A constitutive equation, or constitutive relation, is a mathematical relationship that describes how a specific material responds to external stimuli. In continuum mechanics, it connects الحركية quantities like strain to kinetic quantities like stress. For example, Hooke’s Law, [latex]\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}[/latex], is a constitutive equation for linear elastic solids, relating the stress tensor [latex]\boldsymbol{\sigma}[/latex] to the strain tensor [latex]\boldsymbol{\varepsilon}[/latex].
Constitutive equations are essential because the fundamental laws of continuum الميكانيكا (حفظ الكتلة, momentum, and energy) are universal and apply to all materials. However, different materials behave differently under the same loading conditions. A steel beam, a column of water, and a piece of rubber will all respond uniquely to an applied force. Constitutive equations provide the material-specific information needed to close the system of governing equations and obtain a unique solution for a given problem. They are determined experimentally and represent a mathematical model of a material’s behavior.
The complexity of constitutive equations varies greatly. The simplest models are for linear, isotropic materials. For a linear elastic solid, Hooke’s Law relates stress and strain linearly via a fourth-order stiffness tensor [latex]\mathbf{C}[/latex]. For a Newtonian fluid, the stress is linearly related to the rate of strain. However, many real-world materials exhibit much more complex behavior. Non-linear elasticity is needed for materials like rubber that undergo large deformations. Plasticity models describe permanent deformation after a yield stress is exceeded. Viscoelastic models, used for polymers, exhibit both fluid-like and solid-like characteristics, with their response depending on the rate of loading. Developing accurate constitutive models for advanced materials like composites, biological tissues, or granular materials is a major and ongoing area of research in mechanics.
