Oscillating fracture paths in rubber

Programming fracture patterns of thin films

Controlled fracture presents opportunities for the advanced fabrication of thin films. However, programmability analogous to that of Chinese paper cutting is still challenging, where fracture patterns can be created as required without preformed cracks for guidance. Here, we establish a design framework for tearing adhesive thin films from foldable substrates with such programmability. Our analytical model captures the observed crack behavior, demonstrating that the deflection of crack paths can exceed 90^{∘}. Besides, for thick foldable substrates with multiple ridges, we additionally propose a robust method of directional fracture where the cracks are forced to extend along the ridges.

Gel rupture during dynamic swelling

Hydrogels have had a profound impact in the fields of tissue engineering, drug delivery, and materials science as a whole. Due to the network architecture of these materials, imbibement with water often results in uniform swelling and isotropic expansion which scales with the degree of cross-linking. However, the development of internal stresses during swelling can have dramatic consequences, leading to surface instabilities as well as rupture or bursting events. To better understand hydrogel behavior, macroscopic mechanical characterization techniques (e.g. tensile testing, rheometry) are often used, however most commonly these techniques are employed on samples that are in two distinct states: (1) unswollen and without any solvent, or (2) in an equilibrium swelling state where the maximum amount of water has been imbibed. Rarely is the dynamic process of swelling studied, especially in samples where rupture or failure events are observed. To address this gap, here we focus on rupture events in poly(ethylene glycol)-based networks that occur in response to swelling with water. Rupture events were visualized using high-speed imaging, and the influence of swelling on material properties was characterized using dynamic mechanical analysis. We find that rupture events follow a three-stage process that includes a waiting period, a slow fracture period, and a final stage in which a rapid increase in the velocity of crack propagation is observed. We describe this fracture behavior based on changes in material properties that occur during swelling, and highlight how this rupture behavior can be controlled by straight-forward modifications to the hydrogel network structure.

Crack-Tip Strain Field in Supershear Crack of Elastomers

We characterize the crack-tip strain field in the high-speed (supershear) crack in the elastomers propagating faster than the shear wave speed of sound (C_s). The dependence of steady-state crack velocity (V) on input tearing energy exhibits a crossover at V ≈ C_s between the subsonic (V < C_s) and supershear cracks (V > C_s). Several features of the crack-tip strain field such as strain-magnitude, extent boundary, and singularity exponent also change substantially accompanying the transition from subsonic to supershear cracks. The definite crossover of these characteristics at V ≈ C_s reflects the variations in the crack-growth mechanism: The inertia effect comes into play in the supershear crack. We also demonstrate that the azimuthal distribution of the local crack-tip strain has a close correlation with the macroscopic crack-tip shape, regardless of the regime of V.

Nature of Crack Path Instabilities in Thin Sheets Cut by Blunt Objects

Cutting a brittle thin sheet with a blunt object leaves an oscillating crack that seemingly violates the principle of local symmetry for fracture. We experimentally find that at a critical value of a well chosen control parameter the straight propagation is unstable and leads to an oscillatory pattern whose amplitude and wavelength grow by increasing the control parameter. We propose a simple model that unifies this instability with a related problem, namely that of a perforated sheet, where through a similar bifurcation a series of radial cracks spontaneously spiral around each other. We argue that both patterns originate from the same instability.

Popping Balloons: A Case Study of Dynamical Fragmentation

Understanding the physics of fragmentation is important in a wide range of industrial and geophysical applications. Fragmentation processes involve large strain rates and short time scales that take place during crack nucleation and propagation. Using rubber membranes, we develop an experimental analysis that enables us to track the fragmentation process in situ in both time and space. We find that bursting a highly stretched membrane yields a treelike fragmentation network that originates at a single seed crack, followed by successive crack tip-splitting events. We show that a dynamic instability drives this branching mechanism. Fragmentation occurs when the crack tip speed attains a critical velocity for which tip splitting becomes the sole available mechanism of releasing the stored elastic energy. Given the general character of the fragmentation processes, this framework should be applicable to other crack networks in brittle materials.

The transition from subsonic to supersonic cracks

We present the full analytical solution for steady-state in-plane crack motion in a brittle triangular lattice. This allows quick numerical evaluation of solutions for very large systems, facilitating comparisons with continuum fracture theory. Cracks that propagate faster than the Rayleigh wave speed have been thought to be forbidden in the continuum theory, but clearly exist in lattice systems. Using our analytical methods, we examine in detail the motion of atoms around a crack tip as crack speed changes from subsonic to supersonic. Subsonic cracks feature displacement fields consistent with a stress intensity factor. For supersonic cracks, the stress intensity factor disappears. Subsonic cracks are characterized by small-amplitude, high-frequency oscillations in the vertical displacement of an atom along the crack line, while supersonic cracks have large-amplitude, low-frequency oscillations. Thus, while supersonic cracks are no less physical than subsonic cracks, the connection between microscopic and macroscopic behaviour must be made in a different way. This is one reason supersonic cracks in tension had been thought not to exist.

