Using a multi-resolution technique, we analyze large in-plane fracture fronts moving slowly between two sintered Plexiglas plates. We find that the roughness of the front exhibits two distinct regimes separated by a crossover length scale $δ^*$. Below $δ^*$, we observe a multi-affine regime and the measured roughness exponent $ζ_{\parallel}^{-} = 0.60\pm 0.05$ is in agreement with the coalescence model. Above $δ^*$, the fronts are mono-affine, characterized by a roughness exponent $ζ_{\parallel}^{+} = 0.35\pm0.05$, consistent with the fluctuating line model. We relate the crossover length scale to fluctuations in fracture toughness and the stress intensity factor.
We study the scaling of two-dimensional crack roughness using large scale beam lattice systems. Our results indicate that the crack roughness obtained using beam lattice systems does not exhibit anomalous scaling in sharp contrast to the simulation results obtained using scalar fuse lattices. The local and global roughness exponents ($ζ_{loc}$ and $ζ$, respectively) are equal to each other, and the two-dimensional crack roughness exponent is estimated to be $ζ_{loc} = ζ= 0.64 \pm 0.02$. Removal of overhangs (jumps) in the crack profiles eliminates even the minute differences between the local …
We measure the roughness exponent and the correlation length exponent of a stress-weighted percolation process in the central force model in 2D. The roughness exponent is found to be zeta = 0.75 \pm 0.03 and the correlation length exponent is found to be nu = 1.7 \pm 0.3. This result supports a conjecture that the fracture roughness for large scales is controlled by a stress weighted percolation process, and the fracture roughness can by calculated from the correlation length exponent by zeta = 2*nu/(1+2*nu). We also compare global and local measurements of the fracture roughness and do not fi…
We analyze the effect of disorder and notches on crack roughness in two dimensions. Our simulation results based on large system sizes and extensive statistical sampling indicate that the crack surface exhibits a universal local roughness of $ζ_{loc} = 0.71$ and is independent of the initial notch size and disorder in breaking thresholds. The global roughness exponent scales as $ζ= 0.87$ and is also independent of material disorder. Furthermore, we note that the statistical distribution of crack profile height fluctuations is also independent of material disorder and is described by a Gaussian…
We study the fracture surface of three dimensional samples through a model for quasi-static fractures known as Born Model. We find for the roughness exponent a value of 0.5 expected for ``small length scales'' in microfracturing experiments. Our simulations confirm that at small length scales the fracture can be considered as quasi-static. The isotropy of the roughness exponent on the crack surface is also shown. Finally, considering the crack front, we compute the roughness exponents for longitudinal and transverse fluctuations of the crack line (both 0.5). They result in agreement with exper…
We analyze the scaling of the crack roughness and of avalanche precursors in the two dimensional random fuse model by numerical simulations, employing large system sizes and extensive sample averaging. We find that the crack roughness exhibits anomalous scaling, as recently observed in experiments. The roughness exponents ($ζ$, $ζ_{loc}$) and the global width distributions are found to be universal with respect to the lattice geometry. Failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size. While the characteristic avalanche size scales as $s_0 \…
Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold, or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the cr…
Roughness of i-AlPdMn cleaved surfaces are presently analysed. From the atomic scale to 2-3 nm, they are shown to exhibit scaling properties hiding the cluster (0.45 nm) aperiodic structure. These properties are quantitatively similar to those observed on various disordered materials, albeit on other ranges of length scales. These properties are interpreted as the signature of damage mechanisms occurring within a 2-3 nm wide zone at the crack tip. The size of this process zone finds its origin in the local temperature elevation at the crack tip. For the very first time, this effect is reported…
We report results on the interrelation between driving force, roughness exponent, branching and crack speed in a finite element model. We show that for low applied loadings the crack speed reaches the values measured in the experiments, and the crack surface roughness is compatible with logarithmic scaling. At higher loadings, the crack speed increases, and the crack roughness exponent approaches the value measured at short length scales in experiments. In the case of high anisotropy, the crack speed is fully compatible with the values measured in experiments on anisotropic materials, and we a…
Discrete element modelling (DEM) is one of the most efficient computational approaches to the fracture processes of heterogeneous materials on mesoscopic scales. From the dynamics of single crack propagation through the statistics of crack ensembles to the rapid fragmentation of materials DEM had a substantial contribution to our understanding over the past decades. Recently, the combination of DEM with other simulation techniques like Finite Element Modelling further extended the field of applicability. In this paper we briefly review the motivations and basic idea behind the DEM approach to …
Crack propagation is studied numerically using a continuum phase-field approach to mode III brittle fracture. The results shed light on the physics that controls the speed of accelerating cracks and the characteristic branching instability at a fraction of the wave speed.
