Carbon Fiber Composites: from manual layup to FEA optimization

A laminate is not sheet metal

A carbon fiber composite is not an isotropic material like aluminum. It is a stack of unidirectional plies, each with extreme stiffness in the fiber direction (E1 ~ 120-180 GPa) and two orders of magnitude less in the transverse direction (E2 ~ 7-10 GPa). The stacking sequence determines everything: global stiffness, strength, failure mode, and weight.

The composite designer has too much freedom: choosing the number of plies, the orientation of each (0, 90, +45, -45, or any intermediate angle), the material of each ply, and the stacking sequence. Manual optimization becomes unfeasible beyond 10-15 plies. FEA allows systematic exploration of this design space.

Classical Lamination Theory (CLT)

CLT assumes each ply is in a plane-stress state and that strains are continuous through the thickness. The constitutive relationship of the complete laminate is:

{N}   [A  B] {epsilon_0}
{ } = [    ] {         }
{M}   [B  D] {kappa}

Where:

• A: membrane stiffness matrix (responds to in-plane loads)
• B: membrane-bending coupling matrix. If B != 0, a tensile load produces unwanted curvature
• D: bending stiffness matrix

A symmetric laminate ([0/90/45/-45]s) cancels B, eliminating parasitic couplings. This is why symmetric laminates are the industry standard.

Failure criteria

Unlike metals (Von Mises), composites require specific criteria that distinguish fiber direction, matrix, and compression/tension:

• Tsai-Wu: quadratic interactive criterion. Simple but does not distinguish failure modes
• Tsai-Hill: generalization of the Von Mises criterion to anisotropic materials
• Hashin 3D: distinguishes 4 failure modes (fiber/tension, fiber/compression, matrix/tension, matrix/compression). The most widely used in the aerospace industry
• Puck: based on fiber-matrix interfacial failure physics. More accurate, more parameters

# Hashin criterion for fiber failure in tension
def hashin_fiber_tension(sigma_11, sigma_12, sigma_13, Xt, S12, S13):
    return (sigma_11 / Xt)**2 + (sigma_12**2 + sigma_13**2) / S12**2

# where Xt = tensile strength in fiber direction
#       S12, S13 = interlaminar shear strengths

Delamination and interlaminar failure

The Achilles' heel of laminated composites: the resin between plies is weak in perpendicular tension (Z). Interlaminar stresses — shear tau_xz, tau_yz and normal stress sigma_zz — can initiate delamination even under low in-plane loads.

To capture this phenomenon we need 3D elements (solid-shell or brick) with at least one element per ply. Modeling a 20-ply laminate with solid elements implies meshes of easily 500k-1M elements for a simple part.

Alternative: cohesive elements (cohesive zone model, CZM) inserted between plies, enabling interlaminar crack initiation and propagation using bilinear traction-separation curves.

Topology + fiber orientation optimization

The state of the art combines two nested optimizations:

1. Topology: where to place material and where not (element density variables)
2. Fiber orientation: for each point with material, what is the optimal angle (continuous variable per element)

This generates biological-looking designs — with fibers flowing along principal load paths — reminiscent of wood, bone, or tendons. Biomimetics emerges spontaneously from the mathematical algorithm.

Experimental verification

We validate the FEA model with 4-point bending tests on 8-ply coupons (CFRP T700/M21, autoclave cured at 180C). The force-displacement correlation between FEA and test has R2 = 0.97 up to first failure load. The first-ply failure (FPF) load prediction is within +-5% of the experimental value.

Conclusions

FEA in composites is not an optional add-on: it is the only way to navigate the enormous design space that anisotropic materials offer. CLT gives a fast first approximation; 3D modeling with Hashin criteria and CZM captures real failure modes. Joint topology-orientation optimization promises designs no engineer would draw by hand.