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 at time 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 such that in which is the material rate of the Cauchy stress tensor, and 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. 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 exists. As explained below, specific formulations of hypoelastic models typically employ a so-called objective stress rate so that the function exists only implicitly. Hypoelastic material models frequently take the form where is an objective rate of the Kirchhoff stress, is the deformation rate tensor, and is the so-called elastic tangent stiffness tensor, which varies with stress itself and is regarded as a material property tensor. In hyperelasticity, the tangent stiffness generally must also depend on the deformation gradient in order to properly account for distortion and rotation of anisotropic material fiber directions.
In many practical problems of solid mechanics, it is sufficient to characterize material deformation by the small strain tensor where are the components of the displacements of continuum points, the subscripts refer toCartesian coordinates , and the subscripts preceded by a comma denote partial derivatives. But there are also many problems where the finiteness of strain must be taken into account. These are of two kinds:
large nonlinear elastic deformations possessing a potential energy, , in which the stress tensor components are obtained as the partial derivatives of 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 formulation is necessary and must be used in every load or time step of a finite elementcomputer 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, one may use the deformation rate tensor or increment representing the linearized strain increment from the initial state in the step. Here the superior dot represents the material time derivative, denotes a small increment over the step, = time, and = material point velocity or displacement rate. However, it would not be objective to use the time derivative of the Cauchy stress. This stress, which describes the forces on a small material element imagined to be cut out from the material as currently deformed, is not objective because it varies with rigid body rotations of the material. The material points must be characterized by their initial coordinates because different material particles are contained in the element that is cut out before and after the incremental deformation. Consequently, it is necessary to introduce the so-called objective stress rate, or the corresponding increment. The objectivity is necessary for to be functionally related to the element deformation. It means that that must be invariant with respect to coordinate transformations and must characterize the state of the same material element as it deforms.
