Dynamic stability of a propagating crack

In this work we investigate the stability of a straight two-dimensional dynamically propagating crack to small perturbation of its path. Willis and Movchan (J. Mech. Phys. Solids 43 (1995) 319; J. Mech. Phys. Solids 45 (1997) 591) constructed formulae for the perturbations of the stress intensity factors induced by a small three-dimensional dynamic perturbation of a nominally plane crack. Their solution is exploited here to derive equations for the in-plane and out-of-plane perturbations of the crack path making use of the Griffith fracture criterion and the principle of “local symmetry” (i.e the crack propagates so that local K_II=0). We consider a crack propagating in a body loaded by a pair of point body forces and subjected to a remote uniaxial stress, aligned with the direction of the unperturbed crack. We assume that the loading follows the crack as the crack advances and is such that the unperturbed crack is subjected to Mode I loading. We perform an analysis of the stability of the dynamic crack in a similar way as in earlier work (Obrezanova et al., J. Mech. Phys. Solids 50 (2002) 57) on the quasistatically advancing crack. We present numerical results illustrating the influence of the crack velocity on the crack stability. Numerical computations of the possible crack paths have been performed which show that at velocities of crack propagation exceeding about one-third of the speed of Rayleigh waves the crack may admit one or more oscillatory modes of instability.     

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.     

Dynamic crack growth along a polymer composite-Homalite interface

Dynamic crack growth along the interface of a fiber-reinforced polymer composite-Homalite bimaterial subjected to impact shear loading is investigated experimentally and numerically. In the experiments, the polymer composite-Homalite specimens are impacted with a projectile causing shear dominated interfacial cracks to initiate and subsequently grow along the interface at speeds faster than the shear wave speed of Homalite. Crack growth is observed using dynamic photoelasticity in conjunction with high-speed photography. The calculations are carried out for a plane stress model of the experimental configuration and are based on a cohesive surface formulation that allows crack growth, when it occurs, to emerge as a natural outcome of the deformation history. The effect of impact velocity and loading rate is explored numerically. The experiments and calculations are consistent in identifying discrete crack speed regimes within which crack growth at sustained crack speeds is possible. We present the first conclusive experimental evidence of interfacial crack speeds faster than any characteristic elastic wave speed of the more compliant material. The occurrence of this crack speed was predicted numerically and the calculations were used to design the experiments. In addition, the first experimental observation of a mother-daughter crack mechanism allowing a subsonic crack to evolve into an intersonic crack is documented. The calculations exhibit all the crack growth regimes seen in the experiments and, in addition, predict a regime with a pulse-like traction distribution along the bond line.     

Moving crack for functionally grated material in an infinite length strip under antiplane shear

Considered is a Yoffe crack in an infinite strip of functionally grated material (FGM) subjected to antiplane shear. The shear moduli in two directions of FGM are assumed to be of exponential form. The dynamic stress intensity factor and strain energy density factor at the crack tip are obtained by using integral transforms and dual-integral equations. The numerical results show that the decrease of the strain energy density factor varies with the shear moduli gradient, and the increase of the strain energy density factor varies with the increase of the moving crack speed. The ratio of shear moduli in material vertical orientation has a great influence on the strain energy density factor.     

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.     

Dynamic crack growth under stress wave loading

The crack growth process has been analysed on the basis of a fracture criterion of a dynamic stress intensity factor when a crack in an infinite plate was subjected to a pulse type of stress wave. The crack velocity and the amount of crack extension were related to the constant amplitude and the duration of the stress pulse. The calculated amount of crack extension was well in agreement with the experimental one for the polymer material Acrylite (which is similar to polymethylmethacrylate) found by the authors, thus indicating the validity of the present approach.     

表面裂纹准静态扩展的有限元模拟计算

由于现有断裂准则未能考虑表面裂纹前沿各点实际断裂阻力的变化，所作的表面裂纹准静态扩展模拟计算不能很好地预测几何形貌的真实变化规律。本文提出了一个考虑应力状态不同对实际断裂阻力影响的变阻力断裂准则；并以该准则作为表面裂纹前沿局部断裂扩展的判据，通过三维有限元法进行了表面裂纹准静态扩展的模拟计算；还将模拟预测值与用多试件法测得的实验值进行比较，验证了变阻力断裂准则作为裂纹稳态扩展判据的有效性     

Crack in a 2D beam lattice: Analytical solutions for two bending modes

We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass-spring lattices. Considering the static problems we use the discrete Fourier transform and the Wiener-Hopf technique. Monotonically distributed bending moments ahead of the crack are determined for the symmetric mode, and a self-equilibrated transverse force distribution is found for the antisymmetric mode. It is shown that in the latter case only the crack-front beam resists to the fracture development, whereas the forces in the other beams facilitate the fracture. In this way, the macrolevel fracture energy is determined in terms of the material strength. The macrolevel energy release is found to be much greater than the critical strain energy of the beam, especially in the hyper-singular mode. In both problems, it is found that among the beams surrounding the crack the crack-front beam is maximally stressed, and hence its strength defines the strength of the structure.     

Electric-field-induced fatigue crack growth in ferroelectric ceramics

Electric-field-induced fatigue crack growth in pre-cracked PZT ferroelectric ceramics is experimentally investigated in this work. It is found that the crack open and close under an alternating electric field is a major mechanism of crack propagation. The experimental results also show that the frequency, waveform, as well as the amplitude ratio, of the electric loading, play important roles in electric-field-induced fatigue cracking. Empirical formulations of fatigue crack propagation rates are obtained based on the experimental results. It is revealed that the crack grows at a nearly constant rate when the loading frequency is below 100 Hz. However, with the increase of the loading frequency over 125 Hz, the crack propagation rate diminishes rapidly.     

Dynamic crack growth behaviour during explosive treatment

On the basis of mixed mode fracture as well as dynamic fracture mechanics crack growth behaviour related to the stress wave during the explosive treatment is studied experimentally and theoretically. The effects of the incident wave, the reflected wave and the stress wave duration on the crack growth behaviour are discussed. A method for restricting crack growth by using an energy absorber has been tested and recommended.     

Dynamic fracture analysis of adhesive bonded material under normal loading

Dynamic fracture behavior of a Griffith crack along the interface of an adhesive bonded material under normal loading is studied. The singular integral equations are obtained by employing integral transformation and introducing dislocation density functions. By adopting Gauss-Jacobi integration formula, the problem is reduced to the solution of algebraic equations, and by collocation dots method. their solutions can be obtained Based on the parametric discussions presented in the paper, the following conclusions can be drawn： （1） Mode I dynamic stress intensity factor （DSIF） increases with increasing initial crack length and decreasing visco-elastic layer thickness, revealing distinct size effect; （2） The influence of the visco-elastic adhesive relaxation time on the DSIF should not be ignored.     

基于圆弧底试件的动态裂纹扩展及止裂规律研究

为了研究脆性材料的动态裂纹扩展及止裂规律,设计了一种带圆弧形底边的梯形开口边裂纹（trapezoidal opening crack with arc bottom,TOCAB）构型的试件。在落锤冲击设备加载下,对圆心角为0°、60°、90°和120°的TOCAB试件进行了冲击实验,并采用裂纹扩展计(crack propagation gauge,CPG)监测裂纹起裂和扩展时间,从而获得裂纹扩展速度。采用有限差分软件AUTODYN对落锤冲击设备和试件进行数值模拟,研究了裂纹的动态扩展过程及止裂规律。还基于实验和数值方法,计算了裂纹的临界动态应力强度因子。实验和数值结果均表明：3种弧度的TOCAB试件都可以实现运动裂纹止裂,该构型可用于研究动态裂纹止裂问题;数值计算的裂纹扩展路径与实验结果基本一致,验证了数值模型的有效性;裂纹起裂和止裂时刻的临界动态应力强度因子大于裂纹动态扩展过程中的临界动态应力强度因子。

     

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.     

