Intersonic Crack Research Articles

Subsonic and intersonic shear rupture of weak planes with a velocity weakening cohesive zone

A substantial effort has been devoted in the past toward modeling earthquake source mechanisms as dynamically extending shear cracks. Most of the attention was focused on the subsonic crack speed regime. Recently, a number of reports have appeared in the seismological literature citing evidence of intersonic rupture speeds during shallow crustal earthquakes. In the first part of this paper, we discuss direct experimental observations of intersonic in‐plane shear crack growth along a weak plane joining two homogeneous, isotropic, linear elastic plates. Associated with the primary intersonic crack and at locations behind the propagating shear crack tip, a series of secondary tensile cracks, at a steep angle to the shear crack plane, were also observed. Motivated by these observations, subsonic and intersonic mode II crack propagation with a velocity weakening cohesive zone is analyzed in the main body of the paper. A cohesive law is assumed wherein the cohesive shear traction is either a constant or decreases linearly with the local slip rate, the rate of decrease governed by a slip rate weakening parameter. The cohesive shear traction is assumed to vanish when the crack tip sliding displacement reaches a characteristic breakdown slip. It is shown that a positive energy flux into the rupture front is possible in the entire intersonic regime. The influence of shear strength and of the weakening parameter on the crack propagation behavior is investigated. Crack tip stability issues are also addressed, and favorable speed regimes are identified. Estimates of the slip rate weakening parameter are obtained by using the theoretical model to predict the angle of the secondary cracks. The rest of the parameters are subsequently estimated by comparing the isochromatic fringe patterns (contours of maximum in‐plane shear stress) predicted by the solution with those recorded experimentally.

Intersonic crack propagation in homogeneous media under shear-dominated loading: theoretical analysis

The mechanics of cohesive failure under mixed-mode loading is investigated for the case of a steadily propagating subsonic and intersonic dynamic crack subjected to a follower tensile and shear distributed load. The cohesive failure model chosen in this study is rate independent but accounts for the coupling between normal and tangential damage. Special emphasis is placed here on mixed-mode cases with predominantly shear loading. The analysis shows that the size of the mixed-mode cohesive zone is smaller than that obtained in the pure shear case. The relative extent of the shear and tensile cohesive damage zones depends on the crack speed and the mode mixity. In the intersonic regime, the failure process takes place exclusively in shear, even under remote mixed-mode loading conditions.

Effect of loading and geometry on the subsonic/intersonic transition of a bimaterial interface crack

An experimental investigation was conducted to study the nature of intersonic crack propagation along a bimaterial interface. A single edge notch/crack oriented along a polymer/metal interface was loaded predominantly in shear by impacting the specimen with a high velocity projectile fired from a gas gun. The stress field information around the propagating crack tip was recorded in real time by two different optical techniques––photoelasticity and coherent gradient sensing, in conjunction with high speed photography. Intersonic cracks on polymer/metal interfaces were found to propagate at speeds between the shear wave speed ( c s) and 2 c s of the polymer. The nature of the crack tip fields during subsonic/intersonic transition and the conditions governing this transition were examined. Experimental observations showed the formation of a crack face contact zone as the interfacial crack speed exceeds the Rayleigh wave speed of the polymer. Subsequently, the contact zone was observed to expand in size, shrink and eventually collapse onto the intersonic crack tip. The recorded isochromatic fringe patterns showed multiple Mach wave formation associated with such a scenario. It is found that the nature of contact zone formation as well as its size and evolution differ substantially depending on the sign of the opening component of loading.

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.

Feeding and dissipative waves in fracture and phase transition. III. Triangular-cell lattice

Wave configurations for modes I and II of crack propagation in an elastic triangular-cell lattice are studied. [Mode III was considered in Part I of the paper: Slepyan, L.I. Feeding and dissipative waves in fracture and phase transition. I. Some 1D structures and a square-cell lattice. J. Mech. Phys. Solids 49 (2001) 469.] A general solution incorporates a complete set of the feeding and dissipative waves. The solution is based on the wave dispersion dependences obtained in an explicit form. Also some general properties and the long-wave asymptotes of the corresponding Green function are found. This results in the determination of the wavenumbers and modes. The macrolevel-associated solutions exist as the sub-Rayleigh crack speed regime for both modes and as a shear-longitudinal wave-speed intersonic regime for mode II only. In particular, it is shown that any intersonic crack speed is possible, whereas only the speed (shear wave speed multiplied by 2 ) corresponds to a positive energy release in the cohesive-zone-free homogeneous-material model. This is a manifestation of the fact that the local energy release in the lattice is not connected with the singularity of the macrolevel field. Microlevel solutions, corresponding to a nonzero feeding wavenumber, exist for both modes, at least from the energy point of view, for any, sub- and super-Rayleigh, intersonic and supersonic crack speed regimes. In particular, in the super-Rayleigh regime, a high-frequency wave delivers energy to the crack, while the macrolevel wave carries energy away from the crack.

