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Fracture Mechanics of Soft Materials

Rong Long's picture

Rong Long

Department of Mechanical Engineering, University of Colorado Boulder

Research on the fracture mechanics of soft materials is pioneered by Rivlin and Thomas [1] in 1950s when rubber became an important engineering material in applications such as tires and sealants.  As mentioned by Prof. Wolfgang Knauss in his 2010 Timoshenko Medal speech, explosion of the Challenger space shuttle in 1986 was due to the failure of two rubber O-rings under the low temperature environment on the launch day.  Interests in the fracture of soft materials were also driven by the invention of pressure sensitive adhesives, and were recently renewed due to the developments of tough hydrogels [2-4] and elastomers [5].  An excellent overview of the current understandings and challenges in this area can be found in a review article by Creton and Ciccotti [6]. The most prominent feature of soft material fracture is the large deformation around the tip of a sharp crack (see Fig.1 for example). This may lead to crack tip blunting and significant nonlinear effects at the crack tip region. Here, I would like to focus on the following three topics, which hopefully can provide some background information to initiate discussions.

Figure 1 Severe crack blunting in a pure shear fracture test specimen of a tough hydrogel (image from [7]).

Crack tip fields

In linear elastic fracture mechanics (LEFM), the Williams’ solutions revealing the square root singularity of the crack tip strain and stress fields are of foundational importance.  However, these solutions are based on the kinematic assumption of infinitesimal strain, and thus are not expected to be useful for soft materials with large deformation. Indeed, Livne et al. [8] demonstrated that the LEFM solutions break down near the tip of a dynamically propagating crack in a soft polyacrylamide gel (see Fig.2A). Similar observations were found for quasi-static cracks in an Agar gel by Lefranc and Bouchaud [9] (see Fig.2B).  To accommodate large deformation, the crack tip field analysis must be based on nonlinear constitutive relations and finite strain kinematics.  This was pioneered by Knowles and Sternberg [10] in 1973 where asymptotic crack tip solutions for plane strain Mode-I cracks in a class of compressible hyperelastic materials were solved. Since then many asymptotic crack tip solutions were derived for different geometrical conditions (e.g. plane stress versus plane strain) or constitutive relations, some of which were reviewed recently by Long and Hui [11].  Alternatively, Bouchbinder et al. [12] developed a weakly nonlinear solution for crack tip fields based on perturbation analysis of the LEFM crack tip solutions.

Most of the existing asymptotic solutions are for hyperelastic solids and Mode-I cracks, with a few exceptions where Mode-III cracks were considered [13].  The crack tip fields for more complex loadings (e.g. mix-mode) have not been sufficiently studied. Note that under large deformation, the mix-mode crack tip field can no longer be written as the superposition of three basic fracture modes due to nonlinear coupling effects. More importantly, for complex material behaviors such as anisotropic (e.g. biological tissue or fiber reinforced elastomer) or dissipative (e.g. viscoelastic), large deformation solutions of the crack tip field are not yet available; such solutions can offer greatly insights towards understanding the failure and dissipative processes around the crack tip.

Figure 2 (A) Top: crack opening profile for a dynamically propagating crack in polyacrylamide gel; the LEFM crack tip field solution is unable to capture the experimental data very close to the crack tip. Bottom: schematic of the nonlinear zones around the crack tip due to large deformation effects.  Images from [8]. (B) Top: schematic of the microfluidic device used to apply tensile loading to a pure shear crack specimen of Agar gel (red) through capillary forces. Bottom: crack opening displacement versus the distance to crack tip; the red symbols represent experimental data and the black solid line is obtained from LEFM crack tip field solution. Images from [9].

