Shear Modulus Inplane⚠ unverified
Mechanical / Composites · Compute the in-plane shear modulus of a unidirectional composite
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Gf | Gf | Pa | 1.0 | Fibre shear modulus |
| Gm | Gm | Pa | 1.0 | Matrix shear modulus |
| Vf | Vf | — | 1.0 | Fibre volume fraction (dimensionless), between 0 and 1 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | G12 | Pa | In-plane shear modulus G12, in pascals (Pa) |
The science & history
Understanding the Parameters
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Matrix shear modulus $G_m$ dominates — in series the softest phase controls, and $G_m \ll G_f$, so $G_{12}$ stays close to the matrix value and rises only weakly with $V_f$.
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Series/iso-stress logic — identical to the transverse-modulus case (Rule of Mixtures transverse): fibers and matrix carry the same shear stress and their shear compliances ($1/G$) add by volume fraction.
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Lower bound, and its inaccuracy — this is the Reuss lower bound; it under-predicts the true $G_{12}$ for the same fiber-constraint reason that afflicts transverse modulus. The Halpin-Tsai form (Micromechanics Shear) corrects it.
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Why $G_{12}$ matters — in-plane shear governs the response of $\pm45^\circ$ plies and torsion of laminated tubes/shafts; a low $G_{12}$ makes off-axis laminates compliant and drives fiber-orientation choices.
Derivation (Approaching a Proof)
Apply an in-plane shear stress $\tau_{12}$ to the ply. Fiber and matrix regions shear in series across the load, so they carry the same shear stress $\tau$ (iso-stress) but different shear strains. Each strains by $\gamma = \tau/G$, and the strains add weighted by volume fraction:
$$\gamma_{12} = \gamma_f V_f + \gamma_m(1-V_f) = \tau\left(\frac{V_f}{G_f} + \frac{1-V_f}{G_m}\right).$$
The shear modulus is $G_{12} = \tau/\gamma_{12}$, so its reciprocal (shear compliance) is the volume-weighted sum of constituent compliances:
$$\frac{1}{G_{12}} = \frac{V_f}{G_f} + \frac{1-V_f}{G_m}.$$
As with transverse modulus, the iso-stress assumption neglects the fibers' restraint of the matrix, so the real $G_{12}$ is higher than this bound.
Dimensional check. $\dfrac{1}{G_{12}} = \dfrac{V_f}{G_f} + \dfrac{1-V_f}{G_m}$ has units $1/\text{Pa}$, so $G_{12} = \text{Pa}$ — a modulus, as required.
History and Development
The inverse rule of mixtures for $G_{12}$ is the Reuss shear analog (1929), the lower Voigt–Reuss bound. Its systematic under-prediction is exactly what drove Halpin-Tsai (Halpin-Tsai Equations, Micromechanics Shear) and rigorous elasticity models. The four constants $E_1$, $E_2$, $G_{12}$, $\nu_{12}$ it helps complete are the inputs to classical laminate theory (Composite Laminate Theory).
Related Concepts: Micromechanics Shear, Rule of Mixtures transverse, Rule of Mixtures longitudinal, Poisson Ratio Nu12, Halpin-Tsai Equations, Composite Laminate Theory
Notes: Reuss series lower bound — under-predicts real $G_{12}$ (ignores fiber constraint); use Micromechanics Shear (Halpin-Tsai $\xi=1$) instead. Matrix-dominated, weak $V_f$ dependence. Governs $\pm45^\circ$ plies and laminate torsion.