Research Papers

Mechanical Response of Two-Dimensional Polymer Networks: Role of Topology, Rate Dependence, and Damage Accumulation

[+] Author and Article Information
Konik Kothari, Yuhang Hu

Mechanical Science and Engineering,
University of Illinois at Urbana-Champaign,
Champaign, IL 61801

Sahil Gupta

Computer Science,
University of Illinois at Urbana-Champaign,
Champaign, IL 61801

Ahmed Elbanna

Civil and Environmental Engineering,
University of Illinois at Urbana-Champaign
Champaign, IL 61801
e-mail: elbanna2@illinois.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 18, 2017; final manuscript received December 21, 2017; published online January 24, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(3), 031008 (Jan 24, 2018) (11 pages) Paper No: JAM-17-1582; doi: 10.1115/1.4038883 History: Received October 18, 2017; Revised December 21, 2017

The skeleton of many natural and artificial soft materials can be abstracted as networks of fibers/polymers interacting in a nonlinear fashion. Here, we present a numerical model for networks of nonlinear, elastic polymer chains with rate-dependent crosslinkers similar to what is found in gels. The model combines the worm-like chain (WLC) at the polymer level with the transition state theory for crosslinker bond dynamics. We study the damage evolution and the force—displacement response of these networks under uniaxial stretching for different loading rates, network topology, and crosslinking density. Our results suggest a complex nonmonotonic response as the loading rate or the crosslinking density increases. We discuss this in terms of the microscopic deformation mechanisms and suggest a novel framework for increasing toughness and ductility of polymer networks using a bio-inspired sacrificial bonds and hidden length (SBHL) mechanism. This work highlights the role of local network characteristics on macroscopic mechanical observables and opens new pathways for designing tough polymer networks.

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Grahic Jump Location
Fig. 1

Schematic of the polymer chains with dynamic bonds (including sacrificial bonds and crosslinkers)

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Fig. 2

Schematic of the transition state theory. Application of force decreases the required energy for forward reaction and increases the same for the backward reaction, favoring bond breaking.

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Fig. 3

Representative graphs of network topologies with different co-ordination numbers: (a) Z ≈ 4 topology and (b) Z ≈ 7 topology

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Fig. 4

Sample setup for a Z ≈ 7 network

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Fig. 5

The force stretch plot of a representative network. Uncertainty (depicted by inter-quartile range) is shown around a median force value. In the top-right inset, the max. error depicts the mean normalized spread between the maximum and minimum force values of ten realizations of an experiment on a given graph (network). The decrease in maximum mean normalized error is shown as the number of edges in the network increases. Note that similar plots were seen for all network topologies.

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Fig. 6

Effect of pulling velocities for rate-dependent damage. The two insets show the variations in peak force and ductility at different stretch rates indicated in the legend of the main plot. We see that the peak force and ductility follow almost a linear relationship with the logarithm of strain rates.

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Fig. 7

Effect of different regimes (due to inherent timescales) of pulling experiments. Peak force and energy absorbed are plotted with an estimated error based on central limit theorem arguments.

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Fig. 8

Localization of damage at high Cλ˙ is evident from the figures (a) Cλ˙=5×10−3 and (b) Cλ˙=5. The time scale is not linear as indicated by the broken lines on the x-axis. Simulations were run at constant λ˙, and hence, the y-axis may be used as a substitute for time.

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Fig. 9

Effect of the crosslinker concentration on the observed peak force of the network. Increasing concentration is simulated by increasing the number of nodes in the graph. Each side of an error bar indicates the typical maximum one-sided spread in the peak force from our simulations. The crosslinker concentration is measured as the ratio of the number of nodes in the graph to the area of the sample.

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Fig. 10

Comparison of force responses when the available chain length is drawn from a random distribution: (a) uniform distribution [(1−k)*μ,(1+k)*μ] and (b) Gaussian distribution with standard deviation σ = k * μ, k ∈ (0, 1). In all simulations, the weight of the network is kept constant.

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Fig. 11

Effect of coordination number

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Fig. 12

Sacrificial bonds decrease the initial available length of the chains. Hence, they have a stiffer initial response. Energy is absorbed in breaking the sacrificial bonds which leads to increasing toughness.

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Fig. 13

Clockwise from top-left: left-hand side surfaces for Eq. (15) at Δxf = 0.5, 0.05, 0.005 nm

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Fig. 14

Sacrificial bonds causing an increase in stretch and toughness over a bare network. m is the number of sacrificial bonds per chain of the network. The numbers in the plot indicate energy absorbed (area under the curve) in femto-joules (10−15).



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