An experimental investigation into dynamic fracture: IV. On the interaction of stress waves with propagating cracks

This is the last in a series of four papers dealing with experimental observations of dynamic crack propagation in thin, large sheets of a Homalite 100 plastic material which allow simulation of crack growth in unbounded plates. In the first paper crack initiation resulting from stress wave loading was examined as well as crack arrest. It was found that for increasing rates of loading in the microsecond range the stress intensity factor required for initiation rises markedly. Crack arrest occurs abruptly without any deceleration phase at a stress intensity lower than that which causes initiation under quasi-static loading.In the second paper we analyze the occurrence of micro cracks at the front of the running main crack which control the rate of crack growth. The micro cracks are recorded by real time photography. By the same means it is shown that these micro cracks grow and turn away smoothly from the direction of the main crack in the process of branching.The third paper questions the existence of a unique relation between the instantaneous stress intensity factor and the instantaneous crack speed. It is shown that the crack propagates with constant velocity from the instant of initiation, even though the stress intensity factor varies considerably during crack propagation. The limiting crack speed is viewed as a consequence of crack tip micro-fracturing discussed in the second paper of the series. The crack branching is considered as a natural evolution of the micro fracturing process with certain statistical features.In this last paper we examine the effect of stress waves impinging on the tip of a running crack. The stress waves are induced by (a) reflection from the boundary through a judicious choice of specimen geometry and (b) through use of a second stress wave generator. It is found that these waves affect the direction of propagation, the crack speed, and the process of branching. However, we conclude that while stress waves modify the process of crack branching they do not by themselves constitute the reason for crack branching. Also, it is demonstrated how wave reflections in small specimens may complicate dynamic crack propagation behavior and thereby the interpretation of results.

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Résumé Ceci est le dernier mémoire d'une série de quatre consacrés à des observations expérimentales sur la propagation dynamique d'une fissure dans des feuilles minces et longues d'un matériau plastique — l'Homalite 100 — qui permet de simuler une propagation de fissure dans des tôles non limitées. Dans le premier mémoire, on a examiné l'amorçage d'une fissure résultant d'une sollicitation en onde, ainsi que son arrêt. On a trouvé que pour des vitesses de mise en charge atteignant l'ordre de la microseconde, le facteur d'intensité de contrainte nécessaire à l'amorçage s'accroît de manière marquante. L'arrêt de la rupture survient brutalement sans phase de décélération et correspond à un facteur d'intensité de contrainte plus faible que celui qui est nécessaire pour créer l'amorçage en condition quasi statique.Dans le deuxième mémoire, on s'attache à analyser l'apparition de microfissures en avant d'une fissure principale en propagation, et qui contrôlent la vitesse de cette propagation. On procède, pour ce faire, à leur enregistrement par photographie en temps réel et on montre, par le même procédé, que ces microfissures s'étendent et s'écartent peu à peu de l'axe de la fissure principale, induisant un processus d'arborescence.La troisième étude s'interroge sur l'existence d'une relation unique entre le facteur d'intensité de contrainte instantanée et la vitesse instantanée de la fissure. On montre que la fissure se propage à une vitesse constante dés son initiation, même si le facteur d'intensité de contrainte évolue de manière marquante au cours de la phase de propagation. La vitesse limite de la fissure est considérée comme une conséquence de la microfissuration survenant à la pointe de la fissure, ainsi qu'il en est discuté dans l'étude précédente. L'arborescence de la fissure est, quant à elle, considérée comme une évolution naturelle du processus de micro-rupture, tenant compte de considérations statistiques.Dans le présent mémoire, on examine l'effet d'ondes de contraintes appliquées à l'extrémité d'une fissure en propagation. Ces contraintes sont induites soit par une réflection sur une surface libre, dans une configuration judicieuse de géométrie d'éprouvette, soit par l'utilisation d'un générateur d'ondes.On établit que ces ondes ont une influence sur la direction de la propagation, sur la vitesse de la fissure, et sur le processus d'efflorescence.Toutefois, on conclut que si des ondes de contraintes modifient le processus d'efflorescence, elles n'en constituent par pour autant la cause. On démontre également dans quelle mesure des réflexions d'ondes dans des petites éprouvettes peuvent singulièrement compliquer le comportement à la propagation d'une fissure, ainsi que l'interprétation des résultats.

