INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 38, 1681-1 701 (1995)

HYPERSINGULAR QUARTER-POINT BOUNDARY ELEMENTS FOR CRACK PROBLEMS

A. SAEZ*, R. GALLEGO AND J. DOM~NGUEZ

Eseuela Superior de lngenieros Industriales, Uniuersidad de Seuifla, Au. Reina Mercedes sjn, 41012-Seuifla, Spain

SUMMARY The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratores, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.

KEY WORDS: Boundary Element Method; fracture mechanics; hypersingular; Stress Intensity Factor

INTRODUCTION

There are two particular fields of research within the Boundary Element Method (BEM) that play an important role in the solution of linear elastic fracture mechanic problems. One is the study of different procedures for the evaluation of the stress and displacement fields near the tip of a crack and in particular of the Stress Intensity Factor (SIF); the other is the study of hypersingular boundary integral equations and the subsequent numerical approaches. This paper is at the merging point of those two research fields. It is dedicated to the study and effective implementa- tion of an approach for the analysis of crack problems and the computation of SIF, based on a mixed formulation including standard and hypersingular integral equations. On the way of doing so, a general method for the evaluation of hypersingular line integrals along curved quadratic elements is presented. This method allows the integration of hypersingular kernels along quarter-point elements at the crack tips from which the SIF are readily computed.

The standard BEM formulation has been successfully applied by several authors to the solution of linear elastic fracture mechanic problems.' - However, only symmetric crack prob- lems can be studied by simple analysis of a single region using the BEM. More complicated problems can be studied by the subregion method.' This method requires of artificial boundaries,

* Presently, on leave at Northwestern University

CCC 0029-5981/95/101681-21 0 1995 by John Wiley & Sons, Ltd.

Received 14 April 1994 Revised 28 August 1994

1682 A. SAEZ. R. GALLEGO AND J. DOM~NGUEZ

which are not uniquely defined, and may become rather complicated for crack propagation problems, where these boundaries have to be changed with time.4

The need for artificial subregioning or other more involved techniques for the solution of non-symmetric crack problems disappears when the hypersingular integral equations, arising when the derivative of the standard integral equation is taken, are used for analysis. This advantage and the accuracy of the results have made the hypersingular boundary element method (HBEM) a very fast growing technique5-' in spite of the fact that the integration of hypersingular kernels has required of a lot of attention from the researchers. After some pioneering work by Rudolphi et aI." it has been recently when Guiggiani et uL9*" and Guiggiani' 2,1 have presented a general approach to transform the hypersingular integrals into regular integrals and simple singular integrals which are computed analytically. A similar integration approach was followed by Mi and Aliabadi.I4 The basic ideas of that general method are followed in this paper to derive the actual expressions for the hypersingular integral appearing for the general two-dimensional (2-D) curved quadratic element.

Problems with the crack surfaces under equal pressure conditions can be solved with the sole use of hypersingular integral equations; however, in case of cracks under general loading conditions, a mixed formulation is required. The standard displacement integral equation is applied on one of the crack surfaces and the hypersingular traction equation on the other.I5

The present paper deals with the effective use of the mixed boundary element formulation for linear elastic crack problems. The actual expressions for the regularization and analytical integration of the hypersingular integrals appearing for the general curved quadratic elements are presented. Discontinuous elements are used to satisfy the C' continuity requirement of the hypersingular integral equations. General fracture mechanic problems containing curved cracks can be solved with the proposed approach. The elements may have any general quadratic geometry defined by three nodes and the collocation points can be set at any position within the element. For the first time in the context of a hypersingular formulation, straight-line quar- ter-point elements, which reproduce the f i displacement along the crack, are used to model the vicinity of the tips.

The SIF are computed from the crack opening displacements at nodes which are extremely close to the tip. Several problems with different crack geometries are analysed. It is shown that the present approach is simple and produces very accurate results of very little mesh dependence.

BASIC EQUATIONS

The mixed formulation for the BE solution of crack problems is based on both the displacement and traction integral representations. In the case of zero body forces, the displacement integral representation for a point with coordinates 5 can be written as

where f stands for the Cauchy Principal Value (CPV) integral, 1 and k are Cartesian components, cI, = d,, when 5 is an internal point, cIk = 1/2 d,, when 5 is a smooth boundary point, pk(x) and uk(x) are the k component of the traction and displacement, respectively, at boundary point x, and &x, 6) and u:(x, 5) represent the fundamental solution traction and displacement, respectively, in the k direction at a boundary point x when a point load in the I direction has been applied at

CRACK PROBLEMS 1683

point 5.

where r = Ix - GI, n is the unit normal at boundary point x, and p and v are the material shear modulus and Poisson's ratio, respectively.

