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How does Hooke’s law relate stress to strain for small deformations?

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mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

In the context of the mechanics of solids, Hooke’s law establishes that for relatively small deformations, the stress applied to a material is directly proportional to the resulting strain.12

This linear relationship is expressed mathematically as:

$σ = E e$

In this equation, $σ$ represents the tensile stress, $e$ is the strain (or relative deformation), and $E$ is the constant of proportionality known as Young’s modulus.2

This law holds true only within the material's proportional limit, the range where the material maintains its elastic properties and returns to its original shape and size once the load is removed. While Hooke’s law is fundamentally one-dimensional, it can be generalized to three-dimensional deformations to account for complex behaviors like shearing, twisting, and volume changes, forming the basis for the linear theory of elasticity.12