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Computers & Structures, Vol. 4 pp. 141-754. Rrgnmon Press 1914.Printed in Great Britain AN IMPROVED BOUNDARY-INTEGRAL EQUATION METHOD FOR THREE DIMENSIONAL ELASTIC STRESS ANALYSIS THOMAS A. CRUSE? Department of Mechanical Engineering, Carnegie Institute of Technology, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A. Abstract-An improved numerical implementation of the boundary-integral equation method for three dimensional stress analysis is reported. The new implementation models the boundary data as piecewise- linear variations over the boundary segments. As with all boundary-integral equation models, a system of equations relating unknown boundary data to known boundury data is obtained. The new implementation is described mathematically and verified on several simple test problems. In addition the method is used to study a finite fracture specimen used in material testing. The numerical results and computer run times are compared to an earlier version of the boundary-integral equation method. The results show significant improvement in accuracy for comparable run times for most problems. 1. INTRODUCTION THE report concerns the development of an improved version of the Boundary-Integral Equation (BIE) method of stress analysis for three dimensiona elastic bodies. The report deals with the mathematical basis of the improvement, but also compares the improved version of the BIE method to an earlier version [l]. As described in [l], the BJE method of elastic stress analysis is an efficient, yet general method of analysis, The basis of the BIE method is the development of a boundary con- straint equation which relates all boundary displacements to all boundary tractions. The boundary constraint equation applies regardless of the boundary shape (for well defined boundaries) and boundary conditions. For many exterior boundary value problems (infinite body), only the interior boundary need be treated. Thus the dimension of the problem is reduced by one, allowing for efficient modelling schemes for analysis. Due to the reduced dimensionality of the problem, modelling is considerably simpler than for finite element models which discretize the volume of the body. In the previous numerical implementation of the BIE method (referred to herein as 3D1, [I]) the boundary is modelled by piecewise flat triangular segments. Over each segment the boundary data is assumed to be represented by a constant value, referred to the segment centroid. The model corresponds to using the first term in a Taylor series expansion of the boundary data. The improved model, a second generation effort referred to herein as 3D2, makes use of the linear terms from the Taylor series expansion in the plane of the triangular boundary segment. As shown below, the data is by necessity referred to the nodes of the boundary segment map. The next section develops the mathematical basis of the 3D2 model using standard indicial notation with implied summation on repeated indices. The third section describes a series of test problems, many solved by both BIE models: 3Dl and 3D2. The problems t Currently, Pratt and Whitney Aircraft, East Hartford, Connecticut 06108, U.S.A. 741
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  • Computers & Structures, Vol. 4 pp. 141-754. Rrgnmon Press 1914. Printed in Great Britain

    AN IMPROVED BOUNDARY-INTEGRAL EQUATION METHOD FOR THREE DIMENSIONAL ELASTIC STRESS ANALYSIS

    THOMAS A. CRUSE?

    Department of Mechanical Engineering, Carnegie Institute of Technology, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.

    Abstract-An improved numerical implementation of the boundary-integral equation method for three dimensional stress analysis is reported. The new implementation models the boundary data as piecewise- linear variations over the boundary segments. As with all boundary-integral equation models, a system of equations relating unknown boundary data to known boundury data is obtained. The new implementation is described mathematically and verified on several simple test problems. In addition the method is used to study a finite fracture specimen used in material testing. The numerical results and computer run times are compared to an earlier version of the boundary-integral equation method. The results show significant improvement in accuracy for comparable run times for most problems.

    1. INTRODUCTION

    THE report concerns the development of an improved version of the Boundary-Integral Equation (BIE) method of stress analysis for three dimensiona elastic bodies. The report deals with the mathematical basis of the improvement, but also compares the improved version of the BIE method to an earlier version [l].

    As described in [l], the BJE method of elastic stress analysis is an efficient, yet general method of analysis, The basis of the BIE method is the development of a boundary con- straint equation which relates all boundary displacements to all boundary tractions. The boundary constraint equation applies regardless of the boundary shape (for well defined boundaries) and boundary conditions. For many exterior boundary value problems (infinite body), only the interior boundary need be treated. Thus the dimension of the problem is reduced by one, allowing for efficient modelling schemes for analysis. Due to the reduced dimensionality of the problem, modelling is considerably simpler than for finite element models which discretize the volume of the body.

