A path-independent integral for the calculation of stress intensity factors in three-dimensional bodies C.M. Bainbridge," M.H. Aliabadi," D.P. Rooke* "Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton SO40 7AA, UK ''Structural Materials Centre, Defence Research Agency, Farnborough GUN 6TD, UK Abstract A path-independent surface integral is derived for the calculation of the stress intensity factor in three dimensions; it is similar to an existing contour integral developed for two dimensional problems. The integral is based upon a type of Green's function, which is used in conjunction with Betti's reciprocal theorem. The proposed integral is used to evaluate stress intensity factors for different benchmark problems, and the results are found to be in good agreement with the values found in previous analyses. 1 Introduction In the field of aerospace engineering it is imperative to be able to characterise a safe working lifetime for the various components. Inevitably, all components must be assumed to contain small micro-mechanical flaws which may initiate small cracks. Various loading regimes may cause these cracks to grow in size, until catastrophic failure occurs at some critical crack size. Linear elastic fracture mechanics postulates that the component lifetime is dependent upon a single quantity, the stress intensity factor (SIF), which is a function of geometry and of loading. This dependence has resulted in much work being done to find the SIF for various different geometries and loadings. A different stress intensity factor may be defined for each of the three basic modes of loading one symmetric, and two anti-symmetric modes denoted by KN where N = /,//or ///. A separate issue, which must be emphasised, is that many important structures are fully three dimensional, and admit no realistic two dimensional simplifications. Therefore, efficient methods to calculate the SIF for three dimensional bodies are of great benefit to the engineer. Traditionally, direct methods of calculating the stress intensity factor, such as from the crack opening displacement, rely heavily on accurate representation of the stress field near the crack front. This often proves difficult to achieve in practice, due to the unbounded nature of the stress field along the crack front. For this reason, interest has developed in using path-independent integrals, such as the J-mtegral developed by Rice [1]. Path-independent integrals offer the advantage that the elastic field values need not be calculated in the immediate
48 Boundary Element Technology vicinity of the crack front. However, the J-integral has the disadvantage that the three stress intensity factors are not given directly, but are coupled together. This necessitates the adoption of various special decoupling techniques. The presence of a third dimension further complicates this coupling of the SIFs given by the J-integral. Another approach which has gained currency over the last decade is that of weight functions, and the related topic of fundamental fields, as developed by Bueckner [2]. Fundamental fields are defined by Bueckner [2] to be elastic fields which satisfy the governing equations of elasticity theory, and also have a certain asymptotic behaviour on approaching the crack front. It should be noted that these fundamentalfields,although satisfying the elasticity equations, are unphysical, due to their unbounded strain energy along the crack front. The usual physical elastic field, with the standard asymptotic behaviour along the crack front, is termed a regularfield.unfortunately, there is a limitation to the direct applicability of fundamental fields to finite geometries, arising from the fact that they are defined only for an infinite elastic medium. Adopting the philosophy of both of the above approaches, it is possible to obtain a path-independent integral which uses directly the fundamental fields. A similar approach has been adopted before, in two dimensions, as for example by Stern et al. [4] and also by Zhang and Mikkola [3]. The approach followed here is derived from that used by Zhang and Mikkola in [3], and is a straightforward application of Betti's reciprocal theorem. 2 Derivation of the Path-Independent Integral Consider an arbitrary three dimensional cracked domain O, under two states of loading denoted by B and S in Figure 2.1. 0 may be a subregion of larger elastic R B State B State S Figure 2.1: The two loading states Figure 2.2: Point forces applied in state S domains R& and R^ which need not be identical, as long as 12 C R& and 12 C R^. State B is the actual physical loading state, consisting of a displacement field u, and a stressfieldresulting from a body force f, and boundary tractions t. Now define state 5 to be an auxiliary loading state in which 12 is considered to be cut from an infinite elastic body R^. The subregion 12 is subject to the restriction of the (Mode I) regular elastic field that acts throughout the infinite body; this regular field consists of stresses a' and displacements u'. Thus, 12
Boundary Element Technology 49 may be considered to be subject to boundary conditions u' (displacement), and F (traction), which are those values of u' and t' ((/ = a'n, where n is the normal to the boundary P) on the boundary I\ This regular field will first be used to derive a path independent integral, then later be subject to a limiting procedure, and its limit will be equated to a fundamental field (defined by Bueckner in [2], and known analytically) with a condensation point. The above cut-out procedure eliminates the difficulties caused by the fact that the fundamental field is defined only for an infinite elastic body. Since it is known that fundamentalfieldsmay be obtained by means of a limiting process (see for example [5]), the following procedure may be developed. Considerfirstthe loading of state S to consist of a doublet of equal and opposite opening point forces of magnitude F', one applied perpendicularly to each crack face at distance d from the crack front in a subregion 0 of an infinite elastic space RS, as in Figure 2.2. The displacement field in state S is denoted by u', and the stress field by </ (boundary tractions