Fatigue of double-network hydrogels

: The discovery of tough hydrogels of many chemical compositions, and their emerging applications in medicine, clothing, and engineering, has raised a fundamental question: How do hydrogels behave under many cycles of stretch? This paper initiates the study of the fatigue behavior of the classic PAMPS/PAAM double network hydrogels discovered by Gong and her co-workers (Advanced Materials 15, 1155, 2003). We reproduce the hydrogels, and prepare samples of two types, with or without a crack cut before the test. When an uncut sample is subject to cyclic stretches, internal damage accumulates over thousands of cycles until a steady state is reached. When a cut sample is subject to cyclic stretches, the crack extends cycle by cycle if the amplitude of stretch is above a certain value. A threshold of energy release rate exists, below which the crack remains stationary as the sample is cycled. We find a threshold around 400 J/m 2 for hydrogels containing PAAM networks of a low density of crosslinkers, and around 200 J/m 2 for hydrogels containing PAAM networks of a high density of crosslinkers. The experimental findings are compared to the Lake-Thomas model adapted to the double-network hydrogels.


Introduction
A hydrogel is an aggregate of water and a three-dimensional polymer network ( Fig. 1). That is, the hydrogel is a molecular composite: polymer-reinforced water.
The mesh of the polymer network is on the order of 10 nm, much larger than an individual water molecule.
Consequently, water in the hydrogel retains its molecular properties. In particular, water in the hydrogel is a solvent of small molecules, and transports them. The polymer network gives the hydrogel elasticity.
The hydrogel is compliant under a mechanical force, and recovers its shape after the force is removed. Most tissues of animals and plants are hydrogels. Many synthetic hydrogels are compatible with living tissues chemically, mechanically, and electrically.

Fig. 1
A hydrogel is an aggregate of water and a polymer network. The mesh of the polymer network is much larger than an individual water molecule, so that water retains its molecular properties.
10/21/2017 5 crosslinked. Reported attributes include high toughness, high strength, large stretchability, and self-recovery [41,47]. For example, alginate/polyacrylamide hydrogels achieve toughness above 10,000 J/m 2 , comparable to that of natural rubber [38,48]. Tough hydrogels have also led to strong adhesion between hydrogels and other materials [49]. Moreover, tough hydrogels have been used as matrices to develop composites [51][52][53]. We call these composites water matrix composites (WMCs), in which water is the matrix, and the reinforcements are in two scales, as individual molecular chains and macroscopic fibers and particles. WMCs are, of course, matters of life-tissues of plants and animals.
A new question is being asked, what can we do when water is a tough solid?
Tough materials derive their high toughness from large stress-stretch hysteresis ( Fig. 3). This basic principle of fracture mechanics has long been established for metals [54,56], elastomers [56,57], transformation-toughened ceramics [58], ceramic-matrix composites [59], and elastomer-toughened plastics [60]. In the DN hydrogel, the hysteresis is due to the rupture of the short-chain network [26,61,62]. to the four states marked in the stress-stretch plane. The fracture energy equals the work done to create the two surfaces of the crack and the hysteresis in the two layers.
In the schematic of the stress-stretch curve (Fig. 3a), the elastic modulus during unloading is the same as the elastic modulus during loading, and a residual stretch is left after the stress reduces to zero. Both features are observed in metals, but need not be true for other materials. In particular, for a PAMPS/PAAM hydrogel, a large stretch breaks the PAMPS network, but leaves the PAAM network intact, so that the hydrogel has no residual stretch upon unloading to zero stress. Furthermore, the elastic modulus during unloading is lower than the elastic modulus during loading. 10 In load-bearing applications, hydrogels are often subject to cyclic stretches.
The hysteresis is much smaller in subsequent cycles than in the initial cycle [72]. That the DN hydrogels are prone to fatigue fracture often comes up in conversations, but has not been studied. Indeed, fatigue fracture of hydrogels of any kind has not been reported until recently. Three likely reasons for this lack of interest have been suggested [73]. First, synthetic hydrogels are a relatively new type of materials, and interest in their mechanical behavior started only after the commercialization of contact lenses in the 1960s and superabsorbent diapers in the 1980s. Second, fatigue fracture was not ''mission-critical'' in the initial applications of hydrogels. Third, hydrogels in use may degrade or dehydrate before they rupture under cyclic load.
The discovery of tough hydrogels, as well as their emerging applications in medicine, clothing, and engineering, strongly indicates that hydrogels will be used in applications that require many cycles of loads. Two sets of data on fatigue fracture of hydrogels have just appeared in the literature, one set for the polyacrylamide single-network hydrogel [73], and the other set for the alginate/polyacrylamide hydrogel [74] (Fig. 4). The threshold for fatigue fracture in the alginate/polyacrylamide hydrogel is much below the fracture energy under a monotonic load. Nonetheless, the extension of crack per cycle in the alginate/polyacrylamide hydrogel is much smaller than that in a single-network polyacrylamide hydrogel. The available data of fatigue fracture in the alginate/polyacrylamide hydrogel is obtained using samples prepared with particular 10/21/2017 8 ratios of water, polymers, and crosslinkers. Given that fracture energy is sensitive to these ratios, it is important to study their effects on fatigue behavior.

Fig. 4
The extension of crack per cycle as a function of the energy release rate for a polyacrylamide hydrogel and a particular composition of alginate/polyacrylamide hydrogel. Adapted from Refs [73] and [74].
Incidentally, the phrase "fatigue damage" designates degradation under cyclic load. Examples include cycle-by-cycle change in elastic modulus [75], in hysteresis of stress-stretch curves [43,76], and in functional characteristics of devices [77]. By contrast, the phrase "fatigue fracture" designates rupture under cyclic load.
Here we initiate a study of the fatigue fracture of the PAMPS/PAAM double-network hydrogel discovered by Gong and her co-workers [25]. concentrations, and keep all other parameters fixed. We measure the stress-stretch curves using the uniaxial tensile test, and measure the toughness by using the pure shear test. We apply cyclic loads on uncut samples to observe the shakedown of stress-stretch curves. We apply cyclic loads on cut samples to observe the propagation of fatigue cracks. We compare the experimental findings to the Lake-Thomas model. We further synthesize a DN hydrogel with shorter PAAM chains, and find that the threshold of fatigue fracture is reduced.

Synthesis of double-network hydrogels
We synthesized DN hydrogels using the method of sequential formation of the two networks developed by Gong and her co-workers [25]. The ingredients were obtained from sources different from those for the DN gels reported in [25]. To achieve similar stress-stretch curves as in [25], we have altered the concentrations of the ingredients.
We purchased from Aladdin the following substances: We finally immersed the DN hydrogels in the deionized water for 3 days to remove the residual unreacted substances, and to equilibrate the hydrogels into the fully swollen state. The average thickness in the fully swollen state was 1.3mm for the hydrogels prepared using the solution of 2M AAM, 1.4 mm for 3 M AAM, and 1.5 mm for 4M AAM.
We determined the volume fraction of the PAAM network in a DN hydrogel as follows. We prepared two identical PAMPS gels. We dehydrated one PAMPS gel at 60°C for 12h and measured the dried mass of PAMPS network

Uniaxial tensile test
We used the as-prepared hydrogels for tensile tests. The hydrogels were cut using a razor blade into dumbbell-shaped samples with effective length 12 mm and width 2 mm (GB/T 528-2009-4 standard) (Fig. 5a). The samples were clamped using the two metallic grippers of a tensile tester (SHIMADZU AGS-X). The grippers were displaced by a pneumatic power source (JINGYIN QD1212). We used a load cell of 1000 N to apply a uniaxial monotonic load to each sample at a rate of 100 mm/min until the sample ruptured. The tests were carried out at 25℃ in the open air. Before rupture, the sample elongated significantly (Fig. 5b). The force and displacement for each sample were recorded by the tensile tester. We plotted the stress-stretch curves 10 The shear modulus of each sample is obtained by fitting the beginning portion of its stress-stretch curve to the neo-Hookean model [78]: where s is the nominal stress, λ is the stretch, and µ is the shear modulus. The shear modulus for all hydrogels prepared in this work is roughly 100 kPa (See neo-Hookean model. Our intention here is to obtain a rough value for the small-strain shear modulus for the materials prepared. This shear modulus will not be used in any of the following calculations.

Fracture energy
To measure toughness (also called tearing energy and fracture energy) of the PAMPS/PAAM hydrogels, we adopt an established method for rubber, the pure shear test [79]. Following [38], we prepared two sets of samples with the same geometry.
One set is uncut, and the other set is precut with a crack.
The pure shear test usually takes several minutes for static fracture and at least 1 day for fatigue fracture. Thus, we sealed the samples in a homemade acrylic chamber ( Fig. 6a). During the tests, we sprayed droplets of deionized water on the inner surface of the chamber. The difference between the weight of samples before and after testing was less than 5% of the initial weight.
The uncut sample was of a long rectangular sheet (10 mm × 50 mm) and was clamped to the two grippers of the tensile tester (Fig. 6b). A monotonic load was applied at a rate of 30mm/min. In the undeformed state, the height of the sample is H . In the deformed state, the height of the sample becomes H λ , where λ is the applied stretch. The horizontal deformation was constrained by the rigid grippers.
We plotted the stress-stretch curves of the uncut samples with three AAM concentrations in Fig. 6d-f.
The cut sample with the same geometry was prepared. A 20-mm crack was cut at the edge by using a razor blade. We applied the same load as that for the uncut samples and observed the opening and propagation of the crack (Fig. 6c)  For the pure shear test, the stress-stretch relation with the neo-Hookean model is given by Fitting the beginning portion of the curves in Fig. 6(d)-(f) gives the shear modulus of all the hydrogels roughly 100 kPa (See Appendix for fitting).
For the pure shear test, the energy release rate takes the form [79].
( ) where H is the distance between the two grippers when the sample is undeformed,

( )
W λ is the energy per volume of the uncut samples, and λ is the vertical stretch.
The energy density ( ) W λ is obtained by integrating the area below the stress-stretch curves of the uncut samples (Fig. 7a).
The toughness (i.e., the fracture energy) is defined as the critical energy release rate when the crack propagates. The toughness is calculated by replacing the stretch λ in ( ) W λ with the critical stretch of the cut samples, C λ . The calculated toughness of the hydrogels prepared using the solutions of different AAM concentrations are shown in Fig. 7b. Each error bar shows the three repeated measurements. By comparison, the toughness of single-network polyacrylamide hydrogel is about 50J/m 2 [73], and the toughness of natural rubber is about 10,000J/m 2 [74].

Fatigue damage of uncut samples
The stress-stretch curve of a DN hydrogel degrades under cyclic load because the short-chain network partially ruptures [72]. We now test how the state of rupture evolves over a large number of cycles. Using the pure shear setup, we applied cyclic loads to the uncut samples with a triangular loading profile of a frequency 0.4 Hz.
The minimum load and mean load may affect the fatigue properties of the DN hydrogel. Such effects have been studied for the fatigue fracture of elastomers [80].
Here we set the loads cycle between λ = 1 MIN and a maximum stretch MAX λ for simplicity (Fig. 8a).
The stress-stretch curve of the PAMPS/PAAM DN hydrogel lowers cycle by cycle (Fig. 8). Apparent residual stretch is found for each cycle when we unload the sample to zero stress. When we further unload the sample to λ = 1 MIN , the sample suffers compressive stress and buckles. The parts of stress-stretch curve with negative stress are not shown in Fig.8. The maximum stress at the imposed stretch keeps decreasing over cycles. The hysteresis in the first cycle is much larger than the subsequent cycles.
The change of the stress-stretch loops is negligible after 2000 cycles, and we say that the sample has shaken down to a steady state. By comparison, the polyacrylamide hydrogel shows little hysteresis, and the stress-stretch curves remain nearly unchanged cycle by cycle [73]. The alginate/polyacrylamide hydrogel shows even more significant hysteresis and shakedown than the PAMPS/PAAM hydrogel [74]. The shakedown phenomenon has been observed and studied in elastic-plastic materials, such as metals [83][84][85][86][87][88][89]. The shakedown of hydrogels over a large number cycles was reported for the alginate/polyacrylamide hydrogel [74], and now for the The maximum stress drops greatly in the beginning cycles and reaches a steady state after thousands of cycles (Fig. 9a-c). Following [74], we use the stress-stretch curve of the 2000 th cycle to represent the deformation behavior of the PAMPS/PAAM hydrogels under cyclic load in the steady state ( Fig. 9d-f).

Fatigue facture of cut samples
To study the fatigue fracture of the DN hydrogels, we made a thin sheet of the same rectangular shape (10 mm × 50 mm), and cut a 20-mm crack using a razor blade. The cyclic loads with different maximum stretches MAX λ were applied to the cut samples. The crack propagated cycle by cycle and the extension of crack was recorded by a digital camera (Canon EOS5D) (Fig. 10a). We plotted the extension of crack ∆c as a function of the number of cycles N (Fig. 10b-d).

Table1 Experimental data of the fatigue fracture tests
To compare the data of fatigue fracture of different materials, it is a common practice to plot the extension of crack per cycle, / dc dN , as a function of the energy release rate, G . The practice was first reported for elastomers by Thomas in 1958 [81]. A similar practice using the stress intensity factor was first reported for metals by Paris et al. in 1961 [90]. For metals, fatigue fracture is commonly studied when deformation is small, except for a small zone around the tip of the crack. Outside the small zone, the material is well described by the linear theory of elasticity, and the stress intensity factor is a measure of the applied load. In the linear theory of elasticity, for a crack in a thin sheet, the stress intensity factor K relates to the energy release rate G as K EG = , where E is Young's modulus. Thus, the energy release rate and the stress intensity factor are alternative, but equivalent, measures of the applied load.
For elastomers, fatigue fracture is commonly studied when deformation is large in the entire sample. Still, for a highly elastic elastomer, outside a small zone around the tip of the crack, the material is well described by the nonlinear theory of elasticity, and the energy release rate is a measure of the applied load.
Consider a crack in an elastic material, linearly or nonlinearly elastic. So long as the inelastic zone around the tip of the crack is small compared to the size of the sample, the energy release rate is a measure of the applied load. Near the crack tip, the only length scale is the distance R of a material particle in the sample from the tip of the crack. From a dimensional consideration, the elastic energy per unit volume with the unit J/m 3 is proportional to the energy release rate with the unit J/m 2 by / G R. For tough hydrogels, the entire sample is inelastic during fracture. This large-scale inelasticity is evident in the pronounced stress-stretch hysteresis and subsequent shakedown. Because no where in the sample does the theory of elasticity apply, it is unclear if the energy release rate is an adequate measure of the applied load.
Still, it is desirable to use a parameter to compare different materials.
Following previous studies of fatigue fracture of hydrogels [73,74] and elastomers [81], we calculate an effective energy release rate using a procedure described below.
We use the stress-stretch curve of the 2000 th cycle of the uncut samples, when the hydrogels approach the steady state (Fig. 9d-f). We integrate the loading part of the stress-stretch curve to calculate the energy density The steady-state extension of crack per cycle / dc dN is plotted as a function of the effective energy release rate G (Fig. 11). In testing established materials, such as a commercial metallic alloy, a commonly adopted operational definition for fatigue threshold is the energy release rate at which the crack extends at a particular rate, say 10 -11 m/cycle [83,91]. If the method used to measure crack length is accurate to 10 -4 m, and if the extension of the crack is undetected for 10 7 cycles, a threshold is declared. This large number of cycles takes several months of nonstop testing at the rate of 1 second per cycle. The time is too long to be practical during the development of new materials. Instead, we adopt the following procedure to estimate threshold [74].
We linearly extrapolate the data points in the plane of G and  (Table 1).

