Hygrothermal Strain⚠ unverified
Mechanical / Composites · Compute the combined thermal and moisture (hygrothermal) free strain
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| alpha | α | 1/K | 1.0 | Coefficient of thermal expansion, per kelv |
| delta_T | ΔT | K | 1.0 | Temperature change |
| beta | β | — | 1.0 | Coefficient of moisture expansion (per unit moisture concentration) |
| delta_M | ΔM | — | 1.0 | Moisture concentration change (dimensionless mass fraction) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | strain | — | Total hygrothermal strain (dimensionless) |
The science & history
Understanding the Parameters
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Thermal term $\alpha\,\Delta T$ — ordinary thermal expansion. In a unidirectional ply $\alpha$ is small (even slightly negative) along the fibers — carbon fibers barely expand — but large across them, driven by the matrix. This anisotropy is what generates thermal stress between plies.
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Moisture term $\beta\,\Delta M$ — polymer matrices absorb water over time; the resulting swelling is the hygroscopic analog of thermal expansion, with $\beta$ (per unit moisture uptake) and $\Delta M$ (the moisture mass fraction gained). Like $\alpha$, $\beta$ is much larger transversely.
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Free vs constrained — this is the free (unconstrained) strain. In a laminate, plies of different orientation want to strain differently; forcing them to stay bonded creates residual hygrothermal stresses even with no external load, which add to service stresses and can cause matrix microcracking.
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The "stress-free" reference — $\Delta T$ and $\Delta M$ are measured from the state where the laminate had no residual stress, typically the cure temperature (hot) and dry condition — so a cooled, moist part carries built-in strain.
Derivation (Approaching a Proof)
Both effects are, to first order, linear in their driving change, by analogy with thermal expansion. Thermal expansion gives a free strain proportional to temperature change, $\varepsilon_T = \alpha\,\Delta T$. Moisture swelling is treated identically: absorbed moisture produces a free strain proportional to the moisture change, $\varepsilon_H = \beta\,\Delta M$.
Because both are small free strains acting on the same material, they superpose (linear kinematics):
$$\varepsilon = \varepsilon_T + \varepsilon_H = \alpha\,\Delta T + \beta\,\Delta M.$$
The formal parallel is exact — moisture concentration plays the role of temperature, and $\beta$ the role of $\alpha$ — which is why the two are combined into a single "hygrothermal" analysis in laminate theory, carrying the same mathematics as thermal residual stress.
Dimensional check. $\alpha\,\Delta T = (1/\text{K})\cdot\text{K}$ = dimensionless; $\beta\,\Delta M$ = (per unit moisture)·(moisture fraction) = dimensionless. Their sum $\varepsilon$ is dimensionless — a strain, as required.
History and Development
Hygrothermal analysis became essential as polymer-matrix composites entered aerospace service, where parts cure hot, cool to ambient (locking in thermal residual stress), and then slowly absorb atmospheric or in-service moisture (adding swelling and, often, relieving some of the thermal stress). Combining thermal and moisture strains into one framework — the "hygrothermal" extension of classical laminate theory (Composite Laminate Theory) — is standard practice, drawing directly on the thermal-stress methods of Thermal Stress and Thermal Stress Index.
Related Concepts: Thermal Stress, Thermal Stress Index, Composite Laminate Theory, Rule of Mixtures transverse, Interlaminar Shear Stress, Poisson Ratio Nu12
Notes: Free (unconstrained) strain; constraint in a laminate → residual hygrothermal stress. Both $\alpha,\beta$ strongly anisotropic (larger transversely). Moisture is the hygroscopic analog of thermal expansion. Reference state = cure temperature, dry. Superposition of two linear free strains.