Stress intensity factors for curved and kinked cracks in plane extension

A singular integral equation containing the crack opening displacement (COD) is developed for solving plane elasticity problems. The crack may contain any number of kinks at different intervals and orientations, such as a saw-tooth shape. Cracks in the form of a sine wave can also be treated. The crack tip stress intensity factors are evaluated for a variety of crack shapes and the results are displayed graphically. The distance between the crack tips is found to be a dominant factor on the crack tip stress intensity while the angle between the tangent to the crack tip and load direction determines the proportion of Mode I and II stress intensity factors.     

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.     

Rayleigh waves emitted by a propagating crack in a strain-rate dependent elastic medium

A simple model was proposed for the interpretation of the non-circular form of the Rayleigh wavefronts emitted by a fast running crack in a plate. The surface deformation around the crack tip, due to the high stress concentration there, propagated as a surface wave after fracture of this zone. On the other hand, the moving singularity of the crack tip created a dynamic stress field of varying intensity with time all over the specimen. This dynamic stress field resulted in a significant change of the mechanical properties of a strain-rate dependent material and therefore it influenced the velocity of propagation of fracture-Rayleigh wavefronts. An analysis of this varying dynamic strain field explained the non-circular form of Rayleigh waves, accompanying the propagating crack. For the experimental evaluation of the K₁-factor the method of dynamic caustics was used in conjunction with the high-speed photography technique.     

Crack propagation in an elastic solid subjected to general loading— IV. Obliquely incident stress pulse

The stress intensity factors of a half-plane crack extending nonuniformly in an isotropic elastic solid subjected to stress wave loading are considered. A plane stress pulse is obliquely incident on the crack, and the wavefront strikes the crack at some initial time. At some arbitrary later time, the crack begins to extend at a nonuniform rate. It is found that the mode I and mode II stress intensity factors each have the form of the product of a universal function of instantaneous cracktip speed with the stress intensity factor for an equivalent stationary crack. An energy-rate balance fracture criterion is applied to obtain an equation of motion for the crack tip and to determine the actual delay time between the arrival of the incident wave and the onset of fracture as a function of angle of incidence of the loading wave.     

Stress intensity factor determination error by the method of caustics

The caustics created at a crack tip under biaxial loading are examined using both the closed form and the approximate solution. Particular attention is given to the caustics and their initial curves created by rays reflected from the rear and front surfaces of the specimen in addition to those in the transverse direction of the cracked specimen. These quantities can differ by more than 30% depending on whether the exact or the approximate solution is used for determining the stress intensity factors which can deviate as much as 80%. The crack length also affects the outcome. Larger errors are committed by the approximate solution for caustics that are closer to the crack tip. The advantage of using the normalized distance from the crack tip is discussed in contrast to using the absolute distance. Incorrect experimental measurements are also discussed in relation to the resulting error for evaluating the crack tip stress intensity factor.     

弹性波与单侧界裂纹相互作用问题的边界元法

措助边界元法设计了一种迭代修正方法来求解单侧界面裂纹模型与弹性波的相互作用问题,作为对算法的检验,用这种方法我们具体地分析了平面简谐弹性波对一个则界面裂纹的入射,给出了裂纹面的接触形态及应力场。     

The edge dislocation in a three-quarter plane. Part II: Application to an edge crack

This paper presents the solution for the crack tip stress intensity factors for a short crack emanating from the corner of a three-quarter plane. The solution employs the influence functions for a dislocation along one of the projection lines of a three-quarter plane together with the Williams semi-infinite wedge solutions, and the result is valid for any remote loading configuration. The sole proviso is that the crack is short enough to be considered well within the region in which the asymptotes govern the stress state. This general solution is then applied to a specific geometry and is compared with a calibration for the stress intensity factors for a similar geometry.     

各向异性双材料界面裂纹扩展裂尖场

应用界面断裂力学理论和Stroh方法,研究了广义平面变形下动态裂纹沿着各向异性双材料界面扩展时的裂尖奇异应力及动态应力强度因子.双材料界面的动态裂尖区域特性主要由两个实矩阵W和D确定,且裂尖奇异应力和动态应力强度因子可以由包含这两个矩阵的柯西奇异积分方程确定,同时给出了动态应力强度因子和能量释放率的显示表达式.算例得出当裂纹以小速度扩展时,裂尖振荡因子ε与静态时几乎相同,当界面裂纹扩展速度接近瑞利波速时,ε趋于无穷大;同时得出应力强度因子及能量释放率随裂纹扩展速度的变化关系.     

