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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 approximates the response of that material to external forces. It is combined with other equations governing 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.

Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, much more elaborate constitutive equations often are necessary to account for tensor properties, the rate of response of materials and their non-linear behavior.^[1] See the article Linear response function.

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. Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used. Walter Noll advanced the use of constitutive equations, clarifying their classification and the role of invariance requirements, constraints, and definitions of terms like "material", "isotropic", "aelotropic", etc. The class of "constitutive relations" of the form stress rate = f (velocity gradient, stress, density) was the subject of Walter Noll's dissertation in 1952 under Clifford Truesdell.^[2]

In modern condensed matter physics, the constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions.^[3]

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).

These complex theories provide detailed formulas for material parameters such as permittivities, permeabilities, conductivities and so forth.

[edit] Other examples

[edit] Mechanics

Friction

$F_f = F_p \mu_f \,$

Drag equation

$D={1 \over 2}C_d \rho A v^2 \,$

Linear elasticity (Hooke's law)

$F=-k x \,$

$\sigma = C \, \epsilon \,$ and in tensor form, $\sigma_{ij} = C_{ijkl} \, \epsilon_{kl} \,$ or inversely, $\epsilon_{ij} = S_{ijkl} \, \sigma_{kl} \,$

Newtonian fluid mechanics:

$\tau = \mu \frac {\partial u} {\partial y} \,$

Diffusion (Fick's law)

$J_j=-D_{ij} \frac{\partial C}{\partial x_i}$

Flow in porous media (Darcy's law)

$q = -\frac{\kappa}{\mu} \Delta P$

[edit] Thermodynamics

Heat capacity

$q=c_p T \,$

Thermal conductivity

$p_j=- k_{ij}\frac{\partial T}{\partial x_i} \,$

[edit] Electromagnetism

Electric susceptibility (Permittivity)

$P_j = \epsilon_0 \chi_{ij} E_i \,$

$D_j = \epsilon_{ij} E_i \,$

Magnetic susceptibility (Permeability (electromagnetism))

$M_j = \mu_0 \chi_{m,ij} H_i \,$

$B_j = \mu_{ij} H_i \,$

Ohm's law

${V \over I} = R \,$ or $J_j = \sigma_{ij} E_i \,$

[edit] See also

Principle of material objectivity

[edit] References

^ Clifford Truesdell & Walter Noll; Stuart S. Antman, editor (2004). The Non-linear Field Theories of Mechanics. Springer. p. 4. ISBN 3540027793. http://books.google.com/books?id=dp84F_odrBQC&pg=PR13&dq=%22Preface+%22+inauthor:Antman&lr=&as_brr=0.
^ See Truesdell's account in Truesdell The naturalization and apotheosis of Walter Noll. See also Noll's account and the classic treatise by both authors: Clifford Truesdell & Walter Noll - Stuart S. Antman (editor) (2004). "Preface" (Originally published as Volume III/3 of the famous Encyclopedia of Physics in 1965). The Non-linear Field Theories of Mechanics (3rd ed.). Springer. p. xiii. ISBN 3540027793. http://books.google.com/books?id=dp84F_odrBQC&pg=PR13&dq=%22Preface+to+the+Third%22+inauthor:Antman&lr=&as_brr=0.
^ Jørgen Rammer (2007). Quantum Field Theory of Nonequilibrium States. Cambridge University Press. ISBN 9780521874991. http://books.google.com/books?id=A7TbrAm5Wq0C&pg=PR1&dq=isbn:9780521874991#PPA151,M1.

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