Can crack front waves explain the roughness of cracks?

We review recent theoretical progress on the dynamics of brittle crack fronts and its relationship to the roughness of fracture surfaces. We discuss the possibility that the small scale roughness of cracks, which is characterized by a roughness exponent ?0.5, could be caused by the generation, during local instabilities by depinning, of diffusively broadened corrugation waves, which have recently been observed to propagate elastically along moving crack fronts. We find that the theory agrees plausibly with the orders of magnitude observed. Various consequences and limitations, as well as alternative explanations, are discussed. We argue that another mechanism, possibly related to damage cavity coalescence, is needed to account for the observed large scale roughness of cracks that is characterized by a roughness exponent ?0.8.     

Brittle fracture dynamics with arbitrary paths I. Kinking of a dynamic crack in general antiplane loading

We present a new method for determining the elasto-dynamic stress fields associated with the propagation of anti-plane kinked or branched cracks. Our approach allows the exact calculation of the corresponding dynamic stress intensity factors. The latter are very important quantities in dynamic brittle fracture mechanics, since they determine the crack path and eventual branching instabilities. As a first illustration, we consider a semi-infinite anti-plane straight crack, initially propagating at a given time-dependent velocity, that changes instantaneously both its direction and its speed of propagation. We will give the explicit dependence of the stress intensity factor just after kinking as a function of the stress intensity factor just before kinking, the kinking angle and the instantaneous velocity of the crack tip.     

Modern topics and challenges in dynamic fracture

The field of dynamic fracture has been enlivened over the last 5 years or so by a series of remarkable accomplishments in different fields—earthquake science, atomistic (classical and quantum) simulations, novel laboratory experiments, materials modeling, and continuum mechanics. Important concepts either discovered for the first time or elaborated in new ways reveal wider significance. Here the separate streams of the literature of this progress are reviewed comparatively to highlight commonality and contrasts in the mechanics and physics.Much of the value of the new work resides in the new questions it has raised, which suggests profitable areas for research in the next few years and beyond. From the viewpoint of fundamental science, excitement is greatest in the struggle to probe the character of dynamic fracture at the atomic scale, using Newtonian or quantum mechanics as appropriate (a qualifier to be debated!). But lively interest is also directed towards modeling and experimentation at macroscales, including the geological, where the science of fracture is pulled at once by fundamental issues, such as the curious effects of friction, and the structural, where dynamic effects are essential to proper design or certification and even in manufacture.     

Scaling behavior of thermal shock crack patterns and tunneling cracks driven by cooling or drying

Cracks driven by shrinkage due to cooling or drying arrange themselves via mutual interaction. For parallel straight crack arrays driven by idealized transient shrinkage fields the scaling behavior in an infinite half-space is derived analytically by means of fracture mechanics bifurcation analysis with two plausible scaling assumptions. Crack spacing in thermal shock crack patterns has been found to be approximately proportional to the crack length and inversely proportional to the crack velocity. The spacing of tunneling cracks formed in a drying layer between plates scales as the 2/3rd power of layer thickness as a consequence of the specific interaction between the tunneling cracks. The difference in scaling behavior in the two cases is explained by the dimensionality of the geometrical setup determined by the boundary condition rather than by different physical processes. In either case, good agreement between theory and experiments is found.     

Brittle fracture dynamics with arbitrary paths. II. Dynamic crack branching under general antiplane loading

The dynamic propagation of a bifurcated crack under antiplane loading is considered. The dependence of the stress intensity factor just after branching is given as a function of the stress intensity factor just before branching, the branching angle and the instantaneous velocity of the crack tip. The jump in the dynamic energy release rate due to the branching process is also computed. Similar to the single crack case, a growth criterion for a branched crack is applied. It is based on the equality between the energy flux into each propagating tip and the surface energy which is added as a result of this propagation. It is shown that the minimum speed of the initial single crack which allows branching is equal to 0.39c, where c is the shear wave speed. At the branching threshold, the corresponding bifurcated cracks start their propagation at a vanishing speed with a branching angle of approximately 40°.     

On the dynamics of cracks in three dimensions

We introduce a three-dimensional dynamic crack propagation law, which is derived from Hamilton's principle. The result is an extension of a previous one obtained, corresponding to the two-dimensional case. As a matter of fact, in an orthogonal plane to the crack front, the geometric condition to be satisfied over the path is the same as in two dimensions. The third mode enters only through the energy release rate. The fact that the physics of the problem is locally two dimensional is a consequence of the virtual motions allowed in the set of admissible crack configurations.     