This paper explores the global Mode I fracture energy of a carbon fiber composite subject to a biaxial stress state at a crack tip, specifically in which one stress component is compressive and parallel to the crack. Based on an experimental technique previously coined as The Gap Test and Bazant's Type II Size Effect Law, it is found that there is a monotonic decrease in the Mode I fracture energy as the crack parallel compressive stress increases. Compared to the nominal value of fracture energy, where no crack parallel compression is applied, the fracture energy is observed to decrease by up…
This paper presents an experimental study of the fretting crack nucleation threshold, expressed in terms of loading conditions, with a cylinder/plane contact. The studied material is a damage tolerant aluminium alloy widely used in the aerospace application. Since in industrial problems, the surface quality is often variable, the impact of a unidirectional roughness is investigated via varying the roughness of the counter body in the fretting experiments. As expected, experimental results show a large effect of the contact roughness on the crack nucleation conditions. Rationalisation of the cr…
Dynamic fracture in a wide class of materials reveals "fracture energy" $Γ$ much larger than the expected nominal surface energy due to the formation of two fresh surfaces. Moreover, the fracture energy depends on the crack velocity, $Γ=Γ(v)$. We show that a simple dynamical theory of visco-plasticity coupled to asymptotic pure linear-elasticity provides a possible explanation to the above phenomena. The theory predicts tip blunting characterized by a dynamically determined crack tip radius of curvature. In addition, we demonstrate velocity selection for cracks in fixed-grip strip geometry acc…
We study the dynamics of cracks in brittle materials when the velocity of the crack is comparable to the sound velocity by means of lattice simulations. Inertial and damped dynamics are analyzed. It is shown that dissipation strongly influences the shape of the crack. While inertial cracks are highly unstable, dissipation can stabilize straight cracks. Our results can help to explain recent experiments on PMMA.
The relation between fracture surface morphology and the three-dimensional structure of crack fronts is investigated through direct observation of brittle cracks in gels. A key notion in this investigation is the discontinuity of the crack front, whose advancement creates the fracture surface. We discuss the significance of our findings in the studies of general three-dimensional brittle fractures.
Numerous measurements showed that the crack tip opening angle (CTOA) is nearly constant upon stable ductile fracture in Al alloys which widely used in modern transportation industry. The atomic structure of the very tip of a crack front has remained unknown, however. We have carried out a first-principles density functional theory study to reveal the precise alignment of atoms near the crack tip in single-crystalline Al. The calculations demonstrate that the CTOA increases with the opening displacement, thus the observed constant CTOA in millimeter scale is an entirely plastic effect during du…
A dynamic crack will travel in a straight path up to a material-dependent critical speed beyond which its path becomes erratic. Predicting this critical speed and discovering the origin of this instability are two outstanding problems in fracture mechanics. We recently discovered a simple scaling model based on an effective elastic modulus that gives successful predictions for this critical speed by transforming the nonlinear crack dynamics problem into a linear elasticity representation. We now show that a simple atomic picture based on broken-bond relaxation at the dynamic crack tip provides…
To clarify effects of crack speed and cross-link density on fracture energy of acrylamide gels, we evaluated the roughness of the fracture surface and measured the fracture energy taking into account the roughness. The fracture energy increases linearly with crack speed $V$ in a fast crack speed region, and the increasing rate of fracture energy with $V$ decreases with increasing cross link density in the gels. In a slow crack speed region the fracture energy depends on crack speed more strongly than in the fast crack speed region. This indicates that a qualitative change exists in fracture pr…
The scaling laws describing the roughness development of crack surfaces are incorporated into the Griffith criterion. We show that, in the case of a Family-Vicsek scaling, the energy balance leads to a purely elastic brittle behavior. On the contrary, it appears that an anomalous scaling reflects a R-curve behavior associated to a size effect of the critical resistance to crack growth in agreement with the fracture process of heterogeneous brittle materials exhibiting a microcracking damage.