     

Interactions between inclusions and various types of cracks

The problems of a crack inside, outside, penetrating or lying along the interface of an anisotropic elliptical inclusion are considered in this paper. Because the crack may be represented by a distribution of dislocation, integrating the analytical solutions of dislocation problems along the crack and applying the technique of numerical solution on the singular integral equation, we can obtain the general solutions to the problems of interactions between cracks and anisotropic elliptical inclusions. Since there are no analytical solutions existing for the general cases of interactions between cracks and inclusions, the comparison is made with the numerical results obtained by other methods or with the analytical results for the special cases which can be reduced from the present problems. These results show that our solutions are correct and universal     

Interaction between cracks and rigid-line inclusions by an integral equation approach

 An integral equation approach is presented to investigate the interaction between cracks and rigid-line inclusions embedded in an infinite isotropic elastic matrix subject to remote loading. The relevant fundamental solutions in the integral formulation are presented. Special tip elements are used to simulate the variation of the discontinuous displacements over the crack surfaces, and the axial and shear forces along the rigid-line inclusions. The stress intensity factors at the crack tips and at the ends of the rigid-line inclusions are computed and compared with available solutions. Received: 6 August 2002 / Accepted: 3 February 2003 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7011/01E). The authors would like to thank two reviewers for their constructive comments and suggestions to the paper. The comments of Professor H.P. Hong at the Department of Civil and Environmental Engineering of the University of Western Ontario of Canada are also appreciated.     

The interaction of multiple rows of periodical cracks

In this paper, the interaction of multiple rows of periodical cracks contained in an infinite elastic plate with far-field stress loaded is studied. An extremely accurate and efficient numerical method for solving the problem is presented. The method is mainly by means of the crack isolating analysis technique, stress superposition principle, the Chebyshev polynomial expansion of the pseudo-traction as well as the segmental average collocation technique. This method can be used to compute the stress intensity factors of multiple cracks, periodical cracks, and multiple rows of periodical cracks. In the process of dealing with the superposition of interaction of infinite number of periodic cracks, a key series summation technique is used, which aims at numerical results with extremely high accuracy but with less computation work. Many complex computing examples are given in this paper, and, for some typical examples, numerical results are compared with analytic solutions and with previous numerical solutions. For the problem of the one periodical collinear cracks, the accuracy given by this method reaches to 6 significant digits if a/d 0.9 (where a is the half crack length, and d is the half crack spacing). And even if a/d=0.99, the error is still less than 0.5%. The computer results for multiple rows of periodic collinear and echelon cracks show that the interaction effect between two rows rapidly decline with exponential law as the array pitch increases. This method has filled the gaps in the research field on the interaction of multiple rows of general periodical cracks.     

Determination of dynamic stress intensity factors of multiple cracks

The dynamic stress field around a crack or cracks embedded in an infinite isotropic elastic medium subjected to a SH wave are determined. Based on the qualitatively similar features of crack and dislocation, the stress wave emitted from a vibrating screw dislocation can be regarded as a Green's function. By superimposing an array of dislocation and adjusting the distribution density to fulfill the boundary condition, a singular integral equation with kernels containing Bessel functions is derived, which can be solved by the Galerkin method. Dynamic stress intensity factors, which can be as much as 28 percent higher than the static value, are found to be the same as those results obtained by other investigators. The stress intensity factors of a set of infinite cracks of equal length are also calculated as an application of this method.