Brittle fracture dynamics with arbitrary paths III. The branching instability under general loading

The dynamic propagation of a bifurcated crack under arbitrary loading is studied. Under plane loading configurations, it is shown that the model problem of the determination of the dynamic stress intensity factors after branching is similar to the anti-plane crack branching problem. By analogy with the exact results of the mode III case, the energy release rate immediately after branching under plane situations is expected to be maximized when the branches start to propagate quasi-statically. Therefore, the branching of a single propagating crack under mode I loading should be energetically possible when its speed exceeds a threshold value. The critical velocity for branching of the initial single crack depends only weakly on the criterion applied for selecting the paths followed by the branches. However, the principle of local symmetry imposes a branching angle which is larger than the one given by the maximum energy release rate criterion. Finally, it is shown that an increasing fracture energy with the velocity results in a decrease in the critical velocity at which branching is energetically possible.     

Subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone

A recent experimental study has demonstrated the attainability of intersonic shear crack growth along weak planes in otherwise homogeneous, isotropic, linear elastic solids subjected to remote loading conditions (Rosakis et al., Science 284 (5418) (1999) 1337). The relevant experimental observations are summarized briefly here and the conditions governing the attainment of intersonic crack speeds are examined. Motivated by experimental observations, subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone is subsequently analyzed. A cohesive law is assumed, wherein the cohesive shear traction is either a constant or varies linearly with the local sliding rate. Complete decohesion is assumed to occur when the crack tip sliding displacement reaches a material-specific critical value. Closed form expressions are obtained for the near-tip fields. With a cohesive zone of finite size, it is found that the dynamic energy release rate is finite through out the intersonic regime. Crack tip stability issues are addressed and favorable speed regimes are identified. The influence of shear strength of the crack plane and of a rate parameter on crack propagation behavior is also investigated. The isochromatic fringe patterns predicted by the analytical solution are compared with the experimental observations of Rosakis et al. (1999) and comments are made on the validity of the proposed model.     

Crack movement in layered composites under electric field affect

The variation principle is applied for defining a crack in the solid body. The methods proposed in [G. Sih, C. Chen, Non-self-similar crack growth in elastic–plastic finite thickness plate, Theoretical and Applied Fracture Mechanics 3 (1985) 125–139] extend to presence of electromagnetic fields in material. Crack propagation in non-homogeneous media has been considered. It is shown that electromagnetic fields in the material are essentially affecting the trajectory. The crack trajectory stability has been studied as function of fracture energy, phase portraits of the trajectory in different media have been built, and various attractor types have been revealed. Different crack morphologies from single straight and oscillating crack propagation to straight double crack propagation were theoretically founded. In compliance with the experimental data of [R. Niefanger, V.-B. Pham, G. Schneider, H.-A. Bahr, H. Balke, U. Bahr, Quasi-static straight and oscillatory crack propagation in ferroelectric ceramics due to moving electric field: experiments and theory, Acta Materialia 52 (1) (2004) 117–127], it has been demonstrated that periodic electromagnetic field results in trajectory stochastization. This can be used for switching the crack over from the mode of mainline propagation into the mode of development of the field of diffused microcracks.     

Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis

Piezoelectric systems like multilayer actuators are susceptible to damage by crack propagation induced by strain incompatibilities. These can arise under electric fields for example between the electroded and external regions. Such incompatibilities have been realised in thin rectangular model specimens from PZT-piezoelectric ceramics with top and bottom electrodes only close to one edge. Under an electric field, controlled crack propagation has been observed in situ in an optical microscope. The crack paths are reproducible with very high accuracy. Small electrode widths lead to straight cracks with two transitions between stable and unstable crack growth regions, while large electrode widths result in curved cracks with four transitions. Fracture mechanics analysis is able to explain the different crack paths. An iteration method is developed to simulate the curved crack propagation also for strong curvature of the crack paths using the finite element method. The computed crack contours exhibit excellent quantitative agreement with the experiment with respect to their shape, the stages of stable and unstable crack propagation and the transitions between them. Finally, also the crack length as a function of the electric field can be predicted.     

Adiabatic decohesion in a thermoplastic craze thickening at constant or increasing rate

When a crack in a thermally non-diffusive material is impact loaded—or propagates at high speed—a cohesive process which resists slow crack extension may itself cause decohesion by adiabatic heating. By assuming that decohesion ultimately occurs by low-energy disentanglement within a melt layer of critical thickness, the fracture resistance of craze-forming crystalline polymers can be estimated quantitatively. Previous estimates used a simple, thermomechanically linear representation of craze fibril drawing. This paper presents a more physically realistic, numerical formulation, and demonstrates it for constant craze thickening rate (as imposed by an ideal full-notch tension test) and for linearly increasing thickening rate (as at the tip of an impact-loaded or rapidly propagating crack). For a linear material, the numerical formulation gives results which asymptotically approach those from analytical solutions, as craze density approaches zero. In more realistic model polymers, the enthalpy of fusion increasingly delays decohesion as impact speed increases, although the temperature distribution of an endotherm appears to have little effect. Increasing molecular weight, heuristically associated with decreasing craze density and increasing structural dimension, increases the predicted impact fracture resistance. In every case, fracture resistance passes through a minimum as impact speed increases. The conclusions encourage the use of impact fracture tests, and discourage the use of the full-notch tension test, to assess the dynamic fracture resistance of a craze-forming polymer.     