resulting from the stress field are represented by t'). Application of Betti's reciprocal theorem to the domain H bounded by a surface F, gives when both loading states are applied, F'UB + 1 1'. udf = j f. u'dft + / 1. ii'df, (2.1) r o r where UB is the crack opening displacement in state B at the position where the point force F' was applied in state 5. If d -> 0 and F' -» oo such that = RI = constant, (2.2) then the constant RI is known as Bueckner's field strength (by analogy to a similar constant in 2-D work, see for example [3]), and is defined in [6]. As d -» 0 the displacement UB (strictly UB is the difference in the displacements on the two crack faces) approaches the well-known limiting value on the crack front. Following Hartranft and Sih in [7], this is known to be given by the plane strain value: lim us = lim 2v(r = d,0 =,) = Kr, (2.3) where KI is the mode I stress intensity factor; E is Young's modulus; f/ is Poisson's ratio; and v is the theoretical displacement in the direction perpendicular to the crack face. Also, u' and t' approach the values for a fundamental field with a condensation point, denoted by u* and t* which can easily be derived for certain simple crack front configurations in the infinite cracked space R$. These are given by Bains, Aliabadi and Rooke [8] for the case of an rectilinear crack front in an infinite space, and by Sham and Zhou [9] for a penny shaped crack embedded in an infinite space. Without loss of generality, Bueckner's fundamental field strength may be chosen to be unity, that is,
50 Boundary Element Technology This gives, on neglecting body forces in state J9, (t.u* - t*. u)df. (2.5) ^ ' As the surface F is chosen arbitrarily, the above integral should be path independent; the above integral may then be evaluated for a given problem by any convenient method of quadrature. Note that, as implied above, the fields u* and t* should be chosen to be suitable for the problem at hand. For a problem in a finite body with a rectilinear crack front, it is natural to choose those fields derived for a rectilinear crack front. 3 Test Problems 3.1 Model I : Rectangular bar with a straight crack front The first test problem was that of a planar edge crack with a rectilinear crack front in a rectangular bar, which has a well established solution. In this test, the aim was to establish the validity of the integral for a simple well-known problem before extending the technique to more complex cases. The problem considered is that of a rectangular bar of width 6, thickness t and height 2/&, with a planar rectilinear edge crack of length a lying in the (z,z)-plane, as shown in Figure 3.1. The bar is subject to uniaxial tensile stress a along the y axis. In the example presented here the following ratios hold: a/6 = 0.5, t/b = 1.0 and h/b = 3.0. The numerical solution to the elastic boundary value problem was obtained using the commercial boundary element package BEASY [10]. The surface T was modelled using internal points, which were distributed to form a BEM-like discretisation of F into elements, and the stress and displacement field values were calculated at these internal points. The integral in equation (2.5) was then calculated using Gaussian quadrature over this discretisation of the surface F. The surface F was taken to be a sphere of radius r centred on the point P, the mid-point of the crack front; thus the SIF is obtained for that point. Results obtained using the above formulation are compared with the plane strain value obtained by Keer and Freedman [11], and are shown graphically in Figure 3.3. Note that the normalised SIF, Ki/(cr^/ a) is plotted. As can be seen, the agreement is acceptable, the difference being less than 2% for 0.1 < r/a < 0.8. It should be noted that for r/a > 0.6 there appears to be a slight divergence of the calculated stress intensity factor. This is probably due to the fact that as r/a increases, the contour F includes contributions from the surface- breaking region of the crack, where the stress intensity factor does not exist in the traditional sense. The source of the error for r/a < 0.1 is likely to be caused by the difficulty of calculatingfieldvalues accurately for internal points which lie close to the boundary.
Boundary Element Technology 51 3.2 Model II: Penny-shaped crack in a large cylindrical bar This problem consists of a cylindrical bar of cross-sectional radius R and height 2/i, containing a penny-shaped crack of radius a lying in the (x,y) plane, as shown in Figure 3.2. The bar is subject to uniaxial tensile stress a directed a -y Figure 3.1: Bar with straight fronted crack Figure 3.2: Bar with pennyshaped crack along the z axis. In order to approximate a penny shaped crack in an infinite bar, the following ratios were taken, a/r = 0.1 and h/r = 3. The surface T was taken to be a sphere of radius r centred on the point P of the crack front, which lies on the positive x axis. Results obtained for the above model are again compared in Figure 3.4 with the results given by Cartwright and Rooke in [12]. As can be seen, the agreement is acceptable, differing by less than 2% with Reference [12] for r/a > 0.2. 3.3 Model III: Elliptical crack in a large cylindrical bar The problem consists of a cylindrical bar of radius R and height 2/t, containing an elliptical crack of semi-major axis a, and semi-minor axis 6. The bar is subject to uniaxial tensile stress a along the z axis. The following ratios were used for computation: a/r = Q.l,h/R = 3. Two aspect ratios were considered for the ellipse: b/a = 0.5 and b/a = 0.9. The contour surface F was taken to be a sphere, centred on the various points P,Q etc. of the crack front, where the SIF was to be calculated. The ratio used for the integration contour was r/a = 0.1. A technique known as approximation of fundamental fields was used to model the fundamental field of an elliptical crack using that of a penny shaped crack. The penny shaped crack chosen was that which had the same curvature as the elliptical crack at the point where the SIF was required. Results obtained for the above model for both b/a = 0.5 and b/a = 0.9 are given in Figure 3.6, compared with results from Rooke and Cartwright [12].
