The hyperelastic model for rubberlike materials provides a general capability for
modeling the behavior of nearly incompressible elastomers under large elastic
deformations.
The hyperelastic material model:
is isotropic and nonlinear;
is valid for materials that exhibit instantaneous elastic response up
to large strains (such as rubber, solid propellant, or other elastomeric
materials); and
requires that geometric nonlinearity be accounted for during the
analysis step (General and Perturbation Procedures),
since it is intended for finite-strain applications.
Most elastomers (solid, rubberlike materials) have very little
compressibility compared to their shear flexibility. This behavior does not
warrant special attention for plane stress, shell, membrane, beam, truss, or
rebar elements, but the numerical solution can be quite sensitive to the degree
of compressibility for three-dimensional solid, plane strain, and axisymmetric
analysis elements. In cases where the material is highly confined (such as an
O-ring used as a seal), the compressibility must be modeled correctly to obtain
accurate results. In applications where the material is not highly confined,
the degree of compressibility is typically not crucial; for example, it would
be quite satisfactory in
Abaqus/Standard
to assume that the material is fully incompressible: the volume of the material
cannot change except for thermal expansion.
Another class of rubberlike materials is elastomeric foam, which is elastic
but very compressible. Elastomeric foams are discussed in
Hyperelastic Behavior in Elastomeric Foams.
We can assess the relative compressibility of a material by the ratio of its
initial bulk modulus, ,
to its initial shear modulus, .
This ratio can also be expressed in terms of Poisson's ratio,
,
since
The table below provides some representative values.
Poisson's ratio
10
0.452
20
0.475
50
0.490
100
0.495
1000
0.4995
10,000
0.49995
Compressibility in Abaqus/Standard
In
Abaqus/Standard
it is recommended that you use solid continuum hybrid elements for almost
incompressible hyperelastic materials with initial Poisson's ratio greater than
0.495 (i.e., the ratio of
greater than 100) to avoid potential convergence problems. Otherwise, the
analysis preprocessor will issue an error. Except for fully incompressible
hyperelastic materials, you can use the “nonhybrid incompressible” diagnostics
control to downgrade this error to a warning message.
In plane stress, shell, and membrane elements the material is free to deform
in the thickness direction. Similarly, in one-dimensional elements (such as
beams, trusses, and rebars) the material is free to deform in the lateral
directions. In these cases special treatment of the volumetric behavior is not
necessary; the use of regular stress/displacement elements is satisfactory.
Input File Usage
Use the following option to downgrade an error message to a
warning message:
Except for plane stress and uniaxial cases, it is not possible to assume
that the material is fully incompressible in
Abaqus/Explicit
because the program has no mechanism for imposing such a constraint at each
material calculation point. Instead, we must provide some compressibility. The
difficulty is that, in many cases, the actual material behavior provides too
little compressibility for the algorithms to work efficiently. Thus, except for
plane stress and uniaxial cases, you must provide enough compressibility for
the code to work, knowing that this makes the bulk behavior of the model softer
than that of the actual material. Some judgment is, therefore, required to
decide whether or not the solution is sufficiently accurate, or whether the
problem can be modeled at all with
Abaqus/Explicit
because of this numerical limitation.
If no value is given for the material compressibility in the hyperelastic
model, by default
Abaqus/Explicit
assumes
20, corresponding to Poisson's ratio of 0.475. Since typical unfilled
elastomers have
ratios in the range of 1,000 to 10,000 (
0.4995 to
0.49995) and filled elastomers have
ratios in the range of 50 to 200 (
0.490 to
0.497), this default provides much more compressibility than is available in
most elastomers. However, if the elastomer is relatively unconfined, this
softer modeling of the material's bulk behavior usually provides quite accurate
results. Unfortunately, in cases where the material is highly confined—such as
when it is in contact with stiff, metal parts and has a very small amount of
free surface, especially when the loading is highly compressive—it may not be
feasible to obtain accurate results with
Abaqus/Explicit.
If you are defining the compressibility rather than accepting the default
value, an upper limit of 100 is suggested for the ratio of
.
Larger ratios introduce high frequency noise into the dynamic solution and
require the use of excessively small time increments.
Isotropy Assumption
In
Abaqus
all hyperelastic models are based on the assumption of isotropic behavior
throughout the deformation history. Hence, the strain energy potential can be
formulated as a function of the strain invariants.
