Perhaps a good book would be

http://www.amazon.com/Mathematics-Missed-Need-Graduate-School/dp/0521792...

I haven't read it, but I've heard a friend say good things about it.  There's a cheaper paperback version as well, but I can't seem to find it ..

 -Nachiket

Siddharth,

Typically a textbook covering a particular area in mechanics has the required mathematics covered at the beginning of the text or at least the references could be found in the introductory chapters.

Can you describe the mathematical equations you are talking about? Because there are plenty of references for a wide variety of mathematical topics.

~ Nimish.

Complex Variables - Carrier, Crook and Peterson

Tensors - Synge and Schild's text is good though physics oriented

Real Analysis - Not too many 'engineer oriented' books here - e.g. Rudin is fairly rough going

Boundary Value Problems - Gakhov

Calculus of Variations -  Gelfand's book is concise and thorough

Fourier Transforms - Sneddon

Integral Equations - Tricomi is a great introductory text

Partial Differential Equations - Sobolev's text is cheap, but written for mathematicians & physicists

Linear Algebra - Strang's textbook is usually adequate

Hope that helps.

-NS

Thank you everyone,

 The main issue I face is the conversion of a particular phenomenon into a mathematical description of the same. Since engineering mathematics is more oriented towards solving problems with givens and unknowns and the interplay of equations,  when the time comes to come up with your knowns and unknowns and approximations, I have a hard time. 

I attended a lecture by Prof. Zhigang Suo at Georgia Tech, and while I could follow the mathematics, I had no idea how he came up with equations themselves[ though I unerstood them].

 I believe a lot of practice is  the only thing  that can really help you intuitively get at a problem.

 Regards,

 Siddharth

Dear Siddharth,

I presume you are from India. Knowing your undergraduate engineering college would have been helpful a bit in fine-tuning the answer. (... I guess I can't help but assume University of Pune in answering you.)

As a general comment, for fracture mechanics, what you need is not so much of mathematics as, first of all, mechanics.

1. If it was University of Pune (or, to a lesser extent, COEP), you need to strengthen your UG mathematics by quickly going over Kreyszig---the parts missing from your UG syllabus.

Greenberg's explanations are actually better for the topics he covers, and his text really is written for engineers.

Make sure you go over the complex potential theory. I see no need to solve problems, but don't skip over the illustrative examples either.

2. Go through a good elasticity book. The analytical solutions in fracture mechanics almost always assume only plain (i.e. isotropic homogeneous linear) elasticity. Here, Sadd's book is especially recommended over others. But, for a slightly different flavor (more or less at the same level), also see Solecki et al. (http://www.imechanica.org/node/4067).

Make sure to know how potential functions of various sorts are used in elasticity. To quickly develop a good sense about this, it's actually better to go through the experimental stress analysis book by Riley and Dalley. Really. You will get to know where to put Airy and Westergaard and others far quicker this way. And, those photoelasticity photographs will help fix ideas in a unique way.

Get comfortable with the idea that the "ansatz" or the assumed form for the solution/potential form is purely arbitrary, even though no self-respecting author of mechanics/analytical mathematics would ever highlight this particular part about the analytical theory. What that means is simply that most of analysis really boils down just to twiddling with parameters/coefficients to make the assumed solution behave right with the (assumed) BCs. BTW, this, precisely, is the reason why the detailed expressions for the crack-tip singular field differs from one author to another (if you have noticed that they actually do).

And while swimming in those equations, make sure to know what no book, whether text or reference, will tell you: Make sure to know the correct structure of the entire theoretical edifice of solid mechanics. Essentially, it is this: displacements <--> strains <--> stresses <--> traction/force boundary conditions. The gradient part arises in the first pair of arrows. I found it painfully, after wasting months and years about getting just this part right.(Yes, I did!) Shames, in his introductory text (UG) now-a-days names these relations, but in a slightly more complex way. Similarly, Carlos Fellippa, in his FEM notes, gives more details about these relations but those extra details only serve to make the issue more complex (in my humble opinion). The one I gave here works for me (and worked for my students). See what works better for you. But keep that structure at the back of the mind while going through any book having mathematics of mechanics... It is easy to get lost. (For example, you can define a stress function too and that makes the order of the theory seem to go haywire.)