Dissipation and toughness

As nicely illustrated in a recent review by Zhao [14], the underlying mechanism for various tough gels and elastomers is to introduce energy dissipation into stretchy polymer networks, which can be achieved through different physical mechanisms [14]. From a mechanics perspective, energy dissipation, manifested in the loading-unloading hysteresis, leads to the formation of a dissipation zone that shields the crack tip from energetic driving forces supplied by external loading (see Fig.3).  This principle can be expressed mathematically as follows:

G = G_0 + G_D,

where G is the energy release rate supplied by external loading and G_0 is the energy release rate available to drive the material failure processes at crack tip.  The difference between G and G_0 is the energy release rate G_D required to develop and maintain a dissipation zone at crack tip. 

Figure 3 Schematic of the crack tip dissipation zone with loading/unloading hysteresis and the crack tip cohesive zone (image from [18]).

For viscoelastic materials, the dissipation term G_D is a function of temperature and crack propagation velocity (or the global loading rate), as shown in the experimental data of Gent [15] (see Fig.4A). Theoretically, the relation between G_D and viscoelastic relaxation can be understood based on the “viscoelastic trumpet” model proposed by de Gennes [16] (see Fig.4B). This model has enabled qualitative estimate of G_D and scaling relations between G_D and crack propagation velocity.  However, to precisely predict G_D, computational models incorporating i) nonlinear bulk viscoelasticity models and ii) cohesive zone models that describe the failure processes at crack tip are needed. Both are extremely challenging to obtain, and as a result quantitative prediction of G_D and hence G in viscoelastic materials has not been satisfactory [17]. 

On the other hand, Zhang et al. [18] recently made a very encouraging progress in the fracture of soft dissipative materials with rate-independent damage. They used a Mullin’s effect model to phenomenologically capture the damage mechanism in a tough hydrogel due to the breaking of sacrificial bonds, implemented a bilinear cohesive zone model at the fracture plane, and successfully predicted the critical condition for crack propagation (see Fig.4C). 

Figure 4 (A) Fracture energy of viscoelastic materials determined from tearing and peeling tests in a viscoelastic elastomer, where R is the crack propagation velocity and aT is the Williams-Landel-Ferry (WLF) time-temperature shift factor of viscoelasticity (image from [15]). (B) Illustration of the viscoelastic trumpet model consisting of a hard solid region at crack tip, a soft solid region far away from crack tip and a “liquid” zone in between where viscoelastic dissipation occurs (image from [16]). (C) Comparison between computational predictions and experimental results for crack propagation in a tough hydrogel (image from [18]).

Fatigue fracture

Fatigue fracture refers to crack growth under prolonged loading cycles with relatively low amplitudes. Understanding fatigue fracture in soft materials is of central importance to the reliability analysis of emerging soft material based robotic, electronic and biomedical devices. In comparison to fracture behaviors under monotonic loading, fatigue fracture of soft materials has been much less studied. 

I would like to highlight a few recently works in this area: Mzabi et al. [19], Bai et al. [20], and Fan et al. [21]. In these works, the fatigue resistance was measured using the pure shear crack geometry, and was characterized using the extension of crack length per loading cycle dc/dN as a function of energy release rate G. In particular, Mzabi et al. [19] investigated elastomers with different filler particle content and different crosslinking density. These elastomers exhibited different degrees of dissipation, and hence distinct fatigue resistance curves (i.e. dc/dN versus G curves). Interestingly, Mzabi et al. [19] demonstrated that the dc/dN results, when plotted against the local energy release rate G_0 instead of the global energy release rate G, collapsed into a single master curve. This result suggested the possibility of using the local energy release rate G_0 as a fatigue facture criterion for soft dissipative materials.  Bai et al. [20] reported the first study of fatigue fracture in tough hydrogels with sacrificial bonds. They found that the threshold energy release rate required for fatigue crack growth is much lower than that for crack growth under monotonic loading. It was also found that the tough gel exhibited a “shakedown” behavior, i.e. gradual softening under cyclic loading, which underlies the fatigue crack growth.  Fan et al. [21] studied fatigue crack growth in a stretchable acrylic elastomer and found that dc/dN follows a power law dependence on the energy release rate G.  These works are all experimental in nature. Theoretical frameworks to understand fatigue facture of soft materials, especially the role of energy dissipation, are yet to be established. 