Strain Energy Potentials
Hyperelastic materials are described in terms of a “strain energy potential,” , which defines the strain energy stored in the material per unit of
reference volume (volume in the initial configuration) as a function of the strain at that
point in the material. There are several forms of strain energy potentials available in Abaqus to model approximately incompressible isotropic elastomers: the Arruda-Boyce form, the
Marlow form, the Mooney-Rivlin form, the neo-Hookean form, the Ogden form, the polynomial
form, the reduced polynomial form, the Yeoh form, the Valanis-Landel form, and the Van der
Waals form. As will be pointed out below, the reduced polynomial and Mooney-Rivlin models
can be viewed as particular cases of the polynomial model; the Yeoh and neo-Hookean
potentials, in turn, can be viewed as special cases of the reduced polynomial model. Thus,
we will occasionally refer collectively to these models as “polynomial models.”
Generally, when data from multiple experimental tests are available (typically, this requires at
least uniaxial and equibiaxial test data), the Ogden and Van der Waals forms are more
accurate in fitting experimental results. If limited test data are available for
calibration, the Arruda-Boyce, Van der Waals, Yeoh, or reduced polynomial forms provide
reasonable behavior. When only uniaxial test data is available, the Marlow or the
Valanis-Landel form is recommended, and the Marlow form is recommended if only equibiaxial
or planar test data is available. In this case, a strain energy potential is constructed
that reproduces the test data exactly and that has reasonable behavior in other deformation
modes.
Evaluating Hyperelastic Materials
You can use single-element test cases to evaluate the
strain energy potential.
Arruda-Boyce Form
The form of the Arruda-Boyce strain energy potential is
where U is the strain energy per unit of reference
volume; ,
,
and D are temperature-dependent material parameters;
is the first deviatoric strain invariant defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus,
,
is related to
with the expression
A typical value of
is 7, for which .
Both the initial shear modulus, ,
and the parameter
are printed in the data (.dat) file if you request a
printout of the model data from the analysis input file processor. The initial
bulk modulus is related to D with the expression
Marlow Form
The form of the Marlow strain energy potential is
where U is the strain energy per unit of reference
volume, with
as its deviatoric part and
as its volumetric part;
is the first deviatoric strain invariant defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The deviatoric part of the potential is defined by
providing either uniaxial, equibiaxial, or planar test data; while the
volumetric part is defined by providing the volumetric test data, defining the
Poisson's ratio, or specifying the lateral strains together with the uniaxial,
equibiaxial, or planar test data.
Mooney-Rivlin Form
The form of the Mooney-Rivlin strain energy potential is
where U is the strain energy per unit of reference
volume; ,
,
and
are temperature-dependent material parameters;
and
are the first and second deviatoric strain invariants defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus and bulk modulus are
given by
Neo-Hookean Form
The form of the neo-Hookean strain energy potential is
where U is the strain energy per unit of reference
volume;
and
are temperature-dependent material parameters;
is the first deviatoric strain invariant defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus and bulk modulus are
given by
Ogden Form
The form of the Ogden strain energy potential is
where
are the deviatoric principal stretches ;
are the principal stretches; N is a material parameter;
and ,
,
and
are temperature-dependent material parameters. The initial shear modulus and
bulk modulus for the Ogden form are given by
The particular material models described above—the Mooney-Rivlin and
neo-Hookean forms—can also be obtained from the general Ogden strain energy
potential for special choices of
and .
Polynomial Form
The form of the polynomial strain energy potential is
where U is the strain energy per unit of reference
volume; N is a material parameter;
and
are temperature-dependent material parameters;
and
are the first and second deviatoric strain invariants defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus and bulk modulus are
given by
For cases where the nominal strains are small or only moderately large (<
100%), the first terms in the polynomial series usually provide a sufficiently
accurate model. Some particular material models—the Mooney-Rivlin, neo-Hookean,
and Yeoh forms—are obtained for special choices of .
Reduced Polynomial Form
The form of the reduced polynomial strain energy potential is
where U is the strain energy per unit of reference
volume; N is a material parameter;
and
are temperature-dependent material parameters;
is the first deviatoric strain invariant defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus and bulk modulus are
given by
Valanis-Landel Form
The form of the Valanis-Landel strain energy potential is
where U is the strain energy per unit of reference volume, with as its deviatoric part and as its volumetric part. For the Valanis-Landel model it is further
assumed that the deviatoric part of the strain energy potential, , is expressed as three separable but identical functions of principal
deviatoric stretches
where the deviatoric stretches , J is the total volume ratio, is the elastic volume ratio as defined below in Thermal Expansion, and are the principal stretches. The deviatoric part of the potential is
defined by providing uniaxial test data; while the volumetric part is defined either by
providing volumetric test data, by providing the uniaxial test data with lateral strains
specified, or by defining the Poisson's ratio.
