Poisson Ratio Nu12⚠ unverified
Mechanical / Composites · Compute the major (in-plane) Poisson's ratio of a unidirectional composite
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| nuf | nuf | — | 1.0 | Fibre Poisson's ratio (dimensionless) |
| num | num | — | 1.0 | Matrix Poisson's ratio (dimensionless) |
| Vf | Vf | — | 1.0 | Fibre volume fraction (dimensionless), between 0 and 1 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | ν12 | — | Major Poisson's ratio nu12 (dimensionless) |
The science & history
Understanding the Parameters
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Constituent ratios $\nu_f$, $\nu_m$ — carbon/glass fibers $\sim 0.2$, polymer matrix $\sim 0.35$–$0.4$. The composite value lands between them, matrix-leaning at low $V_f$, fiber-leaning at high $V_f$.
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"Major" and the subscript $12$ — $\nu_{12}$ is the contraction in the transverse (2) direction per unit extension in the fiber (1) direction. It is the large one; the minor ratio $\nu_{21} = \nu_{12}E_2/E_1$ is much smaller because $E_2 \ll E_1$. The two are linked by the orthotropic reciprocity relation.
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Accurate blend — like the longitudinal modulus, the iso-strain condition holds for this loading, so the simple rule of mixtures is reliable (unlike transverse modulus or shear).
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Design defaults — $\nu_f$ default $1.0$ here is a placeholder; physical Poisson's ratios are $\le 0.5$ for isotropic constituents.
Derivation (Approaching a Proof)
Apply a longitudinal stress along the fibers. As in the longitudinal modulus derivation (Rule of Mixtures longitudinal), fiber and matrix share the same longitudinal strain $\varepsilon_1$ (iso-strain). Each phase contracts transversely by its own Poisson's ratio: fiber contraction $\nu_f \varepsilon_1$, matrix contraction $\nu_m\varepsilon_1$.
The overall transverse contraction is the volume-weighted average of the two (the transverse dimension is shared across the phases):
$$\varepsilon_2 = -\left[\nu_f V_f + \nu_m(1-V_f)\right]\varepsilon_1.$$
By definition $\nu_{12} = -\varepsilon_2/\varepsilon_1$, so
$$\nu_{12} = \nu_f V_f + \nu_m(1-V_f).$$
The shared-strain condition is what makes this average accurate, mirroring the longitudinal modulus.
Dimensional check. $\nu_{12} = \nu_f V_f + \nu_m(1-V_f)$ is a sum of dimensionless ratios weighted by a dimensionless fraction → dimensionless, as a Poisson's ratio must be.
History and Development
The rule of mixtures for $\nu_{12}$ completes the set of engineering constants needed to build a ply's stiffness matrix in classical laminate theory (Composite Laminate Theory). Together with $E_1$ (Rule of Mixtures longitudinal), $E_2$ (Halpin Tsai Modulus), and $G_{12}$ (Micromechanics Shear), it defines the orthotropic lamina — the building block from which multi-directional laminate properties and the coupling of extension, bending, and twist are assembled.
Related Concepts: Rule of Mixtures longitudinal, Rule of Mixtures transverse, Halpin Tsai Modulus, Micromechanics Shear, Shear Modulus Inplane, Composite Laminate Theory
Notes: Major (in-plane) ratio; accurate (iso-strain, like $E_1$). Minor ratio $\nu_{21} = \nu_{12}E_2/E_1$ (much smaller) via reciprocity. One of the four ply constants ($E_1,E_2,G_{12},\nu_{12}$). Physical constituent $\nu \le 0.5$ (defaults are placeholders).