# Enhanced Rupture Force in a Cut-Dispersed Double-Network Hydrogel

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## Abstract

**:**

## 1. Introduction

_{b}of 1∼10 MPa, strain at break ε

_{b}of 1000–2000%), and high toughness (fracture energy Γ of 10

^{3}–10

^{4}J m

^{−2}) [35]. These mechanical performances are greatly superior to the SN gels of their individual components and even comparable to human load-bearing tissues and some tough industrial rubbers [32,35]. The double-network strategy has recently been extended to multiple-network elastomers exhibiting extremely reinforced mechanical strength and toughness without sacrificing stretchability [33,47,48,49].

## 2. Results and Discussion

#### 2.1. Effect of Dispersed Cuts on the Fracture Behavior of DN Hydrogel Specimens

_{1}is defined as the horizontal size of the triangle-shaped cuts, while h

_{2}represents the horizontal spacing between adjacent triangle-shaped cuts. In this work, the size of h

_{1}is kept constant at 1.5 mm for simplicity, and the spacing of h

_{2}is tuned from 0.75 mm to 6.0 mm. According to the different spacing h

_{2}, the D-cut gel specimens were prepared from these different structural patterns with dispersed cuts at different spacing ratios h

_{1}:h

_{2}(1:0.5, 1:1, 1:2, 1:3, and 1:4) and their counterpart C-cut specimens with aligned continuous cuts, as shown in Figure 1. Note that for comparison, the C-cut gel specimens are prepared to have the same number of aligned continuous cuts as the D-cut gel specimens such that the overall lengths of the uncracked ligaments can be kept the same between these specimens. The “continuous cuts” geometry in C-cut gels is more like “pure shear” geometry with a long crack and triangle-shaped crack front.

_{y,tens}= 0.57 MPa), which corresponds to the onset of necking (highly deformed region) in the tensile sample [56]. As elucidated in the previous study, the brittle first network is considered to severely rupture into fragments, and the stretchable second network is highly deformed in the necking region [35]. Such stress-yielding occurs ahead of the crack tip, reducing the stress concentration at the crack tip, thereby blunting the crack tip and enhancing the fracture resistance of the material [56].

^{−1}). Compared with samples without cuts shown in Figure S2, the mechanical properties of the sample with cuts showed significant deterioration. Figure 2a–e further show the force (F)–displacement (x) curves of D-cut and C-cut DN hydrogel specimens during loading under different structural patterns, containing dispersed and continuous cuts at different spacing ratios h

_{1}:h

_{2}of 1:0.5, 1:1, 1:2, 1:3, and 1:4. Here, we needed to emphasizes that the force–displacement curves of samples with C-cut and “pure shear” geometry are almost overlapped, indicating that the crack shape in the crack front has no obvious influence on the mechanical properties.

_{1}:h

_{2}, the force curves of D-cut gel specimens exceed those of C-cut specimens from the initial stage of loading. Taking spacing ratio h

_{1}:h

_{2}of 1:0.5 as an example, for the D-cut specimen, the force increases with increasing displacement between clamps until reaching a critical value (rupture force F

_{rupture}= 10.84 ± 0.11 N occurring at x

_{rupture}= 9.83 ± 0.38 mm), at which point the specimen begins to rupture, as seen in the snapshots during loading shown in birefringence experiment in the later section. A drop in load occurs during specimen rupture, as some regions sustain load ruptures to lose load-bearing capability. By contrast, for the C-cut specimen, the force increases in a relatively slow manner, with increasing displacement between clamps until reaching a critical value (rupture force F

_{rupture}= 9.97 ± 0.60 N occurring at x

_{rupture}= 15.73 ± 2.54 mm), at which point the specimen begins to rupture.

_{1}:h

_{2}of 1:0.5, 1:1, 1:2, 1:3, and 1:4. For the D-cut specimens, by increasing the horizontal spacing between adjacent triangle-shaped cuts h

_{2}, both the critical rupture force F

_{rupture}and critical displacement x

_{rupture}increase (Figure 3a). This is reasonable because the cross-sectional area of the uncracked ligament in the specimens that sustain the load increases with the spacing between adjacent cuts h

_{2}, giving rise to a large load-bearing capability.