Comments on ‘‘Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution’’ by Shaofan Li, Peter A. Mataga [J. Mech. Phys. Solids 44 (1996) 1799-1830]

In this comment, it is pointed out that the paper [Li and Mataga, 1996. J. Mech. Phys. Solids 44, 1799-1830], which presents original and valid solution strategy for an important problem of dynamic crack propagation in piezoelectric materials, contains ultimate quantitatively and qualitatively incorrect expressions, conclusions and plots due calculation errors. The correct calculations and corresponding correct conclusions and plots are presented.     

Measurement of energy release rate and energy flux of rapidly bifurcating crack in Homalite 100 and Araldite B by high-speed holographic microscopy

High-speed holographic microscopy is applied to take three successive photographs of fast propagating cracks in Homalite 100 or in Araldite B at the moment of bifurcation. Crack speed at bifurcation is about 540 m/s on Homalite 100, and about 450 m/s on Araldite B. From the photographs, crack speeds immediately before and after bifurcation are obtained, and it is found that discontinuous change of crack speed does not exist at the moment of bifurcation in the case of Homalite 100, but exists in the case of Araldite B. From the photographs, crack opening displacement (COD) is also measured along the cracks as a function of distance r from the crack tips. The measurement results show that the CODs are proportional to √r before bifurcation. After bifurcation, the CODs of mother cracks are proportional to √r, though the CODs of branch cracks are not always proportional to √r. The energy release rate is obtained from the measured CODs, and it is found that energy release rate is continuous at bifurcation point in both cases of Homalite 100 and Araldite B. Energy flux that shows the energy flow toward a crack tip is also obtained.     

The adhesive contact of viscoelastic spheres

We have formulated the restricted self-consistent model for the adhesive contact of linear viscoelastic spheres. This model is a generalization of both the Ting (J. Appl. Mech. 33 (1966) 845) approach to the viscoelastic contact of adhesionless spheres and the restricted self-consistent model for adhesive axisymmetric bodies. We also show how the model can be used in practice by giving a few examples of numerical solutions.     

A fracture criterion for finitely deforming crystalline solids—The dynamic fracture of single crystals

The major objective of this work has been to develop, within a continuum framework, a microstructurally-based computational theory to investigate dynamic failure in metals. To model the nucleation and propagation of failure surfaces at the microstructural scale, under large deformations and dynamic loading conditions, general finite-deformation theory, as relating to the decomposition of the deformation gradient, was tailored to monitor displacement incompatibilities and fracture in crystalline solids subjected to large deformations. Based on this proposed decomposition, a general fracture criterion for finitely deforming crystals, using the integral law of incompatibility, was developed. The analyses indicate that this newly proposed fracture formulation and criterion can be validated with experimental results, and can be used to accurately predict brittle and ductile failure modes for the large deformation of single crystals. As part of the newly proposed decomposition of the deformation gradient, sub-problems can also be solved for lattice distortions, such as twinning and geometrically necessary dislocation (GND) densities. Accordingly, the interactions of GND densities with cracks were investigated for single crystals. GND densities were shown to form as loops for stationary crack tips, but no loops formed for propagating cracks.     

Molecular modeling of cracks at interfaces in nanoceramic composites

Toughness in Ceramic Matrix Composites (CMCs) is achieved if crack deflection can occur at the fiber/matrix interface, preventing crack penetration into the fiber and enabling energy-dissipating fiber pullout. To investigate toughening in nanoscale CMCs, direct atomistic models are used to study how matrix cracks behave as a function of the degree of interfacial bonding/sliding, as controlled by the density of C interstitial atoms, at the interface between carbon nanotubes (CNTs) and a diamond matrix. Under all interface conditions studied, incident matrix cracks do not penetrate into the nanotube. Under increased loading, weaker interfaces fail in shear while stronger interfaces do not fail and, instead, the CNT fails once the stress on the CNT reaches its tensile strength. An analytic shear lag model captures all of the micromechanical details as a function of loading and material parameters. Interface deflection versus fiber penetration is found to depend on the relative bond strengths of the interface and the CNT, with CNT failure occurring well below the prediction of the toughness-based continuum He–Hutchinson model. The shear lag model, in contrast, predicts the CNT failure point and shows that the nanoscale embrittlement transition occurs at an interface shear strength scaling as _τs~_εf,CNTσCNT${\tau }_s~{\epsilon }_{f,CNT}{\sigma }_{CNT}$ rather than _τs~_σCNT${\tau }_s~{\sigma }_{CNT}$ typically prevailing for micron scale composites, where _εf,CNT${\epsilon }_{f,CNT}$ and _σCNT${\sigma }_{CNT}$ are the CNT failure strain and stress, respectively. Interface bonding also lowers the effective fracture strength in SWCNTs, due to formation of defects, but does not play a role in DWCNTs having interwall coupling, which are weaker than SWCNTs but less prone to damage in the outerwall.     

