Investigation_of_a_Crack_Detachment_Problem

1. Investigation of a Crack Detachment Problem

INVESTIGATION OF A
CRACK DETACHMENT
PROBLEM
COURSE PROJECT
AUTHOR: ARTEM RUBANOV

2. What is the problem?

WHAT IS THE PROBLEM?
We have an elastic layer.
It vibrates.
There is a crack on the bottom.
We apply a load on the top.
We want to know: how does the crack open?

3. Why is it difficult?

WHY IS IT DIFFICULT?
The lower surface has mixed conditions:
• Most parts are fixed (no movement).
• The crack is free (no stress).
The math gives an integral equation for the crack
opening.
But the equation has a problem — the kernel does not
converge.

4. How to fix it?

HOW TO FIX IT?
We split the kernel into two parts:
• Main part (handled exactly)
• Remaining part (nice and smooth)
Then we get a well-behaved equation.
This is called operator splitting or regularization.

5. Solution methods

SOLUTION METHODS
Two simple ways to solve it:
1.Analytical — invert the main part, get a Fredholm
equation of the 2nd kind.
2.Numerical — replace integrals with sums (direct
discretization).
Both work because the bad part has been removed.

6. Conclusion

CONCLUSION
We have:
• A clear mathematical model of crack detachment.
• A way to handle the non-converging kernel.
• Two practical methods to find the crack opening.
Result: The problem is solved — at least on paper!

7. Thank you for your attention! If you have any questions, please ask.

THANK YOU FOR YOUR ATTENTION!
IF YOU HAVE ANY QUESTIONS,
PLEASE ASK.