Ch. Zhang

Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM

Abstract This paper presents a transient dynamic crack analysis for a functionally graded material (FGM) by using a hypersingular time-domain boundary integral equation method. The spatial variations of the material parameters of the FGM are described by an exponential law. A numerical solution procedure is developed for solving the hypersingular time-domain traction BIE. To avoid the use of time-dependent Green’s functions which are not available for general FGM, a convolution quadrature formula is adopted for approximating the temporal convolution, while a Galerkin method is applied for the spatial discretization of the hypersingular time-domain traction BIE. Numerical results for the transient dynamic stress intensity factors for a finite crack in an infinite and linear elastic FGM subjected to an impact anti-plane crack-face loading are presented and discussed. The effects of the material gradients of the FGM on the transient dynamic stress intensity factors and their dynamic overshoot over the corresponding static stress intensity factors are analyzed.

Transient elastodynamic antiplane crack analysis of anisotropic solids

Abstract Transient elastodynamic analysis of an antiplane crack in anisotropic solids is presented. A time-domain traction boundary integral equation (BIE) method is applied for this purpose. The traction BIE is hypersingular and has the crack-opening-displacement as its fundamental unknown quantity. Unlike the usual time-domain BIE method the present formulation applies a convolution quadrature developed by Lubich (Lubich, C., 1988a,b. Convolution quadrature and discretized operational calculus. Numer. Math. 52, 129–145 (Part I), 413–425 (Part II)) which requires only the Laplace-domain instead of the time-domain Green’s functions. The spatial variation of the crack-opening-displacement is approximated by an infinite series of Chebyshev polynomials which take the local behavior of the crack-opening-displacement at crack-tips into account. By using a Galerkin method, the time-domain BIE is converted into a system of linear algebraic equations which can be solved step by step. Special attention is devoted to the computation of dynamic stress intensity factors of an antiplane crack in generally anisotropic solids. Numerical results for isotropic solids are presented and compared with the well-known analytical results of Thau and Lu (Thau, S.A., Lu, T.H., 1970. Diffraction of transient horizontal shear waves by a finite crack and a finite rigid ribbon. Int. J. Enggn. Sci. 8, 857–874), to check the accuracy and efficiency of the present time-domain BIE method. The effect of the material anisotropy on the dynamic stress intensity factors is analyzed via several numerical examples.

A new boundary integral equation formulation for elastodynamic and elastostatic crack analysis

Abstract : An elastodynamic conservation integral, the Jk integral, is employed to derive boundary integral equations for crack configurations, in a direct and natural way. These equations immediately have lower order singularities than the ones obtained in the conventional manner by the use of the Betti-Rayleigh reciprocity relation. This is an important advantage for the development of numerical procedures for solving the BIEs, and for an accurate calculation of the strains and stresses at internal points close to the crack faces. For curved cracks of arbitrary shape the BIEs presented here have simple forms, and they do not require integration by parts, as in the conventional formulation. For the dynamic case, the unknown quantities are the crack opening displacements and their derivatives (dislocation densities), while for the static case only the dislocation densities appear in the formulation. For plane cracks the boundary integral equations reduce to the ones obtained by other investigators.

A novel derivation of non-hypersingular time-domain BIEs for transient elastodynamic crack analysis

Abstract A novel application of conservation integrals in transient elastodynamic crack analysis is presented to derive non-hypersingular time-domain boundary integral equations (BIEs). The new derivation is based on an elastodynamic conservation integral which is employed to express the displacement gradients in terms of an integral over the surface of the crack. BIEs are obtained by substituting this representation integral into Hookes law and by taking a limit process. The BIEs obtained in this manner are valid for arbitrary crack configurations, and they are immediately nonhypersingular. The Laplace and the Fourier transforms of these BIEs agree with those obtained by other authors by using the conventional derivation in conjunction with regularization techniques. Numerical examples show that the time-domain BIEs presented here can yield highly accurate results for elastodynamic stress intensity factors.

Time-Domain Boundary Element Analysis of Dynamic Near-Tip Fields for Impact-Loaded Collinear Cracks.

Abstract A time-domain boundary integral equation method has been developed to calculate elastodynamic fields generated by the incidence of stress (or displacement) pulses on single cracks and systems of two collinear cracks. The system of boundary integral equations has been cast in a form which is amenable to solution by the boundary element method in conjunction with a time-stepping technique. Particular attention has been devoted to dynamic overshoots of the stress intensity factors. Elastodynamic stress intensity factors for pulse incidence on a single crack have been computed as function of time, and they have been compared with results of other authors. For collinear macrocrack-microcrack configurations the stress intensity factors at both tips of the macrocrack have been computed as functions of time for various values of the crack spacing and the relative size of the microcrack.