Prediction of threshold using the Lake-Thomas model
The threshold of fatigue fracture in an elastomer is commonly interpreted in terms of the Lake-Thomas model [82]. The model assumes that, at a load approaching the threshold, the extension of a crack only activates a single dissipative process in the elastomer: breaking polymer chains ahead the crack. That is, the model assumes that the hysteresis in the bulk of the elastomer does not contribute to the threshold. The chemical bonds along each polymer chain are much stronger than the physical interactions between different polymer chains. Consequently, when a chain is pulled nearly to the breaking point, all the chemical bonds of the entire chain is pulled to the same state. When the chain breaks at a single bond, the energy in the entire chain dissipates. The Lake-Thomas model estimates the threshold by the chemical energy in a unit area of a single layer of chains.
Incidentally, the Lake-Thomas model is analogous to a lower-bound model for the threshold of fatigue fracture in a metal. In a metal, bonds between any two atoms are comparably strong, and the lower-bound model estimates the threshold by the energy needed to break a single layer of atomic bonds [50]. That is, the model assumes that the hysteresis in the bulk of the sample does not contribute to the threshold, and estimates the threshold by the Griffith limit: The Lake-Thomas model has been adapted to a single-network PAAM hydrogel [73]. For the PAMPS/PAAM double-network hydrogels, we now hypothesize that the rupture of the PAMPS network does not contribute to the threshold, and estimate the threshold by the chemical energy in one layer of PAAM chains: The Lake-Thomas model predicts that if the crosslinker density of PAAM increases by ten times, the number of monomers between two crosslinks decreases to one tenth, resulting in a decrease of threshold by a factor of 10 . This prediction is captured in our experiments qualitatively, but not quantitatively.
"The number of monomers between two crosslinks is roughly = 7000 n for both PAAM gel and alginate/PAAM gel in Ref [73] and [74], and is

Concluding remarks
This paper is the first study of the fatigue of the classic PAMPS/PAAM double network hydrogels. We apply cyclic stretch to samples with or without cuts. For an uncut sample, damage accumulates over thousands of cycles, until a steady state is reached. For a cut sample, the crack extends cycle by cycle if the amplitude of the load is above a threshold. We prepare all samples of the DN hydrogels using PAMPS hydrogels of a fixed composition, but using aqueous solutions of various concentrations of AAM monomers and MBAA crosslinkers. Within the samples prepared for this work, the effect of the concentration of AAM is modest. The concentration of MBAA has a small effect on fracture energy, but significant effect on threshold energy. For the DN hydrogels prepared in the aqueous solution of 2M AAM and 10 -5 number fraction of MBAA relative to AAM, the fracture energy is 3779 J/m 2 , and the fatigue threshold is 418 J/m 2 . For the DN hydrogels prepared in the aqueous solution of 2M AAM and 10 -4 number fraction of MBAA relative to AAM, the fracture energy is 3066 J/m 2 , and the fatigue threshold is 220 J/m 2 .
We adapt the Lake-Thomas model under the hypothesis that the fatigue threshold of a DN hydrogel corresponds to the chemical energy stored in one layer of PAAM chains. This hypothesis predicts the experimentally measured fatigue thresholds qualitatively.
The available data do not allow a comprehensive comparison of the fatigue-resistance of hydrogels of different chemistries, such as the alginate/PAAM (Fig. 4) and PAMPS/PAAM (Fig. 11), given that fatigue resistance depends on the concentrations of ingredients. It is hoped that researchers worldwide will report the fatigue behavior of their own hydrogels under development for load-bearing applications, and that an understanding will soon emerge to link the behavior of fatigue to the chemistry of hydrogels.