The traction integral representation at point 5 with normal N is given by

where stands for the Hadamard Finite Part (FP) integral, CIk = 6 l k when 5 is an internal point, Clk = 1/2 6 l k when G is a smooth boundary point, and stms and d L k are linear combinations of derivatives of pk and u k , respectively, with the following expressions:

+ 2 v ( n l r.rn r,k + nrn r,l r,&) + (l - 2v) (2nk r, l r,m + nm 81k + n1 & n k )

(6) 1 - ( 1 - 4v) n, 61,

Equation (4) has been obtained by differentiation of the displacement integral representation for an internal point 5 (equation (1 ) ) to derive the stress tensor al,(5), in accordance with Hooke's law, and from this tensor, the tractions on a surface defined by its normal components Nrn((). A limit process is carried out to obtain equation (4) for a smooth boundary point.

The Finite Part integral can be defined (Figure 1) as

fr s ~ ( x , 5) uk(x) dUx) = S k k ( X , 6) u&(x) dr(x) -k u k ( 5 ) 7 where

Additional information on the derivation of equation (4) and the Finite Part integral definitions (7) and (8) can be found in Guiggiani et aL9 Those authors, however, prefer to leave the limits of the integrals explicit not using the Hadamard Finite Part notation.

Figure 1. Integration contour for a smooth boundary point

1684 A. SAW R. GALLEGO AND J. DOM~NGUEZ

Several aspects related to the above traction representation should be pointed out as follows.

(1) The displacement must be differentiable at 6 with its first derivatives satisfying a Holder condition: uk E C' at 6. This is an important condition in the selection of the boundary discretiz- ation and collocation points when using the traction representation (4) for a BE implementation.

(2) At non-smooth boundary points, it is not possible to obtain an integral representation of the stress tensor only in terms of boundary displacements and tractions following the usual limit process from internal However, such a process could be followed for the displace- ments derivatives and after inverting the c matrix (which is possible in general) obtain an integral representation of the stress tensor by linear combination of the displacement derivatives repres- entation.

(3) The use of the Finite Part and Cauchy Principal Value concepts is not really necessary for the hypersingular formulation. In fact the integral representation (4) can be written in a general form in terms of limits of integrals when E + O . A proper limiting process assures that no unbounded terms ultimately arise. Singular terms may result from integrals over small parts of the boundary close to the collocation point, but those terms are always cancelled out when the rest of the boundary is taken into account. The Finite Part and the Cauchy Principal Value concepts are used in this paper only for the sake of brevity in the equations.

(4) The curve l-, used to remove point 6 from the boundary may have any shape provided it is a regular curve. A circle is selected because of its simplicity. By doing so e, is symmetric around 6. Should not e, be symmetric around 6, the Finite Part definition (7) should be reviewed to include a logarithmic singular term which appears in the limit p r o c e ~ s . ~ ~ , ' ~ ~ ~ ~ In such event of a curve r, producing two different values of e, at both sides of 6 also the integrals on the left-hand side of equation (1) and on the right-hand side of equation (4), resulting after the limiting process should be reviewed. They would not match the Cauchy Principal Value definition and equations (1) and (4) could only be written in the above form if the CPV definition for those integrals were also reviewed to include a logarithmic singular term. l o

BOUNDARY ELEMENT DISCRETIZATION

The mixed BE formulation for crack problems is based on setting the displacement integral representation on one of the crack surfaces and the traction integral representation on the other. By doing so one obtains two different equations for each couple of points with the same co-ordinates and no splitting of the domain into subregions is required to solve the problem. Only the displacement integral representation is written for the remaining external or internal bound- aries of the body. The resulting mixed integral equation can be solved numerically after a BE discretization.

