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DOI: 10.1039/D0SM01063D
(Paper)
Soft Matter, 2020, Advance Article

Justin Tauber,
Aimée R. Kok,
Jasper van der Gucht* and
Simone Dussi

Physical Chemistry and Soft Matter, Wageningen University, Stippeneng 4, 6708 WE, Wageningen, The Netherlands. E-mail: jasper.vandergucht@wur.nl

Received
8th June 2020
, Accepted 1st October 2020

First published on 2nd October 2020

We study the influence of thermal fluctuations on the fracture of elastic networks, via simulations of the uniaxial extension of central-force spring networks with varying rigidity. Studying their failure response, both at the macroscopic and microscopic level, we find that an increase in temperature corresponds to a more homogeneous stress (re)distribution and induces thermally activated failure of springs. As a consequence, the material strength decreases upon increasing temperature, the microscopic damage spreads over a larger area and a more ductile fracture process is observed. These effects are modulated by network rigidity and can therefore be tuned via the network connectivity and the rupture threshold of the springs. Knowledge of the interplay between temperature and rigidity improves our understanding of the fracture of elastic network materials, such as (biological) polymer networks, and can help to refine design principles for tough soft materials.

However, these athermal network models completely ignore thermal fluctuations. While this may be justified for networks composed of very stiff fibers, it is highly questionable for softer networks. For example, the mechanics of flexible polymer networks, such as elastomers and hydrogels, are known to be governed by thermal fluctuations and entropy, while network connectivity is usually not taken into consideration.^{24,25} This raises the question how connectivity and thermal fluctuations interplay for networks consisting of fibers of intermediate stiffness, as found, for example, in many biological materials. Recent Monte Carlo simulations for such networks suggest that in the presence of thermal fluctuations the linear modulus is dependent on both rigidity and temperature. In particular, it is shown that thermal fluctuations can stabilize (sub-isostatic) central-force spring networks at the network level in a similar way to bending interactions.^{26,27} Also, when looking at failure, it is predicted that the average external force required to break a single element or bond is typically reduced in presence of thermal fluctuations.^{28,29} However, experiments on polymer networks^{30} that demonstrate the presence of this thermally activated failure process in network materials also imply that in networks the thermally activated failure is enhanced. These results suggest that, when thermal fluctuations are present, the elastic and failure response can be controlled by temperature, but also by the structure of the network. However, it remains unclear how these two parameters together govern the failure process.

In this paper we explore to what extent network rigidity controls the influence of thermal fluctuations on the failure behaviour of an elastic material. To this end, we study the response of diluted central-force spring networks (see Fig. 1), similar to previous studies.^{12–14,26,27} To introduce thermal fluctuations into these systems we perform Langevin dynamics simulations, which means that the networks are effectively embedded in an implicit solvent. We find that the strength of the networks is dependent on temperature and that the effect of the thermal fluctuations is coupled to the rigidity of the network. The simple structure of the model allows us to highlight the interplay between rigidity and temperature, and to provide insight in the underlying microscopic mechanisms of stress homogenization and diffuse failure.

Simulations are performed using LAMMPS^{31} and nodes follow Langevin dynamics:

(1) |

(2) |

(3) |

In fracture simulations bonds are broken every 100 steps (i.e. 0.1τ) when the connected nodes are separated by a distance more than _{0} + λ_{0}. While the interval chosen for the evaluation of rupture events influences the stress–strain curves somewhat, it does not affect the conclusions of our paper (see ESI,† Fig. S8 and S9). From the measured stress–strain curves we extract several quantities (see Fig. 1). All quantities are averaged over several configurations and expressed in reduced units. The peak stress σ_{p} is defined as the highest measured stress, and the peak strain ε_{p} is its corresponding strain value. The maximum stress drop Δσ_{max} is calculated according to a procedure^{13,32} where we (i) calculate the derivative of the stress–strain curve, (ii) make a list of consecutive data points which have a negative derivative and note the initial and final strain of each interval, (iii) calculate the stress drops by subtracting the stress at the final strain from the stress at the initial strain, (iv) identify the largest stress interval, which corresponds to maximum stress drop Δσ_{max}. For the stress distribution analysis, we make instantaneous histograms of the bond lengths _{i} during the simulation at every percent strain. Based on these histograms, we calculate the excess kurtosis