Geometric control of failure behavior in perforated sheets

Adding perforations to a continuum sheet allows new modes of deformation, and thus modifies its elastic behavior. The failure behavior of such a perforated sheet is explored, using a model experimental system: a material containing a one-dimensional array of rectangular holes. In this model system, a transition in failure mode occurs as the spacing and aspect ratio of the holes are varied: rapid failure via a running crack is completely replaced by quasistatic failure, which proceeds via the breaking of struts at random positions in the array of holes. I demonstrate that this transition can be connected to the loss of stress enhancement, which occurs as the material geometry is modified.

The dynamics of rapid fracture: instabilities, nonlinearities and length scales

The failure of materials and interfaces is mediated by cracks, almost singular dissipative structures that propagate at velocities approaching the speed of sound. Crack initiation and subsequent propagation-the dynamic process of fracture-couples a wide range of time and length scales. Crack dynamics challenge our understanding of the fundamental physics processes that take place in the extreme conditions within the almost singular region where material failure occurs. Here, we first briefly review the classic approach to dynamic fracture, namely linear elastic fracture mechanics (LEFM), and discuss its successes and limitations. We show how, on the one hand, recent experiments performed on straight cracks propagating in soft brittle materials have quantitatively confirmed the predictions of this theory to an unprecedented degree. On the other hand, these experiments show how LEFM breaks down as the singular region at the tip of a crack is approached. This breakdown naturally leads to a new theoretical framework coined 'weakly nonlinear fracture mechanics', where weak elastic nonlinearities are incorporated. The stronger singularity predicted by this theory gives rise to a new and intrinsic length scale, ℓnl. These predictions are verified in detail through direct measurements. We then theoretically and experimentally review how the emergence of ℓnl is linked to a new equation for crack motion, which predicts the existence of a high-speed oscillatory crack instability whose wavelength is determined by ℓnl. We conclude by delineating outstanding challenges in the field.

Microbranching in mode-I fracture in a randomly perturbed lattice

We study mode-I fracture in lattices using atomistic simulations with randomly distributed bond lengths. By using a small parameter that measures the variation of the bond length between the atoms in perfect lattices and using a three-body force law, simulations reproduce the qualitative behavior of the beyond-steady-state cracks in the high-velocity regime, including reasonable microbranching. In particular, the effect of the lattice structure on the crack appears minimal, even though in terms of the physical properties such as the structure factor g(r) and the radial or angular distributions, these lattices share the physical properties of perfect lattices rather than those of an amorphous material (e.g., the continuous random network model). A clear transition can be seen between steady-state cracks, where a single crack propagates in the midline of the sample, and the regime of unstable cracks, where microbranches start to appear near the main crack, in line with previous experimental results. This is seen in both a honeycomb lattice and a fully hexagonal lattice. This model reproduces the main physical features of propagating cracks in brittle materials, including the total length of microbranches as a function of driving displacement and the increasing amplitude of oscillations of the electrical resistance. In addition, preliminary indications of power-law behavior of the microbranch shapes can be seen, potentially reproducing one of the most intriguing experimental results of brittle fracture. There was found to exist a critical degree of disorder, i.e., a sharp threshold between the cleaving behavior characterizing perfect lattices and the microbranching behavior that characterizes amorphous materials.

Oscillatory instability in slow crack propagation in rubber under large deformation

We performed experiments to investigate slow fracture in thin rubber films under uniaxial tension using high-viscosity oils. In this system we observed an oscillating instability in slowly propagating cracks for small applied strains. The transition between oscillatory and straight patterns occurred near the characteristic strain at which rubber exhibits a nonlinear stress-strain relation. This suggests that nonlinear elasticity plays an important role in the formation of the observed pattern. This was confirmed by numerical simulation for neo-Hookean and linear elasticity models.

Patterning by controlled cracking

Crack formation drives material failure and is often regarded as a process to be avoided. However, closer examination of cracking phenomena has revealed exquisitely intricate patterns such as spirals, oscillating and branched fracture paths and fractal geometries. Here we demonstrate the controlled initiation, propagation and termination of a variety of channelled crack patterns in a film/substrate system comprising a silicon nitride thin film deposited on a silicon substrate using low-pressure chemical vapour deposition. Micro-notches etched into the silicon substrate concentrated stress for crack initiation, which occurred spontaneously during deposition of the silicon nitride layer. We reproducibly created three distinct crack morphologies--straight, oscillatory and orderly bifurcated (stitchlike)--through careful selection of processing conditions and parameters. We induced direction changes by changing the system parameters, and we terminated propagation at pre-formed multi-step crack stops. We believe that our patterning technique presents new opportunities in nanofabrication and offers a starting point for atomic-scale pattern formation, which would be difficult even with current state-of-the-art nanofabrication methodologies.

