The boron fibers are produced by chemical vapor deposition of boron onto fine tungsten wires of dW=10μm. The final wire diameter is dB=75μm. The boron coated wires are then added to an aluminum matrix with VW+B=0.35 fiber volume fraction. The Youngs Moduli of the materials are given below:
EW=410GPa
EB=379GPa
EAl=69GPa
Assuming that the rules of mixtures for binary mixtures applies to ternary systems:
Give a numerical value for the elastic modulus (Ecomp) of the composite in GPa, for loading along the length of the fibers
Ecomp=
E=E1*V1+E2*V2+E3*V3 V1=Boron volume fraction,V2= Tungsten Volume fr. V3=AL vol. fr. V1=pi*d^2/4 *h=pi*75^2/4*h V2=pi*10^2/4*h V1/V2=56.25 V1+V2= 0.35 so V1=0.343, V2=0.007 and V3=1-0.35=0.65, E1=EB , E2=EW, E3=EAL
E=379*0.343 + 410*0.007 + 69*0.65 = 177.71 GPA
To calculate the elastic modulus of the composite (Ecomp) for loading along the length of the fibers, you can use the rule of mixtures for binary mixtures. However, in this case, we have a ternary system with boron fibers, tungsten wires, and an aluminum matrix.
The rule of mixtures states that the overall composite modulus can be calculated as the weighted average of the moduli of the individual components, taking into account the volume fraction of each component.
In this case, we have boron fibers, tungsten wires, and aluminum matrix. Let's assume the volume fractions of the components are:
- Volume fraction of boron fibers (Vfibers)
- Volume fraction of tungsten wires (Vwires)
- Volume fraction of aluminum matrix (Valuminum)
We are given that the fiber volume fraction (VW+B) is 0.35, and we need to calculate the volume fractions of the other components.
To find the volume fraction of the tungsten wires, we can subtract the fiber volume fraction from 1:
Vwires = 1 - VW+B
= 1 - 0.35
= 0.65
To find the volume fraction of the aluminum matrix, we can subtract the sum of the fiber and wire volume fractions from 1:
Valuminum = 1 - (VW+B + Vwires)
= 1 - (0.35 + 0.65)
= 1 - 1
= 0
Since the volume fraction of the aluminum matrix is 0, it means there is no aluminum matrix present in the composite. Therefore, we only need to consider the boron fibers and tungsten wires.
Now, let's calculate the composite elastic modulus (Ecomp) using the rule of mixtures equation for ternary systems:
Ecomp = VB * EB + VW * EW
where VB and VW are the volume fractions of the boron fibers and tungsten wires, respectively, and EB and EW are the corresponding moduli.
Since we don't have an aluminum matrix in this case, we only need to consider the boron fibers and tungsten wires. The volume fractions will be:
VB = VW+B = 0.35 (given)
VW = Vwires = 0.65 (calculated)
Now, substitute these values into the equation:
Ecomp = 0.35 * EB + 0.65 * EW
Given that EB = 379 GPa and EW = 410 GPa, we can substitute these values into the equation:
Ecomp = 0.35 * 379 + 0.65 * 410
Ecomp = 132.65 + 266.5
Ecomp = 399.15 GPa
Therefore, the numerical value for the elastic modulus (Ecomp) of the composite, for loading along the length of the fibers, is approximately 399.15 GPa.