Hey there! It's great that you're working on deriving the stress intensity factor for a crack in a plate with a circular hole. It sounds like a complex problem, but there are various methods you can use to solve it.

If you're looking for innovative approaches or combinations with emerging theoretical methods, you might want to consider using numerical methods like the boundary element method or finite element method. These can be powerful tools for solving the integral equation you mentioned.

Alternatively, you could try using asymptotic methods like matched asymptotic expansions or singular perturbations. These methods involve expanding the solution in terms of a small parameter and matching the asymptotic expansions at different regions of the plate.

Hope this helps

There are several numerical and analytical methods that can be used to solve for the stress intensity factor for a crack at the edge of a circular hole in a plate. One of the commonly used numerical methods is the finite element method (FEM), which is widely available in commercial software packages like ANSYS, Abaqus, and COMSOL.

In terms of analytical methods, one approach is to use the method of complex potentials, which reduces the problem to the solution of a system of singular integral equations. Another approach is to use the technique of eigenfunction expansion, which involves the expansion of the stress field in terms of a series of eigenfunctions. As for recommended reference books and papers on this topic, "Elasticity: Theory, Applications, and Numerics" by Martin H. Sadd and "Fracture Mechanics: Fundamentals and Applications" by T.L. Anderson are two widely used textbooks that cover stress intensity factor calculations. Additionally, the Journal of Applied Mechanics and the International Journal of Fracture are both excellent resources for research articles on this topic.

Thank you for attention!

Omar Sabah Al-Dahiree I apologize for the delayed response. Thank you for mentioning the asymptotic and perturbation methods, which have been very helpful to me. I plan to further explore the path you suggested and continue my research in that direction. I'm also in the process of seeking relevant reference books and literature to enhance my foundational knowledge. Thank you once again for your guidance!

Thank you for providing valuable information. I appreciate the suggestions regarding numerical methods like the finite element method (FEM) and analytical approaches such as the method of complex potentials and eigenfunction expansion. I will explore these methods further and consider the recommended books "Elasticity: Theory, Applications, and Numerics" by Martin H. Sadd and "Fracture Mechanics: Fundamentals and Applications" by T.L. Anderson. Your guidance is greatly appreciated. Mehdi Ebadi-Jamkhaneh

Hello, you can solve the problem using finite element software such as Abaqus. The two mentioned books recommended by friends are also good guides. You can also use other books on fracture mechanics.

I hope I have been able to help you.

There exists an analytical solution by Bowie for two symmetrical cracks around a circular hole see attachment.

Heidar Fakouri Lyavoli Thank you for your response. I am currently studying the content of the two recommended books, and they are indeed excellent resources. I appreciate your guidance!

George Xiroudakis Thank you for providing the reference material. It has been helpful in my analysis of general crack forms or other geometries of hole shapes in thin plates. I appreciate your assistance.