In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, a hypoelastic material is an
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
material that has a
constitutive model
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially Kinetics (physics), kinetic quantities as related to Kinematics, kinematic quantities) that is specific to a ma ...
independent of
finite strain
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal stra ...
measures except in the linearized case. Hypoelastic material models are distinct from
hyperelastic material
A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density ...
models (or standard elasticity models) in that, except under special circumstances, they cannot be derived from a
strain energy density function.
Overview
A hypoelastic material can be rigorously defined as one that is modeled using a
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
satisfying the following two criteria:
# 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 In physics, a Cauchy-elastic material is one in which the stress at each point is determined only by the current state of deformation with respect to an arbitrary reference configuration.R. W. Ogden, 1984, ''Non-linear Elastic Deformations'', Dover, ...
, for which the current stress depends only on the current configuration rather than the history of past configurations.
# 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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
tensor.
If only these two original criteria are used to define hypoelasticity, then
hyperelasticity
A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density ...
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
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
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 300px, Predictions from three objective stress rates under shear
In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a ...
so that the
function exists only implicitly.
Hypoelastic material models frequently take the form
where
is an objective rate of the
Kirchhoff stress In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply ''the'' stress tensor or "true stress". However, several alternative measures of stress can be defined:
#The Kirchhoff stress (\bolds ...
(
),