Intersonic Crack Propagation Research Articles

Non steady-state intersonic cracks in elastomer membranes under large static strain

Dynamic crack propagation in elastomer membranes is investigated; the focus is laid on cracks reaching the speed of shear waves in the material. The specific experimental setup developed to measure crack speed is presented in details. The protocol consists in (1) stretching an elastomer membrane under planar tension loading conditions, then (2) initiating a small crack on one side of the membrane. The crack speed is measured all along the crack path in both reference and actual configurations, including both acceleration and deceleration phases, i.e. non steady-state crack propagation phases. The influence of the prescribed stretch ratio on crack speed is analysed in the light of both these new experiments and the few previously published studies. Conclusions previously drawn for steady-state crack growth are extended to non steady-state conditions: stretch perpendicular to the crack path governs crack speed in intersonic crack propagation regime, and the role of the stretch in crack direction is minor.

Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

Modeling of dynamic crack propagation using rate dependent interface model

Influence of crack tip separation rate-dependent cohesive zone modeling at mesoscale and macroscale behaviors are investigated for sub-Rayleigh, intersonic and sub-Rayleigh to intersonic dynamic delamination growth speeds. A simple rate-dependency factor, k, is proposed for bilinear cohesive zone model (CZM) using the interface model of Corigliano et al. (2006). The factor is defined by the ratio of dynamic to static fracture toughness which is found to be related to the square of the ratio of pure-mode interfacial strengths for the selected bilinear CZM. Experimental cases from the literature with different loading rates are parametrically studied by varying the rate-dependency factor from unity, which is rate-independent, to infinity, a non-physical but theoretical condition. The first case is the three-point impact bending test that exemplifies the low-speed mode-I dynamic fracture. Next, asymmetric impact loading of polymer–composite experiment providing a mode-II dominated high speed intersonic propagation is simulated with various k. After comparing with mode-I low speed and mode-II dominated high-speed dynamic fracture experiments, an parametric analysis is carried out for a mixed-mode sub-Rayleigh to intersonic dynamic delamination in composite L-beams and compared with the experiments from Gozluklu et al. (2015). For this case, it is observed that the macroscopic aspects of fracture do not change with the rate-dependency factor. The effect of rate-dependency on crack growth kinetics is also found to be negligible for crack initiation and early stages of propagation, although the energy release rate increases. For one of the crack tips growing at intersonic speeds, the crack tip slows to sub-Rayleigh as the rate-dependency factor goes to infinity. The crack tip speeds for k>1 provides slightly better results compared with the experiments. In conclusion, rate-dependency is observed to be necessary in accurately modeling low speed, and low-to-high speed crack propagation cases but not in high-speed intersonic crack propagation.

Dynamic behaviors of mode III interfacial crack under a constant loading rate

In an earlier study on intersonic crack propagation, Gao et al. (J. Mech. Phys. Solids 49: 2113–2132, 2001) described molecular dynamics simulations and continuum analysis of the dynamic behaviors of a mode II dominated crack moving along a weak plane under a constant loading rate. The crack was observed to initiate its motion at a critical time after the onset of loading, at which it is rapidly accelerated to the Rayleigh wave speed and propagates at this speed for a finite time interval until an intersonic daughter crack is nucleated at a peak stress at a finite distance ahead of the original crack tip. The present article aims to analyze this behavior for a mode III crack moving along a bi-material interface subject to a constant loading rate. We begin with a crack in an initially stress-free bi-material subject to a steadily increasing stress. The crack initiates its motion at a critical time governed by the Griffith criterion. After crack initiation, two scenarios of crack propagation are investigated: the first one is that the crack moves at a constant subsonic velocity; the second one is that the crack moves at the lower shear wave speed of the two materials. In the first scenario, the shear stress ahead of the crack tip is singular with exponent −1/2, as expected; in the second scenario, the stress singularity vanishes but a peak stress is found to emerge at a distance ahead of the moving crack tip. In the latter case, a daughter crack supersonic with respect to the softer medium can be expected to emerge ahead of the initial crack once the peak stress reaches the cohesive strength of the interface.

