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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, a constitutive equation or constitutive relation is a relation between two
physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...
(especially
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...
quantities as related to
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s or
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s. They are combined with other equations governing
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
s to solve physical problems; for example in
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
the 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 stresses or loads to
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
s or
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
s. Some constitutive equations are simply phenomenological; others are derived from
first principle In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
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 In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
. However, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
. Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior. See the article
Linear response function A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information t ...
.


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 Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
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 Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography ...
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 Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography ...
's dissertation in 1954 under Clifford Truesdell.See Truesdell's account i
Truesdell
''The naturalization and apotheosis of Walter Noll''. See als
Noll's account
and the classic treatise by both authors:
In modern condensed matter physics, the constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions.


Definitions


Deformation of solids


Friction

Friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
is a complicated phenomenon. Macroscopically, the
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
force ''F'' between the interface of two materials can be modelled as proportional to the
reaction force As described by the third of Newton's laws of motion of classical mechanics, all forces occur in pairs such that if one object exerts a force on another object, then the second object exerts an equal and opposite reaction force on the first. The th ...
''R'' at a point of contact between two interfaces through a dimensionless coefficient of friction ''μ''f, which depends on the pair of materials: :F = \mu_\text R. 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 Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
s is commonly known as
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
. In its simplest form, the law defines the
spring constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
(or elasticity constant) ''k'' in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted)
displacement Displacement may refer to: Physical sciences Mathematics and Physics * Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
''x'': :F_i=-k x_i meaning the material responds linearly. Equivalently, in terms of the
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
''σ'',
Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...
''E'', and
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
''ε'' (dimensionless): :\sigma = E \, \varepsilon 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 tensor: :\sigma_ = C_ \, \varepsilon_ \, \rightleftharpoons \, \varepsilon_ = S_ \, \sigma_ where ''C'' is the
elasticity tensor In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
and ''S'' is the compliance tensor.


Solid-state deformations

Several classes of deformations in elastic materials are the following: ;
Plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
: The applied force induces non-recoverable deformations in the material when the stress (or elastic strain) reaches a critical magnitude, called the yield point. ;
Elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
: The material recovers its initial shape after deformation. :;
Viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
: 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: 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: The applied force induces displacements in the material following a strain energy density function.


Collisions

The
relative speed The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A. Classical mechanics In one dimension (non-relativistic) We begin with relative motion in the classi ...
of separation ''v''separation of an object A after a collision with another object B is related to the relative speed of approach ''v''approach by the coefficient of restitution, defined by Newton's experimental impact law: : e = \frac which depends on the materials A and B are made from, since the collision involves interactions at the surfaces of A and B. Usually , in which for completely elastic collisions, and for completely
inelastic collisions An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energ ...
. It is possible for to occur – for superelastic (or explosive) collisions.


Deformation of fluids

The
drag equation In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is: F_\, =\, \tfrac12\, \rho\, u^2\, c_\, A where *F_ is the drag fo ...
gives the drag force ''D'' on an object of cross-section area ''A'' moving through a fluid of density ''ρ'' at velocity ''v'' (relative to the fluid) :D=\fracc_d \rho A v^2 where the
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
(dimensionless) ''cd'' depends on the geometry of the object and the drag forces at the interface between the fluid and object. For a
Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
of
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
''μ'', the
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
''τ'' is linearly related to the
strain rate In materials science, strain rate is the change in strain ( deformation) of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change ...
(transverse
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
) ∂''u''/∂''y'' (units ''s''−1). In a uniform
shear flow The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate: * a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960) ...
: :\tau = \mu \frac, 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''''ij'' of the shear stress tensor and the deformation of the fluid is given by :\tau_ = 2 \mu \left( e_ - \frac13 \Delta \delta_ \right) with e_=\frac12 \left( \frac + \frac \right) and \Delta = \sum_k e_ = \text\; \mathbf, where ''v''''i'' are the components of the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
vector in the corresponding ''x''''i'' coordinate directions, ''e''''ij'' are the components of the strain rate tensor, Δ is the
volumetric strain In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
rate (or dilatation rate) and ''δ''''ij'' is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. The ''
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
'' 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: :pV = nRT where ''R'' is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
(J⋅K−1⋅mol−1).


