Unique stick–slip crack dynamics of double-network hydrogels under pure-shear loading

Significance

Tough soft materials exhibit fundamentally distinct fracture characteristics compared to brittle materials, making them an intriguing area of research. Understanding the unique fracture dynamics of tough soft materials is crucial for their applications. Although double-network gels are a specific class of tough soft materials, the concept of a damage zone in soft materials and the differences in kinetics between the fast crack growth of brittle fracture and the controlled growth of the damage zone, influenced by external energy input, are broadly applicable. This work provides insights into how local dissipative processes near the crack tip prevent brittle fracture, guiding the future design of tough soft materials.

Abstract

In this work, we have found that a prenotched double-network (DN) hydrogel, when subjected to tensile loading in a pure-shear geometry, exhibits intriguing stick–slip crack dynamics. These dynamics synchronize with the oscillation of the damage (yielding) zone at the crack tip. Through manipulation of the loading rate and the predamage level of the brittle network in DN gels, we have clarified that this phenomenon stems from the significant amount of energy dissipation required to form the damage zone at the crack tip, as well as a kinetic contrast between the rapid crack extension through the yielding zone (slip process) and the slow formation of a new yielding zone controlled by the external loading rate (stick process).

Soft materials play a pivotal role in a wide array of applications across classical and emerging engineering domains, including the rubber industry, bioengineering, drug delivery, wearable electronics, and soft robotics, among others (1–14). These applications often impose stringent demands on soft materials, necessitating characteristics such as high stretchability, flexibility, mechanical strength, and toughness to withstand repeated stretching, bending, and twisting during operational conditions. The operational lifespan of these materials hinges significantly on their ability to withstand fracture and resist crack propagation under both monotonic and cyclic loading conditions (13, 15–28). Consequently, comprehending the fracture behaviors of soft materials is of paramount importance to engineers and scientists, both industrially and fundamentally (29–31). However, research on this subject remains relatively scarce compared to hard materials such as metals and ceramics.

In recent decades, much research has been concentrated on designing soft materials with maximal resistance to initial crack initiation, leveraging fracture toughness as a crucial design parameter. However, it is important to note that beyond crack initiation, the subsequent dynamics and morphology of the crack display unique characteristics, contributing to a deeper understanding of fracture mechanisms. For example, Greensmith observed distinctive crack dynamics in carbon-black-filled rubbers, where under monotonic loading, propagated cracks veered from the initial crack direction by curving or bifurcating into two cracks, influenced by the presence of rigid fillers (32). Lee and Phar recently observed a fascinating form of crack dynamics wherein cracks propagated perpendicular to the initial precut in a silicone elastomer (33). They attributed this sideways and stable crack propagation to microstructural anisotropy ahead of the crack tip induced by strain-induced crystallization (33). These studies strongly suggest that the unusual crack dynamics under monotonic loading are closely linked to dissipative processes (or fracture processes) in the near-tip region. However, the mechanisms by which local dissipative processes influence crack dynamics and their relationship to the material’s microstructure remain poorly understood.

When subjected to mode-I (tensile) loading in a pure-shear geometry, a prenotched brittle soft material undergoes rapid and continuous crack propagation once the energy release rate exceeds a critical value (34–36). Such fracture of a brittle material, known as brittle fracture or dynamic fracture, can be explained by the classical fracture theory, where the crack initiates once the supplied energy release rate G exceeds the crack resistance Γ. In a pure-shear geometry, since the uncrack region ahead of the crack tip experiences deformation, the supplied energy release rate from the uncrack region can continuously propel crack propagation without the need of further energy input from external sources. In contrast, a tough soft material usually requires additional loading or energy input to sustain continuous dynamic fracture. The mechanism for the suppression of brittle fracture in tough soft materials remains incompletely understood.

Double-network (DN) gels belong to a category of tough soft materials composed of two interpenetrated elastic networks. The exceptional strength and toughness of DN gels stem from the highly contrasting properties of the two networks: The densely cross-linked first network is stiff and brittle, whereas the sparsely cross-linked second network is soft and stretchable (37). Investigations into the origin of the high fracture toughness of DN gels have revealed that sacrificial covalent bond scission within the brittle network serves as the dissipative mechanism. This dissipation notably occurs at the crack tip, facilitated by the formation of a large internal damage zone (37–48).

In this study, we uncovered a significant finding concerning prenotched DN hydrogels under mode-I (tensile) loading in a pure-shear geometry. Our observations revealed fascinating stick–slip crack dynamics occurring during continuous loading. The stick–slip crack growth observed in DN hydrogels presents an opportunity to investigate mechanisms for effectively suppressing dynamic fracture in tough soft materials. For this purpose, we conducted retardation observations to track changes of the damage zone (dissipation zone) during the stick–slip processes, and we found that the stick–slip behavior synchronizes with the oscillation of the damage (yielding) zone size at the crack tip. We also conducted crack tests for a wide range of loading velocities, and we observed that the stick time controlling the formation of the damage zone is inversely proportional to the loading velocity. From these findings, we concluded that the stick–slip phenomenon is due to a significant kinetic disparity between fast fracture through the yielding zone (slip process) and the slow formation of a new yielding zone (which suppresses dynamic fracture during stick). The stick process underscores the influence of local dissipative processes ahead of the crack tip on the crack dynamics of soft materials. To further explore the role of the brittle first network as the dissipative component in crack dynamics, we investigated the crack dynamics of DN gels subjected to various extents of internal damage prior to crack testing. We noted that the stick–slip phenomenon vanished and transitioned to continuous crack propagation, similar to conventional single network (SN) materials, when the brittle first network experienced significant predamage. This implies that the ruptured brittle first network plays a diminished role as the dissipative component in suppressing the dynamic fracture.

While the molecular mechanism of energy dissipation by brittle network rupture at the crack tip is unique to DN gels, the conceptual framework for crack arrest presented in this study provides insights into how the dissipative process around the crack tip suppresses dynamic fracture in tough soft materials.

Results and Discussion

Stick–Slip Crack Dynamics.

We adopted a typical DN gel consisting of poly(2-acrylamido-2-methylpropane sulfonic acid sodium salt) (PNaAMPS) as the brittle first network and polyacrylamide (PAAm) as the soft second network (SI Appendix) (37, 46). The DN gel was synthesized at a composition to exhibit yielding and strain hardening behaviors under tensile deformation (SI Appendix, Fig. S1). The fracture tests were performed in a pure-shear geometry with the designed configurations (sample width L₀ = 50 mm, height H₀ ~ 10 mm, and initial crack length c₀ = 10 mm, Fig. 1A). Tensile loading was applied at a controlled velocity (v). To visualize the internal damage and record the crack growth during loading, we performed real-time imaging of birefringence adopting the experimental setup reported in previous studies (SI Appendix, Fig. S2) (46, 49). This experimental setup allows for capturing both crack growth behavior and the damage zone, also known as the yielding or dissipative zone. Within this zone, the brittle first network undergoes extensive breakage, and the soft second network strands become highly oriented along the tensile direction (46). Briefly, the samples were positioned between two crossed circular polarized films, with these films placed between a white lamp and a video camera (46, 49). The determination of the damage zone is based on birefringence images reflecting the pronounced orientation of the second network strands in the damage zone (SI Appendix, Fig. S2) (46, 49).

Fig. 1.

Stick–slip crack dynamics of DN gels in the mode-I pure shear fracture test. (A) Illustration (i) and sample image (ii) of the experimental configuration for the fracture test (mode-I loading). Sample width L₀ = 50 mm, sample height H₀ ~ 10 mm, initial crack length c₀ = 10 mm, and loading velocity v = 50 mm/min. (B) Birefringence images at the start of the 1st slip, 1st stick, and 2nd slip during crack propagation. The hollow color circles on the images correspond to the positions in (C). The dashed lines indicate the crack tip positions. (C) Nominal tensile stress σ, accumulative crack extension Δc, and yielding zone area S against the tensile stretch ratio λ ($\lambda =1+vt/H_0$). The enlarged figures on the Right show the 1st slip (i) and stick (i) process and the 2nd slip (ii) and stick (ii) process. The hollow circles in different colors indicate the start and end points of these stick–slip processes. Δc and S are at deformed configurations.

Under a constant loading velocity (v = 50 mm/min, corresponding to a stretch rate of 0.083 s⁻¹), the DN gel exhibited distinct stick–slip crack dynamics. This phenomenon is evident in the in situ recording videos (Movie S1) and the representative birefringence images (Fig. 1B). Specifically, crack growth (slip) and crack arrest (stick) appeared alternatively above the critical stretch ratio, ${\lambda }_c$. The crack velocity during slip was remarkably rapid, completing a single crack growth within one frame of the movie captured by an ordinary camera (1/24 s), whereas crack arrest typically lasted for seconds. The area exhibiting strong birefringence at the crack tip, indicative of the extent of the local yielding zone, decreased during slip and gradually increased during stick.

To analyze crack dynamics, we plotted the evolution of nominal stress (σ), crack extension length (Δc), and birefringence area (S) at the crack tip as a function of stretch ratio (λ) in Fig. 1C. The values of Δc and S, representing the quantities in deformed configuration, were determined from in situ recording videos by analyzing images with ImageJ Software. To determine S, the threshold gray value was set to establish the boundary of the birefringent area ahead of the crack tip, and the pixel numbers within the strongly birefringent area were counted (see SI Appendix for details). Representative birefringence images at onsets of crack propagations (slips) and arrests (sticks) shown in Fig. 1B corresponded to different colored dots in the stress–stretch ratio curves of Fig. 1C. Fig. 1C clearly demonstrates synchronization among stress, yielding zone area, and stick–slip crack propagation. During slip, there is an abrupt reduction in stress, accompanied by a rapid crack extension (with a dynamic crack propagation speed in the order of 10⁰ m/s, see later section) and a notable decrease in the yielding zone area. In contrast, during the stick phase, both stress and yielding zone area gradually increase with additional loading. It is worth noting that during stick phases, there are instances where an ultraslow crack extension occurs (with a propagation speed in the order of 10⁻⁴ m/s). The amplitudes of each single slip or stick are quite random (Fig. 1B).

The stick–slip crack dynamics observed during measurements in water closely resembled those observed in air (SI Appendix, Fig. S3). This similarity suggests that the distinctive crack dynamics of DN gels is not due to poroelasticity (50–52). Conversely, the PAAm SN gel (a typical example of brittle soft materials) exhibited continuous dynamic fracture without any sticking behaviors (SI Appendix, Fig. S4). These observations indicate that the stick–slip crack dynamics of the DN gel are associated with the internal fracture of the brittle first network.

Physical Origin of Crack Arrest.

In the pure-shear geometry, for a fixed applied stretch ratio applied at the grips, the energy release rate to drive crack growth is a constant independent of the amount of crack extension. Therefore, the crucial question is why the crack arrests, or why dynamics fracture is suppressed after the crack has been propagating over a certain distance. To answer this question, we used a polarized high-speed camera operating at a speed of 500 frames per second to capture the retardation images during the fracture process. This allowed us to capture the rapid slip process and observe the growth of the yielding zone during the subsequent stick process. Fig. 2 (i, ii, iii) show, respectively, the retardation images of DN gel at three representative points: start of the 1st slip (i), end of the 1st slip (ii), and start of the 2nd slip (iii). The duration of 1st slip, denoted as Δt_slip,1 (=0.002 s), is much shorter than that of the first stick, denoted as Δt_stick,1 (=2.566 s). From the crack extension length (Δc₁ = 3.7 mm) and the duration (Δt_slip,1 = 0.002 s) of the 1st slip, the crack propagation speed v_slip is estimated as 1.85 m/s. This value closely aligns with the shear wave speed (c_s = 7.6 m/s) of the DN gel (47). This result suggests that the slip process corresponds to dynamic fracture, characteristic of brittle fracture observed in typical SN gels (47, 48). The prominent retardation area observed ahead of the crack tip corresponds to the yielding zone as depicted in Fig. 1B (47). At the onset of the 1st slip, the retardation zone appears large and strong, with a maximum retardation of R_max = 122 nm. This indicates a large yielding zone where the first network is severely damaged and the 2nd network strands are highly oriented along the tensile direction [Fig. 2 (i)]. At the end of the 1st slip (onset of 1st stick), only a weak and small retardation zone with R_max = 77 nm is visible at the crack tip [Fig. 2 (ii)]. This indicates that the crack arrests when it encounters a weak damage zone. During the subsequent stick process, the weak retardation zone gradually grows until the retardation reaches R_max = 122 nm again, beyond which the 2nd slip starts [Fig. 2 (iii)].

