View source for Constitutive equation

{{Short description|Relation between  two physical quantities which is specific to a substance}}

In [[physics]] and [[engineering]], a '''constitutive equation''' or '''constitutive relation''' is a relation between two or more [[physical quantities]] (especially [[Kinetics (physics)|kinetic]] quantities as related to [[Kinematics|kinematic]] quantities) that is specific to a material or [[Matter|substance]] or [[field (physics)|field]], and approximates its response to external stimuli, usually as applied [[field (physics)|field]]s or [[force]]s. They are combined with other equations governing [[physical law]]s to solve physical problems; for example in [[fluid mechanics]] the [[Pipe flow|flow of a fluid in a pipe]], in [[solid state physics]] the response of a crystal to an electric field, or in [[structural analysis]], the connection between applied [[stress (physics)|stresses]] or [[Structural load|loads]] to [[Strain (materials science)|strain]]s or [[Deformation (engineering)|deformation]]s.

Some constitutive equations are simply [[Empirical relationship|phenomenological]]; others are derived from [[first principle]]s. 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, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a [[tensor]]. Constitutive relations are also modified to account for the rate of response of materials and their [[non-linear]] behavior.<ref name=Truesdell>{{cite book |title=The Non-linear Field Theories of Mechanics |author=Clifford Truesdell & Walter Noll; Stuart S. Antman, editor |page=4 |url=https://books.google.com/books?id=dp84F_odrBQC&dq=%22Preface+%22+inauthor:Antman&pg=PR13|isbn=3-540-02779-3 |publisher=Springer |year=2004}}</ref> See the article [[Linear response function]].

==Mechanical properties of matter==

The first constitutive equation (constitutive law) was developed by [[Robert Hooke]] and is known as [[Hooke's law]]. It deals with the case of [[linear elastic material]]s. 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", "aeolotropic", 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 1954 under [[Clifford Truesdell]].<ref name=Noll>See Truesdell's account in [http://www.math.cmu.edu/~wn0g/noll/TL.pdf Truesdell] ''The naturalization and apotheosis of Walter Noll''. See also [http://www.math.cmu.edu/~wn0g/noll/GEN.pdf Noll's account] and the classic treatise by both authors: {{cite book 
|chapter-url=https://books.google.com/books?id=dp84F_odrBQC&dq=%22Preface+to+the+Third%22+inauthor:Antman&pg=PR13|title=The Non-linear Field Theories of Mechanics |author=Clifford Truesdell & Walter Noll – Stuart S. Antman (editor) |isbn=3-540-02779-3 |publisher=Springer |year=2004 |page=xiii |edition=3rd |chapter-format= Originally published as Volume III/3 of the famous ''Encyclopedia of Physics'' in 1965 |chapter=Preface }}</ref>  <!--[[Walter Noll]]'s thesis is now quoted in the Oxford English Dictionary. THE CONTEXT SHOULD BE EXPLAINED. IF IT IS CITED IN THE OED AS THE SOURCE OF "CONSTITUTIVE EQUATION" THAT SHOULD BE STATED EXPLICITLY; a history of Noll's thesis development by [http://209.85.173.132/search?q=cache:0mM42Q3uA2EJ:www.math.cmu.edu/~wn0g/noll/TL.pdf+constitutive+1955+%22Walter+Noll%22&hl=en&ct=clnk&cd=2&gl=us Truesdell] attributes the idea to "Zaremba had published the basic ideas in 1903" and "frame invariance" to "In fact such a principle had been
enunciated by Oldroyd in 1950, but we did not perceive it."  -->

In modern [[condensed matter physics]], the constitutive equation plays a major role. See [[Green–Kubo relations#Linear constitutive relations|Linear constitutive equations]] and [[Green–Kubo relations#Nonlinear response and transient time correlation functions|Nonlinear correlation functions]].<ref name=Rammer>{{cite book |title=Quantum Field Theory of Nonequilibrium States |author=Jørgen Rammer |url=https://books.google.com/books?id=A7TbrAm5Wq0C&pg=PR1 |isbn=978-0-521-87499-1 |year=2007 |publisher=Cambridge University Press}}</ref>

===Definitions===
{| class="wikitable"
|-
! scope="col"  | Quantity (common name/s) 
! scope="col"  | (Common) symbol/s 
! scope="col"  | Defining equation 
! scope="col"  | SI units 
! scope="col"  | Dimension
|-
| General [[stress (mechanics)|stress]], <br/>[[pressure]]
| ''P'', ''σ''
| <math> \sigma = F/A </math><br/>''F'' is the perpendicular component of the force applied to area ''A''
| Pa = N⋅m<sup>−2</sup>
| [M][L]<sup>−1</sup>[T]<sup>−2</sup>
|-
| General [[Deformation (mechanics)|strain]]
| ''ε''
| <math> \varepsilon = \Delta D / D </math>
{{plainlist|1=
*''D'', dimension (length, area, volume)
*Δ''D'', change in dimension of material
}}
| 1
| Dimensionless
|-
| General [[elastic modulus]] || ''E''<sub>mod</sub>
| <math> E_\text{mod} = \sigma / \varepsilon </math>
| Pa = N⋅m<sup>−2</sup>
| [M][L]<sup>−1</sup>[T]<sup>−2</sup>
|-
| [[Young's modulus]]
| ''E'', ''Y''
| <math> Y = \sigma /(\Delta L/ L) </math>
| Pa = N⋅m<sup>−2</sup>
| [M][L]<sup>−1</sup>[T] <sup>−2</sup>
|-
| [[Shear modulus]]
| ''G''
| <math> G = (F/A)/(\Delta x/L)</math>
| Pa = N⋅m<sup>−2</sup>
| [M][L]<sup>−1</sup>[T]<sup>−2</sup>
|-
| [[Bulk modulus]]
| ''K'', ''B''
| <math> B = P/( \Delta V / V) </math>
| Pa = N⋅m<sup>−2</sup>
| [M][L]<sup>−1</sup>[T]<sup>−2</sup>
|-
| [[Compressibility]]
| ''C''
| <math> C = 1/B </math>
| Pa<sup>−1</sup> = m<sup>2</sup>⋅N<sup>−1</sup>
| [M]<sup>−1</sup>[L][T]<sup>2</sup> 
|}

