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.
Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior.
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", "aeolotropic", etc.
[2] In modern condensed matter physics, the constitutive equation plays a major role.
For a Newtonian fluid of viscosity μ, the shear stress τ is linearly related to the strain rate (transverse flow velocity gradient) ∂u/∂y (units s−1).
In general, for a Newtonian fluid, the relationship between the elements τij of the shear stress tensor and the deformation of the fluid is given by where vi are the components of the flow velocity vector in the corresponding xi coordinate directions, eij are the components of the strain rate tensor, Δ is the volumetric strain rate (or dilatation rate) and δij is the Kronecker delta.
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.
In the context of electromagnetism, 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).
These complex theories provide detailed formulas for the constitutive relations describing the electrical response of various materials, such as permittivities, permeabilities, conductivities and so forth.
It is necessary to specify the relations between displacement field D and E, and the magnetic H-field H and B, before doing calculations in electromagnetism (i.e. applying Maxwell's macroscopic equations).
These equations specify the response of bound charge and current to the applied fields and are called constitutive relations.
Examples are: As a variation of these examples, in general materials are bianisotropic where D and B depend on both E and H, through the additional coupling constants ξ and ζ:[11] In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects.
For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant when frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths for which a material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration).
Some man-made materials such as metamaterials and photonic crystals are designed to have customized permittivity and permeability.
In general, the constitutive equations are theoretically determined by calculating how a molecule responds to the local fields through the Lorentz force.
A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates) are based upon approximation of an inhomogeneous material by a homogeneous effective medium[12][13] (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).
[14][15][16][17] The theoretical modeling of the continuum-approximation properties of many real materials often rely upon experimental measurement as well.
Absolute is for one material, relative applies to every possible pair of interfaces; As a consequence of the definition, the speed of light in matter is for special case of vacuum; ε = ε0 and μ = μ0, The piezooptic effect relates the stresses in solids σ to the dielectric impermeability a, which are coupled by a fourth-rank tensor called the piezooptic coefficient Π (units K−1): There are several laws which describe the transport of matter, or properties of it, in an almost identical way.
In every case, in words they read: In general the constant must be replaced by a 2nd rank tensor, to account for directional dependences of the material.