Van Der Waals Form
The form of the Van der Waals strain energy potential is
where
Here, U is the strain energy per unit of reference
volume;
is the initial shear modulus;
is the locking stretch; a is the global interaction
parameter;
is an invariant mixture parameter; and D governs the
compressibility. These parameters can be temperature-dependent.
and
are the first and second deviatoric strain invariants defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus and bulk modulus are
given by
Yeoh Form
The form of the Yeoh strain energy potential is
where U is the strain energy per unit of reference
volume;
and
are temperature-dependent material parameters;
is the first deviatoric strain invariant defined as
where the deviatoric stretches ,
J is the total volume ratio,
is the elastic volume ratio as defined below in
Thermal Expansion,
and
are the principal stretches. The initial shear modulus and bulk modulus are
given by
Thermal Expansion
Only isotropic thermal expansion is permitted with the hyperelastic material
model.
The elastic volume ratio, ,
relates the total volume ratio, J, and the thermal volume
ratio, :
is given by
where
is the linear thermal expansion strain that is obtained from the temperature
and the isotropic thermal expansion coefficient (Thermal Expansion).
Defining the Hyperelastic Material Response
The mechanical response of a material is defined by choosing a strain energy
potential to fit the particular material. The strain energy potential forms in
Abaqus
are written as separable functions of a deviatoric component and a volumetric
component; i.e., .
Alternatively, in
Abaqus/Standard
you can define the strain energy potential with user subroutine
UHYPER, in which case the strain energy potential need not be
separable.
Generally for the hyperelastic material models available in Abaqus, you can either directly specify material coefficients or provide experimental test data
and have Abaqus automatically determine appropriate values of the coefficients. An exception is the
Marlow form: in this case, the deviatoric part of the strain energy potential must be
defined with test data. The different methods for defining the strain energy potential are
described in detail below.
The properties of rubberlike materials can vary significantly from one batch
to another; therefore, if data are used from several experiments, all of the
experiments should be performed on specimens taken from the same batch of
material, regardless of whether you or
Abaqus
compute the coefficients.
To define the instantaneous response, the experiments outlined in
Experimental Tests have
to be performed within time spans much shorter than the characteristic
relaxation times of these materials.
If the long-term elastic response is used, data from experiments have to be collected after
time spans much longer than the characteristic relaxation times of these materials.
Long-term elastic response is the default elastic material behavior.
Compressibility can be defined by specifying nonzero values for (except for the Marlow and Valanis-Landel models), by setting the
Poisson's ratio to a value less than 0.5, or by providing test data that characterize the
compressibility. The test data method is described later in this section. If you specify
the Poisson's ratio for hyperelasticity other than the Marlow or the Valanis-Landel
models, Abaqus computes the initial bulk modulus from the initial shear modulus
For the Marlow model and the Valanis-Landel model the specified Poisson's ratio represents a
constant value, which determines the volumetric response throughout the deformation
process. If is equal to zero, all of the must be equal to zero. In such a case the material is assumed to be
fully incompressible in Abaqus/Standard, while Abaqus/Explicit assumes compressible behavior with (Poisson's ratio of 0.475).
The parameters of the hyperelastic strain energy potentials can be given directly as functions of
temperature for all forms of the strain energy potential except the Marlow and
Valanis-Landel forms.
Using Test Data to Calibrate Material Coefficients
The material coefficients of the hyperelastic models can be calibrated by Abaqus from experimental stress-strain data. In the case of the Marlow and Valanis-Landel
models, the test data directly characterize the strain energy potential (there are no
material coefficients for these models); these models are described in detail below. The
value of N and experimental stress-strain data can be specified for
up to four simple tests: uniaxial, equibiaxial, planar, and, if the material is
compressible, a volumetric compression test. Abaqus will then compute the material parameters. The material constants are determined
through a least-squares-fit procedure, which minimizes the relative error in stress. For
the n nominal-stress–nominal-strain data pairs, the relative error
measure E is minimized, where
is a stress value from the test data, and
comes from one of the nominal stress expressions derived below (see
“Experimental tests”).
Abaqus
minimizes the relative error rather than an absolute error measure since this
provides a better fit at lower strains. This method is available for all strain
energy potentials and any order of N except for the
polynomial form, where a maximum of
is allowed. The polynomial models are linear in terms of the constants
;
therefore, a linear least-squares procedure can be used. The Arruda-Boyce,
Ogden, and Van der Waals potentials are nonlinear in some of their
coefficients, thus necessitating the use of a nonlinear least-squares
procedure.
Fitting of hyperelastic and hyperfoam constants
contains a detailed derivation of the related equations.
