SH-wave propagation and resonance phenomena in a periodically layered composite structure with a crack

This paper presents a dynamic analysis of time-harmonic plane SH-waves propagating in periodically multilayered elastic composites with a strip-like crack. The total wave field in the multilayered elastic structure is described as a sum of incident wave field modeled by the transfer matrix method and the scattered wave field governed by an integral representation containing the crack-opening-displacement. The integral equation derived from the boundary conditions on the crack-faces is solved numerically by a Galerkin method. The paper focuses on resonant and non-resonant regimes of anti-plane wave motion in a stack of elastic layers weakened by a single strip-like crack and wave localization in the vicinity of the crack. The scattered extra displacement induced by the presence of the crack is investigated in detail for both situations of high and low contrast in material properties. Numerical results for the average crack-opening-displacement, the transmission coefficient, the stress intensity factor and the average energy flow are presented and discussed to reveal wave resonance and localization phenomena within the band-gaps and the pass-bands.     

Computing the far field scattered or radiated by objects inside layered fluid media using approximate Green's functions

A numerically efficient technique is presented for computing the field radiated or scattered from three-dimensional objects embedded within layered acoustic media. The distance between the receivers and the object of interest is supposed to be large compared to the acoustic wavelength. The method requires the pressure and normal particle displacement on the surface of the object or on an arbitrary circumscribing surface, as an input, together with a knowledge of the layered medium Green's functions. The numerical integration of the full wave number spectral representation of the Green's functions is avoided by employing approximate formulas which are available in terms of elementary functions. The pressure and normal particle displacement on the surface of the object of interest, on the other hand, may be known by analytical or numerical means or from experiments. No restrictions are placed on the location of the object, which may lie above, below, or across the interface between the fluid media. The proposed technique is verified through numerical examples, for which the near field pressure and the particle displacement are computed via a finite-element method. The results are compared to validated reference models, which are based on the full wave number spectral integral Green's function.     

Resonance blocking of traveling waves by a system of cracks in an elastic layer

Wave processes that occur in an elastic layer when waves traveling in it are diffracted by a system of horizontal cracks are investigated. Integral representations of wave fields are constructed in terms of the convolution of Green’s matrices and unknown jumps of displacements at the cracks. The displacement jumps are determined from the boundary integral equations, which are obtained from the initial boundary-value problem with the boundary conditions at crack faces being satisfied. The spectrum of the integral operator is studied for different variants of mutual crack arrangement and is compared with the spectrum of the corresponding operators for individual cracks; the relationship between the spectrum and the blocking effects is analyzed. The possibility of obtaining an extended frequency band of waveguide blocking in the case of groups of cracks is demonstrated.     

Basic solution of four parallel non-symmetric permeable Mode-III cracks in a piezoelectric/piezomagnetic composite plane

The interaction of four parallel non-symmetric permeable cracks in a piezoelectric/piezomagnetic composite plane subjected to anti-plane shear stress loading was studied by the Schmidt method. The problem was formulated through a Fourier transform into four pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relationships among the electric displacement, magnetic flux and stress fields near the crack tips were obtained. The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the lengths and spacing of cracks. It was also revealed that the crack shielding effect is present in piezoelectric/piezomagnetic composites.     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

激光激发瑞利波检测表面倾斜缺陷的研究

研究激光激发瑞利波检测样品的表面倾斜缺陷。基于频域热弹耦合方程,采用有限元方法建立激光激发瑞利波检测倾斜缺陷的数值模型,研究倾斜缺陷的检测机理。数值计算含不同的长度及倾斜角度缺陷的样品中瞬态位移波形,分析瑞利波在倾斜缺陷处模式转换的过程,研究各种瑞利波的传播路径。在此基础上,研究缺陷宽度与材料黏性对瑞利波传播及缺陷检测的影响。结果表明:瑞利波在缺陷处产生的反射及透射瑞利波的到达观测点的时间可以检测缺陷位置和长度,瑞利波在缺陷的底部发生模式转换产生的切变波可以检测缺陷倾斜角度。数值结果和已有的实验结果一致,从而为表面倾斜缺陷的检测提供有效的理论依据。      