(4) |

In Fig. 2(a) we plot the linear modulus E of the network as a function of the network connectivity factor p for several reduced temperatures T*. The linear modulus E describes the resistance of a network to deformation and we observe that networks below the isostatic point of mechanical stability^{16} (i.e. networks below p_{iso} ≈ 0.66) display a finite linear modulus E, which would be absent for athermal systems (in the limit of T = 0). This finite E is an effect of entropic stiffness, a temperature-dependent phenomenon. Please note that this entropic stiffness is a network effect and is not the same as the entropic elasticity arising from individual polymer chains. As reported in literature,^{26,27} the scaling of the linear modulus with temperature E ∝ T^{α} depends on both connectivity and temperature itself. By plotting E as a function of T* we extract the scaling exponent α from a power-law fit for three different values of p, as shown in Fig. 2(b). For a sub-isostatic network with a connectivity parameter p = 0.55 the linear modulus scales with α = 0.84, which roughly corresponds to the dependence found in the anomalous regime as defined in ref. 26, where a shear deformation was instead considered. It was argued that the disordered network structure causes this sub-linear dependence. Whilst there is a clear dependence of the linear modulus on the temperature below the isostatic point, the curves for the different temperatures start to converge when approaching a structurally rigid network (Fig. 2(a)). Accordingly, stiff networks display temperature insensitivity (α ≈ 0), as can be seen for a network with p = 0.70 in Fig. 2(b). As T* increases, however, the network connectivity becomes less important as the energetic contribution arising from the structural rigidity becomes negligible compared to the entropic elasticity. This is noticeable in Fig. 2(a) where the curve for T* = 10^{−2} is roughly flat for the entire p-range, and also in Fig. 2(b) where for p = 0.70, E increases for T* > 10^{−3}. As predicted in ref. 26, we also find a different scaling for networks close to the isostatic point, see, e.g., the curve for p = 0.62 in Fig. 2(b). Although the exponent α is slightly different from the findings of ref. 26 (where shear deformation and different simulation methods were employed), we were also able to obtain critical rescaling as shown in Fig. 2(c). We can conclude that there are different regimes of dependence for the linear modulus on the temperature based on both rigidity and temperature.

Fig. 2 Characterization of the linear elastic response for diluted triangular networks of fixed system size L = 128. (a) Young's modulus E as a function of the connectivity parameter p for different temperatures T*. (b) Temperature dependence of E for networks below, around, and above the isostatic point (value of p indicated in the legend). The dashed lines indicate the power-law fit T^{α}. (c) Rescaling of Young's modulus according to ref. 26 with a = 1.4, b = 2.8, and z_{c} = 3.78. (d) The non-affinity parameter Γ_{mech} at 1.5% strain as a function of p for different temperatures, same legend as (a). Every data point is based on simulations of at least 10 independent configurations. |

Furthermore, we find similar rigidity-dependent behaviour of the thermal fluctuations in the non-affinity parameter Γ_{mech} (eqn (2)), reported in Fig. 2(d) as a function of p for different T*. The non-affinity of the network describes how much the time-averaged local deformation differs from the global (externally imposed) deformation. At low T* we find a peak in non-affine deformation around the isostatic point (p ≈ 0.66). This peak arises from the tendency of the spring network to minimize internal stress upon deformation. If the spring network is far below the isostatic point, the stress can be reduced significantly by a small amount of non-affine rearrangements while at the isostatic point many non-affine rearrangements are required. At the isostatic point, an increase in T* decreases Γ_{mech}, which suggests that thermal fluctuations act as a stabilizing field, similar to the bending rigidity in fiber networks.^{10} However, we note that the effect of thermal fluctuations is always present, even without external deformations, leading to structural rearrangements in the rest state (see ESI,† Fig. S1). Above the isostatic point, we observe that the non-affinity converges for most values of T* (see ESI,† Fig. S1 for details), which indicates that above the isostatic point the network rigidity dominates the non-affine response. Only if T* > 10^{−4}, we see that thermal fluctuations affect the non-affine response, increasing Γ_{mech}. This is in contrast to fiber networks, where the non-affinity decreases with an increase in bending rigidity. We hypothesize that this difference occurs because in the case of fiber networks the fibers have a preference to remain straight to minimize stress caused by fiber bending, while in the case of thermal fluctuations an affine displacement of the nodes will not minimize the stress caused by the randomly oriented thermal fluctuations. Below the isostatic point, the effect of thermal fluctuations on the non-affine response is significant. We observe that at T* = 10^{−8} the non-affine response is the smallest and that a moderate increase in T* up to T* = 10^{−6} leads to an increase in the non-affine response, corresponding to what is observed for fiber networks. However, we also observe a decrease in Γ_{mech} if the temperature is increased beyond T* = 10^{−6}, which is not observed in fiber networks. It is unclear if this deviation is caused by a fundamental difference between thermal fluctuations and bending rigidity as a stabilizing field or that longer equilibration times are required to gain quantitative information on the non-affine response in this regime (see ESI,† Fig. S1 and S2 for details).