Electromagnetism


Constitutive equations in electromagnetism and related areas

In both classical and quantum physics, the precise dynamics of a system form a set of
coupled ''Coupled'' is an American dating game show that aired on Fox from May 17 to August 2, 2016. It was hosted by television personality, Terrence J and created by Mark Burnett, of '' Survivor'', ''The Apprentice'', '' Are You Smarter Than a 5th G ...
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, which are almost always too complicated to be solved exactly, even at the level of
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. 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 The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
or the Fokker–Planck equation or the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. For example, see
magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
, fluid dynamics,
electrohydrodynamics Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids. It is the study of the motions of ionized particles or molecules and their interactions with ...
, superconductivity,
plasma modeling Plasma modeling refers to solving equations of motion that describe the state of a plasma. It is generally coupled with Maxwell's equations for electromagnetic fields or Poisson's equation for electrostatic fields. There are several main types of p ...
. An entire physical apparatus for dealing with these matters has developed. See for example, linear response theory,
Green–Kubo relations The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot ...
and
Green's function (many-body theory) In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from ...
. 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. 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: :\begin \mathbf(\mathbf, t) &= \varepsilon_0 \mathbf(\mathbf, t) + \mathbf(\mathbf, t) \\ \mathbf(\mathbf, t) &= \frac \mathbf(\mathbf, t) - \mathbf(\mathbf, t), \end where P is the 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: :\mathbf = \varepsilon_0\mathbf ,\quad \mathbf = \mathbf/\mu_0 where ''ε''0 and ''μ''0 are two universal constants, called the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
and permeability of free space, respectively.


Isotropic linear materials

In an ( isotropic) 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: :\mathbf = \varepsilon_0\chi_e\mathbf ,\quad \mathbf = \chi_m\mathbf, where ''χ''e and ''χ''m are the
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and magnetic susceptibilities of a given material respectively. In terms of D and H the constitutive relations are: :\mathbf = \varepsilon\mathbf ,\quad \mathbf = \mathbf/\mu, where ''ε'' and ''μ'' are constants (which depend on the material), called the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
and permeability, respectively, of the material. These are related to the susceptibilities by: :\varepsilon/\varepsilon_0 = \varepsilon_r = \chi_e + 1 ,\quad \mu / \mu_0 = \mu_r = \chi_m + 1


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.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. These relations may be empirical (based directly upon measurements), or theoretical (based upon
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, transport theory or other tools of condensed matter physics). The detail employed may be
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena a ...
or
microscopic The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale be ...
, depending upon the level necessary to the problem under scrutiny. In general, the constitutive relations can usually still be written: :\mathbf = \varepsilon\mathbf ,\quad \mathbf = \mu^\mathbf 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: 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'': : \mathbf=\varepsilon \mathbf + \xi \mathbf \,,\quad \mathbf = \mu \mathbf + \zeta \mathbf. 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 Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
; material absorption can be neglected for wavelengths for which a material is transparent; and
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s with finite conductivity often are approximated at
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero
skin depth Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the co ...
of field penetration). Some man-made materials such as
metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
s and
photonic crystal A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic ...
s are designed to have customized permittivity and permeability.


Calculation of constitutive relations

The theoretical calculation of a material's constitutive equations is a common, important, and sometimes difficult task in theoretical
condensed-matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
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
continuous media Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis Cauchy was the first to fo ...
; 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 mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
analysis such as quantum field theory as applied to condensed matter physics. See, for example,
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
,
Green–Kubo relations The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot ...
and
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
. 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'' Aspnes, D.E., "Local-field effects and effective-medium theory: A microscopic perspective", ''Am. J. Phys.'' 50, pp. 704–709 (1982). (valid for excitations with
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
s much larger than the scale of the inhomogeneity). The theoretical modeling of the continuum-approximation properties of many real materials often rely upon experimental measurement as well. For example, ''ε'' of an insulator at low frequencies can be measured by making it into a
parallel-plate capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
, and ''ε'' at optical-light frequencies is often measured by
ellipsometry Ellipsometry is an optical technique for investigating the dielectric properties (complex refractive index or dielectric function) of thin films. Ellipsometry measures the change of polarization upon reflection or transmission and compares it t ...
.


Thermoelectric and electromagnetic properties of matter

These constitutive equations are often used in crystallography, a field of solid-state physics.


Photonics


Refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

The (absolute)
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of a medium ''n'' (dimensionless) is an inherently important property of geometric and physical optics defined as the ratio of the luminal speed in vacuum ''c''0 to that in the medium ''c'': : n = \frac = \sqrt = \sqrt where ''ε'' is the permittivity and ''ε''r the relative permittivity of the medium, likewise ''μ'' is the permeability and ''μ''r are the relative permeability of the medium. The vacuum permittivity is ''ε''0 and vacuum permeability is ''μ''0. In general, ''n'' (also ''ε''r) are
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. 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; : n_ = \frac


Speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in matter

As a consequence of the definition, the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in matter is :c = \frac for special case of vacuum; and , :c_0 = \frac


Piezooptic effect The piezooptic effect is manifest as a change in refractive index, n, of a material caused by a change in pressure on that material. Early demonstrations of the piezooptic effect were done on liquids. The effect has since been demonstrated in so ...

The
piezooptic effect The piezooptic effect is manifest as a change in refractive index, n, of a material caused by a change in pressure on that material. Early demonstrations of the piezooptic effect were done on liquids. The effect has since been demonstrated in so ...
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): :a_ = \Pi_\sigma_


Transport phenomena


Definitions


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.


See also

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Principle of material objectivity Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography ...
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Rheology Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with Plasticity (physics), plastic flow rather than deforming Elasticity (phy ...


Notes


References

{{Reflist, 30em Elasticity (physics) Equations of physics Electric and magnetic fields in matter