===Deformation of solids===

====Friction====

[[Friction]] is a complicated phenomenon. Macroscopically, the [[friction]] force ''F'' between the interface of two materials can be modelled as proportional to the [[Reaction (physics)|reaction force]] ''R'' at a point of contact between two interfaces through a dimensionless coefficient of friction ''μ''<sub>f</sub>, which depends on the pair of materials:

:<math>F =  \mu_\text{f} R. </math>

This can be applied to static friction (friction preventing two stationary objects from slipping on their own), kinetic friction (friction between two objects scraping/sliding past each other), or rolling (frictional force which prevents slipping but causes a torque to exert on a round object).

====Stress and strain====

The stress-strain constitutive relation for [[linear material]]s is commonly known as [[Hooke's law]]. In its simplest form, the law defines the [[spring constant]] (or elasticity constant) ''k'' in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) [[Displacement (vector)|displacement]] ''x'':

:<math>F_i=-k x_i </math>

meaning the material responds linearly. Equivalently, in terms of the [[Stress (mechanics)|stress]] ''σ'', [[Young's modulus]] ''E'', and [[Deformation (mechanics)|strain]] ''ε'' (dimensionless):

:<math>\sigma = E \, \varepsilon </math>

In general, forces which deform solids can be normal to a surface of the material (normal forces), or tangential (shear forces), this can be described mathematically using the [[stress (mechanics)|stress tensor]]:

:<math>\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl} \, \rightleftharpoons \, \varepsilon_{ij} = S_{ijkl} \, \sigma_{kl} </math>

where ''C'' is the [[elasticity tensor]] and ''S'' is the [[Elasticity_tensor#Definition|compliance tensor]].

====Solid-state deformations====

Several classes of deformations in elastic materials are the following:<ref>Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>
; [[Plastic deformation|Plastic]]: The applied force induces non-recoverable deformations in the material when the stress (or elastic strain) reaches a critical magnitude, called the yield point.
; [[Elasticity (physics)|Elastic]]: The material recovers its initial shape after deformation.
:; [[Viscoelastic]]: If the time-dependent resistive contributions are large, and cannot be neglected. Rubbers and plastics have this property, and certainly do not satisfy Hooke's law. In fact, elastic hysteresis occurs.
:; [[Anelastic attenuation factor|Anelastic]]: If the material is close to elastic, but the applied force induces additional time-dependent resistive forces (i.e. depend on rate of change of extension/compression, in addition to the extension/compression). Metals and ceramics have this characteristic, but it is usually negligible, although not so much when heating due to friction occurs (such as vibrations or shear stresses in machines).
:; [[Hyperelastic material|Hyperelastic]]: The applied force induces displacements in the material following a [[strain energy density function]].

====Collisions====

The [[relative speed]] of separation ''v''<sub>separation</sub> of an object A after a collision with another object B is related to the relative speed of approach ''v''<sub>approach</sub> by the [[coefficient of restitution]], defined by [[coefficient of restitution|Newton's experimental impact law]]:<ref>Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, {{ISBN|0 7195 3382 1}}</ref>

:<math> e = \frac{|\mathbf{v}|_\text{separation}}{| \mathbf{v}|_\text{approach}} </math>

which depends on the materials A and B are made from, since the collision involves interactions at the surfaces of A and B. Usually {{nowrap|0 ≤ ''e'' ≤ 1}}, in which {{nowrap|1=''e'' = 1}} for completely elastic collisions, and {{nowrap|1=''e'' = 0}} for completely [[inelastic collisions]]. It is possible for {{nowrap|''e'' ≥ 1}} to occur – for [[superelastic]] (or explosive) collisions.

===Deformation of fluids===

The [[drag equation]] gives the [[Drag (physics)|drag force]] ''D'' on an object of [[Cross section (geometry)|cross-section area]] ''A'' moving through a fluid of density ''ρ'' at velocity ''v'' (relative to the fluid)

:<math>D=\frac{1}{2}c_d \rho A v^2 </math>

where the [[drag coefficient]] (dimensionless) ''c<sub>d</sub>'' depends on the geometry of the object and the drag forces at the interface between the fluid and object.