_{rupture}in Figure 3b. It is also found that regardless of different spacing ratios h

_{1}:h

_{2}, the critical rupture forces F

_{rupture}of the D-cut specimens are higher than these of the C-cut specimens, and the deviation between F

_{rupture}of the D-cut specimens and C-cut specimens increases with h

_{2}(Figure 3b). Specifically, in the case of spacing ratio h

_{1}:h

_{2}of 1:0.5, the D-cut specimens show a critical rupture force F

_{rupture}of 10.84 ± 0.11 N, slightly higher than that of the C-cut specimens (rupture force F

_{rupture}= 9.97 ± 0.60 N). While in the case of spacing ratio h

_{1}:h

_{2}of 1:4, the D-cut specimens demonstrate the critical rupture force F

_{rupture}of 28.69 ± 2.48 N, much higher than that of the C-cut specimens (rupture force F

_{rupture}= 21.79 ± 1.76 N). Note that at the same spacing ratios h

_{1}:h

_{2}, the D-cut specimens and C-cut specimens all have the same cross-sectional area of the uncracked ligament (the overall cross-sectional area of the whole specimen subtracted by the area occupied by cuts). Because the C-cut geometry is more like the conventional “pure shear” geometry with a precut, this result also indicates that the dispersed cuts can produce an enhanced rupture force than the pure shear specimen with a precut.

_{rupture}is further analyzed in Figure 3c. In the case of spacing ratio h

_{1}:h

_{2}of 1:0.5, the C-cut specimens show a much higher critical stretch ratio at rupture point λ

_{rupture}(= 1.87 ± 0.04) than that of the D-cut specimens (λ

_{rupture}= 2.13 ± 0.11). With further increasing spacing h

_{2}until reaching spacing ratio h

_{1}:h

_{2}of 1:2, the D-cut specimens begin to exhibit comparable λ

_{rupture}with these of the C-cut specimens. In particular, at a spacing ratio h

_{1}:h

_{2}of 1:4, the D-cut specimens exhibit λ

_{rupture}of 2.50 ± 0.08, while the C-cut specimens show λ

_{rupture}of 2.43 ± 0.13.

_{r,bulk}) of different specimens. We should note here that critical bulk stress at the rupture point (σ

_{r, bulk}) is defined as the rupture force F

_{rupture}divided by the cross-sectional area of the uncracked ligament (the overall cross-sectional area of the whole specimen subtracted by the area occupied by cuts). It is clearly shown in Figure 3d that the σ

_{r,bulk}of all the samples is located in the narrow range of 0.40 MPa to 0.55 MPa, which is close to the yielding stress of DN hydrogel (σ

_{y,tens}= 0.57 MPa). Additionally, it is found that regardless of spacing ratios h

_{1}:h

_{2}, the D-cut specimens show higher σ

_{r,bulk}than the C-cut specimens.

_{dis}/F

_{con}) and normalized rupture stretch ratio (λ

_{dis}/λ

_{con}) as functions of spacing ratios h

_{1}:h

_{2}in Figure 4. Here, the F

_{dis}and F

_{con}represent the rupture forces in fracture specimens containing dispersed cuts and continuous cuts at the same spacing ratios h

_{1}:h

_{2}, respectively. The λ

_{dis}and λ

_{con}denote the rupture stretch ratios in fracture experiments containing dispersed cuts and continuous cuts at the same spacing ratios h

_{1}:h

_{2}, respectively. The rupture force ratio F

_{dis}/F

_{con}also can be seen as the enhancement ratio of rupture force of fracture specimens by the dispersed cuts. As shown in Figure 4, the enhancement ratio increases from 1.08 to 1.32 with changing spacing ratios h

_{1}:h

_{2}from 1:0.5 to 1:4. In the meantime, the normalized rupture stretch ratio λ

_{dis}/λ

_{con}increases from 0.8 to 1.03 with changing spacing ratios h

_{1}:h

_{2}from 1:0.5 to 1:4. This suggests that by increasing spacing h

_{2}, the dispersed cuts can enhance the rupture force without sacrificing the stretchability of the bulk materials, thereby increasing the fracture resistance.