Fig. 2.

Retardation images of DN gel during the stick–slip process. Fast slip process and damage zone growth of DN under a loading velocity v = 50 mm/min captured by polarized high-speed camera. Snapshots of retardation images at the start of the 1st slip (i), at the end of the 1st slip (ii), and at the start of the 2nd slip (iii). The color bar represents the retardation R range. Crack velocity is estimated as 1.85 m/s from the slip time Δt_slip,1 (~0.002 s) and crack extension length (Δc₁ = 3.7 mm). The stick time Δt_stick,1 is ~2.566 s. The extension of the yielding zone width along the tensile direction ($\mathrm{\Delta }y$) during the stick process is denoted as $\mathrm{\Delta }y=y_2-y_1$, where $y_2$ and $y_1$ represent the width of the yielding zone at the end and start of the stick process, respectively.

The above experimental observations indicate that the slip phase corresponds to rapid crack growth through the yielding zone by dynamic fracture, while the stick phase (crack arrest) occurs because a new damage zone that can enable crack growth has not yet formed in the crack vicinity. The question that arises is, why the formation of a new damage zone is delayed? Is it due to the viscoelasticity of the networks in the DN gel, possibly coming into play at the crack tip during rapid crack growth, or is it due to the requirement of additional external energy supply for inducing internal damage to the first network at the crack tip?

The damage zone develops through a continuous process involving deformation and load transfer. This process includes loading and breakage of the first network, transfer of load to the second network, and deformation and strain hardening of the second network, subsequently causing further breakage of the first network. This process repeats until the tension on the crack tip becomes sufficiently high to activate the breakage of second network strands at the crack tip. The larger the damage zone, the greater the time and energy flow required for the process (46). If viscoelasticity is important, then crack propagation is anticipated to accelerate after initiation, since less energy is needed for the formation of a smaller damage zone during dynamic fracture (47). Conversely, if energy supply is the determining factor, then the formation of a new yielding zone is governed by external loading, meaning that the stick time would be governed by the loading rate.

To verify this hypothesis, crack tests were conducted on DN gels with varying loading velocities (v) (Fig. 3A). As the loading velocity increases from 5 to 500 mm/min, the stress (σ)–stretch ratio (λ) curves depicted in Fig. 3A exhibit several notable systematic changes. First, there is a systematic increase in stress with the loading velocity in the regime before the 1st slip. Second, the critical stretch ratio for the first slip (λ_c,1) decreases, as depicted in Fig. 3B. Third, the stress–stretch ratio curves become smoother in the stick–slip regime, indicating a less abrupt reduction in stress during slips and a smaller increment in the stretch ratio between two consecutive slips (Δλ_c) (SI Appendix, Fig. S6A).

Fig. 3.

Effect of loading velocity on the stick–slip crack dynamics. (A) Typical stress–stretch ratio curves of DN gels subject to pure shear fracture at various loading velocity v. (B) Dependence of the critical stretch ratio λ_c,1 at the onset of 1st slip on the loading velocity v. The λ_c,1 for each loading velocity v were from three measurements. (C and D) Dependence of (C) the critical yielding zone area (S_c) at the onset of slips (crack initiations), and (D) the increment $\mathrm{\Delta }{S}_c$ of yielding zone area during the stick process on the loading velocity v. (E) Yielding zone width increment $\mathrm{\Delta }y$ versus bulk loading displacement $\mathrm{\Delta }{t}_{\mathit{stick}}v$ during each stick process. (F) Dependence of the observed (${\mathrm{\Delta }S}_c^{\mathit{obs}}$) and calculated (${\mathrm{\Delta }S}_c^{\mathit{calc}}$) increments of yielding zone area during the stick process on the increment of stretch ratio ($\mathrm{\Delta }{\lambda }_c$) in two consecutive slips. The values of ${\mathrm{\Delta }S}_c^{\mathit{calc}}$ were calculated from the energy balance (SI Appendix, Eq. S12). In (C–F), each data point corresponds to one stick process, and the data for each loading velocity v were collected from different stick events of one measurement.

