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

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

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.     

Sudden deceleration or acceleration of an intersonic shear crack

Sudden jumps in the crack tip velocity were revealed by numerical simulation (in both continuum/cohesive element and molecular dynamics approaches) and experiments for rapid shear cracking. The cracking velocity may accelerate from a sub-Rayleigh speed to the intersonic range, or from an intersonic speed to a higher one, when the reflected impact wave reloads the crack tip. On the other hand, the cracking velocity may decelerate from an intersonic speed to a lower one or recede to the sub-Rayleigh range when the fracture driving force declines. The velocity change encountered during intersonic cracking plays a different role from that in the acceleration or deceleration of a subsonic crack. A crack propagating at an intersonic speed would leave a shear wave trailing behind. When the crack decelerates or accelerates, the effect of the trailing wave will lead to a transition period from one steady-state solution of crack tip singularity to another. This investigation aims at quantifying these processes. The full field solution of an intersonic mode II crack whose speed changed suddenly from one velocity (intersonic or subsonic) to another (intersonic or subsonic) is given in closed form. The solution is facilitated via superposing a series of propagating crack problems that are loaded by dislocations to seal the unwanted crack-face sliding or by concentrated forces moving at various speeds to negate the crack-face traction. In contrast to the subsonic solution, the results in the intersonic case indicate that the elastic fields around the crack tip depend on the deceleration or acceleration history that is traced back over a long time. Singularity matching dictates the jump that may actually take place.     

Stability of an intersonic shear crack to a perturbation of its edge

A weight function matrix is developed for obtaining the stress singularity coefficients at the edge of a plane crack, moving uniformly at an intersonic speed while subjected to arbitrary shear loading. This is then utilised for deriving, to first order, the perturbations of these coefficients associated with a small spatially and temporally varying perturbation of its edge. The perturbation solution is employed, in conjunction with a simple fracture criterion, to investigate the stability of a uniformly moving intersonic crack, subjected to following loads.     

Moving line crack accompanied with damage zone subject to remote tensile loading

In the 1920s, a closed-form solution of the moving Griffith crack was first obtained by Yoffe. Based on Yoffe's solution, the Dugdale model for the moving crack case gives a good result. However, the Dugdale model fails when the crack speed is closed to the Rayleigh wave speed because of the discontinuity occurred in the crack opening displacement (COD). The problem is solved in this paper by introducing a restraining stress zone ahead of the crack tip and two velocity functions. The restraining stresses are linearly distributed and related to the velocity of the moving crack. An analytical solution of the problem is obtained by use of the superposition principle and a complex function method. The final result of the COD is continuous while the crack moves at a Rayleigh wave speed. The characteristics of the strain energy density (SED) and numerical results are discussed, and conclusions are given.     

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behavior of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localized surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

Effect of T-stress on branch angle of moving cracks

The effects of the T-stress on Yoffe crack propagation are analyzed. Using a maximum k_I fracture criterion near the kink of a moving crack tip, a branch angle is determined via asymptotic crack-tip field containing two fracture parameters related to singular and constant terms. Results indicate that crack speeds decrease the T-stress. The crack-tip field and the branch angle depend on the T-stress, especially for higher crack velocities. The critical speed for crack bifurcation is independent of remote transverse loading if neglecting the T-stress. Otherwise, the crack branch speed is reduced or raised, depending on positive or negative transverse loading, respectively.     

Contact Zone Approach for a Moving Interface Crack in a Piezoelectric Bimaterial under Thermoelectromechanical Loading

An interface crack of a finite length moving with a constant subsonic speed v along an interface of two semi-infinite piezoelectric spaces is considered. It is assumed that the bimaterial compound is loaded by a remote mixed mode mechanical loading and a thermoelectrical field and that a frictionless contact zone arises at the leading crack tip. Electrically permeable and electrically insulated cases of the open part of the crack are involved into the consideration. By introducing a moving coordinate system at the crack tip the problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved exactly. For both cases of the electrical conditions the transcendental equations are obtained for the determination of the real contact zone length, and moreover, the associated closed form asymptotic formulas are found for small values of this parameter. Variations of the contact zone length and the stress intensity factor with respect to the crack speed and the loading have been investigated both for electrically permeable and electrically insulated cases.     

Crack propagation in wedged double cantilevered beam specimens

A closed form analytical solution of crack propagation in double cantilevered beam specimens opened at a constant rate has been found. Hamilton's principle for non-conservative systems was applied to describe the crack motion, under the assumption of a Bernoulli-Euler beam. The criterion of crack propagation is a critical bending moment at the crack tip. The calculations of beam motion take into account wave effects in the Bernoulli-Euler theory of elastic beams. The beam shape during the crack motion is found with a similarity transformation and expressed by Fresnel integrals. The boundary conditions satisfied are the fixed ones of zero bending moment and constant beam opening rate at the load end of the specimen and the moving ones of zero deflection and zero slope of the deflected beam at the tip of the moving crack. The fracture represents a moving critical bending moment. The analytical results show that the specific fracture surface energy is a unique function of the ratio of the crack length squared to the time subsequent to loading and this is computed from the recorded time-dependence of the crack length.     