It is generally best to obtain data from several experiments involving
different kinds of deformation over the range of strains of interest in the
actual application and to use all of these data to determine the parameters.
This is particularly true for the phenomenological models; i.e., the Ogden and
the polynomial models. It has been observed that to achieve good accuracy and
stability, it is necessary to fit these models using test data from more than
one deformation state. In some cases, especially at large strains, removing the
dependence on the second invariant may alleviate this limitation. The
Arruda-Boyce, neo-Hookean, and Van der Waals models with
= 0 offer a physical interpretation and provide a better prediction of general
deformation modes when the parameters are based on only one test. An extensive
discussion of this topic can be found in
Hyperelastic material behavior.
This method does not allow the hyperelastic properties to be temperature
dependent. However, if temperature-dependent test data are available, several
curve fits can be conducted by performing a data check analysis on a simple
input file. The temperature-dependent coefficients determined by
Abaqus
can then be entered directly in the actual analysis run.
Optionally, the parameter
in the Van der Waals model can be set to a fixed value while the other
parameters are found using a least-squares curve fit.
As many data points as required can be entered from each test. It is
recommended that data from all four tests (on samples taken from the same piece
of material) be included and that the data points cover the range of nominal
strains expected to arise in the actual loading. For the (general) polynomial
and Ogden models and for the coefficient
in the Van der Waals model, the planar test data must be accompanied by the
uniaxial test data, the biaxial test data, or both of these types of test data;
otherwise, the solution to the least-squares fit will not be unique.
The strain data should be given as nominal strain values (change in length
per unit of original length). For the uniaxial, equibiaxial, and planar tests
stress data are given as nominal stress values (force per unit of original
cross-sectional area). These tests allow for entering both compression and
tension data. Compressive stresses and strains are entered as negative values.
If compressibility is to be specified, the
or D can be computed from volumetric compression test
data. Alternatively, compressibility can be defined by specifying a Poisson's
ratio, in which case
Abaqus
computes the bulk modulus from the initial shear modulus. If no such data are
given,
Abaqus/Standard
assumes that D or all of the
are zero, whereas
Abaqus/Explicit
assumes compressibility corresponding to a Poisson's ratio of 0.475 (see
“Compressibility in
Abaqus/Explicit”
above). For these compression tests the stress data are given as pressure
values.
Input File Usage
Use one of the following options to select the strain energy
potential:
The Marlow model assumes that the strain energy potential is independent of the second
deviatoric invariant . This model is defined by providing test data that define the deviatoric
behavior, and, optionally, the volumetric behavior if compressibility must be taken into
account. Abaqus will construct a strain energy potential that reproduces the test data exactly, as
shown in Figure 1.
Figure 1. The results of the Marlow model with test data.
The interpolation and extrapolation of stress-strain data with the Marlow model is
approximately linear for small and large strains. For intermediate strains in the range
0.1 to 1.0 a noticeable degree of nonlinearity may be observed in the
interpolation/extrapolation with the Marlow model; for example, some nonlinearity is
apparent between the 4th and 5th data points in Figure 1. To minimize undesirable nonlinearity, make sure that enough data points are specified
in the intermediate strain range.
The deviatoric behavior is defined by specifying uniaxial, biaxial, or planar test data.
Generally, you can specify either the data from tension tests or the data from compression
tests because the tests are equivalent (see Equivalent Experimental Tests). However, for
beams, trusses, and rebars, the data from tension and compression tests can be specified
together. Volumetric behavior is defined by using one of the following three methods:
Specify nominal lateral strains, in addition to nominal stresses and nominal strains,
as part of the uniaxial, biaxial, or planar test data.
Specify Poisson's ratio for the hyperelastic material.
Specify volumetric test data directly. Both hydrostatic tension and hydrostatic
compression data can be specified. If only hydrostatic compression data are available,
as is usually the case, Abaqus will assume that the hydrostatic pressure is an antisymmetric function of the
nominal volumetric strain, .
If you do not define volumetric behavior, Abaqus/Standard assumes fully incompressible behavior, while Abaqus/Explicit assumes compressibility corresponding to a Poisson's ratio of 0.475.
Material test data in which the stress does not vary smoothly with increasing strain may
lead to convergence difficulty during the simulation. It is highly recommended that smooth
test data be used to define the Marlow form. Abaqus provides a smoothing algorithm, which is described in detail later in this section.
The test data for the Marlow model can also be given as a function of temperature and
field variables. You must specify the number of user-defined field variable dependencies
required.
Uniaxial, biaxial, and planar test data must be given in ascending order of the nominal
strains; volumetric test data must be given in descending order of the volume ratio.
