Although it is common to regard fracture mechanics as the study of brittle materials with low yield strains, the fracture mechanics of highly deformable materials such as elastomers and gels is itself a fascinating and important area of study. An important distinguishing feature of these soft materials is that linear elasticity can no longer be used to describe the elastically deformed region of the material. The stress concentration ahead of an advancing crack tip ensures that the material in this region is in a state of very large elastic strain. The fracture properties of the material are determined by the polymer response in this small, highly strained region; that is, fracture is a problem of large strains at small length scales. The breakdown of the defining assumption of linear elastic fracture mechanics has some interesting implications for the fracture behavior of elastomers and gels. In this viewpoint, I describe some of these implications and list a few outstanding problems that are ripe for additional experimental and theoretical work. This article represents my own point of view on a few relatively specific topics and is relatively brief for that reason. Readers interested in a more in-depth version of this article are encouraged to consult a pair of recent review articles.1, 2
Many of the important concepts in this area have been known for quite some time and represent the building blocks on which further progress will be made. As with much of fracture mechanics, stress-based and energy-based approaches both play important roles. A useful starting point is the work of Gent and Wang,3 who showed that bonded rubber samples fail in hydrostatic tension for values of the hydrostatic stress that are comparable to the elastic modulus. This result can be understood in terms of an energy balance. The overall deformation free energy of the system is minimized by the concentration of the strain within a small volume, thus providing a high driving force for failure. The result can be quantified by the expression of the energy-release rate, , which is the energetic driving force for crack propagation, as a function of the far-field hydrostatic tension, p. For an incompressible material under small-strain conditions, for a penny-shaped (circular) interfacial crack is given by the following expression:4
where R is the crack radius and E is the modulus of the material. Gent and Wang and others have shown that eq 1 greatly underestimates for values of p/E that are close to 1. This breakdown is illustrated in Figure 1, in which the value of /ER obtained from a Neohookean model of large-strain elasticity5 is compared to the prediction of eq 1.
A result of this relationship between the far-field stress and the local energy-release rate is that the far-field stress cannot exceed the elastic modulus of the material. The divergence in the crack driving force ensures that the material will fracture before this happens. The nature of the crack-tip region in a soft material is qualitatively different from what is observed in rigid materials, however. Before fracture, an initially sharp, penny-shaped crack in a soft material will become blunted to form a more spherical cavity. Many of the consequences of this idea were pointed out by Hui et al.6 in a very insightful paper that appeared in 2003. In an adhesive situation, the appropriate stress is the microscopic cohesive stress, which is defined as the interaction potential between the materials of interest. If this stress exceeds the modulus of a material, the crack tip will eventually become blunt, thereby limiting further increases in the maximum stress in front of the crack tip. The cohesive strength is generally quite large (hundreds of megapascals even for van der Waals interactions),7 so crack blunting is expected in most situations involving elastomers and gels. The picture is actually a bit more complicated because thermal fluctuations in soft materials can reduce the effective cohesive stress to a value that is much lower than the value that would be expected from a continuum picture.8 Nevertheless, crack blunting in soft materials can be clearly observed in a wide variety of situations.
One result of this crack-blunting picture, as pointed out by Hui et al.,6 is that the region ahead of the blunted crack tip can often be treated as an effective cohesive zone with a cohesive stress comparable to the elastic modulus of the material. This approach is useful in describing the overall performance of a pressure-sensitive adhesive, for example.9 In these cases, the deformation energy in the cohesive zone is dissipated irreversibly, and the total adhesion energy is roughly equal to its elastic modulus times the applied deformation, typically a few times the original film thickness. The detailed means by which the energy is dissipated requires the release of lateral constraints in the adhesive layer, typically by internal cavitation.2, 10
Some important questions remain, though, that are not directly addressed by this crack-blunting picture. In situations in which adhesive failure or fracture does occur, a stress exceeding the true, microscopic cohesive stress must eventually be overcome. The definition of a zone with a cohesive stress comparable to the elastic modulus of the undeformed material simply moves the true adhesion problem closer to the crack tip. Hui et al.6 suggested that strain hardening in this crack-tip region increases the modulus of the material to a value that is comparable to the microscopic cohesive stress. Although it is difficult to quantitatively specify the appropriate local modulus characterizing the strain-hardened material, this basic argument involving strain hardening must be at least qualitatively correct. The central issue is that the fracture behavior is ultimately determined by the large-strain deformation behavior of the material. To understand the actual adhesive performance of a material, however, one needs to determine how much of the elastic energy associated with these molecular deformations in the crack-tip region is dissipated irreversibly. The internal fracture of an elastomer and the contact behavior of silicone elastomers are two examples illustrating extreme possibilities.