    In the previous numerical implementation of the BIE method (referred to herein as 3D1, [I]) the boundary is modelled by piecewise flat triangular segments. Over each segment the boundary data is assumed to be represented by a constant value, referred to the segment centroid. The model corresponds to using the first term in a Taylor series expansion of the boundary data.

    The improved model, a second generation effort referred to herein as 3D2, makes use of the linear terms from the Taylor series expansion in the plane of the triangular boundary segment. As shown below, the data is by necessity referred to the nodes of the boundary segment map.

    The next section develops the mathematical basis of the 3D2 model using standard indicial notation with implied summation on repeated indices. The third section describes a series of test problems, many solved by both BIE models: 3Dl and 3D2. The problems

    t Currently, Pratt and Whitney Aircraft, East Hartford, Connecticut 06108, U.S.A.

    741

  • 742 THOMAS A. CRUSE

    range from simple cube verification problems to the significant geometry of a finite fracture specimen. Aspects of numerical accuracy and computer run times are given for all problems.

    2. LINEAR VARIATION MODEL

    2. I. Review of the basic integral equations The basic integral equations for the linear elastostatic problem have been reported for

    the three dimensional problem [I] but are reviewed here for completeness. The problem that is posed is the determination of the stresses and/or displacements at any field point p(x)&, due to imposed boundary conditions. The boundary conditions may be of the Dirichlet type,

    or of the Neumann type

    Lri(p)=gi(p)3 pEs,, (1)

    li(P)=Bij(P)n,j(P)=l?i(P), PES, (2)

    where S,+S,=S; or some physically plausible mixture of the two types at the same boundary point.

    It can be shown that the Somigliana identity for the interior displacement

    I&)= - s

    uj(Q)Tj(Pt 0) dS(Q) + s

    t,i(Q)Uij(p, Q) dS(Q) (3) .i s

    satisfies the Navier equations of equilibrium

    l/(1 -2V)Ui, ij+U,j, ii=O (4)

    for interior points per. The indentity (3) is based on the Betti reciprocal work theorem and the solutions to the fundamental problems of three orthogonal point loads in an infinite body. These solutions are given by the kernels Uii(p, Q) and Tij(p, Q) as described in [l].

    While (3) is a solution to (4) the entirety of boundary data required by (3) is not known, a priori. The strategy of the Boundary-Integral Equation (BIE) method is to let p(x)+P(x) in (3). Due to the discontinuous nature of the first integral in (3) the limiting form for p(x)-+P(x) is given symbolically by

    .

    (Cij +Sij)uj(P) + J Ilj(Q)Tij(P, Q) d'(Q)= p lj(Q)Uij(Pv Q) dS(Q). (5) s J s The integrals in (5) are Cauchy principal value integrals [2]; the tensor Cij is equal to - 6ij/2 if P(x) is at a smooth surface. The case when P(X) is at an edge or a corner is dis- cussed below.

    The BIE (5) has been shown [I] (see also [3-51) to be a suitable basis for the solution of very general boundary value problems in elasticity. The scheme is to replace the continuous definition of the boundary data in (5) by some discrete variation that will reduce (5) to a set of linear algebraic equations. For example, in earlier reports, the boundary is replaced by N segments over each of which the boundary data is assumed to be constant.

  • An Improved Boundary-Integral Equation Method 143

    In what follows the piece-wise constant model of the boundary data is replaced by a higher order representation. Specifically, the boundary is replaced by N piecewise flat triangles over which the boundary data varies linearly. A square system of equations is obtained and can be used to solve the same general boundary value problems. The solution method is verified on a number of test problems and shows an improved accuracy over the earlier model.