Summary

A better understanding in soft material fracture will lead to 1) principles to guide the development of tough and fatigue resistant soft materials; 2) theoretical models to predict the critical conditions for crack growth in soft materials; 3) new experimental methods to efficiently characterize soft material fracture. However, the nonlinear effects associated with large deformation at the crack tip have posed significant challenges towards a complete understanding of soft material fracture. There are many open questions in this area and vast opportunities for future progresses on both experimental and theoretical fronts.

References

1.Rivlin, R.S. and Thomas, A.G., “Rupture of rubber. I. Characteristic energy for tearing”, Journal of Polymer Science Part A: Polymer Chemistry, 1953, 10, 291-318.

2.Gong, J.P., Katsuyama, Y., Kurokawa, T. and Osada, Y., “Double network hydrogels with extremely high mechanical strength”, Advanced Materials, 2003, 15, 1155-1158.

3.Sun, J.Y., Zhao, X., Illeperuma, W.R., Chaudhuri, O., Oh, K.H., Mooney, D.J., Vlassak, J.J. and Suo, Z., “Highly stretchable and tough hydrogels”, Nature, 2012, 489, 133-136.

4.Sun, T.L., Kurokawa, T., Kuroda, S., Ihsan, A.B., Akasaki, T., Sato, K., Haque, M.A., Nakajima, T. and Gong, J.P., “Physical hydrogels composed of polyampholytes demonstrate high toughness and viscoelasticity”, Nature Materials, 2013, 12, 932-937.

5.Ducrot, E., Chen, Y., Bulters, M., Sijbesma, R.P. and Creton, C., “Toughening elastomers with sacrificial bonds and watching them break”, Science, 2014, 344, 186-189.

6.Creton, C. and Ciccotti, M., “Fracture and adhesion of soft materials: a review”. Reports on Progress in Physics, 2016, 79, 046601.

7.Haque, M.A., Kurokawa, T., Kamita, G. and Gong, J.P., 2011. “Lamellar bilayers as reversible sacrificial bonds to toughen hydrogel: hysteresis, self-recovery, fatigue resistance, and crack blunting”, Macromolecules, 2011, 44, 8916-8924.

8.Livne, A., Bouchbinder, E., Svetlizky, I. and Fineberg, J., “The near-tip fields of fast cracks”. Science, 2010, 327, 1359-1363.

9.Lefranc, M. and Bouchaud, E., “Mode I fracture of a biopolymer gel: Rate-dependent dissipation and large deformations disentangled”, Extreme Mechanics Letters, 2014, 1, 97-103.

10.Knowles, J.K. and Sternberg, E., “An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack”, Journal of Elasticity, 1973, 3, 67-107.

11.Long, R. and Hui, C.Y., “Crack tip fields in soft elastic solids subjected to large quasi-static deformation - A review”, Extreme Mechanics Letters, 2015, 4, 131-155.

12.Bouchbinder, E., Livne, A. and Fineberg, J., “The 1/r singularity in weakly nonlinear fracture mechanics”, Journal of the Mechanics and Physics of Solids, 2009, 57, 1568-1577.

13.Knowles, J.K., “The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids”, International Journal of Fracture, 1977, 13, 611-639.

14.Zhao, X., “Multi-scale multi-mechanism design of tough hydrogels: building dissipation into stretchy networks”, Soft Matter, 2014, 10, 672-687.

15.Gent, A.N., “Adhesion and strength of viscoelastic solids. Is there a relationship between adhesion and bulk properties?” Langmuir, 1996, 12, 4492-44.

16.de Gennes, P.G., “Soft adhesives”, Langmuir, 1996, 12, 4497-4500.

17.Knauss, W.G., “A review of fracture in viscoelastic materials”, International Journal of Fracture, 2015, 196, 99-146.

