Need Help Determining Math Prerequisites of Contact Mechanics

[megatyler30]

Hi, I'm going to be going into 12th grade.

I have taken (out of applicable classes):
Calculus 1-3
Differential Equations
AP Physics
Statics
Physical Chemistry 1 (Classical Thermodynamics)

Currently Taking:
Linear Algebra
Number Theory
Thermodynamics
Statistical Mechanics (graduate level, chemistry-based, independent study course)

Anyways, I went looking for research and a (math) professor changed his mind about allowing me to do the research topic that would be more interesting to me (due to the math prerequisites). So my question is, for research in contact mechanics, what math would be necessary? The professor ended up mentioning it required partial differential equations and operator theory. So, for operator theory, what books would I have to work through to go through for a sufficient knowledge to be able to use for contact mechanics? What topics would I have to work on beforehand before operator theory and any book suggestions?

His book is Models and Analysis of Quasistatic Contact Variational Approach in the Springer Lecture Notes on Physics Series (Vol. 655), it would be at a little bit more advanced level then this.

 

[TheKracken]

How the heck are you that far ahead in 12th grade? You live in the US op?

 

[megatyler30]

Through extensive dual enrollment, this past school year I have taken 9 dual enrollment classes. Anyways any suggestions? And for additional info, I looked through the preface of his monograph again, and it meantions only functional analysis, so maybe a book on functional analysis that covers some operator theory, or would a full operator theory book be necessary?

 

[romsofia]

How much group/ring theory have you done?

 

[megatyler30]

I have not done any, but I have not seen it as a requirement in any of the functional analysis/operator theory books I have come across.

While looking I have found,
Partial Differential Equations for Scientists and Engineers by Farlow (definitely will be the easier one out of the required ones)
Introductory Functional Analysis with Applications by Erwin Kreyszig (prerequisites as stated, calculus, linear algebra)
The Elements of Operator Theory by Carlos S. Kubrusly (prerequisites, states formal analysis and complex variables would be "helpful", graduate text)
Mathematical Models in Contact Mechanics by by Mircea Sofonea, Andaluzia Matei (not sure if treatment at this level would be necessary, prerequisites: linear algebra, general topology, functional analysis, mechanics of continua)

 

[megatyler30]

Any suggestions? Would these books be good for my purposes? Any better books? Any insight into if the Math. Models in Contact Mechanics would be needed?

 

[verty]

Why not try this book: https://www.amazon.com/dp/0486677486/?tag=pfamazon01-20. This, together with what you are learning in linear algebra, may be enough for now about operators.

The wikipedia page about Heinrich Hertz, under the section "Contact mechanics", says that contact mechanics is based on three theories: the Hertzian theory, the JKR theory and the DMT theory. Perhaps there are more theories but that is where I would start, learning all about those theories.

 

[megatyler30]

Have you read the book? What would the prerequisites for it be? It looks like it would be good, what topics would be missing though due to its age? Also, I need to know tensors, where should I go to learn about them? And I will look into those 3 theories. Thank you very much.

 

[AlephZero]

The essential prerequisites for understanding the math models of the physics would be continuum mechanics and classical mechanics using variational methods.

Beyond that, it depends very much what sub-topic you are actually going to work on. For research into efficient computational methods, you would need some computer science (data structures and algorithms), graph theory, etc. For analytical closed-form solutions you probably need complex analysis, some basic ideas about functional analysis, Fourier series, etc. For research into the physics of the contact mechanism, I don't know - I'm an engineer not a physicist.

 

[megatyler30]

Any book suggestions for the topics? Especially about continuum mechanics since I have not heard of any good ones (as in I haven't come across people saying a specific text on continuum mechanics is good).

For more information, here is the abstract of one of his papers "We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modeled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. "

 

[verty]

Have you read the book? What would the prerequisites for it be? It looks like it would be good, what topics would be missing though due to its age? Also, I need to know tensors, where should I go to learn about them? And I will look into those 3 theories. Thank you very much.

This was about the linear operators book I linked to earlier. I haven't read it but I've read the notes by W.L. Chen as well as Axler's linear algebra book, so I have some idea of the content.

I didn't look inside it till now (this is not correct, I did look inside the book to see what it covered, it is a well-tested russian book with glowing reviews, I didn't look at the writing style because someone who is ready for research is not going to be picky about writing style), I see it is written like a grad book which should not be a problem, but there will be some prerequisites. I'm trying to choose books that aren't too heavy, obviously the following books are not reference books but each should be light and quick to go through, and affordable as well.

https://www.amazon.com/dp/0486650383/?tag=pfamazon01-20

This one has the set theoretic language (one-one correspondence, etc) and the topological terminology (compactness, etc). There may be enough here for you not to need a proof book, but I can't say for sure.

https://www.amazon.com/dp/0201335964/?tag=pfamazon01-20

This one looks to have the right scope for what you need, not too involved and hopefully quick to go through.

https://www.amazon.com/dp/0486623181/?tag=pfamazon01-20

A second book on linear algebra (to follow what you are learning now, although this won't be much more advanced). This should be good preparation for that linear operators book, it has an algebraic flavor as well, I believe.

