Assuming a uniform deformation, a point located at will be shifted to , which leads to the displacement defined as
Further generalization leads to a description for non-uniform deformation, introducing a position dependent vector function ,
Restricting to small displacements from , the displacement function can be developed into a Taylor series and truncated after the linear term at , leading to a relation between the local displacement tensor and the displacement function,
Therefore, the displacement can be expressed as . Frequently, the displacements are expressed via the strain tensor to describe the deformation of a body in three dimensions. In the limit of small deformations, the strain tensor is known as the Green tensor or Cauchy's infinitesimal strain tensor,
The relative length change in the direction is described by the diagonal coefficients , while the off-diagonal elements denote the angular distortions by shear strains.
Also very common are the engineering strains , which are linked with the strain tensor as follows: