18.14: Generalized Hooke’s Law

The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the coordinate axes. The strain components are deduced from the stress components. This process involves considering the impact of each stress component individually and then integrating these effects.

This method uses the superposition principle, which assumes that each effect is linearly related to its load and the resulting deformations are minor. These conditions hold true for multiaxial loading if the stresses do not exceed the material's proportional limit. Additionally, the stress applied on any given face should not cause significant deformations that could impact stress calculation on other faces. Each stress component induces strain in its respective direction and strains in the other two directions. The strain components corresponding to the multiaxial loading can be derived by amalgamating these individual effects. These derived components represent the generalized Hooke's law.

The generalized Hooke's Law extends Hooke's Law to all stress types and directions, aiding in understanding materials under multiaxial stress states.

Consider an isotropic cube subjected to multiaxial loading, where normal stresses act along three coordinate axes.

The cube deforms into a rectangular parallelepiped, with equal sides and normal strain in the direction of the coordinate axes.

Strain components are expressed in terms of stress components by separately considering the effect of each stress component and then combining these effects, using the superposition principle, assuming each effect is linearly related to its load and deformations are minimal.

For multiaxial loading, the conditions are satisfied if the stresses do not exceed the material's proportional limit and the stress on any given face doesn't cause significant deformations to affect stress computations on other faces.

The stress components in each direction cause strain in their respective directions and strains in the other two directions.

By combining the individual effects, the strain components corresponding to the multiaxial loading are derived which are termed as the generalized Hooke's law.