Using a Generalized Hooke’s Law for Stress and Strain

By James H. Allen, III

Part of Mechanics of Materials For Dummies Cheat Sheet

In mechanics of materials, Hookes law is the relationship that connects stresses to strains. Although Hooke’s original law was developed for uniaxial stresses, you can use a generalized version of Hooke’s law to connect stress and strain in three-dimensional objects, as well. Eventually, Hooke’s law helps you relate stresses (which are based on loads) to strains (which are based on deformations).

For a three-dimensional state of stress, the normal strain in a given direction (such as x) is a function of the stresses in all three orthogonal directions (usually the Cartesian x, y-, and z-directions), as shown by this equation:


where E is the modulus of elasticity and ν is Poisson’s ratio for the material. For a uniaxial stress, two of the stresses in the equation are zero. For a biaxial stress condition, one of the stresses in this equation is zero.

The generalized relationship for Hooke’s law for shear in the XY plane can be given as