Linear elasticity:
As noted above, for small deformations, most elastic materials such as springs exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known as Hooke's law. A geometry-dependent version of the idea[4] was first formulated by Robert Hooke in 1675 as a Latin anagram, "ceiiinosssttuv". He published the answer in 1678: "Ut tensio, sic vis" meaning "As the extension, so the force", a linear relationship commonly referred to as Hooke's law. This law can be stated as a relationship between force F and displacement x,where k is a constant known as the rate or spring constant. It can also be stated as a relationship between stress σ and strain :
where E Is known as the elastic modulus or Young's modulus.Although the general proportionality constant between stress and strain in three dimensions is a 4th order tensor, systems that exhibit symmetry, such as a one-dimensional rod, can often be reduced to applications of Hooke's law.
