Constitutive equation

In physics and engineering, 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 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 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 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 = Template:Mvar (velocity gradient, stress, density) 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

Quantity Template:Nobr	(Common) symbol/s	Defining equation	SI units	Dimension
General stress, pressure	Template:Math	$P=\sigma =F_{\perp }/A,$ $\tau =F_{\Vert }/A$	Template:Nobr	[M][L]−1[T]−2
General strain	Template:Math	$\epsilon =\mathrm{\Delta }L/L,$ $\gamma =\mathrm{\Delta }x/L$	1	Dimensionless
General elastic modulus	Template:Math	$E_{\text{mod}}=\frac{\text{stress}}{\text{strain}}$	Template:Nobr	[M][L]−1[T]−2
Young's modulus	Template:Math	$Y=\sigma /\epsilon$	Template:Nobr	[M][L]−1[T] −2
Shear modulus	Template:Mvar	$G=\tau /\gamma$	Template:Nobr	[M][L]−1[T]−2
Bulk modulus	Template:Math	$B=\frac{P}{\mathrm{\Delta }V/V}$	Template:Nobr	[M][L]−1[T]−2
Compressibility	Template:Mvar	$C=1/B$	Template:Nobr	[M]−1[L][T]2

Where:

Template:Mvar is the component of the force applied to area Template:Mvar (either parallel or perpendicular)

Template:Mvar and Template:Mvar are the dimension (length, volume)

Template:Math, Template:Math, and Template:Math are the change in dimension due to the applied force

Deformation of solids

Friction

Friction is a complicated phenomenon. Macroscopically, the friction force Template:Math at the interface of two materials can be modelled as proportional to the reaction force Template:Mvar at a point of contact between two interfaces through a dimensionless coefficient of friction Template:Math, which depends on the pair of materials:

$$F_{\text{f}}={\mu }_{\text{f}}R.$$

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 friction (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) Template:Mvar in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) displacement Template:Mvar:

$$F_i=-k{x}_i$$

meaning the material responds linearly. Equivalently, in terms of the stress Template:Mvar, Young's modulus Template:Mvar, and strain Template:Mvar (dimensionless):

$$\sigma =E\epsilon$$

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:

$${\sigma }_{ij}=C_{ijkl}{\epsilon }_{kl}\rightleftharpoons {\epsilon }_{ij}=S_{ijkl}{\sigma }_{kl}$$

where Template:Mvar is the elasticity tensor and Template:Mvar 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 Template:Math of an object A after a collision with another object B is related to the relative speed of approach Template:Math by the coefficient of restitution, defined by Newton's experimental impact law:

$$e=\frac{|𝐯{|}_{\text{separation}}}{|𝐯{|}_{\text{approach}}}$$

which depends on the materials A and B are made from, since the collision involves interactions at the surfaces of A and B. Usually Template:Math, in which Template:Math for completely elastic collisions, and Template:Math for completely inelastic collisions. It is possible for Template:Math to occur – for superelastic (or explosive) collisions.

Deformation of fluids

The drag equation gives the drag force Template:Mvar on an object of cross-section area Template:Mvar moving through a fluid of density Template:Mvar at velocity Template:Mvar (relative to the fluid)

$$D=\frac{1}{2}{c}_d\rho A{v}^2$$

where the drag coefficient (dimensionless) Template:Mvar 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 Template:Mvar, the shear stress Template:Mvar is linearly related to the strain rate (transverse flow velocity gradient) Template:Math (units s⁻¹). In a uniform shear flow:

$$\tau =\mu \frac{\partial u}{\partial y},$$

with Template:Math the variation of the flow velocity Template:Mvar in the cross-flow (transverse) direction Template:Mvar. In general, for a Newtonian fluid, the relationship between the elements Template:Mvar of the shear stress tensor and the deformation of the fluid is given by

$${\tau }_{ij}=2\mu (e_{ij}-\frac{1}{3}\mathrm{\Delta }{\delta }_{ij})$$

where:

Template:Mvar is the Kronecker delta;

Template:Math is the volumetric strain rate (or dilatation rate)$$\mathrm{\Delta }=\sum _k{e}_{kk}=\text{div}𝐯,$$

Template:Mvar are the components of the strain rate tensor $$e_{ij}=\frac{1}{2}(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})$$ where Template:Mvar are the components of the flow velocity vector in the corresponding Template:Mvar coordinate directions.