There is a C' continuity restriction of the displacement components which must be satisfied by the elements along the crack. To do so, the crack boundaries are discretized with discontinuous or non-conforming quadratic elements; i.e. quadratic elements where the two extreme collocation nodes are shifted towards inside of the element (Figure 2). A similar approach was followed by Mi

collocation node geometrical node

N1 NC1 N2ENC2 NC3 N3 (3 >< 0

#=--1 #=t, (20 # = / F : f = 1

Figure 2. Discontinuous (non-conforming) quadratic element (5 = dimensionless co-ordinate; - 1 d 5 < 1)

CRACK PROBLEMS

- -

I 8

1685

I I I I

continuous

Figure 3. BE discretization for a cracked domain

and Aliabadi for three-dimensional problems. There are alternatives to discontinuous elements in order to satisfy the continuity requirement; for instance, Hermitian elementsI6 and Overhouser elements.” The crack tip element used in this work is a discontinuous, quarter-point quadratic element which is able to represent the (r)lI2 displacement variation along the crack.

Standard continuous quadratic elements are used for the rest of the boundaries, where only the displacement integral representation is used, except for the cases when there is an intersection between a crack and a normal boundary (Figure 3). In such case a semi-discontinuous element is used on the normal boundary to avoid a common node at the intersection.

The geometry of the discontinuous quadratic elements is represented using the same shape functions of the standard quadratic elements (Figure qa)), i.e.

xk = x: 41G + x: 4 2 G + x,” 4 3 G (9)

where xk is the k Cartesian component of a point along the element, x:, x:, x: the k Cartesian component of nodes N1, N2 and N3, respectively and

(10) 4 -1 1G - 2t(< - 4 2 6 = (l - t2b 4 3 G = tr(r + 1)

with t being an element dimensionless co-ordinates (t = - 1 at N1, < = 0 at N2, t = 1 at N3). The boundary variables U k and pk are represented in terms of their values at the collocation

nodes NCI, NC2, NC3 and the shape functions c$l, 42 and 43 (Figure qb)). For instance,

U k = u : $ l +u,242 + .:43 (1 1)

41 = a 1 t 2 + h t , 42=a2t2+b25+1, 4 3 = a 3 t 2 + b 3 5 (12) where

with al = K / 5 , ; u2 = l/(tl t3); a3 = - K / t 3 : b, = - Kt3/ t l ; b2 = - l/tl - l/t3; b3 = K t1/t3, K = l/(tl - t3); and tl and t3 the values of 5 at NC1 and NC3, respectively.

The elements containing the crack tip are simple straight line discontinuous quadratic elements with the mid-node located at one quarter of the element length (Figure 5). It can be easily shown by substitution of this geometry into equation (9) that in such case there is the following relation between the internal dimensionless coordinate t and the distance to the crack tip along the element F:

(13)

1686 A. SAEZ. R. GALLEGO AND J. DOM~NGUEZ

I

#=-1 #=#, #=O #=C3 #=1 Ni N3

#=-1 [= 1

@ Figure 4. Quadratic interpolation functions for discontinuous elements. (a) Interpolation functions for the geometry.

(b) Interpolation functions for the boundary variables uk, pk

3L/4 1 L'4 P I N1 NCl N2 NC3 N3 ++-+

#= 1

Crack tip - Figure 5. Discontinuous quarter-point element

which substituted into equation (1 1) gives a representation for the displacements of the type

where dl, d2 and d 3 depend on the geometry of the element and the uk nodal values. This displacements representation contains the term able to reproduce the displacement behavior at the crack surfaces near the tip.

Displacements and tractions along a three-node quadratic element j are written as usual

CRACK PROBLEMS 1687

The discretized boundary integral representations at a point i can be written as

where equation (17) is written for all the boundary collocation nodes including the nodes along the crack and equation (18) only for the nodes along the crack. The integrals along the elements must be interpreted in the sense of CPV or FP when node i belongs to element j .

EVALUATION OF THE HYPERSINGULAR INTEGRALS

The hypersingular integral in equation (18) for the element j to which node i belongs is

where s: = sLk N,. Taking into account equation (16) and calling A = p/[2n(l - v)], one can write

sf1 = - 2 -[Nl r,l - 4r;l N,r,,] + 2nl rVl N,r,, r2 "I an ar

The integral of the first term is regular since nl r,l + - N1 r,l , &/an = O(r) and N, r , , = O(r) as r + 0. Therefore, it can be carried out using a standard Gaussian quadrature. The expression corresponding to s f 2 is the same as equation (20) with only a change in the subscripts, where 2 must be written instead of 1. The off-diagonal terms are

r2 " { ;: sf2 = sfl = - - 8 - r,l r,2 N,r., + N2 nl + N1 n2 + 2r,l r,2 N,nm } (21)

which behave regularly as r + 0 and can be integrated using a standard Gaussian quadrature. After substitution of equations (20) and (21) into equation (19) the only part which needs of

a special treatment corresponds to the second part of equation (20) and is of the form

where Nmnm = 1 + O(r2).