Crack growth of an interface crack between two dissimilar magneto-electro-elastic materials under anti-plane mechanical and in-plane electric magnetic impact

This paper examines the dynamic response of an interface crack between two dissimilar magneto-electro-elastic materials subjected to the mechanical and electric magnetic impacts. The magneto-electric impermeable boundary conditions are adopted. Laplace and Fourier transforms and dislocation density functions are employed to reduce the mixed boundary value problem to Cauchy singular integral equations in Laplace transform domain, which are solved numerically. Lots of numerical results are given graphically in time domain. The effects of electric impact loading and magnetic impact loading on dynamic energy density factors are discussed. Crack growth and propagation is predicted. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

Toughness increment by crack growth in toughened structural materials

Material toughening could be furnished by the energy dissipating wakes and bridging segments during crack growth. According to their contributions to the energy integral applicable to a growing crack, the toughening mechanisms are categorized as: dilatational plasticity and induced shear yielding in the crack wakes, bridging due to second inclusion phases, and the matrix bridging caused by wavy crack front. Detailed toughening analysis is pursued for structural polymers and composite materials reinforced by short aligned fibers. Sponsored by the State Education Commission of China and by the Fok Ying-Tung Education Foundation     

Bifurcation of collinear crack system under dynamic compression

The evolution pattern of collinear crack array plays a very important role in the final failure pattern of rock and predicting earthquake. Crack interactions lead to the nonhomogeneous pseudo-traction, then result in bifurcation of crack growth pattern. Bifurcation condition of crack growth pattern can be expressed by the crack growth length/spacing ratio. For collinear cracks loaded by dynamic compressive loads, uniform crack growth pattern yields to non-uniform crack growth pattern when the crack growth length/spacing ratio is larger than a critical value. In this paper, crack interactions are studied using stress superposition principle and the Chebyshev polynomials expansion of the pseudo-traction. The analytical solution of the critical value for two collinear cracks and infinite collinear cracks is given out. The critical value is sensitive to pre-existing crack length, the friction coefficient, the orientation of pre-existing crack, crack growth velocity.     

爆炸应力波作用下动、静裂纹相互作用的实验研究

采用数字激光动态焦散线测试系统,研究爆炸应力波作用下动裂纹与预制静裂纹(水平夹角为90°、150°)相互作用机理,以及裂纹扩展的动态行为。结果表明:(1)在动、静裂纹贯通之前,静裂纹两端便出现焦散斑,动、静裂纹贯通以后,静裂纹沿爆炸应力波传播方向扩展,并且扩展速度小于动裂纹扩展速度,也小于无静裂纹时动裂纹扩展速度; (2)静裂纹存在时,动裂纹扩展的总体长度减小。动裂纹起裂时间缩短,扩展速度基本不受静裂纹的影响,裂纹应力强度因子值大于静裂纹两端值; (3)随着静裂纹水平夹角的增大,动、静裂纹贯通时动裂纹沿水平方向偏转距离增大,静裂纹B端反向扩展与动裂纹相互“咬合”,C端裂纹扩展位移和速度增大。

     

Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

Dynamic anti-plane shear of a cracked piezoelectric ceramic

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a line crack subjected to horizontally polarized shear waves in an arbitrary direction. The problem is formulated by means of integral transforms and reduced to the solution of a Fredholm integral equation of the second kind. The path-independent integral G is extended here to include piezoelectric effects, and is evaluated at the crack tip to obtain the dynamic energy release rate. Numerical calculations are carried out for the dynamic stress intensity factor and energy release rate. The material is piezoelectric ceramic.     

Stability of an advancing crack to small perturbation of its path

In this work we investigate the stability of a nominally straight two-dimensional quasistatically growing crack to a small perturbation of its path. Formulae for perturbations of stress intensity factors induced by slight deviation of the crack trajectory were developed by Movchan et al. (Int. J. Solids Struct. 35, 3419) Their solution is exploited to derive an equation for the perturbation of the crack path on the assumption that the crack advances in pure “opening” mode (i.e. local K_II=0). Various types of loading conditions are considered, including a cracked body loaded by a pair of point body forces and a crack whose faces are subjected to given tractions acting in the direction normal to the crack boundary. The body is also subjected to a remotely maintained uniaxial stress, aligned with the direction of the unperturbed crack. The loading is assumed to advance as the crack advances, to maintain the critical value of Mode I stress intensity factor. Numerical computations of possible crack paths have been performed, extending results on crack stability obtained by Cotterell and Rice (Int. J. Fract. 16, 155). The results show that in the case of loading by point body forces the stability of the crack path depends on the positions of the points of application of the applied forces and the magnitude of the applied stress acting parallel to the crack. There exists a critical value of this stress such that the crack path is stable for values less than critical and unstable otherwise. It is shown that the crack is always unstable in the case of point force tractions applied normal to the crack faces.