Hooke's Law

Hooke's Law states that:

"The extension of a spring is directly proportional to the load applied, provided the limit of proportionality is not exceeded."

The stiffness (k) of a spring, measured in Nm^-1 (Newtons per Meter) is the force required to stretch it by 1 meter.

So: F=ke (Force = Stiffness x Extension)

F=kx

where F is the force required to push the spring, k is the spring constant (stiffness), different for every spring and x is the distance the spring is pushed or pulles. If we are pushing, though, the spring will be pushing back on us with an equal and opposite force. This can obviously be expressed as:

F=-kx

where F is the force of the spring pushing back on whatever force is pushing it. This law is accurate for all linear springs so long as x is not too great.

In order to use this in most cases, we must put this force in terms of work, which should be:

W=kx

Yet 'F' is not a single value, but increases linearly with x.
We must use the average force, which is 0.5(0+kx), or 0.5kx. In other words, we have W=F_avgx=(0.5kx)(x)=0.5kx². Since the change in potential energy is equivalent to the work done, we can write the elastic potential is this:

U_p=0.5kx².

This equation works equally well if you're pulling the spring.

The above formulations of Hooke's Law are all well and good; also, there's an extention of Hooke's law used in engineering to relate the stress within a uniform object to the strain. It uses some concepts from tensor analysis, but in it's full form:

τjk = λσiiδjk + 2μσjk
from [2]

Or, going the other way,

σij = Cijklτkl
from [1]

With summation convention assumed throughout.

Here, λ, μ and C_ijkl are analogous to the spring constant; that is, they are unique to the material being analyzed.

These formulae say a lot of important things about the symmetry of stress and strain. A tensor of rank four in space usually takes 3⁴ = 81 components, but [1] says 36 components are necessary.

Short, sweet, and to the point. How odd for a node on physics.

Sources:
1: en.wikipedia.org/wiki/Hooke's_Law
2: Tensors, Differential Forms, and Variational Principles, Lovelock & Rund. Dover Press - ISBN 0-486-65840-6, p. 17.