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''<sup>−1</sup>). In a uniform [[shear flow]]:

:<math>\tau = \mu \frac{\partial u}{\partial y},</math>

with ''u''(''y'') the variation of the flow velocity ''u'' in the cross-flow (transverse) direction ''y''. In general, for a Newtonian fluid, the relationship between the elements ''τ''<sub>''ij''</sub> of the shear stress tensor and the deformation of the fluid is given by

:<math>\tau_{ij} = 2 \mu \left( e_{ij} - \frac13 \Delta \delta_{ij} \right)</math> {{pad|1em}} with {{pad|1em}} <math>e_{ij}=\frac12 \left( \frac {\partial v_i}{\partial x_j} + \frac {\partial v_j}{\partial x_i} \right)</math> {{pad|1em}} and {{pad|1em}} <math>\Delta = \sum_k e_{kk} = \text{div}\; \mathbf{v},</math>

where ''v''<sub>''i''</sub> are the components of the [[flow velocity]] vector in the corresponding ''x''<sub>''i''</sub> coordinate directions, ''e''<sub>''ij''</sub> are the components of the strain rate tensor, Δ is the [[volumetric strain]] rate (or dilatation rate) and ''δ''<sub>''ij''</sub> is the [[Kronecker delta]].<ref>{{Cite book
| publisher = Cambridge University Press
| isbn = 9780521316248
| last = Kay
| first = J.M.
| title = Fluid Mechanics and Transfer Processes
| year = 1985
| pages = 10 & 122–124
}}</ref>

The ''[[ideal gas law]]'' is a constitutive relation in the sense the pressure ''p'' and volume ''V'' are related to the temperature ''T'', via the number of moles ''n'' of gas:

:<math>pV = nRT</math>

where ''R'' is the [[gas constant]] (J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>).

==Electromagnetism==

===Constitutive equations in electromagnetism and related areas===

{{see also|Permittivity|Permeability (electromagnetism)|Electrical conductivity}}

In both [[classical physics|classical]] and [[quantum physics]], the precise dynamics of a system form a set of [[simultaneous equations|coupled]] [[differential equation]]s, 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). 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 function|linear response theory]], [[Green–Kubo relations]] and [[Green's function (many-body theory)]].

These complex theories provide detailed formulas for the constitutive relations describing the electrical response of various materials, such as [[permittivity|permittivities]], [[Permeability (electromagnetism)|permeabilities]], [[Electrical conductivity|conductivities]] and so forth.

It is necessary to specify the relations between [[Electric displacement field|displacement field]] '''D''' and '''E''', and the [[Magnetic field#H-field and magnetic materials|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.

Determining the constitutive relationship between the auxiliary fields '''D''' and '''H''' and the '''E''' and '''B''' fields starts with the definition of the auxiliary fields themselves:
:<math>\begin{align}
  \mathbf{D}(\mathbf{r}, t) &= \varepsilon_0 \mathbf{E}(\mathbf{r}, t) + \mathbf{P}(\mathbf{r}, t) \\
  \mathbf{H}(\mathbf{r}, t) &= \frac{1}{\mu_0} \mathbf{B}(\mathbf{r}, t) - \mathbf{M}(\mathbf{r}, t),
\end{align}</math>

where '''P''' is the [[polarization density|polarization]] field and '''M''' is the [[magnetization]] field which are defined in terms of microscopic bound charges and bound current respectively. Before getting to how to calculate '''M''' and '''P''' it is useful to examine the following special cases.

====Without magnetic or dielectric materials====
In the absence of magnetic or dielectric materials, the constitutive relations are simple:

:<math>\mathbf{D} = \varepsilon_0\mathbf{E} ,\quad \mathbf{H} = \mathbf{B}/\mu_0</math>

where ''ε''<sub>0</sub> and ''μ''<sub>0</sub> are two universal constants, called the [[electric constant|permittivity]] of [[Vacuum|free space]] and [[magnetic constant|permeability]] of free space, respectively.

====Isotropic linear materials====
In an ([[isotropic]]<ref>The generalization to non-isotropic materials is straight forward; simply replace the constants with [[tensor]] quantities.</ref>) linear material, where '''P''' is proportional to '''E''', and '''M''' is proportional to '''B''', the constitutive relations are also straightforward. In terms of the polarization '''P''' and the magnetization '''M''' they are:

:<math>\mathbf{P} = \varepsilon_0\chi_e\mathbf{E} ,\quad \mathbf{M} = \chi_m\mathbf{H},</math>

where ''χ''<sub>e</sub> and ''χ''<sub>m</sub> are the [[electric susceptibility|electric]] and [[magnetic susceptibility|magnetic]] susceptibilities of a given material respectively. In terms of '''D''' and '''H''' the constitutive relations are:

:<math>\mathbf{D} = \varepsilon\mathbf{E} ,\quad \mathbf{H} = \mathbf{B}/\mu,</math>

where ''ε'' and ''μ'' are constants (which depend on the material), called the [[permittivity]] and [[permeability (electromagnetism)|permeability]], respectively, of the material. These are related to the susceptibilities by:

:<math>\varepsilon/\varepsilon_0 = \varepsilon_r = \chi_e + 1 ,\quad \mu / \mu_0 = \mu_r = \chi_m + 1</math>

