 Constitutive equation

A constitutive equation specifies a material's response beyond universal balances such as mass conservation and momentum conservation. In a simple elastic test, Hooke's law $\sigma =E\epsilon$ relates axial stress to strain, with Young's modulus $E$ describing the specimen's elastic response. A Newtonian fluid instead relates viscous stress to the rate-of-strain tensor.

A useful constitutive equation states its variables and regime: temperature, deformation rate and loading history can change the response. Plasticity in continuum mechanics therefore uses internal variables as well as current strain, whereas an ideal elastic material stores and returns mechanical energy. Experiments identify parameters and test whether the model predicts other load paths. Physical consistency also imposes restrictions, including invariance under changes of observer and the Clausius-Duhem inequality. Merely fitting one measured curve does not establish validity for arbitrary deformation.