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Mokhtar Adda-bedia - One of the best experts on this subject based on the ideXlab platform.
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Stability and roughness of tensile cracks in disordered materials.
Physical Review E, 2013Co-Authors: Eytan Katzav, Mokhtar Adda-bediaAbstract:We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of linear elastic fracture mechanics, based on the Griffith Criterion and the principle of local symmetry. This result allows us to extend the stability analysis of Cotterell and Rice [B. Cotterell and J. R. Rice, Int. J. Fract. 16, 155 (1980)] to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics, we obtain the power spectrum of these paths and their probability distribution function and conclude that they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel a systematic bias in such real-space methods and thus contribute to the general discussion of reliability of self-affine measurements.
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Thermal fracture as a framework for quasi-static crack propagation
International Journal of Fracture, 2009Co-Authors: F. Corson, Mokhtar Adda-bedia, H. Henry, Eytan KatzavAbstract:We address analytically and numerically the problem of crack path prediction in the model system of a crack propagating under thermal loading. We show that one can explain the instability from a straight to a wavy crack propagation by using only the principle of local symmetry and the Griffith Criterion. We then argue that the calculations of the stress intensity factors can be combined with the standard crack propagation criteria to obtain the evolution equation for the crack tip within any loading configuration. The theoretical results of the thermal crack problem agree with the numerical simulations we performed using a phase field model. Moreover, it turns out that the phase-field model allows to clarify the nature of the transition between straight and oscillatory cracks which is shown to be supercritical.
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Roughness of moving elastic lines: crack and wetting fronts.
Physical Review E, 2007Co-Authors: Eytan Katzav, Mokhtar Adda-bedia, M. Ben Amar, Arezki BoudaoudAbstract: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 =1/2 and adynamic exponent of z=2. When including the irreversibility, we conclude that the front propagation can become history dependent, and thus we consider the value =1/2 as a lowerbound for the roughness exponent. Interestingly, for propagating contact line in wetting, where irreversibility is weaker than in fracture, the experimental results are close to 0.5, while for fracture the reported values of 0.55‐0.65 are higher.
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Fracture surfaces of heterogeneous materials: a 2D solvable model
Europhysics Letters (EPL), 2007Co-Authors: Eytan Katzav, Mokhtar Adda-bedia, Bernard DerridaAbstract:Using an elastostatic description of crack growth based on the Griffith Criterion and the principle of local symmetry, we present a stochastic model describing the propagation of a crack tip in a 2D heterogeneous brittle material. The model ensures the stability of straight cracks and allows for the study of the roughening of fracture surfaces. When neglecting the effect of the non singular stress, the problem becomes exactly solvable and yields analytic predictions for the power spectrum of the paths. This result suggests an alternative to the conventional power law analysis often used in the analysis of experimental data.
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Generalized Griffith Criterion for dynamic fracture and the stability of crack motion at high velocities.
Physical Review E, 1999Co-Authors: Mokhtar Adda-bedia, M. Ben Amar, R Arias, F LundAbstract:We use Eshelby's energy momentum tensor of dynamic elasticity to compute the forces acting on a moving crack front in a three-dimensional elastic solid [Philos. Mag. 42, 1401 (1951)]. The crack front is allowed to be any curve in three dimensions, but its curvature is assumed small enough so that near the front the dynamics is locally governed by two-dimensional physics. In this case the component of the elastic force on the crack front that is tangent to the front vanishes. However, both the other components, parallel and perpendicular to the direction of motion, do not vanish. We propose that the dynamics of cracks that are allowed to deviate from straight line motion is governed by a vector equation that reflects a balance of elastic forces with dissipative forces at the crack tip, and a phenomenological model for those dissipative forces is advanced. Under certain assumptions for the parameters that characterize the model for the dissipative forces, we find a second order dynamic instability for the crack trajectory. This is signaled by the existence of a critical velocity ${V}_{c}$ such that for velocities $Vl{V}_{c}$ the motion is governed by ${K}_{\mathrm{II}}=0,$ while for $Vg{V}_{c}$ it is governed by ${K}_{\mathrm{II}}\ensuremath{\ne}0.$ This result provides a qualitative explanation for some experimental results associated with dynamic fracture instabilities in thin brittle plates. When deviations from straight line motion are suppressed, the usual equation of straight line crack motion based on a Griffiths-like Criterion is recovered.
Eytan Katzav - One of the best experts on this subject based on the ideXlab platform.
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Stability and roughness of tensile cracks in disordered materials.
