Dynamic penny-shaped crack in a transversely isotropic material

The scattering of a harmonic longitudinal wave by a penny-shaped crack in a transversely isotropic material is investigated using the techniques of Hankel transform. The wave impinges normally on the crack surfaces. A complete contour integration is employed to simplify the expressions of the results. An exact expression of the dynamic stress-intensity factor is obtained as a function of the frequency factor and the anisotropic material constants. The normalized dynamic stress-intensity factor is shown to have different maximum values at different wave frequencies for the sample composite and metallic materials. The distortion of the dynamic crack shape and the displacement at the crack center are also shown to be dependent of the wave frequency and the anisotropy of the material.     

用圆孔内单边裂纹平台巴西圆盘和实验-数值-解析法确定砂岩的动态起裂和扩展韧度

在I型(张开型)动态断裂实验中,利用大直径(?100 mm)分离式霍普金森压杆径向冲击圆孔内单边裂纹平台巴西圆盘试样。考虑了材料惯性效应和裂纹扩展速度对动态应力强度因子的影响,用实验-数值-解析法确定了高加载率和高裂纹扩展速度情况下,砂岩的动态起裂韧度和动态扩展韧度。由动态实验获取试样的动荷载历程,采用裂纹扩展计(Crack Propagation Gauge,CPG)测定试样断裂时刻和裂纹扩展速度,获得裂纹扩展速度对应的普适函数值。然后将动荷载历程带入到有限元软件中进行动态数值模拟,求出静止裂纹的动态应力强度因子历程,再用普适函数值对其进行近似修正。最后根据试样的起裂时刻和穿过CPG中点的时刻,由相应的动态应力强度因子历程分别确定砂岩的动态起裂和动态扩展韧度,它们分别随动态加载率和裂纹扩展速度的提高而增加。     

Dynamic stress distribution around the tip of a running crack

The stress distribution is obtained around the tip of a crack running in a brittle material. The stresses are written as the sum of the associated static solution and the wave-effect terms which depend upon the crack speed. The results obtained clearly reduce to the associated static solutions if the crack speed vanishes.Near the tip of the crack, the dynamic stress-intensity factor for the circumferential stress, $\text{σ}{}_{\mathrm{\theta \theta }}$, is written as the product of the associated static stress-intensity factor and the dynamic correction factor which is a nondimensional function of the crack speed, $\text{V}$, the angle from the crack plane, θ, and Poisson's ratio, ν. The value of the correction factor is computed for various values of $\text{V}$ and θ at $\text{ν = 0.25}$. It is shown that the maximum tensile value of $\text{σ}{}_{\mathrm{\theta \theta }}$, occurs on the crack plane for $\text{V}$ less than 0.7 time shear wave speed, $\text{c}{}_2$, and suddenly shifts to the plane of $\text{θ = 55°}$ for $\text{V}$ slightly larger than 0.7 $\text{c}{}_2$. For $\text{V > 0.7}$$\text{c}{}_2$, the angle θ for the maximum $\text{σ}{}_{\mathrm{\theta \theta }}$, θ being larger than 55°, varies continuously with the crack speed, $\text{V}$. The results obtained are used to discuss the growth of branching crack.     

Diffraction of normal compression waves by a penny-shaped crack in the presence of an axial magnetic field

The problem of diffraction of normally incident compressional waves by a penny-shaped crack located in a perfectly conducting, infinite, isotropic, elastic solid permeated by an uniform magnetostatic field is considered. Using an integral transform technique, the problem is reduced to that of solving a Fredholm integral equation of the second kind having a finite integral kernel. The dynamic singular stress distributions near the crack tip are obtained in closed form and the effects on the dynamic stress-intensity factors due to the presence of the magnetic field are shown graphically. For low frequencies, the dynamic stress-intensity factors are expressed in series of ascending powers of the normalized frequency. The approximate solutions are compared with exact solutions.     

