Strain

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

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

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

After deformation, the position of a point has changed to

$ 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

$ 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

$ \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

$ \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" .

Strain is related to elastic stress by Hooke's law .