Because a response predicted by a hyperelastic
W = C(I1 3) + f (I2 3) ,
(1)
model is independent of the previous state of
where I1 and I2 are strain invariants for an isotropic,
strain, the predicted stresses are independent of
incompressible material, C is a constant, and f de-
the strain history and deformation rate, and the
2
2
2
notes "a function of." Specifically, I1 = λ1 + λ 2 + λ 3
response is that of a conservative system. As such,
2
2
2
and I2 = 1/ λ1 + 1/ λ 2 + 1/ λ 3 , where the λs are the
hyperelasticity cannot model viscous or plastic be-
extension ratios, i.e., ratios of the current length to
havior, such as hysteresis, creep, stress relaxation
the original length, in the principal directions. The
and permanent deformation (Green and Adkins
first term of eq 1 has been found by relating the
1960). However, for rubber-like time-dependent
change in molecular dimensions to macroscopic
materials, McKenna and Zapas (1986), citing Rivlin
strain measurements from homogeneous strain-
(1956), describe the validity of using isochronal
loading experiments, and the last term has been
stressstrain data from stressrelaxation tests, mea-
called a "correction" term (Treloar 1974). A large
sured at relatively short times after a rapid strain
body of work has been conducted to establish im-
application, as an estimate of the equilibrium data
proved or alternative forms of the strain energy
for calculating strain energy functions for use in
function for isotropic rubber materials from a phe-
finite elasticity calculations. Thus, if the data of
nomenological perspective (see Treloar 1975, for
the time-dependent material response are treated
a review). For example, the following polynomial
correctly, the time dependence can be ignored for
form of the strain energy density function (e.g.,
purposes of the finite deformation calculations, and
Rivlin and Saunders 1951) is often used for phe-
the results could be viewed as the response at a
nomenological modeling:
given time, e.g., the long-term response.
∞
∑ Cij (I1 - 3) (I2 - 3) , C00 = 0.
i
j
W=
(2)
Examples of
i = 0, j= 0
sealant behavior
Previous experimental studies of the mechani-
The experimental determination of the strain en-
cal behavior of joint and crack sealants have re-
ergy function is typically a detailed task. Rivlin
lied primarily on structural configuration tests
and Saunders (1951) described a technique using
rather than material property tests to obtain mea-
biaxial loading experiments of rubber sheets that
sures of the sealant response. Although structural
has been adapted by many workers. In this tech-
configuration tests can reveal "apparent" system
nique W/ I1 and W/ I2 are calculated from mea-
properties (Gent and Lindley 1959), material prop-
surements as functions of I1 and I2, and the func-
erties cannot readily be found from these tests. Re-
tions are used to define eq 1. The conditions dur-
sults from structural configuration tests will be
ing the experiments dictate the thermodynamic na-
reviewed in the following section.
ture of the strain energy function that is found,
Catsiff et al. (1970a) have described material test
e.g., that the strain energy function is the function
data from several sealant materials, including poly-
for isothermal and constant pressure conditions
sulfide, silicone and asphalt-polyurethane sealants.
(Ward 1983). Other techniques have been devel-
They presented measurements indicating that an
oped as well. For example, Penn and Kearsley
incompressibility assumption is valid for analyz-
(1976) describe a data reduction technique for tor-
ing structures formed from these materials, and
sion experiments that allows W/ I1 and W/ I2
emphasized the validity of the incompressibility
to be calculated from torque and normal force mea-
assumption for the sealant formulations, not just
surements at different torsion angles.
the base-elastomer. They further suggested that
Implementations of strain energy functions for
stress and elongation data in an isochronous form,
large deformation response predictions of rubber
e.g., stress and extension data for a given time af-
structures are described for several homogeneous
ter the loads are applied in stress-relaxation tests,
and inhomogeneous deformation problems by
are appropriate for structural analysis techniques
Rivlin (1956). Numerical implementations of hyper-
in which time-dependent behavior is not incorp-
elastic models, using the finite element method
orated. This suggestion is consistent with the meth-
with large deformation capabilities, are described,
od suggested by McKenna and Zapas (1986) for
for example, by Hibbitt et al. (1989) and Finney
calculating the strain energy function for quasi-
and Kumar (1988). The effect of temperature on
elastomeric materials, as mentioned above. Catsiff
the structural response can be incorporated in the
et al. (1970a) presented data illustrating the valid-
analysis by measuring the strain energy as a func-
ity of this technique for sealants.
tion of temperature as well as strain.
7
Previous Page