Dynamic fracture of a functionally gradient material having discrete property variation

A functionally gradient material (FGM) with discrete property variation is prepared, and the dynamic fracture in this material is studied using the technique of photoelasticity combined with high-speed photography. Transparent sheets required for the study are made by casting a polyester resin mixed with varying amounts of plasticizer. The mechanical (quasi-static and dynamic) and optical properties of the material are evaluated as a function of the plasticizer content. Results of material characterization show that the fracture toughness increases with increasing plasticizer content, whereas the Young's modulus decreases. The material fringe constant and the dynamic modulus are observed to be relatively insensitive to plasticizer content. The FGM is then prepared by casting together thin strips having different plasticizer content. The dynamic crack propagation phenomenon is studied for four different property variations along the crack propagation direction, and the effects of these property variations on crack speed, crack jump distance and dynamic stress intensity factor are investigated. Results of this investigation show that increasing the toughness in the direction of crack growth reduces the crack jump distance as compared to on increasing-decreasing toughness variation for the same initial energy.     

Hold-time effects on high temperature fatigue crack growth in Udimet 700

Crack growth behaviour under creep-fatigue conditions in Udimet 700 has been studied, and the crack growth data were analysed in terms of the stress intensity factor as well as theJ-integral parameter. Crack growth behaviour is shown to depend on the initial stress intensity level and the duration of hold-time at the peak load. For stress intensities that are lower than the threshold stress intensity for creep crack growth, the crack growth rate decreases with increase in hold time even on a cycle basis, da/dN, to the extent that complete crack arrest could occur at prolonged hold times. This beneficial creep-fatigue interaction is attributed to the stress relaxation due to creep. For stress intensities greater than the threshold stress intensity for creep crack growth, the growth rate on a cycle basis increases with increase in hold time. For the conditions where there is no crack arrest, the crack growth appears to be essentially cycle-dependent in the low stress intensity range and time-dependent in the high stress intensity range. Both the stress intensity factor and theJ-integral are shown to be valid only in a limited range of loads and hold-times where crack growth rate increases continuously.     

Fast fracture and crack arrest according to the Dugdale model

Dynamic effects near a propagating crack tip in a ductile material have been investigated on the basis of a model with a strip-zone of yielding. In the analysis of fast fracture the unknown variables are the speeds of the leading and trailing edges of the yield zone, where the latter defines the position of the actual crack tip. Propagation of the crack tip is governed by two conditions: the usual one that the cleavage stress is bounded at the leading edge of the yield zone, and a second condition which involves the yield stress and the stretch of the fiber at the trailing edge of the yield zone. By combining well-known results for transient dynamic stress-intensity factors and crack-opening displacements corresponding to external loads, with steady-state dynamic results for the fields corresponding to the cohesive tractions in the yield zone, the dynamic problem of fast fracture has been analyzed for both the Mode III and the Mode I case. The results can be used to investigate crack arrest when a propagating crack tip enters a region of higher ductility.     

Interaction of shear waves with a griffith crack situated in an infinitely long elastic strip

This paper deals with the propagation of shear waves in a wave guide which is in the form of an infinite elastic strip with free lateral surfaces. This strip contains a Griffith crack. An integral transform method is used to find the solution of the equation of motion from the linear theory for a homogeneous, isotropic elastic material. This method reduces the problem into an integral equation. It has been observed that only shear waves with frequencies less than a parameter-value, depending on the width of the wave guide, can propagate. The integral equation is solved numerically for a range of values of wave frequency and the width of the strip. These solutions are used to calculate the dynamic stress intensity factor, displacement on the surface of the crack and crack energy. The results are shown graphically.     

Nonlinear adhesive behavior effects in a cracked orthotropic sheet stiffened by a semi-infinite orthotropic sheet

The stress-intensity factors are determined for a cracked orthotropic sheet adhesively bonded to an orthotropic stringer where the adhesive layer is modeled with a nonlinear stress-strain curve. Since the stringer is modeled as a semi-infinite sheet, the solution is most appropriate for a crack tip located near a stringer edge. By the use of Green's functions and the complex variable theory of orthotropic elasticity developed by Lekhnitskii, a set of integral equations is obtained. The integral equations are replaced by an equivalent set of algebraic equations, which are solved to obtain the shear stress distribution in the adhesive layer. With these adhesive stresses, the crack-tip stress-intensity factors are found.When the adhesive was modeled with a nonlinear stress-strain curve, the peak shear stresses in the adhesive were considerably reduced in comparison to the solution for the linear elastic adhesive. This resulted in increases in the stress-intensity factors for the nonlinear adhesive solution compared to the linear adhesive solution. The nonlinear adhesive did not have a significant effect on the stress-intensity factor unless the near crack tip was beneath the stringer. The present investigation assumes that the adhesive bond remains intact. Onset of adhesive failure is predicted to occur at decreasing levels of applied stress as the crack propagates beneath the stringer.     

