The Cauchy–Born rule or Cauchy–Born approximation is a basic hypothesis used in the mathematical formulation of
solid mechanics
Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...
which relates the movement of atoms in a
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
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.
The rule first appears in
Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German-British theoretical physicist who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics, and supervised the work of a ...
and
Huang Kun
Huang Kun (; September 2, 1919 – July 6, 2005) was a Chinese physicist and an academician of the Chinese Academy of Sciences. He was awarded the State Preeminent Science and Technology Award (the highest science award in China) by President J ...
's
''Dynamical Theory of Crystal Lattices'', a refinement of
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
's relations which were used to derive the equations satisfied by the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor (symbol \boldsymbol\sigma, named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor, completely defines the state of stress at a point inside a material in the d ...
.
Description
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
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an or ...
vectors
. The Cauchy–Born rules states that if the body is deformed by a deformation whose
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
is
, the lattice of the deform body can be described by
The rule only describes the lattice, not the atoms.
The approximation generally holds for face-centered and body-centered
cubic crystal system
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties o ...
s. For complex lattices such as
diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Diamond is tasteless, odourless, strong, brittle solid, colourless in pure form, a poor conductor of e ...
, 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.
Extensions
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, a surface Cauchy–Born rule has been proposed.
Several modified forms of the Cauchy–Born rule have also been proposed to cater to crystalline bodies having special shapes.
Marino Arroyo and
Ted Belytschko
Ted Bohdan Belytschko (January 13, 1943 – September 15, 2014) was an American mechanical engineer. He was Walter P. Murphy Professor and McCormick Professor of Computational Mechanics at Northwestern University. He worked in the field of comput ...
proposed a exponential Cauchy Born rule for modeling of mono-layered crystalline sheets as two-dimensional continuum shells has been proposed. A helical Cauchy–Born rule for modeling slender bodies (such as nano and continuum rods) as special Cosserat continuum rods has been proposed.
See also
*
Born–Huang approximation
The Born–Huang approximation is an approximation closely related to the Born–Oppenheimer approximation. It takes into account diagonal nonadiabatic effects in the electronic Hamiltonian better than the Born–Oppenheimer approximat ...
References
Crystallography
Max Born
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