In
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
and
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, a constitutive equation or constitutive relation is a relation between two or more
physical quantities (especially
kinetic quantities as related to
kinematic quantities) that is specific to a material or
substance or
field, and approximates its response to external stimuli, usually as applied
fields or
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s. They are combined with other equations governing
physical laws to solve physical problems; for example in
fluid mechanics 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
strains or
deformations.
Some constitutive equations are simply
phenomenological; 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 volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity in ...
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.
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.
[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 is a complicated phenomenon. Macroscopically, the
friction force ''F'' at 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 ''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 deformation
Several classes of deformation in elastic materials are the following:
;
Plastic: The applied force induces non-recoverable deformation 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: 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, 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. It is possible for to occur – for
superelastic (or explosive) collisions.
Deformation of fluids
The
drag equation 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)
:
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
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
''μ'', the
shear stress ''τ'' is linearly related to the
strain rate (transverse
flow velocity gradient) ∂''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 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.
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:
:
where ''R'' is the
gas constant (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 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 or the
Fokker–Planck equation or the
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
. For example, see
magnetohydrodynamics,
fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
,
electrohydrodynamics,
superconductivity,
plasma modeling. An entire physical apparatus for dealing with these matters has developed. See for example,
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
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:
:
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:
:
where ''ε''
0 and ''μ''
0 are two universal constants, called the
permittivity of
free space 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
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic 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 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 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). The detail employed may be
macroscopic or
microscopic, 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 () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
s with finite conductivity often are approximated at
microwave or longer wavelengths as
perfect metals with infinite conductivity (forming hard barriers with zero
skin depth of field penetration).
Some man-made materials such as
metamaterials and
photonic crystals are designed to have customized permittivity and permeability.
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
Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries.
The intellectual origins of materials sci ...
. 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; 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 as applied to
condensed matter physics. See, for example,
density functional theory,
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
wavelengths 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 branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
, a field of
solid-state physics.
Photonics
Refractive index
The (absolute)
refractive index 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'':
:
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.
The relative refractive index is defined as the ratio of the two refractive indices. Absolute is for one material, relative applies to every possible pair of interfaces;
:
Speed of light in matter
As a consequence of the definition, the
speed of light 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
*
Defining equation (physical chemistry)
*
Governing equation
*
Principle of material objectivity
*
Rheology
Notes
References
{{Reflist, 30em
Elasticity (physics)
Equations of physics
Continuum mechanics
Electric and magnetic fields in matter