Continuum and atomistic studies of intersonic crack propagation

Mechanisms of intersonic crack propagation along a weak interface under shear dominated loading are studied by both molecular dynamics and continuum elastodynamics methods. Part of the objective is to test if continuum theory can accurately predict the critical time and length scales observed in molecular dynamics simulations. To facilitate the continuum-atomistic linkage, the problem is selected such that a block of linearly isotropic, plane-stress elastic solid consisting of a two-dimensional triangular atomic lattice with pair interatomic potential is loaded by constant shear velocities along the boundary. A pre-existing notch is introduced to represent an initial crack which starts to grow at a critical time after the loading process begins. We observe that the crack quickly accelerates to the Rayleigh wave speed and, after propagating at this speed for a short time period, nucleates an intersonic daughter crack which jumps to the longitudinal wave speed. The daughter crack emerges at a distance ahead of the mother crack. The challenge here is to test if a continuum elastodynamics analysis of the same problem can correctly predict the length and time scales observed in the molecular dynamics simulations. We make two assumptions in the continuum analysis. First, the crack initiation is assumed to be governed by the Griffith criterion. Second, the nucleation of the daughter crack is assumed to be governed by the Burridge–Andrew mechanism of a peak of shear stress ahead of the crack tip reaching the cohesive strength of the interface. Material properties such as elastic constants, fracture surface energy and cohesive strength are determined from the interatomic potential. Under these assumptions, it is shown that the predictions based on the continuum analysis agree remarkably well with the simulation results.

Intersonic Crack Propagation—Part II: Suddenly Stopping Crack

In Part I of this series, we have obtained the fundamental solution for a mode II intersonic crack which involves a crack moving uniformly at a velocity between the shear and longitudinal wave speeds while subjected to a pair of concentrated forces suddenly appearing at the crack tip and subsequently acting on the crack faces. The fundamental solution can be used to generate solutions for intersonic crack propagation under arbitrary initial equilibrium fields. In this paper, Part II of this series, we study a mode II crack suddenly stopping after propagating intersonically for a short time. The solution is obtained by superposing the fundamental solution and the auxiliary problem of a static crack emitting dynamic dislocations such that the relative crack face displacement in the fundamental solution is negated ahead of where the crack tip has stopped. We find that, after the crack stops moving, the stress intensity factor rapidly rises to a finite value and then starts to change gradually toward the equilibrium value for a static crack. A most interesting feature is that the static value of stress intensity is reached neither instantaneously like a suddenly stopping subsonic crack nor asymptotically like a suddenly stopping edge dislocation. Rather, the dynamic stress intensity factor changes continuously as the shear and Rayleigh waves catch up with the stopped crack tip from behind, approaches negative infinity when the Rayleigh wave arrives, and then suddenly assumes the equilibrium static value when all the waves have passed by. This study is an important step toward the study of intersonic crack propagation with arbitrary, nonuniform velocities.

Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composite plates

Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composite plates

Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composite plates

Abstract Some recent experimental observations of highly dynamic crack growth events in thick unidirectional composites are presented. The specimens used in this study were 48-ply thick unidirectional graphite–epoxy composite plates which were either symmetrically (mode I) or asymmetrically (mode II) loaded by impact in a one-point bend configuration with an edge pre-notch machined in the fibre direction. Moderate impact speeds of up to 57 ms−1 were used. The lateral shearing interferometric technique of coherent gradient sensing in conjunction with high-speed photography was used to visualize the failure process in real time. Mode-I cracks propagated subsonically with crack speeds increasing to the neighbourhood of the Rayleigh wave speed. For asymmetric mode-II types of loading the results revealed highly unstable and intersonic shear-dominated crack growth along the fibres. These cracks propagated with unprecedented speeds reaching 7400 ms−1, a speed which is more than three times the shear wave speed ...

