Cauchy relations

The Cauchy relations describe symmetry properties of the elasticity tensor of a material.

description

The Cauchy relations often approximate existing symmetry properties of the elasticity tensor:

${\displaystyle c_{iijk}=c_{ijik}}$

with . The elasticity tensor is defined as follows using a generalized Hooke's law , which links stresses (stress tensor ) with deformations (strain tensor ): ${\displaystyle i\neq j,k}$ ${\displaystyle c_{ijkl}}$ ${\displaystyle \sigma _{ij}}$ ${\displaystyle \varepsilon _{ij}}$

${\displaystyle \sigma _{ij}=\sum _{kl}c_{ijkl}\varepsilon _{kl}}$

with . In general, the elasticity tensor is a 4th order tensor with 81 components. The number of components to be determined can be reduced by the Cauchy relations. According to the (crystal) lattice theory, the Cauchy relations are valid if the following conditions are more or less fulfilled (see restrictions ): ${\displaystyle i,j,k,l=1,2,3}$

1. pure central forces act between the building blocks,
2. the individual building blocks sit within the crystal lattice in a center of symmetry,
3. there are no anharmonicities, d. i.e., when the grid potential is formed after the movements / displacements of the building blocks, elements of a higher order (greater than the 2nd order) disappear,
4. the thermal energy of the substances is negligible and
5. missing initial stresses.

restrictions

A strict validity of the Cauchy relations is not to be expected, since especially point 4 cannot be fulfilled exactly. Even at low temperatures, the Cauchy relations are not fulfilled by any substance.

The deviations from the Cauchy relations are a 2nd order tensor with nine components and can be described as ${\displaystyle g_{mn}}$

${\displaystyle c_{iijk}-c_{ijik}=g_{mn}(-1)^{m\cdot n(m-n)/2}}$

with as well as and describe. The transformation of the tensor directly provides the proof of this when changing the Cartesian reference system with the basic vectors . So is ${\displaystyle m\neq i,j}$ ${\displaystyle n\neq i,k}$ ${\displaystyle m,n,i,j,k=1,2,3}$ ${\displaystyle e_{p}}$

${\displaystyle g'_{mn}=\sum _{p,q}a_{mp}a_{nq}g_{pq}}$

with as components of this tensor in a new frame of reference with the basic vectors that are on the ground vectors based on ${\displaystyle g'_{mn}}$ ${\displaystyle e'_{m}}$ ${\displaystyle e_{p}}$

${\displaystyle e'_{m}=\sum _{p}a_{mp}e_{p}}$

surrender. From this, physically important invariance properties follow from the deviations of the Cauchy relations. The components with behave z. B. invariant to a rotation of the Cartesian reference system around the axis . is therefore a characteristic quantity of the plane that is perpendicular to . Consequently, the bond characteristics of the atomic building blocks of a homogeneous and quasi-homogeneous substance find expression in the scalar invariant . ${\displaystyle g_{mm}}$ ${\displaystyle m=1,2,3}$ ${\displaystyle e_{m}}$ ${\displaystyle g_{mm}}$ ${\displaystyle e_{m}}$ ${\displaystyle G=g_{11}+g_{22}+g_{33}}$

For example, crystalline materials with cubic symmetry and isotropic substances (such as glass) are only described by an independent component . In the case of substances with predominant ion, metal, van der Waals and hydrogen bonds , the transverse contraction coefficient is greater than the corresponding shear resistance and thus . These substances often have structures with the maximum packing density of the atomic building blocks. Show directed bond components, i.e. covalent bonds or a considerable direction-dependent overlap of the electron clouds in substances . Examples of this are tectosilicates , magnesium oxide , beryllium , aluminum oxide ( corundum ), glasses rich in silicon dioxide (the higher the silicon dioxide content, the smaller ) and lithium fluoride . Covalent bonds contribute to higher shear resistances and low transverse contraction coefficients. Structurally and chemically related types of crystals and isotypic substance series show the following characteristics: ${\displaystyle g}$ ${\displaystyle c_{iijj}}$ ${\displaystyle c_{ijij}}$ ${\displaystyle g_{mm}>0}$ ${\displaystyle g_{mm}<0}$ ${\displaystyle g}$

1. Substitution of main group elements by subgroup elements or by building blocks with less symmetry within structures of a substance increases . This is caused by increasing lateral contraction coefficients and decreasing shear resistance. ${\displaystyle g_{mm}}$
2. If the polarizability of the substituted building blocks is greater, it increases slightly. ${\displaystyle g_{mm}}$
3. Since is often positive for substances with a simple structure , the deviation of the Cauchy relations increases with increasing temperature. Substances with asymmetrical building blocks do not follow this behavior. ${\displaystyle \mathrm {d} \log(g_{mm})/\mathrm {d} T}$

It should also be noted that even only subtle differences in the structure of some substances are reflected in the values. These values ​​thus serve as an indicator for structural differences. ${\displaystyle g}$

literature

• G. Leibfried: Mechanical and thermal properties of crystals. In: S. Flügge (Ed.): Handbuch der Physik. Volume VII, 1st edition, Springer, Berlin / Göttingen / Heidelberg 1955.
• S. Haussühl: The deviations from the Cauchy relations . In: Physics of Condensed Matter . tape 6 , no. 3 , 1967, ISSN  0722-3277 , p. 181-192 , doi : 10.1007 / BF02422715 .