 |
||||||||||||||||||||||||||||||||||||||||
|
||||||||||||||||||||||||||||||||||||||||
|
Chapter 3
Constitutive
Models
3.5 Hyperelasticity
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 sponge).
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
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 4. The material is isotropic 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.
Polymeric foams (e.g. a sponge) share some of these properties: 1. They are close to reversible, and show little rate or history dependence. 2. In contrast to rubbers, most foams are highly
compressible 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 cell walls. 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
All hyperelastic 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 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 finite elasticity
Suppose that a solid is subjected to a displacement field
1. Let
2. Let 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)
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 volume. 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 3. Therefore, for any deformation gradient Fij,
This
must hold for all possible
4. Finally, the formula for Cauchy stress follows from
the equation relating
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: Then,
and
When using a strain energy density of the form We find that
Thus,
Next, we derive the stress-strain relation in terms
of a strain energy density so that the chain rule gives
Using this and the expression that relates the stress components to the derivatives of U,
we find that the
principal stresses
The spectral
decomposition for B in terms of
its eigenvalues
3.5.4 A note on perfectly incompressible materials
The preceding formulas assume that the material has
some (perhaps small) compressibility
The hydrostatic stress p is an unknown variable, which must be calculated by solving the boundary value problem.
3.5.5 Specific forms of the strain energy density
where
The fully incompressible limit can be obtained by
setting
where
where
where
where
where
where
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
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 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 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)
3.5.7 Representative values of material properties for rubbers
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
Material parameters fit to this data for several constitutive laws are listed below.
|
||||||||||||||||||||||||||||||||||||||||
|
(c) A.F. Bower, 2008 |
||||||||||||||||||||||||||||||||||||||||
