A ply of aramid fibres in epoxy resin has 75 percent fibre by volume. Use the inverse rule of mixtures to find the in-plane shear modulus G12 if the fibre shear modulus is 27 GPa and the matrix shear modulus is 6 GPa. Give the answer to two decimal places.

Step-by-step solution

1. 1

   Choose the series (inverse) form. In-plane shear does not load the fibres along their length, so fibre and matrix carry the same shear stress and their shear strains add. That is a series arrangement, and compliances — not moduli — add:

   $\frac{1}{G_{12}}=\frac{V_f}{G_f}+\frac{V_m}{G_m}.$

2. 2

   Write down the volume fractions. The aramid occupies $75\%$, so

   $V_f=0.75,\qquad V_m=1-0.75=0.25.$

3. 3

   Compute the two compliance terms.

   $\frac{V_f}{G_f}=\frac{0.75}{27}=0.0277778\ \text{GPa}^{-1},\qquad \frac{V_m}{G_m}=\frac{0.25}{6}=0.0416667\ \text{GPa}^{-1}.$

   Notice that the matrix contributes the larger term despite occupying only a quarter of the volume — the soft phase dominates a series model.

4. 4

   Add and invert.

   $\frac{1}{G_{12}}=0.0277778+0.0416667=0.0694444\ \text{GPa}^{-1},$
   $G_{12}=\frac{1}{0.0694444}=14.40\ \text{GPa}.$

   The reciprocal comes out exactly: $0.0694444=\tfrac{1}{14.4}$.

5. 5

   Check the result lies where physics requires. $G_{12}$ must sit between $G_m=6$ and $G_f=27$, and closer to the smaller value. $14.40$ satisfies both. For contrast, the parallel (rule-of-mixtures) formula would give $0.75(27)+0.25(6)=21.75\ \text{GPa}$ — much stiffer, and the wrong model for shear.