====General case====

For real-world materials, the constitutive relations are not linear, except approximately. Calculating the constitutive relations from first principles involves determining how '''P''' and '''M''' are created from a given '''E''' and '''B'''.<ref name=bound_free group="note">The ''free'' charges and currents respond to the fields through the [[Lorentz force]] law and this response is calculated at a fundamental level using mechanics. The response of  ''bound'' charges and currents is dealt with using grosser methods subsumed under the notions of magnetization and polarization. Depending upon the problem, one may choose to have ''no'' free charges whatsoever.</ref> These relations may be empirical (based directly upon measurements), or theoretical (based upon [[statistical mechanics]], [[Transport phenomena (engineering & physics)|transport theory]] or other tools of [[condensed matter physics]]). The detail employed may be [[continuum mechanics|macroscopic]] or [[Green–Kubo relations|microscopic]], depending upon the level necessary to the problem under scrutiny.

In general, the constitutive relations can usually still be written:
:<math>\mathbf{D} = \varepsilon\mathbf{E} ,\quad \mathbf{H} = \mu^{-1}\mathbf{B}</math>
but ''ε'' and ''μ'' are not, in general, simple constants, but rather functions of '''E''', '''B''', position and time, and tensorial in nature. Examples are:

{{bulleted list
| ''[[dispersion (optics)|Dispersion]] and [[Absorption (electromagnetic radiation)|absorption]]'' where ''ε'' and ''μ'' are functions of frequency. (Causality does not permit materials to be nondispersive; see, for example, [[Kramers–Kronig relation]]s.) Neither do the fields need to be in phase, which leads to ''ε'' and ''μ'' being [[complex number|complex]]. This also leads to absorption.
| ''[[nonlinear optics|Nonlinearity]]'' where ''ε'' and ''μ'' are functions of '''E''' and '''B'''.
| ''[[Crystal optics#Anisotropic media|Anisotropy]]'' (such as ''[[birefringence]]'' or ''[[dichroism]]'') which occurs when ''ε'' and ''μ'' are second-rank [[tensor]]s,
<math display="block">D_i =  \sum_j \varepsilon_{ij} E_j ,\quad B_i =  \sum_j \mu_{ij} H_j.</math>
| Dependence of '''P''' and '''M''' on '''E''' and '''B''' at other locations and times. This could be due to ''spatial inhomogeneity''; for example in a [[Magnetic domains|domained structure]], [[heterojunction bipolar transistor|heterostructure]] or a [[liquid crystal]], or most commonly in the situation where there are simply multiple materials occupying different regions of space. Or it could be due to a time varying medium or due to [[hysteresis]]. In such cases '''P''' and '''M''' can be calculated as:<ref name="Halevi">{{cite book | last = Halevi | first = Peter | title = Spatial dispersion in solids and plasmas | publisher = North-Holland | year = 1992 | location = Amsterdam | isbn = 978-0-444-87405-4 }}</ref><ref name="Jackson">{{cite book | author=Jackson, John David |author-link=John David Jackson (physicist) | title=Classical Electrodynamics | edition=3rd | location=New York | publisher=Wiley | year=1999 | isbn=0-471-30932-X}}</ref>
<math display="block">\begin{align}
  \mathbf{P}(\mathbf{r}, t) &= \varepsilon_0 \int {\rm d}^3 \mathbf{r}'{\rm d}t'\;
\hat{\chi}_e \left(\mathbf{r}, \mathbf{r}', t, t'; \mathbf{E}\right)\, \mathbf{E}\left(\mathbf{r}', t'\right) \\
  \mathbf{M}(\mathbf{r}, t) &= \frac{1}{\mu_0} \int {\rm d}^3 \mathbf{r}'{\rm d}t' \;
\hat{\chi}_m \left(\mathbf{r}, \mathbf{r}', t, t'; \mathbf{B}\right)\, \mathbf{B}\left(\mathbf{r}', t'\right),
\end{align}</math>
in which the permittivity and permeability functions are replaced by integrals over the more general [[electric susceptibility|electric]] and [[magnetic susceptibility|magnetic]] susceptibilities.<ref>Note that the 'magnetic susceptibility' term used here is in terms of '''B''' and is different from the standard definition in terms of '''H'''.</ref>  In homogeneous materials, dependence on other locations is known as [[spatial dispersion]].
}}

As a variation of these examples, in general materials are [[Bi-isotropic material|bianisotropic]] where '''D''' and '''B''' depend on both '''E''' and '''H''', through the additional ''coupling constants'' ''ξ'' and ''ζ'':<ref name=Bianisotropy>{{cite book |author1=TG Mackay |author2=A Lakhtakia |publisher=World Scientific |url=http://www.worldscibooks.com/physics/7515.html |title=Electromagnetic Anisotropy and Bianisotropy: A Field Guide |year=2010 |access-date=2012-05-22 |archive-url=https://web.archive.org/web/20101013004900/http://www.worldscibooks.com/physics/7515.html |archive-date=2010-10-13 |url-status=dead }}</ref>
: <math>\mathbf{D}=\varepsilon  \mathbf{E} + \xi \mathbf{H} \,,\quad \mathbf{B} = \mu \mathbf{H} + \zeta \mathbf{E}.</math>

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 (signal processing)|bandwidth]]; material absorption can be neglected for wavelengths for which a material is transparent; and [[metal]]s with finite conductivity often are approximated at [[microwave]] or longer wavelengths as [[perfect conductor|perfect metals]] with infinite conductivity (forming hard barriers with zero [[skin depth]] of field penetration).

