On the use of the finite element method for the solution of a cracked strip under thermal shock

The finite element method is used to solve the problem of a strip containing a crack and subjected to a thermal shock on one edge. The solution of the edge crack problem is compared to the exact solution and it is shown that the FEM yields results with less than 1% error. The dependence of the accuracy of the solution on the time increment is examined. The problem of a strip containing an internal crack is then solved using the FEM. The effects of the Biot number, the length of the crack and the distance of the crack to the edge of the strip, on the transient stress intensity factors are analyzed.     

An analytical solution for a cracked piezoelectric plate according to the PKHS electric boundary condition

Summary Based on the Stroh formalism, an analytical solution for a central crack normal to the poling direction in an infinite plane piezoelectric solid subjected to electromechanical loads at infinity is presented according to the crack model suggested by Parton and Kudryavtsev [1] and used by Hao and Shen [2] (abbreviated as the PKHS type crack [3]) without any treatment of the mapping technique. Under generalized plane deformation, the variable tendency of the energy release rate (ERR) is found to be load-dependent, but the variable tendency of the mechanical strain energy release rate (MSERR) is completely independent of the applied electric loads. It is concluded that the present numerical results under high applied mechanical and electrical loads provide a better understanding for the PKHS type crack, which will benefit to its practical applications.     

Effect of circular holes on the ratchet limit and crack tip plastic strain range in a centre cracked plate

In this paper a centre cracked plate subjected to cyclic tensile loading and cyclic bending moment is considered. The effect of circular holes drilled in the region of the crack tip on the ratchet limit and crack tip plastic strain range is studied. Direct evaluation of the ratchet limit and crack tip plastic strain range is solved by employing the new Linear Matching Method (LMM). Parametric studies involving hole diameter and location are investigated. The optimum hole location for reducing the crack tip plastic strain range with the least reduction in ratchet limit is identified, and located at a distance 10% of the semi-crack length from the crack tip on the side opposite the ligament for both cyclic tensile loading and cyclic bending moment cases. It is also observed that the optimum location is independent of the hole size for both cyclic loading cases.     

Analytical calculation of the stress intensity factor in a surface cracked plate submitted to thermal fatigue loading

A stainless steel specimen with a pre-existing surface notch is exposed to a convective medium of cyclic temperature. The history of stress intensity factor of the cracked body for different crack lengths is obtained by a closed-form integration of the stress field, using Duhamel’s theory with principle of superposition and appropriate weight functions. The obtained results are compared with numerical simulations performed with ABAQUS and they appear to be in very good agreement. The stress intensity factor history shows that fatigue behavior does not depend only on temperature amplitude ΔT=T_max-T_min, quenching rate, and duration of thermal shock but also on heating rate and duration.     

Finite width effects in a cracked plate

The expression for the strain energy of a plate with an elliptical hole has been derived using the Inglis solution, in an attempt to trace the missing details between the Griffith Solutions of 1921 and 1924. The results obtained permit, to a first approximation, the evaluation of the finite width effect on the stress-intensity factor of a cracked plate.

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Résumé On calcule l'énergie de déformation d'une plaque ayant un trou elliptique en utilisant la solution d'Inglis, en vue de tenter de reconsituer les détails manquant entre les deux solutions de Griffith de 1921 et 1924. Les résultats obtenus permettent de prévoir à une première approximation, les effets de la largeur finie sur le facteur d'intensité des contraintes d'une plaque fissurée.

     

On the strain energy release rate for a cracked plate subjected to out-of-plane bending moment

For a through-the-thickness crack in an infinite plate subjected to out-of-plane uniform bending moment, the strain energy release rate is determined using the virtual crack extension and the variation of potential energy. It is shown that the strain energy release rate for the Reissner's plate approaches the classical plate solution as the ratio of plate thickness to crack size becomes infinitesimally small. By using this result, the limiting expression of the stress intensity factor can be explicitly obtained. For general problems, the modified crack closure method is shown to be an efficient tool for evaluating the strain energy release rates from which the stress intensity factor can be calculated. Both the classical plate element and the Mindlin plate element are investigated, and the applicability of the classical plate element is evaluated.Because the stress-free conditions along the crack face lead to inter-penetration of the plate, a line contact model is assumed to investigate the closure effect using Reissner plate theory. Closure at the compressive side is shown to reduce crack opening displacement and consequently the stress intensity factors. When closure is considered, the strain energy rate based on the Reissner plate theory converges to the classical plate solution. This is similar to the nonclosure case.     

