To calculate the stress intensity factor $K_I$ for the crack in the welded plate, we use the following formula for a crack in an infinite plate subjected to a tensile stress:

$K_I = \sigma \sqrt{\pi a}$

Where:

• $K_I$ is the stress intensity factor,
• $\sigma$ is the applied tensile stress,
• $a$ is half the crack length for a center crack,
• $\pi$ is a constant (approximately 3.1416).

Step 1: Find the Applied Stress $\sigma$

The applied stress $\sigma$ is the tensile load divided by the cross-sectional area of the plate:

$\sigma = \frac{P}{A} = \frac{P}{w t}$

Where:

• $P = 20.0 \, \text{kN} = 20,000 \, \text{N}$ (applied tensile load),
• $w = 75.0 \, \text{mm}$ (width of the plate),
• $t$ is the thickness of the plate (not provided, but it’s a necessary parameter).

If the thickness $t$ is not provided, we cannot calculate the exact stress intensity factor $K_I$ without knowing it. However, let's continue assuming you will provide this value.

Step 2: Find Half Crack Length $a$

For a center crack, the crack length is $2a$. Given that the crack length is 40.0 mm, half the crack length $a$ is:

$a = \frac{40.0}{2} = 20.0 \, \text{mm}$

Step 3: Calculate the Stress Intensity Factor $K_I$

Once we have the thickness and hence the applied stress, we will plug the values into the formula for $K_I$:

$K_I = \sigma \sqrt{\pi a}$

If you could provide the thickness $t$ of the plate, I can proceed with the exact calculation.

Would you like to provide that information or ask any questions about the process? Here are some related questions to explore:

1. How does the fracture toughness $K_{\text{IC}}$ relate to the failure criteria in this case?
2. What is the significance of adhesive shear strength in the design of this assembly?
3. How does the thickness of the plate affect the applied stress and ultimately $K_I$?
4. What happens if $K_I$ exceeds $K_{\text{IC}}$, the fracture toughness of aluminium?
5. Why is the crack assumed to behave as a sharp center crack in this context?

Tip: Stress intensity factor $K_I$ is a critical parameter in fracture mechanics to predict the growth of cracks in materials under stress.