Hygrothermal Effects

Why temperature and absorbed moisture load a laminate even with nothing pushing on it, how those effects enter CLT as equivalent forces and moments, and why a symmetric laminate can be stressed without warping.

"Hygrothermal" is just the two words stuck together: hygroscopic, meaning moisture-related, and thermal. They are treated as one topic because the mathematics is identical. Both are cases of a material wanting to change size, and being stopped from doing so freely by the plies around it.

Why a laminate is loaded with no load applied

Heat a free, unconstrained bar of aluminium and it gets longer. No stress develops, because nothing resists the expansion. Clamp both ends first and the same temperature rise produces a large compressive stress instead. Nothing about the material changed; only the constraint did.

A laminate provides that constraint to itself. A ply expands very little along its fibres and a great deal across them, so a 0° ply and a 90° ply bonded together want to change size by different amounts in the same direction. They cannot both get their way. Each ply ends up stressed by its neighbours, and the whole stack may bend. This happens with no external load at all, which is what makes hygrothermal effects easy to forget and expensive to forget.

The two expansion coefficients

A ply's response to its environment is captured by two coefficients in each case, one along the fibres and one across them. For temperature these are the coefficients of thermal expansion, α₁ and α₂. The free strain they produce is simply the coefficient times the change:

There is no thermal shear strain in material coordinates: a uniform temperature change makes a ply grow or shrink, it does not skew it. Typical values show how strongly anisotropic the response is:¹

Two things stand out. The transverse coefficient is an order of magnitude larger than the longitudinal one, because transverse expansion is governed by the resin while longitudinal expansion is held in check by the fibres. And α₁ can be negative for carbon and aramid: those fibres get slightly shorter as they get hotter. A carbon laminate can therefore be laid up to have almost no expansion in a chosen direction, which is why carbon is used for optical benches and satellite structures.

Moisture works the same way, with coefficients of moisture expansion β₁ and β₂ in place of α:

Fibres do not absorb water, so β₁ is close to zero and β₂ carries almost all of the swelling. That makes the moisture response even more lopsided than the thermal one.

From ply expansion to a laminate load

CLT already knows how to solve a laminate given a set of forces and moments. So rather than rewriting the theory, hygrothermal effects are folded in as an equivalent load: the set of forces and moments that would produce the same effect as the temperature or moisture change. Written into the laminate constitutive law:²

The equivalent resultants are built ply by ply. For each ply, take its free hygrothermal strain, rotate it from the fibre frame into the laminate frame, multiply by that ply's transformed stiffness, and integrate over the ply's share of the thickness:

The shape of those two sums is worth noticing, because it is the same shape as the A and B matrices themselves: the force resultant weights each ply by its thickness, and the moment resultant weights it by its distance from the midplane. Everything the ABD matrix knows about symmetry carries straight over, which is the subject of the next section.

Once N^HT and M^HT are known they are simply added to the mechanical load, and the solve proceeds unchanged. That is the whole trick, and it is why the contribution breakdown on the load analysis page can show thermal and moisture as separate columns: each is a genuine load vector in its own right.

Symmetry removes the warping, not the stress

In a symmetric laminate, every ply has a mirror twin at the opposite z. In the M^HT sum the two twins contribute equal and opposite terms, so they cancel and M^HT comes out zero. There is no thermal moment, and so no thermal curvature. This is the same cancellation that makes B = 0 for a symmetric layup, and it is the main practical reason symmetric laminates are the default choice: they come out of the autoclave flat.

N^HT does not cancel, because it weights every ply by thickness rather than by position. And even where the laminate-level resultants both vanish, the individual plies are still fighting each other. Each ply is held at the laminate's average strain rather than the strain it wanted, and the difference is real stress in that ply. A flat laminate is not an unstressed laminate.

This is where residual cure stress comes from. Aerospace epoxies cure somewhere around 120 to 180 °C and are then used at room temperature, so every part carries a built-in ΔT of roughly -100 °C before it ever sees service.¹ The resulting stress acts across the fibres, in the direction where a ply is weakest, and it can consume a meaningful fraction of the transverse strength budget before any external load arrives. It is also the usual explanation when matrix cracking is observed in a part that the numbers said was safe.

Moisture is slow, and that changes how you use it

Temperature changes propagate through a thin laminate in seconds. Moisture does not. Absorption follows Fick's second law of diffusion,¹ which for through-thickness diffusion reads:

with c the moisture concentration and D the diffusion coefficient. Diffusion coefficients in epoxies are small enough that a thick laminate can take months or years to reach equilibrium with its environment, and the equilibrium level itself depends on the surrounding humidity.

The design consequence is that moisture is treated as a long-term end-of-life condition rather than a load case that comes and goes. The usual approach is to analyse the fully saturated laminate, since that is the state the structure will spend most of its life closest to, and to combine it with the temperature extreme that makes matters worst. "Hot and wet" is a standard certification condition for exactly this reason: raised temperature softens the resin while absorbed moisture has already swollen it.

CLT as implemented here assumes the moisture content is uniform through the thickness, which is the saturated end state. It does not model the partly-absorbed transient, where the outer plies have taken up water and the inner ones have not.

Where the coefficients come from

You can measure α and β on a ply, or you can predict them from the fibre and matrix properties. The micromechanics models do the latter. The longitudinal coefficient is a stiffness-weighted average rather than a plain volume average, because the stiffer phase gets more of a say in how much the ply is allowed to move:

That weighting is why α₁ for a carbon laminate sits so close to the fibre value: with E_f perhaps fifty times E_m, the fibre term dominates the fraction even at moderate fibre volume fraction. The transverse coefficient needs a Poisson correction as well, since a ply restrained along the fibres bulges across them.³

In the ABD Composites dashboard

CTE and CME live on the lamina material, either entered directly or predicted by the UD lamina builder from the fibre and matrix. The Laminate Load Analysis tool takes ΔT in °C and ΔM as a percentage, and reports the thermal and moisture responses as their own columns alongside the mechanical one, so you can see immediately how much of the deformation is environmental.

A contribution with no data behind it is marked unavailable rather than shown as zero, so a missing CTE never reads as "no thermal effect".

Downstream, ply stress recovery subtracts each ply's free hygrothermal strain before computing stress, because free expansion produces no stress. Only the part of the strain the ply was prevented from achieving does.

References

1. Roylance, D. "Laminated Composite Plates," in Mechanics of Materials, MIT / Engineering LibreTexts. Open access
2. Jones, R.M. Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999. Eq. 4.38-4.39 (hygrothermal force and moment resultants), Eq. 2.89 (expansion coefficient transformation).
3. Schapery, R.A. "Thermal Expansion Coefficients of Composite Materials Based on Energy Principles," Journal of Composite Materials, vol. 2, no. 3, pp. 380-404, 1968. doi:10.1177/002199836800200308
4. Springer, G.S. and Tsai, S.W. "Thermal Conductivities of Unidirectional Materials," Journal of Composite Materials, vol. 1, no. 2, pp. 166-173, 1967. doi:10.1177/002199836700100206
5. Shen, C.-H. and Springer, G.S. "Moisture Absorption and Desorption of Composite Materials," Journal of Composite Materials, vol. 10, no. 1, pp. 2-20, 1976. doi:10.1177/002199837601000101

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