The ideal gas law is a constitutive relation in the sense the pressure Template:Mvar and volume Template:Mvar are related to the temperature Template:Mvar, via the number of moles Template:Mvar of gas:

$$pV=nRT$$

where Template:Mvar 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. 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 (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 Template:Math and Template:Math, and the magnetic H-field Template:Math and Template:Math, 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 Template:Math and Template:Math and the Template:Math and Template:Math fields starts with the definition of the auxiliary fields themselves: $$\begin{array}{cc}𝐃(𝐫,t)& ={\epsilon }_0𝐄(𝐫,t)+𝐏(𝐫,t)\\ 𝐇(𝐫,t)& =\frac{1}{{\mu }_0}𝐁(𝐫,t)-𝐌(𝐫,t),\end{array}$$

where Template:Math is the polarization field and Template:Math is the magnetization field which are defined in terms of microscopic bound charges and bound current respectively. Before getting to how to calculate Template:Math and Template:Math 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:

$$𝐃={\epsilon }_0𝐄,𝐇=\frac{1}{{\mu }_0}𝐁,$$

where Template:Math and Template:Math 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 Template:Math is proportional to Template:Math, and Template:Math is proportional to Template:Math, the constitutive relations are also straightforward. In terms of the polarization Template:Math and the magnetization Template:Math they are:

$$𝐏={\epsilon }_0{\chi }_e𝐄,𝐌={\chi }_m𝐇,$$

where Template:Math and Template:Math are the electric and magnetic susceptibilities of a given material respectively. In terms of Template:Math and Template:Math the constitutive relations are:

$$𝐃=\epsilon 𝐄,𝐇=\frac{1}{\mu }𝐁,$$

where Template:Mvar and Template:Mvar are constants (which depend on the material), called the permittivity and permeability, respectively, of the material. These are related to the susceptibilities by:

$$\frac{\epsilon }{{\epsilon }_0}={\epsilon }_r={\chi }_e+1,\frac{\mu }{{\mu }_0}={\mu }_r={\chi }_m+1.$$

General case

For real-world materials, the constitutive relations are not linear, except approximately. Calculating the constitutive relations from first principles involves determining how Template:Math and Template:Math are created from a given Template:Math and Template:Math. 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: $$𝐃=\epsilon 𝐄,𝐇={\mu }^{-1}𝐁$$ but Template:Mvar and Template:Mvar are not, in general, simple constants, but rather functions of Template:Math, Template:Math, position and time, and tensorial in nature. Examples are:

As a variation of these examples, in general materials are bianisotropic where Template:Math and Template:Math depend on both Template:Math and Template:Math, through the additional coupling constants Template:Mvar and Template:Mvar: $$𝐃=\epsilon 𝐄+\xi 𝐇,𝐁=\mu 𝐇+\zeta 𝐄.$$

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 (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

Template:See also 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 Template:Math and Template:Math 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 (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, Template:Mvar of an insulator at low frequencies can be measured by making it into a parallel-plate capacitor, and Template:Mvar 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.

Electromagnetic properties of solids

Property/effect	Stimuli/response parameters of system	Constitutive tensor of system	Equation
Hall effect	Template:Nobr	Template:Mvar, electrical resistivity (Ω⋅m)	$E_k={\rho }_{kij}{J}_i{H}_j$
Direct Piezoelectric Effect	Template:Plainlist	Template:Mvar, direct piezoelectric coefficient (C⋅N−1)	$P_i=d_{ijk}{\sigma }_{jk}$
Converse Piezoelectric Effect	Template:Plainlist	Template:Mvar, direct piezoelectric coefficient (C⋅N−1)	${\epsilon }_{ij}=d_{ijk}{E}_k$
Piezomagnetic effect	Template:Plainlist	Template:Mvar, piezomagnetic coefficient (A⋅N−1⋅m)	$M_i=q_{ijk}{\sigma }_{jk}$

Thermoelectric properties of solids

Property/effect	Stimuli/response parameters of system	Constitutive tensor of system	Equation
Pyroelectricity	Template:Plainlist	Template:Mvar, pyroelectric coefficient (C⋅m−2⋅K−1)	$\mathrm{\Delta }{P}_j=p_j\mathrm{\Delta }T$
Electrocaloric effect	Template:Plainlist	Template:Mvar, pyroelectric coefficient (C⋅m−2⋅K−1)	$\mathrm{\Delta }S=p_i\mathrm{\Delta }{E}_i$
Seebeck effect	Template:Nobr	Template:Mvar, thermopower (V⋅K−1)	$E_i=-{\beta }_{ij}\frac{\partial T}{\partial x_j}$
Peltier effect	Template:Plainlist	Template:Math, Peltier coefficient (W⋅A−1)	$q_j={\mathrm{\Pi }}_{ji}{J}_i$