18.Zhang, T., Lin, S., Yuk, H. and Zhao, X., “Predicting fracture energies and crack-tip fields of soft tough materials”, Extreme Mechanics Letters, 2015, 4, 1-8.

19.Mzabi, S., Berghezan, D., Roux, S., Hild, F. and Creton, C., “A critical local energy release rate criterion for fatigue fracture of elastomers”, Journal of Polymer Science Part B: Polymer Physics, 2011, 49, 1518-1524.

20.Bai, R., Yang, Q., Tang, J., Morelle, X.P., Vlassak, J., Suo, Z., “Fatigue fracture of tough hydrogels”, Extreme Mechanics Letters, 2017, 15, 91-96. 

21.Fan, W., Wang, Y. and Cai, S., “Fatigue fracture of a highly stretchable acrylic elastomer”, Polymer Testing, 2017, 61, 373-377.

 

 

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Comments

JDI's picture

A very timely discussion topic! I have two puzzles on this topic:

(1) It seems that traditional fracture criteria such as the critical SIF, J-integral may not be applicable to soft materials since large deformation and deformation hysteresis are common for soft materials. I wonder if there are any suitable criteria for crack propagation in soft materials. 

(2) In the literature, the pure shear test is usually adopted to determine the so called fracture toughness. According to my understanding, the pure shear test is applicabel to hyperelastic materials. For soft materials, such as tough hydrogels, hysteresis exists in the loading and unloading process. And the fracture toughness is usually rate dependent. How to understand the fracture toughness measured from the pure shear test then? Is it really a material parameter?

 

Rong Long's picture

Dear JDI,

Thanks for bringing up these two issues. Here are my thoughts:

(1) For soft materials under large deformation, the crack tip field may no longer be described by the linear elastic solutions. Instead it may depend on the nonlinear consititutive relation of the soft material. Therefore, the stress intensity factor (SIF) in linear elastic fracture mechanics is no longer applicable. For hyperelastic materials, one can obtain asymptotic solutions of the crack tip field and identify the amplitude of such asympotic solutions as the "SIF". But such definitions depend on the detailed form of the hyperelastic model, and thus may not be very useful. 

On the other hand, J-integral is still well defined for hyperelastic materials. Because of large deformation, J-integral should be written with respect to the reference configruation. It is equal to the energy release rate if the crack propagates straightly ahead.

If hysteresis is present, J-integral becomes tricky to interpret. The common practice is to use the critical energy release rate as a fracture criterion. Things can get more complicated for dissipative materials because the critical energy release rate to initiate crack growth and to to maintain a steady-state crack growth may be different. In this case, one may measure the energy release rate as a function of crack extension length, i.e. the crack growth resistance curve. 

(2) This is a great point. In a pure shear fracture test, the energy release rate can be calculated as G = W*H where H is the height of the sample. For hyperelastic materials, W is the strain energy density for material points far ahead of the crack tip. For dissipative materials with hysteresis, W should be interpreted as the work per unit volume doen to material points far ahead of the crack tip (e.g. see Section 3 of our recent paper). The critical value of G upon steady-state crack growth is the fracture toughness. 

With this definition, the fracture toughness is in general not a material parameters. It includes two terms: an intrinsic toughness associated with material failure processes at the crack tip and a dissipative toughness for the energy release rate consumed in a dissipation zone surronding the crack tip. The instrinsic toughness may be a material parameter (i.e. a local fracture criterion) but the dissipative one is not. For viscoelastic materials, the latter depends on crack velocity and temperature. If the dissipation zone size is comparable to the sample size, the dissipative toughness may depend on sample size too. For material with rate-independent damage (e.g. Mullin's effect), the dissipative toughness also depends on prestretch (e.g. see Zhang et al.). However, it is very challenging to separately measure the intrinsic and dissipative toughness in experiments. Most experimental measurements so far can only provide a total fracture toughness. On the other hand, there have been many theoretical and computational efforts to solve this problem as discussed in the Journal Club. 

