18.14 :  一般化されたフックの法則

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.