I meant this to be about operators only, but I suppose there must be nonlinear operators that are more complicated. But I think it would be better to wait until you find something like that, to look it up at the time.

I don't think age is a problem, there are functional analysis books from 1955 still in print, so probably not a lot has changed.

I didn't choose encyclopedic books but you could always follow them with graduate books if you want to, that seems better to me than the other way around.

 

[megatyler30]

Thank you very much for taking the time to find what books would be needed. What topics in Fraleigh and Stoll should I focus on/will need? Do you think any of the 4 books I came across would be necessary? Also would Pinter be appropriate as a replacement to Fraleigh (due to price)? My number theory class (surprisingly) is serving me very well in getting me accustom to proofs. By the way, the textbook used in my Linear Algebra class is "Linear Algebra: A Modern Introduction" by David Poole. Rosenlicht looks good for analysis.

Edit: Also, I have been looking for a good intro to Hertzian Theory on the web, haven't found much. So, I went to look at Hertz's orginal paper, only to find out it's not in English.

 

[verty]

Thank you very much for taking the time to find what books would be needed. What topics in Fraleigh and Stoll should I focus on/will need? Do you think any of the 4 books I came across would be necessary? Also would Pinter be appropriate as a replacement to Fraleigh (due to price)? My number theory class (surprisingly) is serving me very well in getting me accustom to proofs. By the way, the textbook used in my Linear Algebra class is "Linear Algebra: A Modern Introduction" by David Poole. Rosenlicht looks good for analysis.

Edit: Also, I have been looking for a good intro to Hertzian Theory on the web, haven't found much. So, I went to look at Hertz's orginal paper, only to find out it's not in English.

I was hoping you could make it easier by learning all about operators, perhaps that would mean you could shortcut your way to reading that models book you mentioned.

Chapters 1-3 and 5 of Fraleigh is probably all you really want to know at the moment. Stoll, probably the whole book.

Pinter should be fine. I can't give you an answer about tensors, I just don't know them myself. Fischer writes a book called Introduction to Contact Mechanics, perhaps you can find a cheap edition somewhere.

Best of luck.

 

[verty]

Why not get this book on elasticity? As one of the reviewers of a different and more mathematical book said, how difficult can elasticity be? This is an older book and the dyadic notation is apparently obsolete, the index notation (included) and differential forms (not included) are I guess the notations used now. But notation is not a big deal, you can always learn a new one later.

https://www.amazon.com/dp/0486669580/?tag=pfamazon01-20

The benefit of this is that it'll teach you tensors and elasticity, which otherwise [strike]would need two large books[/strike] (some books on elasticity assume continuum mechanics knowledge, and some books on continuum mechanics assume tensor knowledge, but it seems that some CM books contain both, it is very confusing).

PS. As you have said, it is difficult to decide what continuum mechanics book is good. And good is relative, the aim for me is for you to quickly be able to read advanced stuff with some understanding.

 

[megatyler30]

I ending up ordering Rosenlicht, PDE book I mentioned, a operator theory book by Arlen Brown (looks good but not as good as Kubrusly, who knows though, it might end up being better) last night. Was going to Pinter and Stoll but figured that I was already spending too much (as bought 2 books for reference for my statistical mechanics class). I read the preface of the elasticity book, it requires strength of materials which I have not taken. I have found http://www.math.odu.edu/~jhh/counter2.html, I don't know if it would be easy enough to go through quickly, I'll look at it tomorrow.

Edit: Mathematics Applied to Continuum Mechanics by Segel looks promising too. I'm going to try and start on Analysis and PDE (and also operator theory book if initially accessible) as soon as I get them in the mail.

 

[verty]

I ending up ordering Rosenlicht, PDE book I mentioned, a operator theory book by Arlen Brown (looks good but not as good as Kubrusly, who knows though, it might end up being better) last night. Was going to Pinter and Stoll but figured that I was already spending too much (as bought 2 books for reference for my statistical mechanics class). I read the preface of the elasticity book, it requires strength of materials which I have not taken. I have found http://www.math.odu.edu/~jhh/counter2.html, I don't know if it would be easy enough to go through quickly, I'll look at it tomorrow.

Edit: Mathematics Applied to Continuum Mechanics by Segel looks promising too. I'm going to try and start on Analysis and PDE (and also operator theory book if initially accessible) as soon as I get them in the mail.

Ok, no worries. I only focused on operators because I bought Axler's book and didn't like how he covered transformations, it was too dry. So I wanted to make it easier for you with that Stoll book, to ease you into it.

I think a possible difficulty for you is that college education is pretty standardized; subjects that are typically learned in grad school have advanced books that assume the usual graduate knowledge. I'm thinking of John B. Conway's books for example, and perhaps Brown's book is like that. "Operator theory" sounds like a very standard title, it could represent a body of knowledge and style of presentation suited to grad students.

I think you may need to make a choice, whether to try to cut through it all or to go wide and deep in your studies.