Hypoelastic material

In continuum mechanics, a hypoelastic material is an elastic material that has a constitutive model independent of finite strain measures except in the linearized case. Hypoelastic material models are distinct from hyperelastic material models (or standard elasticity models) in that, except under special circumstances, they cannot be derived from a strain energy density function. In continuum mechanics, a hypoelastic material is an elastic material that has a constitutive model independent of finite strain measures except in the linearized case. Hypoelastic material models are distinct from hyperelastic material models (or standard elasticity models) in that, except under special circumstances, they cannot be derived from a strain energy density 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 σ {displaystyle {oldsymbol {sigma }}} at time t {displaystyle t} 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. 2. There is a tensor-valued function G {displaystyle G} such that σ ˙ = G ( σ , L ) , {displaystyle {dot {oldsymbol {sigma }}}=G({oldsymbol {sigma }},{oldsymbol {L}}),,} in which σ ˙ {displaystyle {dot {oldsymbol {sigma }}}} is the material rate of the Cauchy stress tensor, and L {displaystyle {oldsymbol {L}}} 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 G {displaystyle G} exists. As explained below, specific formulations of hypoelastic models typically employ a so-called objective stress rate so that the G {displaystyle G} function exists only implicitly.