Mode I intersonic crack propagation in poroelastic media

Intersonic crack propagation takes place in an elastic fluid-saturated porous solid under Mode I loading conditions. The crack tip speeds of interest c are constant and bounded below by the slower between the slow longitudinal wave-speed and the shear wave-speed, and above by the fast longitudinal wave-speed. Biot’s theory of poroelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities n ∈ [ 0 , 1 ] is investigated. Three characteristic regions in the plane ( n , c ) are delineated, depending on the relative order of the body wave-speeds. The crack surface is considered to be permeable. Cracks with and without a process zone are envisaged. In each region of the plane ( n , c ) , the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressure and velocity fields near the tip of the crack. The effective length scale introduced by the process zone is found to depend strongly on both the actual length of the process zone and the singularity exponent. Intersonic crack propagation may occur with square-root singularity of the stress and velocity fields for crack tip speeds slower than the shear wave-speed and faster than the slow longitudinal wave-speed. In this region of the ( n , c ) -plane, the energy release rate is finite and positive, for cracks with and without a process zone. As an uncommon feature, linked to the fact that the effective singularity exponent is 1/2 for both crack types, the stress and velocity fields are continuous across the Mach ray.

Mode II intersonic crack propagation in poroelastic media

A crack is steadily running in an elastic isotropic fluid-saturated porous solid at an intersonic constant speed c. The crack tip speeds of interest are bounded below by the slower between the slow longitudinal wave-speed and the shear wave-speed, and above by the fast longitudinal wave-speed. Biot’s theory of poroelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities n ∈ [0, 1] is investigated. Solids are obtained as the limit case n = 0, and the continuity of the energy release rate as the porosity vanishes is addressed. Three characteristic regions in the plane (n, c) are delineated, depending on the relative order of the body wave-speeds. Mode II loading conditions are considered, with a permeable crack surface. Cracks with and without process zones are envisaged. In each region, the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressure and velocity fields near the tip of the crack. For subsonic propagation, the asymptotic crack tip fields are known to be continuous in the body [Loret and Radi (2001) J Mech Phys Solids 49(5):995–1020]. In contrast, for intersonic crack propagation without a process zone, the asymptotic stress and pore pressure might display a discontinuity across two or four symmetric rays emanating from the moving crack tip. Under Mode II loading condition, the singularity exponent for energetically admissible tip speeds turns out to be weaker than 1/2, except at a special point and along special curves of the (n, c)-plane. The introduction of a finite length process zone is required so that 1. the energy release rate at the crack tip is strictly positive and finite; 2. the relative sliding of the crack surfaces has the same direction as the applied loading. The presence of the process zone is shown to wipe out possible first order discontinuities.

Implementation aspects of the bridging scale method and application to intersonic crack propagation

AbstractThe major purpose of this work is to investigate the performance of the bridging scale method (BSM), a multiscale simulation framework for the dynamic, concurrent coupling of atomistics to continua, in capturing shear‐dominant failure. The shear‐dominant failure process considered in this work is intersonic crack propagation along a weak plane in an elastic material, similar to the seminal molecular dynamics (MD) simulations by Abraham and Gao (Phys. Rev. Lett. 2000; 84(14):3113–3116). We show that the BSM simulations accurately capture the essential physics of the intersonic crack propagation, including the formation of a daughter crack and the sudden acceleration of the crack to a velocity exceeding the material shear wave speed. It is also demonstrated that the non‐reflecting boundary condition can adequately dissipate the strongly localized wave formed by the Mach cone after the crack accelerates beyond the material shear wave speed. Finally, we provide the algorithm for our implementation of the BSM, as well as the code used to determine the damping kernels via a newly adopted technique which is less expensive than previous methods. Copyright © 2007 John Wiley & Sons, Ltd.

