Question

The Shear stresses in the fiber and matrix are 200GPa and 20GPa respectively. If the fiber volume fraction is 70%, then the longitudinal compressive strength of the lamina is

• A. 112 GPa
• B. 64.8 GPa
• C. 146 GPa
• D. 292 GPa

Hint:

The rule of mixtures is especially useful in predicting the mechanical properties of composite materials by considering the properties of each component and their volume fraction. For mechanical engineers, it provides a simple yet effective method to estimate the overall characteristics of composites without detailed numerical modeling.

Answer 1

The Correct Option is C

Given: 
Shear stresses in the fiber and matrix are given as:  Fiber (\(\sigma_f\)) = 200 GPa \item Matrix (\(\sigma_m\)) = 20 GPa Fiber volume fraction (\(V_f\)) is \(70\%\), and thus matrix volume fraction (\(V_m\)) is \(30\%\) \((100\% - 70\%)\). 

To find: 
The longitudinal compressive strength of the lamina. 

Solution: 
Using the rule of mixtures for composite materials, which states that the properties of the composite can be calculated based on the volume fractions and properties of the individual constituents, the longitudinal compressive strength (\(\sigma_c\)) of the lamina is given by: \[ \sigma_c = \sigma_f V_f + \sigma_m V_m \] Substituting the given values: \[ \sigma_c = (200 \, \text{GPa} \times 0.7) + (20 \, \text{GPa} \times 0.3) = 140 \, \text{GPa} + 6 \, \text{GPa} = 146 \, \text{GPa} \] 

Conclusion: 
The longitudinal compressive strength of the lamina is \(146 GPa\), which matches option (c).