A neo-Hookean solid
is 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 ...
model, similar to
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
s. The model was proposed by
Ronald Rivlin
Ronald Samuel Rivlin (6 May 1915 in London – 4 October 2005) was a British-American physicist, mathematician, rheologist and a noted expert on rubber.''New York Times'' November 25, 2005 "Ronald Rivlin, 90, Expert on Properties of Rubber, Dies ...
in 1948. In contrast to
linear elastic materials, the
stress-strain curve of a neo-Hookean material is not
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the
dissipative
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
release of energy as heat while straining the material and perfect elasticity is assumed at all stages of deformation.
The neo-Hookean model is based on the statistical thermodynamics of cross-linked polymer chains and is usable for
plastic
Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
s and
rubber
Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, an ...
-like substances. Cross-linked polymers will act in a neo-Hookean manner because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain point the polymer chains will be stretched to the maximum point that the covalent cross links will allow, and this will cause a dramatic increase in the elastic modulus of the material. The neo-Hookean material model does not predict that increase in modulus at large strains and is typically accurate only for strains less than 20%.
[Gent, A. N., ed., 2001, Engineering with rubber, Carl Hanser Verlag, Munich.] The model is also inadequate for biaxial states of stress and has been superseded by the
Mooney-Rivlin model.
The
strain energy density function for an
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
neo-Hookean material in a three-dimensional description is
:
where
is a material constant, and
is the
first invariant (
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
), of the
right Cauchy-Green deformation tensor, i.e.,
:
where
are the
principal stretches.
[
For a ]compressible
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
neo-Hookean material the strain energy density function is given by
:
where is a material constant and is 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 ...
. It can be shown that in 2D, the strain energy density function is
:
Several alternative formulations exist for compressible neo-Hookean materials, for example
:
where is the first invariant of the isochoric
Isochoric may refer to:
*cell-transitive, in geometry
*isochoric process
In thermodynamics, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which ...
part of the right Cauchy–Green deformation tensor.
For consistency with linear elasticity,
:
where is the first Lamé parameter and is the shear modulus or the second Lamé parameter. Alternative definitions of and are sometimes used, notably in commercial finite element analysis software such as Abaqus
Abaqus FEA (formerly ABAQUS) is a software suite for finite element analysis and computer-aided engineering, originally released in 1978. The name and logo of this software are based on the abacus calculation tool.
The Abaqus product suite consis ...
. [ Abaqus (Version 6.8) Theory Manua]
Cauchy stress in terms of deformation tensors
Compressible neo-Hookean material
For a compressible Ogden neo-Hookean material the Cauchy stress is given by
:
where is the first Piola-Kirchhoff stress. By simplifying the right hand side we arrive at
:
which for infinitesimal strains is equal to
:
Comparison with Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
shows that and .
For a compressible Rivlin neo-Hookean material the Cauchy stress is given by
:
where is the left Cauchy-Green deformation tensor, and
:
For infinitesimal strains ()
:
and the Cauchy stress can be expressed as
:
Comparison with Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
shows that and .
:
Incompressible neo-Hookean material
For an incompressible neo-Hookean material with
:
where is an undetermined pressure.
Cauchy stress in terms of principal stretches
Compressible neo-Hookean material
For a compressible neo-Hookean 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 ...
, the principal components of the Cauchy stress are given by
:
Therefore, the differences between the principal stresses are
:
:
Incompressible neo-Hookean material
In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by
:
For an incompressible neo-Hookean material,
:
Therefore,
:
which gives
:
Uniaxial extension
Compressible neo-Hookean material
For a compressible material undergoing uniaxial extension, the principal stretches are
:
Hence, the true (Cauchy) stresses for a compressible neo-Hookean material are given by
:
The stress differences are given by
:
If the material is unconstrained we have . Then
:
Equating the two expressions for gives a relation for as a function of , i.e.,
:
or
:
The above equation can be solved numerically using a Newton–Raphson
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-va ...
iterative root-finding procedure.
Incompressible neo-Hookean material
Under uniaxial extension, and . Therefore,
:
Assuming no traction on the sides, , so we can write
:
where is the engineering 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 ...
. This equation is often written in alternative notation as
:
The equation above is for the true stress (ratio of the elongation force to deformed cross-section). For the engineering stress
In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strai ...
the equation is:
:
For small deformations we will have:
:
Thus, the equivalent Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...
of a neo-Hookean solid in uniaxial extension is , which is in concordance with linear elasticity ( with for incompressibility).
Equibiaxial extension
Compressible neo-Hookean material
In the case of equibiaxial extension
:
Therefore,
:
The stress differences are
:
If the material is in a state of plane stress then and we have
:
We also have a relation between and :
:
or,
:
This equation can be solved for using Newton's method.
Incompressible neo-Hookean material
For an incompressible material and the differences between the principal Cauchy stresses take the form
:
Under plane stress conditions we have
:
Pure dilation
For the case of pure dilation
:
Therefore, the principal Cauchy stresses for a compressible neo-Hookean material are given by
:
If the material is incompressible then and the principal stresses can be arbitrary.
The figures below show that extremely high stresses are needed to achieve large triaxial extensions or compressions. Equivalently, relatively small triaxial stretch states can cause very high stresses to develop in a rubber-like material. The magnitude of the stress is quite sensitive to the bulk modulus but not to the shear modulus.
Simple shear
For the case of simple shear the deformation gradient in terms of components with respect to a reference basis is of the form
:
where is the shear deformation. Therefore the left Cauchy-Green deformation tensor is
:
Compressible neo-Hookean material
In this case . Hence, . Now,
:
Hence the Cauchy stress is given by
:
Incompressible neo-Hookean material
Using the relation for the Cauchy stress for an incompressible neo-Hookean material we get
:
Thus neo-Hookean solid shows linear dependence of shear stresses upon shear deformation and quadratic dependence of the normal stress difference on the shear deformation. The expressions for the Cauchy stress for a compressible and an incompressible neo-Hookean material in simple shear represent the same quantity and provide a means of determining the unknown pressure .
References
See also
* 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 ...
* Strain energy density function
* Mooney-Rivlin solid
* 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 ...
* Stress measures 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 (\boldsy ...
Continuum mechanics
Elasticity (physics)
Non-Newtonian fluids
Rubber properties
Solid mechanics