Constitutive theory for mechanics of amorphous thermoplastic polymers under extreme dynamic loading

TL;DR

This work develops a geometrically nonlinear constitutive framework for amorphous thermoplastics that unifies thermoelasticity, viscoelasticity, viscoplasticity, and fracture with phase-field damage and multiple order parameters. By coupling internal state variables for configurational changes, volume changes, melting, and shock decomposition with a phase-field description of damage, the model spans glassy and molten states under extreme pressures and strain rates. Applied to PMMA, the framework reproduces the principal Hugoniot up to over 120 GPa, captures the onset of shock decomposition near $P^H_D = 25.7$ GPa and $\theta_D = 1289$ K, and describes spall strength trends across temperature and loading rate regimes. The approach provides thermodynamically consistent predictions for release waves, steady waves, and 1-D spall, offering a path toward robust 3-D implementations for ballistic and high-rate impacts on amorphous polymers.

Abstract

A geometrically nonlinear continuum mechanical theory is formulated for deformation and failure behaviors of amorphous polymers. The model seeks to capture material response over a range of loading rates, temperatures, and stress states encompassing shock compression, inelasticity, melting, decomposition, and spallation. Thermoelasticity, viscoelasticity, viscoplasticity, ductile failure with localized shear yielding, and brittle fracture with crazing can all emerge under this ensemble of intense loading conditions. Known prior theories have considered one or more, but not all, such physical mechanisms. The present coherent formulation invokes thermodynamics with internal state variables for dynamic molecular and network configurational changes affecting viscoelasticity and plastic deformation, and it uses order parameters for more abrupt structural changes across state-dependent glass-transition and shock-decomposition thresholds. A phase-field order parameter captures material degradation from ductile or brittle fracture, including evolving porosity from crazing. The theory is applied toward polymethyl methacrylate (PMMA) under intense dynamic loading. The high-pressure equilibrium response, with shear strength and temperature over known ranges, is well represented along the principal Hugoniot to pressures far exceeding shock decomposition. Predicted release wave velocities agree with experiment. A semi-analytical solution for steady waves describes the relatively lower-pressure viscoelastic setting, providing insight into relaxation times. One-dimensional calculations assess suitability of the model for representing spall fracture strengths seen in experiments over a range of initial temperatures and loading rates.

Figures (9)

• Figure 1: Hugoniot response of PMMA to extreme pressure: (a) shock stress or shock pressure $P$ vs. strain from model and experiments hauver1964carter1995vanthiel1977reinhart2006 (b) shock velocity $\mathcal{U}$ vs. particle velocity $\upsilon$ from model and experiments hauver1964barker1970schuler1974christman1972carter1995vanthiel1977reinhart2006 (c) temperature $\theta$ vs. stress from model, experiments rosenberg1984bordz2016kormer1968, and DFT coe2022 (d) Lagrangian sound speed $C_L$ from model and release experiments barker1970schuler1974pavlov1976reinhart2006
• Figure 2: Hugoniot response of PMMA: (a) shock velocity $\mathcal{U}$ vs. particle velocity $\upsilon$ from model (mixture, reactants, and decomposed products) and relevant experiments hauver1964carter1995 in the vicinity of decomposition (b) predicted decomposition volume fraction $\Xi$ vs. shock stress $P$
• Figure 3: Hugoniot response of PMMA to moderate pressure: (a) shock stress or shock pressure $P$ vs. strain from model and experiments barker1970schuler1974 (b) shock velocity $\mathcal{U}$ vs. particle velocity $\upsilon$ from model and experiments barker1970schuler1974 (c) temperature $\theta$ vs. stress from model and experiments rosenberg1984 (d) shear strength (Mises stress) $\bar{\sigma}$ from model and experiments gupta1980abatkov1996millett2000jordan2020
• Figure 4: Predicted Hugoniot response of PMMA: (a) ratio of Hugoniot to initial bulk modulus, shear modulus, Grüneisen parameter, and specific heat (b) entropy per unit reference volume (c) minimum values of resistance functions $R_\omega$ and $R_\xi$ needed to suppress melting and fracture in high-pressure Hugoniot states
• Figure 5: Low-pressure shock response of PMMA: (a) instantaneous and relaxed shock pressure $P$ from model and experiments barker1970christman1972 (b) instantaneous and relaxed shock velocity $\mathcal{U}$ from model and experiments barker1970christman1972 (c) instantaneous and relaxed Hugoniot temperature $\theta$ from model and experiments rosenberg1984 (d) shock $\mathcal{U}$ and sound $C_L$ velocity from model and experiments barker1970 (e) steady profiles of particle velocity $\upsilon$ vs. time from model and experiments schuler1970 (f) ratio of local sound speed $C_L$ to steady wave velocity $\mathcal{U}$ (model)