LAW OF HOOKE

In physics, Hooke's law is the simplest constitutive relationship of behavior of elastic materials. It is formulated by saying that the elongation undergone by an elastic body is directly proportional to the force applied to it. The proportionality constant is called an elastic constant and depends on the nature of the material itself. Materials for which Hooke's law is a useful approximation of real behavior are called elastic-linear materials. The classical model of linear elasticity is the perfect or ideal spring, ie a spring without weight, mass, in the absence of friction and other dissipative phenomena.
Precisely in the study of the behavior of springs, the law was first formulated by Robert Hooke in 1675, in the form of the Latin anagram "ceiiinosssttuv", whose solution was by Hooke published in 1678 as "Ut tensio, sic vis" which means " as the extension ", so the force", that is the lengthening produced (in the spring) is directly proportional to the force impressed:
The constant represents the elastic coefficient of the spring, expressed in N / m.
The modern representation of Hooke's law refers to the concepts of tension and deformation and is provided in the one-dimensional case by the relation:
where is Young's modulus of elasticity. In the case of multi-axial tensions and deformations, the law is instead represented in tensor terms by the relation (generalized Hooke law):
where the linear operator (a fourth-order tensor) is called the elasticity tensor. Of its 81 scalar coefficients, it generally has 36 independent coefficients, which are reduced to 21 in the case of hyperelastic material, and only two in the case of isotropic material. In the latter case the constitutive link is given by the relation:

in terms of only two elastic scalar parameters called Lamé constants. The inverse expression of the constitutive link is as follows:

The inverse form of the bond is more usually given in terms of the Young's module and the Poisson module.