In general, we find that at a global level thermal fluctuations act as a stabilizing field in central-force spring networks, dampening rigidity-dependent behaviour around the isostatic point. However, our results suggest that the random nature of the thermal fluctuations causes significant differences in the local response with respect to stabilizing fields in athermal systems such as bending.

Signatures of strain-stiffening can also be observed in the non-affine response of the network. In Fig. 3(b), we report the instantaneous non-affinity parameter Γ, that intrinsically includes both the non-affine contributions from instantaneous thermal fluctuations and structural rearrangements. As a result, high non-affinity values can be observed at low strains, where the size of the non-affine thermal fluctuations is large compared to the applied strain. At low temperatures, a peak can be observed in the non-affine response around the onset strain. At high temperatures, this peak is overshadowed by the non-affine thermal fluctuations. At high strain, the network elasticity is controlled by stretching of the bonds and the network response becomes increasingly affine for most temperatures. Only at T* = 10^{−2}, the non-affine fluctuations are still visible.

To disentangle the effects of temperature and network connectivity, we normalize the stress–strain curve with the ones obtained in the athermal energy-dominated limit. In particular, we plot the stress ratio σ/σ_{ath} in Fig. 3(c), where we used the data obtained at T* = 10^{−8} for σ_{ath}. At this temperature the network behaves according to a network in the athermal limit, but still a small amount of stress is observed i.e. the stress is not zero at small strains even below the isostatic point. A ratio of σ/σ_{ath} ≈ 1 implies that the mechanical behaviour is basically insensitive to variations in temperature. As can be seen in Fig. 3(c) for a network with p = 0.56, there is a regime of strain in which the stress ratio depends on T* and decreases upon stretching the network more and more. At increasing temperature, this stress ratio is both higher at the start and approaches temperature insensitivity at a higher strain. The start of the decrease in stress ratio for all temperatures occurs at approximately the same strain value, corresponding to the onset of strain-stiffening. This transition could therefore be interpreted as a transition between a regime dominated by thermal fluctuations to a regime dominated by bond stretching. This is analogous to the bending-to-stretching transition observed in fiber networks.^{11,33,37} We summarize these observations in a mechanical phase diagram sketched in Fig. 3(d), where we can distinguish two regimes: a mechanically-dominated regime (blue) where structural rigidity overpowers the effect of thermal fluctuations and a temperature-controlled regime (orange) where thermal fluctuations play a more important role in the elastic behaviour. The transition between these regimes depends on the reduced temperature T* (and therefore both on the actual temperature T and the bond stiffness μ), the connectivity parameter p and the strain ε. This transition is in general very gradual as can be seen in the two cross-sections of the mechanical phase diagram reported in Fig. 3(e and f) where we show the stress ratio obtained by some of our simulations. When T* is fixed (Fig. 3(e)) and we increase p, we observe a steep decrease in the strain associated to the thermal-stretching transition. Above the isostatic point, the mechanics of the rigid networks is barely affected by thermal fluctuations at this temperature. In Fig. 3(f), we observe that with increasing temperature the stress ratio increases but the strain characterizing the transition seems to reach a limiting value. This limiting value is a result of the onset of strain-stiffening, which is independent of temperature and corresponds to the transition to the elastic regime.

In summary, we identified a rigidity-dependent transition between two regimes where thermal fluctuations are or are not important. In the following sections, we will investigate whether this underlying transition also influences the fracture of these elastic networks.