In order to transform equation (22) one may take into account the following (Figure 6):

which can be easily checked by taking into account that

. dr lim - = 1 r - 0 l d r l

and this derivative has a maximum at r = 0.

1688 A. SAEZ, R . GALLEGO AND J. DOM~NGUEZ

Natural Coordinate # along the element

# Collocation point "i"

Figure 6. Element geometry around the collocation point i"'7: ,g- $1

N2GNC2 N2ENC2 N2ENC2

NC1 N1 NC1 N1 NCl N1

Figure 7. Distance from the collocation point to the extremes of the elements. (a) Collocation point NCl. (b) Collocation point NC2. (c) Collocation point NC3

The singular integral is now written as

The first integral 1, is regular for r + 0. The second l 2 can be transformed with the aid of the first term of the Taylor's series expansion of the shape function 4q; i.e.

Thus, l 2 can be written as

= 1 2 1 + I 2 2 + I 2 3

where Izl is regular,

CRACK PROBLEMS 1689

(see Figure 7) and 1 2 3 can be easily computed after the following manipulation

where I# = (d4q/d<y and J ’ is the Jacobian of the transformation of the boundary co-ordinate r into E at node i. After equation (29), the last integral of equation (27) can be written as

EVALUATION OF THE SINGULAR INTEGRALS

In addition to the above s:,,,, there are other three singular kernels to be integrated in the mixed BE formulation. One is the Kelvin fundamental solution u:k which has a weak singularity of O(ln r), when r + 0. The other two p i and dkk have a strong singularity of O(r- l ) , r being the distance between the collocation point and the integration point. The integration of those kernels over the element which contains the collocation point can be done as follows.

The integral of the Kelvin fundamental solution in equation (1 7) for the elementj to which node i belongs is

(31) uf l 4 1 u:2 41 u:l 42 U f 2 4 2 U T l 4 3 u?Z 4 3 dT 1

It is immediate to see that after substitution of equation (2) into equation (31) the only singular terms are of the type

1 r

I’. = f, In - 4,dT

when 4, # 0 at the collocation point. This integral can be evaluated numerically using a special quadrature including a logarithm or analytically after the following regularization:

The first integral is regular and the second one I ; can be written as (see Figures 6 and 7):

Z; = f, In f sgn ($) dr = R 1 [l + In $1 + ,,[I + In $1 The first integral in equation (17) consists of terms of the type

(34)

where the first integral is regular and can be done numerically. The second one is only different from zero when 1 # k.

1690 A. SAEZ, R. GALLEGO AND J. DOMiNGUEZ

Figure 8. Co-ordinates at boundary point

If k = 1, I = 2, then (Figure 8):

x2 dx2 x1 dx, 1 dr2 dr nz) = - - + - - = - - = - r dT r dT 2 r d T dF ( r ,2nl - r

If k = 2, I = 1, then:

Therefore, the singular integrals, which come out from the last term of equation (35) when k # 1, are of the type

1 I" = jr, $q dr

which can be regularized as follows:

where 1; is regular and 1; is computed analytically.

the mid-node of continuous elements; see Figure 7). When the collocation node is inside the element (the three nodes of discontinuous elements and

R3 = In - R1

When the collocation node is an extreme node of a continuous element (Figure 9).

1'; =In R (41)

Note that the above expressions of Ti are valid for e, (Figure 1) symmetric around i. The third type of singular integral appearing in the formulation is (see equation (18))

where, according to equation (3,

All the terms in equation (43) including the product r,,,, N , can be integrated numerically since this product r,,, N, = O(r) as r + 0 and the corresponding integral is regular. The only singular

CRACK PROBLEMS 1691

N C1

Figure 9. Continuous element

integral remaining is

when the collocation node is the ‘qth’ of the element. Taking into account equations (36) and (37), it can be regularized as follows:

where the last integral, which is the only singular one, has been previously studied (equations (40) and (41)).

