In
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 rel ...
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 examp ...
(especially
kinetic quantities as related to
kinematic quantities) that is specific to a material or
substance, and approximates the response of that material to external stimuli, usually as applied
fields 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 ...
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) a ...
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
Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and thei ...
, the connection between applied
stresses or
loads to
strains or
deformations.
Some constitutive equations are simply
phenomenological
Phenomenology may refer to:
Art
* Phenomenology (architecture), based on the experience of building materials and their sensory properties
Philosophy
* Phenomenology (philosophy), a branch of philosophy which studies subjective experiences and a ...
; others are derived from
first principles. 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
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
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
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 tens ...
. 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.
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 materials. 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
Clifford Ambrose Truesdell III (February 18, 1919 – January 14, 2000) was an American mathematician, natural philosopher, and historian of science.
Life
Truesdell was born in Los Angeles, California. After high school, he spent two years in Eur ...
.
[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
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 s ...
, 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 (motion), sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative la ...
is a complicated phenomenon. Macroscopically, the
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative la ...
force ''F'' between the interface of two materials can be modelled as proportional to the
reaction force ''R'' at a point of contact between two interfaces through a dimensionless coefficient of friction ''μ''
f, which depends on the pair of materials:
:
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 materials 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 ''x'':
:
meaning the material responds linearly. Equivalently, in terms of the
stress ''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 ...
''E'', and
strain ''ε'' (dimensionless):
:
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:
:
where ''C'' is the
elasticity tensor 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: 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 linear ...
: 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 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
The coefficient of restitution (COR, also denoted by ''e''), is the ratio of the final to initial relative speed between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfect ...
, defined by
Newton's experimental impact law:
:
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 gives the
drag force
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
''D'' on an object of
cross-section area ''A'' moving through a fluid of density ''ρ'' at velocity ''v'' (relative to the fluid)
:
where the
drag coefficient (dimensionless) ''c
d'' 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
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 ''τ'' is linearly related to the
strain rate (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 ...
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:
:
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
:
with
and
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 ...
vector in the corresponding ''x''
''i'' coordinate directions, ''e''
''ij'' are the components of the strain rate tensor, Δ is the
volumetric strain 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 st ...
'' 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:
:
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 p ...
(J⋅K
−1⋅mol
−1).
Electromagnetism
Constitutive equations in electromagnetism and related areas
In both
classical and
quantum physics
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, q ...
, 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). 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. Ler ...
or the
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, a ...
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 G ...
. 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
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
,
electrohydrodynamics,
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlik ...
,
plasma modeling. An entire physical apparatus for dealing with these matters has developed. See for example,
linear response theory 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 Synapse, synaptic input into a response. Because of its many applications in infor ...
,
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
permittivities
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 r ...
,
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:
:
where P is the
polarization
Polarization or polarisation may refer to:
Mathematics
*Polarization of an Abelian variety, in the mathematics of complex manifolds
*Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
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:
:
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 i ...
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 di ...
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:
:
where ''χ''
e and ''χ''
m are the
electric and
magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
susceptibilities of a given material respectively. In terms of D and H the constitutive relations are:
:
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 i ...
and
permeability, respectively, of the material. These are related to the susceptibilities by:
:
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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an ele ...
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,
transport theory or other tools of
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 s ...
). 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 ...
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:
:
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'':
:
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
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 typi ...
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 frequency, frequencies between 300 MHz and 300 GHz respectively. Different sources define different fre ...
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
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 ato ...
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 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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an ele ...
. 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; 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 analysis such as
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
as applied to
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 s ...
. 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-bo ...
,
Green–Kubo relations and
Green's function.
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, tr ...
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, 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
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
, 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, o ...
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, o ...
of a medium ''n'' (dimensionless) is an inherently important property of
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
and
physical optics defined as the ratio of the luminal speed in vacuum ''c''
0 to that in the medium ''c'':
:
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 for ...
.
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;
:
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 fo ...
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 fo ...
in matter is
:
for special case of vacuum; and ,
:
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
−1):
:
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
*
Principle of material objectivity
*
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 plastic flow rather than deforming elastically in response to an appli ...
Notes
References
{{Reflist, 30em
Elasticity (physics)
Equations of physics
Electric and magnetic fields in matter