Laminate Engineering Constants Reference

Q: Why does the laminate have two Poisson's ratios?

Because it is generally stiffer in one direction than the other. The major ratio νxy is the contraction in y when pulling along x, the minor ratio νyx is the reverse, and they are linked by νxy/Ex = νyx/Ey. Only for a laminate with Ex = Ey (quasi-isotropic, or a balanced 0/90 with equal ply counts) are they the same number.

Q: When should I use the restrained values instead of the normal moduli?

Use the free-contraction values (Ex, Ey, ...) when the piece of laminate can contract sideways as it is loaded: coupon tests, narrow strips, and most datasheet comparisons. Use the restrained values (Ex,r, Ey,r, ...) when the surrounding structure holds the transverse strain at zero: a strip in the middle of a wide panel, or a wide-beam bending calculation. The restrained values are never lower, and the difference grows with the laminate's Poisson's ratio.

Q: Why are my flexural moduli different from my membrane moduli?

Because bending weights the plies by the cube of their distance from the midplane, while stretching counts every ply equally. Outer plies dominate bending, so the flexural constants depend on the stacking order. A laminate with its 0° plies on the outside has a higher Exf than the same plies stacked with the 0s in the middle, even though Ex is identical for both.

Q: What does it mean when γD or δD is above 0.2?

It means bending-twisting coupling is strong enough that the specially orthotropic plate formulas (the closed-form solutions used by most textbooks and by the plate tool) are no longer trustworthy for that laminate. The published error at that level is on the unconservative side for buckling, which is why the tool warns rather than silently continuing. Cross-ply laminates have exactly zero; typical quasi-isotropic laminates sit well below the threshold.

Every constant on the full engineering constants readout, one table per group: what it is, where it comes from in the ABD matrix, and when it is the number you actually need.

Laminate engineering constants are the effective laminate properties, the elastic constants (E_x, E_y, G_xy, Poisson's ratios, flexural moduli) and expansion coefficients an equivalent homogeneous sheet would need to respond the way the laminate does. A laminate is not a material, but it is often convenient to pretend it is one, and the engineering constants are that pretence made precise. They all fall out of the ABD matrix, and none of them contain information the matrix does not already hold, but they are the numbers you compare against a datasheet, feed into a hand calculation, or quote to a colleague who thinks in moduli rather than matrix entries.

This page is the lookup key for the whole readout. If you want the theory behind the ABD matrix itself, start with CLT Theory and come back.

Membrane (in-plane) constants

The membrane constants describe the laminate stretched in its own plane. They come from the inverse of the extensional stiffness matrix A, divided through by the thickness t so they land in familiar modulus units:

a = A^{-1}, \qquad E_x = \frac{1}{t \, a_{11}}, \quad E_y = \frac{1}{t \, a_{22}}, \quad G_{xy} = \frac{1}{t \, a_{66}}

Symbol	Name	From the ABD matrix	What it tells you
E_x	Young's modulus, x	$\dfrac{1}{t\,a_{11}}$	Stiffness of the laminate pulled along x, free to contract in y. The number a tensile coupon cut along x would measure.
E_y	Young's modulus, y	$\dfrac{1}{t\,a_{22}}$	Same, pulled along y.
G_xy	Shear modulus	$\dfrac{1}{t\,a_{66}}$	In-plane shear stiffness. For a [0/90] laminate this is just the ply's G₁₂; adding ±45° plies is what raises it.
ν_xy	Major Poisson's ratio	$-\dfrac{a_{12}}{a_{11}}$	Contraction in y per unit extension in x. Can exceed 0.5, and even 1.0, for angle-ply laminates. That is real, not an error.
ν_yx	Minor Poisson's ratio	$-\dfrac{a_{12}}{a_{22}}$	Contraction in x per unit extension in y. Tied to the major one by ν_xy/E_x = ν_yx/E_y, so only one of them is independent.
η_xy,x	Tension-shear coupling	$\dfrac{a_{16}}{a_{11}}$	Shear strain per unit normal strain when pulled along x. Zero for balanced laminates; nonzero means pulling the laminate also skews it.
η_xy,y	Tension-shear coupling	$\dfrac{a_{26}}{a_{22}}$	Same coupling, for pulling along y.

Flexural (bending) constants

The flexural constants are the same idea for bending, taken from the inverse of the bending stiffness matrix D. They are not the membrane values repeated: plies far from the midplane dominate D (their contribution grows with z³), so the flexural constants depend on the stacking order, not just the ply mix. A [0/90/90/0] and a [90/0/0/90] laminate have identical membrane constants and very different flexural ones.

Symbol	Name	From the ABD matrix	What it tells you
E_xf	Flexural modulus, x	$\dfrac{12}{t^3\,d_{11}}$	The modulus an equivalent homogeneous plate would need to match the laminate's bending stiffness about the y-axis.
E_yf	Flexural modulus, y	$\dfrac{12}{t^3\,d_{22}}$	Same, bending about the x-axis.
G_xyf	Flexural shear modulus	$\dfrac{12}{t^3\,d_{66}}$	Twisting stiffness equivalent.
ν_xyf	Flexural Poisson's ratio	$-\dfrac{d_{12}}{d_{11}}$	Anticlastic curvature: how much the plate curls the other way across its width when bent along x.

Unsymmetric laminates: two conventions

For an unsymmetric laminate (B ≠ 0) there are two defensible ways to define the membrane and flexural constants, and they can differ by more than a factor of two, so it matters which one you are reading.

The standard convention (Jones, and the app's default) inverts A and D on their own, deliberately ignoring the membrane-bending coupling. It answers "how stiff would this ply stack be if the coupling were suppressed", which is the number most layup-comparison charts and textbooks quote, but it overstates the stiffness the real laminate shows when it is free to deform, and in the unconservative direction. That is why the app shows a warning banner for unsymmetric laminates.

The coupled convention takes the same formulas from the quadrants of the full inverse of the 6x6 ABD matrix instead. Because a free unsymmetric laminate bends when stretched, part of the work goes into curvature, and the laminate reads softer along every free-deformation constant. This is the stiffness a real free coupon of that laminate would exhibit. The Coupled constants toggle on the Engineering Constants page switches the membrane and flexural groups to this convention; for symmetric laminates the two conventions are identical, so the toggle only applies when B ≠ 0.

The gap is not academic. For a plain [0/90] two-ply carbon laminate the standard convention reports E_x = 96.0 GPa, while the coupled value is 38.6 GPa: sixty percent of the apparent stiffness disappears into membrane-bending coupling.

Two groups are unaffected by the toggle. The restrained moduli below are exact for any laminate, coupled or not: with every other strain and curvature held at zero, N_x = A₁₁ε_x regardless of B. And the effective CTE/CME values keep the standard convention either way, because a fully coupled thermal response would also need the coupling quadrant of the inverse, not just its membrane block. The exact response of an unsymmetric laminate under a specific load is the full 6x6 solve on the Laminate Response page.

Free vs. restrained contraction

The free-contraction moduli (E_x, E_y, G_xy and their flexural counterparts) assume the laminate is free to contract sideways while it is loaded, which is the situation of a tensile coupon in a test machine. But a strip in the middle of a wide panel is not free: the material around it holds the transverse strain at zero, and the strip reads stiffer than the coupon said it was. Both numbers are correct; they answer different questions.

The restrained values come straight from the stiffness matrices instead of their inverses, and they are never softer than the free values. The gap grows with the laminate's Poisson's ratio, so it is largest for angle-ply layups. If you are back-calculating a modulus from a panel test, or feeding a wide-panel hand calculation, the restrained value is usually the one you want.

Symbol	Name	From the ABD matrix	Free-contraction counterpart
E_x,r	Restrained modulus, x	$\dfrac{A_{11}}{t}$	E_x
E_y,r	Restrained modulus, y	$\dfrac{A_{22}}{t}$	E_y
G_xy,r	Restrained shear modulus	$\dfrac{A_{66}}{t}$	G_xy
E_xf,r	Restrained flexural modulus, x	$\dfrac{12\,D_{11}}{t^3}$	E_xf
E_yf,r	Restrained flexural modulus, y	$\dfrac{12\,D_{22}}{t^3}$	E_yf
G_xyf,r	Restrained twisting modulus	$\dfrac{12\,D_{66}}{t^3}$	G_xyf

A useful sanity check: for a single ply or a unidirectional stack, E_x,r equals the ply's reduced stiffness Q₁₁, which is E₁/(1 - ν₁₂ν₂₁), the familiar plane-stress correction. The whole free-vs-restrained distinction is that correction, generalised to a laminate.