Physical Review E, 2013Co-Authors: Eytan Katzav, Mokhtar Adda-bediaAbstract:We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of linear elastic fracture mechanics, based on the Griffith Criterion and the principle of local symmetry. This result allows us to extend the stability analysis of Cotterell and Rice [B. Cotterell and J. R. Rice, Int. J. Fract. 16, 155 (1980)] to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics, we obtain the power spectrum of these paths and their probability distribution function and conclude that they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel a systematic bias in such real-space methods and thus contribute to the general discussion of reliability of self-affine measurements.
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Thermal fracture as a framework for quasi-static crack propagation
International Journal of Fracture, 2009Co-Authors: F. Corson, Mokhtar Adda-bedia, H. Henry, Eytan KatzavAbstract:We address analytically and numerically the problem of crack path prediction in the model system of a crack propagating under thermal loading. We show that one can explain the instability from a straight to a wavy crack propagation by using only the principle of local symmetry and the Griffith Criterion. We then argue that the calculations of the stress intensity factors can be combined with the standard crack propagation criteria to obtain the evolution equation for the crack tip within any loading configuration. The theoretical results of the thermal crack problem agree with the numerical simulations we performed using a phase field model. Moreover, it turns out that the phase-field model allows to clarify the nature of the transition between straight and oscillatory cracks which is shown to be supercritical.
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Roughness of moving elastic lines: crack and wetting fronts.
Physical Review E, 2007Co-Authors: Eytan Katzav, Mokhtar Adda-bedia, M. Ben Amar, Arezki BoudaoudAbstract: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 =1/2 and adynamic exponent of z=2. When including the irreversibility, we conclude that the front propagation can become history dependent, and thus we consider the value =1/2 as a lowerbound for the roughness exponent. Interestingly, for propagating contact line in wetting, where irreversibility is weaker than in fracture, the experimental results are close to 0.5, while for fracture the reported values of 0.55‐0.65 are higher.
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Fracture surfaces of heterogeneous materials: a 2D solvable model
Europhysics Letters (EPL), 2007Co-Authors: Eytan Katzav, Mokhtar Adda-bedia, Bernard DerridaAbstract:Using an elastostatic description of crack growth based on the Griffith Criterion and the principle of local symmetry, we present a stochastic model describing the propagation of a crack tip in a 2D heterogeneous brittle material. The model ensures the stability of straight cracks and allows for the study of the roughening of fracture surfaces. When neglecting the effect of the non singular stress, the problem becomes exactly solvable and yields analytic predictions for the power spectrum of the paths. This result suggests an alternative to the conventional power law analysis often used in the analysis of experimental data.
Ian Main - One of the best experts on this subject based on the ideXlab platform.
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a modified Griffith Criterion for the evolution of damage with a fractal distribution of crack lengths application to seismic event rates and b values
Geophysical Journal International, 2007Co-Authors: Ian MainAbstract:SUMMARY The Griffith Criterion for dynamic crack growth results from a calculation of a minimum in the Gibbs free energy of an infinite medium containing a single elliptical crack. However rock failure in the laboratory or during large earthquakes is usually preceded by the evolution of an aureole of damage in the form of subsidiary microcracks or faults. A characteristic of such precursory damage is that it is fractal, having a power-law crack length distribution, and also a power-law spatial and temporal correlation. Such fractal damage is consistent with the concept of faulting or cracking as a self-organized critical phenomenon, and has been widely confirmed by empirical observation of faults, laboratory fractures and indirect seismic monitoring in the laboratory and field. Here we consider the free energy change ΔF associated with an ensemble of NT weakly interacting, aligned, elliptical cracks of different lengths, with semilength expectation value 〈c〉, under a constant tensile stress s applied at the boundary of each element. The crack ensemble represents a state of damage which may evolve in a quasi-static way due to subcritical crack growth. By considering ∂(ΔF)/∂〈c〉= o, a modified strain energy release rate G′=∂U/∂Ad=f(〈c〉, 〈c2〉) is defined, where U is the potential strain energy, and Ad is the total surface area of the array of cracks. For a given NT, G′ is proportional to the rate of change of the total volume of damage with respect to the total area of damage Ad. This reflects the fact that mechanical energy is stored in a volume, and released on a surface. As the number of cracks tends to 1, G′ tends naturally to the strain energy release rate G for a single crack. For a fractal distribution of crack lengths Nc(c) =NT(c/co)-D, limited to a range (co, c1), it can be shown that G′ is negatively correlated to D for a given constant value of NT. Also, the curves for higher NT are associated with higher G′. A similar result can be obtained more simply by using the expectation value 〈G〉 directly as an appropriate parameter, with the advantage that both NT and D can be allowed to vary independently. These predictions are respectively consistent with the positive (negative) correlation established between acoustic emission event rates (seismic b-values) and G in the laboratory for quasi-static tensile failure by mode I subcritical crack growth due to stress corrosion reactions in double torsion loading. The theory also correctly predicts the order of magnitude of the stress corrosion index for these experiments, and the observation that more heterogeneous materials have higher stress corrosion indices. However the correlations established between event rate, D and G' (or 〈G〉) are completely general, and can apply in principle to other forms of fault development or crack growth with weak long-range interactions. Most studies of the statistics of damage evolution are by their very nature indirect or posthumous, unless the material of interest is optically transparent. Seismic monitoring of damage in the form of small earthquakes or acoustic emissions can be used to measure the parameters a and b of the earthquake frequency-magnitude relation log N, =a - bm, where a = log N, b = CD/3. C is the slope of the scaling relation between magnitude and the common algorithm of seismic moment, and the event rate N is assumed proportional to NT. Thus changes in a and b can in principle be used to infer G′ (or 〈G〉) during the evolution of damage in the intermediate term prior to dynamic failure of laboratory rock samples or the Earth.