Intrinsic nonlinear scale governs oscillations in rapid fracture

When branching is suppressed, rapid cracks undergo a dynamic instability from a straight to an oscillatory path at a critical velocity v(c). In a systematic experimental study using a wide range of different brittle materials, we first show how the opening profiles of straight cracks scale with the size ℓ(nl) of the nonlinear zone surrounding a crack's tip. We then show, for all materials tested, that v(c) is both a fixed fraction of the shear speed and, moreover, that the instability wavelength is proportional to ℓ(nl). These findings directly verify recent theoretical predictions and suggest that the nonlinear zone is not passive, but rather is closely linked to rapid crack instabilities.

Self-oscillating regime of crack propagation induced by a local phase transition at its tip

We report an analytical study of propagation of a straight crack with a stress-induced local phase transition at the tip. We obtain its contribution to the dynamic fracture energy in explicit form and demonstrate that it nonmonotonically depends upon the crack tip velocity. We show that its descending part gives rise to the instability of the steady propagation regime. We obtain the dynamic phase diagram and indicate those domains where self-oscillating regimes of the crack motion take place.

Solidification and ordering during directional drying of a colloidal dispersion

During drying, colloidal dispersions undergo processes such as solidification, cracking, and the draining of interstitial pores. Here we show that the solidification of polystyrene and silica dispersions, during directional drying, occurs in two separate stages. These correspond to the initial ordering and subsequent aggregation of the colloidal particles. Transitions between these stages are observed as changes in transparency and color that propagate as distinct fronts along the drying layer. The dynamics of these fronts are shown to arise from a balance between compressive capillary forces and the electrostatic and van der Waals forces described by DLVO theory. This suggests a simple method by which the maximum interparticle repulsion between particles can be measured through the optical inspection of the dynamics of a drying dispersion, under a microscope.

Acquisition of inertia by a moving crack

We experimentally investigate the dynamics of "simple" tensile cracks. Within an effectively infinite medium, a crack's dynamics perfectly correspond to inertialess behavior predicted by linear elastic fracture mechanics. Once a crack interacts with waves that it generated at earlier times, this description breaks down. Cracks then acquire inertia and sluggishly accelerate. Crack inertia increases with crack speed v and diverges as v approaches its limiting value. We show that these dynamics are in excellent accord with an equation of motion derived in the limit of an infinite strip [M. Marder, Phys. Rev. Lett. 66, 2484 (1991)].

A convective instability mechanism for quasistatic crack branching in a hydrogel

Experiments on quasistatic crack propagation in gelatin hydrogels reveal a new branching instability triggered by wetting the tip opening with a drop of aqueous solvent less viscous than the bulk one. We show that the emergence of unstable branches results from a balance between the rate of secondary crack growth and the rate of advection away from a non-linear elastic region of size G/E , where G is the fracture energy and E the small strain Young modulus. We build a minimal, predictive model that combines mechanical characteristics of this mesoscopic region and physical features of the process zone. It accounts for the details of the stability diagram and lends support to the idea that non-linear elasticity plays a critical role in crack front instabilities.

Nanoscale periodic morphologies on the fracture surface of brittle metallic glasses

Out-of-plane, nanoscale periodic corrugations are observed in the dynamic fracture surface of brittle bulk metallic glasses with fracture toughness approaching that of silica glasses. A model based on the meniscus instability and plastic zone theory is used to explain such dynamic crack instability. The results indicate that the local softening mechanism in the fracture is an essential ingredient for controlling the formation of the unique corrugations, and might provide a new insight into the origin of fracture surface roughening in brittle materials.

Motion of a viscoelastic micellar fluid around a cylinder: flow and fracture

We present an experimental study of the motion of a viscoelastic micellar fluid around a moving cylinder, which ranges from fluidlike flow to solidlike tearing and fracture, depending on the cylinder radius and velocity. The observation of crack propagation driven by the cylinder indicates an extremely low tear strength, approximately equal to the steady state surface tension of the fluid. At the highest speeds a driven crack is observed in front of the cylinder, propagating with a fluctuating speed equal on average to the cylinder speed, here as low as 5% of the elastic wave speed.

Oscillatory instability in two-dimensional dynamic fracture

The stability of a rapid dynamic crack in a two-dimensional infinite strip is studied in the framework of linear elastic fracture mechanics supplemented with a modified principle of local symmetry. It is predicted that a single crack becomes unstable by a finite wavelength oscillatory mode at a velocity vc, 0.8cR<vc<0.85cR, where cR is the Rayleigh wave speed. The relevance of this theoretical calculation to the oscillatory instability reported in the companion experimental Letter is discussed.

