Twisting of an elastic plate containing a crack

The stress distribution caused by twisting an infinite plate containing a finite crack is analyzed in terms of Reissner's theory for the bending of thin plates. The singular character and the detailed structure of the stresses near the ends of the crack are determined in closed form. Numerical results are given for the magnitudes of the stress couples and stress resultants for a range of plate thicknesses.

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Zusammenfassung Due Spannungsvertcilung, hervorgerufen durch die Torsion einer unendlichen Platte mit einem Ri\ begrenzter Länge, wird mit Hilfe der Reissner-Theorie für die Biegebeanspruchung dünner Platten untersucht. Der singulare Charakter und die genauc Verteilung der Spannungen in Nähe der Ri\enden werden bestimmt. Zahlenmä\ige Ergebenisse für die Gro\e der Spannungsparre und ihrer Resultanten werden für eine Reihe von Plattenstärken angegeben.

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Résumé La distribution des contraintes dans une plague infinie comportant une fisure finic et soumise à torsion est analysée au moyen de la théorie de Reissner pour la flexion des tôles minces.Le caractère singulier, et la structure de détail des contraintes au voisinage des extrémités de la fisure sont explicités.Des résultats numériques sont fournis en ce qui regarde les grandeurs des couples de contraintes et de leurs résultantes, pour une certaine gamme d'éspaisseur de tôles.

     

A finite element stress analysis of a crack in a bi-material plate

A stress analysis is presented for the problem of a crack in one material of a bi-material plate located perpendicular to the material interface. A numerical solution using the finite element techniques to determine the force displacement relationships is used. Knowing this, a work integral method is used to determine the stress intensity factors for the crack. Since the work integral is independent of path, the path of integration can be chosen far enough away from the crack tip to avoid the complications of the crack tip singularity. The problem is studied for a number of cases where the crack length to plate width ratio, distance from crack tip to material interface, and the ratio of material constants were varied as parameters.     

Experimental and theoretical study of Rayleigh-Lamb waves in a plate containing a surface-breaking crack

Scattering of Rayleigh-Lamb waves by a normal surface-breaking crack in a plate has been studied both theoretically and experimentally. The two-dimensionality of the far field, generated by a ball impact source, is exploited to characterize the source function using a direct integration technique. The scattering of waves generated by this impact source by the crack is subsequently solved by employing a Green's function integral expression for the scattered field coupled with a finite element representation of the near field. It is shown that theoretical results of plate response, both in frequency and time, are similar to those obtained experimentally. Additionally, implications for practical applications are discussed.     

Plastic zones in an orthotropic plate of finite width containing a Griffith crack

The problem of an orthotropic infinite plate of finite width containing a centrally located stressed Griffith crack is considered. The crack is located perpendicular to the edges of the orthotropic plate. The crack tips are fully yielded and in the inelastic zones the material carries only constant normal stresses equal to the yield stress. Dugdale's model is employed to find the effects of the material anisotropy on the size of the plastic zones around the crack tips. Graphical results showing the effects of anisotropy on the length of the plastic zone are also presented.     

Elasto-plastic analysis for a finite thickness rectangular plate containing a through-thickness central crack

The objective of this work is to present a three-dimensional analysis of the elasto-plastic response of a center-cracked rectangular panel by using an alternate method, the method of lines (MOL). The problem of a finite thickness rectangular plate containing a through-thickness central crack under uniaxial tension is studied in detail. An incremental procedure is used to seek the 3-D elasto-plastic stress and strain distribution. The Prandtl-Reuss equations and the associated von Mises flow rule are employed. The material is assumed to be elastic-plastic with nonlinear work-hardening. Effective stress and effective strain concepts are used. The MOL is used to reduce the governing equations on displacements to a coupled set of ordinary differential equations (ODE) along x, y, z lines. These are integrated by Peano-Baker method. Since the ODE are of constant coefficient the coefficient matrix is diagonalized and the matrizon is easily evaluated. Numerical results on stresses, displacements, the growth of plastic zone and the crack surface opening displacements are reported and their variation along the thickness are presented in graphs. The variation of the stress intensity factor K ₁, along the thickness direction is given. It is found that the computation time needed for determining the distribution of stresses and strains is much less than what has been reported by using other methods. A comparison of results on SIF with those obtained by other methods is also given.

