Constitutive Models Relations between Stress and Strain
3.5 Hyperelasticity time independent behavior of rubbers and foams
subjected to large strains
Hyperelastic constitutive laws are used to model materials that respond
elastically when subjected to very large strains. They account both for
nonlinear material behavior and large shape changes. The main applications of the theory are (i)
to model the rubbery behavior of a polymeric material, and (ii) to model
polymeric foams that can be subjected to large reversible shape changes (e.g. a
In general, the response of a typical polymer is strongly dependent on
temperature, strain history and loading rate.
The behavior will be described in more detail in the next section, where
we present the theory of viscoelasticity.
For now, we note that polymers have various regimes of mechanical
behavior, referred to as `glassy,’ `viscoelastic’ and `rubbery.’ The various regimes can be identified for a
particular polymer by applying a sinusoidal variation of shear stress to the
solid and measuring the resulting shear strain amplitude. A typical result is illustrated in the figure,
which shows the apparent shear modulus (ratio of stress amplitude to strain
amplitude) as a function of temperature.
At a critical temperature known as the glass transition temperature,
a polymeric material undergoes a dramatic change in mechanical response. Below this temperature, it behaves like a
glass, with a stiff response. Near the glass transition temperature, the stress
depends strongly on the strain rate. At
the glass transition temperature, there is a dramatic drop in modulus. Above this temperature, there is a regime
where the polymer shows `rubbery’ behavior the response is elastic; the stress does not
depend strongly on strain rate or strain history, and the modulus increases
with temperature. All polymers show
these general trends, but the extent of each regime, and the detailed behavior
within each regime, depend on the solid’s molecular structure. Heavily cross-linked polymers (elastomers)
are the most likely to show ideal rubbery behavior. Hyperelastic constitutive laws are intended
to approximate this `rubbery’ behavior.
Features of the behavior of a solid rubber:
1. The material is close to ideally elastic. i.e. (i)
when deformed at constant temperature or adiabatically, stress is a function
only of current strain and independent of history or rate of loading, (ii) the
behavior is reversible: no net work is done on the solid when subjected to a
closed cycle of strain under adiabatic or isothermal conditions.
2. The material strongly resists volume changes. The bulk modulus (the ratio of volume change
to hydrostatic component of stress) is comparable to that of metals or
covalently bonded solids;
3. The material is very compliant in shear shear modulus is of the order of times that of most metals;
4. The material is isotropic its stress-strain response is independent of
5. The shear modulus is temperature dependent: the
material becomes stiffer as it is heated, in sharp contrast to metals;
6. When stretched, the material gives off heat.
(e.g. a sponge) share some of these properties:
1. They are close to reversible, and show little rate or
2. In contrast to rubbers, most foams are highly
compressible bulk and shear moduli are comparable.
3. Foams have a complicated true stress-true strain
response, generally resembling the figure to the right. The finite strain response of the foam in
compression is quite different to that in tension, because of buckling in the
4. Foams can be anisotropic, depending on their cell
structure. Foams with a random cell
structure are isotropic.
The literature on stress-strain relations for finite elasticity can be
hard to follow, partly because nearly every paper uses a different notation,
and partly because there are many different ways to write down the same stress-strain
law. You should find that most of the
published literature is consistent with the framework given below but it may take some work to show the
models are constructed as follows:
1. Define the stress-strain relation for the solid by
specifying its strain energy density W as a function of deformation gradient tensor: W=W(F).
This ensures that the material is perfectly elastic, and also means that
we only need to work with a scalar function.
The general form of the strain energy density is guided by experiment;
and the formula for strain energy density always contains material properties
that can be adjusted to describe a particular material.
2. The undeformed material is usually assumed to be isotropic i.e the behavior of the material is
independent of the initial orientation of the material with respect to the
loading. If the strain energy density is
a function of the Left Cauchy-Green deformation tensor the constitutive equation is automatically
isotropic. If B is used as the deformation measure, then the strain energy must
be a function of the invariants of B to ensure that the constitutive
equation is objective (recall that the invariants of a tensor remain constant
under a change of basis)
3. Formulas for stress in terms of strain are calculated
by differentiating the strain energy density as outlined below.
