Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses might depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "hyperelastic material" models (for which stress can be derived from a scalar "elastic potential" function). A hypoelastic material can be rigorously defined as one that is modeled using a constitutive equation satisfying the following two criteria:  1. The Cauchy stress formula_5 at time formula_6 depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a Cauchy elastic material, for which the current stress depends only on the current configuration rather than the history of past configurations. There is a tensor-valued function formula_7 such that formula_8 in which formula_9 is the material rate of the Cauchy stress tensor, and formula_10 is the spatial velocity gradient tensor. 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 formula_7 "exists". As detailed in the main Hypoelastic material article, specific formulations of hypoelastic models typically employ so-called objective rates so that the formula_7 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. Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a strain energy density function ("W"). A model is hyperelastic if and only if it is possible to express the Cauchy stress tensor as a function of the deformation gradient via a relationship of the form This formulation takes the energy potential ("W") as a function of the deformation gradient (formula_14).