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 = (velocity gradient, stress, density) was the subject of Walter Noll's dissertation in 1954 under Clifford Truesdell.See Truesdell's account in Truesdell The naturalization and apotheosis of Walter Noll. See also 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 Quantity (Common)symbol/s Defining equation SI units Dimension General stress, pressure [M][L]−1[T]−2 General strain 1 Dimensionless General elastic modulus [M][L]−1[T]−2 Young's modulus [M][L]−1[T] −2 Shear modulus [M][L]−1[T]−2 Bulk modulus [M][L]−1[T]−2 Compressibility [M]−1[L][T]2
Where: is the component of the force applied to area (either parallel or perpendicular) and are the dimension (length, volume) , , and are the change in dimension due to the applied force
Deformation of solids
Friction
Friction is a complicated phenomenon. Macroscopically, the friction force at the interface of two materials can be modelled as proportional to the reaction force at a point of contact between two interfaces through a dimensionless coefficient of friction , 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 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) in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) displacement :
meaning the material responds linearly. Equivalently, in terms of the stress , Young's modulus , and strain (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 is the elasticity tensor and is the compliance tensor.
Solid-state deformation
Several classes of deformation in elastic materials are the following:Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3 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 of an object A after a collision with another object B is related to the relative speed of approach by the coefficient of restitution, defined by Newton's experimental impact law:Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray,
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 on an object of cross-section area moving through a fluid of density at velocity (relative to the fluid)
where the drag coefficient (dimensionless) 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 shear stress is linearly related to the strain rate (transverse flow velocity gradient) (units s−1). In a uniform shear flow:
with the variation of the flow velocity in the cross-flow (transverse) direction . In general, for a Newtonian fluid, the relationship between the elements of the shear stress tensor and the deformation of the fluid is given by
where: is the Kronecker delta; is the volumetric strain rate (or dilatation rate) are the components of the strain rate tensor where are the components of the flow velocity vector in the corresponding coordinate directions.
The ideal gas law is a constitutive relation in the sense the pressure and volume are related to the temperature , via the number of moles of gas:
where 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 and , and the magnetic H-field and , 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 and and the and fields starts with the definition of the auxiliary fields themselves:
where is the polarization field and is the magnetization field which are defined in terms of microscopic bound charges and bound current respectively. Before getting to how to calculate and 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 and are two universal constants, called the permittivity of free space and permeability of free space, respectively.
Isotropic linear materials In an (isotropicThe generalization to non-isotropic materials is straight forward; simply replace the constants with tensor quantities.) linear material, where is proportional to , and is proportional to , the constitutive relations are also straightforward. In terms of the polarization and the magnetization they are:
where and are the electric and magnetic susceptibilities of a given material respectively. In terms of and the constitutive relations are:
where and 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 and are created from a given and .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 and are not, in general, simple constants, but rather functions of , , position and time, and tensorial in nature. Examples are:
As a variation of these examples, in general materials are bianisotropic where and depend on both and , through the additional coupling constants and :