    2.2. Linear variation of boundary data

    In order to evaluate the integrals in (5) it is convenient to make use of a simple local coordinate system illustrated in Fig. 1. For the current discussion let G(x) be any scalar function which is to vary linearly over the mth triangular segment shown in the Figure. For such a variation it will be shown that the function G(x) is known for all points Q(x)EA& if the nodal values G(k), k=l, 2, 3, are known.

    FIG. 1. Local coordinate system for mth segment.

    Let G(x) be expanded in a double Taylor series in the plane of the mth segment. For the assumed variation of G(x) all derivatives of higher order than the first order are zero. Taking node 1 to be the reference value

    aG aG G(x)=G(l)+ z(&Q-cll)+ ~(~ZQ-~21) (6)

    where clQ, czQ are the local in-plane coordinates of x; Cl 1, Czl are the local in-plane co- ordinates of the first node of the segment.

  • 744 THOMAS A. CRUSE

    These differences in the nodal values of G(x) may be determined by the vector operations

    dG -1 G(2)-G(l)=VG*F,= ili,F,, + $f,,

    2

    *

    G(3)-G(l)= -VG.F,= - $,F,, +$I,, -2

    (7)

    In the local coordinate system, equation (7) may be solved for the two derivates of G(x) to obtain

    -2Ag =[G(2)-G(l)]F,,-[G(3)-G(l)&,

    2/lg =[G(2)-G(l)]F,,-[G(3)-G(l)]&, (8) 2

    where A is the area of the lath triangular segment. Substitution of (1) into (6) results in G(x) being known in terms of the nodal values G(k) and the geometry of the segment. A similar series of steps may be followed using G(2) and G(3) as reference points in (6); these results may be added to (8) to obtain the balanced result

    G(x)= i (i3+[FnzI,,-Fl,lizn,]/2A-[I,,i,a-F,,I,,]/2A)G(k)) k=l

    (9)

    where cl,,,, cl,,, are the local in-plane coordinates of the centroid of the mth segment. The result in (9) can be applied to each component of the boundary displacement and

    boundary traction vector over the rnth triangular segment. At this point it is convenient to introduce the following abridged notation

    Substitution of (9) into (5) results in a new approximation for the BIE; letting m = I, , . . , M for M nodes and N segments we obtain

    (Cij +~ij)u~kn) + tl k$l ~,(n)uj(k.)] us, JTj(kn, Q) WQ) 3

    4 Uj(k) s Ck(OT,j(knv Q) dS(Q) k=l AS, II N 3

    =

    + Bk(n)~j(kn) 1 s Uij(kntv QldS(Q) n=l k=l AS.

    tj(k,) s

    Ck(~)~,,(km Q> WQ) . A.% II (11)

  • An Improved Boundary-Integral Equation Method 745

    The points k,, k, in (11) refer to nodes where adjoining segments meet. As in the earlier approximation it is possible to evaluate the integrals in (11) in closed form.

    While the form of (11) is more complex than the previous model [l] little extra effort is required to compute the additional integrals due to the linear variation. However several non-trivial questions arise upon further study of (11). Any body which is not a smooth surface necessitates placing nodes at corners and edges; thus C, must be computed. Further, while we appear to retain the notion of equal numbers of nodal values of traction and displacements in (1 l), only the displacements are continuous over the entire surface. That is, there is one unique value of u,(k,) for each node. However, the tractions are generally not continuous, particularly when the body contains edges. Each of these two items, plus the evaluation of the principal value integrals in (11) will be discussed separately below.

    2.3. Algorithm for discontinuous boundary tractions At any given node joining boundary segments there will be a single unique vector for

    the nodal displacement, uj(k,). To properly account for discontinuities of nodal tractions t,(k,) the following alternative boundary conditions are taken: Either,

    (1) at a node shared by several boundary segments only three traction components may be unknown; these components need not act on a single segment, however all other tractions at that corner must be known; or

    (2) Where more than one segment at a node has the same component of traction unknown the nodal traction components are assumed to be equal. This is equivalent to forcing continuity of the unknown tractions between adjacent seg- ments.

