The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics

Summary A new method is proposed, by using some special quadrature rules, for the numerical evaluation of the general type of finite-part singular integrals and integral equations with logarithmic singularities. In this way the system of such equations can be numerically solved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation. An application is given, to the determination of the intensity of the logarithmic singularity in a simple crack inside an infinite, isotropic solid.With 1 Figure     

Closed-form solutions for finite-part integrals occurring in shear-mode fracture mechanics

Summary This contribution is a continuation of a recent paper by K. Mayrhofer and F. D. Fischer [1], deriving an analytical solution for a general two-dimensional finite-part integral appearing in tension-mode fracture mechanics. This paper corresponds to shear-mode fracture mechanics where three further types of finite-part integrals occur. Their closed-form solutions will be introduced on the basis of an inclined, elliptically shaped crack. In general they can be expressed in terms of powered input values and by means of the Gauss hypergeometric function₂ F ₁. Formulae for different special cases are listed in the Appendices. Their correctness has been checked by means of two different numerical methods.     

New aspects for the generalization of the Sokhotski—Plemelj formulae for the solution of finite-part singular integrals used in fracture mechanics

New aspects for the generalization of the Sokhotski-Plemelj formulae are investigated, in order to show the behaviour of the limiting values of the finite-part singular integrals, defined over a smooth closed or open contour. The new formulae are more complicated when some corner points are further included in the contour. Beyond the above, when the contour is infinite, then the limiting values of the finite-part singular integrals are calculated by using an additional method. An application of two-dimensional fracture mechanics is finally given, to the determination of the stress intensity factors near a straight crack in a bimaterial infinite and isotropic solid under antiplane shear.     

Calculation of fracture mechanics parameters for a general corner

The Reciprocal Work Contour Integral Method of Stern, et al. is extended so that computation can be made of fracture mechanics parameters at the corner formed by arbitrarily oriented stress free surfaces. Two numerical examples are given. In one, results are compared to those obtained using an overdetermined collocation technique.

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Résumé On procède à une extension de la méthode d'étude par intégrale de contour du travail réciproque, proposée par Stern et Al, en vue de calculer les paramètres de mécanique de rupture correspondant à l'angle formé par deux surfaces d'orientation arbitraire et libres de contrainte. Deux exemples numériques sont fournis, avec, pour l'un, une comparaison des résultats avec ceux que fournit une technique de collocation redondante.

     

The singular function boundary integral method for a two-dimensional fracture problem

The singular function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.     

Using the method of boundary integral equations for the numerical solution of volume problems of nonlinear fracture mechanics

A. A. Blagonravov Institute of Engineering, Academy of Sciences of the USSR, Moscow. Translated from Problemy Prochnosti, No. 7, pp. 3–8, July, 1988.     

A general solution procedure for fracture mechanics weight function evalution based on the boundary element method

This paper reports on the development of an efficient and accurate means for the direct computation of crack surface weight functions for two dimensional fracture mechanics analysis. Weight functions are mathematical representations which can be used to efficiently calculate stress intensity factors for a variety of crack loading and boundary conditions. The method is inherently capable of handling mixed-mode problems. The weight function capability is especially important for problems of fatigue crack growth modeling where the efficient calculation of stress intensity factors is crucial.The basis of the new formulation and numerical solution method is the boundary element method (BEM), as implemented for two dimensional fracture mechanics analysis. The paper will review the analytical formulation of the new BEM, the numerical solution algorithm, and a limited number of validation examples.     

Effect of thickness on fracture criterion in general yielding fracture mechanics

In this research work, the effect of thickness on fracture criterion is studied for extra deep drawn (EDD) steel sheets. Experimental results are generated on fracture toughness of EDD steel sheets using compact tension specimens and a ‘maximum load’ as a fracture criterion. Critical crack tip opening displacement (CTOD) is found with the help of three methods: plastic hinge model (PHM), crack flank opening angle (CFOA) and finite element model (FEM). The fracture toughness is found to increase with increase in thickness of specimens. The fracture behaviour exhibited characteristics of general yielding fracture mechanics.     

Evaluation of fracture mechanics parameters for a general corner using a weight function method

Summary The weight function method (WFM) has been used recently as a reliable tool for evaluation of fracture mechanics parameters, where cracks are represented by zero opening traction free surfaces. The purpose of this paper is to extend this technique to general opening corner problem. The two dimensional singular fields for displacements and stresses are introduced in terms of generalized Bueckner's strength. By means of eigenvalue analysis the stress intensity factors (SIF) are then formulated after appropriate splitting the regular stress and displacement fields into symmetric and antisymmetric modes. Using Betti's reciprocal theorem, a new expression in a more general closed form is derived for Bueckner's strength consisten with the given nonzero opening case. The potentiality of the method is demonstrated by a numerical example for =/2 corner problem. The stress intensity factor for the symmetric mode is evaluated by WFM and by a simple collocation procedure using both boundary element (BE) and finite element (FE) discretization.     

