Relaxation in Polymer Networks under Uniaxial Extension and Biaxial Compression

TL;DR

Addresses the challenge of predicting time- and temperature-dependent viscoelastic response of permanent polymer networks under combined deformation modes. It integrates the van der Waals network theory for equilibrium behavior with irreversible thermodynamics to incorporate a substance-specific relaxation spectrum driving time-dependent stress via $f(t) = G\,w'(t)\,[1 + (\Gamma/G)\,(1 - M(t))]$, where $M(t) = \frac{2}{\Gamma}\int_0^{t} m(t-t')\,\sqrt{w(t')/w(t)}\,dt'$, and $m(t-t') = \sum_i \frac{h_i}{\tau_i} e^{-(t-t')/\tau_i}$. Temperature effects are described by the Williams-Landel-Ferry law $\log a_T(T) = \frac{-C_1\,(T-T_S)}{C_2+T-T_S}$, enabling time-temperature superposition above Tg. The framework is validated on cross-linked PMMA and PVAc across uniaxial extension and biaxial compression, using a small set of physically interpretable parameters and predicting both rate- and temperature-dependent behavior. These results extend rubber-like theories to practical elastomer design, providing a unified predictive tool for complex loading above Tg.

Abstract

Predicting the time and temperature dependent behavior of polymer networks under complex loading is essential for the design of advanced elastomeric materials. Many practical applications involve combinations of deformation modes, such as uniaxial extension and biaxial compression, yet a unified description of their mechanical response remains challenging. In this study, we apply a consistent theoretical framework to describe both uniaxial and biaxial deformation modes, using the same constitutive formalism based on van der Waals network theory. The time dependence of the material response in both cases is governed by a substance specific relaxation spectrum, introduced through irreversible thermodynamics as a linear coupling to the quasi static reference state of the permanent network. The temperature dependence of the relaxation times is well described by the Williams Landel Ferry (WLF) equation in the high temperature or low strain rate regime, demonstrating that the same physical mechanisms underlie time dependent behavior across different loading geometries. Experimental results are presented for cross linked poly(methyl methacrylate) (PMMA) and polyvinyl acetate (PVAc), validating the theoretical model across both materials and deformation modes.

Figures (3)

• Figure 1: A quasi-static stress-strain curve of PMMA (polymethyl methacrylate), with stress (f) plotted in MPa versus strain $\lambda$ under elongation and compression with parameters as indicated. The calculation (red line) is based on the van der Waals strain energy function eqn.(\ref{['vdw1']}) in uniaxial mode for elongation and biaxial mode for compression.
• Figure 2: Measurements of stress-strain curves of PVAc under elongation with different strain rates $\dot\epsilon$ ($\triangledown: 0,236$ 1/s, $\Delta: 0,120$ 1/s, $\circ: 0,06$ 1/s, $\square: 0,001$ 1/s) at a temperature of 333 K). Solid lines with the same color represent theory.
• Figure 3: Stress-strain curves of PMMA under biaxial compression at different strain rates $\dot \epsilon$ ($\square: 0,002$ 1/s, $\triangledown: 0,009$ 1/s $\circ: 0,035$ 1/s) at a temperature of $T=403 K$. Solid lines represent theory.