Intersonic crack propagation in homogeneous media under shear-dominated loading: numerical analysis

The transition from subsonic to intersonic propagation of a planar crack subjected to mixed-mode loading is investigated numerically using a spectral form of the elastodynamic boundary integral equations. The spontaneous motion of the propagating crack is simulated with the aid of a quasi-linear rate-independent cohesive failure model coupling normal and shear failure modes. It is observed that, when the loading amplitude is shear-dominated and represents a substantial fraction of the fracture plane strength, the dynamic crack undergoes a rapid transition to intersonic regime. In some cases, the transition takes place through the temporary creation ahead of the main crack of a secondary cohesive zone which quickly coalesces with the primary cohesive zone. In most cases, however, the subsonic-to-intersonic transition takes place through a rapid but smooth acceleration of the main cohesive failure zone. Intersonic crack propagation can be achieved under mixed-mode conditions when the shear component of the external loading is sufficiently large. However, under steady-state intersonic conditions, cohesive failure occurs exclusively in shear, even when the remote loading of the crack is of mixed-mode nature.

On the intersonic crack propagation in an orthotropic medium

Motivaded by recent theoretical studies the elastodynamic response of an orthotropic material with a semi-infinite line crack, which propagates intersonically. is revisited through an approach which differs from those used in previous studies. The near tip stress and displacement fields are obtained for Mode I and Mode II of steady state crack propagation. The strain energy release rate analysis confirms that the Mode I is physically impossible due to the order of stress singularity, which is larger then one half. For Model II the order of stress is less than one half and it is shown that a steady state intersonic propagation is allowed only for a particular crack tip velocity which is a function of the material orthotropy.

Intersonic Crack Propagation—Part I: The Fundamental Solution

Recent experiments of Rosakis et al. have clearly shown that the crack-tip velocity can exceed the shear wave speed for a crack tip propagating between two weakly bonded, identical and isotropic solids under shear-dominated loading. This has motivated recent theoretical and numerical studies on intersonic crack propagation. We have obtained analytically the fundamental solution for mode-II intersonic crack propagation in this paper. This fundamental solution can provide the general solutions for intersonic crack propagation under arbitrarily initial equilibrium fields. We have also developed a cohesive zone model to determine the crack-tip energy release for an intersonic shear crack.

A unified method for subsonic and intersonic crack growth along an anisotropic bimaterial interface

This paper provides a unified method to solve the problem of an interfacial crack propagating subsonically and intersonically along a general anisotropic bimaterial interface. The asymptotic structure of the interfacial crack-tip field propagating subsonically and intersonically in a general anisotropic bimaterial system is investigated. Using the extended version of the Eshelby–Stroh formulation and the method of analytic continuation, the problem is recast into a Riemann–Hilbert problem of vector form. Upon solving the Riemann–Hilbert equation, the near-tip asymptotic fields are obtained, which reveals characteristics of the intersonic crack growth in an anisotropic bimaterial system. Some important cases are listed. The procedure presented in this paper will provide a theoretical tool for studying intersonic crack growth in a broader range of materials systems. The range of the powers of the crack-tip singularity is reported for four different material combinations.

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (cR) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between cR and the shear wave speed, cs, do not exclude intersonic (cs<υ<cl) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above cs and subsequently accelerate rapidly to the longitudinal wave speed (cl) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2cs. The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2cs is discussed.

Intersonic crack growth on an interface

Evidence has accumulated recently that a crack can propagate on an interface between dissimilar solids at speeds between the smallest and the largest sonic speeds of the constituent solids. Such an intersonic crack has posed several challenges to the existing theory. Assuming that the crack tip is a structureless point, and the solids are linearly elastic all the way to the crack tip, the theory shows that the stress field is singular not only at the crack tip, but also along the shock front. Furthermore, the singularity exponents differ from one half, so that the energy release rate is either zero or infinite. The relation of this theory to the experimental observations has been obscure. Specifically, it is unclear what crack speeds are forbidden by the theory. In this paper, we first introduce a unified method to analyse the crack tip field. The crack can be static, subsonic or intersonic; and the two constituent solids can be isotropic or anisotropic. To address the problem of forbidden crack speeds, we extend a cohesive zone model to intersonic cracks. In this model, the crack tip is no longer a structureless point; rather, a distributed stress represents bonding or friction. The model removes both the singular crack tip and the singular shock front. The length of the cohesive zone also characterizes the thickness of the shock front. This length depends on the crack speed. A crack speed is forbidden if it results in a negative cohesive zone length. The predictions of the model are discussed in the light of the experimental observations.

