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.     

Axisymmetric boundary integral equation analysis of interface cracks between dissimilar materials

The numerical boundary integral equation (BIE) method with quadratic quarter-point crack-tip singular elements is used to analyse interface cracks between dissimilar material in axisymmetry. Such crack problems present modelling difficulties using conventional procedures for obtaining the stress intensity factors. This is because of the oscillatorily singular nature of the stresses in the vicinity of the bimaterial interface crack-tip. Analytical expressions for the direct evaluation of the fracture characterising parameters from the BIE numerical results of displacements or tractions are derived. Three different crack problems are investigated, two of which have known solutions in the literature. Excellent agreement between the BIE results and these other established solutions are obtained even with relatively coarse mesh discretisations. The present study illustrates the ease with which the BIE method may be used in the fracture analysis of both straight and curved binaterial interface cracks.     

Anti-plane Shear Cracks Approaching a Bi-material Interface

A novel procedure is proposed for evaluation of stress intensity factors of planar Mode III shear cracks perpendicular to a nearby interface between two isotropic elastic solids. Shear cracks traversing a flat layer bonded to two different elastic solids are also analyzed. The method is based on superposition of singular near tip stress and displacement fields generated by both the main crack and certain image cracks. Both the main and the image cracks are loaded by self-equilibrating shear tractions of different magnitude, such that matching parts of the said fields are made to satisfy traction and displacement continuity conditions at the interface. Selected comparisons with results obtained by different methods show good agreement. Applications of the method to other crack problems are discussed.     

Analysis of Asymmetric Kinked Cracks of Arbitrary Size, Location and Orientation – Part II. Remote Tension

In this paper, the stress intensity factors of interacting kinked cracks in a solid and the overall strains of the solid under uniaxial tension are determined numerically. The kinked cracks are in general asymmetric, unequal, and arbitrarily oriented and located in the solid. Each kinked crack, assumed to be traction free, consists of a main crack and kinks. The analysis makes use of the dislocation modeling of kinks, and the superposition of problems of straight cracks subjected to dislocation and traction loadings. The model is used to investigate the dependence of the stress intensity factors and the overall strains on crack geometry (straight, Z-shaped and U-shaped cracks) and crack configuration (collinear and stacked cracks, periodic and random crack arrays). This revised version was published online in August 2006 with corrections to the Cover Date.     

A technique for studying interacting cracks of complex geometry in 2D

A method for studying brittle fracture in an infinite plate containing interacting cracks of complex shape under general loading conditions is developed and studied for accuracy and potential applications. This technique is based on superposition and dislocation theory and can be used to determine the full stress and displacement fields in a cracked body. In addition, stress intensity factors at both crack tips and wedges, created by crack kinking and branching, are calculated so that crack growth and initiation can be analyzed at these locations of possible crack propagation. Such information can then be used to study damage accumulation in structures containing a large number of interacting cracks.     

Determination of stress intensity factors in cracked plates by the finite element method

Stress fields near crack tips in an elastic body can be specified by the stress intensity factors which are closely related to the stress singularities arising from the crack tips. These singularities, however, cannot be represented exactly by conventional finite element models. A new method for the analysis of stresses around cracks is proposed in this paper on the basis of the superposition of analytical and finite element solutions. This method is applied to several two-dimensional problems whose solutions are obtained analytically, and it is shown that their numerical results are in excellent agreement with analytical ones. Sufficiently accurate results can be obtained by the conventional finite element analysis with rather coarse mesh subdivision. Computational efforts are then considerably reduced compared with other methods.     

Fracture analysis in 2D magneto–electro–elastic media by the boundary element method

An integral formulation for 2D cracked infinite anisotropic magneto–electro–elastic media is presented. Based on the method proposed by Garcia-Sanchez et al. (Comput Struct 83: 804–820, 2005), the hypersingular kernels are analytically transformed into weakly singular and regular integrals with known analytical solution. Special quadratic discontinuous crack tip elements are employed to model the singular characteristics of the stresses, electric displacements and magnetic inductions. The extended stress intensity factors at the crack tips are calculated using the extended discontinuous displacements at crack tip elements based on one point extended displacement formulation. Some results for curved cracks in magneto–electro–elastic media are also presented.     

Elastostatic interaction of multiple arbitrarily shaped cracks in plane inhomogeneous regions

A numerical technique has been developed for the determination of stress fields associated with multiple arbitrarily shaped cracks in plane inhomogeneous regions. The procedure allows the elastostatic analysis of cracks interacting with one or more straight bimaterial interfaces; of cracks located near, or emanating from, circular inclusions; and of cracks that emanate from single or multiple origins. The cracks may be branched or blunted, and may be subjected to arbitrarily applied stresses. The technique employs an efficient surface integral method, using distributions of edge dislocations to represent the cracks. The resulting singular integral equations are solved using a Gauss-Chebyshev integration formula; appropriate conditions are developed for closing the set of equations governing cracks intersecting inhomogeneity boundaries, based on a consideration of the stresses and displacements at the points of intersection. Crack-tip stress intensity factor results are presented for several crack configurations. The overall scheme provides a more general, direct, and convenient approach than other available schemes. A computer program has been developed to implement the various formulations in a single framework.     

