Driving force analysis in an infinite anisotropic plate with multiple crack interactions

The methodology and a rigorous solution formulation are presented for stress intesity factors (SIF's, k) and total strain energy release rates (SERR, G _T ) of a multicracked plate, that has fully interacting cracks and is subjected to a far-field arbitrary stress state. The fundamental perturbation problem is derived, and the steps needed to formulate the system of singular integral equations whose solution gives rise to the evaluation of the SIF's are identified. Parametric studies are conducted for two, three and four crack problems. The sensitivity and characteristics of the model is demonstrated.The U.S. Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Numerical solution for multiple crack problem in an infinite plate under compression

This paper investigates a numerical solution for multiple crack problem in an infinite plate under remote compression. The influence of friction is taken into account. In the first step of the solution, we make a full contact assumption on the crack faces. The full contact assumption means that one component of the dislocation distribution vanishes, and the first mode stress intensity factors (K ₁) at the crack tips become zero. On the above-mentioned assumption, the problem can be solved by using integral equation method, and the second mode stress intensity factors (K ₂) at the crack tips can be evaluated. Meantime, after solving the integral equation the normal contact stress on the crack faces can be evaluated. The next step is to examine the full contact assumption. If the contact stresses on the crack faces are definitely negative, the solution is true. Otherwise, the obtained solution is not true. It is found from present study that in most cases the full contact condition is satisfied, and only in a few cases the full contact condition is violated. Numerical examples are given. It is found that the friction can lower the stress intensity factors at crack tips in general.     

Hypersingular integral equation approach for the multiple crack problem in an infinite plate

Summary In this paper, a hypersingular integral equation for the multiple crack problem in an infinite plate is formulated. The unknown functions involved in the equation are the crack opening displacements (CODs) while the right hand terms are the tractions applied on the crack faces. Some particular hypersingular integrals are quadratured in a closed form. After the CODs are approximated by a weight function multiplied by a polynomial, the hypersingular integrals in the equation can be evaluated in a closed form, and the regular integrals can be integrated numerically. Numerical examples with the calculated stress intensity factors (SIFs) at the crack tips are given.     

Multiple Zener-Stroh crack problem in an infinite plate

Summary. In this paper, the multiple Zener-Stroh crack problem is studied. We choose the distributed dislocations as unknown functions in the integral equation. The crack faces are assumed to be traction free. The applied generalized loading for cracks is the initial displacement jump (abbreviated as IDJ), which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. A system of singular integral equations is obtained. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between Zener-Stroh cracks are quite different from those for the Griffith cracks, in the qualitative and quantitative aspects.AcknowledgementThe research project is supported by National Natural Foundation of China.     

Solutions of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach

Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper.     

Elasto-plastic finite element analysis of a crack in an infinite plate

The elastic support method was recently developed to simulate the effects of unbounded solids in the finite element analysis of stresses and displacements. The method eliminates all the computational disadvantages encountered in the use of `infinite' elements or coupled finite element boundary element methods while retaining all the computational advantages of the finite element method. In this paper, the method is extended to the elasto-plastic analysis of fracture in infinite solids by using the load increment approach and including the effects of strain hardening. Numerical tests and parametric study are conducted by analysing a straight crack in an infinite plate. Present results for J integrals and plastified zones are compared, respectively, with analytical solutions and available results obtained by using the body force method. The agreement between the results is found to be very good even if the truncation boundary of the finite element model is located very close to the crack tip or the plastified zone.     

Multiple crack problems of antiplane elasticity in an infinite body

Twe elementary solutions are presented for case of a pair of normal or tangential concentrated unit forces acting at a point of both edges of a single crack in an infinite plane isotropic elastic medium. Using these two elementary solutions and the principle of superposition, we found that the multiple crack problems can be easily converted into a system of Fredholm integral equations. Finally, the system obtained is solved numerically and the values of the stress intensity factors at the crack tips can be easily calculated. Two numerical examples are given in this paper. A system of Fredholm integral equations is complex form is also presented. We found that the system of Fredholm integral equations can be easily reduced from the system of singular integral equations given by Panasyuk[1]     

Thermal stresses in an infinite non-isotropic plate having a penny-shaped crack

Of concern in the paper is the distribution of thermal stresses in the vicinity of a penny-shaped crack in a thick elastic plate made of a non-isotropic material. The problem pertains to the situation where the crack is opened by a prescribed normal pressure and a prescribed heat-flux or a prescribed temperature.     

General solutions to an infinite plate weakened by a bent crack for the opening and sliding modes

Making use of the basic theorem of the Mellin transform, the general solution of a bent crack with finite lengths d and b in an isotropic infinite plate subjected to arbitrary normal or inplane shear stress is found in this article. It is easily found that the results of a central crack of mode I and II may be deduced from the general solutions in this study. All of the results in this paper are formulated in simply and closed forms. We can also see that the new method recommended in this study is more convenient than the old one (H. Kitagawa et al., Engng Fracture Mech. 7, 515, 1975) for solving the bent crack problem.     