    As a result of the two restrictions above, with the specification of the three displace- ment components at a node (the maximum number), only three unknown nodal tractions may result. The physics of the problem dictate which part of the boundary contributes the unknown tractions. The integrals in (11) are evaluated by picking a node k,, integrating segment by segment to assemble a row in the coefficient array, and then performing the same operation for node k,+ 1 until all rows have been assembled, The resulting system of equations from (11) will appear as

    [A~jl{xjI={Yi)* (12)

    The vector {xi> will contain 3M unknown nodal values of traction or displacement. The vector {y,} contains the product of the coefficients from (12) and the known boundary data. In assembling [Aij] the columns are scaled using the shear modulus in order to maintain the same order of magnitude for the unknowns. The resulting system of equations is square and is solved by standard methods.

    2.4. Evaluation of jumps in the double layer potential As discussed above, the double layer potential, given for reference here as

    Xi@) = s

    s PAQ)T,j(P, Q)dS(Q), (13)

    is discontinuous as p(x)+P(x), at the surface. In particular it has been shown [l] that for P(x) at a !lat surface

    X*(P)= -AQ)P+ s s~~(Q)T,j(P* Q)WQh (14)

  • 746 THOMAS A. CRUX

    In the current context where P(x) is placed at nodes (corners) of the boundary segments, the surface at P(x) is often discontinuous; i.e. is at a corner or an edge. When the surface is not flat at P(x), (14) is no longer valid.

    The evaluation of the jump in (13) for corners has been obtained in closed form for two dimensional problems [6]. However, in three dimensions the necessary integrals for evaluating the jumps in (13) are intractable. Thus, some indirect means must be employed for evaluating the jump in (13) at corners and edges.

    It has been shown [l] that the traction free problem

    .

    (Cii +dij)Uj(P) f s lij(Q)&j(P, Q) dS(Q)=O (15)

    admits non-trivial solutions which are rigid body motions of the form

    UT(P) = Dj + EjikWiS,~ (16)

    where Dj is rigid translation and Oi is a small rigid body rotation. If oi=O and Dj is taken to be 6, j, 6,j, 6,j in turn (15) gives the result

    7ij(P, Q) JS(Q)=(j (17)

    which reduces to

    The integrals in (18) are already being calculated in ( I I ) for each P(X) at a node; thus Cij(k,) is obtained by summing the term

    Cij(k",) = - f s T&,,, Q) dS(Q); n # nt II = 1 AS, as the coefficient array in (12) is being assembled.

    2.5. Evaluation of the principal value integrals The integrals in (5) are to be evaluated in the sense of the Cauchy Principal Value [2];

    that is, the integral is evaluated by deleting a vanishingly small disk at the singularity. In earlier models of the BE method with piecewise constant boundary conditions these principal value integrals were calculated in a straight forward manner [l]. However, in the linear case, the singularity is at corners (nodes) of the boundary segments and not at the segment centroid. Thus some greater care must be exercised to properly evaluate the principal value integrals.

  • An Improved Boundary-Integral Equation Method 747

    The kernel of the double layer potential, T,#, Q), can be separated into two parts; the first part proportional to &/an; the second part given by

    T~(P, Q)=(l--2V)(njr, i-n,r, j)/[8~nr2(1 - v)] (20)

    For flat segments tJr/&z=O for P, Q in the same segment and then only the second part of the kernel (20) must be considered.

    Substituting (6) into the double layer potential for the mth segment we can obtain, using node 3 for reference

    -F&-j Ti:.(Ps, Q) ] dS@)- (21) As shown in [I] the first integral in (21) is given simply by the path integral around the segment shown in Fig. 2

    x:(m)=pi(3){(1-2v)i[8n(l-vn)E,J~dx,. (22) Figure 3 shows several segments joining at P(x); these segments need not be in a plane.

    For each radial line from P(x), (22) contributes two terms for the two adjoining segment and each term will be equal but opposite in sign. Because the density p,(Q) is continuous the contributions to (24) along radial lines for adjacent segments will cancel.