3. Don't get intimidated by the size of the multivolume handbook on Fracture ed. by Liebowitz. If you have covered the point 2. above, it's actually better to go through this handbook than attempt to understand the fracture mechanics theory through textbooks on FM proper. Really. The handbook does give you step-by-step derivations of analytical expressions with very detailed explanations, something that texts often only skim over.

4. It would be useful to do a course on FEM because that way, the idea of using "ansatz" seems to begin to become more "natural."

5. The other books recommended by others are pretty fine, but more from the technical mathematical viewpoint.

Yet, when it comes to me, I think that to really understand mathematics, it is enormously helpful to read books about mathematics rather than only those that are on it. In this spirit, I would like to recommend a few books, though they hardly have anything directly to do with mechanics/fracture mechanics as such.

I would recommend Morris Kline's books.

Also, a 3-volume book, orig. in Russian and translated by the American profs in '60s of the title: "Mathematics: Its Content, Methods and Meaning," MIT Press. Caveat. It tries to brainwash one into "dialectical materialism." Yet, one can read the book keeping that part aside. As a fact of the matter, the authors are so brilliant mathematicians, they actually end up delivering very nice insights about the actual physical nature of the mathematical abstractions despite their Hegelian intentions. (Sometimes they give philosophically wrong sort of clues. But it's OK. The advantages outweigh the possibility of internalizing errors. And, errors, everyone makes---whether Russians or Americans.)

If you wish to develop a really good understanding of Calculus of Variations, the one by Lanczos is the best of the lot. Indeed no other book comes even close. However, (i) even Lanczos is still not good enough for what I was looking for, and (ii) I have not finished all pages of it myself (but enough of it that I can recommend). Lanczos is good but not as good as Feynman is in his tiny book for the layman on QED. (And then, I don't believe Feynman got QM right either!! But he was the best when it came to explaining QM/QED in simple terms.)

6. I wish I could recommend one or two books about energy methods of mechanics above others. Unfortunately, no mechanician I know of measures up to task---or half the task---or even the quarter of it. Whether in books or technical articles or class notes. (I have been comprehensive in my search.)

With that said, perhaps, J. N. Reddy is more accessible than the rest (I mean his energy methods book, not the FEM ones). Again, I have not finished it but made occasional reference to it.

7. Finally, permit me to be vague. There was one book on mechanics or fracture mechanics which I once saw in the British Library in Pune. This book would help bridge the gap between elasticity books and the FM books. But I no longer remember its title or author. From what I vaguely remember, it was a Chinese name, a professor probably from Australia (or New Zealand---I spotted the book in the British Library.) This book was valuable from one distinctive angle: the author very directly gives a comparative table of the various "ansatz"s that can be used for different potential/stress functions. No other author ever does that. (My hypothesis is that the reason for this is that it would make the arbitrary nature of these functions too plain and open, and thereby, take away the prestige of "exactness" accorded to the analytical theory.) So, that book was different.

8. Hope this helps. But then, again, I am not an expert in the mathematics of fracture either... Just wrote down what all has helped me more than others, even though, of course, there is a valuable place for the more direct books and textbooks as well...

- - - - -
I remain jobless---and I remain being targeted by the Americans on a daily basis, including psychically, but to a noticeably lesser extent since I began talking about it here at iMechanica.

Respected Ajit Sir,

Your 8 points on Mathematics for Fracture Mechanics are really goodone.

But I am on the other side of the river...From the mathematics department(IITR) and pursuing Phd in the field of fracture mechanics.So, please suggest few points regarding people like us who don't have a strong background of mechanics/elasticity. 

one more thing i want to ask is ...what's the scope of the people (from mathematics background)in the field of fracture mechanics?

 kuldeep

Here is a method that has worked for me. I'm not part of an academic institution and learn through independent study.
1. Settle on a specific problem - something modest, but beyond your current abilities and in the direction you want to go.
2. Read everything you can find about it.
3. Identify the parts you can't understand - it's almost always the math.
4. Dig around in the library and on the net until you find reference materials that seem to start at your level of understanding. You may have to backtrack to the next lower tier of complexity if everything seems beyond your grasp.
5. Read. Work on the problem and then reread when you get stuck.
6. Get some presonal math software tools to help you explore the math - things like Mathcad or FlexPDE.
7. Focus on getting a useful answer - something that can be published. There is nothing like committing to write a paper to make you learn.

can someone help me to show that the volume is constant "in plastic region" and not constant "in elastic reigon "from first principle"