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Résumé On détermine le champ des contraintes dynamiques entourant une fissure ou des fissures noyées dans un milieu élastique infini et isotrope soumis à une onde SH. En se basant sur les caractéristiques qualitativement similaires que présentent une fissure et une dislocation, on peut considérer que l'onde de tension émise par une dislocation-vis en vibration constitue une fonction de Green. En superposant une famille de dislocation et en ajustnt la densité de la distribution de manière à satisfaire une condition aux limites, on dérive une équation intégrale singulière comportnt comme kernels des fonctions de Bessel, qui peut être résolue par la méthode de Galerkin. Les facteurs d'intensité dynamique, qui peuvent atteindre une valeur 28 pour cent plus élevée qu'une valeur statique, sont trouvés être les mêmes que ceux identifiés par d'autres chercheurs. Le facteur d'intensité de contraintes correspondant une série de fissures infinies de longueur égale est également calculé en application de la méthode.

     

Experimental investigation of crack growth direction in multiple cracks

In this paper, the interaction between multiple cracks in crack growth direction is studied in an aluminium alloy under static and fatigue loading. Self similar as well as non‐self‐similar crack growth has been observed which depends on the relative crack positions defined by crack offset distance and crack tip distance. On the basis of experimental observations, the criterion for crack coalescence and crack growth direction are expressed in terms of the crack positions defined by crack offset and crack tip distances. The criterion presented in this study can be used to determine the limiting value of crack tip and crack offset distance and to determine the mode in which cracks coalesce during their growth process. Experimental results and crack interaction criterion presented under various crack positions and size conditions could be used to derive a new evaluation method of crack growth in multiple crack geometry.     

Interaction between periodic cracks and periodic rigid-line inclusions in piezoelectric materials

Piezoelectric materials with a doubly periodic array of cracks and rigid-line inclusions under far-field antiplane mechanical load and inplane electric load are investigated. By employing the conformal mapping technique and the elliptical function theory, an exact solution of the whole-field stress and electrical displacement is obtained. The closed-form formulae for the stress and electrical displacement intensity factors at the tips of cracks and rigid-line inclusions and the effective electroelastic moduli of the composites are presented. Some existing solutions can be regarded as degenerated cases of the present results. Numerical examples are provided to show the interesting electroelastic interaction between multiple cracks and multiple rigid-line inclusions.     

Transient analysis of multiple parallel cracks under anti-plane dynamic loading

The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading is considered in this work. The cracks are simulated by distributions of dislocations and an integral equation relating tractions on the crack planes and the dislocation densities is derived. The integral equation in the Laplace transform domain is solved by the Gaussian–Chebyshev integration quadrature. The dynamic stress intensity factor associated with each crack tip is calculated by a numerical inverse Laplace scheme. Numerical results are given for one crack and two or three parallel cracks under normal incidence of a plane horizontally shear stress wave.     

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.     

Elastostatic interaction of multiple arbitrarily shaped cracks in plane inhomogeneous regions

A numerical technique has been developed for the determination of stress fields associated with multiple arbitrarily shaped cracks in plane inhomogeneous regions. The procedure allows the elastostatic analysis of cracks interacting with one or more straight bimaterial interfaces; of cracks located near, or emanating from, circular inclusions; and of cracks that emanate from single or multiple origins. The cracks may be branched or blunted, and may be subjected to arbitrarily applied stresses. The technique employs an efficient surface integral method, using distributions of edge dislocations to represent the cracks. The resulting singular integral equations are solved using a Gauss-Chebyshev integration formula; appropriate conditions are developed for closing the set of equations governing cracks intersecting inhomogeneity boundaries, based on a consideration of the stresses and displacements at the points of intersection. Crack-tip stress intensity factor results are presented for several crack configurations. The overall scheme provides a more general, direct, and convenient approach than other available schemes. A computer program has been developed to implement the various formulations in a single framework.     

Interaction between coplanar cracks under dynamic shock loading

Summary A study is made on the interaction between coplanar cracks in an unbounded body when dynamic loads act on thier surfaces that are described as Heaviside or Dirac delta functions. Graphs are given relating the stress intensity coefficients when there are two disk cracks in the body whose surfaces are loaded by shock external forces.Translated from Fiziko-khimicheskaya Mekhanika Materialov, Vol. 27, No. 3, pp. 50–55, May–June, 1991.     