Elastodynamic analysis of nonplanar cracks in a half-space

Summary The interaction of time harmonic antiplane shear waves with nonplanar cracks embedded in an elastic half-space is studied. Based on the qualitatively similar features of crack and dislocation, with the aid of image method, the problem can be formulated in terms of a system of singular integral equations for the density functions and phase lags of vibrating screw dislocations. The integral equations, with the dominant singular part of Hadamard's type, can be solved by Galerkin's numerical scheme. Resonance vibrations of the layer between the cracks and the free surface are observed, which substantially give rise to high elevation of local stresses. The calculations show that near-field stresses due to scattering by a single crack and two cracks are quite different. The interaction between two cracks is discussed in detail. Furthermore, by assuming one of the crack tips to be nearly in contact with the free surface, the problem can be regarded as the diffraction of elastic waves by edge cracks. Numerical results are presented for the elastodynamic stress intensity factors as a function of the wave number, the incident angle, and the relative position of the cracks and the free surface.     

Fracture mechanics weight functions in three dimensions: Subtraction of fundamental fields

A new procedure is presented for the determination of the fracture mechanics weight functions that are required for the evaluation of stress intensity factors in cracked solids. The procedure can be used with a standard three-dimensional boundary element code. The weight functions are proportional to the displacements on the boundary of the solid when the only loading is a pair of self-equilibrated point forces at the crack front. In previous work, the highly singular crack-tip fields that this loading produces have been modelled by replacing the crack front by a cylindrical cavity with appropriate displacement boundary conditions on the cavity walls. It is shown here that results are dependent on the cavity radius and that convergence of the results cannot be guaranteed. An alternative procedure, based on the substraction of fundamental fields (SFF), is demonstrated herein. The high-order singularities are removed from the field before the reduced problem is solved numerically using a standard boundary element method. Since the reduced problem is equivalent to an unloaded crack in a solìd subjected to boundary tractions, the usual quarter-point displacement elements and quarter-point traction singular elements can be used to improve the accuracy. Weight functions, so obtained, are used to evaluate stress intensity factors as a function of position on the crack front for a straight-fronted crack in a rectangular bar subjected to various loadings. Both edge and central cracks are considered and the validity of the technique is demonstrated by comparing the results with previously published values.     

Some applications of singular fields in the solution of crack problems

This paper reviews some recent developments in superposition methods for calculating linear elastic stress intensity factors and eigenvalues for cracks and notches, presents some new results for pairs of edge cracks and provides new insights into the nature of the errors in these processes. The procedure requires a numerical solution to the full cracked problem and a second solution on the same mesh using the known form of the singularity in an infinite region. This is equivalent to the well-known Subtraction of Singularity (SST) method. The advantages of this procedure over conventional SST are: (1) no modifications need to be made to a standard computer program; (2) multiple crack tips may be analysed without the difficulty of unknown rigid body displacements at the crack tips; (3) solutions with different boundary conditions on the same mesh may be obtained simply in one step by re-using one singular field solution; The singular crack tip field may also be studied independently leading to estimates of the eigenvalues and some insight into mesh-induced errors. The additional computational cost of a two-step procedure is minimal since the solution matrix from step one may be re-used with a new right-hand side. Numerical experiments using the boundary element method demonstrate the accuracy and simplicity of the superposition approach for notches, simple cracks, mixed-mode cracks, two edge cracks of different lengths and eigenvalues under various boundary conditions. © 1998 John Wiley & Sons, Ltd.     

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.     

Strong interactions of morphologically complex cracks

Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps.     

Evaluation of the stress intensity factors and the T-stress in periodic crack problem

This paper investigates the T-stress at crack tips in the periodic crack problem. Remote tension in the y-direction is applied to cracks with an arbitrary inclined angle. The original stress field can be considered a superposition of a uniform stress field and a perturbation stress field. The problem of evaluating the stresses in the perturbation field can be considered a superposition of many single crack problems. A Fredholm integral equation is suggested for the solution of the perturbation stress field. In the equation, the loading on the crack face is chosen as unknown quantity. Once the integral equation is solved, the stress intensity factors and the T-stress at the crack tip can be evaluated immediately. For solving the integral equation and evaluating stresses in the perturbation field, the remainder estimation technique is suggested for evaluating the influences on the central crack from infinite cracks. The technique can considerably improve convergence in computation. Many results for the stress intensity factors and the T-stresses in periodic cracks are presented. It is shown that the interaction is significant for the closer cracks.     

