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
2. There is a tensor-valued function
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
Hypoelastic material models frequently take the form
where
Hypoelasticity and objective stress rates
In many practical problems of solid mechanics, it is sufficient to characterize material deformation by the small (or linearized) strain tensor
where
- large nonlinear elastic deformations possessing a potential energy,
W ( F ) (exhibited, e.g., by rubber), in which the stress tensor components are obtained as the partial derivatives ofW with respect to the finite strain tensor components; and - inelastic deformations possessing no potential, in which the stress-strain relation is defined incrementally.
In the former kind, the total strain formulation described in the article on finite strain theory is appropriate. In the latter kind an incremental (or rate) formulation is necessary and must be used in every load or time step of a finite element computer program using updated Lagrangian procedure. The absence of a potential raises intricate questions due to the freedom in the choice of finite strain measure and characterization of the stress rate.
For a sufficiently small loading step (or increment), one may use the deformation rate tensor (or velocity strain)
or increment
representing the linearized strain increment from the initial (stressed and deformed) state in the step. Here the superior dot represents the material time derivative (
However, it would not be objective to use the time derivative of the Cauchy (or true) stress
Consequently, it is necessary to introduce the so-called objective stress rate