Derive expressions for the transvers modulus and the longitudinal modulus of

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Homework Statement

A composite material is made from a matrix with a tensile modulus of 5 GPa and two different continuous fibres with moduli of 360 GPa (type !) and 450 GPa (type B) respectively.

a) Derive expressions for the transverse modulus and the longitudinal modulus of a 3-component composite.

b) Use the expressions to calculate the volume fraction of type B fibre required to obtain a longitudinal modulus of 200 GPa for a composite material with the above fibres and matrix given that the volume fraction of type A fibres is 0.25.

Homework Equations

Transverse modulus:

$\epsilon$$_{C}$=$\epsilon$$_{f}$=$\epsilon$$_{m}$

$\frac{\sigma_{C}}{\epsilon_{C}}$=$\frac{\sigma_{f}}{\epsilon_{f}}$V$_{f}$+$\frac{\sigma_{m}}{\epsilon_{m}}$V$_{m}$

E$_{C}$=E$_{f}$V$_{f}$+E$_{m}$V$_{m}$

etc, etc.

The Attempt at a Solution

See above. I know what the result should be for a composite with one fibre, but how do I alter the formulae for a composite with two fibres? Do I just replace 'f' with 'f1+f2' or is it more complicated than that? Nowhere in my notes or textbooks or the internet does it tell you.

Please help.

 

[mark726]

Thank you for your question. To derive expressions for the transverse and longitudinal moduli of a 3-component composite, we will modify the formulae you have provided to account for the two different types of fibres.

a) Transverse modulus:

\epsilon_{C}=\frac{\sigma_{C}}{E_{C}}=\frac{\sigma_{f1}}{E_{f1}V_{f1}}+\frac{\sigma_{f2}}{E_{f2}V_{f2}}+\frac{\sigma_{m}}{E_{m}V_{m}}

Longitudinal modulus:

E_{C}=E_{f1}V_{f1}+E_{f2}V_{f2}+E_{m}V_{m}

b) To calculate the volume fraction of type B fibre required to obtain a longitudinal modulus of 200 GPa, we can set the longitudinal modulus equation equal to 200 GPa and solve for V_{f2}:

200 GPa = 360 GPa V_{f1} + 450 GPa V_{f2} + 5 GPa V_{m}

Since we are given that V_{f1} = 0.25, we can substitute that value into the equation:

200 GPa = 360 GPa (0.25) + 450 GPa V_{f2} + 5 GPa V_{m}

Solving for V_{f2}, we get:

V_{f2} = 0.4

Therefore, the volume fraction of type B fibre required to obtain a longitudinal modulus of 200 GPa is 0.4.

I hope this helps. Let me know if you have any further questions.