Cauchy–Born rule

Not to be confused with the Born rule in quantum mechanics.

The Cauchy–Born rule or Cauchy–Born approximation is a basic hypothesis used in the mathematical formulation of solid mechanics which relates the movement of atoms in a crystal to the overall deformation of the bulk solid. A widespread simplified version states that in a crystalline solid subject to a small strain, the positions of the atoms within the crystal lattice follow the overall strain of the medium. 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 $\mathbf {e} ^{0}$. The Cauchy-Born rules states that if the body is deformed by a deformation whoes gradient is $\mathbf {F}$, the lattice of the deform body can be described by $\mathbf {e} =\mathbf {F} \mathbf {e} ^{0}$. The rule only describes the lattice, not the atoms.

The currently accepted form is Max Born's refinement of Cauchy's original hypothesis which was used to derive the equations satisfied by the Cauchy stress tensor. 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. The approximation can then be used to obtain bulk properties of crystalline materials such as stress-strain relationship.

For crystalline bodies of finite size, the effect of surface stress is also significant. However, the standard Cauchy–Born rule cannot deduce the surface properties. To overcome this limitation, Park et al. (2006) proposed a surface Cauchy–Born rule. 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. Kumar et al. (2015) proposed a helical Cauchy–Born rule for modeling slender bodies (such as nano and continuum rods) as special Cosserat continuum rods.

References

• Ericksen, J. L. (May 2008), "On the Cauchy–Born Rule", Mathematics & Mechanics of Solids, 13 (3–4): 199–220, doi:10.1177/1081286507086898, S2CID 120624506.
• Arroyo, M.; Belytschko, T. (Sep 2002), "An atomistic-based finite deformation membrane for single layer crystalline films", Journal of the Mechanics and Physics of Solids, 50 (9): 1941–1977, Bibcode:2002JMPSo..50.1941A, doi:10.1016/S0022-5096(02)00002-9, hdl:2117/8545.
• Park, H.S.; Klein, P.A.; Wagner, G.J. (May 2008), "A surface Cauchy–Born model for nanoscale materials", International Journal for Numerical Methods in Engineering, 68 (10): 1072–1095, CiteSeerX 10.1.1.595.3261, doi:10.1002/nme.1754, S2CID 41694571.
• Kumar, A.; Kumar, S.; Gupta, P. (Dec 2015), "A helical Cauchy-Born rule for special Cosserat rod modeling of nano and continuum rods", Journal of Elasticity, 122.