Toughness of double network hydrogels: the role of reduced stress propagation

TL;DR

The paper addresses why double network hydrogels can be stiff and tough at once, despite low crosslink density. It presents a mesoscale two-network model that resolves local plastic bond breakage and Eshelby stress redistribution, showing how load sharing between a stiff sacrificial network and a soft matrix network delocalizes stress. The core mechanism is a reduction in the Eshelby stress propagator between sacrificial bonds due to matrix load sharing, which suppresses cascades of bond breakage and brittle macroscopic cracking, producing ductile deformation via diffusely distributed microcracks; the double network inherits $G_D \approx 1.47 G_S = 3.11 G_M$ and $\epsilon^*_D \approx \epsilon^*_M \gg \epsilon^*_S$, with a fracture-fate fraction $f \approx 0.41$. Parameter exploration over $M$, $\mu_m$, and $\lambda_m$ reveals a stiffness–ductility trade-off; in the chosen regime, the double network achieves stiffness close to the sacrificial component while retaining matrix-like toughness, with an acknowledged limitation of a 2D, central-force model and simplified connectivity.

Abstract

Double network hydrogels show remarkable mechanical performance, combining high strength and fracture toughness with sufficient stiffness to bear load, despite containing only a low density of cross-linked polymer molecules in water. We introduce a simple mesoscale model of a double network material, detailed enough to resolve the salient microphysics of local plastic bond breakage, yet simple enough to address macroscopic cracking. Load sharing between the networks results in a delocalisation of stress such that the double network inherits both the stiffness of its stiff-and-brittle sacrificial network and the ductility of its soft-and-ductile matrix network. The underlying mechanism is a reduction in the Eshelby stress propagator between sacrificial bonds, inhibiting the tendency for the plastic failure of one sacrificial bond to propagate stress to neighbouring sacrificial bonds and cause a follow-on cascade of breakages. The mechanism of brittle macroscopic cracking is thereby suppressed, giving instead ductile deformation via diffusely distributed microcracking.

Figures (5)

• Figure 1: Portion of an a) single and b) double network, with sacrificial bonds in red and matrix bonds blue. c) Stretched single network at a location on the stress-strain curve shown by the triangle in Fig. \ref{['fig:stressStrain']}a). A Macroscopic crack has propagated across the network, causing catastrophic material failure. d) Stretched double network at a location on the stress-strain curve shown by the circle in Fig. \ref{['fig:stressStrain']}b). Damage has arisen diffusely via many microcracks in the sacrificial network, which do not propagate macroscopically. This allows the double network to stretch much further than the single network without failing.
• Figure 2: Stress-strain curves. a) Single sacrificial network (red solid line) and single matrix network (blue solid line). b) Composite double network (black solid line), with the contributions of the component sacrificial network (red dashed line) and matrix network (blue dotted line) shown separately. Inset: modulus $G=d\Sigma/d\epsilon$ at small strain. Triangle and circle show the location of the snapshots in Fig. \ref{['fig:networks']}. Parameters: $M=3.0, \mu_{\rm m}=0.2, \lambda_{\rm m}=3.0$. c) and d) show the effect of reducing the matrix network connectivity to $z_{\rm m}=3.5$, as discussed in the penultimate paragraph of the main text.
• Figure 3: Stress propagation to neighbouring sacrificial bonds from the breakage of a sacrificial bond at the origin: (a+c) in a single sacrificial network and (b+d) in a double network with $M=5.0, \mu_{\rm m}=0.2$. Colourscales show the change due to stress propagation of (a+b) the $yy$ component of the Kirkwood stress $\delta\Sigma_{yy}$; and (c+d) the sacrificial bond stress $\delta\sigma_{\rm sb}$, with $\sigma_{\rm sb}\equiv \mu_{\rm s}(l-l_0)/l_0$. Each quantity is averaged over all bonds at each grid square. e+f) show the change in bond stress as a function of distance along $x$ at $y=0$ for e) several $M$ at fixed $\mu_{\rm m}=0.2$, and f) several $\mu_{\rm m}$ at fixed $M=3.0$. Lengths are expressed in units such that $\Delta x=1$ corresponds to the typical sacrificial bond length. The propagator of a single sacrificial network is recovered for $M=0$ or $\mu_{\rm m} = 0$.
• Figure 4: Top: failure strain of double network (black solid lines) as a function of a) the amount of matrix $M$ and b) matrix bond stiffness $\mu_{\rm m}$, shown in each case for matrix bond breakage thresholds $\lambda_{\rm m}=1.0, 2.0, 3.0$ in curves bottom to top, marked by circles, triangles and squares respectively. Failure strains of the corresponding single sacrificial network and single matrix network are shown by red dashed and blue dotted lines respectively. Middle: as in a+b) but now showing failure stresses. Bottom: modulus of the sacrificial network relative to that of the double network. Matrix bond stiffness $\mu_{\rm m}=0.2$ in a,c,e). Amount of matrix $M=3.0$ in b,d,f).
• Figure 5: Stress-strain curves for a double network stretched at a finite rate $\dot{\epsilon}>0$, showing also the approach to the quasistatic limit $\dot{\epsilon}\to 0$ explored in the main text. Parameters otherwise as in Fig. \ref{['fig:stressStrain']}a).