Newmark algorithm for dynamic analysis with Maxwell chain model

TL;DR

This paper develops a stable and low-damping Newmark-type time-stepping algorithm for dynamic analysis of viscoelastic structures modeled by a generalized Maxwell chain, based on a discretized Hamilton variational principle.

Contribution

It extends existing algorithms to a generic Maxwell chain and derives it from a variational principle, enhancing stability and reducing artificial damping.

Findings

Algorithm exhibits excellent stability.

Low artificial damping confirmed through examples.

Applicable to distributed systems and fracture simulations.

Abstract

This paper investigates a time-stepping procedure of the Newmark type for dynamic analyses of viscoelastic structures characterized by a generalized Maxwell model. We depart from a scheme developed for a three-parameter model by Hatada et al. in 2000, which we extend to a generic Maxwell chain and demonstrate that the resulting algorithm can be derived from a suitably discretized Hamilton variational principle. This variational structure manifests itself in an excellent stability and a low artificial damping of the integrator, as we confirm with a mass-spring-dashpot example. After a straightforward generalization to distributed systems, the integrator may find use in, e.g., fracture simulations of laminated glass units, once combined with variationally-based fracture models.