. Calculate the following:
a. the modulus of elasticity,
b. the tensile strength,
c. the fraction of the load carried by the fiber,
for the following composite material stressed under isostrain conditions.
The composite consists of a continuous glass‐fiber‐reinforced‐epoxy resin produced by
Using 60 percent by volume of E-glass fibers having a modulus of elasticity of
Ef =7.24× 104 MPa and a tensile strength of 2.4 GPa and a hardened epoxy resin with a
Modulus of Em = 3.1 × 103 MPa and a tensile strength of 0.06 GPa.
To calculate the modulus of elasticity, tensile strength, and the fraction of the load carried by the fiber for the given composite material under isostrain conditions, we can use the following formulas:
a) Modulus of Elasticity (Ec):
Ec = Ef * Vf + Em * Vm
Where:
Ef = Modulus of elasticity of the fiber = 7.24 × 10^4 MPa
Em = Modulus of elasticity of the matrix = 3.1 × 10^3 MPa
Vf = Volume fraction of the fiber = 60% = 0.60
Vm = Volume fraction of the matrix = 100% - 60% = 40% = 0.40
Ec = (7.24 × 10^4 MPa * 0.60) + (3.1 × 10^3 MPa * 0.40)
Ec = 43440 MPa + 1240 MPa
Ec = 44680 MPa
b) Tensile Strength (σc):
σc = σf * Vf + σm * Vm
Where:
σf = Tensile strength of the fiber = 2.4 GPa = 2400 MPa
σm = Tensile strength of the matrix = 0.06 GPa = 60 MPa
σc = (2400 MPa * 0.60) + (60 MPa * 0.40)
σc = 1440 MPa + 24 MPa
σc = 1464 MPa
c) Fraction of the load carried by the fiber (η_fiber):
η_fiber = Ef * Vf / Ec
η_fiber = (7.24 × 10^4 MPa * 0.60) / 44680 MPa
η_fiber = (43440 MPa) / 44680 MPa
η_fiber = 0.972
The fraction of the load carried by the fiber is approximately 0.972, or 97.2%.
In summary:
a) Modulus of Elasticity (Ec) = 44680 MPa
b) Tensile Strength (σc) = 1464 MPa
c) Fraction of the load carried by the fiber (η_fiber) = 0.972, or 97.2%
a. That's a complicated question! It's like asking a clown to calculate the square root of a banana. But fear not, I have my juggling balls ready. To calculate the modulus of elasticity for our composite material, we need to consider the volume fraction of each component. Since we're using 60 percent by volume of E-glass fibers and 40 percent by volume of hardened epoxy resin, we can use the rule of mixtures to calculate the composite modulus.
The composite modulus of elasticity (Ec) can be calculated using the following formula:
Ec = Vf * Ef + Vm * Em
Where:
Vf = Volume fraction of the fiber = 0.6
Ef = Modulus of elasticity of the fiber = 7.24 × 10^4 MPa
Vm = Volume fraction of the matrix = 0.4
Em = Modulus of elasticity of the matrix = 3.1 × 10^3 MPa
Let's plug in the numbers and juggle them around:
Ec = (0.6 * 7.24 × 10^4 MPa) + (0.4 * 3.1 × 10^3 MPa)
Now, let me gather my juggling balls and calculate this for you!
*begins juggling*
Ec = (0.6 * 7.24 × 10^4 MPa) + (0.4 * 3.1 × 10^3 MPa)
*more juggling*
Okay, I think I've got it! The modulus of elasticity for our composite material under isostrain conditions is (drumroll, please) approximately 4.94 × 10^4 MPa.
Tadaaa!
To calculate the required values, we can use the rule of mixtures for composites:
a. Modulus of elasticity (E):
The modulus of elasticity for the composite material can be calculated using the rule of mixtures:
E = Vf * Ef + Vm * Em
Where:
E = Modulus of elasticity of composite material
Vf = Volume fraction of fiber
Ef = Modulus of elasticity of fiber
Vm = Volume fraction of matrix
Em = Modulus of elasticity of matrix
Given the volume fraction of fiber (Vf = 0.6), the modulus of elasticity of the fiber (Ef = 7.24 × 10^4 MPa), and the volume fraction of matrix (Vm = 0.4), and the modulus of elasticity of the matrix (Em = 3.1 × 10^3 MPa), we can substitute these values into the equation to calculate the modulus of elasticity for the composite material:
E = 0.6 * 7.24 × 10^4 MPa + 0.4 * 3.1 × 10^3 MPa
b. Tensile strength (σ):
The tensile strength for the composite material can also be calculated using the rule of mixtures:
σ = Vf * σf + Vm * σm
Where:
σ = Tensile strength of composite material
Vf = Volume fraction of fiber
σf = Tensile strength of fiber
Vm = Volume fraction of matrix
σm = Tensile strength of matrix
Given the volume fraction of fiber (Vf = 0.6), the tensile strength of the fiber (σf = 2.4 GPa), and the volume fraction of matrix (Vm = 0.4), and the tensile strength of the matrix (σm = 0.06 GPa), we can substitute these values into the equation to calculate the tensile strength for the composite material:
σ = 0.6 * 2.4 GPa + 0.4 * 0.06 GPa
c. Fraction of load carried by the fiber:
The fraction of the load carried by the fiber can be calculated using the rule of mixtures as well. In this case, it is the same as the volume fraction of the fiber:
Fraction of load carried by the fiber = Vf = 0.6
Substituting the given values, we can calculate the fraction of the load carried by the fiber.
Please note that the units need to be consistent for accurate calculations.
To calculate the modulus of elasticity (E), the tensile strength (σ), and the fraction of the load carried by the fiber (f), we can use the rule of mixtures for composite materials.
For the modulus of elasticity (E), the rule of mixtures states that it can be calculated by taking the weighted average of the moduli of the individual components based on their volume fractions.
a. Modulus of Elasticity (E):
E = Vf * Ef + Vm * Em
where:
Vf = Volume fraction of fiber
Ef = Modulus of elasticity of fiber
Vm = Volume fraction of matrix
Em = Modulus of elasticity of matrix
In this case, Vf = 0.60 (60% by volume of E-glass fibers) and Vm = 0.40 (40% by volume of epoxy resin), Ef = 7.24 × 10^4 MPa, and Em = 3.1 × 10^3 MPa. Plug in these values to get the modulus of elasticity (E) for the composite material.
b. Tensile Strength (σ):
The tensile strength of a composite material is primarily determined by the strength of the weakest component. In this case, the tensile strength of the composite will be the tensile strength of the epoxy resin because it is lower than the tensile strength of the glass fibers. Therefore, the tensile strength of the composite material is 0.06 GPa.
c. Fraction of Load Carried by the Fiber (f):
The fraction of the load carried by the fiber is determined by dividing the modulus of elasticity of the fiber by the modulus of elasticity of the composite.
f = Ef / E
Plugging in the values, we can calculate the fraction of the load carried by the fiber.
Note: The given values for the modulus of elasticity and tensile strength are not in the same units. Make sure you convert them to a consistent unit (e.g., MPa or GPa) for accurate calculations.