Input File Usage
To define the Marlow test data as a function of temperature and/or field variables,
use the following option:
Specifying the Valanis-Landel Model in Abaqus/Standard
In general, the deviatoric part of the strain energy potential of the Valanis-Landel
model depends on both the first, , and the second, , deviatoric invariants. You define this model by providing test data
and, optionally, the Poisson's ratio. Abaqus constructs a strain energy potential that reproduces the test data exactly.
You define the deviatoric behavior by specifying uniaxial test data. You must specify
data from both tension tests and compression tests together for this model. Volumetric
behavior is defined by using one of the following methods:
Specify nominal lateral strains, in addition to nominal stresses and nominal strains,
as part of the uniaxial test data.
Specify Poisson's ratio for the hyperelastic material.
Specify volumetric test data directly. You can specify both hydrostatic tension and
hydrostatic compression data. If only hydrostatic compression data are available, as
is usually the case, Abaqus assumes that the hydrostatic pressure is an antisymmetric function of the nominal
volumetric strain, .
If a Poisson's ratio of 0.5 is specified, the material behavior is assumed to be fully
incompressible.
Material test data in which the stress does not vary smoothly with increasing strain can
lead to convergence difficulty during the simulation. It is highly recommended that you
use smooth test data to define the Valanis-Landel form. Abaqus provides a smoothing algorithm, which is described in detail later in this section.
The test data for the Valanis-Landel model can also be given as a function of temperature
and field variables. You must specify the number of user-defined field variable
dependencies required.
Uniaxial test data must be given in ascending order of the nominal strains.
Input File Usage
To define the Valanis-Landel test data as a function of temperature and/or field
variables, use both of the following options:
An alternative method provided in Abaqus/Standard for defining the hyperelastic material parameters allows the strain energy potential to
be defined in user subroutine UHYPER or in user subroutine UHYPER_STRETCH. You can specify
either compressible or incompressible behavior. Optionally, you can specify the number of
property values needed as data in the user subroutine. If needed, you can specify the
number of solution-dependent variables (see About User Subroutines and Utilities).
User subroutine UHYPER requires that the values of
the derivatives of the strain energy density function of the hyperelastic material are
defined with respect to the strain invariants.
User subroutine UHYPER_STRETCH assumes that the
strain energy potential uses the Valanis-Landel form. It requires that the values of the
derivatives of the strain energy density function of the hyperelastic material are defined
with respect to the principal deviatoric stretches, , and elastic volume ratio, .
Input File Usage
Use one of the following options to specify the strain energy potential in user subroutine
UHYPER:
For a homogeneous material, homogeneous deformation modes suffice to
characterize the material constants.
Abaqus
accepts test data from the following deformation modes:
Uniaxial tension and compression
Equibiaxial tension and compression
Planar tension and compression (also known as pure shear)
Volumetric tension and compression
These modes are illustrated schematically in
Figure 2
and are described below. The most commonly performed experiments are uniaxial
tension, uniaxial compression, and planar tension.
Figure 2. Schematic illustrations of deformation modes.
Combine data from these three test types to get a good characterization of
the hyperelastic material behavior.
For the incompressible version of the material model, the stress-strain
relationships for the different tests are developed using derivatives of the
strain energy function with respect to the strain invariants. We define these
relations in terms of the nominal stress (the force divided by the original,
undeformed area) and the nominal, or engineering, strain defined below.
The deformation gradient, expressed in the principal directions of stretch,
is
where ,
,
and
are the principal stretches: the ratios of current length to length in the
original configuration in the principal directions of a material fiber. The
principal stretches, ,
are related to the principal nominal strains, ,
by
Because we assume incompressibility and isothermal response,
and, hence,
= 1. The deviatoric strain invariants in terms of the principal stretches are
then
and
Uniaxial Tests
The uniaxial deformation mode is characterized in terms of the principal
stretches, ,
as
where
is the stretch in the loading direction. The nominal strain is defined by
To derive the uniaxial nominal stress ,
we invoke the principle of virtual work:
so that
The uniaxial tension test is the most common of all the tests and is usually
performed by pulling a “dog-bone” specimen. The uniaxial compression test is
performed by loading a compression button between lubricated surfaces. The
loading surfaces are lubricated to minimize any barreling effect in the button
that would cause deviations from a homogeneous uniaxial compression
stress-strain state.