In this paper we first obtain the order of stress singularity for a dynamically propagating self-affine fractal crack. We then show that there is always an upper bound to roughness, i.e. a propagating fractal crack reaches a terminal roughness. We then study the phenomenon of reaching a terminal velocity. Assuming that propagation of a fractal crack is discrete, we predict its terminal velocity using an asymptotic energy balance argument. In particular, we show that the limiting crack speed is a material-dependent fraction of the corresponding Rayleigh wave speed.
We present an intuitive physical picture for fractures in nacre-type stratified materials via scaling arguments: strain distributions around a fracture are rather different depending on directions and size of fractures. We thus observe that viscoelastic effects are important for parallel fractures. These effect are taken into account via the simplest viscoelastic model for weakly cross-linked polymer. Within a certain limit, we find a trumpet crack shape similar to that in certain polymers and make predictions for the fracture energy of certain stratified materials in this mode.
We investigate propagating fronts in disordered media that belong to the universality class of wetting contact lines and planar tensile crack fronts. We derive from first principles their nonlinear equations of motion, using the generalized Griffith criterion for crack fronts and three standard mobility laws for contact lines. Then we study their roughness using the self-consistent expansion. When neglecting the irreversibility of fracture and wetting processes, we find a possible dynamic rough phase with a roughness exponent of $ζ=1/2$ and a dynamic exponent of z=2. When including the irrever…
To obtain the probability distribution of 2D crack patterns in mesoscopic regions of a disordered solid, the formalism of Paper I requires that a functional form associating the crack patterns (or states) to their formation energy be developed. The crack states are here defined by an order parameter field representing both the presence and orientation of cracks at each site on a discrete square network. The associated Hamiltonian represents the total work required to lead an uncracked mesovolume into that state as averaged over the initial quenched disorder. The effect of cracks is to create m…
A series of experimental results on the in-plane fracture of a fiber reinforced laminated composite panel is analyzed using the variational multi-scale cohesive method (VMCM). The VMCM results demonstrate the influence of specimen geometry and load distribution on the propagation of large scale bridging cracks in the fiber reinforced panel. Experimentally observed variation in fracture resistance is substantiated numerically by comparing the experimental and VMCM load-displacement responses of geometrically scaled single edge-notch three point bend (SETB) specimens. The results elucidate the s…
A planar crack generically segments into an array of "daughter cracks" shaped as tilted facets when loaded with both a tensile stress normal to the crack plane (mode I) and a shear stress parallel to the crack front (mode III). We investigate facet propagation and coarsening using in-situ microscopy observations of fracture surfaces at different stages of quasi-static mixed-mode crack propagation and phase-field simulations. The results demonstrate that the bifurcation from propagating planar to segmented crack front is strongly subcritical, reconciling previous theoretical predictions of line…
Recently the scaling laws describing the roughness development of fracture surfaces was proposed to be related to the macroscopic elastic energy released during crack propagation [Mor00]. On this basis, an energy-based asymptotic analysis allows to extend the link to the nominal strength of structures. We show that a Family-Vicsek scaling leads to the classical size effect of linear elastic fracture mechanics. On the contrary, in the case of an anomalous scaling, there is a smooth transition from the case of no size effect, for small structure sizes, to a power law size effect which appears we…
In this paper, we'll answer several abstract, formal questions about the nature of crack growth and nucleation. Bringing a field theory point of view to fracture illuminates things in what I hope will be an entertaining way. Formally, what is the crack nucleation rate? Fracture is an instability of elastic theory under tension. For thermally nucleated cracks, there is an analogy with supercooled liquids. Here the crack nucleation rate can be thought of as an imaginary part of the free energy - giving the decay rate of the metastable, stretched material. As an amusing consequence, we can formal…
This work presents a critical overview of the effects of different aspects of model formulation on crack path selection in quasi-static phase field fracture. We consider different evolution methods, mechanics formulations, fracture dissipation energy formulations, and forms of the irreversibility condition. The different model variants are implemented with common numerical methods based on staggered solution of the phase-field and mechanics sub-problems via FFT-based solvers. These methods mix standard approaches with novel elements, such as the use of bound-constrained conjugate gradients for…
We present high resolution measurements of the displacement and strain fields near the tip of a dynamic (Mode I) crack. The experiments are performed on polyacrylamide gels, brittle elastomers whose fracture dynamics mirror those of typical brittle amorphous materials. Over a wide range of propagation velocities ($0.2-0.8c_s$), we compare linear elastic fracture mechanics (LEFM) to the measured near-tip fields. We find that, sufficiently near the tip, the measured stress intensity factor appears to be non-unique, the crack tip significantly deviates from its predicted parabolic form, and the s…
We study a lattice model for mode III crack propagation in brittle materials in a stripe geometry at constant applied stretching. Stiffening of the material at large deformation produces supersonic crack propagation. For large stretching the propagation is guided by well developed soliton waves. For low stretching, the crack-tip velocity has a universal dependence on stretching that can be obtained using a simple geometrical argument.