Wave attenuation and dispersion in randomly cracked solids—I. Slit cracks

Abstract Attenuation and dispersion of elastic waves in randomly cracked solids are investigated by using both the theory of Foldy and the causal approach based on Kramers-Kronig relations. In the theory of Foldy, the forward scattering amplitude is first computed numerically via a boundary element method. The complex effective wave number is then determined by using the equation of Foldy. The effective wave velocity and the attenuation coefficient are immediately obtained by taking the real and the imaginary part of the complex effective wave number. In the causal approach, the scattering cross section of a single crack is first calculated numerically by adopting a boundary element method. Then, the attenuation coefficient is determined by a simple relation which stems from energy considerations. The effective wave velocity is subsequently computed via Kramers-Kronig relations. Both the random and the non-random orientation of micro-cracks are considered. In particular, two micro-crack systems are analysed, namely, aligned slit cracks and randomly oriented slit cracks in plane strain. Special attention is devoted to explore the effects of the micro-crack density, the micro-crack orientation or the direction of wave incidence, and the wave frequency on the attenuation coefficient and the effective wave velocity. A comprehensive parametrical study is carried out. Numerical results are presented and discussed. The present analysis requires only a few parameters to describe the statistical distribution and orientation of the micro-cracks, and it puts no limitation on frequencies. The analysis is limited to small crack densities, and it has some direct relevance to the detection and characterization of damages in randomly cracked solids by ultrasonics.

Diffraction of SH waves by a system of cracks: Solution by an integral equation method

Abstract A set of straight cracks in an infinite elastic medium under time harmonic SH-wave loading is considered. Using the representation theorem for the displacements the problem is described by a system of integral equations. Numerical solutions for dynamic stress intensity factors of various crack configurations are presented and crack interaction phenomena are discussed.

Scattering of body waves by an inclined surface-breaking crack

Abstract Parametrical studies are presented which display the variation of the scattered displacement field with angles of incidence and observation, angle of inclination and the frequency of the incident wave, for scattering of elastic body waves by a surface-breaking crack which is inclined under an arbitrary angle with the free surface of an elastic half-space. By using the elastodynamic representation integral for the scattered displacement field, a system of (traction) boundary integral equations for the crack-opening displacement has been obtained. These boundary integral equations are hyper singular, and their solution, therefore, requires a special numerical procedure which was recently developed by the authors. Numerical results for the scattered far field are displayed in several figures. Both back scattering and forward scattering have been investigated. The results suggest the optimal choices of angles of incidence and observation for a maximum response in an ultrasonic test to detect and characterize surface-breaking cracks.

Wave attenuation and dispersion in randomly cracked solids—II. Penny-shaped cracks

Abstract In this paper, attenuation and dispersion of elastic waves in a solid permeated by a random distribution of penny-shaped micro-cracks are analysed. The Foldys equation is applied for computing the complex effective wave number of the cracked solid. By taking the real and the imaginary part of the complex effective wave number, the effective wave velocity and the attenuation coefficient are subsequently obtained. Both the aligned and the randomly oriented penny-shaped micro-cracks are investigated. Numerical results for the attenuation coefficient and the effective wave velocity are presented, as functions of the crack density parameter, the crack orientation or the direction of wave incidence, and the wave frequency. Results obtained in this paper for penny-shaped micro-cracks are compared with those for slit micro-cracks given in a previous paper by the authors, to explore the effect of the micro-crack type on the attenuation coefficient and the effective wave velocity.

Wave propagation in damaged solids

Abstract In this paper, a theoretical model is presented for investigating elastic wave propagation in damaged solids. This model is suited for damaged solids with dilutely distributed defects, and it may aid in the design of experimental configurations and in the proper interpretation of measured data from ultrasonic non-destructive evaluation (NDE) for detecting and characterizing the damage states of the solid. The problem of wave scattering by a single defect of arbitrary shape is first formulated as a set of boundary integral equations, whose solution yields the unknown quantities on the boundary of the defect. The scattering cross-section is then introduced as a measure of the overall effects of the defect on the energy withdrawal from the incident wave. The damaged solid is approximated by an equivalent effective medium which is thought of as statistically homogeneous and linearly viscoelastic. By introducing a complex wave number, neglecting interaction effects among individual defects, and using energy considerations, a simple equation is obtained for calculating the attenuation coefficient from the average scattering cross-section and the number density of the defects. Kramers-Kronig relations are subsequently applied to compute the effective wave (phase) velocity from which the group velocity can be immediately calculated. A method for finding the dynamic effective stiffness of the damaged solid from the attenuation coefficient and the effective wave velocity is proposed. Numerical results are presented for a damaged solid permeated by a distribution of completely randomly oriented penny-shaped microcracks.