An Analysis of Intersonic Crack Growth Under Shear Loading

Crack growth in a homogeneous elastic solid under impact shear loading conditions is analyzed numerically, with the crack constrained to grow along a weak plane directly ahead of the initial crack tip. The configuration analyzed is a plane-strain model of that used in the experiments of Rosakis et al. (1999). A cohesive surface constitutive relation is specified along the weak plane that relates the tractions and displacement jumps across it and that allows for the creation of new free surface. The resistance to crack initiation and the crack speed history are predicted without invoking any additional failure criterion. The effect of cohesive strength and impact pulse time on the response is explored. In a certain parameter regime, the calculations reproduce, at least qualitatively, the type of crack speed histories seen in the experiments. For other parameter values, an abrupt transition from crack growth at the Rayleigh wave speed to a value above 2 times the shear wave speed is seen. This transition involves microcrack nucleation ahead of the main crack. At intersonic crack speeds, shock-like gradients in the near-tip stress field are found as seen in the experiments.

The effect of bond strength and loading rate on the conditions governing the attainment of intersonic crack growth along interfaces

Dynamic crack growth along a bimaterial interface under impact shear loading is analyzed numerically. The material on each side of the bond line is characterized by an isotropic hyperelastic constitutive relation. A cohesive surface constitutive relation is also specified that relates the tractions and displacement jumps across the bond line and that allows for the creation of new free surface. The resistance to crack initiation and the crack speed history are predicted without invoking any additional failure criterion. Full finite strain transient analyses are carried out. A plane strain model of the configuration used in experiments of Rosakis and co-workers is analyzed. Calculations are carried out for parameters characterizing a steel-PMMA bimaterial. For a sufficiently low impact velocity, the crack speed increases smoothly to the PMMA Rayleigh wave speed, whereas above a sharply defined transition impact velocity, the crack speed reaches a value somewhat less than the PMMA dilational wave speed. This high speed crack growth is associated with multiple crack face contact, separated by discrete micro-crack like openings behind the main shear crack. The calculations reproduce, at least qualitatively, the type of crack speed histories and crack tip fields seen in the experiments. They are also consistent with optical observations of finite multi-site contact occurring at intersonic crack speeds.

Analysis of intersonic crack growth in unidirectional fiber-reinforced composites

Recent experiments on dynamic fracture of unidirectional fiber-reinforced graphite/epoxy composite materials showed that, in Mode I, the crack tip velocity could never exceed the shear wave speed, while the crack tip velocity in Mode II not only exceeded the shear wave speed but also approached a stable velocity at which the crack grew for a substantial period of time in experiments. The experimentally obtained fringe patterns also clearly showed the existence of shear shock waves when the crack tip velocity exceeded the shear wave speed. In the present study, we have obtained the asymptotic fields near an intersonically propagating crack tip. It is shown that Mode-I intersonic crack propagation is impossible because the crack tip energy release rate supplied by the elastic asymptotic field is negative and unbounded, which is physically unacceptable since a propagating crack tip cannot radiate out energy. For Mode II, however, it is established that there exists a single crack tip velocity (higher than the shear wave speed) that gives a finite and positive crack tip energy release rate. At all other intersonic crack tip speeds the energy release rate supplied by the elastic asymptotic field is identically zero. This critical crack tip velocity agrees well with the stable crack tip velocity observed in experiments. The synthetically obtained fringe patterns based on the asymptotic field also agree with experimentally obtained fringe patterns, particularly on the existence of the shock waves.

Intersonic Crack Propagation in an Orthotropic Material

Intersonic crack propagation is found to exhibit essentially the same features in orthotropic and isotropic materials, provided that the crack propagates along a plane of elastic symmetry. Thus the stress and strain singularity at the crack edge is weaker than the inverse square root singularity in the sub-Rayleigh case, except at one distinct velocity. The energy flux into the process region is determined by using the Barenblatt model. It depends on the crack velocity and on the size of the process region, approaching zero with this size.

Effect of elastic mismatch in intersonic crack propagation along a bimaterial interface

Recent experiments showed that the speed of a crack tip propagating along a bimaterial interface can exceed the shear wave speed of the more compliant constituent in the bimaterial. This experimental observation has motivated analytical and numerical investigation on fast crack growth. Among these investigations, Huang et al. obtained a simple, analytic full-field solution for an elastic/rigid bimaterial with crack-face contact. Although this solution compares quite favorably with all available experimental data, it is not clear which bimaterial can be approximated by the elastic/rigid model. In this paper, we use the method of analytical continuation to obtain the asymptotic stress fields near the crack tip and near the trailing end of the contact zone. It is established that the elastic/rigid model is an excellent approximation to all bimaterials that have been used in fast crack growth experiments. Therefore, the simple, analytic solution of elastic/rigid model provides a useful means for analyzing experimental fringe patterns and data. It is shown that, as the elastic mismatch decreases, the elastic/rigid model may become invalid.