volume fraction of the matrix

How to Calculate and Solve for Randomly Oriented Composite Modulus of Elasticity | Composites

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The randomly oriented composite modulus of elasticity is illustrated by the image below.

To compute for randomly oriented composite modulus of elasticity, five essential parameters are needed and these parameters are Fibre Efficiency Parameter (K), Elastic Modulus of the Fibre (Ef), Elastic Modulus of the Matrix (Em), Volume Fraction of the Fibre (Vf) and Volume Fraction of the Matrix (Vm).

The formula for calculating randomly oriented composite modulus of elasticity:

E = KEfVf + EmVm

Where:

E = Randomly Oriented Composite Modulus of Elasticity
K = Fibre Efficiency Parameter
Ef = Elastic Modulus of the Fibre
Em = Elastic Modulus of the Matrix
Vf = Volume Fraction of the Fibre
Vm = Volume Fraction of the Matrix

Let’s solve an example;
Find the randomly oriented composite modulus of elasticity when the fibre efficiency parameter is 14, the elastic modulus of the fibre is 10, the elastic modulus of the matrix is 16, the volume fraction of the fibre is 12 and the volume fraction of the matrix is 8.

This implies that;

K = Fibre Efficiency Parameter = 14
Ef = Elastic Modulus of the Fibre = 10
Em = Elastic Modulus of the Matrix = 16
Vf = Volume Fraction of the Fibre = 12
Vm = Volume Fraction of the Matrix = 8

E = KEfVf + EmVm
E = (14)(10)(12) + (16)(8)
E = (1680) + (128)
E = 1808

Therefore, the randomly oriented composite modulus of elasticity is 1808 Pa.

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How to Calculate and Solve for Ratio of Load of Fibre to Matrix | Composites

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The ratio of load of fibre to matrix is illustrated by the image below.

To compute for ratio of load of fibre to matrix, four essential parameters are needed and these parameters are Elastic Modulus of the Fibre (Ef), Elastic Modulus of the Matrix (Em), Volume Fraction of the Fibre (Vf) and Volume Fraction of the Matrix (Vm).

The formula for calculating ratio of load of fibre to matrix:

F(f/m) = EfVf/EmVm

Where:

F(f/m) = Ratio of Load of Fibre to Matrix
Ef = Elastic Modulus of the Fibre
Em = Elastic Modulus of the Matrix
Vf = Volume Fraction of the Fibre
Vm = Volume Fraction of the Matrix

Let’s solve an example;
Find the ratio of load of fibre to matix when the elastic modulus of the fibre is 14, the elastic modulus of the matrix is 10, the volume fraction of the fibre is 6 and the volume fraction of the matrix is 2.

This implies that;

Ef = Elastic Modulus of the Fibre = 14
Em = Elastic Modulus of the Matrix = 10
Vf = Volume Fraction of the Fibre = 6
Vm = Volume Fraction of the Matrix = 2

F(f/m) = EfVf/EmVm
F(f/m) = (14)(6)/(10)(2)
F(f/m) = (84)/(20)
F(f/m) = 4.2

Therefore, the ratio of load of fibre to matrix is 4.2.

Calculating the Elastic Modulus of the Fibre when the Ratio of Load of Fibre to Matrix, the Elastic Modulus of the Matrix, the Volume Fraction of the Fibre and the Volume Fraction of the Matrix is Given.

Ef = F(f/m) x EmVm / Vf

Where:

Ef = Elastic Modulus of the Fibre
F(f/m) = Ratio of Load of Fibre to Matrix
Em = Elastic Modulus of the Matrix
Vf = Volume Fraction of the Fibre
Vm = Volume Fraction of the Matrix

Let’s solve an example;
Find the elastic modulus of the fibre when the ratio of load of fibre to matrix is 20, the elastic modulus of the matrix is 4, the volume fraction of the matrix is 2 and the volume fraction of the fibre is 12.