The slight stress increase with loading velocity in the regime before the 1st slip aligns with the behavior observed in unnotched samples. The tensile stress (σ)–stretch ratio (λ) curves of unnotched DN gels in the range of 10 to 1,000 mm/min also demonstrate slightly higher stress prior to yielding point with increasing loading velocity over the range of 10 to 1,000 mm/min (SI Appendix, Fig. S5) (46). Considering that DN gels consist of two elastic networks, such a slight loading rate dependence suggests that the internal fracture of the first network exhibits slight loading rate dependence. Within the observed velocity ranges, the bond rupture force of the first network strands slightly increases with the loading rate, a characteristic feature demonstrated in many single-chain force spectroscopy studies (53, 54).

The decrease of the stretch ratio for the first slip (λ_c,1) and the smaller Δλ_c with the increase of v could be understood by the relaxation dynamics of the networks, as revealed in our previous study (46). Indeed, the maximum yielding zone area in the stick process (S_c, Fig. 3C), the increment of yielding zone area (ΔS_c, Fig. 3D), and the opening length of the yielding zone along the tensile direction ($\mathrm{\Delta }y$, SI Appendix, Fig. S6C) during the stick process (from onset to end of the stick process) slightly decrease with increasing v. These observations reveal that the stick–slip crack dynamics are slightly influenced by the loading velocity (v). A high loading velocity results in a smaller yielding zone and a reduced crack resistance.

However, when considering the significant change in loading velocity spanning two orders of magnitude (from 5 to 500 mm/min), the velocity effect on the yielding zone and crack resistance seems to be relatively minor. On the other hand, the stick time ($\mathrm{\Delta }{t}_{\mathit{stick}}$), which correlates with the increment of applied displacement during the stick process for developing the yielding zone (${\mathrm{\Delta }\lambda }_c{H}_0$) and the loading velocity ($v$) by ${\mathrm{\Delta }\lambda }_c{H}_0=\mathrm{\Delta }{t}_{\mathit{stick}}v$, is nearly inversely proportional to v since ${\mathrm{\Delta }\lambda }_c{H}_0$ only weakly depends on v (SI Appendix, Fig. S6B). Furthermore, despite significant deviations in different stick events, $\mathrm{\Delta }y$ is nearly identical to $\mathrm{\Delta }{t}_{\mathit{stick}}v$ for all the stick events at various loading velocity (v) (Fig. 3E). The increment of the yielding zone area ΔS_c also depends on $\mathrm{\Delta }{t}_{\mathit{stick}}v$, as shown by the nearly overlapped data series with various loading velocities (Fig. 3F). These results indicate that the extension of the yielding zone is predominately associated with the increment of bulk loading, while the relaxation dynamics of the networks only exerts a minor influence on the formation of the yielding zone. In other words, the formation of the yielding zone could be roughly regarded as a quasi-static process, relying on the input loading energy rather than the loading rate. The loading rate controls the stick time ($\mathrm{\Delta }{t}_{\mathit{stick}}$) for generating a more extensive new damage zone that enables crack growth.

To further confirm this energy-related mechanism, we analyze the energy expenditure during stick (SI Appendix) and calculated the rate of change of the yielding zone area ${\mathrm{\Delta }{S}_c}^{\mathit{calc}}$ with respect to $\mathrm{\Delta }{\lambda }_c$, for each stick event at various loading velocities (v) (SI Appendix, Eq. S12). The $\mathrm{\Delta }{S}_c^{\mathit{obs}}$, observed from our polarized experimental methods, is nearly identical to the calculated ${\mathrm{\Delta }{S}_c}^{\mathit{calc}}$ for each stick event $\mathrm{\Delta }{\lambda }_c^{\mathit{obs}}$ (Fig. 3F). Importantly, this observation is found to be independent of loading velocity v. This result conclusively confirms that, during the stick phase, additional energy is introduced into the crack tip vicinity to extend the new yielding zone. Based on these findings, we conclude that stick–slip is essentially a consequence of a substantial kinetic difference between the fast crack propagation (dynamic fracture) through the yielding zone (slip process) and the loading energy-dependent formation of a new yielding zone at the crack tip (stick process).

Effect of Predamage on Crack Dynamics.

The above finding indicates that crack arrest occurs because the energy release rate during fast slip is not sufficient to induce formation of a new yielding zone at the crack tip. Consequently, additional energy supply is required to form a more extensive new damage zone that facilitates crack growth. Therefore, if the brittle 1st network is predamaged to some extent, the stick–slip phase may disappear because less energy is needed to damage the brittle network to form the yielding zone ahead of the crack tip. The question is at what level of preexisting damage does the stick phenomenon cease to occur? To answer this question, we tune the damage of the first network and study its influence on the crack dynamics. Specifically, before conducting crack tests, we applied various extents of stretch with a prestretch ratio (λ_pre) to the unnotched DN gels (SI Appendix, Fig. S7A). During this prestretch process, strands of the brittle first network reaching their stretching limits progressively broke, leading to a nonrecoverable hysteresis loop in the loading-unloading stress curves (SI Appendix, Fig. S7B) (55). The extent of predamage can be quantified by the area of the hysteresis loop (U_hys). U_hys increases with λ_pre when λ_pre is above 1.5, indicating a progressively enhanced internal damage (SI Appendix, Fig. S7C). Using this method, we obtained DN gels with various extents of internal damage by varying λ_pre. We denoted these predamaged DN gels as DN-λ_pre, where DN-1.0 corresponds to the virgin DN gel with no prestretch. The tensile behaviors of these DN-λ_pre gels depend on the stretch ratio λ relative to the prestretch ratio λ_pre; When λ ≤ λ_pre, the gel is elastic, without mechanical hysteresis because short strands that break at λ have already broken during the prestretch (SI Appendix, Fig. S8 A, i). When λ > λ_pre, the gel becomes inelastic, exhibiting mechanical hysteresis due to the breaking of additional strands as λ increases beyond λ_pre (SI Appendix, Fig. S8 A, ii). The loading energy density W_load, stored strain energy density W_el (=W_unload), and the dissipated energy density U_diss (=U_hys) as a function of maximum stretch ratio λ_max for various λ_pre are depicted in SI Appendix, Fig. S8 B–D (55).

The influence of λ_pre on crack dynamics is shown in Fig. 4, revealing how internal damage tunes the crack dynamics of DN gels. Two distinct regimes were observed: stick–slip crack dynamics for λ_pre ≤ 3.0 (regime I) and continuous crack dynamics for λ_pre ≥ 3.25 (regime II), as shown by the representative stress and crack extension curves in Fig. 4 A and B, respectively. The continuous crack dynamics at λ_pre ≥ 3.25, exemplified by DN-4.0, are showcased in the in situ recording video (Movie S2). Detailed aspects of the crack dynamics, including the evolution of 1) the stress–stretch ratio profiles, 2) crack extension Δc (mm), and 3) birefringence area S (mm²) at the crack tip during crack propagation for the DN-4.0 gel, are presented in SI Appendix, Fig. S9. The stress decrease, rapid crack extension, and decrease in the yielding zone area all occurred in a catastrophic manner within the time resolution of the camera (1/24 s) (SI Appendix, Fig. S9).

Fig. 4.

Crack dynamics in predamaged DN gels. (A and B) The nominal stress (σ)–stretch ratio (λ) curves (A) and crack extension (Δc) curves (B) for DN-λ_pre gels in pure-shear fracture tests. The dotted lines in (A) show the loading-unloading cycles of unnotched virgin DN gels stretched to the corresponding λ_pre for inducing predamage prior to the fracture test. The loading velocity is v = 50 mm/min. (C) Dependency of critical stretch ratio (λ_c) at crack initiation on λ_pre. Blue circles represent the first slip, and gray circles in regime I represent subsequent slips. (D) Energy release rate required to overcome crack arrest (G_arrest) and energy release rates at slips (G_slip) as a function of λ_pre. Some of curves in (A), λ_c in (C), and G_arrest in (D) are adapted from ref. 55.

