Theory & References
Theoretical Background
Classical Lamination Theory extends single-ply mechanics to multi-ply laminates using Kirchhoff plate assumptions. The key steps are:
Ply Stiffness (Q matrix)
For each ply, the reduced stiffness matrix [Q] is computed from the engineering constants (E₁, E₂, G₁₂, ν₁₂). This is a 3×3 matrix relating in-plane stresses to strains in the material coordinate system.
Transformed Stiffness (Q̄ matrix)
The [Q] matrix is transformed from material coordinates (1-2) to laminate coordinates (x-y) using the fiber orientation angle θ and standard transformation matrices [T].
ABD Assembly
The [A], [B], and [D] matrices are assembled by integrating the transformed stiffness [Q̄] through the thickness. Each ply contributes based on its [Q̄] and its z-coordinates (distance from the mid-plane).
Solve for Strains & Curvatures
The 6×6 ABD system is inverted to obtain the compliance matrix [abd]. Multiplying by the load vector {N, M} gives mid-plane strains {ε⁰} and curvatures {κ}.
Ply Strains & Stresses
Strains at any z-coordinate are computed as ε(z) = ε⁰ + z·κ. Laminate-coordinate stresses are σ = [Q̄]·ε. Material-coordinate values are obtained by rotating back through the ply angle.
Failure Evaluation
Material-coordinate stresses and strains are compared against allowables using the selected failure criteria. Margins of safety are computed for each ply at top, mid, and bottom locations.
Kirchhoff Assumptions
CLT is based on the following assumptions (Kirchhoff plate theory):
- The laminate is thin compared to its in-plane dimensions.
- Normals to the mid-surface remain straight, normal, and inextensible after deformation (no transverse shear deformation).
- Each ply is in a state of plane stress (σ₃ = τ₁₃ = τ₂₃ = 0).
- Ply properties are linearly elastic and uniform within each ply.
- Perfect bonding exists between adjacent plies (no slip or delamination).
Key References
- Jones, R.M., “Mechanics of Composite Materials,” 2nd Edition, Taylor & Francis, 1999.
- Tsai, S.W. and Hahn, H.T., “Introduction to Composite Materials,” Technomic, 1980.
- CMH-17 (Composite Materials Handbook), Volume 3 — Structural Analysis.
- Puck, A. and Schürmann, H., “Failure Analysis of FRP Laminates by Means of Physically Based Phenomenological Models,” Composites Science and Technology, 1998.
- Dávila, C.G., Camanho, P.P., and Rose, C.A., “Failure Criteria for FRP Laminates,” Journal of Composite Materials, Vol. 39, 2005 (LaRC03).
- Christensen, R.M., “The Theory of Materials Failure,” Oxford University Press, 2013.