Non-dimensional plate parameters

Four dimensionless ratios of D-matrix entries characterise how a laminate bends, independent of how stiff or thick it is. Two laminates with the same parameters buckle and deflect in the same pattern, just at different loads. They are the vocabulary of the plate-buckling literature, which is why the app reports them: when a paper says its chart applies for β_D between 1 and 2, these are the numbers to check.

Symbol	Name	Definition	How to read it
β_D	Seydel orthotropy parameter	$\dfrac{D_{12} + 2D_{66}}{\sqrt{D_{11} D_{22}}}$	Exactly 1 for an isotropic plate. Near 1, isotropic plate charts and formulas are a fair approximation; far from 1, the plate is strongly orthotropic in bending and they are not.
ν_D	Bending Poisson term	$\dfrac{D_{12}}{\sqrt{D_{11} D_{22}}}$	The bending analogue of a Poisson's ratio; equals ν exactly for an isotropic plate.
γ_D	Bending-twisting anisotropy	$\dfrac{D_{16}}{\left(D_{11}^{3} D_{22}\right)^{1/4}}$	How strongly bending couples into twisting. Zero for cross-ply laminates; small but nonzero for most quasi-isotropic ones. Above roughly 0.2 in magnitude, the closed-form specially-orthotropic plate solutions stop being reliable. This is the same threshold the plate tool warns on.
δ_D	Bending-twisting anisotropy	$\dfrac{D_{26}}{\left(D_{11} D_{22}^{3}\right)^{1/4}}$

Thermal and moisture expansion

When the laminas carry expansion data, the readout also reports the laminate's effective expansion behaviour: the strain and curvature it takes on per degree of temperature change or per percent of absorbed moisture, with nothing pushing on it. The physics behind these is covered on the Hygrothermal Effects page; the short version of the table below is that membrane terms are expansion and flexural terms are warping.

Symbol	Name	What it tells you
CTE_x, CTE_y	Effective thermal expansion	In-plane strain per °C. Can be near zero, or negative, for carbon laminates. That is a design feature, not a glitch.
CTE_xy	Thermal shear expansion	Shear strain per °C. Zero for balanced laminates; nonzero means heating skews the laminate.
κ_x^T, κ_y^T, κ_xy^T	Thermal curvatures	Curvature per °C. Zero for symmetric laminates; nonzero is why an unsymmetric laminate comes off a hot tool warped.
CME_x, CME_y, CME_xy	Effective moisture expansion	The same three in-plane terms, per unit of absorbed moisture instead of per °C.
κ_x^M, κ_y^M, κ_xy^M	Moisture curvatures	Warping per unit of absorbed moisture. Zero for symmetric laminates.

What the zeros are telling you

Several of these constants are diagnostic: their being exactly zero is a statement about the layup, and a value that should be zero but is not means the layup is not what you thought it was.

If the layup is...	Then these are zero	Which means
Symmetric (mirror image about the midplane)	All κ^T and κ^M terms	No warping from temperature or moisture. The single strongest practical reason symmetric layups are the default.
Balanced (every +θ ply has a -θ partner)	η_xy,x, η_xy,y, CTE_xy, CME_xy	Pulling or heating the laminate does not skew it.
Cross-ply (0° and 90° plies only)	γ_D, δ_D	No bending-twisting coupling; the closed-form plate solutions apply without caveats.

References

Jones, R.M. Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999. Chapter 4 covers the laminate engineering constants.
Nemeth, M.P. "Importance of Anisotropy on Buckling of Compression-Loaded Symmetric Composite Plates," AIAA Journal, vol. 24, no. 11, 1986 (also NASA TP-2563). Defines the bending-twisting anisotropy parameters γ and δ and the 0.2 usability threshold.
Seydel, E. "Über das Ausbeulen von rechteckigen, isotropen oder orthogonal-anisotropen Platten bei Schubbeanspruchung," Ingenieur-Archiv, vol. 4, 1933. Origin of the orthotropy parameter.
Barbero, E.J. Introduction to Composite Materials Design, 3rd ed., CRC Press, 2018. doi:10.1201/9781315296494

Frequently Asked Questions

Why does the laminate have two Poisson's ratios?

When should I use the restrained values instead of the normal moduli?

Why are my flexural moduli different from my membrane moduli?

What does it mean when γD or δD is above 0.2?

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