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From Connectivity to Rupture: A Coarse-Grained Stochastic Network Dynamics Approach to Polymer Network Mechanics

Shaswat Mohanty, Wei Cai

TL;DR

The paper addresses the challenge of linking chain-level entropy-driven elasticity and rupture kinetics to macroscopic failure in disordered polymer networks, while avoiding the computational cost of fully resolved molecular dynamics. It introduces CGSND, a graph-based coarse-graining framework where nodes represent cross-linked beads, edges carry bond counts $w_{ij}$, and bond forces follow an inverse-Langevin stiffened spring with rupture governed by a threshold, complemented by rupture hazard rates and a Gini coefficient to quantify load localization. The results show CGSND reproducing the qualitative nonlinear $\sigma$-$\lambda$ response, a well-defined ultimate tensile strength, a peak in bond-breaking hazard near failure, and post-peak softening, alongside rupture-length statistics that match the initial network distribution and pronounced force localization preceding failure. This approach delivers computational efficiency and mechanistic insight, enabling large-scale statistical studies and extensions to heterogeneity or healing, while clarifying limitations relative to explicit CGMD such as the rate-independent, affine loading and absence of local relaxation or thermal dynamics.

Abstract

We introduce a coarse-grained stochastic network dynamics (CGSND) framework for modeling deformation and rupture in polymer networks. The method replaces explicit molecular dynamics (MD) or coarse-grained molecular dynamics (CGMD) with network-level evolution rules while retaining chain entropic elasticity and force-controlled bond failure. Under uniaxial loading, CGSND reproduces the characteristic nonlinear stress--stretch response of elastomeric networks, including a well-defined ultimate tensile strength and post-peak softening due to progressive bond rupture. Comparison with coarse-grained molecular dynamics (CGMD) simulations shows that CGSND captures the qualitative form of the stress response and the onset of catastrophic damage despite its rate-independent formulation. Analysis of rupture kinetics reveals a pronounced peak in the bond-breaking hazard rate near the ultimate tensile strength in both approaches. In addition, the distribution of broken segment lengths remains statistically indistinguishable from the initial network, indicating that rupture is not biased toward short or long chains. Finally, the evolution of the Gini coefficient of bond force magnitudes reveals strong force localization preceding failure. These results demonstrate that CGSND provides a computationally efficient and physically interpretable framework for connecting force localization and rupture kinetics to macroscopic failure in polymer networks.

From Connectivity to Rupture: A Coarse-Grained Stochastic Network Dynamics Approach to Polymer Network Mechanics

TL;DR

The paper addresses the challenge of linking chain-level entropy-driven elasticity and rupture kinetics to macroscopic failure in disordered polymer networks, while avoiding the computational cost of fully resolved molecular dynamics. It introduces CGSND, a graph-based coarse-graining framework where nodes represent cross-linked beads, edges carry bond counts , and bond forces follow an inverse-Langevin stiffened spring with rupture governed by a threshold, complemented by rupture hazard rates and a Gini coefficient to quantify load localization. The results show CGSND reproducing the qualitative nonlinear - response, a well-defined ultimate tensile strength, a peak in bond-breaking hazard near failure, and post-peak softening, alongside rupture-length statistics that match the initial network distribution and pronounced force localization preceding failure. This approach delivers computational efficiency and mechanistic insight, enabling large-scale statistical studies and extensions to heterogeneity or healing, while clarifying limitations relative to explicit CGMD such as the rate-independent, affine loading and absence of local relaxation or thermal dynamics.

Abstract

We introduce a coarse-grained stochastic network dynamics (CGSND) framework for modeling deformation and rupture in polymer networks. The method replaces explicit molecular dynamics (MD) or coarse-grained molecular dynamics (CGMD) with network-level evolution rules while retaining chain entropic elasticity and force-controlled bond failure. Under uniaxial loading, CGSND reproduces the characteristic nonlinear stress--stretch response of elastomeric networks, including a well-defined ultimate tensile strength and post-peak softening due to progressive bond rupture. Comparison with coarse-grained molecular dynamics (CGMD) simulations shows that CGSND captures the qualitative form of the stress response and the onset of catastrophic damage despite its rate-independent formulation. Analysis of rupture kinetics reveals a pronounced peak in the bond-breaking hazard rate near the ultimate tensile strength in both approaches. In addition, the distribution of broken segment lengths remains statistically indistinguishable from the initial network, indicating that rupture is not biased toward short or long chains. Finally, the evolution of the Gini coefficient of bond force magnitudes reveals strong force localization preceding failure. These results demonstrate that CGSND provides a computationally efficient and physically interpretable framework for connecting force localization and rupture kinetics to macroscopic failure in polymer networks.
Paper Structure (25 sections, 8 equations, 5 figures)

This paper contains 25 sections, 8 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Coarse-Graining of the Polymer Network
  4. Loading and deformation
  5. Normalized extension
  6. Bond force law
  7. Bond rupture
  8. Stress calculation (reported in MPa)
  9. Bond rupture hazard rate
  10. Force localization and Gini coefficient
  11. Loading Algorithm
  12. Results
  13. Stress--stretch response
  14. Bond rupture kinetics
  15. Rupture kinetics and failure onset
  16. ...and 10 more sections

Figures (5)

  • Figure 1: Nominal stress--stretch response from CGMD and CGSND. Macroscopic nominal stress $\sigma$ as a function of applied stretch ratio $\lambda$ obtained from (a) CGMD simulations and (b) CGSND framework. Stress in CGSND is computed using a bulk virial formulation and reported in MPa via the thermal stress scale $k_B T / b^3$.
  • Figure 2: Bond rupture kinetics from CGMD and CGSND. Fraction of broken bonds as a function of stretch for polymer strands and cross-links obtained from (a) the CGMD simulations and (b) the CGSND framework.
  • Figure 3: Rupture kinetics and failure onset. Hazard rate of bond-breaking events as a function of stretch for (a) CGMD and (b) CGSND, resolved into cross-link and backbone contributions.
  • Figure 4: Distributions of broken segment contour lengths from CGMD and CGSND. Probability density functions of segment contour lengths for the initial network and for segments that rupture during deformation, obtained from (a) CGMD and (b) CGSND.
  • Figure 5: Load-path concentration during deformation (CGSND). Gini coefficient of bond force magnitudes as a function of applied stretch obtained from the CGSND framework. The Gini coefficient quantifies the degree of force localization within the network, with $G=0$ corresponding to homogeneous load sharing and $G\to1$ indicating extreme localization onto a small subset of strands.