Some man-made materials such as [[metamaterial]]s and [[photonic crystal]]s are designed to have customized permittivity and permeability.

====Calculation of constitutive relations====
{{See also|Computational electromagnetics}}
The theoretical calculation of a material's constitutive equations is a common, important, and sometimes difficult task in theoretical [[condensed-matter physics]] and [[materials science]]. In general, the constitutive equations are theoretically determined by calculating how a molecule responds to the local fields through the [[Lorentz force]]. Other forces may need to be modeled as well such as lattice vibrations in crystals or bond forces. Including all of the forces leads to changes in the molecule which are used to calculate '''P''' and '''M''' as a function of the local fields.

The local fields differ from the applied fields due to the fields produced by the polarization and magnetization of nearby material; an effect which also needs to be modeled. Further, real materials are not [[continuum mechanics|continuous media]]; the local fields of real materials vary wildly on the atomic scale. The fields need to be averaged over a suitable volume to form a continuum approximation.

These continuum approximations often require some type of [[quantum mechanics|quantum mechanical]] analysis such as [[quantum field theory]] as applied to [[condensed matter physics]]. See, for example, [[density functional theory]], [[Green–Kubo relations]] and [[Green's function (many-body theory)|Green's function]].

A different set of ''homogenization methods'' (evolving from a tradition in treating materials such as [[Conglomerate (geology)|conglomerates]] and [[laminate]]s) are based upon approximation of an inhomogeneous material by a homogeneous ''[[Effective medium approximations|effective medium]]''<ref name=Aspnes>[[David E. Aspnes|Aspnes, D.E.]], "Local-field effects and effective-medium theory: A microscopic perspective", ''Am. J. Phys.'' '''50''', pp. 704–709 (1982).</ref><ref name=Kang>
{{cite book
 |author1=Habib Ammari |author2=Hyeonbae Kang |title=Inverse problems, multi-scale analysis and effective medium theory : workshop in Seoul, Inverse problems, multi-scale analysis, and homogenization, June 22–24, 2005, Seoul National University, Seoul, Korea
 |url=https://books.google.com/books?id=dK7JwVPbUkMC&q=%22effective+medium%22
 |publisher=American Mathematical Society
 |location=Providence RI
 |isbn=0-8218-3968-3
 |page=282
 |year=2006
}}</ref> (valid for excitations with [[wavelength]]s much larger than the scale of the inhomogeneity).<ref name= Zienkiewicz>
{{cite book
|author1=O. C. Zienkiewicz |author2=Robert Leroy Taylor |author3=J. Z. Zhu |author4=Perumal Nithiarasu |title=The Finite Element Method
|year=2005
|edition=Sixth
|page=550 ff
|url=https://books.google.com/books?id=rvbSmooh8Y4C&q=finite+element+inauthor:Zienkiewicz
|publisher=Butterworth-Heinemann
|location=Oxford UK
|isbn=0-7506-6321-9
}}</ref><ref>N. Bakhvalov and G. Panasenko, ''Homogenization: Averaging Processes in Periodic Media'' (Kluwer: Dordrecht, 1989); V. V. Jikov, S. M. Kozlov and O. A. Oleinik, ''Homogenization of Differential Operators and Integral Functionals'' (Springer: Berlin, 1994).</ref><ref name=Felsen>
{{cite journal
 |title=Multiresolution Homogenization of Field and Network Formulations for Multiscale Laminate Dielectric Slabs
 |author1=Vitaliy Lomakin
 |author2=Steinberg BZ
 |author3=Heyman E
 |author4=Felsen LB
 |volume=51
 |issue=10
 |year=2003
 |pages=2761 ff
 |url=http://www.ece.ucsd.edu/~vitaliy/A8.pdf
 |journal=IEEE Transactions on Antennas and Propagation
 |doi=10.1109/TAP.2003.816356
 |bibcode=2003ITAP...51.2761L
 |url-status=dead
 |archive-url=https://web.archive.org/web/20120514204026/http://www.ece.ucsd.edu/~vitaliy/A8.pdf
 |archive-date=2012-05-14
 }}</ref><ref name=Coifman>
{{cite book
 |title=Topics in Analysis and Its Applications: Selected Theses
 |author=[[Anna C. Gilbert|AC Gilbert]] (Ronald R Coifman, Editor)
 |page=155
 |url=https://books.google.com/books?id=d4MOYN5DjNUC&q=homogenization+date:2000-2009
 |publisher=World Scientific Publishing Company
 |location=Singapore
 |isbn=981-02-4094-5
 |date=May 2000
}}</ref>

The theoretical modeling of the continuum-approximation properties of many real materials often rely upon experimental measurement as well.<ref name=Palik>
{{cite book
 |author1=Edward D. Palik |author2=Ghosh G |title=Handbook of Optical Constants of Solids
 |publisher=Academic Press
 |location=London UK
 |isbn=0-12-544422-2
 |url=https://books.google.com/books?id=AkakoCPhDFUC&q=optical+constants+inauthor:Palik
 |page=1114
 |year=1998
}}</ref> For example, ''ε'' of an insulator at low frequencies can be measured by making it into a [[parallel-plate capacitor]], and ''ε'' at optical-light frequencies is often measured by [[ellipsometry]].