Impact response of a cracked soft ferromagnetic medium

A solution is given for the problem of an infinite soft ferromagnetic solid containing a central crack subjected to normal impact load. The solid is permeated by a uniform magnetostatic field normal to the crack surface. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of integral equations in the Laplace transformed plane. A numerical Laplace inversion technique is used to compute the values of the dynamic stress-intensity factor, and the results are compared with the corresponding elastodynamic values to reveal the influence of magnetic field on the dynamic stress-intensity factor. The dynamic stress intensity factor is found to increase with increasing values of the magnetic field.With 4 Figures     

Diffraction of transient horizontal shear waves by a running crack

An analysis of the diffraction of horizontally polarized shear waves of arbitrary profile by a finite crack extending uniformly is investigated. Transform techniques and a generalized Wiener-Hopf method are employed to solve the mixed boundary-value problem exactly from the instant the incident wave first strikes the crack until the diffracted wave reaches the opposite edge, is rediffracted, and then returns the original edge. The dynamic stress-intensity factors for an incident wave with a step function stress profile are obtained as functions of time, the angle of incidence and the speed of crack propagation. The effects of the aforementioned system parameters on the dynamic stress-intensity factors are shown graphically.     

The mechanics of matrix cracking in fiber reinforced ceramic composites containing a viscous interface

A detailed fracture mechanics analysis of matrix cracking in a fiber reinforced ceramic composite is presented for the case where the fiber—matrix interface exhibits viscous flow as can be the case when ceramic composites containing amorphous interfacial layers are subjected to loads at elevated temperatures. The analysis considers the case where matrix cracks are fully bridged by fibers, and the role of the viscous interface is to introduce a time dependence into the stress-intensity formulations. Such time-dependence arises because the bridging fibers are able to pull out of the matrix by viscous interfacial flow, with the result that the crack opening, as well as the actual (or shielded) matrix crack-tip stress-intensity factor, increase with time under the action of a constant externally applied load to the composite. The differential equation governing the mechanics of the fiber pull-out is derived. This is then applied to obtain expressions for the time-dependence of the crack opening and the effective crack-tip stress-intensity factor in terms of material and microstructural factors. These expressions predict that the matrix crack will exhibit stable crack growth, with the crack growth rate being essentially crack length (and time) independent and a function only of the applied stress and of material and microstructural factors. It is also shown that the composite lifetime is independent of the sizes of pre-existing cracks and is dependent only on a critical microstructure dependent flaw size, applied stress and microstructural factors.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Quasi-static and dynamic fracture behaviour of rock materials: phenomena and mechanisms

An experimental investigation is conducted to study the quasi-static and dynamic fracture behaviour of sedimentary, igneous and metamorphic rocks. The notched semi-circular bending method has been employed to determine fracture parameters over a wide range of loading rates using both a servo-hydraulic machine and a split Hopkinson pressure bar. The time to fracture, crack speed and velocity of the flying fragment are measured by strain gauges, crack propagation gauge and high-speed photography on the macroscopic level. Dynamic crack initiation toughness is determined from the dynamic stress intensity factor at the time to fracture, and dynamic crack growth toughness is derived by the dynamic fracture energy at a specific crack speed. Systematic fractographic studies on fracture surface are carried out to examine the micromechanisms of fracture. This study reveals clearly that: (1) the crack initiation and growth toughness increase with increasing loading rate and crack speed; (2) the kinetic energy of the flying fragments increases with increasing striking speed; (3) the dynamic fracture energy increases rapidly with the increase of crack speed, and a semi-empirical rate-dependent model is proposed; and (4) the characteristics of fracture surface imply that the failure mechanisms depend on loading rate and rock microstructure.     

Elastodynamic forms of caustics for running cracks under constant velocity

For the study of elastodynamic problems of propagating cracks it is necessary to evaluate the dynamic stress intensity factor K^d_I, which depends on the form of expressions for the stress components existing at the running crack tip at any instant of the propagation of the crack and the corresponding dynamic mechanical and optical properties of the material of the specimen under identical loading conditions. In this paper the distortion of the form of the corresponding reflected caustic from the lateral faces of a dynamically loaded transparent and optically inert specimen containing a transverse crack running under constant velocity was studied on the basis of complex potential elasticity theory and the influence of this form on the value of the dynamic stress intensity factor was given. The method was applied to the study of a propagating under Mode I crack in a PMMA specimen under various propagation velocities and the corresponding dynamic stress intensity factor K^d_I, evaluated.     