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Résumé L'objet du travail est de présenter une analyse tridimentionnelle de la réponse élastoplastique d'un panneau rectangulaire présentant une fissure centrale, et utilisant une méthode alternative, la méthode des lignes MOL. On étudie dans le détail le problème de la plaque rectangulaire d'épaisseur finie comportant une fissure centrale traversante et soumise à la contrainte uniaxiale. Une procédure incrémentale est utilisée en vue de rechercher la distribution des contraintes et des déformations élastoplastiques suivant les trois dimensions. On utilise les équations de Prandtl-Reuss et la règle d'écoulement de von Mises qui lui est associée. On suppose que le matériau est élastoplastique et qu'il présente un vieillissement non linéaire. On recourt au concept des contraintes effectives et des déformations effectives. Le MOL est utilisé en vue de ramener les équations gouvernant les déplacements à un ensemble d'équations différentielles ordinaires suivant les axes X, Y, Z; on fait appel à la méthode de Peano-Baker pour leur intégration. Comme les équations différentielles ont un coéfficient constant, la matrice de coéfficient peut être résolue par diagonales et le matriciel peut être aisément évalué. Les résultats numériques relatifs aux contraintes, aux déplacements, à la croissance de la zône plastique et aux déplacements d'ouverture de la surface de la fissure sont présentés, et leur variation sur l'épaisseur sont mises en graphiques. On fournit la variation du facteur d'intensité de contrainte K suivant la direction de l'épaisseur. On trouve que le temps de calcul nécessaire à la détermination de la distribution des contraintes et déformations est beaucoup moindre que celui nécessaire par l'utilisation d'autres méthodes. Une comparaison des résultats relatifs aux facteurs d'intensité de contraintes avec ceux obtenus par d'autres méthodes est également fournie.

     

On a plate supported by an elastic foundation and containing a finite crack

Using an integral formulation, the equation for a plate resting on an elastic foundation, of a spring constant k and containing a crack of length 2c, is solved for the Kirchhoff bending stresses. The inverse square root singular behavior of the stresses peculiar to crack problems is obtained. Furthermore, this singularity may be related to that found in an unsupported plate by where a is for small values of the parameter is a positive constant.

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Zusammenfassung Für eine auf einem elastischen Fundament (Elastizitätskonstante k) ruhende Platte, welche einen Riß der Länge 2c aufweist und Kirchhoff Biegebeanspruchungen ausgesetzt ist, konnte die Gleichung mit Hilfe einer Integralformel gelöst werden.Es ergibt sich das den Rißproblemen eigene singulare Gesetz des umgekehrten Verhältnisses zur Quadratwurzle. Außerdem kann these Singularität mit der für eine nicht gestützte Platte ermittelten über die Gleichung: in Zusammenhang gebracht werden, wobei a für kleine Werte des Parameters eine positive Konstante ist.

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Résumé A l'aide d'une formule d'intégration, on a résolu l'équation d'une plaque reposant sur un support de constante élastique k, comportant une fissure de longueur 2e, et soumise à des contraintes de flexion de Kirchhoff. On aboutit à une distribution des contraintes dont l'amplitude varie en raison inverse du carré de la distance, comportement singulier propre aux problèmes de fissuration.En outre, cette singularité pent etre mise en balance avec celle que l'on trouve dans une plaque non supportée, par l'expression: où a est une constante positive pour de petites valeurs du paramètre .

     

An orthotropic sandwich plate containing a part-through crack under mixed mode deformation

An orthotropic sandwich plate containing a part-through crack subjected to in-plane mixed mode tractions is analyzed, and the basic equations for estimating the stress intensity factors $\text{k}{}_1$, $\text{k}{}_2$ and the debonding naion around the crack are presented.     

Evaluating crack tip stress field in a thin glass plate under thermal load

The stress field around a propagating crack tip in a quenched thin glass plate is discussed through experimental and theoretical analyses. Instantaneous phase-stepping photoelasticity using a CCD camera equipped with a pixelated micro-retarder array is used for measuring the crack tip stress field. From the successive phase maps of principal direction, the position and the velocity of the crack tip are evaluated. On the other hand, the fracture parameters, that is, the stress intensity factors and the T-stress are determined from the phase maps of the retardation. Experimental results obtained for a straight crack show good agreement with those obtained by theory of elasticity. The results also indicate that the direction of the crack propagation arising in the quenching process is not determined by the direction of the maximum principal stress. Furthermore, the results show that the T-stress criterion is inappropriate to evaluate the crack path instability in a quenched thin glass plate.     