===Thermoelectric and electromagnetic properties of matter===

These constitutive equations are often used in [[crystallography]], a field of [[solid-state physics]].<ref>{{cite web|url=http://www.mx.iucr.org/iucr-top/comm/cteach/pamphlets/18/node2.html|title=2. Physical Properties as Tensors|website=www.mx.iucr.org|access-date=19 April 2018|archive-url=https://web.archive.org/web/20180419072909/http://www.mx.iucr.org/iucr-top/comm/cteach/pamphlets/18/node2.html|archive-date=19 April 2018|url-status=dead}}</ref>

{| class="wikitable"
|+Electromagnetic properties of solids
! scope="col" | Property/effect 
! scope="col" | Stimuli/response parameters of system
! scope="col" | Constitutive tensor of system
! scope="col" | Equation
|-
| [[Hall effect]]
| {{plainlist|
*''E'', [[electric field strength]] (N⋅C<sup>−1</sup>)
*''J'', electric [[current density]] (A⋅m<sup>−2</sup>)
*''H'', [[magnetic field intensity]] (A⋅m<sup>−1</sup>)
}}
|''ρ'', electrical [[resistivity]] (Ω⋅m)
| <math> E_k = \rho_{kij} J_i H_j </math>
|-
| [[Piezoelectricity|Direct Piezoelectric Effect]]
| {{plainlist|
*''σ'', Stress (Pa)
*''P'', (dielectric) [[polarization density|polarization]] (C⋅m<sup>−2</sup>)
}}
| ''d'', direct piezoelectric coefficient (C⋅N<sup>−1</sup>)
| <math>P_i = d_{ijk}\sigma_{jk} </math>
|-
| [[Piezoelectricity|Converse Piezoelectric Effect]]
| {{plainlist|
*''ε'', Strain (dimensionless)
*''E'', electric field strength (N⋅C<sup>−1</sup>)
}}
| ''d'', direct piezoelectric coefficient (C⋅N<sup>−1</sup>)
| <math>\varepsilon_{ij} = d_{ijk}E_k </math>
|-
| Piezomagnetic effect
| {{plainlist|
*''σ'', Stress (Pa)
*''M'', [[magnetization]] (A⋅m<sup>−1</sup>)
}}
| ''q'', piezomagnetic coefficient (A⋅N<sup>−1</sup>⋅m)
| <math>M_i = q_{ijk}\sigma_{jk} </math>
|}

{| class="wikitable"
|+Thermoelectric properties of solids
! scope="col" | Property/effect 
! scope="col" | Stimuli/response parameters of system
! scope="col" | Constitutive tensor of system
! scope="col" | Equation
|-
| [[Pyroelectricity]]
| {{plainlist|
*''P'', (dielectric) polarization (C⋅m<sup>−2</sup>)
*''T'', temperature (K)
}}
|''p'', pyroelectric coefficient (C⋅m<sup>−2</sup>⋅K<sup>−1</sup>)
| <math> \Delta P_j = p_j \Delta T </math>
|-
| [[Electrocaloric effect]]
| {{plainlist|
*''S'', [[entropy]] (J⋅K<sup>−1</sup>)
*''E'', electric field strength (N⋅C<sup>−1</sup>)
}}
|''p'', pyroelectric coefficient (C⋅m<sup>−2</sup>⋅K<sup>−1</sup>) 
| <math> \Delta S = p_i \Delta E_i </math>
|-
| [[Seebeck effect]]
| {{plainlist|
*''E'', electric field strength (N⋅C<sup>−1</sup> = V⋅m<sup>−1</sup>)
*''T'', temperature (K)
*''x'', displacement (m)
}}
|''β'', thermopower (V⋅K<sup>−1</sup>)
| <math> E_i = - \beta_{ij} \frac{\partial T}{\partial x_j} </math>
|-
| [[Peltier effect]]
| {{plainlist|
*''E'', electric field strength (N⋅C<sup>−1</sup>)
*''J'', electric current density (A⋅m<sup>−2</sup>)
*''q'', [[heat flux]] (W⋅m<sup>−2</sup>)
}}
| Π, Peltier coefficient (W⋅A<sup>−1</sup>)
| <math> q_j = \Pi_{ji} J_i </math>
|}

==Photonics==

===[[Refractive index]]===

The (absolute) [[refractive index]] of a medium ''n'' (dimensionless) is an inherently important property of [[geometric optics|geometric]] and [[physical optics]] defined as the ratio of the luminal speed in vacuum ''c''<sub>0</sub> to that in the medium ''c'':

:<math> n = \frac{c_0}{c} = \sqrt{\frac{\varepsilon \mu}{\varepsilon_0 \mu_0}} = \sqrt{\varepsilon_r \mu_r} </math>

where ''ε'' is the permittivity and ''ε''<sub>r</sub> the relative permittivity of the medium, likewise ''μ'' is the permeability and ''μ''<sub>r</sub> are the relative permeability of the medium. The vacuum permittivity is ''ε''<sub>0</sub> and vacuum permeability is ''μ''<sub>0</sub>. In general, ''n'' (also ''ε''<sub>r</sub>) are [[complex numbers]].