We use classical molecular dynamics (MD) simulations to investigate the mechanical properties of pre-cracked, nano-porous single layer MoS2 (SLMoS2) and the effect of interactions between cracks and pores. We found that the failure of pre-cracked and nano-porous SLMoS2 is dominated by brittle type fracture. Bonds in armchair direction show a stronger resistance to crack propagation compared to the zigzag direction. We compared the brittle failure of Griffith prediction with the MD fracture strength and toughness and found substantial differences that limit the applicability of Griffith criteri…
We combine molecular dynamics simulations of deformations at the submicron scale with a continuum fracture mechanics calculation for the onset of crack propagation to predict the macroscopic fracture toughness of amorphous glassy polymers. Key ingredients in this multiscale approach are the elastic properties of polymer crazes and the stress at which craze fibrils fail through chain pullout or scission. Our results are in quantitative agreement with dimensionless ratios that describe experimental polymers and their variation with temperature, polymer length and polymer rigidity.
In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the regularized crack. The key novelty is the combination of a strain energy split with a partial rank-I relaxation in the vicinity of the diffuse interface. The former is necessary to account for physically meaningful crack kinematics like crack closure, the latter ensures the mechanical jump conditions throughout the diffuse region. The model is verified by a co…
The problem of a crack impinging on an interface has been thoroughly investigated in the last three decades due to its important role in the mechanics and physics of solids. In this investigation, this problem is revisited in view of the recent progresses on the phase field approach to brittle fracture. In this concern, a novel formulation combining the phase field approach for modeling brittle fracture in the bulk and a cohesive zone model for pre-existing adhesive interfaces is herein proposed to investigate the competition between crack penetration and deflection at an interface. The model,…
Soft materials including elastomers and gels are pervasive in biological systems and technological applications. Whereas it is known that intrinsic fracture energies of soft materials are relatively low, how the intrinsic fracture energy cooperates with mechanical dissipation in process zone to give high fracture toughness of soft materials is not well understood. In addition, it is still challenging to predict fracture energies and crack-tip strain fields of soft tough materials. Here, we report a scaling theory that accounts for synergistic effects of intrinsic fracture energies and dissipat…
We investigate dynamic fracture of heterogeneous materials experimentally by measuring displacement fields as a rupture propagates through a periodic array of obstacles of controlled fracture energy. Our measurements demonstrate the applicability of the classical equation of motion of cracks at a discontinuity of fracture energy: the crack speed jumps at the entrance and exit of an obstacle, as predicted by the crack-tip energy balance within the brittle fracture framework. The speed jump amplitude is governed by the fracture energy contrast and by the combination of rate-dependency of fractur…
We prepared thin layers of amorphous silicon by deposition of a liquid-phase polysilane precurser on glass substrate. Raman scattering provides evidence for residual tensile stress in the silicon, which is evaluated quantitatively. Under treatment with hydrofluoric acid, this stress leads to spiral cracks in the silicon. We explain the process of crack formation and examine this phenomenon both analytically and numerically, the latter with the finite element method (FEM). The FEM yields the geometry correction factor for such spiral cracks in terms of the Griffith criterion. This allows for th…
This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in defining the ensemble. Ensembles are obtained by dividing a huge rock sample into many mesoscopic volumes. Because rocks are a disordered collection of grains in cohesive contact, we expect that once shear strain is applied and cracks begin to arrive in the system, the mesoscopic volumes will have a wide distribution of different crack states. These mesoscopic …
We show that the delivery of fracture work to the tip of an advancing planar crack is strongly reduced by surface phonon emission, leading to forbidden ranges of crack speed. The emission can be interpreted through dispersion of the group velocity, and Rayleigh and Love branches contribute as well as other high frequency branches of the surface wave dispersion relations. We also show that the energy release rate which enters the Griffith criterion for the crack advance can be described as the product of the continuum solution with a function that only depends on the lattice geometry and descri…
