Publications Details

Numerical Integration of Viscoelastic Models

Lester, Brian T.; Long, Kevin N.

Accurate modeling of viscoelasticity remains an important consideration for a variety of materials (e.g. polymers and inorganic glasses). As such, over the previous decades a substantial body of work has been dedicated to developing appropriate constitutive models for viscoelasticity ranging from initial considerations of linear thermoviscoelasticity to more complex non-linear formulations incorporating fictive temperatures or potential energy clocks including the use of both internal state variable(ISV) and hereditary integral representations. Nonetheless, relatively limited (in comparison to plasticity) attention has been paid to the numerical integration of such schemes. In terms of integral based formulations, Taylor et al. first considered the problem of the integration of a linear viscoelasticity model. That work focused on the integration of the hereditary integrals and demonstrated improved performance of the new scheme with a custom finite element code over an existing finite difference reference. Chambers and Becker, using a free volume based shift factor, also considered the integration of the hereditary integrals and the impact on the problem of a pressurized thick-walled cylinder and developed an adaptive scheme to bound the error. Chambers later developed three-point Gauss and composite integration schemes for the hereditary integrals and noted improved accuracy. With respect to ISV-based schemes, formulations for the non-linear Schapery model have been proposed. However, in those efforts greater attention was paid to convergence of the non-linear solution scheme than impact of numerical integration. Various authors (e.g. Holzapfel and Simo and Hughes) have also studied the use of convolution integrals with differential forms of ISVs for temperature-independent formulations. Regardless, while the "potential energy clock" (PEC) and "simplified potential energy clock"(SPEC) models have been used to study a variety of non-linear responses (e.g.), limited attention has been paid to the numerical performance. As will be discussed later, the "clock" at the center of the formulations includes temperature and complex history dependence making the numerical integration of such a model even more challenging. Thus, in the current work an initial effort towards characterizing the numerical integration of the constitutive model through simplified problems is performed. To that end, in Section 2 the theory of the model is briefly presented while the numerical integration is discussed in Section 3. Results of various studies characterizing the numerical behavior and performance are then given in Section 4. Finally, some concluding remarks and thoughts for follow on works are provided in Section 5.