Why the stress trajectories in the Dean-Hutchinson plastic sector of the growing mode III crack are an unfocused fan

The plastic zone of the growing mode III crack in an elastic perfectly plastic solid consists of two sectors in contact with each other. The sector closer to the crack plane, first studied analytically by Chitaley and McClintock (CM), consists of a fan of straight maximum shear stress trajectories that are focused on the crack tip. The other sector, first analyzed numerically by Dean and Hutchinson (DH), is a ‘radial’ fan of straight lines that are not focused at the crack tip or at another common point. In this paper it is shown with use of the dislocation density field that the need that the stress magnitude in the plastic wake be below the yield stress requires the existence of an unfocused fan in the DH sector. It appears unlikely that this result can be obtained without explicit use of dislocations.     

Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity

Understanding sub-Rayleigh-to-intersonic transition of mode II cracks is a fundamental problem in fracture mechanics with important practical implications for earthquake dynamics and seismic radiation. In the Burridge-Andrews mechanism, an intersonic daughter crack nucleates, for sufficiently high prestress, at the shear stress peak traveling with the shear wave speed in front of the main crack. We find that sub-Rayleigh-to-intersonic transition and sustained intersonic propagation occurs in a number of other models that subject developing cracks to intersonic loading fields. We consider a spontaneously expanding sub-Rayleigh crack (or main crack) which advances, along a planar interface with linear slip-weakening friction, towards a place of favorable heterogeneity, such as a preexisting subcritical crack or a small patch of higher prestress (similar behavior is expected for a small patch of lower static strength). For a range of model parameters, a secondary dynamic crack nucleates at the heterogeneity and acquires intersonic speeds due to the intersonic stress field propagating in front of the main crack. Transition to intersonic speeds occurs directly at the tip of the secondary crack, with the tip accelerating rapidly to values numerically equal to the Rayleigh wave speed and then abruptly jumping to an intersonic speed. Models with favorable heterogeneity achieve intersonic transition and propagation for much lower prestress levels than the ones implied by the Burridge-Andrews mechanism and have transition distances that depend on the position of heterogeneity. We investigate the dependence of intersonic transition and subsequent crack propagation on model parameters and discuss implications for earthquake dynamics.     

Some surprising phenomena in weak-bond fracture of a triangular lattice

A semi-infinite crack growing along a straight line in an unbounded triangular-cell lattice and in lattice strips is under examination. Elastic and standard-material viscoelastic lattices are considered. Using the superposition similar to that used for a square-cell lattice (J. Mech. Phys. Solids 48 (2000) 927) an irregular stress distribution is revealed on the crack line in mode II: the strain of the crack-front bond is lower than that of the next bond. A further notable fact about mode II concerns the bonds on the crack line in the lattice strip deformed by a ‘rigid machine’. If the alternate bonds, such that are inclined differently than the crack-front bond, are removed, the stresses in the crack-front bond and in the other intact bonds decrease. These facts result in irregular quasi-static and dynamic crack growth. In particular, in a wide range of conditions for mode II, consecutive bond breaking becomes impossible. The most surprising phenomenon is the formation of a binary crack consisting of two branches propagating on the same line. It appears that the consecutive breaking of the right-slope bonds—as one branch of the crack—can proceed at a speed different from that for the left-slope bonds—as another branch. One of these branches can move faster than the other, but with time they can change places. Some irregularities are observed in mode I as well. Under the influence of viscosity, crack growth can be stabilized and crack speed can be low when viscosity is high; however, in mode II irregularities in the crack growth remain. It is found that crack speed is a discontinuous function of the creep and relaxation times.     

A new 2D discrete model applied to dynamic crack propagation in brittle materials

In this work, a 2D discrete model (DM) applied to the dynamic crack propagation in brittle materials is developed and implemented. The proposed model is based on a particular discretization of Navier’s equations, presenting similarities to the Born model, with the advantage that the constants appearing in it are explicitly related to the elastic properties. This model overcomes the limitations in the choice of Poisson’s ratio present in other discrete models. Three numerical examples are presented to show the capability of this method in modelling wave propagation and dynamic fracture problems. The obtained results are in agreement with experimental and numerical results reported by other researchers.