# The first course in continuum mechanics

In response to Zhigang's forum topic on the first course in continuum mechanics, it is so happened that I am also teaching a continuum mechanics course this semester. I shall list our continuum mechanics course outline taught here in Berkeley.

Berkely has its tradition and its special flavour on Continuum Mechanics. The history goes back to Paul Naghdi, Tom Hughes, Jerry Marsden, Juan Simo, David Bogy, Coby Lubliner, Bob Taylor, Karl Pister, James Casey, Geroge Johnson, and some others.

In old days, the emphasis here are finite deformation in accordance with application of differential topolgy on the manifold and applications to nonlinear finite element methods. Throughout the years, the faculty in Berkeley before my time may have molded its own continuum mechanics curriculum --- the so-called ``west coast system''.

We start tensor algebra and analysis first. For the first course, we mainly teach Cartesian tensor in Euclidean space, but we do introduce concepts such as ``push forward'', ``pull-back''. Gateaux derivative, Lie derivative, co-variant and contra-variant tensors, tangent space, etc. , even though we don't require any pre-requsite on this course except advanced mechanics of materials and multivariable calculus.

After priliminary mathematics preparation, we start to discuss Kinematics, which is in the context of finite deformation. We introduce strain measures such as the deformation gradient, the right/left Cauchy-Green tensors, Lagrangian-Green strain, Eulerian-Almansi strain, etc. The emphasis is put on the Polar decomposition of deformation gradient, which brings out rotation tensor and the right/left stretch tensors. We do degenerate finite deformation to infinitesimal case, and discuss infinitesimal deformation in details such as physical meaning of small strain and Saint Venant compatibility conditions, and we don't assume that students already have taken any elasticity course.

After Kinematics, we studies Equilibrium. We introduce concepts such as the Cauchy stress, the 1st and 2nd Piola-Kirchhoff stresses, Biot stress, etc. The emphasis is put on stress transformation and physical meaning of stress measures.

Then, we study Balance laws, material time derivatives, rate of deformation, and varaious stress rates, such as Jaumann rate, Treusdell rate, Oldroyd rate, Green-Naghdi rate etc. Once that is done, we start to discuss material frame-indifference and objectivity in preparation to make a transition to study Constitutive Relations.

The third major part of the course is Constitutive Relations. Currently, we cover four types of constitutive relations:

(1) Elasticity: it includes: linear elasticity, hypo-elasticity, and hyperelasticity; For example, we discuss Neo-Hooken materials, Mooney-Revilin materials, etc.

(2) Quasi-continuum theory: it includes both Embedded Atom Method and Quasi-Continuum Method;

(3) Plasticity: it includes both infinitesimal plasticity theory and large deformation theory. But for finite deformation plasticity, we only discuss rate formulation. We leave E. H. Lee's finite decomposition and hyper-elastic-plastic theory to a subsequent course. This part of the course is intimately connected to non-linear finite element theory. Many materials are taken, for example, from Tom Hughes and Juan Simo's ``Computational Inelasticity'' and Ted Belytschko, Wing Liu and Brian Moran's ``Nonlinear Finite Element Methods''. For instance, we discuss the difference between continuum tangent and Simo-Taylor's algorithm tangent in linearization.

(4) Viscoelasticity

This is the general outline of the course. I give three lectures a week for 14 weeks. We have one midterm examination, one computer project, and one final exam. Each week, students will hand in one HW for about six problems.

In general, this course will be a challenge for many students, but at least I can tell that they enjoy the intellectual challenge.

Because of the time constraint, we cannot cover continuum thermodynamics, variational principles and variational calculus, gradient theories , etc. Those are left for a subsequent advanced continuum mechanics in next spring.

Furthermore, we cannot solve many application problems like Timonsheko and others approaches usually do, but that is left to some other application oriented mechanics courses such as the``Introduction Micromechanics and Nano-mechanics'', which I'll teach next semester. There we shall cover Green's function theory, dislocation theories and dislocation dynamics theory, some fracture mechanics theories, configurational force mechanics, inclusion theories, void growth theory, and multiscale homogenization theories.

### is your lecture note available

Dear Mr. Shaofan Li,

I am looking for a lecture note on continuum mechanics.  I am wondering if there exists any good lecture notes on this topic.

Thank you very much

Phu

### Dear Phu, I have not had

Dear Phu,

I have not had time to organize my own lecture notes. The best book that I would recommend for students self study is:

Nonlinear continuum Mechanics for finite element analysis,

by Bonet and Wood, Cambridge University Press. [1997]

Shaofan Li

### Thanks, Shaofan

Thanks a lot for pointing out this to me.

Phu

### Naghdi's Notes

Is this course taught in ME or CE? Just wondering because when I took it from Paul Naghdi we did not cover nearly as much material in the first continuum mechanics course. Has this changed a great deal, or is this an entirely different course (i.e. not ME185)?

I was fortunate enough to be in the ME version of continuum mechanics the last time Naghdi taught it. His notes/book has been converted to PDF in the subsequent years and I have a copy that I believe I downloaded from the internet. I would be happy to post them but am not sure about copyright issues. At the least, I can send a copy to individuals wanting them.

### Book by Gerhard A. Holzapfel

Hello,

Nonlinear Solid Mechanics: A Continuum Approach for Engineering  by Prof.Gerhard A. Holzapfel.

I used this book and its really reader-friendly!Please send the copy at pingle[at]m.kth.se or please post.I think there may not be any copywrite issue.

### Naghdi's notes

hello ericmock

can you please send me the copies (pdf) of the notes to my email id - karan.rahman@gmail.com ... thank you

### link to Paul Naghdi's lecture notes

The link to Paul Naghdi's lecture notes is here: http://me.berkeley.edu/csml/PMNnotes/ME185Naghdi.pdf

Incidentally a link to the above document is on the imechanica page that lists lecture notes of interest to mechanicians.  See here: http://imechanica.org/node/1551

Many excellent sources of information are provided at the above link.

regards,

Louie

### Introductory book

Hello,

Can anyone suggest me any good Introductory book in Continuum Mechanics? I am going to take the course next september but before doing that I want to study the basics, so I don't want to get involved yet in nonlinearities or other extreme fields.

Thanks a lot

Dimitri

### the book written by Malvern

the book written by Malvern is a comprehensive as well as introductory book in this field.

Best

### Gerhard A. Holzapfel's book

Hi,

I found that Gerhard A. Holzapfel's book (Nonlinear Solid Mechanic - A Continuum Approach for Engineering) is an excellent one. Prof Romesh Batra's book is also nice.

Best regards,