312 16 Constitutive Relations
For zero pressure and at temperatures, where the fluctuation contributions to the
elasticity coefficientsBandGare negligible, (16.37) implies theCauchy relation
3 B= 5 G. (16.41)Upon the assumptions just mentioned, the Cauchy relation holds true for solids
composed of spherical particles interacting with any pairwise additive potential, then
the ratioG/Bis equal to 3/ 5 = 0 .6. Experimental values for this ratio are smaller,
e.g. one hasG/B≈ 0 .5 for the metals copper, nickel, iron,G/B≈ 0 .3 for silver, and
G/B≈ 0 .2 for gold. The deviations from the Cauchy relation are closely associated
with many particle interactions. An efficient way to include the relevant many particle
interactions is provided by theembedded atom method[104, 105]. The basic idea of
this method is stressed in [106]: the interaction of two particles is influenced by the
density of the other particles within a well defined vicinity involving twenty to fifty
particles. When this local density is too high or too low, compared with a prescribed
density, the two particles under consideration feel an extra repulsion or attraction,
respectively. Use of this method does not increase significantly the time needed in
molecular dynamics computer simulations. A variant of the embedded atom method,
realistic enough to model the elastic properties and simple enough to allow extended
non-equilibrium molecular dynamics simulations of the visco-plastic behavior of
metals, is presented in [107].
16.3 Viscosity and Non-equilibrium Alignment Phenomena.
While the elasticity of solids is an equilibrium property, the viscous flow behavior
of fluids is a typical non-equilibrium phenomenon. In both cases, symmetry consid-
erations and the use of tensors play an important role. In this section, the viscosity
in simple and in molecular fluids, the influence of external fields on the viscosity, as
well as flow birefringence, heat-flow birefringence and visco-elasticity are treated.
16.3.1 General Remarks, Simple Fluids.
In thermal equilibrium, the pressure tensor of a fluid is isotropic and it is given by
pνμ=Pδμν, wherePis the hydrostatic pressure. The part of the pressure tensor
linked with non-equilibrium is decomposed into its irreducible isotropic, symmetric
traceless and antisymmetric parts, cf. (7.53), according to
pνμ−Pδμν= ̃pδμν+pνμ+1
2
ενμλpλ,