Use the inverse rule of mixtures to calculate the transverse stiffness E22 of a composite laminate with fibre volume fraction 0.483, fibre modulus 282 GPa and matrix modulus 2.7 GPa. Give the answer in gigapascals to one decimal place.

Step-by-step solution

1. 1

   Know why the transverse case uses a different formula. Loaded along the fibres, fibre and matrix strain together and their stiffnesses add in parallel. Loaded across the fibres they carry the same stress and their strains add, so it is the compliances that add — a series arrangement:

   $\frac{1}{E_{22}}=\frac{V_f}{E_f}+\frac{V_m}{E_m}.$

2. 2

   Find the matrix volume fraction. The two phases fill the volume, so

   $V_m=1-V_f=1-0.483=0.517.$

3. 3

   Compute each compliance term.

   $\frac{V_f}{E_f}=\frac{0.483}{282}=0.00171277\ \text{GPa}^{-1},\qquad \frac{V_m}{E_m}=\frac{0.517}{2.7}=0.19148148\ \text{GPa}^{-1}.$

   Note how lopsided these are: the matrix term is over a hundred times larger, because the soft phase dominates a series arrangement.

4. 4

   Add and invert.

   $\frac{1}{E_{22}}=0.00171277+0.19148148=0.19319425\ \text{GPa}^{-1},$
   $E_{22}=\frac{1}{0.19319425}=5.1761\ \text{GPa}.$

5. 5

   Round to one decimal place.

   $E_{22}\approx 5.2\ \text{GPa}.$

6. 6

   Sanity-check the magnitude. The answer must lie between the two constituent moduli, and much nearer the smaller one — a chain is as stiff as its softest link. Getting a value close to $282$ would mean the parallel (longitudinal) formula was used by mistake; here $5.2$ is only about twice $E_m=2.7$, which is exactly what a series model with roughly half matrix predicts.