Hooke's Law

Robert Hooke, in 1676, first stated the empirical law that governs elastic deformation for small enough stresses: within a certain limiting value of stress, called the proportional limit, the strain produced in a body is directly proportional to the stress applied to it. This is Hooke's law, and it can be written as

or, introducing a constant of proportionality,

The constant $k$ depends on the material of the body and on the kind of deformation involved (longitudinal, shearing, or volume), and is called the modulus of elasticity of the material for that kind of deformation. Since stress has SI unit $\text{Pa}$ and strain is dimensionless, every modulus of elasticity also has SI unit $\text{Pa}$ (equivalently $\text{N/m}^2$), the same as stress and pressure.

Hooke's law is not a fundamental law of nature in the way Newton's laws are -- it is an experimentally observed approximation that holds only over a limited range of stress. Real materials, plotted on a stress-versus-strain graph (the subject of the next section), do trace out a straight line through the origin for small stresses, exactly as Hooke's law predicts -- but that straight-line behaviour eventually ends at the proportional limit, beyond which stress and strain are no longer proportional to each other, even though the body may still, for a little further, return to its original shape once unloaded (this slightly larger range is called the elastic range, and its end point the elastic limit -- for most engineering materials the proportional limit and the elastic limit are so close together that the two are often treated as the same point, a simplification this chapter also follows).

Because there are three distinct kinds of stress and strain (tensile/compressive, shearing, and volume, from the previous section), Hooke's law gives rise to three distinct elastic moduli for a given material:

• Young's modulus ($Y$), the modulus of elasticity for tensile/compressive (longitudinal) stress and strain,
• Shear modulus of rigidity ($\eta$), the modulus of elasticity for shearing stress and strain, and …