Continuum mechanics (NMMO 401)

Webpages of the lecture "Continuum mechanics" (NMMO 401), winter semester 2019/2020.

Introduction

Contact

If you need something, please let me know. You can talk to me before/after the lecture, or we can fix an appointment via email (prusv@karlin.mff.cuni.cz).

Schedule

See the study information system SIS.

Consultations

I am ready to talk to you each Thursday after the lecture/tutorial. If you are not fine with that, please write me an email (prusv@karlin.mff.cuni.cz), and we meet at a time suitable for you. Whenever you think that you do not understand something, please speak up. I really mean it. The main objective of the lectures and tutorials is to explain things that can be hardly read from books, and to help you to understand the subject. Consultations are a natural part of the process of learning!

Do not forget to talk to your peers! If you think that the homework problems are too difficult, you can try to ask your friends for an advice. (Note that advice is something else that "copy and paste".)

Tutorials

If you want to get the credits for the tutorial you must:

• solve the homework problems on time,
• attend the tutorials.

If you are not happy with the rules, we can talk about alternatives. (For example, if you find the tutorial extremely boring, then we can make an agreement that you do not need to attend the tutorials provided that you will solve some extra problems.) I am ready to talk about such things only at the beginning of the semester. There is no chance to discuss these rules at the end of the semester. (Except, of course, of emergency situations such as a long illness.)

Homeworks

Please submit the homeworks on time. You can even scan your solution and send it to me via email (prusv@karlin.mff.cuni.cz). A PDF file with the homeworks is available here.

Problems

You can find some problems in the traditional set of problems. (In Czech.) If you are not fluent in Czech, do not worry, you still have a chance to practice. Many problems can be found for example in the textbook by Gurtin, M. E.; Fried, E.; Anand, L.: The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010.

Literature

The first part of the lecture in mostly covered in the book by Gurtin, M. E.; Fried, E.; Anand, L.: The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010. (I try to follow their notation.) Another reasonable book is the book by Ogden, R.: Nonlinear elastic deformations, Dover, 1997. If you want a Czech textbook, you can try the book by Maršík, F.: Termodynamika kontinua, Academia, Praha, 1999.

Exam

Organisation of the exam

The exam is an oral exam, and it consists of three parts.

• Proof of some simple theorem. You are asked to derive the so-called Beltrami-Michell equations. In presenting the proof, you can use your own notes!
• Present a solution to a problem discussed in a scientific paper, see below. The objective is to show that you know what is the paper about, and what are the used methods and conclusions. In discussing the problem, you can use your own notes!
• During our conversation we will definitely encounter some notions from the field of continuum mechanics. You could be asked to explain some of the notions. (Expect questions of the type "What is the Cauchy stress tensor" and so on.) Detailed list of the definitions/theorems you are expected to know can be found below.

Once you sign up for the exam in the Study Information System, please let me know the title of the chosen scientific paper we shall discuss during the exam.

Scientific paper

You must be able, using your own notes, describe the problem that is being solved and its solution. You must show that you

• understand the problem setting,
• understand the assumptions used in the solution (assumptions concerning boundary conditions, omission of some apparently small terms in the equations and so on),
• can explain in detail the whole solution process,
• can interpret the solution.

I do not want you to reproduce, unless stated otherwise, the numerical computations used in the papers. In this respect you can trust the authors.

List of problems/papers/sources

If you want to present a different paper (something you are interested in) it is not a problem. If it is the case, please ask me in advance to approve your choice.

• Drag acting on a sphere moving with constant velocity in incompressible Navier--Stokes fluid. ("Stokes drag".) The calculation is described in Landau, Lifschitz: Fluid mechanics, Pergamon Press, Oxford, 1966 or in Brdička: Mechanika kontinua, Academia, Praha, 2011. (The approach presented in Brdička is, in my opinion, a better one.)
• Pressure singularity in reentrant corners. The calculation is described in W. R. Dean and P. E. Montagnon (1949). On the steady motion of viscous liquid in a corner. Mathematical Proceedings of the Cambridge Philosophical Society, 45, 389--394. doi:10.1017/S0305004100025019. (Ignore Section 3, Section 5 and Section 6.)
• Finite strain solutions for a compressible elastic solid. The calculation is described in Carroll, M. M., & Horgan, C. O. (1990). Finite strain solutions for a compressible elastic solid. Quarterly of applied mathematics, 48(4), 767--780. http://www.jstor.org/stable/43637678 (Discuss only one of the deformations described in the paper. The choice is yours.)
• Oscillatory shearing of nonlinearly elastic solids. The calculation is described in Carroll, M. M. (1974). Oscillatory shearing of nonlinearly elastic solids, Journal of Applied Mathematics and Physics 25(83). doi: 10.1007/BF01602111.
• Torsion of a right-circular cylinder made of a hyperelastic solid. The calculation is described in Rivlin, R. S. (1948). Large elastic deformations of isotropic materials. III. Some simple problems in cylindrical polar co-ordinates. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 240(823), 509-525. doi: 10.1098/rsta.1948.0004.
• Determination of the stretch and rotation in the polar decomposition of the deformation gradient -- a useful theorem in matrix analysis. The calculation is described in Hoger, A. and Carlson, D. E. (1984). Determination of the stretch and rotation in the polar decomposition of the deformation gradient. Quarterly of applied mathematics, 42(1), 113-117. doi: 10.2307/43637254.
• Yet another expression for the stress power. The calculation is described in Fitzgerald, J. E. (1980). A tensorial Hencky measure of strain and strain rate for finite deformations, J. Appl. Phys. 51, 5111. doi: 10.2307/43637254

All the papers/sources can be downloaded here.

Syllabus

Detailed syllabus that matches the lecture given in 2018/2019 is available here. Updated syllabus for the academic year 2019/2020 will be available at the end of semester.