JDI's picture

Dear Rong,

Thanks for your detailed explainations! They are very informative. The physics behind the fracture of soft materials is really very rich and worthy further investigations. It might be a good method to devide the fracture energy into the "intrinsic" part (local part) and dissipative part. The correponding experiment methods to measure the local fracture energy may be very important for this area. I wonder if there are suitable methods so far.  BTW, the cohesive zone medel may be very suitable to characterize the local  fracture energy as also mentioned in the work by Prof. X. H. Zhao. It becomes difficult again if we consider the rate effect. I guess the local fracture energy in most soft materials is also rate dependent such as physical hydrogels.    

JDI's picture

Dear Rong,

Thanks for your detailed explainations! They are very informative. The physics behind the fracture of soft materials is really very rich and worthy further investigations. It might be a good method to devide the fracture energy into the "intrinsic" part (local part) and dissipative part. The correponding experiment methods to measure the local fracture energy may be very important for this area. I wonder if there are suitable methods so far.  BTW, the cohesive zone medel may be very suitable to characterize the local  fracture energy as also mentioned in the work by Prof. X. H. Zhao. It becomes difficult again if we consider the rate effect. I guess the local fracture energy in most soft materials is also rate dependent such as physical hydrogels.    

Ruobing Bai's picture

Dear JDI,

Thanks for sharing all the good thoughts and questions. Here are some of my understanding.

If we follow the literature and devide the fracture energy into intrinsic part and dissipative part, then the physical picture of the intrinsic fracture energy would be something described by Lake and Thomas in their early work (1967 Lake & Thomas). That is, theoretically the intrinsic fracture energy only depends on the chemical network (such as chain length, crosslink density, water content etc.), but not the loading rate. Such a theoretical picture is hard to verify in practice, and this is related to finding a proper method to measure the local toughness. Rong has indeed mentioned a few methods so far, and I'd like to address some further details of these methods:

(a) in [18], Zhang et. al preload the hydrogel samples to very large stretch, and then treat the samples as new materials to measure the fracture energy. The measured fracture energy is ~300 J/m2, one order of magnitude higher than the usual number that Lake-Thomas model predicts (~10 J/m2). This method is simple to implement, but it is hard to exclude the effect of viscoelastic or other dissipation from the bulk residual, even after several pre-loading cycles.

(b) in [19], Mzabi et. al use digital image correlation to capture the full strain field in the sample during fatigue fracture test, and define a length scale H0 as the size of the highly stretched zone in the undeformed state. They then use the integral of the unloading curve and pure shear test to obtain a local energy release rate. The method is innovative, but is also limited to the requirement of obtaining the visible strain field.

(c) in [20], we use the traditional fatigue fracture test to obtain the intrinsic fracture toughness (or fatigue fracture threshold), below which fatigue fracture never happens. This method is straightforward, and gives consistent number as the Lake-Thomas model (~50 J/m2), but it really requires time-consuming cyclic loading tests under many different values of stretch per cycle.

All these methods have their advantages and shortages. Related to the study of elastomers, there is also one more method to obtain the intrinsic fracture toughness. That is to measure the fracture toughness vs. crack propagation speed (so-called G-v curve). Shaoting and Rong have discussed quite a few of this under the thread already. If one has a solid theoretical model of such behavior, the intrinsic fracture toughness can be readily derived from the experimentally measured curves. A good and successful example is the study on elastomers by Gent et. al (e.g. 1994 Gent & Lai, 1996 Gent). Following this method, there is an excellent work done by Baumberger et. al (2006 Baumberger, Caroli and Martina). They actually studied the G-v curve of a physical hydrogel, and the theoretical model is quite consistent with their experimental results. Indeed, in their paper, the local fracture energy is not rate dependent, but related to the polymer-chain-pulling-out from the network after overcoming the weak interaction between them (hydrogen bonding). I guess this study partially answers your question in the last sentence. For chemically crosslinked hydrogels, however, the study on rate dependency and intrinsic fracture energy is still not quite clear.