The relative refractive index is defined as the ratio of the two refractive indices. Absolute is for on material, relative applies to every possible pair of interfaces;

:<math> n_{AB} = \frac{n_A}{n_B} </math>

===[[Speed of light]] in matter===

As a consequence of the definition, the [[speed of light]] in matter is

:<math>c = \frac{1}{\sqrt{\varepsilon \mu}}</math>

for special case of vacuum; {{nowrap|1=''ε'' = ''ε''<sub>0</sub>}} and {{nowrap|1=''μ'' = ''μ''<sub>0</sub>}},

:<math>c_0 = \frac{1}{\sqrt{\varepsilon_0\mu_0}}</math>

=== Piezooptic effect ===

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<sup>−1</sup>):

:<math>a_{ij} = \Pi_{ijpq}\sigma_{pq} </math>

==Transport phenomena==

===Definitions===

{| class="wikitable"
|+ Definitions (thermal properties of matter)
! scope="col" | Quantity (common name/s)
! scope="col" | (Common) symbol/s
! scope="col" | Defining equation
! scope="col" | SI units
! scope="col" | Dimension
|-
| General [[heat capacity]]
| ''C'', heat capacity of substance
| <math>q = C T</math>
| J⋅K<sup>−1</sup>
| [M][L]<sup>2</sup>[T]<sup>−2</sup>[Θ]<sup>−1</sup> 
|-
| linear [[thermal expansion coefficient]] 
| {{plainlist|
*''L'', length of material (m)
*''α'', coefficient linear thermal expansion (dimensionless)
*''ε'', strain tensor (dimensionless)
}}
| {{plainlist|1=
*<math>\frac{\partial L}{\partial T} = \alpha L </math>
*<math>\varepsilon_{ij} = \alpha_{ij}\Delta T </math>
}}
| K<sup>−1</sup>
| [Θ]<sup>−1</sup>
|-
| [[Thermal expansion#General thermal expansion coefficient|Volumetric thermal expansion coefficient]]
| ''β'', ''γ''
{{plainlist|
*''V'', volume of object (m<sup>3</sup>)
*''p'', constant pressure of surroundings
}}
| <math> \left(\frac{\partial V}{\partial T}\right)_p = \gamma V</math> 
| K<sup>−1</sup>
| [Θ]<sup>−1</sup>
|-
| [[Thermal conductivity]]
| ''κ'', ''K'', ''λ'',
{{plainlist|
* '''q''', [[heat flux]] through material (W/m<sup>2</sup>)
* ∇''T'', [[temperature gradient]] in material (K⋅m<sup>−1</sup>)
}}
| <math>\lambda = - \frac{\mathbf q}{\nabla T}</math>
| W⋅m<sup>−1</sup>⋅K<sup>−1</sup>
| [M][L][T]<sup>−3</sup>[Θ]<sup>−1</sup>
|-
| [[Thermal conduction|Thermal conductance]]
| ''U''
| <math> U = \frac{\lambda}{\delta x}</math> 
| W⋅m<sup>−2</sup>⋅K<sup>−1</sup>
| [M][T]<sup>−3</sup>[Θ]<sup>−1</sup>
|-
| Thermal resistance
| ''R''<br/>Δ''x'', displacement of heat transfer (m)
| <math>R = \frac{1}{U} = \frac{\Delta x}{\lambda}</math>
| m<sup>2</sup>⋅K⋅W<sup>−1</sup>
| [M]<sup>−1</sup>[L][T]<sup>3</sup>[Θ]
|-
|}

{| class="wikitable"
|+ Definitions (electrical/magnetic properties of matter)
! scope="col" | Quantity (common name/s)
! scope="col" | (Common) symbol/s 
! scope="col" | Defining equation
! scope="col" | SI units
! scope="col" | Dimension
|-
| [[Electrical resistance]]
| ''R''
| <math>R = \frac{V}{I}</math>
| Ω, V⋅A<sup>−1</sup> = J⋅s⋅C<sup>−2</sup>
| [M][L]<sup>2</sup>[T]<sup>−3</sup>[I]<sup>−2</sup>
|-
| [[Electrical resistivity and conductivity|Resistivity]]
| ''ρ''
| <math>\rho = \frac{RA}{l}</math>
| Ω⋅m
| [M]<sup>2</sup>[L]<sup>2</sup>[T]<sup>−3</sup>[I]<sup>−2</sup>
|-
| Resistivity [[temperature coefficient]], linear temperature dependence
| ''α''
| <math>\rho - \rho_0 = \rho_0\alpha(T - T_0)</math>
| K<sup>−1</sup>
| [Θ]<sup>−1</sup>
|-
| [[Electrical resistance and conductance|Electrical conductance]]
| ''G''
| <math> G = \frac{1}{R} </math>
| S = Ω<sup>−1</sup>
| [M]<sup>−1</sup>[L]<sup>−2</sup>[T]<sup>3</sup>[I]<sup>2</sup>
|-
| [[Electrical resistivity and conductivity|Electrical conductivity]]
| ''σ''
| <math>\sigma = \frac{1}{\rho}</math>
| Ω<sup>−1</sup>⋅m<sup>−1</sup>
| [M]<sup>−2</sup>[L]<sup>−2</sup>[T]<sup>3</sup>[I]<sup>2</sup>
|-
| [[Magnetic reluctance]]
| ''R'', ''R''<sub>m</sub>, <math>\mathcal{R}</math>
| <math>R_\text{m} = \frac{\mathcal{M}}{\Phi_B}</math>
| A⋅Wb<sup>−1</sup> = H<sup>−1</sup>
| [M]<sup>−1</sup>[L]<sup>−2</sup>[T]<sup>2</sup>
|-
| Magnetic [[permeance]]
| ''P'', ''P''<sub>m</sub>, Λ, <math>\mathcal{P} </math>
| <math>\Lambda = \frac{1}{R_\text{m}}</math>
| Wb⋅A<sup>−1</sup> = H
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|}