STRESS INTENSITY FACTOR COMPUTATIONS

It was mentioned above that in order to have the proper J- displacement representation the elements containing the crack tip are straight line discontinuous with the mid-node at one quarter of the element length (Figure 5). In the present work the coliocation points NC1, NC2 and NC3 are located at = - 3/4, r2 = 0 and t3 = 3/4, respectively. In such case, the distance r to the crack tip, obtained from equation (13), is (Figure 10):

L fl =z for NC1

L 4

f2 =- for NC2

49L 64

f3 =- for NC3

1692

\ / \/ \; / \

A. SAEZ, R. GALLEGO AND J. DOMiNGUEZ

Figure 10. Discontinuous quarter point element with = - 3/4; tz = 0; t3 = 3/4

Defining as usual a polar co-ordinate system, centered at the crack tip, such that 8 = f 71 are the crack surfaces, the normal displacement along the crack surfaces for a plane strain situation can be written as

and the shear displacement as

where only the first term of the series expansion has been considered, the superscripts + and - are used to denote the two different crack surfaces, v is Poisson’s ratio, p the shear modulus, and KI and KI1 the Stress Intensity Factors (SIF) for the modes I and 11, respectively. Particulariz- ing equations (47) and (48) for f = L/64 one may obtain the following one point displacement formulae for SIF evaluation

If one writes equations (47) and (48) for i = L/4 (point NC2) the following SIF evaluation formulae are obtained

It may be worth to use also two-point displacement correlation formulae to compute SIF from the quarter-point hypersingular BE results. Equation (1 1) for the difference between the two crack surfaces (Crack Opening Displacements (COD)) becomes

CRACK PROBLEMS 1693

Substituting equations (12) and (13), with the crack tip at N1 and t3 = - t1 = 3/4, one obtains a COD representation of the type

where a, /? and y are linear combinations of Aui , A u i , and Au;.

COD series expansion (equations (47) and (48)) one obtains Using the condition Auk = 0 for r = 0 (a = 0) and making f l equal to the term of the

' ( 3 2 A u Y ' - 2AuyC2) $; 12(1 - v) KF' -NC2 =

There are several other approaches to compute SIF using a mixed BE formulation. In the present work, in addition to the discontinuous element displacement formulae presented above, two more SIF evaluation procedures have been tested. One is based on the evaluation of the SIF from the stress components computed at an internal point very close to the crack tip once the boundary value problem has been solved. The other one is based on the use of a semi- discontinuous element at the crack tip. Instead of using only discontinuous elements along the crack, the element containing the tip has one of its extreme nodes at the tip (t = 1 or 5 = - 1). At that node only one BE equation is needed; consequently, only the standard displacement integral representation is written. The SIF are computed from the COD of the other two nodes of the tip element. These last two SIF computation approaches were left behind in this work. The first one because requires some additional effort and does not improve the results obtained using the COD according to equations (49)-(56), and the second one because the results were less accurate.

NUMERICAL EXAMPLES

The use of hypersingular quarter-point elements for Stress Intensity Factor computations is tested next. To do so, three crack problems, for which there are solutions accurate within 1 per cent and which have also been analysed by Portela et ~ 1 . ' ~ using the BEM, are studied. The BE analysis, done by these authors, is also based on a mixed formulation but they use discontinuous elements with a geometry defined by three equidistant nodes (so the constant Jacobian is removed from the kernels of the integrals). These authors compute the SIF either from the COD at the nodes of those elements or evaluating the J-integral. Their first approach gives less accurate results than the present one. Evaluating the J-integral, Portela et achieve an accuracy of the same level as that of the approach presented in this paper, however such computation of the J-integral is more involved and requires more effort than the present COD-based approach. In addition, it cannot be used for dynamic problems.

Example 1. Edge crack (Figure 11)

Consider a square plate with an edge crack as shown in Figure 1 1 . The plate is under the effect of a uniform traction at two. opposite sides. In order to compare the present study with those of Civelek and Erdogan" and Portela et ~ 1 . ' ~ the following values of the crack length have been considered: a/w = 0*2,0.3,0.4,0*5 and 0.6. The BE discretization consists of 32 quadratic elements (four on each side of the crack and 24 on the external boundary). The crack discretization was

1694 A. SAEZ, R. GALLEGO AND I. DOMiNGUEZ

Figure 11. Square plate with an edge crack under uniform traction ( h = w)

Table I. Mode-I Stress Intensity Factor for a square plate with an edge crack

a/w K U V KNCI K N C 2 I I I Krc1-Nc2

0.2 1.488 1.496 1.495 1.496 0.3 1.848 1-859 1.863 1.858 0.4 2.324 2.340 2.353 2.336 0-5 3.010 3.034 3.063 3.024 0.6 4,152 4.193 4.255 4.113

graded, towards the tip, like 0*4,0.3,02 and 0.1 (this is the same discretization et al., in Reference 15).

