Numerical methods for determination of stress intensity factors of singular stress field

The asymptotic solution of the singular stress field near a singular point is generally comprised of one or more singular terms in the form of Kr^λ-1f_ij(θ). Based on the asymptotic solution of the singular stress field and the common numerical solution (stresses or displacements) obtained by an ordinary tool such as the finite element method or boundary element method, a simple and effective numerical method is developed to calculate stress intensity factors for one and two singularities. Three examples show that the stress intensity factors evaluated using the method proposed in this paper are very accurate.     

A remark on the direct numerical determination of stress intensity factors at crack tips

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.     

A vibration method for the determination of stress intensity factors

A new experimental method of determining stress intensity factors is presented. It is based upon measurements of natural frequencies of a specimen before and after the crack is introduced. Single edge notched rectangular and axisymmetrically notched cylindrical specimens were tested and the results compared with previous work. It is shown that the new test will often be sufficiently accurate and is considerably easier to carry out than other experimental procedures.     

Simple methods of determining stress intensity factors

A prerequisite for any fracture mechanics analysis of a cracked structure, is a knowledge of the stress intensity factor at the tip of the crack. Many methods are available for evaluating stress intensity factors, but if the structural configuration is complex, they are usually costly in time and money. This paper describes some simpler approximate methods which are both quick and cheap. Their use is illustrated by examples typical of aerospace applications, e.g. cracks at holes and cracks in stiffened sheets. The errors introduced into calculations of residual static strength and fatique lifetimes by the use of such approximations are acceptable for many practical cases: They are usually no greater and often smaller than those due to uncertainties in other parameters such as service loads, material toughness, etc.     

Strain energy release rate determination of stress intensity factors by finite element methods

The determination of the Mode I stress intensity factors for selected crack configurations, using finite element methods and energy release rate principles, is the subject of this study. The crack configurations which were investigated are the double edge crack, the single edge crack and the center crack. The method of analysis utilized was the “Stiffness Derivative Method.” This approach relates the change in strain energy resulting from crack advancement, to the change in the stiffness matrix of the structure containing the crack. The results indicated that through mesh optimization and proper control of certain parameters including the crack advance increment, the crack tip element contour size and mesh refinement, an accurate solution can be calculated with a relatively coarse finite element mesh consisting entirely of contemporary elements. The numerically generated solutions are compared with analytical solutions with the results within 0.001% of each other for the double edge crack, 0.858% for the single edge crack and 2.021% for the center crack.     

Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients

This paper proposed numerical methods to determine the multiple stress singularities (including the oscillatory stress singularities) and the related stress intensity coefficients, by the use of common numerical solutions (stresses or displacements) obtained by an ordinary numerical tool such as finite element method (FEM) or boundary element method (BEM). To verify the efficiency of the present methods, two models of bonded dissimilar materials under the plane strain state are analyzed by BEM, and the orders of the stress singularities and the related stress intensity coefficients are examined numerically. The results show that all the orders of the stress singularities at an interface edge can be determined precisely by the present method, and the related stress intensity coefficients can be determined by the extrapolation method with a very good linearity. It is found that the methods presented in this paper are very simple and efficient. Moreover, they can be easily extended to any singular problem.     

On the photoelastic determination of complex stress intensity factors

An improvement of the one-parameter extrapolation method of photoelastic determination of complex (mixed-mode) stress intensity factors at straight or curvilinear crack tips in a plane isotropic elastic medium due to Smith et al. [12, 13] can be achieved by measuring the absolute value of such a factor on the isochromatic fringes along properly selected polar directions and not at the maxima of the isochromatic fringes. In this way, the unknown value of the constant term of the stress field near the crack tip is taken into account. It is seen that it is always possible to find at least one appropriate polar direction to measure the absolute value of the stress intensity factor. In the case of Mode I stress intensity factors, these polar angles are = ± 120° and not = ± 90° as generally considered previously. Some numerical results are also presented in this special case and show the efficiency of the present method.     

