# Strain

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

Let the displacement (from its initial position ${\displaystyle [x_{1}x_{2}x_{3}]}$ of a point in the body be denoted by ${\displaystyle [u_{1}u_{2}u_{3}]}$. 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 ${\displaystyle [\Delta x_{1}\Delta x_{2}\Delta x_{3}]}$ . The scalar distance between these points is (from the Pythagorean Theorem)

${\displaystyle L^{2}=\sum _{i}^{3}\Delta x_{i}\Delta x_{i}}$

After deformation, the position of a point has changed to

${\displaystyle x'_{i}=x_{i}+\sum _{j}^{3}{\frac {\partial u_{i}}{\partial x_{j}}}\Delta x_{j}}$

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

${\displaystyle L'^{2}-L^{2}=\sum _{i}^{3}\sum _{j}^{3}[{\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}+{\frac {\partial u_{i}}{\partial x_{j}}}{\frac {\partial u_{j}}{\partial x_{i}}}]\Delta x_{j}\Delta x_{i}=\sum _{i}^{3}\sum _{j}^{3}2\epsilon _{ij}\Delta x_{i}\Delta x_{j}}$

which defines the strain tensor

${\displaystyle \epsilon _{ij}={\tfrac {1}{2}}[{\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}+{\frac {\partial u_{i}}{\partial x_{j}}}{\frac {\partial u_{j}}{\partial x_{i}}}]}$

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

${\displaystyle \epsilon _{ij}={\tfrac {1}{2}}[{\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}]}$

(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].