Stress intensity factors for a surface-breaking crack in a half-plane subject to contact loading

Modes I and II stress intensity factors are derived for a crack breaking the surface of a half-plane which is subject to various forms of contact loading. The method used is that of replacing the crack by a continuous distribution of edge dislocations and assume the crack to be traction-free over its entire length. A traction free crack is achieved by cancelling the tractions along the crack site that would be present if the half-plane was uncracked. The stress distribution for an elastic uncracked half-plane subject to an indenter of arbitrary profile in the presence of friction is derived in terms of a single Muskhelishvili complex stress function from which the stresses and displacements in either the half-plane or indenter can be determined. The problem of a cracked half-plane reduces to the numerical solution of a singular integral equation for the determination of the dislocation density distribution from which the modes I and II stress intensity factors can be obtained. Although the method of representing a crack by a continuous distribution of edge dislocations is now a well established procedure, the application of this method to fracture mechanics problems involving contact loading is relatively new. This paper demonstrates that the method of distributed dislocations is well suited to surface-breaking cracks subject to contact loading and presents new stress intensity factor results for a variety of loading and crack configurations.     

Longitudinal Shear of an Elastic Wedge with Cracks and Notches

We study the problem of longitudinal shear of an infinite wedge with cracks and notches. The integral representations of the complex stress potential are constructed in terms of the jumps of displacements and stresses on curvilinear contours identically satisfying the boundary conditions imposed on the faces of the wedge (stresses or displacements are equal to zero). By using these representations, we deduce singular integral equations of the analyzed problem for a wedge weakened by a system of cracks and holes of any shape. In some cases (a crack along the bisectrix of the wedge, a crack along a circular arc whose center is located at the edge of the wedge, and a circular notch near the edge of the wedge), we obtain exact closed solutions.     

Crack opening displacements of Vickers indentation cracks

Vickers indentation cracks are an appropriate tool to determine the crack-tip toughness K_I0 and, possibly, the bridging relation of ceramics with an R-curve behaviour from the total crack opening displacements. Two contributions to the total crack opening displacement field are addressed. First, the residual stresses occurring in the uncracked body are considered and then, the contact stresses generated by preventing crack face penetration are computed. The COD solution resulting from the superposition of residual and contact displacements is given and an analytical expression is provided. Near-tip displacements are represented by the first terms of series expansions. As an example of application, an evaluation of the actual stress intensity factor is presented for a window glass 1 h after Vickers indentation.     

Collinear cracks in a layered half – plane with a graded nonhomogeneous interfacial zone – Part II: Thermal shock response

In Part I of the paper, the problem of collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone was investigated under mechanical loading and the cracking behavior was addressed by evaluating the stress intensity factors as functions of various geometric and material parameters. In Part II, the solution framework is extended to the problem of thermal shock on the basis of uncoupled, quasi-static thermoelasticity. The interfacial zone, in this case, is assumed to have the graded thermoelastic properties. Using the principle of superposition, a system of singular integral equations is solved subjected to equivalent crack surface tractions obtained form the transient thermoelasticity solution for a uncracked medium. Main results presented are the transient thermal stress intensity factors of collinear cracks to illustrate the parametric effects of geometric and material combinations of the layered medium with the thermoelastically graded interfacial zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

裂纹面荷载作用下多裂纹应力强度因子计算

该文基于比例边界有限元法计算了裂纹面荷载作用下平面多裂纹应力强度因子.比例边界有限元法可以给出裂纹尖端位移场和应力场的解析表达式,该特点可以使应力强度因子根据定义直接计算,同时不需要对裂纹尖端进行特殊处理.联合子结构技术可以计算多裂纹问题的应力强度因子.数值算例表明该文方法是有效且高精确的,进而推广了比例边界有限元法的...     

HYBRID-TREFFTZ FINITE ELEMENT FORMULATION FOR SIMULATION OF SINGULAR STRESS FIELDS

An equilibrium hybrid-Trefftz formulation based on the direct approximation of the stress and boundary displacement fields is presented. The general solution of the governing differential equations is used to approximate the stress field and the boundary displacements are represented by polynomial functions. When singular solutions are implemented to model local high stress gradients due to concentrated loads or to the presence of wedges or cracks, rational functions are used to approximate the boundary displacements in the neighbourhood of such singular stress points. The equilibrium conditions and the kinematic boundary conditions are locally satisfied. The remaining fundamental relations—the compatibility conditions, the static boundary conditions and the constitutive relations—are enforced in a weighted residual form so designed as to preserve the duality and constitutive reciprocity. The resulting governing system is symmetric and all intervening structural operators have boundary integral expressions. Numerical applications are presented to illustrate the performance of the formulation.