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 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 isYoung'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. The one downside is the dense thicket of matrix math required to disentangle object deformation from simple rotation. This matrix math is both difficult to write and slow to run: in 2004, around 1 second per frame (!) for just 11K elements.

A paper from France in 2005 by Nesme, Payan, and Faure recommended a QR decomposition to find the rotation portion of the matrix, and so achieve 30fps on small meshes of a few thousand elements.