Constitutive equation

A constitutive equation relates a material’s response, such as stress or heat flux, to variables such as deformation, temperature, and their histories. It closes balance laws by encoding material behavior.

Stress (mechanics): A measure of internal force per unit area acting within a material. Stress is a common response variable in mechanical constitutive equations.
Elasticity: The property of returning toward an original shape after a deforming load is removed. Elastic constitutive laws connect recoverable deformation with stress.
Hooke's law: A linear relation between elastic stress and strain, valid for small deformations in suitable materials. It is the standard simple constitutive equation for linear elasticity.
Classical continuum mechanics: The theory that models matter as continuously distributed fields governed by balance laws. Constitutive equations close its continuum balance equations.
Strain: A measure of deformation expressed as relative changes in length, angle, or volume. Strain often supplies the deformation input paired with stress.
Viscoelasticity: Material behavior combining time-dependent viscous flow and elastic deformation. Its constitutive laws include dependence on loading history and rate.
Generalized Newtonian fluid: A fluid model in which viscosity depends on the rate of deformation but not its history. Its constitutive law describes shear-thinning and shear-thickening flows.
Nonlocal elasticity: An elasticity theory in which stress at a point depends on deformation across a finite region. It contrasts with local constitutive laws based only on conditions at one point.
Temperature: A physical quantity that characterizes thermal state and determines the direction of heat transfer. Temperature can alter material response or serve as a constitutive input.
Plasticity (physics): Permanent deformation that remains after a material is unloaded beyond its elastic range. Plastic constitutive equations describe yielding and irreversible strain.

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