Three-dimensional evaluation of the T-stress in centre cracked plates

Thick three-dimensional (3-D) finite element models of centre cracked plates are used to study the variation in the biaxiality factor with the crack aspect ratio a/w. The use of two widely accepted methods to evaluate the T-stress in two-dimensions, namely the boundary layer method and the displacement field method, to calculate the T-stress in three-dimensions is studied. It is shown that the boundary layer method gives results that compare rather well with the two-dimensional plane strain values (maximum difference of 6 percent), while the displacement field method results are about 15 percent lower. Two parameters are shown to affect the three-dimensional evaluation of the biaxiality factor, namely the material's Poisson's ratio and the specimen's thickness t. The biaxiality factor is directly proportional to and inversely proportional to t. Three-dimensional analysis is required to assess correctly the effect of and t on .     

On the use of singular displacement finite elements for cracked plate in bending

The objective of this work is to assess the performance, convergence and accuracy of four different displacement crack-tip elements used in modelling a cracked plate subjected to out-of-plane bending. A methodology is developed for calculating the singular field from the computed results and optimizing the mesh used in the numerical solution. It was found that using the quarter-node triangular elements surrounded by quadrilateral transition elements yields very accurate estimates of the singular fields of displacements and stresses as well as the stress-intensity factor for various materials and thicknesses of plates. It was also found that when the transition elements are incorporated, the optimum length of the singular element ranges from 0.5 to 1 per cent of half of the crack length.

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Résumé Le but de ce travail est d'établir les possibilités, la convergence et la sécurité de quatre modèles différents d'éléments finis pour extrémités de fissure, utilisés pour représenter une plaque fissurée soumise à flexion hors de son plan. On a développé une méthodologie pour le calcul du champ singulier qui résulte des calculs, et pour l'optimisation du réseau utilisé dans la solution numérique.On trouve qu'en utilisant des éléments triangulaires quart-onde entourés d'éléments quadrilatères de transition, on est conduit à des estimations très précises des champs singuliers de déplacements et de contraintes ainsi que du facteur d'intensité de contrainte, dans le cas de divers matériaux et de diverses épaisseurs de plaque.On trouve également que lorsque sont incorporés les éléments de transition, la longueur optimale d'un élément singulier varie entre 0,5 et 1% de la demi-longueur de la fissure.

     

In-plane bending problem of a double edge cracked plate

In this paper the boundary collocation method is used for evaluating the stress intensity factors (SIF) of a double edge cracked plate under in-plane bending. For the case with a large ratio of the plate height to the width, h/b, the results obtained compare very favorably with existing solutions for an infinite strip. Moreover, this method has been used for different finite plates, and a series of conclusions is provided for application.     

The application of a displacement controlled weight function for a single edge cracked plate to thermal fatigue damage assessment

A weight function based on displacement boundary conditions for the single edge cracked plate is extended and tested against finite element analysis results for the stress intensity factor under a linear end displacement. The stress intensity factor fluctuation, caused by sinusoidal thermal striping on the surface of the plate, is predicted as a function of crack depth and plate aspect ratio by the frequency response model with the new weight function and by the finite element method. Good agreement is found. Sample calculations are presented for the threshold surface temperature fluctuation for propagation of existing surface defects.     

A review of the three-dimensional stress problem for a cracked plate

An analytical study has been carried out to examine the influence of plate thickness on the stress distribution around the crack. The qualitative feature of the three-dimensional solution is first determined by an asymptotic expansion of the stresses and displacements in terms of the cylindrical polar coordinates r, , z for small values of r which is referenced from the border of a semi-infinite crack. It is found that the stresses _rr, , _zz , and _r are singular of the order r , but the transverse shear stresses _rz and _z , are bounded for plates under stretching and bending. The intensity of crack-border stress field becomes a function of the thickness coordinate z. Knowing that the problem prohibits any exact analytical solutions of a quantitative nature, the three-dimensional equations of elasticity will be approximated by appealing to minimum principles in the calculus of variations.Guided by the results obtained from the asymptotic expansions, each one of the six stress components is assumed to be the product of two functions, one being assigned to describe the stress distribution in the plane of the plate and the other across the thickness. The z-distribution of the stresses may either be pre-assigned arbitrarily or determined from the plane strain condition ahead of the crack. On the basis of the principle of minimum complementary energy, a system of three simultaneous differential equation in two variables is obtained and solved for the problem of an infinite plate containing a through crack by means of integral representations. Determined in closed elementary form are the detailed structure of the three-dimensional crack-edge stress field. The stress-intensity factor, which varies in the thickness direction, is shown to be a function of the ratio of plate thickness to crack length and is found to increase rapidly in magnitude as the plate thickness is perturbed slightly from zero. The present analysis suggests a method by which the effect of a finite plate thickness can be incorporated into an examination of the fracture toughness of cracked sheet specimens.

