A Continuum Theory of Dynamically Loaded Polymers

TL;DR

This paper introduces the GAP model, a comprehensive thermo-mechanical continuum theory for glassy polymers that captures behavior across quasi-static and high-rate loading conditions, including shock phenomena.

Contribution

The paper develops the first rate-dependent, continuum-based model for glassy polymers that incorporates a non-equilibrium equation of state and the Hierarchical Flow Stress model.

Findings

The GAP model accurately predicts shock Hugoniot and thermo-mechanical behavior.

The HFS model captures stress plateau, softening, and hardening in PMMA.

Comparison with experiments shows good agreement across loading rates.

Abstract

A thermo-mechanical continuum theory is proposed for dynamically loaded glassy polymers. The theory is based on an ansatz for the Helmholtz free energy where both the deviatoric and the volumetric contributions to the free energy are rate-dependent. The requirement that the free energy is fully rate dependent arises from the need to model the full range of conditions between those found in quasi-static applications to those common in high-rate shock loading scenarios. Using a purely equilibrium equation of state is found to be insufficient. The resulting model, called the Glassy Amorphous Polymer (GAP) model, suitably captures the thermo-mechanical behavior of both equilibrium properties, but also high-rate phenomena like the shock Hugoniot. Polymethylmethacrylate (PMMA) is used as a representative polymer because it is one of the few polymers where sufficient experiments have been done to determine many of the parameters required by the GAP model. An important part of the GAP model is the Hierarchical Flow Stress (HFS) model, which is shown to accurately represent the stress-strain behavior of PMMA over a broad range of strain rates and temperatures. The HFS model replicates the details of stress plateau, softening, and hardening behavior observed in glassy amorphous polymers. The theory is not restricted to isothermal conditions. Analysis is devoted to several issues associated with a non-equilibrium equation of state (EOS). Because data is insufficient for the full determination of the model's non-equilibrium EOS, several plausible conditions called the quasi-equilibrium hypothesis are put forth to complete the model. Comparisons of experimental and theoretical results over a wide-range of loading rates and conditions are reported.