The following results are shown in Table I:

used by Portela

KpV, KI SIF given by Civelek and Erdogan"; K,NcL, KI SIF obtained in the present study from the COD at node NC1 of the tip element (equation (49)); K,Nc2, KI SIF obtained using the COD at node NC2 (equation (51)); KYC'-NCZ, K , SIF obtained from the COD at nodes NC1 and NC2 (equation (55)).

It should be noted that all the results given in Table I show a high level of accuracy. In particular the results obtained from the COD at NC1 (K,"") and those obtained from the COD at NC1 and NC2 (KYC'-NCZ) are always within a 1 per cent difference with respect to those given by Civelek and Erdogan.'*

To check the stability of the above results with respect to the crack tip element length L, the a/w = 0.5 case was studied using the following discretizations for the crack: (1, L/a = 0.1): the crack is discretized into four elements graded towards the tip like 0.4, 0.3, 0.2 and 0 1 ; (2, L/a = 0.2): the crack is discretized into four elements graded like 033, 0.27, 0.2 and 02; (3, L/a = 0.3): three elements graded like 0-37, 0.33 and 0.3; (4, L/a = 0.4): three elements graded like 027,0*33 and 0.4; (5, L/a = 0.5): the crack is discretized into two elements graded like 0-5 and 0.5 and the rest of the boundary into 16 elements; (6, L/a = 06): the crack is discretized into two elements graded like 0.4 and 0.6 and the rest of the boundary into 16 elements.

Figure 12 shows the percentage difference with respect to Civelek and Erdogan's solution ( K , = 3.01). It can be seen in the figure that all the computed SIF are stable as L varies, in

CRACK PROBLEMS 1695

particular the one obtained form the COD at NC1; i.e., at a point L/64 from the tip, is very accurate and remains always within a 1 per cent difference. The less accurate results are obtained, as expected, for the longer tip elements and the computation point furthermost from the tip (NC2). Note that the ($ I2 behaviour of the COD starts to change as one moves away from the tip because of the influence of other terms of the series.

Example 2. Inclined central crack (Figure 13)

Consider a rectangular plate with an inclined central crack as shown in Figure 13. The plate is under the effect of a uniform traction at two opposite sides. The crack has a length 2a and it is inclined 45" with respect to the plate sides. The following ratios between the crack length and the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

L/a Figure 12. Difference (percentage) of the computed mode4 SIF with respect to Civelek and Erodogan's solution''

( K , = 3.01). Square plate with an edge crack. L/a = Length of the crack tip elements/Length of the crack

I I I U I n Figure 13. Rectangular plate with an inclined central crack under uniform traction (h/w = 2)

1696 A. SAEZ, R. GALLEGO AND J. DOM~NGUEZ

plate width are considered alw = 0.2, 0-3, 0-4, 0.5 and 0.6. The BE discretization consist of 36 quadratic elements (six on each side of the crack and 24 on the external boundary). The crack discretization was graded, towards the tips, like 0.5, 03, 0.2. The results obtained are shown in Tables I1 and 111 for the mode-I and mode-I1 SIF, respectively. The present BE results are compared with those given by Murakami" (KrUR and KrUR), which are accurate within 1 per cent. The BE results given in Tables I1 and 111 show a high level of accuracy.

Table 11. Mode-I Stress Intensity Factor for a rectangular plate with an inclined central crack

K M U R K N C ~ K N C 2 I I I K:C'-NC2

0.2 0518 0.519 0-515 0-521 0 3 0.541 0.543 0.538 0.544 0.4 0572 0.575 0.570 0.577 0.5 0612 0617 0.613 0.619 0.6 0661 0.670 0.667 0.671

Table 111. Mode-I1 Stress Intensity Factor for a rectangular plate with an inclined central

crack

UJW KEUR KEc' Kic2 K i C l - N C Z

0 2 0507 0.507 0.502 0508 0.3 0.516 0-515 0.510 0.517 0.4 0529 0.527 0.522 0.529 0.5 0546 0.543 0.537 0.545 0.6 0567 0.564 0.557 0.566

-v

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

L/a Figure 14. Difference (percentage) of the computed mode-I SIF with respect to Murakami's sol~t ion'~ (K, = 0572).