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Zusammenfassung Es wurde eine analytische Untersuchung durchgeführt, um den Einfluß tier Blechstärke auf die Spannungsverteilung in tier Umgebung eines Risses zu bestimmen.Qualitativ gesehen kennzeichnet sich die gefundene dreidimensionale Lösung in erster Linie durch eine asymptotische Ausdehnung tier Spannungen und Verschiebungen, ausgedrückt durch die polaren Koordinaten r, and z für kleine Werte von r, bezogen auf die Grenze eines halb-unendlichen Risses. Man finder, daß die Spannungen _rr , , _zz und _r eine Singularität der Ordnung r aufweisen; im Falle von Blechen unter Zugbiegebeanspruchungen sind die transversalen Scherbeanspruchungen _rz and _z jedoch gebunden. Die Intensität des Spannungsfeldes am Rande eines Risses wird abhängig von der Dickenkoordinaten z.Unter Berücksichtigung der Tatsache, daß das Problem eine exakte analytische Lösung quantitativer Art nicht zuläßt, müssen die dreidimensionalen elastischen Gleichungen durch Annäherung gelöst werden, unter Bezugnahme auf das Minimum-Prinzip der Variationsrechnungen.Auf Grund der von der asymptotischen Ausdehnung erzielten Ergebnisse wird jede der sechs Spannungskomponenten als das Produkt von zwei Funktionen angenommen, wovon eine die Spannungsverteilung in der Ebene des Bleches und die andere diejenige entlang der Dickenkoordinaten beschreibt. Die Spannungsverteilung entlang der z-Achse kann entweder beliebig festgelegt, oder aus den planen Spannungsbedingungen vor dem Riß ermittelt werden.Unter Anwendung des Prinzips der minimalen Komplementarenergie erhält man ein System von drei Differentialgleichungen mit zwei Unbekannten. Diese Gleichungen werden durch eine Integraldarstellung für den Fall eines unendlichen Bleches mit einem sich über die ganze Dicke erstreckenden Riß gelöst. Die endliche Struktur des dreidimensionalen Spannungsfeldes am Rand des Risses kann in Form von endlichen Elementen beschrieben werden.Es wird bewiesen, daß der Spannungsintensitätsfaktor, welcher entlang der Dickenachse ändert, eine Funktion des Verhältnisses von Blechstärke zu rißlänge ist; für Blechstärken leicht öber Null steigt er schon bei nur sehr geringer Erhöhung der Dicke, sehr erheblich an. Die durchgeführte Analyse führt zu einem Verfahren, das es ermöglicht den Einfluß der Dicke eines Bleches endlicher Abmessungen bei der Untersuchung der Zerreißfestigkeit von Feinblechproben mit Rissen, zu berücksichtigen.

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Résumé On a effectué une etude analytique pour examiner l'influence de l'épaisseur d'une tôle sur la distribution ties contraintes au voisinage d'une fissure. Qualitativement, la solution tridimensionnelle trouvée se caractérise au premier chef par une extension asymptotique des contraintes et déplacements, exprimés en coordonnées polaires r, et z, pour les faibles valeurs de r rapportées aux frontières de l'extrémité d'une fissure semi-infinie. On trouve que les contraintes _{rr zz} , _r, et _r présentent une singularitée d'ordre r , mais que les contraintes transversales de cisaillement _rz et _z sont associées dans le cas d'une tôle soumise à traction et flexion. L'intensité du champ de contraintes an bord de la fissure devient une variable dépendante de l'axe z des épaisseurs.Compte tenu que le problème ne permet pas de proposer une solution analytique exacte de caractère quantitatif, il faut résoudre les équations élastiques tridimensionnelles par approximation, en procédant à un calcul des variations et en appliquant le principe des minimas.En se basant sur l'extension asymptotique des contraintes et déplacements, on suppose que chacune des six composantes de la contrainte est le produit de deux fonctions dont l'une décrit la distribution des contraintes dans le plan de la tôle et l' autre selon son épaisseur.La distribution ties contraintes selon l'axe ties z peut être fixée de manière arbitraire, on pent être exprimée à partir des conditions d'état plan de déformation existant devant la fissure.En appliquant le principe de l'énergie complémentaire minimum, on obtient un système de trois équations différentielles à deux inconnues. Ces équations sont résolues par représentation intégrale pour le cas de la tôle infinie comportant une fissure de part en part de son épaisseur. La structure fine du champ tridimensionnel ties contraintes au bord de la fissure peut être décrite sous la forme d'éléments finis.On démontre que le facteur d'intensité ties contraintes, qui est variable selon l'axe ties épaisseurs, est une fonction du rapport de lépaisseur de la tôle à longueur de la fissure; il augmente considérablement pour de légères variations de l'épaisseur aus-dessus de l'épaisseur nulle.L'analyse effectuée laisse déduire une méthode permettant de faire intervenir l'influence de l'épaisseur d'une tôle de dimensions finies dans l'étude de la ténacité à la rupture d'échantillons de tôles minces comportant ties fissures.