Oscillations in rapid fracture

Experiments of pure tensile fracture in thin brittle gels reveal a new dynamic oscillatory instability whose onset occurs at a critical velocity, VC=0.87CS, where CS is the shear wave speed. Until VC, crack dynamics are well described by linear elastic fracture mechanics (LEFM). These extreme speeds are obtained by suppression of the microbranching instability, which occurs when sample thicknesses are made comparable to the minimum microbranch width. The wavelength of these sinusoidal oscillations is independent of the sample dimensions, thereby suggesting that these macroscopic effects are due to an intrinsic microscopic scale that is unrelated to LEFM.

Hyperelasticity, viscoelasticity, and nonlocal elasticity govern dynamic fracture in rubber

Dynamic cracks in rubber can spontaneously oscillate under certain biaxial strain conditions [R. D. Deegan et al., Phys. Rev. Lett. 88, 014304 (2002)]. We have found that this unusual phenomenon can be understood from the unique mechanical properties of rubber: hyperelasticity, viscoelasticity, and nonlocal elasticity. While all these are important, the decisive role of nonlocality needs to be particularly emphasized. Through numerical simulations with a lattice model, we have quantitatively reproduced the experimental results.

Cracks in thin sheets: when geometry rules the fracture path

We study the propagation of brittle fractures coupled to large out-of-plane bending, as when a brittle elastic thin sheet is cut by a moving object. Taking into account the separation of the film's bending and stretching energies and using fracture theory we show that such cracks propagate according to a simple set of geometrical rules in the limit of small thickness. In particular, this provides some insight into the geometrical origin of the oscillatory fracture patterns reported in two recent experiments. Numerical integration of our geometrical rules accurately reproduces both the shape of the fracture pattern and the detailed time evolution of the propagation of the crack tip, for various geometries of the cutting object.

Crack motion in viscoelastic solids: the role of the flash temperature

We present a simple theory of crack propagation in viscoelastic solids. We calculate the energy per unit area, G(v), to propagate a crack, as a function of the crack tip velocity v. Our study includes the non-uniform temperature distribution (flash temperature) in the vicinity of the crack tip, which has a profound influence on G(v). At very low crack tip velocities, the heat produced at the crack tip can diffuse away, resulting in very small temperature increase: in this "low-speed" regime the flash temperature effect is unimportant. However, because of the low heat conductivity of rubber-like materials, already at moderate crack tip velocities a very large temperature increase (of order of 1000 K) can occur close to the crack tip. We show that this will drastically affect the viscoelastic energy dissipation close to the crack tip, resulting in a "hot-crack" propagation regime. The transition between the low-speed regime and the hot-crack regime is very abrupt, which may result in unstable crack motion, e.g. stick-slip motion or catastrophic failure, as observed in some experiments. In addition, the high crack tip temperature may result in significant thermal decomposition within the heated region, resulting in a liquid-like region in the vicinity of the crack tip. This may explain the change in surface morphology (from rough to smooth surfaces) which is observed as the crack tip velocity is increased above the instability threshold.

Dynamic instabilities of fracture under biaxial strain using a phase field model

We present a phase-field model of the propagation of fracture under plane strain. This model, based on simple physical considerations, is able to accurately reproduce the different behavior of cracks (the principle of local symmetry, the Griffith and Irwin criteria, and mode-I branching). In addition, we test our model against recent experimental findings showing the presence of oscillating cracks under biaxial load. Our model again reproduces well observed supercritical Hopf bifurcation and is therefore the first simulation which does so.

Biocide squirting from an elastomeric tri-layer film

Protective layers typically act in a passive way by simply separating two sides. Protection is only efficient as long as the layers are intact. If a high level of protection has to be achieved by thin layers, complementary measures need to be in place to ensure safety, even after breakage of the layer-an important issue in medical applications. Here, we present a novel approach for integrating a biocide liquid into a protective film (about 300-500 microm thick), which guarantees that a sufficient amount of biocide is rapidly released when the film is punctured. The film is composed of a middle layer, containing the liquid in droplet-like compartments, sandwiched between two elastomeric boundary layers. When the film is punctured, the liquid squirts out of the middle layer. A theoretical model was used to determine the size and density of droplets that are necessary to ensure a sufficient quantity of biocide is expelled from an adequately elastic matrix to provide protection at the site of damage. We demonstrate the utility of this approach for the fabrication of surgical gloves.

Does the continuum theory of dynamic fracture work?

We investigate the validity of the linear elastic fracture mechanics approach to dynamic fracture. We first test the predictions in a lattice simulation, using a formula of Eshelby for the time-dependent stress intensity factor. Excellent agreement with the theory is found. We then use the same method to analyze the experiment of Sharon and Fineberg. The data here are not consistent with the theoretical expectation.