Pop-in behavior induced by interaction of cracks

The linear fracture mechanics concept was used to analyze the stability of two and three colinear cracks in infinite planes. It was found that at a particular condition of the initial crack arrangement, long crack can absorb the short one stably resulting the pop-in behavior. The condition of the crack arrangement for the pop-in occurance, amount of load dropping and crack displacement increment were obtained.     

Analyzing the interaction between collinear interfacial cracks by an efficient boundary element-free method

A new traction boundary integral equation is presented for analyzing the interaction effect of any number of collinear interface cracks in a two-dimensional bimaterial. The dislocation densities on every crack surface are expressed in the products of the characteristic terms and the weight functions, and the unknown weight functions are approximated using the moving least-squares technique based on the constructed orthogonal basis functions. An efficient numerical integral method is employed to evaluate the Cauchy principal integrals that appear in the meshless method. The boundary element-free method is established, and a series of numerical results is presented. The interaction between the collinear interfacial cracks is analyzed.     

Effect of electromechanical coupling on the dynamic interaction of cracks in piezoelectric materials

Summary This article provides a comprehensive treatment of the dynamic interaction between two arbitrarily located and oriented cracks in a piezoelectric medium under steady-state inplane electrical and antiplane mechanical loads. Using an impermeable condition along the crack surfaces, a fundamental dynamic solution was developed for the single crack problem. In this fundamental solution, the single crack problem was treated using Fourier transform and the appropriate singular integral equations. The fundamental solution was then implemented into a pseudo-incident wave method to account for the interaction between the cracks. Numerical examples are provided to show the effect of the geometry of the cracks, the material constants, the frequency of the incident wave and the applied electrical field upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon the stress intensity factor at the crack tip.     

A new boundary element method for mixed boundary value problems involving cracks and holes: Interactions between rigid inclusions and cracks

This paper derives a new boundary integral equation (BIE) formulation for plane elastic bodies containing cracks and holes and subjected to mixed displacement/ traction boundary conditions, and proposes a new boundary element method (BEM) based upon this formulation. The basic unknown in the formulation is a complex boundary function H(t), which is a linear combination of the boundary traction and boundary displacement density. The present BIE formulation can be related directly to Muskhelishvili's formalism. Singular interpolation functions of order r ^–1/2 (where r is the distance measured from the crack tip) are introduced such that singular integrand involved at the element level can be integrated analytically. By applying the BEM, the interaction between a rigid circular inclusion and a crack is investigated in details. Our results for the stress intensity factor are comparable with those given by Erdogan and Gupta (1975) and Gharpuray et al. (1990) for a crack emanating from a stiff inclusion, and with those by Erdogan et al. (1974) for a crack in the neighborhood of a stiff inclusion.     

Interactions of cracks and inclusions in homogeneous elastic media

The work is devoted to static problems of elasticity for an infinite homogeneous medium containing planar parallel cracks and heterogeneous inclusions of arbitrary shapes. Cracks and inclusions occupy a finite region of the medium that is subjected to arbitrary external forces. The problem is reduced to a system of surface integral equations for crack opening vectors and volume integral equations for the stress tensor in the region. Gaussian approximating functions are used for discretization and efficient numerical solution of this system. Such functions are centered at the nodes of a regular node grid that covers all the inclusions and the crack surfaces. For Gaussian functions, the elements of the matrix of the discretized system have forms of standard integrals that can be tabulated and calculated fast. The matrix of the discretized system is not sparse but it has Teoplitz’s structure, and the number of independent matrix elements is much smaller than the total number of the elements. In addition, fast Fourier transform technique can be used for calculation matrix-vector products with such matrices. It accelerates substantially the process of iterative solutions of the discretized system. The method is mesh free. Examples of numerical solutions of the problems for planar circular cracks and spherical inclusions are presented and compared with analytical and numerical solutions available in the literature.