The cohesive strength of crosslinked elastomers is an example in which the dissipated energy is quite high. Even in situations in which viscoelastic effects are eliminated completely, the tear energy of these materials exceeds the energy required to break covalent bonds across the interface by a factor between 10 and 100. The origin of this effect, as articulated originally by Lake and Thomas,11 is that the stored elastic energy in a large number of bonds is dissipated irreversibly when a single covalent bond is broken. This argument is appropriate when the failure of an individual covalent bond is a well-defined, discrete event. In the contrasting case involving the self-adhesion of silicone elastomers, the separation of the van der Waals bonds between the materials is a continuous process that proceeds reversibly, at least under very carefully controlled experimental conditions.12 In this case, none of the elastic energy stored in the crack-tip region is dissipated irreversibly. This distinction between reversible and irreversible bonds can be quantified by the consideration of the shape of the bonding potential and the way in which this potential is perturbed by an applied force. This model, refined and applied to single-molecule experiments by Evans and Ritchie,13 has recently been combined with a cohesive zone model to quantify rate-dependent effects in the interfacial failure of elastomeric materials.14 These rate effects will certainly be important in biomedical hydrogels, in which adhesive interactions are often mediated by receptor–ligand interactions that are highly rate-dependent.
These examples illustrate some of the important concepts underlying future work in this area:
- 1A divergence in the energy-release rate for values of the far-field hydrostatic tension comparable to the elastic modulus.
- 2The existence of different stress zones based on proximity to the crack tip:
A far-field zone, in which the stress cannot substantially exceed the elastic modulus.
A blunted zone, in which the stress is comparable to the elastic modulus.
A near-field zone, in which the stress is determined by the stress required to break individual bonds.
- 3Rate-dependent effects within each of the different zones of stress. These include, for, example, rate-dependent bond strengths in the near-field zone and viscoelastic-energy dissipation in the far-field zone.
I include three suggested avenues for further study. The first of these is a very general issue related to model refinement, the second is an important general issue related to two component gels, and the third is a specific issue involving the fracture mechanisms of supertough gels.
The first issue for further study is the experimental verification and refinement of the molecular models. This problem is a very general one requiring input from both experimentalists and theorists. A comprehensive view of the different outlined effects needs to be used. The real challenge is not necessarily to identify specific issues that are of importance but to develop a coherent picture that takes into account many of the features that are already known to be true. How, for example, are the different length scales mentioned previously accounted for in a self-consistent way? The use of well-characterized model systems will certainly play a role in this area. Self-assembling gels formed from block copolymers can be particularly useful in this case. These materials have very reproducible structures, with bond strengths corresponding to the stress required to disrupt the aggregates of solvophobic groups responsible for their elasticity.15
The second issue is the effect of osmotic compressibility in polymer gels. Polymer gels are not one-component systems but are actually elastic solutions of polymers in a small-molecule solvent. The solvent will be enriched in regions of hydrostatic tension that characterize the near-field zone in the immediate vicinity of the crack tip. Constitutive models that can model the effects of osmotic compressibility at high strains need to be developed and applied to understand the role of solvents in the fracture process. In addition, the dynamics of solvent flow at a moving crack tip will have some impact on the fracture properties of these materials, as illustrated recently by Baumberger et al.16 for gelatin gels. Further work in this area with model systems representative of different physical and chemical crosslinking mechanisms is needed to shed additional light on the nature of the fracture process in these types of materials.
The third and final issue is the toughness of double-network gels. Gong and coworkers17, 18 reported on a series of remarkably tough, double-network hydrogels that consist of two interpenetrating networks with very different crosslink densities. These gels mimic many of the favorable features of connective tissues in the body, maintaining very low friction at high normal stresses. They are exciting materials with a variety of potential uses in applications for which low sliding friction is desired. These uses are made possible only by the ability of these materials to support compressive stresses that are several times larger than the elastic moduli of the gels. The mechanism by which the addition of high-molecular-weight chains to a gel increases the energy dissipated during fracture is not completely clear and is quite likely related to the mechanisms responsible for the toughness of naturally occurring materials, including living tissues. I use it as a final example to illustrate the importance of an enhanced understanding of fracture processes in a variety of soft materials, in which energy dissipation is determined by the large-strain elastic response.