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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 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.

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.

Examples

Friction

F f = F p μ f

Drag equation

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

Electric susceptibility (Permittivity)

P j = ε 0 χ i j E i

D j = ε i j E i

Magnetic susceptibility (Permeability (electromagnetism))

M j = μ 0 χ m, i j H i

B j = μ i j H i

Linear elasticity (Hooke's law)

F = - k x

$\sigma = C \, \epsilon$

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 = σ i j E i

Newtonian fluid mechanics:

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

Heat capacity

q = c p T

Thermal conductivity

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

Diffusion (Fick's law)

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

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01-04-2007 01:21:04

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