Consider a rod with initial length L which is stretched to a length L'. The strain measure e, a dimensionless ratio, is defined as the ratio of elongation with respect to the original length,

Consider an arbitrary point in the bar P, which has a position vector x, and its infinitesimal neighbor dx. Point P shifts to P', which has a position vector x', after the stretch. In the meantime, the small "step" dx is stretched to dx'.

The strain at point p can be defined the same as in the global strain measure,

Since the displacement , the strain can hence be rewritten as,

The infinitesimal strain-displacement relationships can be summarized as,

where u is the displacement vector, x is coordinate, and the two indices i and j can range over the three coordinates {1, 2, 3} in three dimensional space.

Expanding the above equation for each coordinate direction gives,

where u, v, and w are the displacements in the x, y, and z directions respectively (i.e. they are the components of u).

where . Called the engineering shear strain, g_xy is a total measure of shear strain in the x-y plane. In contrast, the shear strain e_xy is the average of the shear strain on the x face along the y direction, and on the y face along the x direction.

Engineering shear strain is commonly used in engineering reference books. However, please beware of the difference between shear strain and engineering shear strain, so as to avoid errors in mathematical manipulations.

These compatibility conditions for infinitesimal strain reffered to rectangular Cartesian coordinates are,

In two dimensional problems (e.g. plane strain), all z terms are set to zero. The compatibility equations reduce to,

Note that some references use engineering shear strain () when referencing compatibility equations.