3.5.1 Deformation Measures used in
Suppose that a solid is subjected to a displacement field .
gradient and its Jacobian
The Left Cauchy-Green deformation tensor
Invariants of B (these are the conventional definitions)
An alternative set of invariants of B (more convenient for models of nearly
incompressible materials note that remain constant under a pure volume change)
Principal stretches and principal stretch
1. Let denote the three eigenvalues of B.
The principal stretches are
2. Let denote three, mutually perpendicular unit eigenvectors of B. These define the principal stretch
directions. (Note: since B is symmetric its eigenvectors are
automatically mutually perpendicular as long as no two eigenvalues are the
same. If two, or all three eigenvalues
are the same, the eignevectors are not uniquely defined in this case any convenient mutually
perpendicular set of eigenvectors can be used).
3. Recall that B
can be expressed in terms of its eigenvectors and eigenvalues as
3.5.2 Stress Measures used in finite elasticity
Usually stress-strain laws are given as equations relating Cauchy
stress (`true’ stress) to left Cauchy-Green deformation tensor. For some computations it may be more
convenient to use other stress measures.
They are defined below, for convenience.
The Cauchy (“true”) stress represents the
force per unit deformed area in the solid and is defined by
Nominal (First Piola-Kirchhoff) stress
(Second Piola-Kirchhoff) stress
3.5.3 Calculating stress-strain relations from the
strain energy density
The constitutive law for an isotropic hyperelastic material is defined
by an equation relating the strain energy density of the material to the
deformation gradient, or, for an isotropic solid, to the three invariants of
the strain tensor
The stress-strain law must then be deduced by differentiating the
strain energy density. This can involve
some tedious algebra. Formulas are
listed below for the stress-strain relations for each choice of strain
invariant. The results are derived below
Strain energy density in terms of
Strain energy density in terms of
Strain energy density in terms of
Strain energy density in terms of
Derivations: We start by
deriving the general formula for stress in terms of :
1. Note that, by definition, if the
solid is subjected to some history of strain, the rate of change of the strain energy density W (F) must
equal the rate of mechanical work done on the material per unit reference
2. Recall that the rate of work done per unit undeformed volume by body forces and surface tractions is
expressed in terms of the nominal stress as .
3. Therefore, for any deformation gradient Fij,
must hold for all possible ,so
4. Finally, the formula for Cauchy stress follows from
the equation relating to
For an isotropic material, it is necessary to find derivatives of the
invariants with respect to the components of F in order to compute the stress-strain function for a given strain
energy density. It is straightforward,
but somewhat tedious to show that:
When using a strain energy density of the form , we will have to compute the derivatives of
the invariants with respect to the components of F in order to find
We find that
Next, we derive the stress-strain relation in terms of
a strain energy density that is expressed as a function of the
principal strains. Note first that
so that the chain
Using this and the
expression that relates the stress components to the derivatives of U,
we find that the
principal stresses are related to the corresponding principal
stretches (square-roots of the eigenvalues of B) through
decomposition for B in terms of its
eigenvalues and eigenvectors :
now allows the stress tensor to be written as
3.5.4 A note
on perfectly incompressible materials
The preceding formulas assume that the material has
some (perhaps small) compressibility that is to say, if you load it with
hydrostatic pressure, its volume will change by a measurable amount. Most rubbers strongly resist volume changes,
and in hand calculations it is sometimes convenient to approximate them as
perfectly incompressible. The material
model for incompressible materials is specified as follows:
The deformation must satisfy J=1 to preserve volume.
The strain energy density is therefore only a
function of two invariants;
furthermore, both sets of invariants defined above are identical. We can use a strain energy density of the
Because you can apply any pressure to an
incompressible solid without changing its shape, the stress cannot be uniquely
determined from the strains.
Consequently, the stress-strain law only specifies the deviatoric stress . In problems involving quasi-static loading,
the hydrostatic stress can usually be calculated, by solving the
equilibrium equations (together with appropriate boundary conditions). Incompressible materials should not be used
in a dynamic analysis, because the speed of elastic pressure waves is infinite.
The formula for stress in terms of has the form
The hydrostatic stress p is an unknown variable, which must be calculated by solving the
boundary value problem.
Specific forms of the strain energy density
Neo-Hookean solid (Adapted from Treloar, Proc Phys Soc 60
where and are material properties (for small
deformations, and are the shear modulus and bulk modulus of the
solid, respectively). Elementary statistical mechanics treatments predict that ,
where N is the number of polymer chains per unit volume, k is the
Boltzmann constant, and T is
temperature. This is a rubber elasticity
model, for rubbers with very limited compressibility, and should be used with . The stress-strain relation follows as
The fully incompressible limit can be obtained by
setting in the stress-strain law.