The equibiaxial deformation mode is characterized in terms of the principal
stretches, ,
as
where
is the stretch in the two perpendicular loading directions. The nominal strain
is defined by
To develop the expression for the equibiaxial nominal stress,
,
we again use the principle of virtual work (assuming that the stress
perpendicular to the loading direction is zero),
so that
In practice, the equibiaxial compression test is rarely performed because of
experimental setup difficulties. In addition, this deformation mode is
equivalent to a uniaxial tension test, which is straightforward to conduct.
A more common test is the equibiaxial tension test, in which a stress state
with two equal tensile stresses and zero shear stress is created. This state is
usually achieved by stretching a square sheet in a biaxial testing machine. It
can also be obtained by inflating a circular membrane into a spheroidal shape
(like blowing up a balloon). The stress field in the middle of the membrane
then closely approximates equibiaxial tension, provided that the thickness of
the membrane is very much smaller than the radius of curvature at this point.
However, the strain distribution will not be quite uniform, and local strain
measurements will be required. Once the strain and radius of curvature are
known, the nominal stress can be derived from the inflation pressure.
The planar deformation mode is characterized in terms of the principal
stretches, ,
as
where
is the stretch in the loading direction. Then, the nominal strain in the
loading direction is
This test is also called a “pure shear” test since, in terms of logarithmic
strains,
which corresponds to a state of pure shear at an angle of 45° to the loading
direction.
The principle of virtual work gives
where
is the nominal planar stress, so that
For the (general) polynomial and Ogden models and for the coefficient
in the Van der Waals model this equation alone will not determine the constants
uniquely. The planar test data must be augmented by uniaxial test data and/or
biaxial test data to determine the material parameters.
Planar tests are usually done with a thin, short, and wide rectangular strip
of material fixed on its wide edges to rigid loading clamps that are moved
apart. If the separation direction is the 1-direction and
the thickness direction is the 3-direction, the
comparatively long size of the specimen in the 2-direction
and the rigid clamps allow us to use the approximation
;
that is, there is no deformation in the wide direction of the specimen. This
deformation mode could also be called planar compression if the
3-direction is considered to be the primary direction. All
forms of incompressible plane strain behavior are characterized by this
deformation mode. Consequently, if plane strain analysis is performed, planar
test data represent the relevant form of straining of the material.
The following discussion describes procedures for obtaining
values (or D, for the Arruda-Boyce and Van der Waals
models) corresponding to the actual material behavior. With these values you
can compare the material's initial bulk modulus, ,
to its initial shear modulus (
for the polynomial model,
for Ogden's model) and then judge whether
values that will provide results are sufficiently realistic. For
Abaqus/Explicit
caution should be used;
should be less than 100. Otherwise, noisy solutions will be obtained and time
increments will be excessively small (see “Compressibility in
Abaqus/Explicit”
above). The
and D can be calculated from data obtained in pure
volumetric compression of a specimen (volumetric tension tests are much more
difficult to perform). In a pure volumetric test ;
therefore,
and
(the volume ratio). Using the polynomial form of the strain energy potential,
the total pressure stress on the specimen is obtained as
This equation can be used to determine the .
If we are using a second-order polynomial series for U, we
have ,
and so two
are needed. Therefore, a minimum of two points on the pressure-volume ratio
curve are required to give two equations for the .
For the Ogden and reduced polynomial potentials
can be determined for up to .
A linear least-squares fit is performed when more than N
data points are provided.
An approximate way of conducting a volumetric test consists of using a
cylindrical rubber specimen that fits snugly inside a rigid container and whose
top surface is compressed by a rigid piston. Although both volumetric and
deviatoric deformation are present, the deviatoric stresses will be several
orders of magnitude smaller than the hydrostatic stresses (because the bulk
modulus is much higher than the shear modulus) and can be neglected. The
compressive stress imposed by the rigid piston is effectively the pressure, and
the volumetric strain in the rubber cylinder is computed from the piston
displacement.
Nonzero values of
affect the uniaxial, equibiaxial, and planar stress results. However, since the
material is assumed to be only slightly compressible, the techniques described
for obtaining the deviatoric coefficients should give sufficiently accurate
values even though they assume that the material is fully incompressible.
The superposition of a tensile or compressive hydrostatic stress on a
loaded, fully incompressible elastic body results in different stresses but
does not change the deformation. Thus,
Figure 3
shows that some apparently different loading conditions are actually equivalent
in their deformations and, therefore, are equivalent tests:
Uniaxial tension
Equibiaxial compression
Uniaxial compression
Equibiaxial tension
Planar tension
Planar compression
Figure 3. Equivalent deformation modes through superposition of hydrostatic
stress.
On the other hand, the tensile and compressive cases of the uniaxial and
equibiaxial modes are independent from each other: uniaxial tension and
uniaxial compression provide independent data.
