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In structural analysis, 'constitutive relations' connect applied stresses or forces to strains or deformations. The constitutive relations for linear materials are linear, and termed Hooke's law.
More generally, in physics, a 'constitutive equation' is a relation between two physical quantities (often 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 some physical problem, 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 Hooke which
was later completed in 19th century an is known as 'generalaized' Hooke's law for the case of the linear elastic materials. But the expression 'constitutive law' was first used in the doctoral thesis of Walter Noll on 1954 whose advisor was Clifford Truesdell at Indiana University. Actually the 'concept' of 'constitutive law' was introduced by Walter Noll which now has found very wide acceptance. Walter Noll's thesis is now quoted in the Oxford English Dictionary.
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.

Contents

Examples

See also

Examples

★ Friction
:$F\_f\; =\; F\_p\; mu\_f$

★ Drag equation
:$D=\{1\; over\; 2\}C\_d$
ho A v^2

★ Electric susceptibility (Permittivity)
:$P\_j\; =\; epsilon\_0\; chi\_\{ij\}\; E\_i$
:$D\_j\; =\; epsilon\_\{ij\}\; E\_i$

★ Magnetic susceptibility (Permeability (electromagnetism))
:$M\_j\; =\; mu\_0\; chi\_\{m,ij\}\; H\_i$
:$B\_j\; =\; mu\_\{ij\}\; H\_i$

★ Linear elasticity (Hooke's law)
:$$
F=-k x

or
:$$
sigma = C , epsilon

and in tensor form,
:$$
sigma_{ij} = C_{ijkl} , epsilon_{kl}

or, equivalently,
:$$
epsilon_{ij} = S_{ijkl} , sigma_{kl}


★ Ohm's law
:$$
{V over I} = R
or
:$$
J_j = sigma_{ij} E_i


★ Newtonian fluid mechanics:
:

★ Heat capacity
:$$
q=c_p T


★ Thermal conductivity
:$$
p_j=- k_{ij}rac{partial T}{partial x_i}


★ Diffusion (Fick's law)
:$$
J_j=-D_{ij} rac{partial C}{partial x_i}


★ Flow in porous media (Darcy's law)
:$$
q = -rac{kappa}{mu} Delta P

See also

★ Principle of material objectivity