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application of a modified Griffith Criterion to the evolution of fractal damage during compressional rock failure
Geophysical Journal International, 1993Co-Authors: Ian Main, Peter Sammonds, P G MeredithAbstract:A modified Griffith Criterion for a fractal ensemble of cracks is applied to the interpretation of Acoustic Emission (AE) statistics during the compressional deformation of intact and artificially pre-cut rock specimens in the laboratory. A mean energy release rate per unit crack surface area [G] is recovered from the observed AE event rate N and the seismic b value, by calculating an inferred mean crack length [c] and measuring the differential stress sigma for a range of experimental conditions. Temporal variations in [G] under compressive deformation show very similar trends to those predicted by a synoptic model determined by direct extrapolation from observations of subcritical crack growth under tension. (In the tensile case, deformation is centred on a dominant macrocrack and the stress intensity K, which scales as the square root of G, is the relevant measured variable.)The three independent variables measured during the tests (sigma, N, b) are reduced to points that map out a path through a 3-D phase space ([G], N, b), which depends on the material type and the experimental conditions. 2-D slices through this phase space [([G], N), ([G], b)] are compared with results from the tensile tests [(K, N), (K, b)]. The event rate N is found to scale with square-root [G] according to a power law, with an exponent n' which is smaller than that for tensile fracture, reflecting the greater stability of compressional rock fracture in its early stages. The effective subcritical crack growth index n' is correlated with the material type and degree of apparent 'ductility' on a macroscopic scale, with more brittle behaviour corresponding to higher n'. The value of n' is similar on unloading of the stress after dynamic faulting as on the loading portion, though the curve is systematically offset, most probably due to the material weakening associated with faulting. The model does not apply near the period of dynamic failure, where strong local interactions are dominant. The seismic b value is also found to scale negatively with square root [G], in a manner similar to experiments where K can be measured independently. The acceleration of the mean seismogenic crack length [c] = f(t) has a similar power-law form to that predicted from Charles' law for a single tensile macrocrack, with an implied subcritical crack growth index n smaller than that for fracture in compression. The extra dimension introduced by the time dependence of [c] allows an independent check on the validity of the theory used to calculate [G]. In particular n' from the diagram ([c], t) is found to be similar in magnitude to the exponent obtained from the event rate dependence ([G], N), a phenomenon first discovered by empirical observation of tensile subcritical crack growth.
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Damage mechanics with long-range interactions: correlation between the seismic b-value and the fractal two-point correlation dimension
Geophysical Journal International, 1992Co-Authors: Ian MainAbstract:SUMMARY A modified Griffith Criterion for a two-dimensional array of aligned elliptical cracks with a long-range interaction potential is presented. In accordance with observation, the pattern of cracks is assumed to be fractal, with a two-point correlation dimension Dc indicating a power-law distribution of crack spacings r, and a power-law exponent D of the crack length distribution. From a simple dislocation theory of the seismic source D is proportional to the seismic 6-value if an individual earthquake or acoustic emission is produced by displacement on a specific fault or crack in the population. As a result, the theory is applicable to incremental damage rather than the long-term evolution of crack systems with large displacements. The long-range interaction between cracks is taken to be elastic, implying a positive interaction potential proportional to r-'. Two models are presented for the spatio-temporal evolution of the resulting seismicity due to: (A) progressive alignment of epicentres along an incipient fault plane; and (B) clustering of epicentres around potential nucleation points on an existing fault trace. The modified Griffith Criterion predicts either an increase or a decrease in the potential energy release rate G', depending on the sign of aDc/aD and the nature of the concentration of deformation. For model (A), if aDc/aD > 0 (corresponding to an implied positive correlation between the b-value and Dc), then G' increases in the presence of an interaction potential. In contrast G' increases if aD,/aD O. The mechanical hardening (lowering G ') is associated with geometrically distributed damage in either case. Equivalently this can be seen as a shielding effect, with the zone of damage reducing the local stresses on a particular crack. If there is no correlation the interaction potential has a slight mechanical hardening effect with no strong geometric effect. These predictions are also consistent with the usual tenets of damage mechanics, in which early crack growth is stable, distributed and is associated with mechanical hardening, and material failure occurs later in the cycle due to localized, unstable crack coalescence, associated with mechanical weakening. The main difference between the theory presented here and standard damage mechanics is that crack coalescence is organized, and hence instability can develop at lower crack densities.
