Finite Element Method (FEM) Simulation

• 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. 
  

Practice--Finite Elements

	// zero out node forces
	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 */
	}

Implementation