We will have a further work to study the intrinsic fracture toughness of a same hydrogel with different values of water content, and compare the experimental results to the theoretical prediction by the Lake-Thomas model. The comparison is not perfect, but does capture the trend. The study also indicates that there is more room for the theoretical improvement beyond the Lake-Thomas model, on predicting the intrinsic fracture toughness. We will post the paper on iMechanica once it is published.

Best regards,

Ruobing Bai

Rong Long's picture

Hi Ruobing,

Thanks for the great points, especially on the methods to measure local toughness! I look forward to reading your paper when it comes out. To clarify, for the method based on the G-v curve, do you mean to determine G_0 by extrapolating the G-v curve to the point of v=0? 

I totally agree that there are more questions regarding the intrinsic toughness, especially on how to link the intrinsic toughness to the molecular-level failure mechanisms at crack tip. The Lake-Thomas theory is for a well crosslinked network, but for more complex networks (e.g. physically associated networks) more understandings are needed. For example, in Ref. [9] of the Jounral Club thread, Lefranc and Bouchaud studied the fracture of Agar gel (physically associated via hydrogen bonds, similar to the gelatin gel studied in Baumberger et al. 2006). Interestingly, even though the gel exibhits essentially elastic response under rheological tests (i.e. storage modulus is much larger than the loss modulus and both are rate-independent), the measured fracture toughness is still rate-dependent. This was attributed to the reptation processes of chains when being pulled out of a junction zone (analgous to a crosslinking point), i.e. the chain pull-out is resisted by neighboring chains, which may lead to an effective local viscosity. This physical picture, although still speculative, suggests that the instrinsic toughness may be rate dependent in the Agar gel. 

JDI's picture

Dear Ruobing,

Thanks for sharing your knowledge on this point. I think Rong's comments have expressed my concern quite well. For rupture of polymer networks, it also dissipates energy to pull the chains out of the network. That's why I guess the "intrinsic" fracture energy might be rate-dependent. I am looking forward to reading your new work on this topic.

Best Regards,

Jun Luo 

Ruobing Bai's picture

Dear Rong and JDI,

Thank you both for raising this interesting point! Perhaps the way of artificially deviding the fracture toughness into intrinsic and extrinsic itself is too much for many hydrogels with complex structures. In that sense, studying the local field at the crack tip directly is really appreciated to overcome this complexity.

Then the discussion brings me a question: why is the intrinsic toughness (or local crack tip field, or fatigue threshold) important for real engineering applications? We've already known how to toughen a soft material by introducing energy dissipation. Or put it in this way: why do we want to enhance the intrinsic toughness (or local material strength)? One answer to this, which I'm familiar, is the "terrible" fatigue fracture resistance of the current materials to cyclic loading (it seems that you can do no much better than the number Lake & Thomas predict). Do you have any other cases in mind?

Thanks again,

Ruobing

Rong Long's picture

Hi Ruobing,

My take is that the amount of energy dissipation depends on intrinsic toughness. Higher intrinsic toughness can lead to a larger dissipation zone and hence more energy dissipation. In viscoelastic materials (e.g. Gent 1996 [15]) or materials with rate-independent hysteresis (e.g. Zhang et al. [18]), it was found that the dissipative toughness is proportional to the intrinsic toughness. Even though the magnitude of the intrinsic toughness may be tiny in comparison to the dissipative one, but an increase of the intrinsic toughness can be greatly amplified through energy dissipation and thus leads to a large increase in total toughness. In this sense, I tend to think the energy dissipation mechanism acts as an "amplifier" for the intrinsic toughness.

From a modeling perspective, a question of interest is how can we predict fracture behavior of the soft dissipative materials (e.g. for reliability analysis)? In my opinion, the local fracture criterion (e.g. a cohesive zone model at crack tip) is the best bet, since the dissipative part of the toughness can be affected by loading conditions and sample geometry. What type of traction-separation relation in the cohesive zone model should we use and does it matter? This also requires a good understanding of the local fracture process.