We first focus on macroscopic descriptors and characterize the stress–strain curves obtained from fracture simulations. In Fig. 4(a and b), we show the response of two representative networks with a small rupture threshold λ = 0.03 and different connectivities at several temperatures T*. For the network with p = 0.65 ≃ p_{iso} (panel a), a clear decrease in peak stress σ_{p} for increasing T* is observed, while a variation in the peak strain ε_{p} is less evident as the fracture becomes more ductile and the decrease in stress after the peak is less pronounced. For the very rigid network (p = 0.90, panel b), the decrease in both σ_{p} and ε_{p} is clearly observed. Similarly, the fracture becomes more ductile for higher T*, even though a clear stress drop is still recognizable at the highest temperature simulated. In both cases, the networks become weaker with increasing T*. Furthermore, when approaching the athermal limit (T* → 0) the peak stress becomes less sensitive to variation in temperature. In Fig. 4(c), we show the temperature dependence of σ_{p} for several connectivities with λ = 0.03. The common trend is little variation at low temperatures, almost a plateau that is indicative of approaching the athermal limit, followed by a decrease when temperature is increased, with σ_{p} eventually dropping to zero when a temperature of T* ≃ 10^{−4} is reached. On the one hand, for low T* the peak stress is evidently controlled by the network rigidity, as previously investigated in the athermal limit.^{12,13} On the other hand, when the thermal energy is of the order of the network structure is irrelevant, as springs spontaneously break and the system shows melting behaviour. We will later describe the melting point using the reduced quantity . In between these limits, there is a broad cross-over regime. To better assess the role of rigidity in this intermediate regime, we normalize σ_{p} by its value in the athermal limit σ_{p,ath} and plot this ratio in Fig. 4(d). The transition between the athermal limit, where σ_{p}/σ_{p,ath} = 1, and the melting limit, where such a ratio goes to zero, depends on a subtle coupling between connectivity and temperature itself. Far below (p < 0.60) and far above the isostatic point (p > 0.80) the connectivity plays a small role since at every temperature the curve exhibits two plateaus (at small and large p). However, around the isostatic point rigidity and thermal fluctuations are coupled, since at all the intermediate temperatures we can observe a sharp increase in σ_{p}/σ_{p,ath} upon increasing p, connecting the two limiting plateaus. On passing, we note that the plateau for small p is lower, suggesting that temperature starts to affect failure of very diluted networks earlier than for networks with large p. Furthermore, we speculate that the complex temperature-dependence around the isostatic point arises from locally floppy regions that are rigidified by thermal fluctuations (whose magnitude depends on temperature itself) and are therefore able to sustain and concentrate stress, and break. Since the isostatic point marks the onset of mechanical stability, such an effect is largest for networks close to it.

A direct inspection of the simulation snapshots (Fig. 6) suggests that bonds that break up to the peak strain ε_{p} (red bonds) are dispersed more homogeneously throughout the sample at a higher temperature. The snapshots also reveal a big difference in the response to temperature between networks around (p = 0.65) and far above the isostatic point (p = 0.90). Around the isostatic point, the damage up to ε_{p} is already diffusive in the athermal limit, and its delocalization is enhanced when the temperature is increased. In contrast, the failure response far above the isostatic point shows a clear transition from crack nucleation in the athermal regime to a more diffuse failure response close to the melting point. However, the post peak response at p = 0.90 is clearly still dominated by the propagation of cracks. Nevertheless, at high temperatures we observe the development of multiple cracks, sometimes even not perpendicular to the deformation direction, and evidence of crack merging.

In summary, we show that an increase in temperature leads to an increase in diffuse failure, implying suppression of stress concentration before the peak stress. These observations suggest that thermal fluctuations are responsible for two apparently contrasting effects: on the one hand, they create “instantaneous defects” resulting in more regions with broken bonds, that reduce material strength; on the other hand, the fluctuations allow to delocalize stress away from such defects, delaying the propagation of large cracks. As a result, the damage pattern is diffuse throughout the system.

Therefore, we examine how thermal fluctuations affect the macroscopic fracture descriptors for different system sizes, focusing on the maximum stress drop Δσ_{max} that quantifies fracture abruptness. In Fig. 7(a–d), we plot the size-scaling of the maximum stress drop Δσ_{max} (closed symbols) together with the peak stress σ_{p} (open symbols) for four combinations of p and λ at different temperatures. In all cases, we observe a monotonic decrease of σ_{p} as a function of the system size. These trends can be fitted by a power law σ_{p} = (L/α)^{−β} + σ^{∞}_{p}, where σ^{∞}_{p} is the failure stress in the thermodynamic limit (infinite system size), β the size scaling exponent and α a fitting constant. Values for β are comparable to values found in literature.^{13,22} It is interesting to note that we find a finite value for σ^{∞}_{p}, which is different from many other studies on network failure.^{38} This is because in our work all elements have the same strength and therefore a finite amount of stress is required at all network sizes to start the failure process. In contrast, some studies reported a vanishing σ^{∞}_{p} since they employed a distribution in element strength extending to zero.^{38} In Fig. 7(e), we plot σ^{∞}_{p} normalized by its athermal value σ^{∞}_{p,ath} as a function of T* normalized by (see ESI,† Fig. S6 for the other fitting parameters). The observed trend underlying a transition from low T* to melting is consistent with the data at fixed system size and fixed rupture threshold λ presented earlier in Fig. 4. Here, we can also appreciate the effect of varying λ in the intermediate temperature regime. For example, the largest λ = 0.30 (downward triangles, networks with p = 0.90) shows a steeper decrease in the normalized fracture stress, suggesting that thermal effects kick in at higher temperatures for these very rigid networks.