#### 2.2. Birefringence Observation on the Rupture of Specimens Containing Dispersed Cuts and Continuous Cuts

_{1}:h

_{2}of 1:0.5, 1:2, and 1:4, respectively. As seen in Figure 5a, in the case of spacing ratio h

_{1}:h

_{2}of 1:0.5, for the D-cut specimens, owing to the small spacing h

_{2}between adjacent cuts, all the spacing regions are homogeneously deformed to a high level to exhibit a strong birefringence at the rupture point, just like the strong birefringence observed in the tensile process as reported in the previous study by Gong et al. After reaching the critical value of rupture force, some highly deformed spacing regions suddenly rupture, as seen in Figure 5a. While in the case of C-cut specimens, a large bright triangle-shaped birefringence region occurs ahead of the crack tip (Figure 5b), like that observed in the “pure shear” geometry. As elucidated in the previous work by Gong et al., such a large bright birefringence area corresponds to the stress-yielding occurring ahead of the crack tip accompanied by the formation of a large yielding zone, where the brittle first network is considered to severely rupture into fragments and the stretchable second network is highly deformed. The formation of a large yielding zone reduces the stress concentration at the crack tip, thereby blunting the crack tip and enhancing the fracture resistance of the material [56].

_{2}between adjacent cuts to reach a spacing ratio h

_{1}:h

_{2}of 1:2, the spacing regions of the D-cut specimens are highly deformed to display nearly trapezoid-shaped birefringence areas, each of which is smaller than the triangle-shaped birefringence area in the C-cut specimens (Figure 6). With further increasing of spacing h

_{2}between adjacent cuts to reach a spacing ratio h

_{1}:h

_{2}of 1:4, the spacing regions of the D-cut specimens are highly deformed to display nearly triangle-shaped birefringence areas, some of which are comparable to the triangle-shaped birefringence area in the C-cut specimens (Figure 7). Additionally, it should be mentioned that the C-cut specimens all exhibit the directed crack propagation direction from left to right like the conventional “pure shear” specimen, while the D-cut specimens rupture randomly in the weakest points of the spacing regions.

_{tip}exceeding the critical threshold value σ

_{threshold}. Owing to the stress concentration, the crack tip stress σ

_{tip}is usually amplified from the bulk stress σ

_{bulk}. If we denote a stress concentration factor α as the ratio of crack tip stress σ

_{tip}over the bulk stress σ

_{bulk}, ${\sigma}_{tip}=\alpha {\sigma}_{bulk}$, a severe stress concentration in the crack tip means that the crack tip stress σ

_{tip}will be amplified to α times the bulk stress σ

_{bulk}. Because the yielding dominates the yielding zone area (birefringence area) ahead of the crack tip, we can simply consider that the maximum stress of the material point ahead of the crack tip, which is also the critical threshold value σ