    .si

    2

    \

    a

    I

    0 7 0 pm Fro. 2. Geometry for principal value integrals. FIG. 3. Adjoining segments at principal value node,

    Taking the arc surrounding P(x) to be made up of circular arcs of equal radius, 6, for each boundary segment in Fig. 3, the total contribution to (24) at P(x) for all adjoining segments will be zero as

    f d&=0. s

    (23)

  • 748 THOMAS A. CRUSE

    The result (23) also assumes ar( Q) to be continuous at P(x). The total contribution of (21) for P(x) and Q(x) in the same boundary segment may be obtained simply by integrating (21) from corner 1 to corner 2 for the tnth segment.

    3. NUMERICAL RESULTS

    3.1. VeriJication problems of a loaded cube

    Three verification probiems were studied; the geometry and boundary segments are given in Fig. 4(a), (b); the boundary conditions for the cube are shown in Fig. 4(c). The elasticity solution for Problem 1 shows that the reaction stresses and the u, displacements vary linearly; however the displacement u,, is not linear and the solution will be approximate. The cantilever problems in Fig. 4(c) involve even more serious approximations due to the shear deformations and because a parabolic shear distribution at x=2 in. cannot be modelled exactly.

    (2,0,01 ---_ x

    z/ top, 2 ) (2,031

    Ra. 4(a). Test problem geometry.

    FIG. 4(b). Test problem boundary segments.

    Fra. 4(c). Beam boundary conditions for test problems.

  • An Improved Boundary-Integral Equation Method 749

    y 0.5

    0 Interior points

    4 / 1000 psi

    FIO. 5. Bending stresses---problem 1.

    Figure 5 is a plot of the stresses 6, computed at various locations including interior and boundary points. As expected, the correspondence with the known solution is excellent. Furthermore the displacement uY of the middle point on the x=2 in. face was within 4 per cent of the beam theory result; the displacement U, of the top and bottom edges of the same face are within 3 per cent of the beam theory results.

    ax /IO,000 psi

    - Txy /IO,000 psi

    FIG. 6. Bending and shear stresses-problems 2, 3.

    Figure 6 shows the shear and bending stresses along the vertical center line of specimen at midlength for Problems 2 and 3. Further, results for Problem 3 were also computed using t he earlier piecewise constant model 3Dl [l], for comparison. While Fig. 6 does show some deviations for 3D2 from the strength of material predictions, the greatest source of the deviation is due to the manner of loading.

    The data computed by 3Dl is seriously in error; in fact the results are essentially meaningless. The resultant tractions at the fixed end show shear equilibrium but very poor bending stress reactions. This result is typical of the experience of this user for problems where sign&ant bending is involved, as is discussed further in Section 3.2.

    The displacements predicted for the fixed beam in Problems 2, 3 is given in [8] by the equation

    a=%[ I+@71 (;~+o.lo(y]. (24)

  • 750 THOMAS A. CRUSE

    For the geometry shown the computed result from (24) is 6=0*0129 in. The result for the center point on the x=2 in. face is 6=0.0123 in. for an error less than 5 per cent. The vertical reaction is in error 4.9 and 6.7 per cent respectively; the maximum reaction normal stresses are in error by l-2 and I a8 per cent respectively.

    3.2. Finite fracture specimen The last problem studied is the stress analysis of a cracked specimen, similar to the

    compact tension specimen used in fracture testing [lo]. The same specimen was analyzed using 3Dl [3], however the model used herein differs somewhat in element arrangement from that used in [3]. The basic idealized problem is shown in Fig. (8)a. The purpose of the current study is to compare the accuracy (and run times) of 3D 1 and 3D2 for computing the stress intensity factor ICI (for further details see [3, IO]) along the center plane, z=O.

    As discussed in [ll] the BIE method has a significant modelling problem for crack geometries. If the crack in Fig. 8(a) is modelled exactly the BIE method is unable to distinguish between the two crack surfaces and the numerical problem is singular. Two means for avoiding the difficulty are described in detail in [I 11. One consists of modelling the crack by a gap with some rounded closure at the crack tip. The recommended relative gap between the crack faces is 6 =2a/25. Such a model for 3Dl and 3D2is shown in Fig. 8(c) where the faces of the symmetric quarter of the geometry are shown in one plane.

