Question 3. You wish to produce a fibre reinforced composite (fibres aligned) wi
ID: 1713051 • Letter: Q
Question
Question 3. You wish to produce a fibre reinforced composite (fibres aligned) with a Young's modulus E along the fibre direction of 150 GPa. You have available glass fibres with an E of 75.9 GPa, graphite fibres with an E of 552 GPa, and various polymer matrix materials, all with E 3 GPa. Since the glass fibres are significantly cheaper $10 m-3 than the carbon fibres $1000 m3, your boss wants you to make the composite from glass fibres and polymer matrix if possible. (At least 30% matrix is necessary to prevent the composite from being too brittle). a)Predict the largest E that can be obtained for a non-brittle composite of glass fibres and polymer matrix. ANS: 54 GPa b) Quantitatively suggest a minimum cost solution to obtain a composite with an E of 200 GPa along the fibre direction. Ans: $311 per cubic meterExplanation / Answer
Answer a:
The rule of mixtures predicts that an upper limit of elastic modulus of composites "Ec" is given by:
Ec = Em Vm + Ep Vp
Where,
Em = Elastic Modulus of Matrix,
Vm = Volume Fraction of Matrix
Ep = Elastic Modulus of Particulate
Vp = Volume Fraction of Matrix
Thus Ec = 3 x 0.3 + 75.9 x 0.7 = 54.03 GPa
Thus the largest E that can be obtained is 54.03 GPa.
Answer b:
Here to find the minimum cost solution our target should be to use minimum graphite fibres as its cost is more. Here again this can be obtained by using the rule of mixtures:
Ec = Em Vm + Egl Vgl + Eg Vg
Where gl and g are respectively denotes "Glass" and "Graphite:.
30% matrix needs to be used in the mixture.So Glass and Graphite volume together shall be 70%.
Thus Vgl + Vg = 0.7
Therefore Vgl = 0.7-Vg
Ec = 200 GPa
Substituting in above equation:
200 = 3 x 0.3 + 75.9 x [0.7 - Vg] + 552 x Vg
Thus Vg = 0.3065 i.e. 30.65%
Vgl = 0.7 - 0.3065 = 0.3935 i.e.39.35%
Thus the total cost becomes 0.3065 x 1000 + 0.3935 x 10 = $311.
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