Hi,

I cannot visualize your docx properly (a pdf would be recommendable). You may eventually take a look at

Salvadori, A., Gray, L.: “Analytical integrations and SIFs computation in 2D fracture mechanics” – Int. J. Numer. Meth. Engng 2007; 70:445–495

A. Salvadori, “Analytical integrations in 2D BEM elasticity” - “Int. J. Numer. Meth. Engng.”, 53(7), 1695-1719, (2002)

Hope it helps.

Why do not solve it numericaly, it is much more easy

I have now added a pdf file of my previous attachment. I shall read your papers referenced above and contact again , thanks

To answer Mahmoud Hamadiche, closed form solutions of crack dislocation distributions allow simple expressions of crack-tip stresses and crack extension force where the role of physical parameters appears clearly.

Hi Patrick, your attachment cannot be viewed. First make it visible to enable closed form solution to be reached.

Do you have the recent pdf file form I have attached in a minute?

To answer Alberto Salvadory: at the present you do not give us any closed form solution. We are working to find a closed form solution to our singular integral equation

Patrick,

now that I can see your doc, I guess you are right. My formule have been applied to BEM approximations, where we used polynomial approximations of fi(t). You may find them useful in such a context, perhaps. Sorry, I don't have more than that.

Maybe you may find some literature in classical books for your closed form.

Best wishes,

Hi Patrick,

I think the method in the MUSKHELISHVILI 's book may reduce your integral to some Riemann-Hilbert Problems. As you don't have any logarithmic kernels, it may work for you. (Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Books on Physics))

I work on a system of coupled singular integral equations that contains Cauchy and logarithmic singularities related to mixed boundary value problem in Elasticity. Unfortunately, I cannot find some close solution for that particular problem so far.

Regards,

Thanks, I do have MUSKHELISHVILI's book and I am using it

Please see end of attachment.

Thanks, I have taken a quick look at your suggestion. I have used previously (see relation (11) in: P.N.B. Anongba, A Non-Planar Crack Analysis Using Continuously Distributed Sinusoidal Edge Dislocations and Linear Elasticity, Phys. Stat. Sol. (b), 194 (1996) 443-452. ) an expansion of the unknown function in terms of Chebyshev polynomials of first kind. But this was not a solution in closed form. Could we obtain a closed form solution with this method?

Patrick,

Only certain relatively simple crack problems will have closed form solutions. Most crack problems will require some form of numercial analysis. I like the Chebyshev approach because it is elegant and deals accurately with the crack singularities. Good luck with your problem.

Many thanks

Hi

You may eventually look at fractional derivatives... they look a bit like your operator and can have (intricate) analytical solutions... A lot of work on them and their relation to Fox's H function have been published recently (by Gorenflo, Mainardi, Metzler Klafter,...)

Dunnot know if this will help thought...

Hello Patrick,

In my experience, such problems require considerable effort to attain a solution, especially in closed-form. With that said, I hope the attached paper will have what you need to obtain a solution...

Thanks, Luisiana

I have your paper and I will look at it and see if it can help finding a solution

To Luisiana: I do not see how to use your work

To Rimbert: maybe you could show how "fractional derivatives" makes easier further calculations.

To Patrick

I have been able able to deduce the closed form solution to the singular integral equation. May I know the type of research you are carrying out. How will I benefit by revealing this solution to you?

Great we are delighted to see your solution. May be you could make immediately available your solution (in the form of a short note) to the appreciation of both ResearchGate members and followers of the present Q&A. By doing so your solution will be given a date of publication. Then it will be given a usefulness when I will in due course present the treatment of the associated crack problem with reference to your result. I would like to stress that the expected solution is unbounded at the ends of the line of integration.

I must also say that this work is purely academic with no financial benefit.

I am not talking about financial benefit instead I wish to be assured that I will get the credit for the analytical solution in any subsequent publication (journal or conference) from your research.

My solution has a square singularity at plus or minus 'a'.

Interestingly i have numerically verified it to be exact using Gauss-Legendre quadrature.

It is unfortunate that excellent books from the former Soviet Union are unavailable to scholars from Africa. A good treatment of Integral equations (Volterra and Fredholm types) can be found in

Krasnov, M., Kiselev, A. and Makarenko , G. (1971) Problems and exercises in Integral equations, Mir publishers Moscow (In English)

You should be confident since this forum is known from ResearchGate members. If you are right, you should be associated with a first publication with this crack problem.

I have your solution. May I ask you to check if the first form (attached) of the function f(t0) agrees with the one I have given to you after one integration. I want to be sure that every thing is ok before proceeding further.

I don't see how to reconcile the solution by Amaechi J. Anyaegbunam to a particular solution (attached) given by Bilby and Eshelby

x is to be read as t in the Bilby and Eshelby

Thanks, I have taken a quick look at your result for the integral present in the initial form of f(t0). It will be helpful that a qualified mathematician gives its own expression. Concerning the integral equation itself, the method of solution may be numerical using an expansion involving Chebyshev polynomials as suggested earlier by L.N. McCartney from the National Physical Laboratory. The search of a closed form solution remains open.

