A finite viscoelastic constitutive model for low to high strain rate response of elastomers with application of strain rate-induced glass transition

TL;DR

The paper develops a thermodynamically consistent, mechanistically motivated finite viscoelastic constitutive framework for amorphous elastomers exhibiting strain-rate–induced glass transition, using two concurrent viscous mechanisms: molecular relaxation at low rates and intermolecular rearrangement at high rates. By employing a volume-decoupled free-energy split and multiplicative kinematics, the model yields equilibrium Arruda–Boyce–type responses and non-equilibrium responses with Ree–Eyring and related viscoelastic laws, calibrated against polyborosiloxane PBS data. Numerical results demonstrate rate-sensitive stress, energy dissipation trends with a crossover signaling glass transition, cyclic loading behavior with diminishing dissipation at high rates, and DMA-like moduli transitions across frequency. The framework provides a practical, physically interpretable tool for design-by-analysis of elastomeric components and metamaterials under diverse strain-rate conditions, with future work extending to thermo-mechanical coupling and damage.

Abstract

Amorphous elastomers exhibit significant rate-stiffening and unique viscous flow characteristics across a wide range of strain rates, often undergoing glass transition above a strain rate threshold. We have developed a thermodynamically-consistent and micromechanically-inspired constitutive model for soft elastomeric materials to capture the rate-dependent stress-strain behavior and hysteresis when subjected to low to high strain rates. Our proposed constitutive model encapsulates the viscous flow of materials through molecular motion at low strain rates and local rearrangement and alignment at high strain rates, essentially covering the glass transition. We applied our constitutive model to uniaxial compression experiments performed at low and high strain rates for polyborosiloxane (PBS) to identify the material parameters, and subsequently, performed numerical simulations of single and multi-cycle compression, stress relaxation, and small amplitude oscillatory tension-compression. Our analyses indicate that the model predicts higher total energy dissipation with increasing strain rate; however, dissipation associated with molecular relaxation decreases (forming a cusp) because, beyond a crossover strain rate, molecular rearrangement and alignment become dominant, which is consistent with the onset of the glass transition. For cyclic loading-unloading, we observed that dissipation over a cycle remains constant at low strain rate but decreases non-monotonically at high strain rates before becoming constant with peak stress over the cycle becoming higher, which can be interpreted as more loading being carried elastically by the polymer network as the molecular rearrangement process occurs. Additionally, our model was able to predict the qualitative nature of the storage modulus and loss modulus in the limit of small strain over a wide range of frequency sweeps.

Figures (7)

• Figure 1: (a) Multiplicative decomposition of the deformation gradient into elastic and viscous parts, (b) a simple rheological representation of the generalized Maxwell model.
• Figure 2: Rheological representation of dual-mechanism finite viscoelasticity model for quasi-static and high strain rate response of amorphous elastomers undergoing glass transition.
• Figure 3: Identification of material parameters: (a) calibration of the equilibrium shear modulus, $G_{\mathop{\mathrm{eq}}\nolimits}$, and locking stretch, $\lambda_{\mathrm{L}}$, from lowest strain rate experimental data avaiable ($\dot{\varepsilon} = 10^{-3}$ s$^{-1}$ with $R^2 = 0.9832$, (b) calibration of Bergström-Boyce and Ree-Eyring parameters from compressive loading-unloading experiment performed at a strain rate, $\dot{\varepsilon} = 10^{-2}$ s$^{-1}$ ($R^2 = 0.7093$) and split Hopkinson pressure bar (SHPB) test performed at $\dot{\varepsilon} = 4500$ s$^{-1}$ with a coefficient of determination, $R^2=0.9930$, respectively. Inset shows model fit at $\dot{\varepsilon} = 10^{-2}$ s$^{-1}$. All experimental data were taken from konaleLargeDeformationModel2023.
• Figure 4: Rate sensitivity study of the material model in uniaxial compressive loading-unloading: (a) stress-strain curve at different strain rates, (b) accumulated individual and total dissipation at different strain rates, contribution of non-equilibrium (dissipative) stress from (c) mechanism-I and (d) mechanism-II at different strain rates.
• Figure 5: Stress-strain response under cyclic compressive loading and unloading performed at (a) low ($\dot{\varepsilon} = 5 \times 10^{-2}$ s$^{-1}$), (b) moderate ($\dot{\varepsilon} = 5$ s$^{-1}$), and (c) high ($\dot{\varepsilon} = 5 \times 10^3$ s$^{-1}$) strain rates where $n$ represents the cycle number, and (d) total dissipated energy density per cycle for different strain rates with an inset illustrating the dissipated energy at low and moderate strain rates.

Theorems & Definitions (2)

• Remark 3.1
• Remark 3.2