300 16 Constitutive Relations
16.1 General Principles.
16.1.1 Curie Principle
Constitutive relations are laws of physics where, in general, tensors of rankare
linked with tensors of rankkvia equations like (2.51), viz.
bμ 1 μ 2 ...μ=Cμ 1 μ 2 ...μν 1 ν 2 ...νkaν 1 ν 2 ...νk. (16.1)HereCis a tensor of rank+k.
At about the same time, when Woldemar Voigt invented the notion ‘tensor’ and
presented many applications in physics, Pierre Curie [2] formulated the principle
which bears his name.
In our words, theCurie Principlesays:
the coefficient tensorChas to be in accordance with
the symmetry of the physical system.
The statement can also be reverted: when both the tensorsaandbare known, the
coefficient tensorCreflects or reveals the symmetry of the system. The symmetry
of the underlying physics should not be confused with the symmetry of tensors with
respect to an interchange of indices, although there may be a close interrelation.
When a microscopic physics model exists for the relation under study, the symme-
try properties are usually ‘obvious’. In many applications, however, the quantities
aandbare just phenomenologically defined macroscopic observables. Even when a
microscopic picture of the mechanisms underlying a constitutive relation like (16.1)
are not known, symmetry considerations provide information on the coefficient ten-
sorC. In particular, the number of independent elements needed to quantify theC
tensor is reduced by symmetry considerations.
Symmetry is closely associated with permanent or induced anisotropies. Exam-
ples for the latter are applied electric or magnetic fields, the gradients of these fields,
as well as the normal on a bounding surface which also imposes a preferential direc-
tion. Preferential orientations in liquid crystals and the structure of crystalline solids
are examples for anisotropies which are ‘permanent’ as long they are not partially
destroyed by irreversible processes or bysymmetry breakingnonlinear phenomena.
Further information onCis provided byparityandtime reversalarguments, cf.
Sects.2.6and2.8. As a reminder: letPa,PbandPCbe the parities of the tensors
occurring in (16.1). When one has
Pb=PCPa,cf. (2.57), the parity is not violated by the relation (16.1). Usually, the parities ofa
andbare given by their physical meaning. Since the square of the parity value is