Finally, we focus on the maximum stress drop Δσ_{max}. From Fig. 7(a–d), we observe that a non-monotonic trend is present in basically all cases, consistent with our previous results in the athermal limit.^{13} We speculated that the initial decrease, implying a more ductile fracture upon increasing system size, is associated to the rupture and reformation of locally stressed regions (often consisting of aligned springs, and sometimes called force chains^{12,39–41}). However, upon increasing the system size Δσ_{max} starts to increase, suggesting that stress concentration around defects is present in the system, since it fractures in a more abrupt way. At even larger L, Δσ_{max} decreases again, now following the same trend for the peak stress σ_{p} that sets the upper bound to the possible stress drop. In Fig. 7(c) the entire trend is visible for the system sizes explored in this work, whereas in the other panels only parts of it are captured. Importantly, for all systems, the trend depends on temperature. In particular, in Fig. 7(f) we quantify the effect of temperature by plotting the system size L_{min} corresponding to the minimum Δσ_{max} as a function of T*. We observe that thermal fluctuations increase the value of L_{min}, which can be interpreted as a lengthscale for stress concentration. The role of temperature seems particularly relevant at low connectivity, where the stress is already very delocalized in the athermal limit.

In summary, we find that also in the thermodynamic limit there is a crossover from an athermal regime to a melting regime where the failure behaviour is determined by both rigidity and thermal fluctuations. Moreover, thanks to the analysis of Δσ_{max}, we find evidence that temperature increases the region over which stress is delocalized.

We note that at a first glance central-force spring networks subjected to thermal fluctuations behave like athermal networks in a stabilizing field. However, the instantaneous nature of the thermal fluctuations introduces important differences. It is striking that, without any applied deformation, the thermal fluctuations induce structural rearrangements of the average network structure (see ESI,† Fig. S1). Furthermore, providing enough time, the thermal fluctuations allow the failure of bonds even if they are not intrinsically under tension (activated failure), leading to diffuse damage. A final consequence of introducing thermal fluctuations is that time becomes an important parameter. In our simulations the system was deformed at a constant strain rate, i.e. it was driven at a given speed. If the driving speed is too low, the system will melt due to the process of activated failure. If the driving speed is too high, the system has no time for stress relaxation as it is held back by the viscous surroundings. Therefore, the failure response of an elastic network is generally determined by the coupling between the driving speed, viscosity, rigidity and thermal fluctuations. Our study was focused on a regime in which driving and viscosity effects were small (see ESI† for discussion).

This work provides new insight into the relation between the static network structure, thermal fluctuations and the failure response of central-force spring networks. Above all, it shows that rigidity remains a controlling parameter in the failure response of spring networks in the presence of thermal fluctuations, even close to the melting temperature. This suggests that the failure response of thermal networks in experiment, such as semiflexible polymer networks, could be rigidity-dependent as well. The ratio between the energy of the thermal fluctuations, E_{therm} = k_{B}T, and the energy required to break an elastic element, E_{break}, emerges as a relevant parameter to classify the failure regime of a particular network (either the athermal regime, the cross-over regime, or the melting regime) and could thus be a relevant parameter to classify the failure response of experimental systems. For example, a rough estimate of this ratio for experimental systems shows us that for a collagen network E_{therm}/E_{break} ≈ 1 × 10^{−6}, which corresponds to the athermal limit, while the values for a semiflexible network like actin (E_{therm}/E_{break} ≈ 8 × 10^{−3}) and a flexible polymer network (E_{therm}/E_{break} ≈ 7 × 10^{−3}) are significantly higher (see ESI† for details). These examples show that it is not just the temperature, but also the type of building block that determines the relevant failure regime. Furthermore, the ratio also makes clear that the temperature sensitivity is not only dependent on the element stiffness μ, but also on extensibility λ. Therefore, it is a possibility that a network with a temperature dependent elastic response at the network level (determined by T*), might not be sensitive to thermal fluctuations in the failure regime (determined by ). Our model predicts that networks with weak crosslinkers, i.e. small λ, are most likely to have a failure response as observed in the cross-over regime.

We hope that our simulation results will stimulate further experimental work aimed at mapping out the roles that rigidity and thermal fluctuations play in governing mechanical failure of elastic networks.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm01063d |

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