The dependence of crack dynamics on λ_pre confirms that the crack arrest process diminishes for DN gels with severe internal damage (U_hys ≥ 0.58 MJ/m³ for λ_pre ≥ 3.25), leading to continuous crack propagation, akin to an elastic SN gel (SI Appendix, Fig. S4). It is noteworthy that the prestretch ratio for the transition, λ_{pre, tran} (=3.0 ~ 3.25), from regime I to regime II closely aligns with the yielding stretch ratio, λ_y (~3.20), of DN gels observed from uniaxial tensile curve (SI Appendix, Fig. S1). This observation aligns with the discussion in the previous section, indicating that the stick process corresponds to the formation of a new yielding zone at the crack tip. For a DN gel prestretched beyond its yielding point, the brittle first network undergoes severe breakage, and the primary load-bearing role shifts to the second network (56). In this state, no additional energy is required to form the yielding zone ahead of the propagating crack tip. The stored strain energy in the strands of the second network is sufficient to drive continuous crack propagation once the critical stretch ratio (λ_c) is surpassed. As a result, catastrophic brittle fracture occurs without any crack arrest (regime II).

We then investigate the influence of λ_pre on the onset of crack initiation (the critical stretch ratio λ_c) for DN gels in these two crack dynamics regimes. In regime I where stick–slip crack dynamics occurs, each slip corresponds to a critical stretch ratio λ_c and we get multiple λ_c for each sample. We plot all these λ_c values for each sample against λ_pre in Fig. 4C. For the first slip (depicted as blue open circles in Fig. 4C), λ_c is nearly independent of λ_pre for λ_pre ≤ 2.0 but increases with λ_pre in the range of λ_pre = 2.0 to 3.0. The λ_c gradually increases with the successive slip steps (illustrated as gray open circles in Fig. 4C). When λ_c reaches approximately λ_{pre, tran} (≈3.0 ~ 3.25), the crack propagates in a continuous manner. This observation aligns with the fact that when λ_pre is larger than λ_{pre, tran} ≈ λ_y, the stick phenomenon disappears. When λ_c reaches λ_{pre, tran}, the entire sample reaches the yielding state during the loading, and no additional energy is required to drive the crack propagation (regime II). It is interesting to notice that in regime II, λ_c becomes identical to λ_pre.

To confirm this, we revisit the influence of λ_pre on the energy release rates associated with the initiation of the crack. As the energy applied to the DN gels is partially dissipated during loading due to the breaking of the brittle first network, only the stored elastic energy propels crack propagation. Therefore, the energy release rate for slip G_slip = W_el(λ_c) H₀, where W_el represents the stored elastic energy density, estimated from the area under the unloading curves of unnotched DN-λ_pre gels to the λ_c (SI Appendix, Fig. S8 B, ii). In regime I, λ_c of the 1st slip is used to estimate G_slip. Fig. 4D depicts the dependence of G_slip on the prestretch ratio λ_pre, indicating a modest increase with λ_pre, especially in regime II.

As crack arrest results from the additional energy required to break the first network, the energy release rate needed to overcome the crack arrest (G_arrest) for the 1st slip corresponds to the energy release rate estimated from the loading curves, G_arrest = W_load(λ_c) H₀ (SI Appendix, Fig. S8 B, i). The dependence of G_arrest on the prestretch ratio λ_pre is also depicted in Fig. 4D. The results indicate that the relationship G_slip < G_arrest always hold in regime I where the stick–slip occurs, and G_slip = G_arrest in regime II where continuous crack propagation takes place. This result reaffirms that crack arrest is attributed to an insufficient energy release rate after slip, in agreement with our argument in the preceding section. It is worth noting that these results are consistent with our previous work, which showed that neckable DN hydrogels exhibit an abrupt crack velocity jump from no propagation to fast mode at an increased energy release rate (47). This abrupt crack velocity jump indicates that the crack must either arrest at a low energy release rate or propagate quickly at a high energy release rate. There is no intermediate velocity at which the crack can grow stably.