    Y

    t

    20125 r

    i >L

    8 x 2H

    -ta----l

    Thickness = B

    P +

    FIG. 8(a). Finite fracture specimen-model 1.

    FIG. 8(b). Finite fracture specimen-model 2.

  • An Improved Boundary-Integral Equation Method

    - - - Symmetry lines

    FIG. 8(c). Boundary segments for model 1.

    -- - Symmetry line

    FIG. 8(d). Boundary segments for model 2.

    751

    The alternative modelling scheme is to replace the interior plane ahead of the crack (y =O; x > a) by a boundary with boundary conditions rxr =0, r,,, =0, and u,=O(see Fig. 8(b)). This modelling scheme allows the crack to lie in the plane y=O. However, the continuum model of the plane ahead of the crack has been discretized: Instead of allowing the stresses to vary smoothly as x+u+, the stresses will vary in a piecewise linear fashion. Further,

  • 752 THOMAS A. CRUSE

    while a stress singularity exists at x=u, the numerical model will compute a finite stress for that location.

    The model using symmetry with respect to the crack plane and 6=2a/25 was solved using both versions of the program. The two problems are referred to as 3Dl-1 and 382-2. The second model using the plane ahead of the crack as a physical surface is referred to as 3D2-3 (see Fig. 8(d).

    The geometry for the finite fracture specimen is the same as used in [3] except for 6=a/lOO in [3]; that is, a/w=H/w=B/w=O*S where B is the specimen thickness. The plane z=O is used as a plane of symmetry for both models. The loading is chosen as uniform shear of amount z=P/BH. P was the same for both models. When a loading function of the same form as Problem 2, Fig. 4c was used the change in stresses for the second model was altered a negligible amount. However, for 6=2a/25 the form of the loading function did effect the results by as much as 15 per cent. The results for the uniform loading are reported herein.

    The stress intensity factor for this geometry is given by [lo] as

    K,/rJa=l-2. (25)

    Figure 9 shows a plot of the two center-line stress concentration factors ox/r, 0,/r for the three problems. The asymptotic value of the stress concentration factor for plane strain, using (25), is also plotted in Fig. 9. Due to the large bending stress (a, becomes negative at about r/a=l) the two stress ratios have not converged to the singular term asymptote. However, models 3D2-2 and 3D2-3 are in reasonable agreement although the stresses for 3D2-2 are higher. This result was also found in two dimensional problems [6] where the open crack model elevates the stesses away from the crack tip. The three dimensiona] aspects of the stress field near the crack tip substantiated the results reported in [31.

    Plane strain asympta

    FIG. 9.

    The stresses predicted by 3Dl-1 are seen to be quite in error, on the low side. Since

    the interior stresses are computed from known tractions and unknown displacements, it

  • An Improved Boundary-Integral Equation Method 753

    must be the displacement solution to the BIE which is seriously in error, as seen in Fig. 10.

    r/a (8=2=0)

    a 6 2.0 in.

    DO10 -

    @003- I III 0.01 002Oe3004 010 0.2 03 0405 I.0

    r/a (8=r;Z=O)

    Fro. 10. Crack opening displacements.

    For the center-line nodes in the two models 3D2-2 and 3D2-3 the crack opening dis- placements in Fig. 10 were used to obtain KI estimates based on the limiting relation

    (26)

    The predictions for the stress intensity factors as well as the req uired computer run times are given in Table 1.

    TABU I. Finite fracture specimen

    Model Run times (min) Degrees-of-

    lVllrJa Delint Solver* freedom

    3Dl-1 4.38 14.0 5.2 351 3D2-2 7.05 17.8 1.5 219 3D2-3 7.16 16.5 3.2 291 Exact 7.20 - - -

    * Time used for peripheral storage is deleted; Univac 1108

    Table 1 shows that for the fracture problem much greater accuracy is achieved using the program 3D2 for the same amount of computer time. The reduction in degrees-of- freedom for the 3D2-2 model accounts for run time efficiency. In model 3D2-3 the accuracy is even better, with a very slight increase in run time. Model 3D2-3 has more degrees-of- freedom but only one plane of symmetry which reduces the necessary generation time. Thus the modelling philosophy in 3D2-3 commends itself for future fracture specimen studies.

