image:Yeoh model comp.png, 300px, Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data fro
PolymerFEM.com]
The Yeoh hyperelastic material model
is a phenomenological model for the deformation of nearly incompressible, nonlinear Elasticity (physics), elastic materials such as
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.
Types of polyisoprene ...
. The model is based on
Ronald Rivlin's observation that the elastic properties of rubber may be described using a
strain energy density function
A strain energy density function or stored energy density function is a scalar (mathematics), scalar-valued function (mathematics), function that relates the strain energy density of a material to the deformation gradient.
:
W = \hat(\boldsy ...
which is a power series in the
strain invariants
Strain may refer to:
Science and technology
* Strain (biology), variants of biological organisms
* Strain (chemistry), a chemical stress of a molecule
* Strain (general relativity), measure of spacetime stretching in linearized gravity
* Strai ...
of the
Cauchy-Green deformation tensors. The Yeoh model for incompressible rubber is a function only of
. For compressible rubbers, a dependence on
is added on. Since a polynomial form of the strain energy density function is used but all the three invariants of the left Cauchy-Green deformation tensor are not, the Yeoh model is also called the reduced
polynomial model.
Yeoh model for incompressible rubbers
Strain energy density function
The original model proposed by Yeoh had a cubic form with only
dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as
:
where
are material constants. The quantity
can be interpreted as the initial
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
.
Today a slightly more generalized version of the Yeoh model is used.
[Selvadurai, A. P. S., 2006, "Deflections of a rubber membrane", ''Journal of the Mechanics and Physics of Solids'', vol. 54, no. 6, pp. 1093-1119.] This model includes
terms and is written as
:
When
the Yeoh model reduces to the
neo-Hookean model for incompressible materials.
For consistency with
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
the Yeoh model has to satisfy the condition
:
where
is the
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
of the material.
Now, at
,
:
Therefore, the consistency condition for the Yeoh model is
:
Stress-deformation relations
The Cauchy stress for the incompressible Yeoh model is given by
:
Uniaxial extension
For uniaxial extension in the
-direction, the
principal stretches are
. From incompressibility
. Hence
.
Therefore,
:
The
left Cauchy-Green deformation tensor can then be expressed as
:
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:
Since
, we have
:
Therefore,
:
The
engineering strain is
. The
engineering stress
In engineering, deformation (the change in size or shape of an object) may be ''elastic'' or ''plastic''.
If the deformation is negligible, the object is said to be ''rigid''.
Main concepts
Occurrence of deformation in engineering application ...
is
:
Equibiaxial extension
For equibiaxial extension in the
and
directions, the
principal stretches are
. From incompressibility
. Hence
.
Therefore,
:
The
left Cauchy-Green deformation tensor can then be expressed as
:
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:
Since
, we have
:
Therefore,
:
The
engineering strain is
. The
engineering stress
In engineering, deformation (the change in size or shape of an object) may be ''elastic'' or ''plastic''.
If the deformation is negligible, the object is said to be ''rigid''.
Main concepts
Occurrence of deformation in engineering application ...
is
:
Planar extension
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the
directions with the
direction constrained, the
principal stretches are
. From incompressibility
. Hence
.
Therefore,
:
The
left Cauchy-Green deformation tensor can then be expressed as
:
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:
Since
, we have
:
Therefore,
:
The
engineering strain is
. The
engineering stress
In engineering, deformation (the change in size or shape of an object) may be ''elastic'' or ''plastic''.
If the deformation is negligible, the object is said to be ''rigid''.
Main concepts
Occurrence of deformation in engineering application ...
is
:
Yeoh model for compressible rubbers
A version of the Yeoh model that includes
dependence is used for compressible rubbers. The strain energy density function for this model is written as
:
where
, and
are material constants. The quantity
is interpreted as half the initial shear modulus, while
is interpreted as half the initial bulk modulus.
When
the compressible Yeoh model reduces to the
neo-Hookean model for incompressible materials.
History
The model is named after Oon Hock Yeoh. Yeoh completed his doctoral studies under
Graham Lake at the
University of London
The University of London (UoL; abbreviated as Lond or more rarely Londin in Post-nominal letters, post-nominals) is a collegiate university, federal Public university, public research university located in London, England, United Kingdom. The ...
. Yeoh held research positions at
Freudenberg-NOK,
MRPRA (England),
Rubber Research Institute of Malaysia
The Rubber Research Institute of Malaysia (RRIM; ) is a research center for problems and matters pertaining to rubber and its industry in Malaysia.
History
On 29 June 1925, the bill to incorporate the Rubber Research Institute of Malaya was pass ...
(Malaysia),
University of Akron
The University of Akron is a public university, public research university in Akron, Ohio, United States. It is part of the University System of Ohio. As a STEM fields, STEM-focused institution, it focuses on industries such as polymers, advance ...
,
GenCorp
Aerojet Rocketdyne is a subsidiary of American Arms industry, defense company L3Harris that manufactures rocket, Hypersonic flight, hypersonic, and electric propulsive systems for space, defense, civil and commercial applications. Aerojet traces ...
Research, and
Lord Corporation
LORD Corporation is a diversified technology and manufacturing company that develops adhesives, coatings, motion management devices, and sensing technologies for industries such as aerospace, automotive, oil and gas, and industrial. With world ...
. Yeoh won the 2004
Melvin Mooney Distinguished Technology Award The Melvin Mooney Distinguished Technology Award is a professional award conferred by the ACS Rubber Division. Established in 1983, the award is named after Melvin Mooney, developer of the Mooney viscometer and of the Mooney-Rivlin hyperelastic ...
from the
ACS Rubber Division
ACS or Acs may refer to:
Aviation
* ACS-3, the military version of Raybird-3, a Ukrainian UAV
* Aerial Common Sensor, a Lockheed Martin reconnaissance aircraft airframe for the US Army and Navy
* Air Cess, a cargo airline based in Sharjah, Uni ...
.
[{{cite news , title=Rubber Division names 3 for awards , url=https://www.rubbernews.com/article/20031027/NEWS/310279997/rubber-division-names-3-for-awards , access-date=16 August 2022 , work=Rubber and Plastics News , publisher=Crain , date=27 October 2003]
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 densit ...
*
Strain energy density function
A strain energy density function or stored energy density function is a scalar (mathematics), scalar-valued function (mathematics), function that relates the strain energy density of a material to the deformation gradient.
:
W = \hat(\boldsy ...
*
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 str ...
*
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 (\bold ...
Elasticity (physics)
Rubber properties
Solid mechanics
Continuum mechanics