A simplified analysis of the dynamic stress-intensity factors for the part-through surface crack in an elastic plate

Line-spring model was used for the simplified analysis of dynamic stress-intensity factor for part-through surface crack. Methods were presented for both evaluation of dynamic compliance of the line-spring and of dynamic stress-intensity factor along the crack front. It was found that the value of the dynamic compliance coefficient of the line-spring does not vary with the frequency of applied load and is good agreement with the analytical result given by Rice and Levy under static load conditions. To verify the validity of the present model, the dynamic stress-intensity factors were calculated for edge cracked strips in plane strain. The results obtained from the present model coincide with those obtained from the finite element analysis using quarter-point singularity elements around the crack tip. As an example of the application of the present model, the dynamic response of a plate containing a semi-elliptical surface crack was analysed where the plate is subjected to a vibrating stress at its ends.     

Dynamic initiation and propagation behaviour of sub-interfacial cracks in PMMA/aluminium bi-material system

The paper describes studies on the dynamic initiation and growth of sub‐interfacial cracks in a PMMA/aluminium bi‐material system using high‐speed photography combined with the method of caustics. Dynamic fracture phenomena such as crack propagation trajectories, crack velocity, phase angle and stress intensity factor were determined from recordings of a series of dynamic caustic patterns surrounding the propagating sub‐interfacial crack tip.. There is a considerable influence of the distance between the interface and the crack on fracture characterization.     

Energy-density concept in fracture mechanics

A theory of fracture mechanics is proposed in which attention is focused on the intensity of the energy field in the crack tip region. This energy field possesses a 1/r-type of singularity for both elastic and plastic materials. The strength or amplitude of this field will be referred to as the “energy-density factor”, S. Unlike the stress-intensity factor k in classical fracture mechanics which is only a measure of the local stress amplitude, the energy-density factor is also direction sensitive. The difference between k and S is analogous to the difference between a scalar and vector quantity. In this sense, the critical value S_cr specifies the direction of crack initiation as well as the fracture toughness of the material.     

Fracture toughness prediction of a valve body: Numerical analysis

The crack repair technique by “crack grinding”, due to its efficiency, has been widely used to extend the service life of the cracked structural components. The “crack grinding” technique is carried out by removing the materials which contains the crack. So it is of great importance to study the effects of this repair technique. In this paper, the finite element method is used to analyse the behavior of repaired cracks in mode I by computing the stress-intensity factors at the crack tip. 3D numerical simulation was carried out to model the steam governor valve in a coal-fired thermal power plant with a semi-elliptic crack. Since no extra auxiliary solutions are required for the computation, the proposed method appears to be visual and can be used for evaluation of the associated stress-intensity factors. The plot of the stress-intensity factors according to the crack length in mode I, shows that the stress-intensity factor exhibits an asymptotic behavior as the crack length increases. It’s to be noted that a transition zone allows connecting the refined specific crack tip mesh to the global structural discretization. So this technique can be used for complex industrial structures.     

宏微观双尺度运动裂纹模型面内拉伸下的解析解*

研究裂纹动态扩展中宏微观因素相互作用机制与微观裂尖区的钝化效应。平面拉伸状态下,宏观主裂纹以恒定速度运动。通过一个介观约束应力过渡区,将宏观主裂纹与微观裂尖区相连接,由此建立了一个宏微观双尺度运动裂纹模型。应用弹性动力学与复变函数理论,分别在宏观与微观尺度下对该模型进行解析求解,获得了解析解。通过裂纹张开位移从宏观到微观的连续性条件与宏微观应力场协调条件,将两个不同尺度下的解相耦合,获得了计算宏微观损伤区特征长度的显式表达式。研究表明,运动裂纹的宏观应力场仍具有通常的r&;#61485;1/2的奇异性。由于微观裂尖的钝化,微观应力场奇异性的阶次有所降低,与宏观应力场相比具有弱奇异性。双尺度运动裂纹模型中,可允许裂纹运动速度达到剪切波速,解除了经典运动裂纹理论中裂纹速度不能超过Rayleigh波速的限制。数值结果表明,介观损伤过渡区与裂尖微观损伤区尺寸,及裂纹张开位移等,与裂纹运动速度、材料性质、约束应力比、裂尖钝化角度等因素有关。