Laminate Progressive Failure

Progressive Failure of a Composite Laminate (structures/composites/laminate-progressive-failure)

Use when the task is the multi-event march of a symmetric balanced composite laminate past first-ply failure to last-ply failure by the ply-discount method: at each ply event, degrade the stiffness of the failed ply, reassemble the in-plane laminate stiffness, reapply the load resultant and continue until no load-bearing stiffness remains. This leaf implements the strength-and-stiffness-evolution chain in pure Python, stdlib only. It pairs with structures/composites/laminate-first-ply-failure, which owns the first-event analysis under its own linearized reserve-factor convention, and reuses the sibling's exact Tsai-Wu index closed form for the per-event indices.

Domain quick reference

• Loads: in-plane resultants (Nx, Ny, Nxy) in N/mm; stresses, moduli and allowables in MPa; ply thickness in mm; strains dimensionless; A-block entries in N/mm.
• Plane-stress ply stiffness: q11 = E1/(1 - nu12 nu21), q22 = E2/(1 - nu12 nu21), q12 = nu12 E2/(1 - nu12 nu21), q66 = G12, with nu21 = nu12 E2/E1.
• In-plane laminate block: A_ij = sum over plies of Qbar_ij(theta_k) t_k, over the plies' CURRENT (possibly degraded) constants, with Qbar the standard fourth-power rotation of the ply stiffness. The full 3x3 block (A11, A12, A16, A22, A26, A66) is inverted in closed form for the mid-plane strains {eps} = [A]^-1 {N}. laminate_a_matrix and midplane_strains carry this step.
• Tsai-Wu index: FI = F1 s1 + F2 s2 + F11 s1^2 + F22 s2^2 + F66 t12^2
  • 2 F12 s1 s2 with F1 = 1/Xt - 1/Xc, F2 = 1/Yt - 1/Yc, F11 = 1/(Xt Xc), F22 = 1/(Yt Yc), F66 = 1/S^2, F12 = -0.5 sqrt(F11 F22). FI >= 1.0 marks ply failure. per_ply_failure_indices evaluates it on the intact laminate.
• Event convention: within a load segment every active ply's index is quadratic in the load scale lambda. A ply event is the smallest positive root of FI_k(lambda) = 1, the exact quadratic root, never the sibling's linearized reserve scale k* = 1/max(FI).
• Ply-discount reduction: a matrix-mode failure zeroes E2, G12 and nu12 (E1 retained, the ply becomes a unidirectional tape and can later fail again in fiber mode); a fiber-mode failure zeroes E1 as well. Mode discrimination: fiber mode iff the fiber stress s1 >= Xt or s1 <= -Xc at the event, else matrix mode. ply_discount_mode carries this rule.
• Termination: last-ply-failure when no ply retains stiffness or the reassembled A block loses positive definiteness; strain-limit when an optional maximum active-ply fiber strain binds first; no-further-ply-failure when the load direction never drives the laminate to failure.
• CMH-17 frames the ply allowables and lamina data context, FAR-25 the certification framing; the relations above are standard mechanics, summary-only.

Workflow

1. Fix the material and stack: engineering constants E1, E2, nu12, G12 and the Tsai-Wu allowables Xt, Xc, Yt, Yc, S in MPa, the ply angles and thickness in mm. Build the intact laminate block with laminate_a_matrix.
2. Recover the mid-plane strains and the per-ply Tsai-Wu indices of the intact laminate with per_ply_failure_indices (the sibling-parity oracle surface, used to confirm the event convention against the sibling's own index machinery).
3. Run the ply-discount march with progressive_failure_march at the chosen reference resultant (Nx, Ny, Nxy); internally it re-derives the quadratic event root at every active ply each segment and classifies each failing ply's mode with ply_discount_mode.
4. Read the failure sequence from the report dict: each event's load, failed plies and modes, the degraded laminate stiffness a_after, the first-ply-failure event (fpf_load_nx, fpf_plies, fpf_modes), the ultimate (last-ply-failure) load and the post-FPF reserve factor (ultimate / FPF).
5. For the unidirectional reduction check or a strain-governed design, run the march on a [0]n stack, with or without an optional strain_limit; the march collapses to the sibling's FPF identity when the stack is unidirectional.
6. Confirm the deterministic checks with the contract test scripts/test_laminate_progressive_failure.py.

Worked example

High-strength carbon/epoxy: E1 = 181 GPa, E2 = 10.3 GPa, G12 = 7.17 GPa, nu12 = 0.28, with T800H-class design allowables Xt = 2700 MPa, Xc = 2000 MPa, Yt = 40 MPa, Yc = 246 MPa, S = 68 MPa. QI stack [0/90/45/-45]s, 8 plies at 0.125 mm (h = 1.0 mm), uniaxial resultant Nx only. Real module outputs:

