On the Energy Release Rate at the Crack Tips in a Finite Pre-Strained Strip

The influence of the initial finite stretching or compressing of the strip containing a single crack on the Energy Release Rate (ERR) and on the SIF of mode I at the crack tips is studied by the use of the Three-Dimensional Linearized Theory of Elasticity. It is assumed that the edges of the crack are parallel to the face planes of the strip and the ends of the strip are simply supported. The initial finite strain state arises by the uniformly distributed normal forces acting at the ends of the strip. The additional normal forces act on the edges of the crack. The elasticity relations for the strip material are given by the harmonic type potential. The corresponding boundary-value problem is solved by employing FEM. The numerical results on the influence of the initial finite strain state the values of the ERR and of the SIF of mode I are presented. In particular, it is established that the values of the ERR and of the SIF of mode I decrease (increase) monotonically with an increase (decrease) in the initial stretching (compression).     

Thermal-stress analysis for a strip of finite width containing a stack of edge cracks

The thermal-stress problem of an infinite strip containing an infinite row of periodically distributed edge cracks normal to its edge is investigated. The surrounding temperature adjacent to the crack-containing edge is assumed to be cooled suddenly to simulate the thermo-shock condition. By the superposition principle, the formulation leads to a mixed-boundary-value problem, with the negating tractions derived from the thermal stresses of a crack-free infinite strip. An integral equation is obtained and solved numerically. The effect on the SIFs (stress-intensity factors) due to the presence of periodically distributed cracks in an infinite strip is delineated. The normalized SIFs increase as the stacking cracks separate, due to the reduction of the shielding effect. After a characteristic time period, the negating tractions along the crack faces become almost linear. The SIF solutions under the considered crack geometry are worked out in detail for the case of linear traction loading.     

An elastic strip with periodic surface cracks under thermal shock

The problem of two periodic edge cracks in an elastic infinite strip located symmetrically along the free boundaries under thermal shock is investigated. It is assumed that the infinite strip is initially at constant temperature. Suddenly the surfaces containing the edge cracks are quenched by a ramp function temperature change. Very high tensile transient thermal stresses arise near the cooled surface resulting in severe damage. The degree of the severity for a subcritical crack growth mode is measured by determining the stresses intensity factors. The thermoelastic problem is treated as uncoupled quasi-static. The superposition technique is used to solve the problem. The thermal stresses obtained from the uncracked strip with opposite sign are utilized as the only external loads to formulate the perturbation problem. By expressing the displacement components in terms of finite and infinite Fourier transforms, a hypersingular integral equation is derived with the crack surface displacement as the unknown function. Numerical results for stress intensity factors are carried out and presented as a function of time, cooling rate, crack length, and periodic crack spacing.     

The problem of internal cracks in an infinite strip having a circular hole

Summary. The elastostatic plane problem of an infinite strip having a circular hole and containing two symmetrically located internal cracks perpendicular to the boundary is formulated in terms of triply coupled integral equations. The solution of the problem is obtained for various crack geometries and for uniaxial tension applied to the strip away from the crack region. Quantities of physical interest are displayed in graphical forms.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermal shock of a cracked strip: Effect of temperature-dependent material properties

The problem of a strip containing an edge crack and an interior crack subjected to a thermal shock on one edge and insulated on the other is solved in order to analyze the difference between using temperature-dependent material properties and constant ones. For this purpose two brittle materials, ceramics and glass, are each subjected to a thermal shock. The results show that in general, using constant material properties over large temperature ranges can lead to considerable underestimation of the maximum stress intensity factors. The difference in the results is dependent on the variation of the thermal diffusivity and the thermal expansion coefficient with temperature for a given material. Also, this difference varies for different crack lengths and different thermal shock conditions at the boundary.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.     

含裂纹损伤有限加筋板弹塑性承载力分析

基于含直裂纹问题的复应力函数解法,提出了用Dugdale模型分析和求解弹塑性条件下含中心裂纹的有限加筋板承载力问题的方法。通过将加筋板离散为筋、板的结构,将含裂纹有限加筋板的问题转化为边界受切向力作用的含裂纹有限板的问题进行求解。计算了筋、板相对刚度不同的情况下,含中心裂纹有限加筋板裂纹尖端开口位移CTOD(Crack Tip Opening Displacement)值随裂纹长度及承载力情况变化的系列值。     

Closed-form solution for the size of plastic zone in an edge-cracked strip

This paper is concerned with the problem of plastic zone at the tip of an edge crack in an isotropic elastoplastic strip under anti-plane deformations. By means of complex potential and Dugdale model, the stress intensity factor and the size of plastic zone are obtained in closed-form. Furthermore, the analytic solutions for an edge crack at the free boundary of a half-space and a semi-infinite crack heading towards a free surface are determined as the limiting cases of the strip geometries.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Edge crack in a strip of an elastic solid

The problem of determining the distribution of stress and the deformation of a long strip of an elastic material, damaged by a crack normal to an edge of the strip, is investigated. The strip is deformed by pressure applied to the faces of the crack. The stress intensity factor is calculated and its variation with the depth of the crack, relative to the width of the strip, in the special case of uniform pressure, is illustrated.     

