## A First Course in Continuum Mechanics

Yuan-cheng Fung

Mentioned 1

Revision of a classic text by a distinguished author. Emphasis is on problem formulation and derivation of governing equations. New edition features increased emphasis on applications. New chapter covers long-term changes in materials under stress.

I want to ask about jelly physics ( http://www.youtube.com/watch?v=I74rJFB_W1k ), where I can find some good place to start making things like that ? I want to make simulation of cars crash and I want use this jelly physics, but I can't find a lot about them. I don't want use existing physics engine, I want write my own :)

Something like what you see in the video you linked to could be accomplished with a mass-spring system. However, as you vary the number of masses and springs, keeping your spring constants the same, you will get wildly varying results. In short, mass-spring systems are not good approximations of a continuum of matter.

Typically, these sorts of animations are created using what is called the Finite Element Method (FEM). The FEM does converge to a continuum, which is nice. And although it does require a bit more know-how than a mass-spring system, it really isn't too bad. The basic idea, derived from the study of continuum mechanics, can be put this way:

1. Break the volume of your object up into many small pieces (elements), usually tetrahedra. Let's call the entire collection of these elements the mesh. You'll actually want to make two copies of this mesh. Label one the "rest" mesh, and the other the "world" mesh. I'll tell you why next.

2. For each tetrahedron in your world mesh, measure how deformed it is relative to its corresponding rest tetrahedron. The measure of how deformed it is is called "strain". This is typically accomplished by first measuring what is known as the deformation gradient (often denoted F). There are several good papers that describe how to do this. Once you have F, one very typical way to define the strain (e) is: e = 1/2(F^T * F) - I. This is known as Green's strain. It is invariant to rotations, which makes it very convenient.

3. Using the properties of the material you are trying to simulate (gelatin, rubber, steel, etc.), and using the strain you measured in the step above, derive the "stress" of each tetrahdron.

4. For each tetrahedron, visit each node (vertex, corner, point (these all mean the same thing)) and average the area-weighted normal vectors (in the rest shape) of the three triangular faces that share that node. Multiply the tetrahedron's stress by that averaged vector, and there's the elastic force acting on that node due to the stress of that tetrahedron. Of course, each node could potentially belong to multiple tetrahedra, so you'll want to be able to sum up these forces.

5. Integrate! There are easy ways to do this, and hard ways. Either way, you'll want to loop over every node in your world mesh and divide its forces by its mass to determine its acceleration. The easy way to proceed from here is to:

• Multiply its acceleration by some small time value dt. This gives you a change in velocity, dv.
• Add dv to the node's current velocity to get a new total velocity.
• Multiply that velocity by dt to get a change in position, dx.
• Add dx to the node's current position to get a new position.

This approach is known as explicit forward Euler integration. You will have to use very small values of dt to get it to work without blowing up, but it is so easy to implement that it works well as a starting point.

6. Repeat steps 2 through 5 for as long as you want.

I've left out a lot of details and fancy extras, but hopefully you can infer a lot of what I've left out. Here is a link to some instructions I used the first time I did this. The webpage contains some useful pseudocode, as well as links to some relevant material.

http://sealab.cs.utah.edu/Courses/CS6967-F08/Project-2/

The following link is also very useful:

http://sealab.cs.utah.edu/Courses/CS6967-F08/FE-notes.pdf

This is a really fun topic, and I wish you the best of luck! If you get stuck, just drop me a comment.

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