Cohesive solutions of intersonic moving dislocations

A class of cohesive solutions of moving glide dislocations with intersonic speeds has been derived on the basis of the fundamental equation of a moving dislocation introduced by Weertman in conjunction with a proposed generalized Bilby–Cottrell–Swinden–Dugdale model. In this model we assume a straight weak path within an infinite elastic plate. Two length scales, namely the width (thickness) of the weak path and the material intrinsic length, which scales strain-gradient-induced hardening and energy dissipation, are taken into account by applying the traction–separation law for the decohesion of the weak path. Dislocations propagate along this weak path with a speed higher than the shear wave speed. The accumulation of these moving dislocations forms a macroscale crack growth with a cohesive zone ahead of the crack tip. Similar to the Bilby–Cottrell–Swinden–Dugdale model, the remote enforced stress and/or stress-rate boundary conditions are represented as an equivalent crack surface traction associated with the dislocation distribution. The involved Cauchy integral and corresponding eigenvalue problem are solved using the algorithms introduced by Muskhelishvili and by Weertman. The problems associated with three types of decohesion law are constant traction, traction linearly dependent on separation, and separation- and separation-rate-dependent traction. These problems are solved using three different solution strategies: the direct-integration method, the iteration method and the Jacobi polynomial expansion respectively. The derived solutions provide explicit relations between the remote load propagation speed, the material intrinsic length, the weak path thickness and the strain-rate-hardening parameter. The solutions demonstrate that the intersonic speed region can be divided into two subdomains; steady-state propagation occurs within the subdomain where the propagation speeds are equal to or greater than the Eshelby speed (c s × 21/2, where c s is the shear wave speed). For a weak path with a finite width and the corresponding decohesion law scaled by material intrinsic length, an intersonic crack propagation will not take place if only a constant remote stress is imposed. A ‘steady-state’ crack surface load and/or remote stress-rate boundary condition, which can be considered as a point force or a distributed force with a constant distance to the moving crack tip, is required to maintain steady-state intersonic crack propagation.

Modeling dynamic crack propagation in fiber reinforced composites including frictional effects

Dynamic crack propagation in a unidirectional carbon/epoxy composite is studied through finite element analyses of asymmetric impact (shear loading) of a rod against a rectangular plate. A finite deformation anisotropic visco-plastic model is used to describe the constitutive response of the composite. Crack propagation is simulated by embedding zero thickness interface element along the crack path. An irreversible mixed-mode cohesive law is used to describe the evolution of interface tractions as a function of displacement jumps. Contact and friction behind the crack tip are accounted for in the simulations. The failure of the first interface element at the pre-notch tip models onset of crack extension. Crack propagation is modeled through consecutive failure of interface elements. The dynamic crack propagation phenomenon is studied in terms of crack initiation time, crack speed, mode I and mode II displacement jumps and tractions associated with the failure of interface elements, effective plastic strain at the crack tip and path independent integral J ′. Analyses are carried out at impact velocities of 5, 10, 20, 30 and 40 m/s, assuming the crack wake is frictionless. Moreover, analyses at impact velocities of 30 and 40 m/s are also carried out with a friction coefficient of 0.5, 1, 5 and 10 along the crack surfaces. The analyses show that steady-state intersonic crack propagation in fiber reinforced composite materials occurs when the impact velocity exceeds a given threshold. A steady-state crack speed of 3.9 times the shear wave speed and 83% of the longitudinal wave speed is predicted in the cases in which the impact velocity is above 10 m/s. Detailed discussion is given on the features of sub-sonic and intersonic crack propagation. It is shown that friction effects, behind the crack tip, do not have a significant effect on maximum crack speed; however, they do on characteristics of the shock wave trailing the crack tip. The analyses also show that the contour integral J ′, computed at contours near the crack tip, is indeed path independent and can serve as a parameter for characterizing intersonic crack propagation.

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.

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.

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.

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.

Intersonic crack propagation in bimaterial systems

This paper describes experimental observations of various phenomena characteristic of dynamic intersonic decohesion of bimaterial interfaces. Two separate but complementary optical methods are used in conjunction with high-speed photography to explore the nature of the large-scale contact and mach wave formation at the vicinity of running cracks in two different bimaterial systems. Theoretical predictions of crack tip speed regimes, where large-scale contact is implied, are confirmed. Also, the theoretically predicted mach wave emanating from the intersonically propagating crack tip is observed. Direct visual evidence is also obtained for another traveling mach wave emanating from the end of the intersonically moving contact zone. Subsequently, a physical model for intersonic crack propagation along bimaterial interfaces is presented and ratified in view of recent experimental observations and theoretical developments. Finally, the paper presents very recent experimental evidence that shows crack tip speeds exceeding the intersonic regime and becoming clearly supersonic.