Constitutive equation

Many defining equations are in the form of a constitutive equation, since parameters of a property or effect associated matter are characteristic to the substance in question. A large number of other defining equations not specifically constitutive, but more generally applicable, are in that article.

In physics, a constitutive equation is a relation between two physical quantities (especially kinetic quantities are related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external forces. It is combined with other equations governing physical laws to solve physical problems, like the flow of a fluid in a pipe, or the response of a crystal to an electric field.

As an example, in structural analysis, constitutive relations connect applied stresses or forces to strains or deformations. The stress-strain constitutive relation for linear materials is commonly known as Hooke's law.

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, much more elaborate constitutive equations often are necessary to account for tensor properties, the rate of response of materials and their non-linear behavior.^[1] See the article Linear response function.

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.^[2]

In modern condensed matter physics, the constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions.^[3]

Following are numerous examples of constitutive equations, many in frequent use.

Matter

Definitions

Nomenclature
F = friction force (N) R = reaction force at point of contact (N)

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Static, kinetic, and rolling friction coefficients	μ, μ_f	$F \leqslant \mu_\mathrm{static} R \,$ $F = \mu_\mathrm{kinetic} R \,$ $F = \mu_\mathrm{roll} R \,$ Normal force to a surface: $F_\bot = \mathbf{F}\cdot\mathbf{\hat{n}}\,\!$	dimensionless	dimensionless

Definitive laws

Property/effect	Equation
Hooke's law, defines elasticity/spring constants	Scalar forms: $F_i=-k x_i \,$ or $\sigma = Y \, \epsilon \,$ Tensor forms, $\sigma_{ij} = C_{ijkl} \, \epsilon_{kl} \,$ or inversely, $\epsilon_{ij} = S_{ijkl} \, \sigma_{kl} \,$
F = tensile/compressive force (N) x = extended/contracted displacement (m) σ = stress (Pa) ε = strain (dimensionless) k = (Hooke's Law) spring constant (N m⁻¹) Y = Young's modulus (Pa) C = elasticity tensor (Pa) S = compliance tensor (dimensionless)
Newton's law of experimental impact, defines coefficient of restitution	$e = \frac{\left \| \mathbf{v} \right \| _\mathrm{separation}}{\left \| \mathbf{v} \right \| _\mathrm{approach}} \,\!$ Usually $0 \leqslant e \leqslant 1 \,\!$ , but it is possible that $e \geqslant 1 \,\!$
e = Coefficient of restitution v_separation = relative velocity of separation (m s⁻¹) v_approach = relative velocity of approach (m s⁻¹)

Transport phenomena

Nomenclature
x = displacement of substance(m) A = Cross-section Area(m²) v = Flow velocity (m s⁻¹) μ = Fluid viscosity (Pa s) F_d = Drag force (N) g = Standard gravitational acceleration = 9.81 N kg^-1 ρ = Volume mass density of fluid (kg m⁻³)

Definitions

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Drag equation, drag coefficient	c_d	$F_d = \frac{1}{2}c_d \rho A v^2 \,$	dimensionless	dimensionless

Definitive laws

Property/effect	Equation
Fick's law of diffusion, defines diffusion coefficient D	$J_j = - D_{ij} \frac{\partial C}{\partial x_i}$
D = mass diffusion coefficient (m² s⁻¹) J = diffusion flux of substance (mol m⁻² s⁻¹) ∂C/∂x = (1d)concentration gradient of substance (mol dm⁻⁴)	$J_j = - D_{ij} \frac{\partial C}{\partial x_i}$
Darcy's law for porous flow in matter, defines permeability κ	$q_j = -\frac{\kappa}{\mu} \frac{\partial P}{\partial x_j}$
κ = permeability of medium (m²) q = discharge flux of substance (m s⁻¹) ∂P/∂x = (1d) pressure gradient of system (Pa m⁻¹)	$q_j = -\frac{\kappa}{\mu} \frac{\partial P}{\partial x_j}$

Equations

Property/effect	Equation
Terminal velocity	$v_t = \sqrt{\frac{2 F_d g}{c_d \rho \mathbf{A} \cdot \mathbf{\hat{n}}}} \,$
v_t = terminal velocity within fluid (m s^-1)
Newtonian fluid	$\tau = \mu \frac {\partial u}{\partial y} \,$
τ = shear stress exerted by fluid (Pa) ∂u/∂y = velocity gradient perp. to shear direction, strain rate (s⁻¹)	$\tau = \mu \frac {\partial u}{\partial y} \,$

Thermodynamics

Nomenclature
L = length of material (m) Δx = Displacement of heat transfer (m) A = surface cross section (m²) V = Volume (m³) T = Temperature (K) ∇T = temperature gradient (K m⁻¹) q = heat energy (J) P = thermal current, thermal power (W) ε = strain or emmisivity (both dimensionless)

