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# Weakly Nonlinear Theory of Dynamic Fracture

A fundamental understanding of the dynamics of brittle fracture remains a challenge of great importance for various scientific disciplines. From a theoretical point of view, a major difficulty in making progress in this problem stems from the fact that it intrinsically involves the coupling between widely separated time and length scales. Brittle fracture is ultimately driven by the release of linear elastic energy stored on large scales, while this energy is being dissipated in the very small scales near the front of a crack, where large stresses and deformations are concentrated and material separation is actually taking place. The strongly nonlinear and dissipative dynamics in the near vicinity of a crack's front controls the rate of crack growth and its direction, and hence its resolution seems relevant. Indeed, there are indications that various fast fracture instabilities (micro-branching, oscillations) are intimately related to the small scale physics near a crack's front. On the other hand, the phenomenology of brittle fracture appears to be rather universal, being qualitatively and quantitatively similar in materials with wholly different micro-structures and dissipative mechanisms (e.g. glassy polymers, structural glasses and elastomer gels).

These observations may suggest that a well-established theory of brittle fracture should incorporate a lengthscale that is associated with the near crack front region, but should otherwise be independent of the details of the small scales physics (unless one aims at calculating the fracture energy, i.e. the amount of energy needed to propagate a crack, instead of using it an a phenomenological material parameter). The canonical theory of fracture, linear elastic fracture mechanics (LEFM), is a scale-free theory and hence every lengthscale that appears in this framework is necessarily of a geometrical nature. This immediately implies that the identification of a non-geometric lengthscale entails the extension of LEFM when it breaks down near the front of a crack. As LEFM is based on a linear elastic constitutive behavior, which is only a first term in a more general displacement-gradients expansion, it is expected to break down near the front of a crack, where deformations become large enough to invalidate the linearity assumption. Progress in understanding the physics of this critical, near-front, nonlinear region has been, on the whole, limited by our lack of hard data describing the detailed physical processes that occur within. Due to the microscopic size and near–sound speed propagation of this region, it is generally experimentally intractable.

Recently, this experimental barrier was overcome by using a quasi-2D brittle neo-Hookean material (polyacrylamide gel) in which the fracture phenomenology mirrors that of more standard brittle amorphous materials (e.g. soda-lime glass and Plexiglass), but in which the near-tip (a crack-front becomes a crack-tip in 2D) region is significantly larger and moves significantly more slowly [1, 4]. The latter property allows unprecedented, direct and precise measurements of the near-tip fields of rapid cracks. These experiments revealed in detail how the canonical 1/√r fields and parabolic crack tip opening displacement (CTOD) of LEFM break down as the tip is approached.

To account for these observations, a weakly nonlinear theory of dynamic fracture was developed based on a systematic displacement-gradients expansion [2, 3]. The theory predicts novel, universal, 1/r singular displacement-gradients and log(r) displacements. It was shown to be in excellent quantitative agreement with the direct near-tip measurements of rapid cracks [2, 3]. The theory also resolves various puzzles in LEFM, such as the fact that the normal (to the crack propagation direction) component of the linear strain tensor ahead of a running crack becomes negative at sufficiently high speeds, which is physically unintuitive [2]. The presence of linear and weakly nonlinear terms in the crack tip solution allows the definition of a new lengthscale (basically by taking the ratio of these terms), that is shown to be related to a high-speed crack tip oscillatory instability [5]. This lengthscale may hold the key for unlocking various open questions in dynamic fracture. The special mathematical properties of the 1/r singularity (which is strictly forbidden in LEFM) and its relation to the concept of autonomy are discussed in detail in [3]. It is important to note that the weakly nonlinear theory is universally applicable since elastic nonlinearities must precede any irreversible behavior as the crack tip is approached.

A very recent combined experimental and theoretical study of the large deformation crack-tip region in a neo-Hookean brittle material revealed a hierarchy of linear and nonlinear elastic zones through which energy is transported before being dissipated at a crack’s tip [4]. This result provides a comprehensive picture of how remotely applied forces drive brittle failure and highlight the emergence of a lengthscale associated with nonlinear elastic effects, which are expected to precede near-tip dissipation. The results are corroborated by unprecedented direct measurements of the linear and nonlinear J-integral for cracks approaching the Rayleigh wave speed.

It is important to stress that LEFM works perfectly well for the gels used in the experiments where it should - not too close to the tip. In fact, these experiments provide the most comprehensive validation of LEFM under fully dynamic conditions, as this material has successfully “passed” every “test” that LEFM can throw its way (e.g. functional form of fields not too close to the tip, equations of motions in both an infinite medium and strip – soon to be published in Physical Review Letters); therefore, these results can be considered to be much more general than simply relevant for this class of neo-Hookean materials. Finally, these experiments have also demonstrated that – at least in this class of materials – nonlinear effects entirely dominate the behavior of the fields surrounding the crack’s tip. Dissipation may still be considered “point-like” – but this material shows that the two qualitatively different mechanisms for the breakdown of LEFM (nonlinear elasticity and dissipation) are separated. It remains to be seen if this is also characteristic for other materials. Currently, these materials are the only ones for which we have such detailed data.

[1] A. Livne, E. Bouchbinder, J. Fineberg,

**Breakdown of Linear Elastic Fracture Mechanics near the Tip **

** of a Rapid Crack**,

Phys. Rev. Lett. 101, 264301 (2008).

Also: ArXiv:0807.4866

[2] A. Livne, E. Bouchbinder, J. Fineberg, **Weakly Nonlinear Theory **

** of Dynamic Frcature**

Phys. Rev. Lett. 101, 264302 (2008).

Also: ArXiv:0807.4868

[3] E. Bouchbinder, A. Livne, J. Fineberg,

**The 1/r Singularity in Weakly Nonlinear Fracture Mechanics**

J. Mech. Phys. Solids 57, 1568 (2009).

Also: ArXiv:0902.2121

[4] A. Livne, E. Bouchbinder, I. Svetlizky, J. Fineberg,

**The Near-Tip Fields of Fast Cracks** Science 327, 1359 (2010).

[5] E. Bouchbinder

**Dynamic Crack Tip Equation of Motion: High-speed **

** Oscillatory Instability**

Phys. Rev. Lett. 103, 164301 (2009).

Also: ArXiv:0908.1178

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## Comments

## Re: Weakly Nonlinear Theory of Dynamic Fracture

Dear Eran:

Thanks for your post. I've already read your recent JMPS paper. Excellent work! Look forward to learning more about all this.

Regards,

Arash

## Papers on weakling nonlinear fracture

Hi Eran,

Can you attach preprints of your papers to your post for those of us without access to PRL and Science?

Thanks,

Biswajit

## Re: Papers on weakling nonlinear fracture

Hi Biswajit,

I added links to the free arXiv versions of all of the papers, except for the Science one (I'll ask Science about their policy in this regard and act accordingly).

Regards,

Eran