Our goal is to be able to simulate solid objects, like wood, metal, and bone, as they deform, bend and break. This is actually a problem from materials science, which uses a thick pile of odd specialized terminology:

- Strain
measures how much the material has been deformed, usually measured from
some "rest" configuration. The symbol is usually e or
epsilon. Units are usually in inches of displacement per inch of
material. For Hooke's law on a one-inch spring, strain is the end
displacement x. Strain can be simulated (for example, integrated
from particle forces), or measured with either accurate calipers or a strain gauge.

- Stress is the force the material exerts to fight back against strain. The symbol is usually the Greek letter sigma σ. The units for stress are in pounds of force per square inch of material. In 1D material, stress is a scalar. In
3D material, stress can be different in different directions, but it
can always be represented by a 3x3 matrix, the "Cauchy stress tensor",
which maps each direction to the force the material exerts along that
direction.

- For a purely elastic material, the stress-strain curve is linear, like a spring: push on it, and it deforms (or deform it, and it pushes); let it go, and it returns to its original shape and stops pushing. This linearity is captured in Hooke's Law: F=-kx in physics terminology; or σ=Ee (stress is stiffness time strain) for engineers. The E there is Young's modulus, which measures the stiffness of an elastic material; this varies from a few thousand psi for rubber to a few million psi for metals. Of course, under enough strain all real materials break down and go nonlinear at some point--that is, no material completely follows Hooke's Law (it's more a guideline, really.)
- By contrast, in a plastic material, like wet clay, the stress-strain curve is highly nonlinear--enough strain causes the material to yield and permanently deform. Many materials are work hardening, displaying an increase in stiffness upon plastic deformation.
- A brittle
material, like dried clay, fractures into pieces instead of
deforming. The stress-strain curve breaks down entirely.
Most metals are elastic for small strains, plastic for some distance
outside that, and then brittle.

In real life, material stresses often depend somewhat on all the above factors--real materials are elastoviscoplastic and subject to creep and fracture. Luckily, most of the time only one or two of these phenomena are important at once!

In code, a typical finite element simulation looks a lot like our little spring system:

// zero out node forcesAll the good stuff above is hidden inside the constitutive_model call.

for (n=0;n<nodes.size();n++) nodes[n].F=vec3(0);

// add element forces to nodes

for (e=0;e<elements.size();e++) {

mat3 strain = compute_strain(e); /* looks at node positions */

mat3 stress = constitutive_model(strain,e); /* e.g., elastoplastic */

for (n=0;n<elements[e].numnodes;n++) /* apply our stress to each node */

nodes[n].F+=apply(stress,elements[e].nodes[n]);

}

// move nodes based on net force

for (n=0;n<nodes.size();n++)

{

vec3 A=nodes[n].F/nodes[n].m; /* F=mA */

nodes[n].V+=dt*A; /* dv/dt = A */

nodes[n].P+=dt*V+0.5*dt*dt*A; /* dp/dt = V */

}

I'm excited by Irving, Teran, and Fedkiw's 2004 paper, which proposes a simple but robust matrix-based approach for keeping track of the rest configuration and computing strains, from which you can then calculate stresses according to any constitutive response you like. He's got a bunch of amazing simulation videos on his site.