Continuum Mechanics - Solids and Fluids

[Astronuc]

I started reading a great textbook and thought I would share some of its finer points. It is a great introduction to mechanics of solids and liquids, although the title explicitly states Earth and Environmental Sciences.

http://www.science.mcmaster.ca/~geo/faculty/emeriti/middleton/index.html ," Cambridge University Press, 1994.

1. Introduction
2. Review of elementary mechanics
3. Dimensional analysis and theory of models
4. Stress
5. Pressure, buoyancy and consolidation
6. Flow through porous media
7. Strain
8. Elasticity
9. Viscous fluids
10. Flow of natural materials
11. Turbulence
12. Thermal convection
Appendices
References

From the point of view of classical physics, matter is generally assumed to take one of three forms: a point mass, a rigid body, or a continuum. All three forms represent an idealization of real matter but the level of idealization decreases as we move from a point mass to a continuum. Though the concept of a point mass is an extreme idealization of real bodies of matter, it is a simplification of the real world that works very well in some applications.

It was Newton himself who first showed that one does not need to know the exact distribution of mass within the Earth (or other planet) in order to apply the 'universal' law of gravitation. For most purposes, one can assume the mass is concentrated at a point, the center of mass.

This is the case for a central body force (gravity). In reality, variations in density (e.g. granite or basalt compared to water) or elevation (mountains) cause perturbations in the gravitational force field.

For many other problems concerned with solid bodies, one must be concerned with the size and shape of the material, as well as its mass. For example, a boulder moved by water or a round object rolling down an incline.

When solids or fluids deform in response to applied forces, one must be concerned with the distribution of material properties (e.g., mass, resistance to deformation) with the material. There is still important simplying assumptions to be made, which is the 'continuum hypothesis'.

The useful range in length scale for the continuum hypothesis has two limits: the lower limit is defined by an element of volume that is much bigger than an atom or molecule, and the upper limit is defined by an element of volume that is smaller than any important spatial variation in material properties.

This is an important point to keep in mind, particularly at the beginning of model development.

The issue of element size is more complicated when significant thermal gradients are present since properties like density and strength, or solubility of different phases may be significantly affected by temperature (internal energy). When a radiation field is imposed on a material, the modeling can be even more complex.

The reason one uses the continuum hypothesis is simply that it allows us to use differential calculus to analyze the properties of a material and its motion and deformation, i.e. its behavior. Continuous changes, or gradients, in physical properties and forces define mechanical problems, and differential calculus is the mathematical tool that treats such gradients precisely and efficiently. In using the continuum hypothesis, we are not limited to cases without abrupt boundaries between different materials. In these cases, we have to define the appropriate boundary conditions and then we can use continuum mechanics to describe what happens within those boundaries.

There are relevant topics in the other tutorials in this section and the physics tutorials sections, as well as the forums Mechanical & Aerospace Engineering and Materials & Chemical Engineering

Some definitions:

Statics - study of equilibrium of forces, i.e. there is no acceleration because the net forces and net moments are null.

Kinematics - study of motion, exclusive of masses and forces.

Dynamics - study of the relationship of motion and forces

All three are included within mechanics.

 

[Astronuc]

So notes on Mechanics of Solids

Advanced Mechanics of Solids
http://www.engin.brown.edu/courses/en175/notes_frame.htm

These notes were written by Prof. A. F. Bower during the fall semesters of 1998 and 1999, and updated during the Fall semester of 2005. The notes are intended for individual study in Engineering 175, Advanced Mechanics of Solids at Brown University. Please seek the author's permission before reproducing the notes for any other purpose.

Division of Engineering
Brown University
Providence RI 02912

More advanced courses
EN222 MECHANICS OF SOLIDS
http://www.engin.brown.edu/courses/en222/notes_frame.htm

EN224: Linear Elasticity
http://www.engin.brown.edu/courses/en224/notes.htm


It is very kind of Professor Bower and Brown University to make these notes available.

Bower's own site - http://solidmechanics.org/contents.htm

 

[Astronuc]

Or if you want to jump into it.

Nice introduction to Vector and Tensor Mathematics
http://www.eng.uc.edu/~gbeaucag/RyanBreese/Math/mathematics.htm

which leads to practical applications in continuum mechanics
http://www.eng.uc.edu/~gbeaucag/RyanBreese/Mechanics/mechanics.htm

 

[MathematicalPhysicist]

Astronuc, i had a look in my university library, and i found two books on continuum mechanics:
1. schaum outline.
2. a four volumes book by ivanovich sedov, which is called a course on continuum mechanics.
i think that the second is more advanced than the first book, have you tried any of them?

 

[Astronuc]

I may have Schaum's outline on Continuum Mechanics, but I'd have to dig through my library.

Leonid Ivanovich Sedov (b. 1907 - d. 2000) is a big name in Continuum Mechanics. I am sure the text, Course on CM is very involved, although I am not familiar with it.

 

[uiulic]

Mase's Continuum Mechanics is really a good book. It gives concise concepts before showing concrete examples plus extra exercises. Other good texts are Malvern, Fung, Eringen,...

Currently, not many people are writing continuum mechanics books. Three reasons for this maybe. 1: The old texts were good enough so that there is no need for any updating. 2: Not many people are doing related research so that the speed of the updating is too slow.3: Classical Continuum mechanics
do not have many applications (at least the theory is not necessary) or even some theory are out of date.

On the contrary, fluid mechanics as a part of continuum mechancis, has been a hot topic now.