  • 754 THOMAS A. CRUSE

    3.4. Conclusions The ability to modify the three dimensional BIE program to allow for linear variations

    in boundary data has been demonstrated through a series of problems ranging from trivial to complex. The results indicate that the 3D2 program is much more accurate in analyzing problems with significant bending than the piecewise constant model.

    While the generation of the known coefficient arrays is inherently more expensive in 3D2, referring unknowns to nodes rather than element centroids reduces the number of unknowns. In the fracture specimen study the improvement in solution accuracy is significant, at no rise in computer cost. Further, with the use of nodes as the solution points it is possible to control the location of solution data more systematically, resulting in improved means for studying three dimension variations in the stress intensity factor.

    Finally, due to the reduced number of unknowns inherent in the piecewise-linear model, it is possible to model fracture problems more accurately. Namely the crack plane can be discretized maintaining a flat, ideal crack model. While this results in discretized stresss near the crack tip, the crack surface displacements yield very accurate estimates of the stress intensity factor.

    Future efforts to improve the accuracy of the BIE method should concentrate on even higher order variation of the boundary data and curved boundary segments. Such effort has already had significant pay-off in two dimensional problems [14]. The emphasis, as in finite elements, will shift to efficient algorithms for performing the necessary discrete inte- grations numerically.

    Acknowledgements-The author is indebted to the Army Research Office, Durham for support of this effort through Grant DA-ARO-D-31-124-72-G3; to CarnegieMellon University where this work was performed; and to J. L. Swedlow for many comments and suggestions.

    REFERENCES

    [l] T. A. CRUSE, Numerical solutions in three-dimensional elastostatizs. Znt. J. So/ids Struct. 5, 1259-1274 (1969).

    [2] S. G. MIKHLIN, Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford (1965).

    [3] T. A. CRUST and W. VAN BUREN, Three-dimensional elastic stress analysis of a fracture specimen with an edge crack. ht. J. Fracture Mech. 7, 1-16 (1971).

    [4] T. A. CRUSE, Some classical elastic sphere problems solved numerically by integral equations. J. Appl. Mech. 39,272-274 (1972).

    [S] T. A. CRUSE, Application of the boundary-integral equation method to three-dimensional stress analysis. Computers 81 Structures 3, 509-527 (1973).

    [6] P. C. RICCARDELLA, An implementation of the boundary-integral technique for planar problems in elasticity and elasto-plasticity. Report SM 73-10, Mechanical Engineering Department, Carnegie- Mellon University, Pittsburgh, Pa. (1973).

    [7j V. D. KUPRADZE, Potential Methods in the Theory of EIasticity. Davey, New York (1965). [S] S. P. TIMOSHENKO, Strength of Materials, Vol. I, 3rd Edn., p. 175. Van Nostrand, New York (1955). [9] A. I. LURE, Three Dimensional Problems of the Theory of Elasticity. Interscience, New York (1964).

    [lo] W. F. BROWN and J. E. SRAWLEY, Plane Strain Crack Toughness Testing of High Strength Metallic Materials. ASTM STP 410, American Society for Testing and Materials, Philadelphia (1964).

    [ll] T. A. CRUSE, Numerical evaluation of elastic stress intensity factors by the boundary-integral equation method. In The Surface Crack: Physical Problems and Computational Solutions, (Edited by J. L. SWEDLOW), American Society of Mechanical Engineers, New York (1972).

    [12] T. A. CRUSE and J. L. SWEDLOW, Interactive program for analysis and design problems in advanced composites technology. AFML-TR-71-268 (1971).

    [13] B. GROSS, E. ROBERTS, JR. and J. E. SRAWLEY, Elastic displacements for various edge-cracked plate specimens. Znt. J. Fracture Mech. 4, 267-276 (1968).

    [14] P. hl. BESUNER, Pratt and Whitney Aircraft, East Hartford, Connecticut, Private Communication (1973).

    (Received 24 August 1973)