Because a response predicted by a hyperelastic

(1)

model is independent of the previous state of

where *I*1 and *I*2 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, *I*1 = λ1 + λ 2 + λ 3

response is that of a conservative system. As such,

2

2

2

and *I*2 = 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) , *C*00 = 0.

(2)

i = 0, j= 0

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*/ *I*1 and *W*/ *I*2 are calculated from mea-

properties (Gent and Lindley 1959), material prop-

surements as functions of *I*1 and *I*2, 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*/ *I*1 and *W*/ *I*2

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

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