# Hooke's law

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## Hooke's law:

see elasticity**elasticity,**

the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. All solids are elastic for small enough deformations or strains, but if the stress exceeds a certain amount known as the elastic

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## Hooke’s Law

a basic law expressing the relationship between the stress and strain of an elastic body. It was formulated by the English physicist R. Hooke in 1660 for the simplest case of the elongation or compression of a rod in the following form: the absolute elongation (compression) Δ/ of a cylindrical rod is directly proportional to the tensile stress *N*, that is, Δ/ = *kN*, where *k* = l/*ES* (*l* is the length of the rod, *S* is the area of its cross section, and *E* is the modulus of longitudinal elasticity, which is a mechanical characteristic [constant] of a material). Hooke’s law can also be conveniently represented in the form σ ‗ *E*∈, where σ‗ *N/S* is the normal stress acting in a cross section and ∈‗ Δl/l is the relative elongation (compression) of the rod.

With shearing stress, Hooke’s law is written as τ = *G*/γ, where τ is the tangential stress, γ is the shear, and *G* is the so-called shear modulus. In the presence of shear, the tangential stress is directly proportional to the shear.

The generalized Hooke’s law—for a body of any arbitrary shape—states that six quantities determining the stress at a point are expressed linearly by six quantities determining the strain in the neighborhood of the point under consideration. In these equations the coefficients of proportionality are called elastic moduli. In anisotropic bodies, for example, crystals, the elastic moduli are different in different directions, so that in the general case the elastic properties of a solid are characterized by means of 21 elastic moduli. For isotropic bodies, the number of independent elastic constants reduces to two.

Hooke’s law is not valid when certain stresses (or deformations) attain limiting values characteristic for each material and the body passes into an elastic-plastic state. Hooke’s law is a basic relationship applicable in calculating the strength and deformability of structures and buildings.

### REFERENCE

Il’iushin, A. A., and V. S. Lenskii.*Soprotivlenie materialov*. Moscow, 1959.