• Intact assembly: A11 = A22 = 76368.2177 N/mm, A12 = 22607.3555 N/mm, A66 = 26880.4311 N/mm, with A16 and A26 vanishing to machine precision (5.27e-14 and 2.17e-12 N/mm).
• Event 1 (the FPF event) at Nx = 276.6979 N/mm, plies [1, 6] (the two 90-degree plies), modes matrix, matrix: the transverse stress reaches the Tsai-Wu unity boundary first (a Yt-class transverse failure). Degraded A11 falls to 73781.6780 N/mm while A22 stays near 76165.4330 N/mm, the 90-degree fibers still stiffening the transverse direction as a tape.
• Event 2 at Nx = 347.8073 N/mm, plies [2, 3, 4, 5] (the four +/-45-degree plies), all matrix mode: the angle plies fail under their combined transverse and shear stress, joining the tapes.
• Event 3 at Nx = 925.6800 N/mm, plies [0, 7] (the two 0-degree plies), both fiber mode at s1 = Xt = 2700 MPa: this sets the ultimate load, since only the retained-E1 tapes carry past it.
• Event 4 at the SAME load Nx = 925.6800 N/mm, plies [1, 2, 3, 4, 5, 6] (the six tapes), all fiber mode: a load-controlled cascade, the 90-degree tapes in fiber compression and the 45-degree tapes in fiber tension fail instantly, zeroing the laminate stiffness.
• Summary: fpf_load_nx = 276.6979 N/mm, ultimate_load_nx = 925.6800 N/mm, post_fpf_reserve_factor = 3.3455, fpf/ultimate ratio 0.2989: the QI laminate survives to 3.35 times its first-ply-failure load before the 0-degree fibers break.
• [0]8 reduction identity: a single fiber event at Nx = Xt h = 2700.0 N/mm, FPF = last-ply = ultimate, post-FPF reserve factor 1.0. With strain_limit = Xt/(2 E1) = 0.0074586, the march halts at Nx = 1350.0 N/mm with zero ply events, reason "strain-limit".
• Event-convention fence (T300/5208 QI reprise, Xt = Xc = 1500 MPa): this leaf's exact first event sits at 276.1190 N/mm, not the sibling's linearized k* load of 319.4818 N/mm (ratio 1.1570); the sibling's own index function, re-evaluated at this leaf's event load, returns max index 1.0, confirming the index-oracle equivalence fence.

Verification

• Confirm the QI march (allowables above) returns fpf_load_nx about 276.70 N/mm, ultimate_load_nx about 925.68 N/mm, post_fpf_reserve_ factor about 3.3455 and termination "last-ply-failure" with the four events above.
• Confirm the QI magnitude gate: 0.20 <= fpf_load_nx / ultimate_load_nx <= 0.35 (real value 0.2989).
• Confirm the [0]8 reduction identity (single fiber event at Xt h, post-FPF reserve factor 1.0) and the strain-limit branch (halts at half that load with zero events when the limit binds).
• Confirm the degraded A11, A22 and A66 are non-increasing event to event, and that A16, A26 stay below 1e-6 N/mm absolute on the intact balanced symmetric stack.
• Confirm ply_discount_mode classifies fiber mode at and beyond the Xt or -Xc boundary, matrix mode otherwise.
• Confirm every non-physical input (non-positive engineering constant, singular Poisson product, empty ply stack, non-positive thickness, non-positive allowable, all-zero reference resultant, non-positive strain limit) raises ValueError.
• Run the contract test offline: python3 scripts/test_laminate_progressive_failure.py (29 tests, deterministic).

Related leaves

• structures/composites/laminate-first-ply-failure: the first-event analysis under the linearized k* = 1/max(FI) reserve convention; this leaf reports the same first event only as the first step of its march, at index unity.
• structures/composites/failure-criteria: the single-ply strength criteria verdict from given stresses, no laminate assembly or stiffness evolution.
• structures/composites/delamination-growth: energy-based delamination onset and growth, a different failure class from this leaf's stress-based ply-discount march.
• structures/composites/laminate-stiffness: the symmetric-laminate A synthesis this leaf reassembles after every discount, without any strength content of its own.

Pitfalls

• Reporting the linearized k* load as the march's first event: the sibling's k* = 1/max(FI) is a linearized reserve factor, exact only at the failure boundary; this leaf's event is the exact quadratic root of FI(lambda) = 1, and the two loads differ by a real, measurable factor (1.1570 in the T300/5208 worked case).
• Discounting a matrix failure to zero: a matrix-mode event retains E1, so the failed ply keeps carrying fiber-direction load as a tape and still stiffens the laminate (A11 stays near 73781.68 N/mm after event 1, not zero); only a later fiber-mode event zeroes it completely.
• Treating same-load events as one event: events 3 and 4 of the worked example share the applied load (925.68 N/mm) but are two distinct events, the 0-degree fiber failure followed by the load-controlled cascade of the remaining tapes at the unchanged load.
• Feeding an unbalanced or unsymmetric stack: the full 3x3 in-plane block handles general coupling, but the worked identities (A16, A26 vanishing, the [0]8 reduction) assume a balanced symmetric layup; the report still returns a well-posed march for other stacks as long as the reassembled A block stays positive definite.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_laminate_progressive_failure.py

The test covers the QI worked example (intact A block, the four sequential-ply-failure events with their loads, failed plies and modes, the degraded laminate stiffness after each event, the first-ply-failure load, the ultimate laminate load and the post-FPF reserve factor), the QI magnitude gate, the [0]8 reduction identity and its strain-limit branch, the sibling-oracle per-ply index parity and the event-convention fence, mode discrimination at the fiber boundary, degraded-stiffness monotonicity, the exact report and event key sets, determinism, and ValueError rejection of non-physical inputs.

Compliance

• Standards referenced, not reproduced: CMH-17 (Composite Materials Handbook, SAE) frames the ply allowables and lamina data context; FAR-25 (14 CFR Part 25) frames the airframe certification context; the ply-discount march is standard mechanics, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.