On the numerical evaluation of the anvil force for accurate dynamic stress intensity factor determination

To evaluate precisely the dynamic fracture toughness of a brittle material in the tests with short time-to-fracture, both tup and anvil forces have to be known. Unfortunately, the anvil force is rarely registered by the standard impact testing equipment. The method for numerical evaluation of the support reactions by using registered tup force and the calculated specimen modal parameters is proposed. It assumes that the contact between the specimen and the supports can be described by the quasi-static Hertz’s theory. Both linearized and nonlinear relations for the specimen-support contact compliance are considered. The efficiency of the method has been verified by processing the results of two three-point-bend impact tests reported by Böhme and Kalthoff. The influence of the various calculation parameters (number of eigenmodes taken into account, time step size) and the specimen geometry (length of the specimen overhangs) on the accuracy of determination of the anvil force and dynamic stress intensity factor variation with time is investigated.     

A semi-theoretical and experimental approach for the determination of the stress intensity factors

The stress intensity factors for plexiglass plates containing edge cracks and subjected to either pure bending or tension are determined herein. The method of investigation was based on a semi-theoretical and experimental approach, where the stress intensity factors are expressed in terms of the measured diameter of the caustic, the crack length, and the width of the specimen. First, two basic crack arrangements (single and double edge cracks) were studied and then the method was utilized for the investigation of more complicated crack arrangements which are difficult or maybe impossible to be investigated otherwise. In particular, the stress intensity factor for plates having a sharp V-notch of various angles θ, and semi-infinite plates containing equal parallel edge cracks subjected to pure bending and tension respectively, were investigated in order to verify the validity of this method.     

Finite elements for determination of crack tip elastic stress intensity factors

A new type of finite element is introduced which embodies the inverse square root singularity present near a crack in an elastic medium. Using this element near the tip in two typical cracked configurations, stress intensity factors within 5 per cent of accepted values were obtained with meshes having as few as 250° of freedom.     

Comparison of methods for calculating stress intensity factors with quarter-point elements

The quarter-point quadrilateral element is employed with various methods for calculating the stress intensity factor in order to provide guidelines for a best method. These methods include displacement extrapolation, J-integral and Griffith's energy calculations, and the stiffness derivative technique. Three geometries are considered: a central crack, a single edge crack and double edge cracks in a rectangular sheet. For these cases, it is observed that the stiffness derivative method yields the most accurate results, whereas displacement extrapolation is the easiest method to implement and still yields reasonable accuracy.

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Résumé On utilise les éléments en quadrilatère quart point dans diverses méthodes de calcul du facteur d'intensité de contrainte, afin de servir de guide pour le choix de la meilleure méthode. Il s'agit notamment des méthodes par extrapolation des déplacements, par calcul d'intégrale J ou d'énergie de Griffith, et par dérivée de la raideur. On considère trois géométries: une fissure centrale, une fissure de bord simple et deux fissures de bord dans une feuille rectangulaire. On observe pour ces trois cas que la méthode de la dérivée de la raideur conduit aux résultats les plus précis; par ailleurs, la méthode d'extrapolation des déplacements est la plus aisée à mettre en oeuvre et conduit néanmoins à une raisonnable exactitude.

     

A simple method for the photoelastic determination of mode I stress intensity factors

A new simple method for the photoelastic determination of Mode I stress intensity factors from isochromatics is proposed. This method takes into account the fact that a considerable part of the error committed in the photoelastic determination of Mode I stress intensity factors K_I at crack tips, based on experimentally obtained isochromatic fringe patterns, is due to ignoring the non-singular part of the stress field near the crack tips for the evaluation of these factors. This error can, in most cases, be minimized by an appropriate selection of the polar direction from the crack tip on which the experimental measurements for the subsequent evaluation of the stress intensity factors K_I are made. The suitable polar direction for determining K_I depends in general on the distance of the point where measurements on the isochromatics are made from the crack tip. The method was applied to the problem of a simple crack inside an infinite medium under uniaxial and biaxial loading. A comparison of the present method whith the employed analogous methods shows the superiority of the proposed method.