Permeable cracks between two dissimilar piezoelectric materials

An interfacial crack with electrically permeable surfaces between two dissimilar piezoelectric ceramics under electromechanical loading is investigated. An exact expression for singular stress and electric fields near the tip of a permeable crack between two dissimilar anisotropic piezoelectric media are obtained. The interfacial crack-tip fields are shown to consist of both an inverse square root singularity and a pair of oscillatory singularities. It is found that the singular fields near the permeable interfacial crack tip are uniquely characterized by the real valued stress intensity factors proposed in this paper. The energy release rate is obtained in terms of the stress intensity factors. The exact solution of stress and electric fields for a finite interfacial crack problem is also derived.     

SH波对一维六方准晶中直裂纹的散射

分析了SH波对一维六方准晶中直裂纹的散射问题。利用积分变换技术,结合Copson方法,通过求解对偶积分方程,得到声子场和相位子场应力、位移及裂纹尖端动应力强度因子的解析表达式。通过数值算例讨论了裂纹长度、入射角和入射波频率对标准动应力强度因子的影响,此研究在工程材料应用中有一定的参考价值。     

The determination of dynamic fracture toughness of AISI 4340 steel by the shadow spot method

Dynamic crack propagation experiments have been performed using wedge loaded double cantilever beam specimens of an austenitized, quenched and tempered 4340 steel. Measurements of the dynamic stress intensity factor have been made by means of the optical method of caustics. The interpretation of experimental data, obtained from the shadow spot patterns photographed with a Cranz-Schardin high speed camera, is based on an elastodynamic analysis. The instantaneous value of the dynamic stress intensity factor K^d_I is obtained as a function of crack tip velocity. Finally, the interaction of reflected shear and Rayleigh waves with the moving crack tip stress field is considered.     

The propagation and arrest of an edge crack in anti-plane shear

The motion of an edge crack extending non-uniformly in an elastic half-space under conditions of anti-plane shear is investigated, taking the first reflected stress wave into account. An expression for the stress intensity factor at the crack tip is obtained, and an energy-balance crack propagation criterion is used to find the equation of motion of the tip. On solving this equation numerically, it is found that crack arrest occurs before the second reflected wave reaches the tip.     

Determination of stress intensity factor with direct stress approach using finite element analysis

In this article,a direct stress approach based on finite element analysis to determine the stress intensity fac-tor is improved.Firstly,by comparing the rigorous solution against the asymptotic solution for a problem of an infinite plate embedded a central crack,we found that the stresses in a restrictive interval near the crack tip given by the rigorous solution can be used to determine the stress intensity fac-tor,which is nearly equal to the stress intensity factor given by the asymptotic solution.Secondly,the crack problem is solved numerically by the finite element method.Depending on the modeling capability of the software,we designed an adaptive mesh model to simulate the stress singularity.Thus, the stress result in an appropriate interval near the crack tip is fairly approximated to the rigorous solution of the corre-sponding crack problem.Therefore,the stress intensity factor may be calculated from the stress distribution in the appro-priate interval,with a high accuracy.     

爆炸应力波作用下裂纹与孔洞的动态焦散线分析

将高速摄影技术与动态焦散线方法相结合，研究了爆炸应力波对裂纹、空孔的作用历史，记录了环绕它们的动态焦散斑图，给出了爆炸应力场中裂纹尖端复合应力强度因子的时间依赖关系以及空孔周围应力场分布的瞬态变化过程，为固体介质的爆破机理研究提供了新方法。     

An exact transient analysis of plane wave diffraction by a crack in an orthotropic or transversely isotropic solid

A transient plane strain analysis of diffraction of plane waves by a semi-infinite crack in an unbounded orthotropic or transversely isotropic solid is performed. The waves approach the crack at a general oblique angle, and are of two types, a normal stress pulse and a shear stress pulse, i.e. a P- and an SV-wave, respectively, in the isotropic limit. A class of materials that includes this limit and beryl, cobalt, ice, magnesium and titanium is chosen for illustration, and exact solutions are obtained for the initial/mixed boundary value problems.In contrast to related work, a factorization in the Laplace transform space is used to simplify the solution forms and the Wiener-Hopf component of the solution process, and to yield a more compact expression for the Rayleigh wave speed. Calculations for this speed, the two allowable, direction-dependent, plane wave speeds, and quantities related to the Mode I and Mode II dynamic stress intensity factors are given for the five anisotropic materials mentioned.     

Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Dynamic anti-plane fracture problem of an exponentially graded linear magnetoelectroelastic plane with a finite impermeable crack subjected to time-harmonic SH-waves is solved. Directions of wave propagation and material inhomogeneity are chosen in an arbitrary way. The fundamental solution for the coupled system of partial differential equations with variable coefficients is derived in a closed form by the hybrid usage of both an appropriate algebraic transformation for the displacement vector and the Radon transform. The formulated boundary-value problem is solved by a nonhypersingular traction boundary integral equation method (BIEM). The collocation method and parabolic approximation for the unknown generalized crack opening displacements are used for the numerical solution of the posed problem. Quarter point elements placed next to the crack-tips ensure properly modeling the singular behavior of the field variables around the crack tip. Fracture parameters as stress intensity factor, electric field intensity factor and magnetic field intensity factor are computed. Intensive simulations reveal the sensitivity of the generalized intensity factors (GIF) at the crack-tips to the material inhomogeneity, characteristics of the incident wave, coupling effects, wave-material and wave-crack interaction phenomena.     

On the higher order asymptotic analysis of a non-uniformly propagating dynamic crack along an arbitrary path

Transient mixed-mode elastodynamic crack growth along arbitrary smoothly varying paths is considered. Asymptotically, the crack tip stress field is square root singular with the angular variation of the singular term depending weakly on the instantaneous values of the crack tip speed and on the mode-I and mode-II stress intensity factors. However, for a material particle at a small distance away from the moving crack tip, the local stress field will depend not only on the instantaneous values of the crack tip speed and stress intensity factors, but also on the past history of these time dependent quantities. In addition, for cracks propagating along curved paths the stress field is also expected to depend on the nature of the curved crack path. Here, a representation of the crack tip fields in the form of an expansion about the crack tip is obtained in powers of radial distance from the tip. The higher order coefficients of this expansion are found to depend on the time derivative of crack tip speed, the time derivatives of the two stress intensity factors as well as on the instantaneous value of the local curvature of the crack path. It is also demonstrated that even if cracks follow a curved path dictated by the criterion K ₁₁ ^d =0, the stress field may still retain higher order asymmetric components related to non-zero local curvature of the crack path.     

Interaction of general plane P wave and cylindrical inclusion partially debonded from its viscoelastic matrix

The interaction of a general plane P wave and an elastic cylindrical inclusion of infinite length partially debonded from its surrounding viscoelastic matrix of infinite extension is investigated. The debonded region is modeled as an arc-shaped interface crack between inclusion and matrix with non-contacting faces. With wave functions expansion and singular integral equation technique, the interaction problem is reduced to a set of simultaneous singular integral equations of crack dislocation density function. By analysis of the fundamental solution of the singular integral equation, it is found that dynamic stress field at the crack tip is oscillatory singular, which is related to the frequency of incident wave. The singular integral equations are solved numerically, and the crack open displacement and dynamic stress intensity factor are evaluated for various incident angles and frequencies. The project supported by the National Natural Science Foundation of China (19872002) and Climbing Foundation of Northern Jiaotong University     

A general solution method for an anti-plane shear crack dynamically accelerating along a bimaterial interface

We develop a general solution method for a dynamically accelerating crack under anti-plane shear conditions along the interface between two different homogeneous isotropic elastic materials. The crack is initially at rest, and after loading is applied the crack-tip speed which may accelerate up to the shear wave speed of the more compliant material. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time-dependent crack-face traction, as well as expressions for computing the crack-face displacements and the stress in front of the crack. We also present some numerical examples for fixed loads and for loads moving with the crack tip, using a stress intensity factor fracture criterion, in order to examine the predicted effect of material mismatch on interfacial fracture.     

SH波在正交各向异性功能梯度无限长条中心裂缝处的散射

研究了正交各向异性功能梯度材料无限长条中心裂缝对SH波的散射问题,为方便起见,材料两个方向的剪切模量和密度假定为指数模型.通过Fourier积分变换,将问题转化为对偶积分方程的求解.然后,用Cop-son方法求解对偶积分方程,定义了标准动应力强度因子,通过数值算例,讨论了在SH波作用下,裂缝尖端的标准动应力强度因子与入射波的频率、材料参数之间的关系.