Rectangular plate with an inclined crack. L/a = Length of the crack tip elements/half-length of the crack

CRACK PROBLEMS 1697

6 1 1

€3

NC1-NCZ + K .,. K NC1

n

II

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

L/a Figure 15. Difference (percentage) of the computed mode-I1 SIF with respect to Murakami‘s s o l ~ t i o n ‘ ~ (K, = 0.529).

Rectangular plate with an inclined crack. L/a = Length of the crack tip elements/half-length of the crack

The stability of the computed results with respect to the crack tip element length L is checked in this case for a crack length a/w = 0.4. The following discretizations for the crack are analysed: (1, L/u = 0.1): the crack is discretized into six elements graded, from the centre towards the tips, like 0.6, 0.3 and 0.1; (2, L/a = 0.2): six elements graded like 0.5, 0.3 and 0.2; (3, L/a = 0.3): six elements graded like 04,0-3 and 0.3; (4, L/a = 0.4): four elements along each crack surface graded, toward the tips, like 0.6 and 0.4; (5, L/a = 0.5): four elements graded like 0.5 and 0.5; (6, L/a = 0.6): four elements graded like 0.4 and 0-6. Figures 14 and 15 show the percentage difference with respect to Murakami’s results for the mode-I and mode-I1 SIF (K, = 0572 and KII = 0.529) respectively. Both figures show that the values computed using the proposed approach are stable as L varies. In both cases the SIF computed using the COD at NCl and at NC1-NC2 remain stable within a 1.5 per cent difference with respect to Murakami’s reference value.

Example 3. Kinked crack (Figure 16)

To complete the numerical experiments, a rectangular plate containing a kinked crack is studied. This problem was analyzed by Portela et al.” and Murakami.” The kink of the crack is at the center of the plate which is loaded at the ends with a uniform traction. The height of the plate is twice its width and the length of the horizontal part of the crack is u/w = 0.1. Three possible values of the length of the inclined part of the crack are considered b/a = 0.2,0*4 and 0.6. The quadratic element discretization used consists of 30 continuous elements on the external boundary and nine discontinuous ones on each side of the crack. Of these nine discontinuous elements, five of the same length are on each side of the horizontal part of the crack and four of the same length on each side of the inclined part. The elements at both tips have the mid-node located at one quarter of their length. The rest of the elements have the mid-node at the central-point of the element.

The results obtained from the BE analysis are shown in Tables IV-VII. Tables IV and V correspond to the mode-I and mode-I1 SIF, respectively, for the tip A. Tables VI and VII correspond to the mode-I and mode-I1 SIF., respectively, for the tip B. In those tables the superscript MUR indicates the results taken as a reference from Murakami”, and NCl, NC2 and

1698 A. SAEZ. R. GALLEOO AND J. DOM~NGUEZ

2 w - --

I 2h

Figure 16. Rectangular plate with a kinked central crack under uniform traction (h /w = 2)

Table IV. Mode-I Stress Intensity Factor at crack tip A for a rectangular plate with a kinked

crack

0.2 0.995 0.995 0.978 1.000 0.4 0.990 0990 0.975 0.995 0.6 0.986 0.987 0.974 0.992

Table V. Mode-11 Stress Intensity Factor at crack tip A for a rectangular plate with a kinked

crack

0.2 0.028 0.027 0.028 0.027 0.4 0.033 0033 0.033 0.033 0.6 0.030 0030 0.030 0.030

Table VI. Mode-I Stress Intensity Factor at crack tip B for a rectangular plate with a kinked

crack

0.2 0.598 0.599 0.599 0.600 0.4 0.574 0.574 0.572 0.575 0.6 0.568 0.568 0.564 0.570

CRACK PROBLEMS 1699

Table VIT. Mode-I1 Stress Intensity Factor at crack tip B for a rectangular plate with a kinked

crack

b/a KMLIR K7c2 Kic1- NC2

- I1 K F ' -

0 2 0.557 0.561 0.558 0.562 0.4 0.607 0,609 0.605 0.610 0.6 0.627 0.630 0.625 0.631

K NC1-NC2

4

-2 t -4

t

* * .

