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Chapter 16: Problem 6

For a glass fiber-epoxy matrix combination, the critical ratio of fiber length to fiber diameter is 40. Using the data in Table 16.4, determine the fiber- matrix bond strength.

Short Answer

Expert verified
Answer: The fiber-matrix bond strength for the glass fiber-epoxy matrix combination with a critical ratio of fiber length to fiber diameter of 40 is approximately 16.56 MPa.

Step by step solution

01

Write down the given information

We are given that the critical ratio of fiber length to fiber diameter is 40: \[ \frac{l}{d} = 40 \]
02

Find data in Table 16.4

In Table 16.4, we are given the following values related to glass fiber and epoxy matrix: Fiber tensile strength (σf) = 3450 MPa Fiber diameter (d) = 10 μm Fiber length (l) = 1000 μm Matrix shear strength (τm) = 25 MPa
03

Find the critical length using the given ratio

Since we have the critical ratio of fiber length to fiber diameter, we can find the critical fiber length (lc) using the equation: \[ lc = 40 \times d \] Now, substituting the given fiber diameter (10 μm): \[ lc = 40 \times 10 \] \[ lc = 400 \, \mu m \]
04

Determine the fiber aspect ratio

The fiber aspect ratio is defined as the ratio of fiber length to the critical fiber length: \[ l' = \frac{l}{lc} \] Now, substituting the given fiber length (1000 μm) and the calculated critical fiber length (400 μm): \[ l' = \frac{1000}{400} \] \[ l' = 2.5 \]
05

Calculate the fiber-matrix bond strength

The fiber-matrix bond strength can be calculated using the following equation: \[ \tau_{f-m} = \frac{\sigma_f \, (l'-1)}{2 \, l' \, l} \times lc\] Now, substituting the given fiber tensile strength (3450 MPa), calculated fiber aspect ratio (2.5), fiber length (1000 μm), and critical fiber length (400 μm): \[ \tau_{f-m} = \frac{3450 \, (2.5-1)}{2 \times 2.5 \times 1000} \times 400 \] \[ \tau_{f-m} = \frac{3450 \, (1.5)}{12500} \times 400 \] \[ \tau_{f-m} = 16.56 \, MPa \] The fiber-matrix bond strength for the glass fiber-epoxy matrix combination is approximately 16.56 MPa.

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