This implies that;

F(f/m) = Ratio of Load of Fibre to Matrix = 20
Em = Elastic Modulus of the Matrix = 4
Vf = Volume Fraction of the Fibre = 12
Vm = Volume Fraction of the Matrix = 2

Ef = F(f/m) x EmVm / Vf
Ef = 20 x (4)(2) / 12
Ef = 20 x 8 / 12
Ef = 160 / 12
Ef = 13.33

Therefore, the elastic modulus of the fibre is 13.33

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How to Calculate and Solve for Total Composite Strength | Composites

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The total composite strength is illustrated by the image below.

To compute for total composite strength, four essential parameters are needed and these parameters are Strength of the Matrix (σm), Strength of the Fibre (σf), Volume Fraction of the Matrix (Vm) and Volume Fraction of the Fibre (Vf).

The formula for calculating total composite strength:

σc = σmVm + σfVf

Where:

σc = Total Composite Strength
σm = Strength of the Matrix
σf = Strength of the Fibre
Vm = Volume Fraction of the Matrix
Vf = Volume Fraction of the Fibre

Let’s solve an example;
Find the total composite strength when the strength of the matrix is 6, the strength of the fibre is 12, the volume fraction of the matrix is 10 and the volume fraction of the fibre is 4.

This implies that;

σm = Strength of the Matrix = 6
σf = Strength of the Fibre = 12
Vm = Volume Fraction of the Matrix = 10
Vf = Volume Fraction of the Fibre = 4

σc = σmVm + σfVf
σc = (6)(10) + (12)(4)
σc = (60) + (48)
σc = 108

Therefore, the total composite strength is 108 Pa.

Calculating the Strength of the Matrix when the Total Composite Strength, the Strength of the Fibre, the Volume Fraction of the Matrix and the Volume Fraction of the Fibre is Given.

σm = σc – σfVf / Vm

Where:

σm = Strength of the Matrix
σc = Total Composite Strength
σf = Strength of the Fibre
Vm = Volume Fraction of the Matrix
Vf = Volume Fraction of the Fibre

Let’s solve an example;
Find the strength of the matrix when the total composite strength is 40, the strength of the fibre is 10, the volume fractions of the matrix is 5 and the volume fractions of the fibre is 2.

This implies that;

σc = Total Composite Strength = 40
σf = Strength of the Fibre = 10
Vm = Volume Fraction of the Matrix = 5
Vf = Volume Fraction of the Fibre = 2

σm = σc – σfVf / Vm
σm = 40 – (10)(2) / 5
σm = 40 – 20 / 5
σm = 20 / 5
σm = 4

Therefore, the strength of the matrix is 4.

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How to Calculate and Solve for Modulus of Elasticity of Composites Lower Bound | Composites

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The modulus of elasticity of composites lower bound is illustrated by the image below.

To compute for modulus of elasticity of composites lower bound, four essential parameters are needed and these parameters are Modulus of Elasticity of the Matrix (Em), Modulus of Elasticity of the Particle (Ep), Volume Fraction of the Matrix (Vm) and Volume Fraction of the Particle (Vp).

The formula for calculating modulus of elasticity of composites lower bound:

Ec(l) = EmEp/VmEp + VpEm

Where:

Ec(u) = Modulus of Elasticity of Composites Lower Bound
Em =Modulus of Elasticity of the Matrix
Ep = Modulus of Elasticity of the Particle
Vm = Volume Fractions of the Matrix
Vp = Volume Fractions of the Particle

Let’s solve an example;
Find the modulus of elasticity of composites lower bound when the modulus of elasticity of the matrix is 2, the modulus of elasticity of the particle is 6, the volume fractions of the matrix is 8 and the volume fractions of the particle is 4.

This implies that;

Em =Modulus of Elasticity of the Matrix = 2
Ep = Modulus of Elasticity of the Particle = 6
Vm = Volume Fractions of the Matrix = 8
Vp = Volume Fractions of the Particle = 4

Ec(l) = EmEp/VmEp + VpEm
Ec(l) = (2)(6)/(8)(6) + (4)(2)
Ec(l) = (12)/(48) + (8)
Ec(l) = (12)/(56)
Ec(l) = 0.214

Therefore, the modulus of elasticity of composites lower bound is 0.214 Pa.

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