Elastostatic analysis of edge cracked orthotropic strips

Summary. The elastostatic problem of an edge cracked orthotropic strip is considered. The crack possesses a semi-infinite length. The crack surfaces are subjected to opening mode I fracture, by a concentrated force action, while the strip surfaces are traction free. Fourier transforms and asymptotic analyses are employed to reduce the problem to a first kind singular integral equation. The stress intensity factor is determined in a closed form expression. The effects of geometric and elastic characteristics of the strip on the values of the stress intensity factor are explained.     

Line-spring model for surface cracks in a reissner plate

In this paper the line-spring model developed by Rice and Levy for a surface crack in elastic plates is reconsidered. The problem is formulated by using Reissner's plate bending theory. For the plane strain problem of a strip containing an edge crack and subjected to tension and bending new expressions for stress intensity factors are used which are valid up to a depth-to-thickness ratio of 0.8. The stress intensity factors for a semi-elliptic and a rectangular crack are calculated. Considering the simplicity of the technique and the severity of the underlying assumptions, the results compare rather well with the existing finite element solutions.     

Vibration in mode III of an elastic layer containing a crack perpendicular to the boundary edge

A mathematical formulation of the problem of stresses and displacements in an elastic layer which contains a crack perpendicular to the boundary and subjected to a vibrating stress, in mode III, is developed. The boundary conditions for the case of free loading at the edges of the strip are used to obtain a solution to this problem. The problem is reduced to the solution of a Fredholm integral equation. In Part II the problem describing the case of rigid constraint at the edges of an elastic strip containing a vibrating external crack in mode II is reduced to the Fredholm integral equation of the second kind. In Part III the solution of the problem describing the case of a strip containing a vibrational crack in mode III and laying on a rigid boundary is presented.

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Résumé On développe une formulation mathématique du problème des containtes et déplacements dans une couche élastique comportant une fissure perpendiculaire au bord d'une bande et soumise à une sollicitation vibratoire de mode III. Les conditions aux limites de bord suivantes sont envisagées: (1) sollicitation libre en bordure de bande, (2) bridage sévère des bords de la bande et (3) maintien de la bande dans une fixation rigide. Dans les trois cas, le problème est ramené à la solution d'une intégrale de Fredholm.

     

The solution of a rectangular orthotropic sheet with a central crack under anti-plane shear

Making use of the basic theorem of the Fourier transform and series, the solution of the stress intensity factor of a rectangular orthotropic plate containing a central crack under anti-plane shear, is obtained in this study. The result to the mixed boundary value problem is expressed in terms of a Fredholm integral equation of the second kind. It is easily proved that the problem of a strip with a central crack of mode III, are the special cases of the solution in this article.     

A strip yield model for two collinear cracks in plates of arbitrary thickness

As two cracks grow and approach each other under fatigue loading, a deleterious interaction between them can considerably affect the crack growth rate, making theoretical evaluations and experimental data from a single isolated crack case considerably inaccurate. The aim of the present study is to investigate the interaction between two collinear cracks of equal length, taking into account the plate??s thickness effect, which was demonstrated to have a large effect on fatigue crack growth in the case of a single crack. The obtained solution to the problem is based on the Dugdale strip yield model and the distributed dislocation technique. In addition, a fundamental solution for an edge dislocation in a finite thickness plate is utilised. The present solution shows a very good agreement with previously published results for some limiting cases. The obtained results confirm a significant dependence of the interaction and stress intensity factors on the plate thickness, which can dramatically affect the plastic collapse conditions as well as fatigue crack growth rates.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

On the fracture behavior of a Zener–Stroh crack with plastic zone correction in three-phase cylindrical composite material

With crack tip plastic zone correction, stress investigation on the fracture behavior of a Zener–Stroh crack in three-phase composite was carried out. A Zener–Stroh crack (in the matrix phase) is near a circular inclusion, with the three-phase cylindrical composite model used to represent the composite material. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. The Dugdale model of small scale yielding is used to introduce a thin strip of yielded plastic zone each crack tip. The physical problem is formulated into a set of singular integral equations, using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function. The singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements using Erdogan and Gupta’s method with some iterative numerical procedures.     

Crack tip opening displacement of a Dugdale crack in a three-phase cylindrical model composite material

The analytical investigation of the plastic zone size of a crack in three-phase cylindrical model composite material was carried out. The physical problem is simulated as a crack near a circular inclusion (a single fiber) in the composite matrix, while the three-phase cylindrical composite model is used to represent the composite matrix. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, a thin strip of yielded plastic zone is introduced at each crack tip. Using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function, the physical problem is formulated into a set of singular integral equations. By employing Erdogan and Gupta’s method, as well as iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements.     

Surface and internal cracks in a residually stressed plate

The problem of a surface or an internal crack in a plate which contains residual stresses is examined. The line spring model, which reduces a three-dimensional elasticity problem into a two-dimensional problem in plate theory, is used to model the crack. The Reissner plate theory, which takes into account transverse shear deformations, is used to model the plate. The formulation is based on Fourier Transforms which lead to a pair of singular integral equations that are solved numerically. The line spring method requires the plane strain solution to both the edge and internally cracked strip with crack surface loads representative of tension, bending, and the given residual stress distribution. For general use, plane strain solutions are presented for polynomial loading through the thickness up to the fifth order. Comparisons are made between the results given by the line spring model for the Reissner plate theory and the finite element method.