Laminate Hygrothermal Response

Laminate Hygrothermal Response (structures/composites/laminate-hygrothermal-response)

Use when the task is the hygrothermal response of a composite laminate: equilibrium moisture content from the ambient relative humidity, stiffness-weighted laminate thermal and moisture expansion coefficients by classical lamination theory (CLT), the hygrothermal strain from a temperature change plus a moisture change, and the residual strain from the cure-cooldown temperature drop. This leaf implements the exact CLT free-expansion solution in pure Python, stdlib only, deterministic and offline. Material properties (ply stiffness, ply expansion coefficients, saturation moisture content) are inputs with documented typical bounds; the code path is exact CLT arithmetic with no empirical fits. It pairs with structures/composites/laminate-stiffness, whose Qbar assembly this leaf reuses for the thermal and moisture weighting.

Domain quick reference

• Linear isotherm: equilibrium moisture content M = m_sat * rh, where m_sat is the saturation mass fraction (default 0.015, published CFRP magnitudes 0.01-0.02) and rh the ambient relative humidity fraction.
• Plane-stress reduced stiffness: q11 = e1/d, q22 = e2/d, q12 = nu12e2/d = nu21e1/d, q66 = g12, with nu21 = nu12e2/e1 and d = 1 - nu12nu21.
• Qbar rotation with m = cos(theta), n = sin(theta): qbar11 = q11m^4 + 2(q12 + 2q66)m^2n^2 + q22n^4, qbar22 = q11n^4 + 2(q12 + 2q66)m^2n^2 + q22m^4, qbar12 = (q11 + q22 - 4q66)m^2n^2 + q12(m^4 + n^4), qbar66 = (q11 + q22 - 2q12 - 2q66)m^2n^2 + q66*(m^4 + n^4), qbar16 = (q11 - q12 - 2q66)m^3n + (q12 - q22 + 2q66)mn^3, qbar26 = (q11 - q12 - 2q66)mn^3 + (q12 - q22 + 2q66)m^3n.
• Free-expansion CLT for a symmetric balanced laminate: assemble the 2x2 in-plane stiffness A = sum_k Qbar_k * t_k (rows qbar11, qbar12; qbar12, qbar22; shear row dropped) and the thermal force resultant per unit temperature Nth = sum_k Qbar_k * [alpha_x_k, alpha_y_k]^T * t_k, then solve A * [alpha_x, alpha_y]^T = Nth by the 2x2 determinant inversion. The moisture vector uses [beta_x_k, beta_y_k] the same way with the moisture force resultant Nm.
• The exact 2x2 inversion is REQUIRED: the simplified stiffness-weighted scalar ratio (Nth_x/A11 alone) returns 1.65 ppm for the worked [0/90]s, a wrong value, and fails the unidirectional identity. Only the full inversion returns alpha_1 exactly for a 0-deg unidirectional laminate.
• Hygrothermal strain: eps = alphadelta_t + betadelta_m per laminate axis. Cure-cooldown strain: alpha_x * (t_rt - t_cure), negative for a cooldown with positive alpha.
• Coefficients are raw SI in the module (alpha in 1/K, beta per unit moisture mass fraction); cte_ppm(alpha) = alpha * 1e6 is the reporting helper.
• Transport aeroplane structural context sits in FAR-25 (reference only; the relations above are standard mechanics, summary-only).

Workflow

1. Gather the ambient relative humidity and the ply material properties: e1, e2 (Pa), nu12, g12 (Pa), alpha_1 and alpha_2 (1/K), beta_1 and beta_2 (per unit moisture fraction), ply angle theta_deg and ply thickness t (m). Use equilibrium_moisture_content for M at the given rh with m_sat (default 0.015).
2. Build the plane-stress stiffness of one ply with plane_stress_q and rotate it to the laminate axes with qbar.
3. Assemble the symmetric balanced laminate and solve for the laminate coefficients with laminate_cte_cme, which returns alpha_x, alpha_y, beta_x and beta_y by the exact 2x2 CLT inversion.
4. Apply the temperature and moisture changes with hygrothermal_strain, or report the residual cure strain with cure_cooldown_strain from the cure temperature and room temperature (21 C default).
5. For a one-call assessment run laminate_hygrothermal_response with the ply list, rh_fraction and delta_t_k; delta_m defaults to the equilibrium moisture content and t_cure_c adds the cure branch.
6. Report coefficients with cte_ppm where parts per million read better than raw 1/K, then confirm the deterministic checks with the contract test.

Worked example

T300/5208-style carbon/epoxy [0/90]s symmetric balanced laminate, ply t = 0.125e-3 m each: e1 = 181e9 Pa, e2 = 10.3e9 Pa, nu12 = 0.28, g12 = 7.17e9 Pa, alpha_1 = -0.3e-6/K, alpha_2 = 28.1e-6/K, beta_1 = 0, beta_2 = 0.6 per unit moisture fraction. Ambient rh = 0.6, m_sat = 0.015, cure 177 C, room 21 C. Real module outputs:

• equilibrium_moisture_content(0.6) = 0.009 (0.015 * 0.6).
• plane_stress_q: q11 = 181.81 GPa, q22 = 10.35 GPa, q12 = 2.90 GPa, q66 = 7.17 GPa.
• laminate_cte_cme: alpha_x = alpha_y = 1.59998e-6/K (1.60 ppm, inside the 1.55-1.70 ppm band); beta_x = beta_y = 0.04014 per unit moisture fraction (the CLT moisture solution for beta_1 = 0, beta_2 = 0.6). The simplified q11-only scalar ratio gives 1.65 ppm, outside the exact CLT result, and must not be used.
• Cure-cooldown strain: alpha_x * (21 - 177) = -2.496e-4 (about -2.50e-4) after the -156 K cooldown.
• Moisture branch: hygrothermal_strain at delta_m = 0.009 gives beta_x * 0.009 = 3.613e-4 (about 3.6e-4); published CFRP swelling strain at saturation is roughly 0.1-0.7% when beta_2 ~ 0.1-0.6 and m_sat ~ 1-2%.
• Combined hygrothermal strain at delta_t = -156 K and delta_m = 0.009: alpha_x * delta_t + beta_x * delta_m = -2.496e-4 + 3.613e-4 = +1.117e-4. In this laminate the moisture swelling dominates the cooldown contraction.
• Identity check: a 0-deg unidirectional laminate of the same ply returns alpha_x = alpha_1 = -0.3e-6/K exactly (float-precision difference 0.0), which only the full 2x2 inversion achieves.

Pitfalls

• Using the simplified scalar ratio instead of the exact inversion: the q11-only stiffness-weighted average returns 1.65 ppm for the worked [0/90]s where the exact 2x2 CLT inversion gives 1.60 ppm, and only the full inversion returns alpha_1 exactly for a 0-deg unidirectional laminate.
• Confusing the moisture and thermal branches: the hygrothermal strain is alpha * delta_t + beta * delta_m per axis, and in the worked laminate the moisture swelling (+3.6e-4) dominates the cure cooldown contraction (-2.5e-4); summing them with the wrong sign misses the net +1.12e-4.
• Assuming every laminate is 0-deg: the ply angles enter through the Qbar rotation, so a 45-deg unidirectional ply must return the rotated material coefficient; an angle outside [-90, 90] raises ValueError.
• Quoting raw SI where ppm reads better: cte_ppm(alpha) = alpha * 1e6 is the reporting helper, and mixing 1/K values with ppm values in one comparison misreads the coefficient by 1e6.
• Forgetting the moisture content endpoint domain: the isotherm is M = m_sat * rh with rh in [0, 1] and m_sat > 0; rh outside the unit interval or a non-positive saturation raises ValueError.
• Feeding non-physical ply properties: non-positive moduli, nu12*nu21 >= 1, empty ply lists, and non-positive thickness all raise ValueError; the code path is exact CLT arithmetic, so the material inputs carry the physics.

Verification

• Confirm laminate_cte_cme on the worked [0/90]s returns alpha_x inside 1.55-1.70e-6/K and beta_x inside 0.035-0.045.
• Confirm a 0-deg unidirectional layup returns alpha_1 and beta_1 to float precision, and that a 45-deg unidirectional ply returns the rotated material coefficient.
• Confirm equilibrium_moisture_content maps rh = 0 to 0 and rh = 1 to m_sat, and rejects rh outside [0, 1] and m_sat <= 0 with ValueError.
• Confirm the 1D stiffness-weighted average formula matches a direct ply-by-ply sum for the [0/90]s at about 1.23e-6/K (inside 0-5 ppm), distinct from the exact 1.60 ppm inversion.
• Confirm the cure-strain sign: cooling (delta_t < 0) with positive alpha gives a negative strain.
• Confirm plane_stress_q, qbar and laminate_cte_cme reject non-physical inputs (non-positive moduli, nu12*nu21 >= 1, empty ply list, non-positive thickness, angle outside [-90, 90]) with ValueError.
• Confirm determinism: the module has no RNG and returns identical floats run to run.
• Run the contract test offline: python3 scripts/test_laminate_hygrothermal_response.py (32 tests, deterministic, exits 0).

Related leaves

• structures/composites/laminate-stiffness: the mechanical CLT sibling whose Qbar assembly this leaf reuses for the thermal and moisture weighting; it owns the elastic A matrix, this leaf owns expansion.
• structures/composites/cmh17-allowables: hot/wet statistical strength knockdown factors that consume the moisture content this leaf computes; this leaf computes expansion strain, not strength.
• structures/thermal-structures/thermal-stress-analysis: isotropic constrained-member alpha*dT stress and bimetallic strips, the homogeneous-material counterpart of this laminate expansion leaf.
• structures/composites/sandwich-panels: honeycomb core selection context where core moisture is a caveat, separate from laminate hygrothermal strain.

Behavior contract (gate 3)

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

python3 scripts/test_laminate_hygrothermal_response.py

The test covers the worked [0/90]s example (q11 ~ 181.8 GPa, q22 ~ 10.35 GPa, alpha_x ~ 1.60e-6/K, beta_x ~ 0.040, equilibrium moisture 0.009, cure strain ~ -2.50e-4, moisture strain ~ 3.6e-4), the exact 2x2 inversion identity for a 0-deg unidirectional laminate, Qbar rotation invariants and coupling-term symmetry, equilibrium moisture isotherm endpoints, cure-cooldown sign, ValueError rejection of non-physical inputs, determinism, and the one-call convenience dict keys and values.

Compliance

• Standards referenced, not reproduced: FAR-25 is US government work (public domain); the CLT hygrothermal relations above are standard mechanics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.