The volume fractions for the fiber and matrix in the laminate.

Question

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Answer 1

Question

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

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Answer 2

Question

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

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Answer 3

Question

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

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Answer 4

Question

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

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Answer 5

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

Answer 6

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Answer 7

Chapter 12.15, Problem 97SEP

(a)

To determine

The volume fractions for the fiber and matrix in the laminate.

Answer 8

(a)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Write the equation for the rule of mixture for the composite material.

$Ec=EfVf+EmVm$ (I)

Here, modulus of elasticity of composite is $Ec$ , modulus of elasticity of fiber is $Ef$ , volume fraction of fiber is $Vf$ , modulus of elasticity of matrix is $Em$ and volume fraction of matrix is $Vm$ .

Write the equation to calculate the volume fraction of matrix.

$Vm=1−Vf$ (II)

Conclusion:

It is given that the component is suggested to be made from unidirectional carbon fiber composite. Therefore, the fiber layer will be carbon and the matrix layer will be epoxy.

Take the modulus of elasticity of carbon fiber to be $33×106 psi$ and the modulus of elasticity of epoxy matrix to be $0.5×106 psi$ .

Substitute $10×106 psi$ for $Ec$ , $33×106 psi$ for $Ef$ , $0.5×106 psi$ for $Em$ and $(1−Vf)$ for $Vm$ in Equation (I).

$10×106 psi=(33×106 psi)Vf+(0.5×106 psi)(1−Vf)10×106 psi=(33×106 psi)Vf+(0.5×106 psi)−(0.5×106 psi)Vf10×106 psi−0.5×106 psi=(33×106 psi−0.5×106 psi)Vf9.5×106 psi=(32.5×106 psi)Vf$

$Vf=9.5×106 psi32.5×106 psi=0.29$

Substitute $0.29$ for $Vf$ in Equation (II).

$Vm=1−0.29=0.71$

Thus, the volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Answer 13

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

Answer 14

(b)

To determine

The volume fractions for the fiber and matrix in the laminate.

Answer 15

(b)

Expert Solution

Answer to Problem 97SEP

The volume fractions for the fiber and matrix in the laminate are $0.29$ and $0.71$ respectively.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Write the equation to calculate the percentage in weight reduction.

$%Wt=ρAl−ρCρAl×100$ (III)

Here, density of carbon composite is $ρC$ and density of aluminum composite is $ρAl$ .

Conclusion:

Refer Appendix I, "Important properties of the selected engineering materials" to obtain the densities of aluminum composite and carbon composite.

$ρAl=0.0975 lb/in.3ρC=0.0553 lb/in.3$ .

Substitute $0.0553 lb/in.3$ for $ρC$ and $0.0975 lb/in.3$ for $ρAl$ in Equation (III).

$%Wt=0.0975 lb/in.3−0.0553 lb/in.30.0975 lb/in.3×100=0.43×100=43%$

Therefore, the carbon composite will be 43% lighter when compared to aluminum composite.

Answer 20

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The volume fractions for the fiber and matrix in the laminate.

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The volume fractions for the fiber and matrix in the laminate.

Answer to Problem 97SEP

Explanation of Solution

The volume fractions for the fiber and matrix in the laminate.

Answer to Problem 97SEP

Explanation of Solution

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