Constitutive equation

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In structural analysis, constitutive relations connect applied stresses or forces to strains or deformations. The constitutive relation for linear materials is linear, and is commonly known as Hooke's law.

More generally, in physics, a constitutive equation is a relation between two physical quantities (often described by tensors) that is specific to a material or substance, and does not follow directly from physical law. It is combined with other equations that do represent physical laws to solve physical problems, like the flow of a fluid in a pipe, or the response of a crystal to an electric field.

The first constitutive equation (constitutive law) was discovered by Robert Hooke and is known as Hooke's law. It deals with the case of linear elastic materials. The concept of "constitutive law" was introduced in the doctoral thesis of Walter Noll in 1954.^[1]

Some constitutive equations are simply phenomenological; others are derived from first principles. A constitutive equation frequently has a parameter taken to be a constant of proportionality in ideal systems.

1 Constitutive equations in electromagnetism
2 Other examples
3 See also
4 References

[edit] Constitutive equations in electromagnetism

In both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics. This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents (which enter Maxwell's equations through the constitutive relations, which can be quite complicated, as described above). As a result, various approximation schemes are typically used.

For example, in real materials, complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier-Stokes equations. For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. An entire physical apparatus for dealing with these matters has developed. See for example, linear response theory, Green–Kubo relations and Green's function (many-body theory).