Study on the mechanisms and quantitative law of mode I supersonic crack propagation

Continuum mechanics predicts that the propagation speed of non-equilibrium information in solids is limited by the longitudinal wave speed, so is crack propagation. However, solids are essentially discrete systems. In this paper, via theoretical analysis and numerical simulations, it is demonstrated in a straightforward way that non-equilibrium disturbance (e.g. force, displacement, energy, and so on) can propagate at a supersonic speed in discrete systems, although the magnitude of the disturbance attenuates very quickly. In dynamic fracture, a cascade of atomic-bond breaking events provides an amplification mechanism to counterbalance the attenuation of the disturbance. Therefore, supersonic crack propagation can be realized in a domino way. Another key factor for supersonic crack propagation is to ensure sufficient energy flowing into the crack tip. Since most energy can only be transferred at a speed limited by the longitudinal wave speed, the conditions for the occurrence of supersonic crack propagation are not easily met in most situations, unless there is high pre-stored energy along the crack path or continuous energy supply from the loading concomitantly moving with the crack tip. A quantitative relation between supersonic crack propagation speed and material properties and parameters is given, which implies that knowing all the classical macroscopic quantities is not enough in determining the supersonic crack propagation speed, and the microstructure does play a role. Moreover, it is interesting to note that fracture toughness affects the crack propagation speed in the subsonic regime, but not in the supersonic regime, because the deformation/stress is uniform in front of a supersonic crack where strength criterion dominates.     

压电材料反平面运动裂纹的能量释放率与分叉角

用复变函数方法,研究了压电材料中反平面运动裂纹的动态断裂问题,研究表明：介质内的耦合场与裂纹运动速度有关,在裂纹尖端有奇异。应力强度因子与裂纹运动速度无关,与纯弹性结构一致,沿裂纹延长线扩展的动态能量释放率可用应力强度因子表示,而与电载荷无关,裂纹运动的高速度具有止裂作用,在一定条件下,裂纹有扩展成曲线裂纹或分叉的趋势。     

Propagating semi-infinite crack in fluid-saturated porous medium of finite height

Derived in this work are the Mode I stress intensity factor results for a constant velocity semi-infinite crack moving in a fluid-saturated porous medium with finite height. Two limiting cases are discussed; they correspond to a low and high speed crack propagation. To be expected is that the crack front stress intensification would increase as the medium height is reduced in relation to the segment length in which mechanical pressure is applied. Moreover, the stress intensity factor for the high speed crack is larger than the low speed crack, the magnification of which depends on the material. Dissatisfaction of the crack surface and tip boundary condition is found in the present solution which calls possibly for the additional consideration of a local boundary layer as discussed by other authors.     

Two dimensional numerical simulation of crack kinking from an interface under dynamic loading by time domain boundary element method

The hybrid time-domain boundary element method, together with the multi-region technique, is applied to simulate the dynamic process of propagation and/or kinking of an interface crack in a two-dimensional bi-material. The whole bi-material is divided into two regions along the interface. The traditional displacement boundary integral equations are employed with respect to each region. However, when the crack kinks into the matrix material, the non-hypersingular traction boundary integral equations are used with respect to the part of the crack in the matrix. Crack propagation along the interface is numerically modelled by releasing the nodes in the front of the moving crack-tip controlled by the fracture criterion. Kinking of the interface crack is controlled by a criterion developed from the quasi-static one. Once the crack kinks into the matrix, its propagation is modeled by adding new elements of constant length to the moving crack-tip controlled by a criterion extended from the quasi-static maximum circumferential stress. The numerical results of the crack growth trajectory for different material combinations are computed and compared with the corresponding experimental results. Good agreement between numerical and experimental results implies that the present boundary element numerical method can provide an excellent simulation for the dynamic propagation and deflection of an interface crack.     

Fatigue crack growth under anti-plane shear mode deformation

In this paper, the theory of the steady growth of fatigue crack in an infinite medium under the periodic anti-plane remote shear loading has been examined. The criterion of accumulative plastic work for material failure associated with the slip displacement in the fracture process zone of Dugdale type ahead of the crack tip is employed in the analysis. The effect of the locked dislocation in the fracture process zone is considered. Under the assumption that the speed of fatigue crack propagation remains uniform through the fracture process zone, the steady speed of fatigue crack can be expressed as a function of the range of the applied shearing stress and the maximum shearing stress. The effect of the crack size on the fatigue crack speed is discussed. The effect of the finite width of specimen on the speed of fatigue crack speed is investigated. The differences between the present work and the previous studies on fatigue crack speed are discussed.     

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

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

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature

Viscoplastic crack-tip deformation behaviour in a nickel-based superalloy at elevated temperature has been studied for both stationary and growing cracks in a compact tension (CT) specimen using the finite element method. The material behaviour was described by a unified viscoplastic constitutive model with non-linear kinematic and isotropic hardening rules, and implemented in the finite element software ABAQUS via a user-defined material subroutine (UMAT). Finite element analyses for stationary cracks showed distinctive strain ratchetting behaviour near the crack tip at selected load ratios, leading to progressive accumulation of tensile strain normal to the crack-growth plane. Results also showed that low frequencies and superimposed hold periods at peak loads significantly enhanced strain accumulation at crack tip. Finite element simulation of crack growth was carried out under a constant ΔK-controlled loading condition, again ratchetting was observed ahead of the crack tip, similar to that for stationary cracks.A crack-growth criterion based on strain accumulation is proposed where a crack is assumed to grow when the accumulated strain ahead of the crack tip reaches a critical value over a characteristic distance. The criterion has been utilized in the prediction of crack-growth rates in a CT specimen at selected loading ranges, frequencies and dwell periods, and the predictions were compared with the experimental results.