16.2 Elasticity 305
is determined by the trace of the deformation tensor. The symmetric traceless part is
associated with volume conserving squeeze, stretch or shear deformations.
In linear approximation to be used in the following, the deformation tensor defined
by (16.11), reduces to
uνμ=1
2
(∇νuμ+∇μuν). (16.13)Bydefinition,thedeformation tensor,whichisalsocalledstrain tensor,issymmetric:
uνμ=uμν. An antisymmetric part ofuνμwould induce an infinitesimal rotation but
not a deformation.
Adeformationofasolidcausesastress.Apartfromthesign,thestresstensorσμνis
essentially the deviation of the pressure tensorpμνfrom its value in the undeformed
state. The elastic properties of a solid are expressed in terms of the fourth rank
elasticity tensor, which relates the stress tensor to the deformation tensor, viz.
−(pμν−Pδμν)≡σμν=Gμν,λκuλκ. (16.14)In this linear constitutive relation, which is essentiallyHooke’s law,σμν =σνμ
is presupposed. This assumption is common practice in solid state mechanics. The
symmetry of the stress tensor holds true for substances composed of particles with
a spherical symmetric interaction potential. In general, however, the the pressure
tensor and consequently also the stress tensor can possess an antisymmetric part, cf.
Sect.16.3.5.
The deformation costs energy, this means
σμνuμν=uμνGμν,λκuλκ> 0 , (16.15)i.e. the elasticity tensor is positive definite, in a thermodynamically stable state. The
symmetric tensorsσμνanduμνhave 6 independent components, thus the elasticity
tensor has 36 components, some of which can be zero. In accord with the Curie
principle, the number of independent components of the elasticity tensor is consider-
ably smaller, for certain types of the symmetry of the undeformed solid. In fact, just
two coefficients suffice to characterize the linear elastic properties of an isotropic
solid. For cubic symmetry, three different material coefficients are needed. These
cases are discussed in Sect.16.2.4. Group theoretical methods for the general crystal
symmetries are found in text books devoted to Solid State Physics, in particular to
the Mechanics of Solids.
16.2.2 Voigt Coefficients.
A notation introduced by Voigt replaces the information contained in the fourth
rank elasticity tensorGμν,λκby a six by six elasticity coefficient matrixcij.Tothis