Definitions

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
General heat capacity	C = Heat capacity	$q = C T \,$	J K⁻¹
Linear thermal expansion	α = Linear coefficient of linear thermal expansion	$\partial L/\partial T = \alpha L \,\!$ $\epsilon_{ij} = \alpha_{ij}\Delta T \,$	K⁻¹	[Θ]⁻¹
Volumetric thermal expansion	β, γ	$\partial V/\partial T = \beta V \,\!$	K⁻¹
Thermal conductivity	κ, K, λ	$\lambda = - P/\left (\mathbf{A} \cdot \nabla T \right ) \,\!$	W m⁻¹ K⁻¹
Thermal conductance	U	$U = \lambda/\delta x \,\!$	W m⁻² K⁻¹
Thermal resistance	R	$R=1/U=\Delta x/\lambda\,\!$	m² K W⁻¹

Definitive laws

Property/effect	Equation
Fourier's law of thermal conduction, defines thermal conductivity λ	$q_j= - \lambda_{ij}\frac{\partial T}{\partial x_i} \,$
λ = Thermal conductivity (W m⁻¹ K⁻¹ ) q = heat flux (W m⁻²) ∂T/∂x = (1d) Temperature gradient (K m⁻¹)	$q_j= - \lambda_{ij}\frac{\partial T}{\partial x_i} \,$
Stefan–Boltzmann law of black-body radiation, defines emmisivity ε	For a single radiator: $I = \epsilon \sigma T^4\,$ For a temperature differance: $I = \epsilon \sigma \left ( T_\mathrm{ext}^4 - T_\mathrm{sys}^4 \right ) \,$ $0 \leqslant \epsilon \leqslant 1\,\!$ $\epsilon = 0\,\!$ for perfect reflector $\epsilon = 1\,\!$ for perfect absorber (true black body)
I = radiant intensity (W m⁻²) σ = Stefan–Boltzmann constant (W m⁻² K⁻⁴) T_sys = temperature of radiating system(K) T_ext = temperature of external surroundings (K) T = temperature (K) ε = emissivity (dimensionless)
Ideal gas law, defines parameters of an ideal gas	$pV = nRT\,$
p = pressure (Pa) T = temperature (K) V = volume (m⁻³) R = gas constant (J K⁻¹ mol⁻¹) n = number of moles, amount of substance (mol)	$pV = nRT\,$

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.

Nomenclature
E = electric field (N C⁻¹) D = electric displacement field (Cm⁻²) P = polarization density (C m⁻²) B = magnetic flux density, induction field (T = N A⁻¹ m⁻¹) H = magnetic field intensity (A m⁻¹) M = magnetization field (A m⁻¹)

Definitions

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Electrical resistance	R	$R = V/I \,\!$	Ω = V A⁻¹ = J s C⁻²	[M] [L]² [T]⁻³ [I]⁻²
Resistivity	ρ	$\rho = RA/l \,\!$	Ω m	[M]² [L]² [T]⁻³ [I]⁻²
Resistivity temperature coefficient, linear temperature dependence	α	$\rho - \rho_0 = \rho_0\alpha(T-T_0)\,\!$	K⁻¹	[Θ]⁻¹
Electrical conductance	G	$G = 1/R \,\!$	S = Ω⁻¹	[T]³ [I]² [M]⁻¹ [L]⁻²
Electrical conductivity	σ	$\sigma = 1/\rho \,\!$	Ω⁻¹ m⁻¹	[I]² [T]³ [M]⁻² [L]⁻²
Relative permittivity (aka dielectric constant)	ε_r	can only be found from experiment	F m⁻¹	[I]² [T]⁴ [M]⁻¹ [L]⁻³
(Absolute) dielectric permittivity	ε	$D_j = \epsilon_{ji} E_i \,$	F m⁻¹	[I]² [T]⁴ [M]⁻¹ [L]⁻³
Electric susceptibility	χ_E	$P_j = \epsilon_0 \left ( \chi_E \right )_{ji} E_i \,$	dimensionless	dimensionless
Magnetic reluctance	R, R_m, $\mathcal{R}$	$R_\mathrm{m} = \mathcal{M}/\Phi_B$	A Wb^-1 = H^-1	[M]^-1[L]^-2[T]²
Magnetic permeance	P, P_m, Λ, $\mathcal{P}$	$\Lambda = 1/R_\mathrm{m}$	Wb A^-1 = H	[M][L]²[T]^-2
Relative permeability	μ_r	can only be found from experiment	dimensionless	dimensionless
Absolute magnetic permeability	μ	$B_j = \mu_{ji} H_i \,$	H m⁻¹
Magnetic susceptibility	χ_M	$M_j = \left ( \chi_M \right )_{ji} H_i \,$	dimensionless	dimensionless