The stress intensity factor for a crack perpendicular to a longitudinal weld in a large plate

The paper presents a theoretical analysis which gives the stress intensity factor for a crack that is perpendicular to a weld, the residual stress being tensile in the immediate vicinity of the weld and compressive away from the weld. Whether or not the stress intensity factor becomes zero when the crack tip is at a finite distance within the compressive region, is shown to be dependent on the fine details of the residual stress pattern. The conclusions are relevant to the arrest of a crack in a material with a low stress corrosion crack growth resistance.     

Analytical solution for orthotropic composite plate containing a mode I crack along principle axis

Analytical stress analyses are presented for orthotropic composite materials containing a through crack under uniaxial normal loads (mode I). A harmonic differential equation has been established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry by introducing new complex variables. The complex theory was employed to find stress functions to satisfy the equilibrium equation, compatibility equation and boundary condition at infinite and crack surfaces. An analytical solution was examined in the case of isotropic materials. It is demonstrated that the analytical solution obtained is correct for the orthotropic composite plates.     

Kinked crack in a bending trapezoidal plate

The interaction problem of a kinked crack and the edges of a bending trapezoidal plate which takes the effects of transverse shear deformation into account is presented. The research method is based upon the complex potential technique of Muskhelishvili using conformal mapping. Furthermore, for the analysis of the moment intensities at the tips of the kinked crack, the concept of dislocation distribution is applied. The integral equations for the stress disturbance problem along the line that is the presumed location of the kinked crack are then obtained as a system of singular integral equations with simple Cauchy kernels. As a consequence, the variation of moment intensity factors at the crack-tips is also illustrated.     

Photoelastic analysis of a thick square plate containing central crack and loaded by pure bending

Three-dimensional photoelastic analysis is employed to study the stresses in the neighborhood of the tip of a central crack in a square plate with finite thickness. The plate is being loaded by pure bending moment around its free edges. Both maximum shear stresses and maximum normal stresses were investigated in planes parallel to the crack and through the plate thickness. In addition, the variation of these stresses was investigated along various lines drawn from the crack tip and again in various planes of the plate. The locations of the largest shear stress and the largest normal stress were established.It was found that these variations of the stresses depend not upon the crack length, but upon the ratio of the plate thickness to crack length. Both the theoretical and the experimental analysis show that a hydrostatic state of stress exists at points along the longitudinal axis of the crack originating from the crack tip outward.The experimental photoelastic results are correlated with the existing three-dimensional theoretical solution.

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Résué L'analyse photoélastique à trois dimensions est utilisée pour l'étude des contraintes au voisinage de la pointe d'une fissure centrale traversante dans une plaque carrée d'épaisseur finie. La plaque est sollicitée flexion pure autour de ses bords libres. On examine les tensions tangentielles maximum et les tensions normales maximum dans des tranches parallèles découpées sur l'épaisseur de la plaque. En outre, on étudie la variation de ces contraintes le long de différentes lignes tracées à partir de la pointe de la fissure, et ce dans les différents plans découpés. On situe l'emplacement où apparaissent la plus grande tension tangentielle et la plus grande tension normale. On trouve que ces variations des tensions ne dépendent pas de la longueur de la fissure mais bien du rapport de l'épaisseur de la plaque à la longueur de la fissure. L'analyse théorique et l'expérience montrent qu'il existe un état de tensions hydrostatiques le long de l'axe longitudinal de la fissure prenant naissance en la partie extérieure de l'extrémité de la fissure. Ces résultats expérimentaux obtenus par photoélasticité sont comparés à la solution théorique tridimensionnelle existante.

     

On singular stress at the crack tip of a thick plate under in-plane loading

By superposition of three asymptotic solutions, which collectively satisfy all the equilibrium and compatibility equations as well as boundary conditions, the three-dimensional crack tip stresses of a thick plate with a through-the-thickness cut are shown to be of square root singularity throughout the thickness. The near tip deformation is shown to be plane strain with stress intensity factor changing in the thickness direction. All components of the singular stresses vanish on plate surfaces. The profile of the stress intensity factor in the thickness direction cannot be determined by the asymptotic analysis alone.     

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.     

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.