A strain energy density function or stored energy density function is a
scalar-valued function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that relates the
strain energy
In physics, the elastic potential energy gained by a wire during elongation with a tensile (stretching) force is called strain energy. For linearly elastic materials, strain energy is:
: U = \frac 1 2 V \sigma \epsilon = \frac 1 2 V E \epsilon ...
density of a material to the
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 ...
.
:
Equivalently,
:
where
is the (two-point) deformation gradient
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
,
is the
right Cauchy–Green deformation tensor,
is the
left Cauchy–Green deformation tensor,
and
is the rotation tensor from the polar decomposition of
.
For an anisotropic material, the strain energy density function
depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representation,
must further depend explicitly on the polar rotation tensor
to provide sufficient information to convect the reference texture vectors or tensors into the spatial configuration.
For an
isotropic material, consideration of the principle of material frame indifference leads to the conclusion that the strain energy density function depends only on the invariants of
(or, equivalently, the invariants of
since both have the same eigenvalues). In other words, the strain energy density function can be expressed uniquely in terms of the
principal stretches or in terms of the
invariants of the
left Cauchy–Green deformation tensor or
right Cauchy–Green deformation tensor and we have:
For isotropic materials,
:
with
:
For linear isotropic materials undergoing small strains, the strain energy density function specializes to
:
A strain energy density function is used to define a
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 f ...
by postulating that the
stress
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in the material can be obtained by taking the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of
with respect to the
strain
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* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
. For an isotropic hyperelastic material, the function relates the energy stored in an
elastic material, and thus the stress–strain relationship, only to the three
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
(elongation) components, thus disregarding the deformation history, heat dissipation,
stress relaxation In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing some ...
etc.
For isothermal elastic processes, the strain energy density function relates to the specific
Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ener ...
function
,
:
For isentropic elastic processes, the strain energy density function relates to the internal energy function
,
:
Examples
Some examples of hyperelastic
constitutive equations
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 app ...
are:
[Muhr, A. H. (2005). Modeling the stress–strain behavior of rubber. Rubber chemistry and technology, 78(3), 391–425]
/ref>
* Hyperelastic material#Saint Venant–Kirchhoff model, Saint Venant–Kirchhoff
* Neo-Hookean
* Generalized Rivlin
* Mooney–Rivlin
* Ogden
* Yeoh
*Arruda–Boyce model
In continuum mechanics, an Arruda–Boyce model Arruda, E. M. and Boyce, M. C., 1993, A three-dimensional model for the large stretch behavior of rubber elastic materials,, J. Mech. Phys. Solids, 41(2), pp. 389–412. is a hyperelastic constitu ...
*Gent
Gent is a shortened form of the word gentleman. It may also refer to:
* Ghent (Dutch language, Dutch: Gent), a Belgian city
** K.A.A. Gent, a football club from Ghent
** K.R.C. Gent, a football club from Ghent
** Gent RFC, a rugby club in Ghen ...
See also
{{wikiversity, Continuum mechanics/Thermoelasticity
*Finite strain theory
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 ...
* Helmholtz and Gibbs free energy in thermoelasticity
*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 f ...
*Ogden–Roxburgh model
The Ogden–Roxburgh model is an approach which extends hyperelastic material models to allow for the Mullins effect. It is used in several commercial finite element
The finite element method (FEM) is a popular method for numerically solving ...
References
Continuum mechanics
Rubber properties
Solid mechanics
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