Derive expressions for the transvers modulus and the longitudinal modulus of

Your Name]In summary, we have derived expressions for the transverse and longitudinal moduli of a 3-component composite material with two different types of fibres. We have also calculated the volume fraction of type B fibre required to obtain a longitudinal modulus of 200 GPa using the given information.
  • #1
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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:

[itex]\epsilon[/itex][itex]_{C}[/itex]=[itex]\epsilon[/itex][itex]_{f}[/itex]=[itex]\epsilon[/itex][itex]_{m}[/itex]

[itex]\frac{\sigma_{C}}{\epsilon_{C}}[/itex]=[itex]\frac{\sigma_{f}}{\epsilon_{f}}[/itex]V[itex]_{f}[/itex]+[itex]\frac{\sigma_{m}}{\epsilon_{m}}[/itex]V[itex]_{m}[/itex]

E[itex]_{C}[/itex]=E[itex]_{f}[/itex]V[itex]_{f}[/itex]+E[itex]_{m}[/itex]V[itex]_{m}[/itex]

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.
 
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  • #2




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.


 

FAQ: Derive expressions for the transvers modulus and the longitudinal modulus of

1. What is the difference between transverse modulus and longitudinal modulus?

The transverse modulus is a measure of a material's resistance to deformation when a force is applied perpendicular to its surface. On the other hand, the longitudinal modulus is a measure of a material's resistance to deformation when a force is applied along its length.

2. How do you derive the expression for transverse modulus?

The expression for transverse modulus can be derived by dividing the applied force by the change in length of the material perpendicular to the applied force. This is known as Hooke's law and the resulting expression is the transverse modulus.

3. Can the transverse modulus and longitudinal modulus be equal?

Yes, in some cases the transverse modulus and longitudinal modulus can be equal. This is usually seen in isotropic materials, where the properties are the same in all directions.

4. What factors affect the value of the longitudinal modulus?

The value of the longitudinal modulus is affected by several factors, such as the type of material, its composition, and the temperature at which it is tested. The modulus also varies with the stress-strain relationship of the material.

5. How do you measure the transverse and longitudinal modulus?

The transverse and longitudinal modulus can be measured using a tensile testing machine. The machine applies a force to the material and measures its corresponding change in length. The modulus can then be calculated using the applied force and the change in length.

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