Ultrasonic surface crack characterization on complex geometries using surface waves

The characterization of surface cracks on complex geometries using surface waves is investigated numerically and experimentally. The specimen geometry is implemented in a finite difference code by approximation of the contour using a Cartesian grid. In the experiments the out-of-plane surface displacement is measured by means of a heterodyne laser interferometer. Good agreement is shown by comparison of the calculated out-of-plane displacement with experimental results for both cracked and non-cracked specimens. The crack depth is measured down to a size of 0.7 times the surface wavelength using a time delay approach. The many Rayleigh pulses propagating after the crack can be separated from the other modes by a filtering procedure based on the surface wave propagation velocity. Only a detailed analysis of the scattering phenomenon using the simulation allows an identification of the transmitted pulse required for crack depth measurement. Application of the method to a specimen with a real fatigue crack shows a systematical error possibly due to the inclined crack profile.     

Pulsed Rayleigh wave scattered at a surface crack

This paper investigates Rayleigh wave interaction with machined slots on flat aluminium blocks to simulate surface breaking cracks. Using a finite element method, Rayleigh wave scattering by narrow slots of varied depth ranging from 0.5 mm to 20 mm is calculated. Pulsed wideband Rayleigh waves with a centre frequency of 590 kHz and -6 dB bandwidth of 520 kHz is considered. Reflection and transmission coefficients are calculated and compare well with the published literature. We and other workers have reported enhancement of the measured amplitude or particle velocity of an apparent Rayleigh wave close to a surface defect. In this paper, it is found that the predicted enhancement of in-plane components of particle velocities close to a crack is significantly higher than that of the out-of-plane components of particle velocities which appears to be mainly due to the mode-converted surface skimming longitudinal wave from the crack that has mainly in-plane components near the sample surface. The enhancement of the in-plane particle velocity will be observed regardless of the type of in-plane sensitive ultrasonic detector used. The explanation of the discrepancy of the reflection and transmission coefficients obtained by pulsed and narrow band or pseudo continuous Rayleigh waves is discussed. The later-arriving Rayleigh waves from reverberation along the inside of the crack surface are observed, as has been previously reported by other workers, and this may also be used to gauge slot depth.     

Analysis on cohesive crack opening displacement considering the strain softening effect

In order to investigate the crack propagation in quasi-brittle materials like rock, ceramic and concrete, Hillerborg and his co-researchers abstracted the fracture process zone in front of a stress free crack in terms of a “fictitious crack zone”. On the fictitious crack zone, cohesive stresses distribute following a given softening relationship of stress vs. crack opening. Based on the polynomial or power series expression of cohesive crack opening displacement, the relationship of the cohesive stress vs. the crack opening displacement is established using elastic theory and integral equation, and some unknown physics variables are obtained using variation approach. The calculation results gained in this paper are close to the experimentally test ones.     

On the plastic zone size and crack tip opening displacement of a sub-interface crack in an infinite bi-material plate

Elastic–plastic stress analysis has been carried out for the plastic zone size and crack tip opening displacement of a sub-interface crack with small scale yielding. In our study, the shape of plastic zone is assumed as a long, slim strip at both crack tips. In the plastic zone, both normal stress and shear stress exist and are considered due to the bi-material interface. The values of the plastic zone size, normal stress and shear stress are determined by satisfying the conditions where both Modes I and II stress intensity factors vanish and Von Mises yield criterion is met. In the present paper, the sub-interface crack is simulated by continuously distributed dislocations which will result in singular integral equations. Those singular integral equations can be solved by reducing them to a set of linear equations. The values of the plastic zone size and crack tip opening displacement are obtained through an iterative procedure. Finally, the effect of normalized loading, normalized crack depth (distance to the interface) and Dundurs’ parameters on the normalized plastic zone size and the normalized crack tip opening displacement is discussed.     

