# Strain

Strain is a non-dimensional measure of the deformation of a continuous body.

Let the displacement (from its initial position of a point in the body be denoted by . If the displacement vector **u** is constant in space, this displacement is just a translation; *deformation* corresponds to the spatial *variation* of displacement.

Consider two points close together in the medium, separated by the separation vector
. The scalar distance between these points is (from the Pythagorean Theorem)

After deformation, the position of a point has changed to

(first-order Taylor expansion). The change in the scalar distance (independent of any rotation) is given by

which defines the strain tensor

In the case of seismic wave propagation, the displacements are assumed to be small, so the strain tensor reduces to

(For the large strains considered in geomechanics, the non-linear term should be retained.)

**Note** that shear strains of the strain tensor differ (by a factor 1/2) from the shearing strains defined in the SEG Wiki Dictionary, which are "engineering strains" ^{[1]}.

Strain is related to elastic stress by Hooke's law ^{[2]}.