Constitutive equation


In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. 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 forces 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 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 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 was the subject of Walter Noll's dissertation in 1954 under Clifford Truesdell.
In modern condensed matter physics, the constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions.

Definitions

PressureP, σ
F is the perpendicular component of the force applied to area APa = N⋅m−2−1−2General strainε

  • D = dimension
  • ΔD = change in dimension of material
1dimensionlessGeneral elastic modulusEmodPa = N⋅m−2−1−2Young's modulusE, YPa = N⋅m−2−1 −2Shear modulusGPa = N⋅m−2−1−2Bulk modulusK, BPa = N⋅m−2−1−2CompressibilityCPa−1 = m2⋅N−1−12

Deformation of solids

Friction

is a complicated phenomenon. Macroscopically, the friction 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 μf, which depends on the pair of materials:
This can be applied to static friction, kinetic friction, or rolling.

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 k in a scalar equation, stating the tensile/compressive force is proportional to the extended displacement x:
meaning the material responds linearly. Equivalently, in terms of the stress σ, Young's modulus E, and strain ε :
In general, forces which deform solids can be normal to a surface of the material, or tangential, 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:
The relative speed of separation vseparation of an object A after a collision with another object B is related to the relative speed of approach vapproach by the coefficient of restitution, defined by Newton's experimental impact law:
which depends 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 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
where the drag coefficient cd 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 ∂u/∂y. In a uniform shear flow:
with u the variation of the flow velocity u in the cross-flow direction y. In general, for a Newtonian fluid, the relationship between the elements τij of the shear stress tensor and the deformation of the fluid is given by
where vi are the components of the flow velocity vector in the corresponding xi coordinate directions, eij are the components of the strain rate tensor, Δ is the volumetric strain rate and δij 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.

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, but also the dynamics of bound charges and currents. 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 theory, Green–Kubo relations and Green's function.
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. 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 ε0 and μ0 are two universal constants, called the permittivity of free space and permeability of free space, respectively.

Isotropic linear materials

In an 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 χe and χm are the electric and magnetic susceptibilities of a given material respectively. In terms of D and H the constitutive relations are:
where ε and μ are constants, 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. These relations may be empirical, or theoretical. 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 metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity.
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. 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 are based upon approximation of an inhomogeneous material by a homogeneous effective medium.
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, a field of solid-state physics.
ρ = electrical resistivity Direct Piezoelectric Effect

d = direct piezoelectric coefficient Converse Piezoelectric Effect

d = direct piezoelectric coefficient Piezomagnetic effect

q = piezomagnetic coefficient
p = pyroelectric coefficient Electrocaloric effect

p = pyroelectric coefficient Seebeck effect

β = thermopower Peltier effect

Π = Peltier coefficient

Photonics

;Refractive index
The refractive index of a medium n is an inherently important property of geometric and physical optics defined as the ratio of the luminal speed in vacuum c0 to that in the medium c:
where ε is the permittivity and εr the relative permittivity of the medium, likewise μ is the permeability and μr are the relative permmeability of the medium. The vacuum permittivity is ε0 and vacuum permeability is μ0. In general, n 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;
;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 :

Transport phenomena

Definitions


K−1−1Volumetric thermal expansionβ, γ

K−1−1Thermal conductivityκ, K, λ,

W⋅m−1⋅K−1−3−1Thermal conductanceUW⋅m−2 K−1−3−1Thermal resistanceR
Δx = displacement of heat transfer m2⋅K⋅W−1−13

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:
In general the constant must be replaced by a 2nd rank tensor, to account for directional dependences of the material.
Darcy's law for fluid flow in porous media, defines permeability κ

Ohm's law of electric conduction, defines electric conductivity


Fourier's law of thermal conduction, defines thermal conductivity λ

Stefan–Boltzmann law of black-body radiation, defines emmisivity ε



For a temperature difference: