Brittle fracture dynamics with arbitrary paths I. Kinking of a dynamic crack in general antiplane loading

We present a new method for determining the elasto-dynamic stress fields associated with the propagation of anti-plane kinked or branched cracks. Our approach allows the exact calculation of the corresponding dynamic stress intensity factors. The latter are very important quantities in dynamic brittle fracture mechanics, since they determine the crack path and eventual branching instabilities. As a first illustration, we consider a semi-infinite anti-plane straight crack, initially propagating at a given time-dependent velocity, that changes instantaneously both its direction and its speed of propagation. We will give the explicit dependence of the stress intensity factor just after kinking as a function of the stress intensity factor just before kinking, the kinking angle and the instantaneous velocity of the crack tip.     

A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

Transient dislocation emission from a crack tip under dynamic mode III loading

Summary  Transient dislocation emission from a crack tip under dynamic mode III loading is analyzed. By taking into account the dynamic interaction between the crack and dislocation, the governing equation for the dislocation motion is derived under the quasi-steady assumption. The behavior of dislocation emission is explored in detail by solving this equation numerically. A critical initial speed can be determined, which must be exceeded by dislocations to escape from the crack tip. The dislocation emission process is found to be completed in such a short time period that the applied load may be approximately treated as constant during dislocation emission. Based on this fact, an asymptotic criterion for transient dislocation emission is developed, from which the critical initial speed can be evaluated. In the case that the dislocation is emitted from rest, we recover the quasi-static criterion of dislocation emission. Received 22 November 2000; accepted for publication 20 March 2001     

Kinking of a crack emanating from the free surface of a beam under pure bending

A perturbation technique developed by Karihaloo et al. is employed to obtain the stress intensity factors at the tip of a kinking crack that emanates from the free surface of a beam under pure bending. Under the condition that the kink extends in the direction of vanishing K₁₁ the crack path is obtained as well as a path stability condition. From conditions on K₁ a material parameter r^* akin to that of Ramulu and Kobayashi's r_c is obtained. By analysis of the slope of the kinking crack a stability condition is obtained corroborating the stability condition from consideration of vanishing K₁₁. It is shown that for a beam in pure bending the nonsingular remote stress term T must be greater than some positive critical value for kinking to occur confirming the results of Sayir and Schindler.     

Analytic solutions to crack tip plastic zone under various loading conditions

Based on stress field equations and Hill yield criterion, the crack tip plastic zone is determined for orthotropic materials and isotropic materials under small-scale yielding condition. An analytical solution to calculating the crack tip plastic zone in plane stress states is presented. The shape and size of the plastic zone are analyzed under different loading conditions. The obtained results show that the crack tip plastic zones present “butterfly-like” shapes, and the elastic–plastic boundary is smooth. The size of the plastic zone for orthotropic composites is less at the crack tip for various loading conditions, compared with the case of isotropic materials. Crack inclination angle and loading conditions affect greatly the size and shape of crack tip plastic zone. The mode I crack has a crucial effect on the plastic zone for mixed mode case in plane stress state. The plastic zone for pure mode I crack and pure mode II crack have a symmetrical distribution to the initial crack plane.     

Extension of a cut-induced crack under antiplane loading

The extension of a crack formed by cutting at a high velocity into the surface of an elastic solid is investigated. The solid is assumed to be in a state of uniform antiplane shear before the cut is induced. The anti-plane wave motion which is generated by the cutting process is analyzed through a Green's function technique. This technique leads to an integral equation for the stress in the plane of the crack. The stress intensity and velocity intensity functions are obtained, and the propagation of the crack after the cutting process has been terminated is analyzed by means of the balance-of-rates-of-energy criterion. It is shown that the proclivity towards propagation beyond the length of the cut-induced crack shows a significant dependence on the speed of cutting.

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Zusammenfassung Es wird die Ausdehnung eines Risses untersucht, der bei Durchschneidung der Oberfläche eines elastischen Festkörpers mit hoher Geschwindigkeit erzeugt ist. Hierbei wird angenommen, dass sich der Festkörper vor dem Schnitt in einem Zustand gleichförmiger Schubspannung befindet. Die Wellenbewegung, welche durch den Schnittprozess erzeugt wird, wird mit Hilfe einer GREENschen-Funktion analysiert. Diese Methode führt für die Spannung in der Ebene des Risses auf eine Intergralgleichung. Man erhält Spannungsintensitätsund Geschwindigkeitsintensitätsfunktionen. Die Rissausbreitung nach Beendigung des Schnittprozesses wurde mit Hilfe des Kriteriums vom Gleichgewicht von Energieraten bestimmt. Es wird gezeigt, dass die Tendenz zur Ausbreitung über die Lage des schnittinduzierten Risses hinaus eine bedeutende Abhängigkeit von der Schnittgeschwindigkeit aufweist.

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Now at the Stanford Research Institute, Menlo Park, Cal.     

Mode-III crack kinking under stress-wave loading

A horizontally polarized step-stress wave is incident on a semi-infinite crack in an elastic solid. At the instant that the crack tip is struck, the crack starts to propagate in the forward direction, but under an angle κπ with the plane of the original crack. In this paper a self-similar solution is obtained for the particle velocity of the diffracted cylindrical wave field. The use of Chaplygin's transformation reduces the problem to the solution of Laplace's equation in a semi-infinite strip containing a slit. The Schwarz-Christoffel transformation is employed to map the semi-infinite strip on a half-plane. An analytic function in the half-plane which satisfies appropriate conditions along the real axis, can subsequently be constructed. The Mode-III stress-intensity factor at the tip of the kinked crack has been computed for angles of incidence varying from normal to grazing incidence, for angles of crack kinking defined by -0.5?κ?0.5, and for arbitrary subsonic crack tip speeds.     

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.     

Three-dimensional elliptic crack under impact loading

The dynamic stress intensity factor of a three-dimensional elliptic crack under impact loading is determined with the finite element method. The computation results can take into account the influence of time and the ratio of the wave speeds on the stress intensity factor. The present method is suitable not only for three-dimensional dynamic crack, but also for three-dimensional dynamic contact. Project supported by the National Natural Science Foundation of China (No. K19672007).     

Stability of a viscoelastic plate under dynamic loading

Tashkent Polytechnic Institute. Translated from Prikladnaya Mekhanika, Vol. 27, No. 9, pp. 78–87, September, 1991.     

Effect of contact interaction of the crack faces for a crack under harmonic loading

This paper is concerned with dynamic problems in fracture mechanics for elastic solids having cracks with contacting faces. The contact problem for a penny-shaped crack with a nonzero initial opening under normally incident harmonic wave is solved by the method of boundary integral equations. The solutions are compared with those that neglect the contact interaction of the crack faces. Results are presented for different values of the initial crack opening Presented at the 6th International Conference on Modern Practice in Stress and Vibration Analysis (Bath, United Kingdom, September 5–7, 2006). Published in Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 125–131, July 2007.     

Analytical solutions of dynamic mode I crack under the conditions of displacement boundary

By the approaches of the theory of complex variable functions, the problems of dynamic mode I crack under the condition of displacement boundary are investigated. For this kind of dynamic crack extension problems with arbitrary index of self-similarity, the universal representations of analytical solutions are facilely deduced by the methods of self-similar functions. Analytical solutions of the stresses, displacements and stress intensity factors are readily acquired using the methods of self-similar functions. The problems studied can be very easily translated into Riemann–Hilbert problems and their closed solutions are gained rather straightforward in terms of this technique. According to corresponding material properties, the mutative rule of stress intensity factor was illustrated very well. Using those solutions and superposition theorem, the solutions of arbitrarily complex problems can be attained.     

Dynamic crack growth under stress wave loading

The crack growth process has been analysed on the basis of a fracture criterion of a dynamic stress intensity factor when a crack in an infinite plate was subjected to a pulse type of stress wave. The crack velocity and the amount of crack extension were related to the constant amplitude and the duration of the stress pulse. The calculated amount of crack extension was well in agreement with the experimental one for the polymer material Acrylite (which is similar to polymethylmethacrylate) found by the authors, thus indicating the validity of the present approach.     

Internal circular crack under normal antisymmetric loading

Summary A complete solution is given for the first time to the title problem. Explicit expressions are derived for the field of stresses and displacements in a transversely isotropic space weakened by a penny-shaped crack and subjected to two antisymmetrically applied normal concentrated forces. The method is based on the new results in potential theory obtained by the author earlier. The presented results may be used as Green's functions for a general case of antisymmetric loading so that the complete solution can be presented in quadratures. Several specific applications to fracture mechanics are considered.

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Antisymmetrische Lasten senkrecht zum kreisförmigen Innenriß

Übersicht Für das Titelproblem wird hier erstmals eine vollständige Lösung mit expliziten Ausdrücken für die Spannungen und Verschiebungen im transversal-isotropen Raum mit zwei Einzellasten senkrecht zu einem Kreisriß angegeben. Die Herleitung beruht auf neueren Ergebnissen des Autors in der Potentialtheorie. Die Ergebnisse können als Greensche Funktionen für den allgemeinen Fall antisymmetrischer Relastung benutzt werden, so daß dessen vollständige Lösung durch Quadraturen darstellbar ist. Schließlich werden verschiedene Anwendungen in der Bruchmechanik betrachtet.

     

Fatigue crack propagaion under mixed mode loading

Mixed model fatigue crack propagation is analyzed in this paper, using a centre cracked plate geometry, loaded under un-iaxial cyclic tension. Based on maximum principal stress criterion, a modified Paris expression of fatigue crack growth rate is derived in terms of ΔK and crack angle β_α for an inclined crack. It is also shown that it is more convenient to express the Paris equation by means of crack length projected on the x -axis, α_x rather than the actual length, α itself. The crack trajectory due to cyclic loading is predicted, β is varied from 29° to 90°. Experimental data on Type L3 aluminium agree fairly well with predicted values when β_α exceeds 30°.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.