Thanks,

Rong 

linst06's picture

Hi Rong Long,

Thanks for giving a very nice summary and review of the field in fracture of soft materials. 

1. In addition to the intensive studies focusing on the deformed samples with notches, people studied the delayed fracture by performing creep tests on the samples without notch. I put the two papers I read monthes ago:

Leocmach, M., Perge, C., Divoux, T., Manneville, S., 2014. Creep and fracture of a protein gel under stress. Physical review letters 113, 038303.

 ADDIN EN.REFLIST Karobi, S.N., Sun, T.L., Kurokawa, T., Luo, F., Nakajima, T., Nonoyama, T., Gong, J.P., 2016. Creep Behavior and Delayed Fracture of Tough Polyampholyte Hydrogels by Tensile Test. Macromolecules 49, 5630-5636.

In both papers, the origin of the delayed fracture is asscociated with the thermal activated process which gives the viscoelsaticity I think  . Can you give some comments on these two types of approaches: deforming the sample with cuts; and deforming the sample without cuts?

2. The following paper first reported the time−temperature superposition principle in a polyampholyte hydrogel. I wonder does this principle work for all hydrogels with viscoelasticity? 

Sun, T.L., Luo, F., Hong, W., Cui, K., Huang, Y., Zhang, H.J., King, D.R., Kurokawa, T., Nakajima, T., Gong, J.P., 2017. Bulk Energy Dissipation Mechanism for the Fracture of Tough and Self-Healing Hydrogels. Macromolecules 50, 2923-2931.

Best,

Shaoting

Rong Long's picture

Hi Shaoting,

Thanks for the two very interesting points and the refenreces. 

1. I think delayed fracture should be associated with some kinetic processes within the material. For the two references you cited, it is due to the kinetic process of bond breaking and reformation. As you pointed out, the same process leads to viscoelasticity in the bulk material. For some hydrogels, it can also be due to the kinetic process of solvent transport, as shown in a previous work from Prof. Wei Hong's group

As for the difference between samples with and without cuts, I would think in samples witout cuts, failure can be sensitive to pre-existing defects unless they are smaller than a characteristic length scale (see a recent work from Prof. Zhigang Suo's group). Since these pre-existing defects are random in nature, one may observe a large sample-to-sample variation in the failure of samples without cuts. In contrast, in samples with cuts, the inentionally introduced crack is the dominate defect and one can obtain a much more consistent measurement of fracture toughness in this way. Prof. Wei Hong's paper has a nice discussion on this aspect. Perhaps one can better study the underlying mechanism of delayed fracture using samples with cuts. 

2. I guess for the time-temperature superposition (in terms fracture toughness versus crack propagation velocity) to work, at least three conditions are needed: i) the dissipative toughness dominates over the intrinsic one; ii) at the time-scale of interest, viscoelasticity is the main time dependent mechanism (e.g. no significant effect of poroelasticity); iii) the bulk viscoelasticity follows the time-tempreature superposition within a wide range of strain (from small to large strain) and rates. The third condition is included because conventional characterization of viscoelasticity is done at the small strain regime, but fracture can invovle large strain at the crack tip. Under large strain, the viscoelastic relaxation kinetics may be coupled to strain. In this case, the third condition may not be satisfied and I am not sure if the time-temperature superposition relation will still hold for the fracture toughness. 

Cai Shengqiang's picture

Dear Rong, 

Many thanks for such a timely review. Fracture in soft materials has been a really interesting topic in the community. 

Thanks for citing our work on the test of fatigue fracture of an acrylic elastomer (VHB).

We have another paper on fracture in liquid crystal elastomer (http://caigroup.ucsd.edu/pdf/2016-06.pdf). 

