In 1678, Robert Hooke set down the basis for modern finite element stress analysis with Hooke's law. An elastic body stretches in proportion to the force (stress) on it. Mathematically:
F=kx
Hooke proved the equation by using weights to stretch wires hanging from the ceiling.
Imagine that a coffee cup is sitting on a table. It is broken down into 2,000 little brick elements. Each element has eight corners, or nodes. All nodes on the bottom of the coffee cup are fixed (all translations are constrained), so they cannot move. Press down on just one node near the top of the cup.
That one node moves a little because all materials have some elasticity. F = kx describes the movement for that element except that other elements are in the way. In fact, as the force is transmitted through the first element, it spreads out to other nodes.
In the finite element method, a step occurs called element stiffness formulation. A stiffness, k, is created for the relationship between every node on each element. Every node is connected to every other node on each element by a spring. It behaves like F = kx. We reduce the coffee cup to a large system of springs. A value for the translation, x, and force, F, is determined for each node by the formula F = kx.
In the final step, results evaluation, the stresses are determined by knowing the force at each node and the geometry of each element.
Other physical phenomena, such as heat transfer, fluid flow, and electrical effects, adhere to similar governing equations.