===Definitive laws===

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:

:'''''Flux (density)''' is proportional to a '''gradient''', the constant of proportionality is the characteristic of the material.''

In general the constant must be replaced by a 2nd rank tensor, to account for directional dependences of the material.

{| class="wikitable"
|-
! scope="col"  | Property/effect
! scope="col"  | Nomenclature
! scope="col"  | Equation
|-
|'''[[Fick's law]] of [[diffusion]]''', defines diffusion coefficient ''D''
| {{plainlist|
*''D'', mass [[Mass diffusivity|diffusion coefficient]] (m<sup>2</sup>⋅s<sup>−1</sup>)
*''J'', diffusion flux of substance (mol⋅m<sup>−2</sup>⋅s<sup>−1</sup>)
*∂''C''/∂''x'', (1d)[[concentration]] gradient of substance (mol⋅dm<sup>−4</sup>)
}}
|<math> J_i = - D_{ij} \frac{\partial C}{\partial x_j} </math>
|-
| '''[[Darcy's law]] for fluid flow in porous media''', defines permeability ''κ'' 
| {{plainlist|
*''κ'', [[Permeability (earth sciences)|permeability]] of medium (m<sup>2</sup>)
*''μ'', fluid [[viscosity]] (Pa⋅s)
*''q'', discharge flux of substance (m⋅s<sup>−1</sup>)
*∂''P''/∂''x'', (1d) [[pressure gradient]] of system (Pa⋅m<sup>−1</sup>)
}}
| <math> q_j = -\frac{\kappa}{\mu} \frac{\partial P}{\partial x_j} </math>
|-
| '''[[Ohm's law]] of electric conduction''', defines electric conductivity (and hence resistivity and resistance)
| {{plainlist|
*''V'', [[potential difference]] in material (V)
*''I'', [[electric current]] through material (A)
*''R'', [[Electrical resistance and conductance|resistance]] of material (Ω)
*∂''V''/∂''x'', [[potential gradient]] ([[electric field]]) through material (V⋅m<sup>−1</sup>)
*''J'', electric [[current density]] through material (A⋅m<sup>−2</sup>)
*''σ'', electric [[electrical resistivity and conductivity|conductivity]] of material (Ω<sup>−1</sup>⋅m<sup>−1</sup>)
*''ρ'', electrical [[electrical resistivity and conductivity|resistivity]] of material (Ω⋅m)
}}
|
Simplest form is:<br/><math> V = IR </math>

More general forms are:<br/><math>\frac{\partial V}{\partial x_j} = \rho_{ji} J_i \, \rightleftharpoons \, J_i = \sigma_{ij} \frac{\partial V}{\partial x_j} </math>
|-
| '''[[Fourier's law]] of thermal conduction''', defines [[thermal conductivity]] ''λ''
| {{plainlist|
*''λ'', [[thermal conductivity]] of material (W⋅m<sup>−1</sup>⋅K<sup>−1</sup> )
*''q'', [[heat flux]] through material (W⋅m<sup>−2</sup>)
*∂''T''/∂''x'', [[temperature gradient]] in material (K⋅m<sup>−1</sup>)
}}
| <math> q_i= - \lambda_{ij}\frac{\partial T}{\partial x_j} </math>
|-
| '''[[Stefan–Boltzmann law]] of black-body radiation''', defines emmisivity ''ε''
| {{plainlist|
*''I'', [[radiant intensity]] (W⋅m<sup>−2</sup>)
*''σ'', [[Stefan–Boltzmann constant]] (W⋅m<sup>−2</sup>⋅K<sup>−4</sup>)
*''T''<sub>sys</sub>, temperature of radiating system (K)
*''T''<sub>ext</sub>, temperature of external surroundings (K)
*''ε'', [[emissivity]] (dimensionless)
}}
|
For a single radiator:<br/><math>I = \varepsilon \sigma T^4</math>

For a temperature difference{{ubli
 | <math>I = \varepsilon \sigma \left( T_\text{ext}^4 - T_\text{sys}^4\right) </math>
 | 0 ≤ ''ε'' ≤ 1; 0 for perfect reflector, 1 for perfect absorber (true black body)
 }}
|}

==See also==
* [[Defining equation (physical chemistry)]]
* [[Governing equation]]
* [[Principle of material objectivity]]
* [[Rheology]]

==Notes==
{{Reflist|group=note}}

== References ==
{{Reflist|30em}}

[[Category:Elasticity (physics)]]
[[Category:Equations of physics]]
[[Category:Electric and magnetic fields in matter]]