In the paper, we discovered another interesting toughening mechanism in polydomain LCE, which is stretch-induced polydomain-to-mondomain transition. It is somehow similar to phase transition toughening mechanism in ceramics such as zirconia.  More interestingly, because such transition can cause transparency change, the transitioning process during the crack progation is very easy to be observed as shown below.

 

Rong Long's picture

Dear Shengqiang,

Thanks for brining up your work on the fracture of LCE. It is very interesting! In the polydomain-to-monodomain transition, does the LCE change from isotropic to anisotropic? If so, I am interested in how the transition affects the crack tip field. Also, if one can quantify the change in transparency and correlate it with some measure of stress or strain, perhaps this offers a way to probe the crack tip field as the crack opens up?

Cai Shengqiang's picture

Yes. During the polydomain-to-mondomain transition, LCE changes from isotropic state to anisotropic state with alligned mesogenic monomer. I think one neatness of such transition during fracture is its high visibility. I would like to mention that such transition is reversible with hysteresis. As you can see, after the crack extends through the entire width of the sample, the material near the crack path turns back to turbid becuase of the unloading. The dissipated energy associated with such hysteric loading and unloading contributes significantly to the fracture toughness of polydomain LCE. 

It is indeed a good idea to correlate the transparency to stress/strain state. Careful and detailed calibration may not be easy though. 

Rong Long's picture

Thanks! This is indeed a very interesting system. One more question: in the last three images you posted, the the crack has propagated but some of the material behind the crack tip should be unloaded but is still transparent. Is this some kind kinetic effect? In other words, the phase transition takes time, and has not fully occurred in the unloaded material because of the fast crack propagation?

Teng zhang's picture

Dear Rong,

Excellent and timely review!

Most models you summarized are at the continuum level and assume homogeneous material (my understanding may not be complete), however, these already lead to very complicated and highly nonlinear couplings among several different mechanisms, such as Mullins effect, viscoelasticity and cavitation. One question is whether the material heterogeneity plays a role in determining the crack propagation?  

Many natural biological soft materials exhibit significant material heterogeneity, anisotropy and gradient, which are believed to play important roles in reducing stress concentration and enhancing the robustness of the structures. Here is one recent paper on the material gradient in tendon-bone insertion:

Rossetti, L., L. A. Kuntz, E. Kunold, J. Schock, K. W. Müller, H. Grabmayr, J. Stolberg-Stolberg et al. "The microstructure and micromechanics of the tendon-bone insertion." Nature Materials 16, no. 6 (2017): 664-670.

A quantative understanding of how the micro- and meso-structures detrmine the overall mechanical properties, especially bulk and interface faluire is still missing.

In parallel to the great opportunities in developing accurate models and experiments to quantify the contributions to the toughness from various sources, it will also  be very important and challenging to link the micro- and meso-scale structures of the soft materials with their macro-scale mechanical properties, such as modulus, viscosity, energy dissipation and toughness. Indeed, the success of the double network hydrogel and elastomer originate from the synergistic coupling between polymer networks. Similar to your discussion with Ruobing, a further question may be how can we deliberately design the micro-structures to optimize the mechanical and physical properties of the soft tough materials? 

Thanks.

Teng

 

Rong Long's picture

Hi Teng,

Excellent points about the heterogeneity and microstructure design! In addition to the biological materials, there are many cases where fillers particles (e.g. silica particles, magnetic particles,...) are added to soft elastomers or gels to tune mechanical properties or to functionalize the material. Understanding the fracture of such soft composite materials are important too, but it may be a challenging task. As for the microstructure design, I think mechanics can can play an important role by providing directions or principles for design, and it may invovle modeling and experimental efforts at multiple scales. Continuum level studies can ellucidate what kind of bulk material behaviors (e.g. nonlinear elasticity, viscoelasticity, ...) and crack tip failure process (e.g. reflected in the traction separation relation in a cohesive zone) are optimal for toughness enhancement, which will set the targets for material design.

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