Constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic 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  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 = f (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. Deformation of solids. Friction. 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 "μ", 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 deformations. Several classes of deformations in elastic materials are the following: Collisions. The relative speed of separation "v" of an object A after a collision with another object B is related to the relative speed of approach "v" by the coefficient of restitution, defined by Newton's experimental impact law: