The relaxation response for a step-load experiment is expressed in
equation 221,22,28 where
G(
t) is the time-dependent shear relaxation modulus. Given that ideal instantaneous step loading is not physically attainable, actual rise time (
tR) for ramp loading should be considered in the derivation.
21,23Equation 3 is a viscoelastic integral operator for relaxation where
u is a strain function of time dummy variable τ.
As suggested by Mattice et al.,
21 a Boltzmann integral method
23 is used here. When the time-dependent relaxation modulus in
equation 2 is combined with
equation 3, the resultant Boltzmann integral equation is shown in
equation 4.
For ramp-loading rate
k, displacement for ramp-hold relaxation can be written as
The solution for displacement-controlled relaxation is expressed as the step-loading relaxation solution adjusted by an RCF to take into account the difference in relaxation caused by noninstantaneous ramp loading.
23,28 Since load
P(
t), is exponentially decaying during relaxation, it is expressed as
whereas the material relaxation function has the form,
where τ represents each time constant for each exponential form,
Bn represents a fitting constant, and
Cn represents relaxation coefficients. It is parsimonious and, therefore, desirable to minimize the number of exponential terms for curve fitting. We settled on three terms that captured relaxation behavior with <6.5% error. Once all the fitting parameters (
Bn) have been determined, they can be converted to material parameters (
Cn) using
equations 9 and
10.
The equation for RCF, which compensates for actual ramp versus ideal step loading, is shown in
equation 11.
Instantaneous
G(0) and long-time
G(∞) stiffnesses can be computed from the fitted relaxation coefficients (
Ci), as shown in
equations 12 and
13.