Mooney-Rivlin solid (Adapted
from Mooney, J Appl Phys 11 582 1940)
where and are material properties. For small deformations, the shear modulus and
bulk modulus of the solid are and . This is a rubber elasticity model, and should
be used with .
The stress-strain relation follows as
polynomial rubber elasticity potential
where and are material properties. For small strains the shear modulus and bulk
modulus follow as .
This model is implemented in many finite element codes. Both the neo-Hookean solid and the
Mooney-Rivlin solid are special cases of the law (with N=1 and appropriate choices of ).
Values of are rarely used, because it is difficult to
fit such a large number of material properties to experimental data.
model (Ogden, Proc R Soc Lond A326, 565-84 (1972),
ibid A328 567-83 (1972))
where , and are material properties. For small strains the shear modulus and bulk
modulus follow as .
This is a rubber elasticity model, and is intended to be used with . The stress can be computed using the formulas
in 3.4.3, but are too lengthy to write out in full here.
Arruda-Boyce 8 chain model (J. Mech. Phys. Solids, 41, (2)
where are material properties. For small deformations are the shear and bulk modulus, respectively.
This is a rubber elasticity model, so . The potential was derived by calculating
the entropy of a simple network of long-chain molecules, and the series is the
result of a Taylor
expansion of an inverse Langevin function.
The reference provided lists more terms if you need them. The stress-strain law is
Ogden-Storakers hyperelastic foam
where are material properties. For small strains the shear modulus and bulk
modulus follow as . This is a foam model, and can model highly compressible
materials. The shear and compression
responses are coupled.
Blatz-Ko foam rubber
where is a material parameter corresponding to the
shear modulus at infinitesimal strains. Poisson’s ratio for such a material is
3.5.6 Calibrating nonlinear elasticity models
To use any of these constitutive relations, you will
need to determine values for the material constants. In some cases this is quite simple (the incompressible
neo-Hookean material only has 1 constant!); for models like the generalized
polynomial or Ogden’s
it is considerably more involved.
Conceptually, however, the procedure is
straightforward. You can perform various
types of test on a sample of the material, including simple tension, pure
shear, equibiaxial tension, or volumetric compression. It is straightforward to
calculate the predicted stress-strain behavior for the specimen for each
constitutive law. The parameters can
then be chosen to give the best fit to experimental behavior.
Here are some guidelines on how best to do this:
1. When modeling the behavior of rubber under ambient
pressure, you can usually assume that the material is nearly incompressible,
and don’t need to characterize response to volumetric compression in
detail. For the rubber elasticity models
listed above, you can take MPa. To fit the remaining parameters, you can
assume the material is perfectly incompressible.
2. If rubber is subjected to large hydrostatic stress
(>100 MPa) its volumetric and shear responses are strongly coupled.
Compression increases the shear modulus, and high enough pressure can even
induce a glass transition (see, e.g. D.L. Quested, K.D. Pae, J.L. Sheinbein and
B.A. Newman, J. Appl. Phys, 52, (10) 5977 (1981)). To account for this, you would have to use
one of the foam models: in the rubber models the volumetric and shear responses
are decoupled. You would also have to determine the material constants by
testing the material under combined hydrostatic and shear loading.
3. For the simpler material models, (e.g. the neo-Hookean
solid, the Mooney-Rivlin material, or the Arruda-Boyce model, which contain
only two material parameters in addition to the bulk modulus) you can estimate
material parameters by fitting to the results of a uniaxial tension test. There are various ways to actually do the fit
you could match the small-strain shear modulus
to experiment, and then select the remaining parameter to fit the stress-strain
curve at a larger stretch. Least-squared
fits are also often used. However,
models calibrated in this way do not always predict material behavior under
multiaxial loading accurately.
4. A more accurate description of material response to
multiaxial loading can be obtained by fitting the material parameters to
multiaxial tests. To help in this
exercise, the nominal stress (i.e.
force/unit undeformed area) v- extension
predicted by several constitutive laws are listed in the table below (assuming
perfectly incompressible behavior, as suggested in 1.)
3.5.7 Representative values of material properties for
The properties of rubber are strongly sensitive to its molecular
structure, and for accurate predictions you will need to obtain experimental
data for the particular material you plan to use. As a rough guide, the experimental data of
Treloar (Trans. Faraday Soc. 40, 59.1944) for the behavior of vulcanized
rubber under uniaxial tension, biaxial tension, and pure shear is shown in the
picture. The solid lines in the figure
show the predictions of the Ogden
model (which gives the best fit to the data).
Material parameters fit to this data for several constitutive laws are