Composite beams and young's modulus

[ABoul]

Homework Statement

"A polymer, with Young’s modulus 2GPa and yield stress 50MPa, is reinforced by
uniaxially aligned glass fibres which have Young’s modulus 70GPa and fracture at
1400MPa. The volume fraction of fibres is 40%.


(a) Calculate Young’s modulus for the composite in the direction of the fibres.
(b) The composite is loaded parallel to the fibres. What would be the stress on the
matrix when the composite yields ?"

Homework Equations

The Attempt at a Solution

(a) E = xE_f + (1-x)E_m
E = 0.4(70e9) + 0.6(2e9) = 29.2 GPa

firstly, i don't know if the above is right. also, i have o clue how to do part (b).

 

[Nate810]

i tried using the E from part (a) in the equation for yield stress, but this didn't work. any help would be greatly appreciated.

 

[nlightner]

I would like to clarify and provide additional information about composite beams and Young's modulus. Composite beams are structures made from combining two or more materials with different properties to create a new material with improved performance. Young's modulus is a measure of a material's stiffness or ability to resist deformation under stress. It is also known as the elastic modulus and is represented by the letter E.

In this scenario, a polymer with a Young's modulus of 2GPa and a yield stress of 50MPa is being reinforced with uniaxially aligned glass fibers with a Young's modulus of 70GPa and a fracture stress of 1400MPa. The volume fraction of the fibers is 40%.

To calculate the Young's modulus of the composite in the direction of the fibers, we use the rule of mixtures formula, as shown in the attempt at a solution. This formula takes into account the volume fraction and the Young's modulus of each component. The calculated value of 29.2 GPa for the composite's Young's modulus in the direction of the fibers is a reasonable approximation.

For part (b), we need to determine the stress on the matrix (polymer) when the composite yields. Yielding refers to the point at which a material starts to deform plastically under stress. In this case, the composite will yield when the stress on the matrix reaches 50MPa, which is its yield stress. This stress can be calculated using the following formula:

σ_m = σ_c / (1-x)
Where σ_m is the stress on the matrix, σ_c is the yield stress of the composite, and x is the volume fraction of the fibers. Substituting the given values, we get:

σ_m = 50MPa / (1-0.4) = 83.3 MPa

Therefore, when the composite is loaded parallel to the fibers and reaches its yield point, the stress on the matrix will be 83.3 MPa.

In conclusion, composite beams and Young's modulus are important concepts in material science and engineering. By combining materials with different properties, we can create composites with improved performance. The rule of mixtures and the formula for calculating the stress on the matrix when the composite yields are useful tools in analyzing and predicting the behavior of composite materials.