NON-DESTRUCTIVE EVALUATION OF HORIZONTAL CRACK DETECTION IN BEAMS USING TRANSVERSE IMPACT

Numerical and experimental studies for crack detection in beam employing transverse impact are presented. In the numerical study, a beam model of wave propagation is developed to calculate the time history of beam response before, over and after the crack region. It is expected that the resulting wave in the beam will be scattered by the crack and will carry information on the location and geometry of the crack. Experiments using a scanning laser vibrometer on specimens containing simulated crack are then conducted to verify the numerical results. Comparison study between the numerical results and experimental observations are conducted; good correlation between theory and experiment is observed. The beam model of wave propagation and adaptive multilayer perceptron networks (MLP) are then used for inverse identification of crack parameters (i.e., crack location, depth and length) in the beams. Time-domain displacement responses calculated using the present beam model containing predetermined crack parameters are used as training data for the MLP. Once the MLP is trained, the MLP networks are then employed for inverse determination of an unknown crack in a beam using experimental displacement responses measured with a scanning laser vibrometer. Examples show that the procedure performs well for the determination of a wide range of values for the location, depth and length of the crack.     

Linear time domain model of the acoustic potential field

A new time domain formulation of the acoustic wave is developed to avoid approximating assumptions of the linearized scalar wave equation that limit its validity to low Mach particle velocity modeling or to a smooth potential field in a stationary medium. The proposed model offers precision of the moving frame while retaining the form of the widely used linearized scalar wave equation although with respect to modified coordinates. It is applicable to field calculations involving transient waves with unlimited particle velocity, propagating in inhomogenous fluids or in those with time varying density. The model is based on the exact flux continuity equation and the equation of motion, both using the moving reference frame. The resulting closed-form free space scalar wave equation employing total derivatives is converted back to the partial differential form by using modified independent variables. The modified variables are related to the common coordinates of space and time following integral expressions involving transient particle velocity representing wave radiated by each point of a stationary source. Consequently, transient field produced by complex surface velocity sources can be calculated following existing surface integrals of the radiation theory although using modified coordinates. The use of the proposed model is presented in a numerical simulation of a transient velocity source vibrating at selected magnitudes, leading to the determination of the propagating pressure and velocity wave at any point.     

DYNAMIC BEHAVIOUR OF A CRACKED SOLDERED JOINT

The two-dimensional “in-plane” time-harmonic elastodynamic problem for a multi-layered cracked soldered joint system is studied. This problem is solved by using a hybrid of both displacement and hyper-singular traction boundary integral equation method. The proposed method directly accounts for the effect of the outer boundary of a finite multi-layered body and the interaction between the internal and interface cracks. The open fracture model is used to present the interface crack. Numerical results are shown and discussed to reveal the effect of the existence and sizes of cracks, the crack interaction, the debonding effect, the influence of the wave frequency and the type of the material combination on the crack-tip fracture parameters and the displacement scattered far field.     

Solutions to two or four parallel Mode-I permeable cracks in magnetoelectroelastic composite materials

The solutions to two or four parallel Mode-I permeable cracks in magnetoelectroelastic composite materials are derived using the generalized Almansi's theorem under permeable electric and magnetic boundary conditions. The problem can be solved through the Fourier transform with the help of two pairs of dual integral equations, in which unknown variables were jumps of displacements across crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among the electric displacement intensity factors, the magnetic flux intensity factors and the stress intensity factors at the crack tips. The paper presents the interactions of two or four parallel Mode-I cracks in magnetoelectroelastic composite materials and the crack-shielding effect in magnetoelectroelastic composite materials.     

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions is presented analytically. Mirror method and multi-polar coordinate systems are developed to solve the complex boundary value problem. Considering the displacement and stress continuity conditions, a series of integral equations for unknown coefficients are obtained and solved by truncation. The solution is used to calculate the dynamic stress concentration factor around the edge of the inclusion, and the analysis and numerical results are discussed. The results show that degree of dynamic stress concentration around the circular inclusion is influenced by the incident angle, the incident wave number, and the parameters of medium.     