It is noteworthy that both G_slip and λ_c (Fig. 4C) exhibit a slightly increase as λ_pre increases, particularly in regime II where G_arrest = G_slip, and λ_c = λ_pre. As internal damage continues to rise with λ_pre in regime II (SI Appendix, Fig. S7C), this suggests that the critical energy release rate for the crack initiation slightly increases as the internal damage of the first network increases. The rupture of the brittle network is likely to release the hidden length of the stretchable network strands and the broken fragments of the brittle network could serve as sliding cross-links to delocalize the stress−concentration near the crack tip, consequently delaying the crack initiation (55).

To investigate whether the single crack extension, Δc, correlates with the predamage of DN gels in regime I, we plotted Δc against λ_pre in SI Appendix, Fig. S10B. For statistical significance, data on Δc were collected from three parallel tests for each DN gel in regime I (SI Appendix, Fig. S11). The results show considerable deviations in Δc, suggesting that the crack propagation distance in a single slip is quite random. Moreover, the average value of Δc is nearly identical for DN gels with less damage (λ_pre ≤ 2.0) and monotonically increases with moderate damage (2.0 ≤ λ_pre ≤ 3.0). The increase of Δc could be understood by the decrease between the difference of G_arrest and G_slip at large λ_pre.

In summary, prestretching the DN gel above its yielding point (λ_y) before the crack test eliminates the stick process, and continuous crack dynamics prevails. This observation aligns well with our energy-related mechanism on why crack arrests, highlighting the crucial role of the sacrificial brittle network in influencing crack dynamics.

Conclusion

We have identified a distinctive stick–slip crack dynamics in the fracture behavior of tough and strong DN hydrogels under mode-I loading in a pure-shear geometry. This stick–slip crack dynamics manifests as a peculiar form of fracture of the materials. Through a comprehensive analysis involving retardation observations during the stick–slip process and crack tests conducted across a wide range of loading velocities, we have unveiled that the slip process corresponds to a rapid, dynamic fracture within the yielding zone and the stick process (crack arrest) corresponds to the formation of a new yielding zone at the crack tip. The formation of the yielding zone at the crack tip necessitates cumulative internal fracture of the first network, demanding additional strain energy input which is influenced by external loading velocity. This hypothesis is further substantiated by calculation of energy expenditure during the stick process, revealing a notable agreement between the observed $\mathrm{\Delta }{S}_c^{\mathit{obs}}$ and calculated $\mathrm{\Delta }{S}_c^{\mathit{calc}}$. In instances where the DN gel is extensively prestretched beyond its yielding point, resulting in severe damage of the brittle first network before the crack test, the stick process disappears. Instead, continuous dynamic fracture, reminiscent of conventional single-network materials, takes precedence over stick–slip crack dynamics. Although this stick–slip behavior is observed in the inelastic DN gels, the conceptual model for crack arrest and suppressed dynamic fracture outlined in this work offers valuable insight into how the dissipative process around the crack tip modulates crack dynamics in tough soft materials.

Materials and Methods

Materials.

2-Acrylamido-2-methylpropanesulfonic acid sodium salt (Toagosei Co., Ltd.), acrylamide (Junsei Chemical Co. Ltd.), N,N′-methylenebis(acrylamide) (Wako Pure Chemical Industries, Ltd.), and α-ketoglutaric acid (Wako Pure Chemical Industries, Ltd.) were used as received. Milli-Q water (resistivity: 18.3 MΩ/cm) was used in all experiments.

Synthesis of DN Gels.

The PNaAMPS/PAAm DN hydrogels were synthesized by a two-step sequential network formation technique following the literature (44).

All details associated with sample preparations, uniaxial tensile tests, cyclic tensile tests in pure shear geometry and the calculation of U_hys and W_el, pure-shear fracture tests and real-time birefringence observation, and retardation observation near the crack tip, and energy expenditure during stick, and inducing internal damage to DN gels prior to crack tests are available in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Stick-slip crack dynamics of DN-1.0 hydrogel.

Movie S2.

Continuous crack dynamics of pre-damaged DN-4.0 hydrogel.

Data, Materials, and Software Availability

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