Modern references for continuum mechanics

Question

I'm wondering what some standard, modern references might be for continuum mechanics. I imagine most references are probably more used by mechanical engineers than physicists but it's still a classical mechanics question.

This came up in a conversation with my father (who is a mechanical engineer). I was curious to see the types of mathematics they use in stress analysis. Complex analysis gets used to study 2-dimensional isotropic material here:

N.I. Muskhelishvili. Some basic problems of the mathematical theory of elasticity. 3rd edition, Moscow-Leningrad. 1949. (Translated by J. R. M. Radok. Noordhoff. 1953.)

but that's quite an old reference and my library does not have it. Any favourite more modern and easy-to-find references?

I'm fine with mathematical sophistication (I'm a mathematician) but I'm not particularly seeking it out. I'm looking for the kind of references that would be valued by physicists and engineers. References "for math types" are fine too but that's not really what I'm after.

Thanks!

Answer 1

The extension of elasticity in continuum mechanics would be contact mechanics in the elastic half space. Here are some example books.

1. Contact Mechanics
2. Contact Mechanics III

Answer 2

I understand there is a new book by Michael Spivak called "Physics for Mathematicians" with Vol 1. being on Mechanics. Based on some notes for the first chapters, available here http://www.math.uga.edu/~shifrin/Spivak_physics.pdf it looks pretty wonderful and you might particularly like it since it is aimed at teaching Mechanics to Mathematicians.

Answer 3

The classic reference is Truedell and Noll's "Non-linear field theories of mechanics", published in 1965 and republished (in several languages) 4 or 5 times since. Other books worth using: Mort Gurtin (1981, 2010), R. Bowen (available on his Texas A&M website), L. Malvern (1969 or so, old but good).

Answer 4

My favorite is: "Mechanical Vibrations". I find it interesting because it gives the tensor notation for small vibrations. It's very practical oriented and it's cheap (Dover published):

http://www.amazon.com/Mechanical-Vibrations-Dover-Books-Engineering/dp/0486647854

Answer 5

Mathematical foundations of elasticity by Marsden and Hughes. Published 1983 and then republished by Dover 1994, so while not exactly recent it's very modern compared to the one by Muskhelishvili. It uses differential forms, lie groups, and bifurcation theory and all that.

Answer 6

I walked into my local Mech. E. department and had a chat with a continuum mechanist. Two books he showed me that looked like pretty good answers were:

Marcelo Epstein and Marek Elzanowski. Material inhomogeneities and their evolution: a geometric approach.

Marcelo Epstein. The geometrical language of continuum mechanics.

One thing I like about these references is they're fairly explicit about the kinds of mathematics they're using. I haven't read them in much detail but a possible downside is there appears to be relatively few detailed families of applications.

Answer 7

A good one is J. H. Heinbockel- Introduction to Tensor Calculus and Continuum Mechanics. I guess that's the kind of mathematics an engineer would use, but I don't know for sure.

You can download it here.

Answer 8

I don't know about very modern references, but it's hard to get better than Landau & Lifschitz: Theory of Elasticity, as far as I imagine most physicists are concerned.