Quasi-sudden brittle fracture at both edges of a finite crack

The diffraction of a plane horizontally polarized shear wave by a crack of finite length is analyzed and the extension of both crack edges prior to the arrival of the first diffracted waves, i.e. quasi-sudden fracture, is studied. In light of an energy rate balance criterion it is found that for an incident step-stress pulse, quasi-sudden fracture may occur but always at both crack edges, often initiating at the trailing edge first. For an incident wave whose stress vanishes at the wavefront, however, quasi-sudden fracture may occur only at the leading crack edge, or if at both edges, at the leading edge first. For both waveforms, the rate of crack extension is non-constant and increases rapidly so that crack branching may be expected. Finally instantaneous crack extension at a uniform rate is possible only if the incident wave stress possesses a square-root sinularity at the wavefront. This result agrees with earlier work by Achenbach.     

Diffraction of horizontal shear waves by a moving interface crack

Summary An analysis of the diffraction of horizontally polarized shear waves by a finite crack moving on a bimaterial interface is carried out. Fourier transform method is used to reduce the mixed boundary value problem to the solutions of two pairs of dual integral equations. These equations are further reduced to a pair of Fredholm integral equations of the second kind. The dynamic stress intensity factors are obtained for several values of wave number, incident angle, crack velocity, and material constants.With 7 Figures     

A note on the low frequency diffraction of elastic waves by a Griffith crack

The two dimensional problem of the diffraction of normally incident compressional and antiplane shear waves by a Griffith crack in an infinite isotropic elastic medium is considered. For wavelengths long compared to the crack length, the stress intensity factors as well as the maximum crack openings are expressed in series of ascending powers of the normalized wave number. The approximate solutions are compared with exact solutions obtained in a previous paper[1], for a Poisson's solid. The results indicate that a five term expansion of each of the series solutions is sufficiently accurate for most problems of practical interest.     

The interaction of two inclined cracks with dynamic stress wave loadings

To gain insight into the phenomenon of the interaction of stress waves with material defects and the linkage of two cracks, the transient response of two semi-infinite inclined cracks subjected to dynamic loading is examined. The solutions are obtained by the linear superposition of fundamental solutions in the Laplace transform domain. The fundamental solution is the exponentially distributed traction on crack faces proposed by Tsai and Ma [1]. The exact closed form solutions of stress intensity factor histories for these two inclined cracks subjected to incident plane waves and diffracted waves are obtained explicitly. These solutions are valid for the time interval from initial loading until the first wave scattered at one crack tip returns to the same crack tip after being diffracted by another crack tip. The result shows that the contribution of diffracted waves to stress intensity factors is much less than the incident waves. The probable crack propagation direction is predicted from the fracture criterion of maximum circumferential tensile stress. The linkage of these two cracks is also investigated in detail.     

Crack growth kinetics under impulse loading

The dynamic photoelasticity method has been used to study the effect of the acting wave, and the size and orientation of the incident wave front on crack growth kinetics (under short-term pulse loading). It has also been used to study the mechanism of dynamic stress formation at the crack tip. It has been shown that when compression or tensile waves are incident on the crack at an angle of 0<<80 and 100<<180°, the field of dynamic stress which arises is determined by the transverse shear strain while at an angle of 90±10°, stress formation is caused by strain tearing. When transverse waves act on a crack, there is a considerable amount of stress concentration at the crack tip irrespective of the incident angle. It has been established that one-, two-, or threefold crack growth occurs depending on the pulses and the crack depth.Translated from Problemy Prochnosti, No. 7, pp. 18–21, July, 1992.     

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.     

Evaluation of Fatigue Crack Orientation Using Non-collinear Shear Wave Mixing Method

In this paper, a non-collinear shear wave mixing technique is proposed for evaluation of fatigue crack orientation. Numerical analysis of the nonlinear interaction of two shear waves with crack is performed using two-dimensional finite-element simulations. The simulation results show that the nonlinear interaction of the two shears waves with cracks leads to the generation of transmitted and reflected sum-frequency longitudinal waves (SFLW), moreover the propagation direction of reflected SFLW is correlated with the orientation of crack, which can be used for crack orientation evaluation. Non-collinear wave-mixing experiments were conducted on specimens with fatigue crack. The experimental results show that the directivity of the generated SFLW agrees well with the simulation results, and non-collinear shear wave mixing has potential use in fatigue crack orientation evaluation.     

The propagation and reflection of finite amplitude cylindrically symmetric shear waves in a hyperelastic incompressible solid

Summary We consider axially symmetric shear waves propagating in an incompressible hyperelastic thick-walled cylindrical shell, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. A continuous pulse is initiated at the interior boundary of the cylinder by surface tractions of finite duration. The pulse propagates away from the interior boundary, then reflects from the outer boundary, and subsequently reflects back and forth between the two boundaries of the cylinder. We analyze shock development of the first incident and first reflected wave. The incident pulse can break before it reaches the outer boundary. Using Whitham's nonlinearization technique, we determine conditions under which the incident wave breaks and which shock waves can subsequently occur. Similar calculations are carried out for the first reflection. The formulas obtained for the incident pulse provide accurate estimates of the breaking distance and time, and the location of the shock paths, for any incident shock waves that occur. Results obtained for the reflected wave cannot be used to make similar estimates, but they do reveal that once the pulse has completely left the outer boundary, the possible shock that can occur is the same as for the incident wave. Our analysis is carried out for axial shear waves. A similar analysis can be done for torsional shear waves, but not for combined axial and torsional shear wave propagation. We illustrate the conclusions of our shock analysis with numerical solutions obtained using a relaxation scheme for systems of conservation laws. Numerical results are obtained for axial shear and for combined axial and torsional shear. These results indicate that the shock behavior indicated by our analysis of axial shear is also valid for combined axial and torsional shear wave propagation.     

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.     

Elastodynamic analysis of nonplanar cracks in a half-space

Summary The interaction of time harmonic antiplane shear waves with nonplanar cracks embedded in an elastic half-space is studied. Based on the qualitatively similar features of crack and dislocation, with the aid of image method, the problem can be formulated in terms of a system of singular integral equations for the density functions and phase lags of vibrating screw dislocations. The integral equations, with the dominant singular part of Hadamard's type, can be solved by Galerkin's numerical scheme. Resonance vibrations of the layer between the cracks and the free surface are observed, which substantially give rise to high elevation of local stresses. The calculations show that near-field stresses due to scattering by a single crack and two cracks are quite different. The interaction between two cracks is discussed in detail. Furthermore, by assuming one of the crack tips to be nearly in contact with the free surface, the problem can be regarded as the diffraction of elastic waves by edge cracks. Numerical results are presented for the elastodynamic stress intensity factors as a function of the wave number, the incident angle, and the relative position of the cracks and the free surface.     

Three-Dimensional Ultrasonic Crack Detection in Anisotropic Materials

The scattering of probe-generated ultrasonic fields incident upon a strip-like crack in an anisotropic half-space is discussed. In the situation considered, two possibly coinciding probes are attached to the surface of the half-space. One is transmitting waves incident upon the crack and the other one is receiving the scattered waves. An electric signal response is calculated via an electromechanical reciprocity relation. For a crack far away from the probes and the surface, an approximate expression is calculated. Several numerical examples are presented for an isotropic and a transversely isotropic solid. The results are presented as A-, B-, and C-scans.     

Conditions for crack kinking under stress-wave loading

Kinking of a crack in a prestressed body under the influence of incident stress waves is investigated on the basis of the balance of rates of energies. It is assumed that the crack tip will choose to propagate at a time, in a direction, and at a speed for which the energy flux into the propagating crack tip attains a maximum value with respect to variation of the kinking angle. It is shown that the balance of rates of energies implies that the crack tip speed is zero at the onset of fracture. Consequently, the conditions for the onset of crack kinking and for the computation of the kinking time and kinking angle are completely defined by the elastodynamic field around the original crack tip. Examples of the incidence of step stress waves on a semiinfinite crack in a prestressed body have been investigated. It is shown that for an incident antiplane wave with Mode III fracture, kinking is generally not possible. For an incident inplane wave with mixed Mode I–II fracture, kinking may happen. For that case curves are presented which relate the kinking time and the kinking angle to the state of prestress and to the parameters of the incident wave.     

Dynamics response of complex defects near bimaterials interface by incident out-plane waves

Dynamics response of an elliptical cavity and a crack (on different sides) near bimaterials interface under incident out-plane waves is studied by applying the methods of complex variables and Green’s function. Firstly, based on “conjunction,” the analytical model is divided along the horizontal interface into an elastic half-plane possessing an elliptical cavity and a full elastic half-plane containing a crack. Using complex variables, the scattering displacement field of the half-plane containing an elliptical cavity under incident out-plane waves is then derived. According to the method of Green’s function, the corresponding Green’s functions of two half-planes impacted by an out-plane source load are further deduced. Combined with “crack division,” a crack at the full elastic the half-plane is created, and thus, expressions of displacement and stress are derived while the cavity coexists with the crack. Undetermined antiplane forces are loaded on the horizontal surfaces for conjunction of two sections and then solved by a series of Fredholm integral equations on account of continuity conditions of the interface. Finally, this paper focuses on the discussion of the influence law of different parameters on the dynamics response of complex defects near bimaterials interface by comprehensive numerical results.     

Wave-crack interaction in finite elastic bodies

This paper continues studies in Lalegname et al. (Int J Fract 152:97–125, 2008) on crack propagation in a bounded linear elastic body under the influence of incident waves. In Lalegname et al. (2008) we have considered shear waves, whereas in this paper we discuss the influence of plane elastic waves to a running crack. Actually, the time dependent problem is formulated in a two-dimensional current cracked configuration by a system of linear elasto-dynamic equations. In order to describe the behaviour of the elastic fields near the straight crack tip, we transform these equations to a reference configuration and derive the dynamic stress singularities. Furthermore, we assume that an energy balance law is valid. Exploiting the knowledge on the singular behaviour of the crack fields, we derive from the energy balance law a dynamic energy release rate. Comparing this energy release rate with an experimentally given fracture toughness we get an ordinary differential equation for the crack tip motion. We present first numerical simulations for a Mode I crack propagation.     

Scattering of inhomogeneous wave by viscoelastic interface crack

Summary The reflection, refraction and scattering of inhomogeneous plane waves of SH type by an interface crack between two dissimilar viscoelastic bodies are investigated. The singular integral equation method is used to reduce the scattering problem into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Then, the singular integral equation is solved numerically by Kurtz's piecewise continous function method. The crack opening displacement and dynamic stress intensity factor characterizing the scattered near-field are estimated for various incident angles, frequencies and relaxation times. The differences on crack opening displacement and stress intensity factor between elastic and viscoelastic interface crack are contrasted. And the effects of incident angle, incident frequency and relaxation time of the viscoelastic material are analyzed and explained by the features of phase lag and energy dissipation of the viscoelastic wave.     

Elastic Wave Propagation in a Class of Cracked,Functionally Graded Materials by BIEM

Elastic wave propagation in cracked, functionally graded materials (FGM) with elastic parameters that are exponential functions of a single spatial co-ordinate is studied in this work. Conditions of plane strain are assumed to hold as the material is swept by time-harmonic, incident waves. The FGM has a fixed Poisson’s ratio of 0.25, while both shear modulus and density profiles vary proportionally to each other. More specifically, the shear modulus of the FGM is given as μ (x)=μ ₀ exp (2ax ₂), where μ ₀ is a reference value for what is considered to be the isotropic, homogeneous material background. The method of solution is the boundary integral equation method (BIEM), an essential component of which is the Green’s function for the infinite inhomogeneous plane. This solution is derived here in closed-form, along with its spatial derivatives and the asymptotic form for small argument, using functional transformation methods. Finally, a non-hypersingular, traction-type BIEM is developed employing quadratic boundary elements, supplemented with special edge-type elements for handling crack tips. The proposed methodology is first validated against benchmark problems and then used to study wave scattering around a crack in an infinitely extending FGM under incident, time-harmonic pressure (P) and vertically polarized shear (SV) waves. The parametric study demonstrates that both far field displacements and near field stress intensity factors at the crack-tips are sensitive to this type of inhomogeneity, as gauged against results obtained for the reference homogeneous material case     

Dynamic stress in a semi-infinite solid with a cylindrical nano-inhomogeneity considering nanoscale microstructure

In this paper, the dynamic stress around a cylindrical nano-inhomogeneity embedded in a semi-infinite solid under anti-plane shear waves is investigated. The surface/interface stress effects around the nano-inhomogeneity and at the straight edge of the semi-infinite solid are both considered. The boundary condition at the straight edge of the semi-infinite solid with surface/interface effects is satisfied by the image method. The incident, scattered and refracted displacement fields in the nano-sized composites are expressed by employing the wave function expansion method. The addition theorem for a cylindrical wave function is applied to accomplish the superposition of wave fields in the two semi-infinite solids. Analyzes show that the effect of interface properties, especially that at the straight edge, on the dynamic stress is significant, and the effect increases noticeably due to the nanoscale of the structure. The incident frequency and angle of waves and the shear modulus ratio of the nano-inhomogeneity to matrix also show a pronounced effect on the dynamic stress distribution if the semi-infinite solid shrinks to nanoscale.     

Investigation of the scattering of harmonic elastic shear waves by two collinear symmetric cracks using the non-local theory

Summary In this paper, the scattering of harmonic shear waves by two collinear symmetric cracks is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then, a set of triple integral equations is solved using a new method, namely Schmidt's method. This method is simple and convenient for solving this problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at 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.     

Investigation of the scattering of harmonic elastic waves by two collinear symmetric cracks using non-local theory

In this paper, the scattering of harmonic waves by two collinear symmetric cracks is studied by use of non-local theory. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the dynamic problem to obtain the stress occurring at the crack tips. The Fourier transform is applied and a mixed boundary-value problem is formulated. The solutions are obtained by means of the Schmidt method. This method is more exact and more appropriate than Eringen's for solving this kind of problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at 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 lattice parameter and the circular frequency of the incident wave.