Scattering of surface waves by a sub-surface crack

The scattering of Rayleigh waves by a two-dimensional sub-surface crack, which is perpendicular to the free surface of an elastic half-space, is investigated. The boundary-value problem for the scattered field is stated, and reduced to an uncoupled system of integral equations which are solved numerically. At large distances from the crack the scattered field is shown to consist of outgoing Rayleigh waves and cylindrical body waves. Graphs of the variation of the amplitude and phase of the forward and backward scattered Rayleigh waves with the frequency and the geometrical parameters of the crack are presented. The stress intensity factors at both crack tips have also been obtained.     

Thermal shock fracture of piezoelectric materials

This paper deals with a case study for the piezoelectric materials suddenly exposed to an environmental medium of different temperature. The problem is idealized to a plate containing an edge crack or an embedded crack. The stress and electric displacement histories in an uncracked plate are calculated. These stresses and electric displacements are then added to the crack surface tractions and electric displacements with opposite sign to formulate a mixed boundary value problem. The cracking problem is thus reduced to a singular integral equation of Cauchy type, which is then solved numerically. Both impermeable crack assumption and permeable crack assumption are considered. The results for stress and electric displacement intensity factors are computed as a function of normalized time and crack size. Lower bound solutions are obtained for the maximum thermal shock that the material can sustain without catastrophic failure according to the two distinct criteria: (i) The maximum local tensile stress equals the tensile strength of the medium. (ii) The maximum stress intensity factor for the pre-existing representative crack equals the fracture toughness of the medium. The parameters that control the transient thermal stress and electric displacement are also identified. The method can be used to explore susceptibility to thermal fracture in piezoelectric materials containing pre-cracks.     

弹性力学的重构核粒子边界无单元法

将重构核粒子法(RKPM)和边界积分方程方法结合，提出了一种新的边界积分方程无网格方法——重构核粒子边界无单元法(RKP-BEFM).对弹性力学问题，推导了其重构核粒子边界无单元法的公式，研究其数值积分方案，建立了重构核粒子边界无单元法离散化边界积分方程，并推导了重构核粒子边界无单元法的内点位移和应力积分公式.重构核粒子法形成的形函数具有重构核函数的光滑性，且能再现多项式在插值点的精确值，所以本方法具有更高的精度.最后给出了数值算例，验证了本方法的有效性和正确性. 关键词： 重构核粒子法 弹性力学 边界无单元法     

Fracture of diamond coatings by high velocity sand erosion

This paper describes a study of the behaviour of diamond coatings when subjected to solid particle erosion from sand particles. The coatings were deposited by chemical vapour deposition (CVD) onto tungsten substrates and tested using a high velocity air–sand erosion test facility. The erosion tests were conducted using particle impact velocities of between 33 and 268 m/s. Examination of the eroded test specimens showed that the principal damage features were circumferential cracks and pin-holes. Comparison with Hertz impact theory revealed that the measured circumferential crack diameters were more than double the predicted Hertzian contact diameter. Moreover, a trend of increasing circumferential crack diameter with coating thickness, which is not predicted by Hertz, was found. Instead, the crack diameters showed good agreement with those predicted by the theory of stress wave reinforcement, which is more commonly associated with liquid impact damage of brittle materials. During impact, the bulk compression and shear waves are reflected at the rear surface of debonded regions of the coating to return to the front surface and reinforce the Rayleigh surface wave, which generates a tensile stress. Where this stress exceeds the local tensile strength of the coating, a ring of cracks surrounding the area of impact is created. The results from the present study therefore suggest that stress wave reflection is responsible for the formation of the cracks at locally debonded regions of the coating. This hypothesis was supported by images acquired using scanning acoustic microscopy, which showed that circumferential cracks and pin-holes were only found on areas of the coating that had become delaminated by multiple particle impacts during the erosion tests.     

Elastic analysis of a mode Ⅱ crack in an icosahedral quasicrystal

Based on the displacement potential functions, the elastic analysis of a mode II crack in an icosahedral quasicrystal is performed by using the Fourier transform and dual integral equation theory. By the solution, the analytic expressions for the displacement field and stress field are obtained. The asymptotic behaviours of the phonon and phason stress fields around the crack tip indicate that the stresses near the crack tip exhibit a square root singularity. The most important physical quantities of fracture theory, crack stress intensity factor and energy release rate, are evaluated in an explicit version.