I have not opened your new file. I do not believe that the integral in initial f(t0) you have calculated previously is right. I am talking about the difference in sign you obtained as compared to my own calculation.

Please let me also say that feel free to remove your file or to let your new tentative to the appreciation of ResearchGate followers.Surely I would quote (if you are right) your result in the references but I feel free not to associate you as co-author in a first report on the associated crack problem.

My result of the integral term in f(to) now agrees with your own. That is what I meant by "rectifying the questionable integral." Thus. that term I said may be negative is actually positive in agreement with your own..

I don't expect you to join me as a co-author. However, I expect you to quote as a reference in your paper that Anyaegbunam provided the closed-form solution via an informal communication in Researchgate.

Please note that the denominator in your integrand of f(to) is to - t while the denominator in my Eq.7 is t - to. This accounts for the apparent difference in the results.

Mr. Amaechi

At the moment I am busy doing something else. I haven't yet opened the file of your second solution. I will let you know when I open your file, sincerely.

I am quite amazed! I have derived a closed-form solution to the singular integral equation posed by Monsieur Anongba and 12 days after none of the 11 other followers of this topic has criticized or approved my solution. Why can't somebody prove my solution wrong? The proof is an elementary exercise! Otherwise could the 'panel of judges' approve my solution. What is the essence of asking a question on research gate?

Apologies for my seeming hurry to rush out solutions. I have corrected a typographical error in my last solution and produced a final and correct solution to the problem. Working conditions are very poor in Nigeria because of incessant cuts in electricity power supply. In this very hot period in Nigeria one is bound to be careless and inattentive due to lack of cooling systems. I hope Prof. Anongba will appreciate my effort in his research work and give me the requisite credit.

I am happy to say that the integral equation I have proposed in the present Q&A corresponds to "Example 43", page 158, in the following book: Ricardo Estrada and Ram P. Kanwal, "Singular Integral Equations", Ed. Birkhäuser, Boston (2000). A treatment is proposed there that we shall follow.

I am able to declare that I have a solution (attached) to the proposed integral equation. The associated crack problem should be presented very soon.

Dear Colleague Professor Anogba.

I have reservations about your esoteric solution. The exponent i.delta appears in it. Can you show that eventually the entire solution will be real valued.

Secondly, the solution is not yet in closed-form because the p.v. of the indicated integrals have not been given in closed form.

How do you intend to evaluate the principal values of integrals with branch cuts?

The solution is real (sum of complex and its conjugate). The integration using the given f(to) is apparently easy. Be sure that this is the solution, sincerely

Sorry, a sign minus appears surely in front of the second term in the given solution.

Hello Professor.

You are saying the integral equation lacks a unique solution. I am very sure my own solution is correct. This contradicts all mathematical ideas since the time of Weierstrass.Something is not right somewhere!

This is just a typing error; again, a sign minus appears surely in front of the second term in the only solution I have given. I will change nothing more in this displayed result. Please let other researcher appreciate.

I have followed an "Operational Approach" in the book by Estrada and Kanwal. This requires to solve a pair of Carleman type equations with a complex parameter (i beta). I independently solved them using the treatment by Muskhelishvili (see his book " Singular Integral Equations, Dover, 1992"). "Non-special end points" are absent. I conclude from the calculations that I can hardly see another acceptable solution to the proposed integral equation.

This singular integral equation whose solution is also given, is basic in a class of problems involving the interface crack under load. In order to give proper credit to former endeavors, please let us follow our "Q&A": Did the solution of the attached singular integral equation used in a previous work involving the interface crack?

With the given f(t), we have in closed forms the dislocation densities, relative displacements of the faces of the crack and crack-tip stresses. The calculation of the crack extension force is under way.

The crack extension force is obtained. The presentation of the study will be given in due course.

Bonjour Professor Anongba

Could you please provide a plot of phi(x) versus x using your solution of 10th march and assuming a = 2, beta = 0.3, xo = 0.2, bC = 1, sigma_c =10, alpha-bar =.2. Recall you stated your original question in terms of t but provided a solution as a function of x.

By the way why is to (or xo) absent in your solution for phi(x) of 10th March?

This is very suspicious!

Also, you have not answered my question about how the p.v. of the integral with branch cuts shall be determined analytically.

t and x are identical. The solution phi(x) has to be given as a function of one variable only with an arbitrary constant const.. You should just wait for the paper, on the crack problem, that I am now preparing. Everything is suited, coherent, conform with previous results on the interface crack under load, up to the expression of the crack extension force.

Let me add that I find no difficulty in integrating with the f(t), I have given.

Tres bien Professeur.

Sadly, I have no interest in fracture mechanics. I was only interested in the Mathematics of the singular integral equation.

Adieu

The new crack problem is given a first paper intitled "Interface crack loaded in tension: introducing an internal shear stress promoted by the Poisson's effect"; see our contributions in ResearchGate

Internal shear stresses develop on both sides of the interface loaded in tension as the result of a difference in the Poisson's contractions in the media. Attached is an image of this internal shear stress. The value of the crack extension force is significantly modified by this shear stress: see our contribution " Interface crack loaded in tension: introducing an internal shear stress promoted by the Poisson's effect" for details.