Hooke's law for continuous media
Hooke’s law states that the force needed to extend or compress a spring by some distance is proportional to that distance.The stresses and strains of the material inside a continuous elastic material is mathematically similar to Hooke’s spring law.
A parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.
The relevant state of the medium around a point must be represented by two second-order tensors, the strain tensor and the stress tensor and a fourth-order tensor called the stiffness tensor or elasticity tensor.
|σ||Stress tensor (dimensionless)|
|c||Elasticity tensor (dimensionless)|
|ϵ||Strain tensor (dimensionless)|