 0.99357.
2. Q₁₁ = 140.9 GPa, Q₂₂ = 10.06 GPa, Q₁₂ = 3.02 GPa, Q₆₆ = 5.0 GPa.
3. Two 0° and two 90° plies, each 0.125 mm: A₁₁ = A₂₂ = 0.25 × (140 906 + 10 065) = 37 743 N/mm, A₁₂ = 0.5 × 3019 = 1510 N/mm, A₆₆ = 0.5 × 5000 = 2500 N/mm. Symmetric, so B = 0.
4. Effective modulus: E_x = (37 743² − 1510²)/(37 743 × 0.5) = 75.4 GPa.
5. Strains: ε_x = A₂₂·N_x/(A₁₁·A₂₂ − A₁₂²) = 2.654 × 10⁻³, ε_y = −A₁₂·N_x/(A₁₁·A₂₂ − A₁₂²) = −1.06 × 10⁻⁴.
6. 0° ply: σ₁ = Q₁₁·ε_x + Q₁₂·ε_y = 373.6 MPa, σ₂ = Q₁₂·ε_x + Q₂₂·ε_y = 6.9 MPa.
7. 90° ply (its fibres along y): σ₁ = Q₁₂·ε_x + Q₁₁·ε_y = −6.9 MPa, σ₂ = Q₂₂·ε_x + Q₁₂·ε_y = 26.4 MPa.
8. Check: (373.6 + 26.4) × 2 × 0.125 = 100 N/mm = N_x. Answer: E_x = 75.4 GPa; 0° plies σ₁ = 374 MPa; 90° plies σ₂ = 26.4 MPa transverse tension. With a typical transverse strength of a few tens of MPa (a material allowable), the 90° plies would crack first, long before the fibres fail.

Common mistakes

• Using the series (inverse) rule for E₁ or the parallel rule for E₂.
• Forgetting the reciprocal relation: ν₂₁ is not equal to ν₁₂.
• Using E₁ in place of Q₁₁; the difference is small but Q₁₂ ≠ ν₁₂·E₁.
• Assuming B = 0 for an unsymmetric laminate such as [0/90]: it bends when stretched and warps after cure.
• Confusing ply axes (1, 2) with laminate axes (x, y) when recovering stresses.
• Calling a [0/±45/90]s laminate fully isotropic; only its in-plane stiffness is.

For GATE AE

Expect rule-of-mixtures moduli, reciprocal Poisson's ratio, the meaning of A, B and D, symmetric and balanced layups, reduced stiffness terms, and simple cross-ply stiffness or strain calculations. Practise example 2 until computing A and inverting a 2 × 2 block is quick.

Quick check

1. What does B = 0 mean physically, and which layups give it?
2. A ply has E₁ = 150 GPa, E₂ = 9 GPa, ν₁₂ = 0.3. What is ν₂₁?
3. Which plies are most effective for bending stiffness?
4. Which ply usually fails first in a cross-ply laminate loaded along 0°? Answers: 1. no coupling between stretching and bending, given by symmetric layups; 2. 0.3 × 9/150 = 0.018; 3. the outermost plies (D grows with z³); 4. the 90° plies, by transverse cracking.