Takayuki Kitamura - One of the best experts on this subject based on the ideXlab platform.
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Griffith Criterion for Nanoscale Stress Singularity in Brittle Silicon.
ACS nano, 2017Co-Authors: Takashi Sumigawa, Takahiro Shimada, Shuuhei Tanaka, Hiroki Unno, Naoki Ozaki, Shinsaku Ashida, Takayuki KitamuraAbstract:Brittle materials such as silicon fail via the crack nucleation and propagation, which is characterized by the singular stress field formed near the crack tip according to Griffith or fracture mechanics theory. The applicability of these continuum-based theories is, however, uncertain and questionable in a nanoscale system due to its extremely small singular stress field of only a few nanometers. Here, we directly characterize the mechanical behavior of a nanocrack via the development of in situ nanomechanical testing using a transmission electron microscope and demonstrate that Griffith or fracture mechanics theory can be applied to even 4 nm stress singularity despite their continuum-based concept. We show that the fracture toughness in silicon nanocomponents is 0.95 ± 0.07 MPa√m and is independent of the dimension of materials and therefore inherent. Quantum mechanics/atomistic modeling explains and provides insight into these experimental results. This work therefore provides a fundamental understandi...
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Griffith Criterion for Nanoscale Stress Singularity in Brittle Silicon
2017Co-Authors: Takashi Sumigawa, Takahiro Shimada, Shuuhei Tanaka, Hiroki Unno, Naoki Ozaki, Shinsaku Ashida, Takayuki KitamuraAbstract:Brittle materials such as silicon fail via the crack nucleation and propagation, which is characterized by the singular stress field formed near the crack tip according to Griffith or fracture mechanics theory. The applicability of these continuum-based theories is, however, uncertain and questionable in a nanoscale system due to its extremely small singular stress field of only a few nanometers. Here, we directly characterize the mechanical behavior of a nanocrack via the development of in situ nanomechanical testing using a transmission electron microscope and demonstrate that Griffith or fracture mechanics theory can be applied to even 4 nm stress singularity despite their continuum-based concept. We show that the fracture toughness in silicon nanocomponents is 0.95 ± 0.07 MPa√m and is independent of the dimension of materials and therefore inherent. Quantum mechanics/atomistic modeling explains and provides insight into these experimental results. This work therefore provides a fundamental understanding of fracture Criterion and fracture properties in brittle nanomaterials
Changqing Hong - One of the best experts on this subject based on the ideXlab platform.
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effect of graphite flake on microstructure as well as mechanical properties and thermal shock resistance of zrb2 sic matrix ultrahigh temperature ceramics
Journal of Alloys and Compounds, 2009Co-Authors: Zhi Wang, Sai Wang, Xinghong Zhang, Changqing HongAbstract:Abstract ZrB2–20vol.%SiC containing the various volume fractions of graphite flake (ZSG) composites were investigated to determine the effect of graphite content on the microstructure as well as the mechanical properties and thermal shock resistance. The results revealed that the flexural strength generally decreased as the graphite volume fractions increased. Compared with the fracture toughness of about 4.5 MPa m1/2 for the ZrB2–SiC composites, the toughness of the ZSG composites is essentially higher than that of the ZrB2–SiC composites due to their lower strength relative to the ZrB2–SiC composites. The toughening mechanisms, such as the crack deflection and bridging, were related to the thermal residual stresses, and the thermal residual stresses in interfaces were calculated using Hsueh's formula. Moreover, the critical crack size that represents thermal shock resistance was calculated using the Griffith Criterion. It was found that the thermal shock resistance was improved with the increasing graphite volume fractions.
