Cauchy–Born rule

To give a more precise definition, consider a crystalline body where the position of the atoms can be described by a set of reference lattice vectors

The approximation generally holds for face-centered and body-centered cubic crystal systems.

For complex lattices such as diamond, however, the rule has to be modified to allow for internal degrees of freedom between the sublattices.

Several modified forms of the Cauchy–Born rule have also been proposed to cater to crystalline bodies having special shapes.

Arroyo & Belytschko (2002) proposed an exponential Cauchy Born rule for modeling of mono-layered crystalline sheets as two-dimensional continuum shells.