Smoothing the Test Data
Experimental test data often contain noise in the sense that the test
variable is both slowly varying and also corrupted by random noise. This noise
can affect the quality of the strain energy potential that
Abaqus
derives. This noise is particularly a problem with the Marlow form, where a
strain energy potential that exactly describes the test data that are used to
calibrate the model is computed. It is less of a concern with the other forms,
since smooth functions are fitted through the test data.
Abaqus
provides a smoothing technique to remove the noise from the test data based on
the Savitzky-Golay method. The idea is to replace each data point by a local
average of its surrounding data points, so that the level of noise can be
reduced without biasing the dominant trend of the test data. In the
implementation a cubic polynomial is fitted through each data point
i and n data points to the immediate
left and right of that point. A least-squares method is used to fit the
polynomial through these
points. The value of data point i is then replaced by the
value of the polynomial at the same position. Each polynomial is used to adjust
one data point except near the ends of the curve, where a polynomial is used to
adjust multiple points, because the first and last few points cannot be the
center of the fitting set of data points. This process is applied repeatedly to
all data points until two consecutive passes through the data produce nearly
the same results.
By default, the test data are not smoothed. If smoothing is specified, the
default value is n=3. Alternatively, you can specify the
number of data points to the left and right of a data point in the moving
window within which a least-squares polynomial is fit.
Input File Usage
For the Marlow form, use one of the first three options and, optionally, the fourth option;
for the Valanis-Landel model, use the first option and, optionally, the fourth option;
and for the other potential forms, use one and up to four of the following
options:
Model Prediction of Material Behavior Versus Experimental Data
Once the strain energy potential is determined, the behavior of the
hyperelastic model in
Abaqus
is established. However, the quality of this behavior must be assessed: the
prediction of material behavior under different deformation modes must be
compared against the experimental data. You must judge whether the strain
energy potentials determined by
Abaqus
are acceptable, based on the correlation between the
Abaqus
predictions and the experimental data.
Single-element test cases can be used to derive the nominal
stress–nominal strain response of the material model.
See
Fitting of rubber test data,
which illustrates the entire process of fitting hyperelastic constants to a set
of test data.
Hyperelastic Material Stability
An important consideration in judging the quality of the fit to experimental
data is the concept of material or Drucker stability.
Abaqus
checks the Drucker stability of the material for the first three deformation
modes described above.
The Drucker stability condition for an incompressible material requires that
the change in the stress, ,
following from any infinitesimal change in the logarithmic strain,
,
satisfies the inequality
Using ,
where is the tangent
material stiffness, the inequality becomes
thus requiring the tangential material stiffness to be positive-definite.
For an isotropic elastic formulation the inequality can be represented in
terms of the principal stresses and strains,
As before, since the material is assumed to be incompressible, we can choose
any value for the hydrostatic pressure without affecting the strains. A
convenient choice for the stability calculation is ,
which allows us to ignore the third term in the above equation.
The relation between the changes in stress and in strain can then be
obtained in the form of the matrix
where .
For material stability must be
positive-definite; thus, it is necessary that
This stability check is performed for the polynomial models, the Ogden potential, the Van der
Waals form, the Marlow form, and the Valanis-Landel form. The Arruda-Boyce form is always
stable for positive values of (, ); hence, it suffices to check the material coefficients to ensure
stability.
You should be careful when defining the
or
for the polynomial models or the Ogden form: especially when
,
the behavior at higher strains is strongly sensitive to the values of the
or ,
and unstable material behavior may result if these values are not defined
correctly. When some of the coefficients are strongly negative, instability at
higher strain levels is likely to occur.
Abaqus performs a check on the stability of the material for six different forms of
loading—uniaxial tension and compression, equibiaxial tension and compression, and planar
tension and compression—for (nominal strain range of ) at intervals . If an instability is found, Abaqus issues a warning message and prints the lowest absolute value of for which the instability is observed. Ideally, no instability occurs.
If instabilities are observed at strain levels that are likely to occur in the analysis,
it is strongly recommended that you either change the material model or carefully examine
and revise the material input data. If user subroutine UHYPER or user subroutine
UHYPER_STRETCH is used to define the
hyperelastic material, you are responsible for ensuring stability.
Improving the Accuracy and Stability of the Test Data Fit
Unfortunately, the initial fit of the models to experimental data may not
come out as well as expected. This is particularly true for the most general
models, such as the (general) polynomial model and the Ogden model. For some of
the simpler models, stability is assured by following some simple rules.
For positive values of the initial shear modulus,
,
and the locking stretch, ,
the Arruda-Boyce form is always stable.
