Elasticity (physics)
Hypoelastic materials

A hypoelastic material can be rigorously defined as one that is modeled using a constitutive equation satisfying the following two criteria:

If only these two original criteria are used to define hypoelasticity, then hyperelasticity would be included as a special case, which prompts some constitutive modelers to append a third criterion that specifically requires a hypoelastic model to not be hyperelastic (i.e., hypoelasticity implies that stress is not derivable from an energy potential). If this third criterion is adopted, it follows that a hypoelastic material might admit nonconservative adiabatic loading paths that start and end with the same deformation gradient but do not start and end at the same internal energy.

Note that the second criterion requires only that the function G {\displaystyle G} exists. As detailed in the main hypoelastic material article, specific formulations of hypoelastic models typically employ so-called objective rates so that the G {\displaystyle G} function exists only implicitly and is typically needed explicitly only for numerical stress updates performed via direct integration of the actual (not objective) stress rate.
punctuation:no_comma