SUMMARYThe present paper is concerned with the effective numerical implementation of the two-dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack-edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J-integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed-mode conditions.

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There are several methods of deriving the BEM formulations: Cruse and Rizzo[l3] used Betti's reciprocal theorem; Brebbia and Dominguez[l8] introduced the weighted residual concept into the derivation, and Jeng and Wexler[l9] used a variational formulation similar to that used in the finite element method. The boundary element formulations may be divided into two different but closely related categories. The first and perhaps the most popular is the so-called "direct" formulation in which the unknown functions appearing in the formulation are the actual physical variables of the problem. For example in elasticity these unknown functions are displacement and traction fields. The other approach is called the "indirect" formulation in which the unknown functions are represented by fictitious source densities. Once these source densities are found the values of physical parameters can be obtained by simple integrations. In this book only the direct boundary element formulation will be described and readers interested in the indirect formulation should refer to other text books [20][21][22][23][24][25] on the subject. THE BOUNDARY ELEMENT FORMULATION IN ELASTICITYThe direct boundary element formulation for elastostatic problems can be derived from Betti's reciprocal work theorem for two self-equilibrated states (u,t,b) and (u*,t*,b*): u and u* are displacements; t and t* are tractions (i.e. t.=(J .. n., where n. are the components of the outward normal); and b and b*are body to~~J. Betti's r~ciprocal theorem (see Appendix A) can be expressed as J b;ujdO + J t;ujdr = J bju;dO + J tju;dr , o r o r (4.1) where repeated suffix summation is assumed, the domain 0 with boundary r and the domain 0* with boundary r* encompass the states (u,t,b) and (u*,t*,b*) respectively.As shown in figure 4.1, the problem under consideration in domain 0 is contained within a general region 0*, having the same elastic properties.The state of displacement u;, traction t~ and body force b~ is chosen to correspond to a known solution of the governing Navier's equation (2.7). The solution chosen is that due to a unit point force applied to the body, namely J.lu~ kk + ---l!...-11* ki + 8(X-X')e. = 0 , where X,X'eO*, 8(X-X') is the Dirac delta function and the unit vector component e j corresponds to a unit positive point force in the i direction applied at X'. In twodimensional problems e j is a force per unit thickness and in three-dimensional problems is a pure force. The displacement and traction field corresponding to the solution of (4.2) can be written as are the aisplacements and tractions in the j direction at a point X due fu a unit point force acting in the i direction at X'. The body force component b~ (force per unit volume) corresponds to a point force and is given by b~ = S(X-X')ej .The Dirac delta function S(X-X') has the property J g(X)S(X-X')dQ(X) = g(X') 0.Therefore the rust integral in (4.1) can be written as J b~ujdo. = uj(X')ej . ui(X') = J Uij(X',x)tj(x)dr(x) -J Tij(X',x)uj(x)dr(x) + J Uij(X',X)bj(X)dO(X) , r r 0 ~n wher...

A boundary element formulation, which does not require domain discretization and allows a single region analysis, is presented for steady-state thermoelastic crack problems. The problems are solved by the dual boundary element method which uses displacement and temperature equations on one crack surface and traction and flux equations on the other crack surface. The domain integrals are transformed to boundary integrals using the Galerkin technique. Stress intensity factors are calculated using the path independent ,]-integral. Several numerical problems are solved and the results are compared, where possible, with existing solutions.

SUMMARYThe present paper further develops the boundary element singularity subtraction technique, to provide an efficient and accurate method of analysing the general mixed-mode deformation of two-dimensional linear elastic structures containing sharp notches. The elastic field around sharp notches is singular. Because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields, these singularities are subtracted out of the original elastic field, using the first term of the Williams series expansion. This regularization procedure introduces the stress intensity factors as additional unknowns in the problem; hence extra conditions are required to obtain a solution. Extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the Williams solution; the procedure can take into account any number of terms of the series expansion. The standard boundary element method is modified to handle additional unknowns and extra boundary conditions. Analysis of plates with symmetry boundary conditions is shown to be straightforward, with the modified boundary element method. In the case of non-symmetrical plates, the singular tip-tractions are not primary boundary element unknowns. The boundary element method must be further modified to introduce the boundary integral stress equations of an internal point, approaching the notch-tip, as primary unknowns in the formulation. The accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed-mode problems. New results are presented for the mixed-mode analysis of a non-symmetrical configuration of a single edge notched plate.

the point P'(x,-y) has field parameters ~j' £~j' uj and ~, ~ they [,,11,,11,,11] [ ~' ; l{ 2 } + + 1 [,,11,,11,,11 [,,11,,11,,11] Now, since the integration path is taken to be symmetrical with respect to the crack along the x-axis (y=O), the outward normal components (nx,n» at points p(x,y) and P'(x,-y) have the following relationships (5.86) By using this relationship, the traction fields in (5.84) at point P(x,y) can be related to those at P'(x,-y), that is (5.87)Other field parameters are also related as follows:r;,} = { ~'} {:~}=r~J} Therefore using the above relationships the J-integral can be decoupled into mode I and mode II components, hence ~ and Kn can be evaluated separately from (5.81).The J-integral being a energy approach has the advantage that elaborate representation of the crack tip singular fields is not necessary. This is due to the relatively small contribution that the crack-tip fields make to the total J (Le. strain energy) of the body. Further advantage of the J-integral approach is that it can be used as a post-processing procedure for the evaluation of stress intensity factors with the internal stress components oj' evaluated from (4.9) using the boundary values of the displacements and tractioJs. The strain components £ij are Numerical Fracture Mechanics 171 subsequently evaluated using strain-stress relationships in (2.10) for plane strain or (2.12) for plane stress. The partial derivatives of displacement 8U/8X may be evaluated accurately using the formula in (4.8) or alternatively a simple approximation formula [51] It is worth noting that the denominator in (5.91) may become zero, depending on the integration path; in such cases it is possible to use the following relationship2 xy 8yIn order to assess the accuracy of the J-integral approach described above, the problem of a central slant crack of length 2a in a rectangular sheet of width W, height 2W and angle of inclination a.=fI1' (see figure 5.9) subjected to tensile uniform stress was studied since alternative accurate results are available [52]. Table 5.3 shows the percentage error compared to [52] in the calculated values of stress intensity factors for a/W=0.1 and 0.2 using different size circular integration paths of radius r centered at the crack tip. The results were obtained using 58 quadratic elements to model the boundary and 18 internal points. The partial derivatives of displacements were calculated from equation (4.8) So far all the techniques discussed in this chapter, have been based on attempts to model the singular stress behaviour at the crack tip. In contrast, the subtraction of the singularity technique avoids the need for this difficult task; it removes the singular fields completely. This leaves a non-singular field to be modelled numerically-a much simpler task. This approach was first introduced into the two-dimensional boundary element formulation of potential problems by Symm[53] and Papamichel and Symm[54] who used constant elements to model a symmetrical slit. Later Xanthis et al [ll] used th...

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