For positive values of the coefficient
the neo-Hookean form is always stable.
Given positive values of the initial shear modulus,
,
and the locking stretch, ,
the stability of the Van der Waals model depends on the global interaction
parameter, a.
For the Yeoh model stability is assured if all .
Typically, however,
will be negative, since this helps capture the S-shape feature of the
stress-strain curve. Thus, reducing the absolute value of
or magnifying the absolute value of
will help make the Yeoh model more stable.
In all cases the following suggestions may improve the quality of the fit:
Both tension and compression data are allowed; compressive stresses and
strains are entered as negative values. Use compression or tension data
depending on the application: it is difficult to fit a single material model
accurately to both tensile and compressive data.
Always use many more experimental data points than unknown coefficients.
If
is used, experimental data should be available to at least 100% tensile strain
or 50% compressive strain.
Perform different types of tests (e.g., compression and simple shear
tests). Proper material behavior for a deformation mode requires test data to
characterize that mode.
Check for warning messages about material instability or error messages
about lack of convergence in fitting the test data. This check is especially
important with new test data; a simple finite element model with the new test
data can be run through the
analysis input file processor
to check the material stability.
You can perform one-element simulations for simple
deformation modes and compare the
Abaqus
results against the experimental data.
Delete some data points at very low strains if large strains are
anticipated. A disproportionate number of low strain points may unnecessarily
bias the accuracy of the fit toward the low strain range and cause greater
errors in the large strain range.
Delete some data points at the highest strains if small to moderate
strains are expected. The high strain points may force the fitting to lose
accuracy and/or stability in the low strain range.
Pick data points at evenly spaced strain intervals over the expected
range of strains, which will result in similar accuracy throughout the entire
strain range.
The higher the order of N, the more oscillations
are likely to occur, leading to instabilities in the stress-strain curves. If
the (general) polynomial model is used, lower the order of
N from 2 to 1 (3 to 2 for Ogden), especially if the
maximum strain level is low (say, less than 100% strain).
If multiple types of test data are used and the fit still comes out
poorly, some of the test data probably contain experimental errors. New tests
may be needed. One way of determining which test data are erroneous is to first
calibrate the initial shear modulus
of the material. Then fit each type of test data separately in
Abaqus
and compute the shear modulus, ,
from the material constants using the relations
Alternatively, the initial Young's modulus, ,
can be calibrated and compared with
The values of
or
that are most different from
or
indicate the erroneous test data.
Elements
The hyperelastic material model can be used with solid (continuum) elements,
finite-strain shells (except S4), continuum shells, membranes, and one-dimensional elements
(trusses and rebars). In
Abaqus/Standard
the hyperelastic material model can be also used with Timoshenko beams (B21, B22, B31, B31OS, B32, B32OS, PIPE21, PIPE22, PIPE31, PIPE32, and their “hybrid” equivalents). It cannot be used with
Euler-Bernoulli beams (B23, B23H, B33, and B33H) and small-strain shells (STRI3, STRI65, S4R5, S8R, S8R5, S9R5).
Pure Displacement Formulation Versus Hybrid Formulation in Abaqus/Standard
For continuum elements in
Abaqus/Standard
hyperelasticity can be used with the pure displacement formulation elements or
with the “hybrid” (mixed formulation) elements. Because elastomeric materials
are usually almost incompressible, fully integrated pure displacement method
elements are not recommended for use with this material, except for plane
stress cases. If fully or selectively reduced-integration displacement method
elements are used with the almost incompressible form of this material model, a
penalty method is used to impose the incompressibility constraint in anything
except plane stress analysis. The penalty method can sometimes lead to
numerical difficulties; therefore, the fully or selectively reduced-integrated
“hybrid” formulation elements are recommended for use with hyperelastic
materials.
In general, an analysis using a single hybrid element will be only slightly
more computationally expensive than an analysis using a regular
displacement-based element. However, when the wavefront is optimized, the
Lagrange multipliers may not be ordered independently of the regular degrees of
freedom associated with the element. Thus, the wavefront of a very large mesh
of second-order hybrid tetrahedra may be noticeably larger than that of an
equivalent mesh using regular second-order tetrahedra. This may lead to
significantly higher CPU costs, disk space, and memory requirements.
Incompatible Mode Elements in Abaqus/Standard
Incompatible mode elements should be used with caution in applications
involving large strains. Convergence may be slow, and in hyperelastic
applications inaccuracies may accumulate. Erroneous stresses may sometimes
appear in incompatible mode hyperelastic elements that are unloaded after
having been subjected to a complex deformation history.