Fracture is the ultimate source of failure of amorphous carbon (a-C) films, however it is challenging to measure fracture properties of a-C from nano-indentation tests and results of reported experiments are not consistent. Here, we use atomic-scale simulations to make quantitative and mechanistic predictions on fracture of a-C. Systematic large-scale K-field controlled atomic-scale simulations of crack propagation are performed for a-C samples with densities of $ρ=2.5, \, 3.0 \, \text{ and } 3.5~\text{g/cm}^{3}$ created by liquid quenches for a range of quench rates $\dot{T}_q = 10 - 1000~\te…
To unravel how the microstructure affects the fracture surface roughness in heterogeneous brittle solids like rocks or ceramics, we characterized the roughness statistics of post-mortem fracture surfaces in home-made materials of adjustable microstructure length-scale and porosity, obtained by sintering monodisperse polystyrene beads. Beyond the characteristic size of disorder, the roughness profiles are found to exhibit self-affine scaling features evolving with porosity. Starting from a null value and increasing the porosity, we quantitatively modify the self-affine scaling properties from a…
Fractoluminescence experiments are performed on two kinds of silicate glasses. All the light spectra collected during dynamic fracture reveal a black body radiator behaviour, which is interpreted as a crack velocity-dependent temperature rise close to the crack tip. Crack velocities are estimated to be of the order of 1300 m.s$^{-1}$ and fracture process zones are shown to extend over a few nanometers.
Toughness describes the ability of a material to resist fracture or crack propagation. It is demonstrated here that fracture toughness of a material can be asymmetric, i.e., the resistance of a medium to a crack propagating from right to left can be significantly different from that to a crack propagating from left to right. Such asymmetry is unknown in natural materials, but we show that it can be built into artificial materials through the proper control of microstructure. This paves the way for control of crack paths and direction, where fracture -- when unavoidable -- can be guided through…
On the complementary roles of anisotropic crack density and anisotropic crack driving force in phase-field modeling of mixed-mode fracture
School of Mechanical Engineering, Changwon National University（昌原国立大学机械工程学院）；Department of Smart Shipbuilding Systems, Mokpo National University（木浦国立大学智能造船系统系）
In this work we present the design of a new test geometry inspired by the Tapered Double Cantilever Beam (TDCB) specimen that is shown to provide an improved characterization of the fracture properties of brittle solids. First, we show that our new design results in an exponential increase of the specimen compliance with crack length, leading to an extremely stable crack growth during the test. We determine an analytical description of this behavior, which provides a simple procedure to extract the fracture energy without depending on finite element calculations. Validation tests are done on p…
The dynamics of planar crack fronts in heterogeneous media is studied using a recently proposed stochastic equation of motion that takes into account nonlinear effects. The analysis is carried for a moving front in the quasi-static regime using the Self Consistent Expansion. A continuous dynamical phase transition between a flat phase and a dynamically rough phase, with a roughness exponent $ζ=1/2$, is found. The rough phase becomes possible due to the destabilization of the linear modes by the nonlinear terms. Taking into account the irreversibility of the crack propagation, we infer that the…
The widespread use of sodium aluminosilicate glass in many engineering applications due to its mechanical and optical properties (transparency, dielectric, etc.), has become common in recent years. However, glass, a brittle material, has its vulnerability to fracture. Processes such as heat treatment (heat tempering) or chemical strengthening through ion-exchange have been used to create residual stress profiles on the glass, in a bid to improve its fracture strength. However, failure still occurs, which is mostly catastrophic and expensive to repair. Therefore understanding, predicting, and e…
Cracks, the major vehicle for material failure, tend to accelerate to high velocities in brittle materials. In three-dimensions, cracks generically undergo a micro-branching instability at about 40% of their sonic limiting velocity. Recent experiments showed that in sufficiently thin systems cracks unprecedentedly accelerate to nearly their limiting velocity without micro-branching, until they undergo an oscillatory instability. Despite their fundamental importance and apparent similarities to other instabilities in condensed-matter physics and materials science, these dynamic fracture instabi…
The extent to which time-dependent fracture criteria affect the dynamic behavior of fracture in a discrete structure is discussed in this work. The simplest case of a semi-infinite isotropic chain of oscillators has been studied. Two history-dependent criteria are compared to the classical one of threshold elongation for linear bonds. The results show that steady-state regimes can be reached in the low subsonic crack speed range where it is impossible according to the classical criterion. Repercussions in terms of load and crack opening versus velocity are explained in detail. A strong qualita…
