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Rethinking failure in polymer networks: a probabilistic view on progressive damage

Noy Cohen, Nikolaos Bouklas, Chung-Yuen Hui

Abstract

The mechanics of single-chain stretching and rupture are central to understanding the resilience of biological polymers and designing strong and tough soft materials such as double-network gels and multi-network elastomers. In this work, we develop a statistical mechanics based model that enables one to determine the distribution of forces along the chain segments. By combining the force distribution with a tilted bond potential that captures the stretch energy stored in these bonds, we calculate the corresponding activation energy required for bond dissociation. This allows us to determine the probability of bond (and consequently chain) failure. The proposed approach is simple, direct, and readily adaptable for constructing higher-level coarse-grained descriptions of damage and fracture in polymer networks. We demonstrated this by applying the theory to three problems of practical interest: (1) toughening via sacrificial bond rupture in polymer chains, (2) toughening of double network hydrogels, and (3) incorporation of the local chain model into a 3-dimensional constitutive relation that captures damage in elastomers. The latter was implemented through the micro-sphere framework, which accounts for different chain orientations, as well as the computationally inexpensive eight chain model. The findings from this work provide a physically-based model to quantify the stretching and failure of a single chain and pave the way to the integration of local damage models into 3-dimensional networks.

Rethinking failure in polymer networks: a probabilistic view on progressive damage

Abstract

The mechanics of single-chain stretching and rupture are central to understanding the resilience of biological polymers and designing strong and tough soft materials such as double-network gels and multi-network elastomers. In this work, we develop a statistical mechanics based model that enables one to determine the distribution of forces along the chain segments. By combining the force distribution with a tilted bond potential that captures the stretch energy stored in these bonds, we calculate the corresponding activation energy required for bond dissociation. This allows us to determine the probability of bond (and consequently chain) failure. The proposed approach is simple, direct, and readily adaptable for constructing higher-level coarse-grained descriptions of damage and fracture in polymer networks. We demonstrated this by applying the theory to three problems of practical interest: (1) toughening via sacrificial bond rupture in polymer chains, (2) toughening of double network hydrogels, and (3) incorporation of the local chain model into a 3-dimensional constitutive relation that captures damage in elastomers. The latter was implemented through the micro-sphere framework, which accounts for different chain orientations, as well as the computationally inexpensive eight chain model. The findings from this work provide a physically-based model to quantify the stretching and failure of a single chain and pave the way to the integration of local damage models into 3-dimensional networks.
Paper Structure (16 sections, 40 equations, 11 figures, 1 table)

This paper contains 16 sections, 40 equations, 11 figures, 1 table.

Table of Contents

  1. Keywords:
  2. Introduction
  3. The free energy of a chain
  4. Procedure A
  5. Procedure B
  6. The energy of a repeat unit along the chain
  7. Mechanical response of single chains
  8. The classical chain
  9. A polymer chain with sacrificial bonds
  10. Double network polymers
  11. Implementation to 3-dimensional networks
  12. An isotropic network of chains
  13. The eight-chain model
  14. Conclusions
  15. Derivation of the force on a chain
  16. ...and 1 more sections

Figures (11)

  • Figure 1: The potential energy landscape $u\left(l\right)-f_{b}\left(l-l_{eq}\right)$ as a function of the bond length $l$.
  • Figure 2: (a) The normalized equilibrium potential $u_{eq}/mD_{e}$ and (b) the normalized activation energy $E_{a}/mD_{e}$ as a function of the normalized force a bond experiences $\tau=f_{b}/f_{m}$.
  • Figure 3: (a) The normalized force on the chain $f/f_{m}$ as a function of the normalized end-to-end distance $\rho=r/nl_{0}$. (b) The probability of rupture $p_{c}$ as a function of the normalized chain force $f/f_{m}$.
  • Figure 4: Several sacrificial bonds on single chains that are connected to substrates via bond C. (a) A molecule with four fixed internal loops through the sacrificial bonds A and B. Once a sufficient force is applied, a sacrificial bond breaks to reveal the corresponding hidden length from the free loop. (b) Two chain segments that are interconnected through a sequence of four sacrificial bonds (of types A and B). The application of a sufficiently large force gradually breaks the bonds to extend the length of the chain. Once all bonds dissociate, the chains break.
  • Figure 5: The probability of bond dissociation $p_{b}^{\left(\bullet\right)}$ as a function of the force $f$ for the bonds $\bullet=A$, $\bullet=B$, and $\bullet=C$.
  • ...and 6 more figures