Relations

Property/effect	Equation
Electric field vectors	$E_i = \frac{1}{\epsilon } D_i = \frac{1}{\epsilon_0} \left ( D_i - P_i \right )\,$
Magnetic field Vectors	$B_i = \mu H_i = \mu_0 \left ( H_i %2B M_i \right ) \,$
Permittivity	$\epsilon/\epsilon_0 = \epsilon_r = \left ( \chi_E %2B 1 \right )\,\!$
Permeability	$\mu / \mu_0 = \mu_r = \left ( \chi_M %2B 1 \right ) \,\!$

Laws

Property/effect	Equation
Ohm's law of electric conduction, defines electric conductivity (and so resistivity and resistance)	Simplist form is: $V = IR \,$ More general forms are: $J_j = \sigma_{ji} E_i \,$ $E_j = \rho_{ji} J_i \,$
J = Electric Current Density (A m⁻²) σ = Electric Conductivity (Ω⁻¹ m⁻¹) ρ = Electrical resistivity (Ω m)

Photonics

Nomenclature
c = luminal speed in medium (m s⁻¹) c₀ = luminal speed in vacuum (m s⁻¹) ε = electric permittivity of medium (F m⁻¹) ε₀ = vacuum electric permittivity (F m⁻¹) ε_r = relative electric permittivity of medium (F m⁻¹) μ = magnetic permeability of medium (H m⁻¹) μ₀ = vacuum magnetic permeability (H m⁻¹) μ_r = relative magnetic permeability of medium (H m⁻¹)

Definitions

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Refractive index	n	$n = \frac{c}{c_0} = \sqrt{\frac{\epsilon_0 \mu_0}{\epsilon \mu}} = \sqrt{\epsilon_r \mu_r} \,$	dimensionless	dimensionless

Relations

Property/effect	Equation
Luminal speed in matter	$c = 1/\sqrt{\epsilon \mu} \,$ for special case of vacuum; ε = ε₀ and μ = μ₀, $c_0 = 1/ \sqrt{\epsilon_0\mu_0} \,$

Solid state physics

These constitutive equations are often used in crystallography - a field of solid state physics.^[4]

Nomenclature
x = displacement (m) s = thickness of hall/probe (m) σ = mechanical stress (Pa) ε = mechanical strain (dimensionless) e = electron charge (C) n = charge carrier density E = electric field (N C⁻¹) D = electric displacement field (Cm⁻²) P = dielectric polarization density (C m⁻²) B = magnetic flux density, induction field (T = N A⁻¹ m⁻¹) H = Magnetic Field Intensity (A m⁻¹) M = magnetization field (A m⁻¹) I = electric current (across Hall plate) (A) J = electric current density (A m⁻²) ρ = electrical resistivity (Ω m) a = dielectric impermeability (K) V_H = Hall voltage (V) T = temperature (K) S = entropy (J K⁻¹) q = heat flux (W m⁻²)

Definitions

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Pyroelectricity, pyroelectric coefficient	p	$\Delta P_j = p_{j} \Delta T \,$	C m⁻² K⁻¹
Electrocaloric effect, pyroelectric coefficient	p	$\Delta S = p_{i} \Delta E_i \,$	C m⁻² K⁻¹	[I][T][L]^-2[Θ]^-1
Seebeck effect, thermopower coefficient	β	$E_{i} = - \beta_{ij} \frac{\partial T}{\partial x_j} \,$	V K⁻¹
Hall effect, Hall coefficient	R_H	$R_H =\frac{E_y}{J_xB}= \frac{sV_H}{IB}=-\frac{1}{ne}$	C^-1	[I]^-1[T]^-1
Peltier effect, peltier coefficient	Π	$q_{j} = \Pi_{ji} J_{i} \,$	W A⁻¹
Direct piezoelectric effect, direct piezoelectric coefficient	d, d_e	$P_{i} = \left ( d_\mathrm{e} \right )_{ijk}\sigma_{jk} \,$	K⁻¹
Converse piezoelectric effect, direct piezoelectric coefficient	d, d_e'	$\epsilon_{ij} = \left ( d_\mathrm{e'} \right )_{ijk}E_{k} \,$	K⁻¹
Piezomagnetic effect, piezomagnetic coefficient	q, d_m	$M_{i} = \left ( d_\mathrm{m} \right )_{ijk}\sigma_{jk} \,$	K⁻¹
Piezooptic effect, piezooptic coefficient	Π	$a_{ij} = \Pi_{ijpq}\sigma_{pq} \,$	K⁻¹