A general method for multiple crack problems in a finite plate

A novel method for the multiple crack problems in a finite plate is proposed in this paper. The basic stress functions of the solution consist of two parts. One is the Fredholm integral equation solution for the crack problem in an infinite plate, and the other is that of the weighted residual method for general plane problems. The combined stress functions are used in the analysis and the boundary conditions on the crack surfaces and the boundary are considered. After the coefficients of the functions have been determined, the stress intensity factors (SIF) at the crack tips can be calculated. Some numerical examples are given and it was observed that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the influence of the boundary can be considered. This method can be used for arbitrary multiple crack problems in a finite plate.     

Crack problems for a rectangular plate and an infinite strip

In this paper, the general plane problem for an infinite strip containing multiple cracks perpendicular to its boundaries is considered. The problem is reduced to a system of singular integral equations. Two specific problems of practical interest are then studied in detail. The first is the investigation of the interaction effect of multiple edge cracks in a plate or beam under tension or bending. The second problem is that of a rectangular plate containing an arbitrarily oriented crack in the plane of symmetry. Particular emphasis is placed on studying the problem of a plate containing an edge crack and subjected to concentrated forces. The plate has the dimensions of a standard compact tension specimen and is intended to simulate the CTS.

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Résumé On considère dans ce mémoire le problème général d'une bande infinie comportant des fissurations multiples perpendiculaires à ses bords. Le problème est réduit à un système d'équations intégrales singulières. Deux problèmes spécifiques d'intérêt pratique sont ensuite étudiés dans le détail. Le premier consiste à étudier l'effet d'interaction de fissures de bord multiples dans une tôle ou dans une poutre sous tension ou en flexion. Le second problème est celui d'une plaque rectangulaire comportant une fissure orientée de manière arbitraire dans son plan de symétrie. Un accent particulier est mis sur l'étude du problème d'une plaque comportant une fissure de bord et soumise à des forces concentrées. La plaque a les dimensions d'une éprouvette standard compacte de traction et est conçue de manière à simuler celle-ci.

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This work was supported by NASA-Langley Research Center under the Grant NGR 39-007-011 and by NSF under the Grant ENG 78-09737.     

Evaluation of T‐stresses in multiple crack problems of finite plate

This paper provides a solution for T‐stresses for multiple cracks in a finite plate. The results for stress intensity factors (SIFs) are also presented. The case of two cracks in a rectangular plate is taken as an example. In the problem, the crack faces are applied by some loadings, and tractions are free along edges of a rectangular plate. The whole stress field is considered as a superposition of three particular stress fields. The first and second stress fields are initiated by loadings on the first and second crack faces in an infinite plate. The third field is chosen in a polynomial form of complex potentials. After discretization, the loadings on two cracks and the undetermined coefficients in the complex potentials become the unknowns. The relevant algebraic equations are formulated. The solution of algebraic equations will lead to the results of SIFs and T‐stresses at the crack tips. Several numerical examples are presented, which were not reported previously.     

Rigid body rotation at the tip of a slant crack in an infinite plate

Using Kolosov-Muskhelishvili relations of stresses the rigid body rotation is obtained in the form of complex potentials. The rotation at a point near the tip of a slant crack is expressed in terms of stress intensity factors and the coordinates (r, ) of the point. The relation of rigid body rotation near the crack tip are used to describe some features of mode I and mode II crack tip plastic zone. The rotation field surrounding the tip of a slant crack in infinite plate is obtained and its properties are discussed.

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Résumé En utilisant les relations de Kolosov-Muskhelishvili relatives aux contraintes, on obtient la rotation d'un corps rigide sous forme de potentiels complexes. La rotation en un point près de l'extrémité d'une fissure inclinée est exprimée en fonction du facteur d'intensité d'entaille et des coordonnées (r-) do point. On utilise les relations de rotation d'un corps rigide au voisinage de l'extrémité d'une fissure pour décrire certaines caractéristiques de la zone plastique à l'extrémité d'une fissure de mode I et de mode II. Le champ rotationnel autour de l'extrémité d'une fissure inclinée dans une plaque infinie est obtenue et ses propriétés sont discutées.

     

Dynamic stress intensity factors around a rectangular crack in an infinite plate under impact load

A three-dimensional solution is presented for the transient response of an infinite plate which contains a rectangular crack. The Laplace and Fourier transforms are used to reduce the problem to a pair of dual integral equations. These equations are solved with the series expansion method. The stress intensity factors are defined in the Laplace transform domain, and they are inverted numerically in the physical space.