Review of new formats for fracture mechanics parameters

New methods to formulate fracture mechanics parameters are presented. These include the common format that relates the deformation behavior of a cracked structure or specimen to the deformation behavior of a tensile test and the concise format that suggests a shortened form of common fracture mechanics parameters to replace the usual polynomial formulations. These forms may give an easier way to use the fracture mechanics parameters in analytical calculations. __________ Translated from Problemy Prochnosti, No. 4, pp. 39–45, July–August, 2006.     

Application of conformal mapping for mesh generation in two-dimensional elasto-plastic fracture mechanics problems

Conformal mapping techniques may be used for the automatic generation of finite element meshes in two-dimensional fracture mechanics calculations. Using a Schwartz-Christonel conformal transformation, the conformal mapping leads to an optimum mesh in the crack region. This resultant mesh is refined around the crack tip, and becomes progressively more coarse at areas where regular stress fields are expected. The advantageous characteristics of the mapped mesh are verified by means of elasto-plastic finite-element analysis of a compact tension specimen and a center cracked plate.     

Surface integral finite element hybrid (SIFEH) method for fracture mechanics

An effective surface integral and finite element hybrid (SIFEH) method has been developed to model fracture problems in finite plane domains. This hybridization by (incrementally) linear superposition combines the best features of both component methods. Finite elements are used to model the finite domain (and eventually nonlinearity), while continuous distributions of dislocations (resulting in surface integral equations) are used to model the fracture (i.e. displacement discontinuity). This method has been implemented in a computer program and results of representative problems are presented: these compare very well with known solutions and they demonstrate the computational advantages of SIFEH over other numerical methods (including the individual components).     

Boundary integral equation fracture mechanics analysis of plane orthotropic bodies

In this paper, a quick and efficient means of determining stress intensity factors, K _I and K _II, for cracks in generally orthotropic elastic bodies is presented using the numerical boundary integral equation (BIE) method. It is based on the use of quarter-point singular crack-tip elements in the quadratic isoparametric element formulation, similar to those commonly employed in the BIE fracture mechanics studies in isotropic elasticity. Analytical expressions which enable K _Iand K _II to be obtained directly from the BIE computed crack-tip nodal traction, or from the computed nodal displacements, of these elements are derived. Numerical results for a number of test problems are compared with those established in the literature. They are accurate even when only a very modest number of boundary elements are used.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.     

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

This study presents a novel development of a new semi‐analytical method with diagonal coefficient matrices to model crack issues. Accurate stress intensity factors based on linear elastic fracture mechanics are extracted directly from the semi‐analytical method. In this method, only the boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw–Curtis numerical integration result in diagonal Euler's differential equations. Consequently, when the local coordinates origin is located at the crack tip, the stress intensity factors can be determined directly without further processing. In order to present infinite stress at the crack tip, a new form of nodal force function is proposed. Validity and accuracy of the proposed method is fully demonstrated through four benchmark problems, which are successfully modeled using a few numbers of degrees of freedom. The numerical results agree very well with the analytical solution, experimental outcomes and the results from existing numerical methods available in the literature.     

New analytical approximate solution of the generalised temperature integral for kinetic reactions

Simple closed-form expression of the generalised temperature integral in the basic equation to describe kinetic reactions for solid materials in linear heating process is always suitable for use in determining parameters. Many developed solutions only can give high accuracies on the general conditions. A new analytical approximate solution was deduced in this work. The deviations of this solution from the true value are fully analysed. This solution takes advantage in broader application conditions than other known solutions. The application of the new analytical approximate solution in austenite kinetic reaction in this work reinforces that austenite reaction rate is feasible to be given a priori.     

Experimental and finite element analysis of fracture criterion in general yielding fracture mechanics

Efforts made over the last three decades to understand the fracture behaviour of structural materials in elastic and elasto-plastic fracture mechanics are numerous, whereas investigations related to fracture behaviour of materials in thin sheets or general yielding fracture regimes are limited in number. Engineering simulative tests are being used to characterize formability and drawability of sheet metals. However, these tests do not assure consistency in quality of sheet metal products. The prevention of failure in stressed structural components currently requires fracture mechanics based design parameters like critical load, critical crack-tip opening displacement or fracture toughness. The present attempt would aim to fulfill this gap and generate more information thereby increased understanding on fracture behaviour of sheet metals. In the present investigation, using a recently developed technique for determining fracture criteria in sheet metals, results are generated on critical CTOD and fracture toughness. Finite element analysis was performed to support the results on various fracture parameters. The differences are within 1 to 4%. At the end it is concluded that magnitude of critical CTOD and/or critical load can be used as a fracture criterion for thin sheets.     

Energy-based r-adaptivity: a solution strategy and applications to fracture mechanics

This paper deals with energy based r-adaptivity in finite hyperelastostatics. The focus lies on the development of a numerical solution strategy. Although the concept of improving the accuracy of a finite element solution by minimizing the discrete potential energy with respect to the material node point positions is well-known, the numerical implementation of the underlying minimization problem is difficult. In this paper, energy based r-adaptivity is defined as a minimization problem with inequality constraints. The constraints are introduced to restrict the maximum distortion of the finite element mesh. As a solution strategy for the constrained problem, we use a classical barrier method. Beside the theoretical aspects and the implementation, a numerical experiment is presented. We illustrate the performance of the proposed r-adaptivity in the case of a cracked specimen.