Photonics

Refractive index

The (absolute) refractive index of a medium Template:Mvar (dimensionless) is an inherently important property of geometric and physical optics defined as the ratio of the luminal speed in vacuum Template:Math to that in the medium Template:Mvar:

$$n=\frac{c_0}{c}=\sqrt{\frac{\epsilon \mu }{{\epsilon }_0{\mu }_0}}=\sqrt{{\epsilon }_r{\mu }_r}$$

where Template:Mvar is the permittivity and Template:Math the relative permittivity of the medium, likewise Template:Mvar is the permeability and Template:Math are the relative permeability of the medium. The vacuum permittivity is Template:Math and vacuum permeability is Template:Math. In general, Template:Mvar (also Template:Math) 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;

$$n_{AB}=\frac{n_A}{n_B}$$

Speed of light in matter

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

$$c=\frac{1}{\sqrt{\epsilon \mu }}$$

for special case of vacuum; Template:Math and Template:Math,

$$c_0=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}$$

Piezooptic effect

The piezooptic effect relates the stresses in solids Template:Mvar to the dielectric impermeability Template:Mvar, which are coupled by a fourth-rank tensor called the piezooptic coefficient Template:Math (units K⁻¹):

$$a_{ij}={\mathrm{\Pi }}_{ijpq}{\sigma }_{pq}$$

Transport phenomena

Definitions

Definitions (thermal properties of matter)

Quantity Template:Nobr	(Common) symbol/s	Defining equation	SI units	Dimension
General heat capacity	Template:Mvar, heat capacity of substance	$q=CT$	J⋅K−1	Template:Nobr
linear thermal expansion coefficient	Template:Plainlist	$\begin{array}{cc}\frac{\partial L}{\partial T}& =\alpha L\\ {\epsilon }_{ij}& ={\alpha }_{ij}\mathrm{\Delta }T\end{array}$	K−1	[Θ]−1
Volumetric thermal expansion coefficient	Template:Math	${\left(\frac{\partial V}{\partial T}\right)}_p=\gamma V$	K−1	[Θ]−1
Thermal conductivity	Template:Math	$\lambda =-\frac{𝐪}{\mathrm{\nabla }T}$	W⋅m−1⋅K−1	[M][L][T]−3[Θ]−1
Thermal conductance	Template:Mvar	$U=\frac{\lambda }{\delta x}$	W⋅m−2⋅K−1	[M][T]−3[Θ]−1
Thermal resistance	Template:Math, displacement of heat transfer (m)	$R=\frac{1}{U}=\frac{\mathrm{\Delta }x}{\lambda }$	m2⋅K⋅W−1	[M]−1[L][T]3[Θ]

Definitions (electrical/magnetic properties of matter)

Quantity Template:Nobr	(Common) symbol/s	Defining equation	SI units	Dimension
Electrical resistance	Template:Mvar	$R=V/I$	Template:Nobr	[M][L]2[T]−3[I]−2
Resistivity	Template:Mvar	$\rho =RA/l$	Ω⋅m	Template:Nobr
Resistivity temperature coefficient, linear temperature dependence	Template:Mvar	$\rho -{\rho }_0={\rho }_0\alpha (T-T_0)$	K−1	[Θ]−1
Electrical conductance	Template:Mvar	$G=1/R$	S = Ω−1	[M]−1[L]−2[T]3[I]2
Electrical conductivity	Template:Mvar	$\sigma =1/\rho$	Ω−1⋅m−1	[M]−2[L]−2[T]3[I]2
Magnetic reluctance	Template:Math $ℛ$	$R_{\text{m}}=ℳ/{\mathrm{\Phi }}_B$	A⋅Wb−1 = H−1	[M]−1[L]−2[T]2
Magnetic permeance	Template:Math $𝒫$	$\mathrm{\Lambda }=1/R_{\text{m}}$	Wb⋅A−1 = H	[M][L]2[T]−2

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.

Property/effect	Nomenclature	Equation
Template:Nobr (defines diffusion coefficient Template:Mvar)	Template:Nobr	$J_i=-D_{ij}\frac{\partial C}{\partial x_j}$
Darcy's law for fluid flow in porous media (defines permeability Template:Mvar)	Template:Plainlist	$q_j=-\frac{\kappa }{\mu }\frac{\partial P}{\partial x_j}$
Ohm's law of electric conduction (defines electric conductivity Template:Mvar, and hence resistivity Template:Mvar and resistance Template:Mvar)	Template:Plainlist	Simplest form: $V=IR$ More general forms: $\frac{\partial V}{\partial x_j}={\rho }_{ji}{J}_i\rightleftharpoons J_i={\sigma }_{ij}\frac{\partial V}{\partial x_j}$
Fourier's law of thermal conduction (defines thermal conductivity Template:Mvar)	Template:Plainlist	$q_i=-{\lambda }_{ij}\frac{\partial T}{\partial x_j}$
Stefan–Boltzmann law of black-body radiation (defines emissivity Template:Mvar)	Template:Plainlist	For a single radiator: $I=\epsilon \sigma T^4$ For a temperature difference: $I=\epsilon \sigma (T_{\text{ext}}^4-T_{\text{sys}}^4)$ Template:Math; 0 for perfect reflector, 1 for perfect absorber (true black body)

See also

• Defining equation (physical chemistry)
• Governing equation
• Principle of material objectivity
• Rheology