_{threshold}, should be related to yielding stress σ

_{y,tens}by a factor of β; thus we have ${\sigma}_{threshold}=\beta {\sigma}_{y,tens}$. For the same material, β should be a constant. Then, the rupture criterion will be ${\sigma}_{tip}\ge {\sigma}_{threshold}$, thus $\alpha {\sigma}_{bulk}\ge \beta {\sigma}_{y,tens}$. With increasing the bulk stress until reaching the critical bulk stress ${\sigma}_{r,bulk}$ at the rupture point, the rupture occurs. So, we have $\alpha {\sigma}_{r,bulk}=\beta {\sigma}_{y,tens}$; thus, a normalized stress concentration factor will be $\frac{\alpha}{\beta}=\frac{{\sigma}_{y,tens}}{{\sigma}_{r,bulk}}$. We next plot the $\frac{{\sigma}_{y,tens}}{{\sigma}_{r,bulk}}$ for D-cut specimens and C-cut specimens in Figure 8. It can be seen that the values of $\frac{{\sigma}_{y,tens}}{{\sigma}_{r,bulk}}$ for the D-cut specimens are lower than those of the C-cut specimens, suggesting that the stress concentration due to the crack tip (or cuts) is less severe in the “dispersed cuts” cases. Note that for common DN hydrogels, even in the “continuous cuts” cases, the formation of a large yielding zone already remarkably reduces the stress concentration at the crack tip, thereby blunting the crack tip and enhancing the fracture resistance of the material. Here, we show that the introduction of a “dispersed cuts” pattern can further reduce the stress concentration at the crack tip, enhancing the rupture force. It can also be observed from the birefringence snapshots for D-cut specimens in Figure 5, Figure 6 and Figure 7 that the spacing regions between every two adjacent cuts, which are sustaining loads, are all deformed to a high level ahead of the crack tip, meaning that the dispersed cuts may homogenize the stress around the crack tip surrounding every cut, avoiding stress concentration in one certain cut. Therefore, the “dispersed cuts” specimens can sustain an enhanced rupture force.

#### 2.3. Characteristic Fracture Structure of the DN Gels with Dispersed and Continuous Cuts

## 3. Conclusions

## 4. Materials and Methods

#### 4.1. Materials

#### 4.2. Synthesis of DN Hydrogels

#### 4.3. Preparation of Various Cuts Patterns in DN Hydrogel Specimens

_{1}is defined as the horizontal size of the triangle-shaped cuts, while h

_{2}represents the horizontal spacing between adjacent cuts. The h

_{1}is kept constant at 1.5 mm in this work, and the spacing h

_{2}is controlled in various lengths of 0.75 mm, 1.5 mm, 3.0 mm, and 6.0 mm, respectively. Accordingly, the DN hydrogel specimens with structural cut patterns were prepared at different spacing ratios h

_{1}:h

_{2}(1:0.5, 1:1, 1:2, 1:3, and 1:4).

#### 4.4. Real-Time Birefringence Observation on the Fracture Tests

#### 4.5. Tensile Test

^{−1}).

#### 4.6. Microscopic Observation of Cuts in the DN Gels

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Scheme of the structural models with dispersed cuts and continuous cuts patterns for fracture experiments. (

**a**) Scheme of the structural model containing an array of dispersed cuts investigated in this work. The length scale h

_{1}is defined as the horizontal size of the triangle-shaped cuts, while h

_{2}represents the horizontal spacing between adjacent cuts. In this work, the h

_{1}is kept constant at 1.5 mm for simplicity. (

**b**) The structural models with dispersed cuts at different spacing ratios h

_{1}:h

_{2}(1:0.5, 1:1, 1:2, 1:3, and 1:4) and their counterpart structural models with the same number of aligned continuous cuts. The pure shear model was used as the reference.

**Figure 2.**Representative force curves of highly deformable DN hydrogels under different structural models with dispersed cuts (D-cut gels) and continuous cuts (C-cut gels). The representative force (F)–displacement (x) curves of DN hydrogels under different structural models containing dispersed cuts and continuous cuts at different spacing ratios h

_{1}:h

_{2}of 1:0.5 (

**a**), 1:1 (

**b**), 1:2 (

**c**), 1:3 (

**d**), and 1:4 (

**e**). The force curves of samples with a pure shear geometry were also provided in (

**a**,

**c**,

**e**) for comparison.

**Figure 3.**Effect of spacing ratios h

_{1}:h

_{2}and dispersed/continuous cuts on the mechanical behaviors of DN hydrogel samples in fracture experiments. (

**a**) Summarized force (F)–displacement (x) curves of DN hydrogels. (

**b**) Rupture force (F

_{rupture}), (

**c**) critical stretch ratio at rupture point (λ

_{rupture}), and (

**d**) critical bulk stress at rupture point (σ

_{r,bulk}) for DN hydrogel samples containing dispersed cuts (D-cut gels) and continuous cuts (C-cut gels) at different spacing ratios h

_{1}:h

_{2}(1:0.5, 1:1, 1:2, 1:3, and 1:4).