52 Boundary Element Technology 2.85 CO 1.08 ***** 192 nodes ##### 788 node* Plane Strain 1.04 ** ft).2 1.02 Current Work Rooke * Cartwright oi.oo 2.65 0.00 0.20 0.40 0.60 0.80 r/a ratio i.oo 0.08 0.00 0.20 0.40 0.60 0.80 r/a ratio i.oo Figure 3.3: Normalised Mode I SIF for straight fronted crack Figure 3.4: Normalised Mode I SIF for penny shaped crack The results are seen to be acceptable, being within 2% of Reference [12]. 4 Discussion and Conclusions 4.1 Sources of error It may be noted that generally, as r/a» 0, the solution becomes less accurate. The most likely explanation of this fact is that for small values of r/a, some of the internal points at which u and t are calculated lie very close to part of the external boundary (namely the crack surface). This gives rise to numerical difficulties in calculating the solution at these internal points. 4.2 Possible improvements in accuracy To utilise the technique of approximation of fundamental fields, it is desirable to have small r/a radius contour surfaces. Various options are available to improve the accuracy of the BEM internal point solution on these small radius contour surfaces. The most obvious approach is to use afinerbem mesh for the stress analysis, which allows more accurate internal point results for points somewhat closer to the boundary. However, this causes a large increase in the number of degrees of freedom of the BEM problem, and hence a large increase in solution time. An alternative approach is to use more sophisticated integration techniques in the calculation of elastic field values for those internal points that lie close to the boundary, for example the work of Cruse and Aithal [14], and also Mi and Aliabadi [15].
Boundary Element Technology 53 1.00 -i 0.60 0 so GO QO Parametric angle of ellipse Figure 3.5: Elliptical crack in a large cylindrical bar Figure 3.6: Variation of SIF with angle along crack front for elliptical crack Acknowledgement This work was sponsored by the Defence Research Agency, Farnborough, U.K. British Crown Copyright 1995/DRA. Published with the permission of the Controller of Her Britannic Majesty's Stationery Office. References [1] J. R. Rice (1968), A path independent integral and the approximate analysis of strain concentration by notches and cracks. Trans. ASME, J. Appl. Mech., 35, 379-386 [2] H. F. Bueckner (1987), Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three-space, Int. J. Solids Structures, 23, 57-93 [3] Z. L. Zhang and T. P. J. Mikkola (1992), A simple path-independent integral for calculating mixed mode stress intensity factors. Fatigue Fract. Engng Mater. Struct., 15, 1041-1049 [4] M. Stern, E. B. Becker and R. S. Dunham (1976), A contour integral computation of mixed mode stress intensity factors, Int. J. Fract., 12, 359-368 [5] H. F. Bueckner (1989), Observations on weight functions, Engng Anal. Boundary Elements, 6, 3-18
54 Boundary Element Technology [6]R. Bains, M. H. Aliabadi, and D. P. Rooke, Fracture mechanics weight functions in three dimensions: subtraction of fundamental fields, Int. J. Num. Meth. Engng., 35, 179-202 [7] R. J. Hartranft and G. C. Sih (1977), Stress singularity for a crack with an arbitrarily curved crack front, Engng Frac. Mech., 9, 705-718 [8] R. Bains, M. H. Aliabadi, and D. P. Rooke (1992), Fracture mechanics weight functions in three dimensions: subtraction of fundamental fields, Int. J. Num. Meth. Engng, 35, 179-202 [9] T.-L. Sham and Zhou, Y. (1989), Computation of three-dimensional weight functions for circular and elliptical cracks, Int. J. Fract., 41, 51-75 [10] BEASY, Boundary Element Analysis System Software, Computational Mechanics, Southampton, U.K. [11] L. M. Keer and J. M. Freedman (1973), Tensile strip with edge cracks, Int. J. Engng Sci., 11, 1265-1275 [12] D. P. Rooke and D. J. Cartwright (1976), Compendium of Stress Intensity Factors, HMSO. [13] R. S. Bains, M. H. Aliabadi, D. P. Rooke (1992), Stress intensity factor weight functions for 3-D cracks infinitegeometries, in Advances in Boundary Element Methods for Fracture Mechanics, Eds. M. H. Aliabadi, C. A. Brebbia, Computational Mechanics Publications, Southampton, U.K. [14] T. A. Cruse and R. A. Aithal (1993) Non-singular boundary integral equation implementation, Int. J. Num. Meth. Engng., 36, 237-254 [15] Y. Mi and M. H. Aliabadi (1994), Semi-analytical integration method for 3D near-hypersingular integrals, in Proc!(?** Int. Con}, on Boundary Element Methods, pp 423-430, Ed. C. A. Brebbia, Computational Mechanics Publications, Southampton, U.K