0 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

L,/a

Figure 17. Difference (percentage) of the computed mode-I SIF at crack tip A with respect to Murakami's s o l ~ t i o n ' ~ ( K , = 0.99). Rectangular plate with a kinked crack. L/a = Length of the crack tip elements/length of the horizontal part of

the crack

NC1-NC2 the results computed in the present analysis using equations (49, 50), (51, 52), and (55, 56), respectively. It is worth to notice that all the BE results presented in these tables are very accurate as compared to the results given by Murakami.

Once again a stability analysis is done for this example. To do so, the b/u = 0 4 case is chosen and the following discretizations for the crack are considered (1, L l / a = 0-1, L2/b = 0-15): the horizontal part a is discretized into five elements graded like 0.3,0-2,0.2,0.2,0.1 and the inclined part b into four elements graded like 0.35, 0.30, 0.20 and 0.15; (2, L , / a = 0.2, L2/b = 0.25): part a is discretized into five elements graded like 0.2,0*2,0.2,0.2 and 0.2, and part b into four elements graded like 0.25, 0.25, 025 and 0-25; (3, L l / u = 0.3, L2/b = 0.35): part a is discretized into three elements graded like 0-4, 0-3 and 0-3, and part b into two elements graded like 0-65 and 0.35; (4, L l / a = 04, L z / b = 0-45): part a is discretized into three elements graded like 0-3,0.3 and 04, and part b into two elements graded like 0.55 and 0.45; (5, L l / a = 0.5, L z / b = 0.55): part a is discretized into two elements graded like 0.5 and 0.5, and part b into two elements graded like 0.45 and 055; (6, L l / a = 0.6, Lz/b = 0.65): part a is discretized into two elements graded like 0-4 and 0.6 and part b is discretized into two elements graded like 0.35 and 0.65. Figures 17 and 18 show the percentage difference with respect to Murakami's results for the mode4 SIF at the crack tips A and B, respectively (K: = 0.990 and KF = 0574). In both cases the stability of the results is

1700 A. SAEZ, R. GALLEGO AND J. DOM~NGUEZ

6 1 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 18. Difference (percentage) of the computed mode-I SIF at crack tip B with respect to Murakami's ~o lu t ion '~ ( K , = 0574). Rectangular plate with a kinked crack. L, /b = Length of the crack tip elementshength of the inclined part of

the crack

good. In particular when K , is computed from the node closer to the tip (NC1) or from the two-points recurrence formula (NCl-NC2), the results are very accurate for all the crack tip element length range considered. The less accurate results correspond again to NC2, i.e. to the computation point furthermost from the tip where terms of the COD series other than (F)l'* become significant.

CONCLUSIONS

The mixed BEM for crack problems in elasticity is based on the use of two integral representa- tions on the crack surfaces: one is the standard displacements representation and the other is the tractions representation obtained by a combination of derivatives of the first one. This second representation contains hypersingular integrals. In this paper the mixed BEM formulation is used and an approach for the evaluation of the hypersingular integrals for quadratic curved boundary elements is presented. The integration approach is based on a regularization process and the analytic integration of the remaining basic singular integrals. This idea, a variation of that found in Rudolphi et al." and Guiggiani,12*13 is developed in detail for the case of the tractions integral representation along curved quadratic elements with the collocation points at any location along the element satisfying the C' continuity requirement.

The generality of the integration method allows for the use of straight line quarter-point quadratic elements and collocation points located extremely close to the crack tip. By doing so the J- behaviour of the Crack Opening Displacement (COD) is reproduced and the Stress Intensity Factors (SIF) can be evaluated from the COD at points where this behaviour is highly dominant.

The proposed procedure is tested by analysing three different cracked geometries. The SIF directly computed from the COD at a single collocation point very close to the tip are very accurate and show very little dependence on the size of the quarter-point element containing the tip. The use of a correlation formula in terms of the COD at two points near the tip is also shown to be accurate and stable, however,*it not necessary to use such formula since the one point evaluation equation, in addition. to be the simplest, is very accurate and stable.

CRACK PROBLEMS 1701

These simple and accurate SIF evaluation approaches can also be used for dynamic crack problems in the context of a frequency domain or time domain mixed formulation as will be shown by the authors in a forthcoming paper.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the financial support provided by the Direccion General de Investigacion Cientifica y Tbcnica under the Research Project No. PB90-0892.

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