**Figure 4.**The rupture force ratio (F

_{dis}/F

_{con}) and normalized rupture stretch ratio (λ

_{dis}/λ

_{con}) as functions of spacing ratios h

_{1}:h

_{2}. The F

_{dis}and F

_{con}represent the rupture forces in fracture specimens containing dispersed cuts and continuous cuts, respectively. The λ

_{dis}and λ

_{con}denote the rupture stretch ratios in fracture experiments containing dispersed cuts (D-cut gels) and continuous cuts (C-cut gels), respectively.

**Figure 5.**Birefringence observation during fracture of D-cut and C-cut gels at spacing ratio h

_{1}:h

_{2}of 1:0.5. (

**a**,

**b**) The representative snapshots of crack evolution in DN hydrogel samples under different structural models containing dispersed cuts ((

**a**) D-cut gel) and continuous cuts ((

**b**) C-cut gel) at spacing ratio h

_{1}:h

_{2}of 1:0.5.

**Figure 6.**Birefringence observation during fracture of D-cut and C-cut gels at spacing ratio h

_{1}:h

_{2}of 1:2. (

**a**,

**b**) The representative snapshots of crack evolution in DN hydrogel samples under different structural models containing dispersed cuts ((

**a**) D-cut gel) and continuous cuts ((

**b**) C-cut gel) at spacing ratio h

_{1}:h

_{2}of 1:2.

**Figure 7.**Birefringence observation during fracture of D-cut and C-cut gels at spacing ratio h

_{1}:h

_{2}of 1:4. (

**a**,

**b**) The representative snapshots of crack evolution in DN hydrogel samples under different structural models containing dispersed cuts ((

**a**) D-cut gel) and continuous cuts ((

**b**) C-cut gel) at spacing ratio h

_{1}:h

_{2}of 1:4.

**Figure 8.**The normalized stress concentration ratio (the ratio between tensile yielding stress and critical bulk stress, σ

_{y,tens}/σ

_{r,bulk}) as functions of spacing ratios h

_{1}:h

_{2}.

**Figure 9.**The crack tip structure of DN hydrogels was observed by optical microscopy. (

**a**,

**b**) The representative crack tip structure in DN hydrogel samples before and after loading to a stretch ratio λ of 2 under different structural models containing dispersed cuts (

**a**) and continuous cuts (

**b**) at spacing ratio h

_{1}:h

_{2}of 1:2. Wrinkles-like damaged structure can be observed ahead of the crack tip, corresponding to the damage zone observed by birefringence.

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## Share and Cite

**MDPI and ACS Style**

Zhu, S.; Yan, D.; Chen, L.; Wang, Y.; Zhu, F.; Ye, Y.; Zheng, Y.; Yu, W.; Zheng, Q.
Enhanced Rupture Force in a Cut-Dispersed Double-Network Hydrogel. *Gels* **2023**, *9*, 158.
https://doi.org/10.3390/gels9020158

**AMA Style**

Zhu S, Yan D, Chen L, Wang Y, Zhu F, Ye Y, Zheng Y, Yu W, Zheng Q.
Enhanced Rupture Force in a Cut-Dispersed Double-Network Hydrogel. *Gels*. 2023; 9(2):158.
https://doi.org/10.3390/gels9020158

**Chicago/Turabian Style**

Zhu, Shilei, Dongdong Yan, Lin Chen, Yan Wang, Fengbo Zhu, Yanan Ye, Yong Zheng, Wenwen Yu, and Qiang Zheng.
2023. "Enhanced Rupture Force in a Cut-Dispersed Double-Network Hydrogel" *Gels* 9, no. 2: 158.
https://doi.org/10.3390/gels9020158