     

Fracture criteria of cracked metallic plate

The Griffith energy criterion is equivalent to a critical stress and strain environment criterion for brittle fractures. For quasi-brittle fracture, i. e. the case where small amounts of plastic deformation at the crack tip precedes fracture, the energy criterion is still equivalent to stress and strain environment criterion. Based on the concept of a critical stress and strain environment, it is proposed as an engineering criterion for ductile fracture, that at fracture the size of the highly strained region at crack tip is constant.

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Zusammenfassung Griffith fand das Energiebruch Kriterium fuer sproedes Material. Fuer das Spannungsund Dehnungs Kriterium fuer sproede Brueche ist dieses Energie em fluss Kriterium gleichwertig, wie von der elastischen Loesung fuer sproede teste Koerper ersehen werden kana. Fuer fast sproede Brueche, dass heisst, in dem Falle wo geringe Mengen bildsamer Formgebung an der Bruchspitze dem Bruch vorangehen, ist das Energie Kriterium immernoch dem Spannungs- und Dehnungseinfluss Kriterium gleichwertig, Gegruendet an die Theorie des Spannungs- und Dehnungseinflusses, ein Engenieurs Kriterium fuer formbare Brueche, es wird angenommen, dass die Groesse eines stark gedehnten Bereiches konstant ist.

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Résumé Griffith a établi le critère d' énergie de fracture pour les matériaux fragiles fissurés. Ce critère d' énergie est équivalent au critère d' environnement de contrainte et de déformation pour les fractures fragiles comme indiqué par les solutions élastiques pour les solider fissurés. Pour une fracture quasi-fragile, c' est à dire dans le car oú de petites quantités de déformation plastiques à l' extrémité de la fissure précedent la fracture, le critère d' énergie est également équivalent au critère d'environnement de contrainte et de déformation. Un critère d' engineering pour des fractures ductiles, basé sur le concept d'environnement de contrainte et de déformation, suggére que, à la fracture, la dimension de la région de forte déformation à l' extrémité de la fissure est constante.

     

Stress intensification in a viscoelastic plate on a Winkler foundation and under a large thermal gradient

Summary A thin viscoelastic plate on a Winkler foundation is subjected to vertical loads. Its response is strongly affected by the presence of a vertical temperature gradient which causes a very pronounced change in the viscosity coefficient through the plate thickness. While eventually the entire load will be transferred to the underlying foundation, during the time dependent deformation process it is possible that some bending stresses will actually increase rather than decrease as would normally be expected with this relaxation process. The rationale for this behavior lies in the competition between the two physical processes of a) the load transfer to the foundation causing an overall relief of the bending moment and thus the bending stress and b) the rapid relaxation of bending stress in the hot lower portion of the plate causing the colder upper portion to carry a larger share of that portion of the bending moment still carried by the plate. The simple case of a clamped circular plate of a metallic material and subjected to a uniform load is used to illustrate this behavior. Certain simplifications are made to allow an analytic solution; these simplifications do not alter the basic behavior. In the present case, these simplifications are the neglect of Poisson's ratio effects and the assumption of a linear equivalent viscoelastic material behavior. This latter point is discussed in an appendix. Numerical solutions containing more general behavior indicate that the basic behavior is still well modeled even with these assumptions.With 4 Figures     

Line-spring boundary element method for a surface cracked plate

In the published line-spring boundary element method, the effect of bending is not considered. Therefore it cannot be used to deal with the problem of a surface cracked plate. Taking the advantage of the line-spring model and the boundary element method the authors present a new line-spring boundary element method in which the effect of membrane force and bending moment in a Reissner plate is taken into account. The implementation of the method is discussed in detail. The stress intensity factors for several example problems concerning surface cracked plates are calculated. Comparisons are made with the Newman-Raju solutions. The results show that the proposed method is efficient.     

Dugdale model solutions for a single edge cracked plate

International Journal of Fracture - A technique for improving the convergence rate of the Gauss-Seidel iterative procedure for finite element analysis is discussed. It is noted that this technique...     

Buckling behavior of a central cracked thin plate under tension

The buckling characteristics of cracked plates subject to uniaxial tensile loads are analysed by the aid of the finite element method. Owing to the fact that crack buckling behavior is affected by the in-plane stress distribution around a crack, to get more accurate results, pre-buckling in-plane stress fields are analysed by the finite element method. For the critical loads calculation, the finite element approach adopted is based on Von Karman's linearize theory for buckling of plates subjected to pre-buckling state of plane stress. Several singular elements based on the Willian series are used in this plate bending approach. In this study, the effect of crack length, the effect of boundary condition, the effect of Poison's ratio and the effect of biaxial force on critical loads are analysed and discussed. Furthermore, the effect of initial imperfection is also discussed. There is a good agreement between other researcher's work and present results.