Influence of Retardation Time on Viscoelastic Behavior of Plane Beams Modeled by the Nonlinear Positional Formulation of the Finite Element Method

Abstract

A numerical procedure is presented to avoid the divergence problem during the iterative process in viscoelastic analyses. This problem is observed when the positional formulation of the finite element method is adopted in association with the finite difference method. To do this, the nonlinear positional formulation is presented considering plane frame elements with Bernoulli–Euler kinematics and viscoelastic behavior. The considered geometrical nonlinearity refers to the structural equilibrium analysis in the deformed position using the Newton–Raphson iterative method. However, the considered physical nonlinearity refers to the description of the viscoelastic behavior through the adoption of the stress-strain relation based on the Kelvin–Voigt rheological model. After the presentation of the formulation, a detailed analysis of the divergence problem in the iterative process is performed. Then, an original numerical procedure is presented to avoid the divergence problem based on the retardation time of the adopted rheological model and the penalization of the nodal position correction vector. Based on the developments and the obtained results, it is possible to conclude that the presented formulation is consistent and that the proposed procedure allows for obtaining the equilibrium positions for any time step value adopted without presenting divergence problems during the iterative process and without changing the analysis of the final results.

Shear contribution on viscoelastic behavior of beams modeled using plane frame elements with Reissner kinematics

“…In Mesquita and Coda (2002) a bidimensional mesh with 200 free formulation discrete Kirchhoff triangle finite elements and the viscoelastic Boltzmann model were considered. Results obtained from Becho and Greco (2021), with only 10 plane frame finite elements, are in good agreement with results available in the literature. The second reason, not treated in current the manuscript, is regarding the model calibration and it is also a direct consequence from the first reason.…”

“…Both regarding the number of degrees of freedom and the mathematical operations are simpler. For instance, in Becho and Greco (2021) a cylindrical pressure vessel made by viscoelastic material is simulated. The pressure is applied gradually and the radial displacement evolution along time is analyzed.…”