Alaibadi Portela

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Compurers & S~ruerwes Vol. 4b. No. 2 pp. 237-247, 1993 ~~7~9~3 36.00 + 0.00Printed in Great Britain. 0 1993 Pergamon fks Ltd

DUAL BOUNDARY ELEMENT INCREMENTAL ANALYSISOF CRACK PROPAGATION

A. PoRrnLA,t M. H. ALIABADI~ and D. P. ROOKEJtWessex Institute of Technology, Computational Mechanics Institute, Ashurst,Southampton SO4 2AA, U.K.

$Defence Research Agency, Aerospace Division, RAE, Famborough, Hampshire GUI4 6TD, U.K.Received 14 November 1991)

Abstract-This paper describes an application of the dual boundary element method to the analysisof mixed-mode crack growth in linear elastic fracture mechanics. Crack-growth processes are simulatedwith an incremental crack-extension analysis based on the maximum principal stress criterion which isexpressed in terms of the stress intensity factors. For each increment of the crack extension, the dualboundary element method is applied to perform a single-region stress analysis of the cracked structureand the J-integral technique is used to compute the stress intensity factors. When the crack extension ismodelled with new boundary elements, remeshing is not required because of the single-region analysis,an intrinsic feature of the dual boundary element method. Results of this incremental crack-extensionanalysis are presented for several geometries. The analysis of fatigue crack-growth is introduced as apost-processing procedure on the crack-extension results and an example is presented.

INTRODUCTIONCatastrophic fracture failure of engineering struc-tures is caused by cracks that extend beyond a safeize. Cracks, present to some extent in all structures,either as a result of manufactu~ng fabrication defectsor localized damage in service, may grow by mech-anisms such as fatigue, stress-corrosion or creep. Thecrack growth leads to a decrease in the structuralstrength. Thus, when the service loading cannot besustained by the current residual strength, fractureoccurs leading to the failure of the structure. Frac-ture, the final catastrophic event which takes placevery rapidly, is preceded by crack growth whichdevelops slowly during normal service conditions,mainly by fatigue due to cyclic loading.

Damage tolerance is the structural property thatdefines whether a crack can be sustained safelyduring the projected service life of the structure.Damage tolerance assessment is, therefore, requiredas a basis for any fracture control plan, generating thefollowing information, upon which fracture controldecisions can be made:

The effect of cracks on the structural residualstrength, leading to the evaluation of theirmaximum permissible size.The crack growth as a function of time, leadingto the evaluation of the life of the cracks, toreach their maximum permissible size, fromwhich the safe operational life of the structure isdefined.

Linear elastic fracture mechanics can be used indamage tolerance analyses to describe the behaviourof cracks. The fundamental postulate of linear elastic23

fracture mechanics is that the crack behaviour isdetermined solely by the values of the stress intensityfactors which are a function of the applied loadand the geometry of the cracked structure. The stressintensity factors, thus, play a fundamental role inlinear elastic fracture mechanics applications.

Crack-growth processes are simulated with anincremental crack-extension analysis. For eachincrement of the crack extension, a stress analysisis carried out and the stress intensity factors areevaluated. The crack path, thus predicted on anincremental basis, is computed by a criterion definedin terms of the stress intensity factors.

In general, numerical methods must be usedfor the evaluation of the stress intensity factors inengineering structures, because of their complexgeometry which is continuously changing with theextension of the cracks. The finite element methodhas a long and well-documented history in fracturemechanics applications [I, 21. More recently it hasbeen applied to study crack-growth processes [3-71.An intrinsic feature of the finite element method,common to all these formulations, is the need forcontinuous remeshing to follow the crack extension;this is a practical disadvantage of the method.

The boundary element method (BEM) is a well-established numerical technique in fracture mech-anics [8] which has also been applied in incrementalanalysis of crack-extension problems by Ingraffeaet al. [9]. However, the solution of general crackproblems cannot be achieved with the direct appli-cation of the BEM, in a single-region analysis,because the coincidence of the crack surfaces givesrise to a singular system of algebraic equations. Theequations for a point located at one of the surfaces

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238 A. PORTELA~~ i,of the crack are identical to those equations for and is described by a static load level with a stressthe point on the opposite surface, with the same amplitude ratio. Results of a fatigue crack-growthcoordinates, because the same integral equation is analysis, obtained by the post-processing of a crack-applied with the same integration path, at both extension analysis, are also presented for a simplecoincident points. problem.Some special techniques have been devised toovercome this difficulty. Among these, the mostgeneral are the subregions method, Blandfordet al. (lo] and the dual boundary element method,introduced by Portela ef al. [I 11. The subregionsmethod introduces artificial boundaries into thestructure, which connect the cracks to the boundary,in such a way that the domain is divided into thesubregions without cracks. In an incremental crack-extension analysis, these artificial boundaries mustbe repeatedly introduced for each increment of thecrack extension. The main drawback of this methodis that the introduction of artificial boundaries is notunique, and thus cannot be easily implemented as anautomatic procedure in an incremental analysis ofcrack-extension problems. In addition, the methodgenerates a larger system of algebraic equations thanis strictly required.

THE DUAL BOUNDARY INTEGRAL EQUATIONSThe dual equations, on which the DBEM is

based, are the displa~ment and the traction bound-ary integral equations. The displacement boundaryintegral equation can be derived with the classicalwork theorem [12]. In the absence of body forces andassuming continuity of the displa~ments at a bound-ary point x, the boundary integral representation ofthe displacement components ui is given by

C~(X )~j(X ) f Fj CX, h+ XI dr (xlI-=s,(x, x)t,(x) dr(x), (1)r

The dual boundary element method (DBEM)incorporates two independent boundary integralequations, with the displacement equation applied forcollocation on one of the crack surfaces and thetraction equation on the other. As a consequence,general mixed-mode crack problems can now besolved in a single-region formulation. Althoughthe integration path is still the same for coincidentpoints on the crack surfaces, the respective boundaryintegral equations are now distinct. The single-region analysis, characteristic of the DBEM, caneliminate the remeshing problems which are typical ofthe finite element and multi-region boundary elementmethods.

where i and j denote Cartesian components; To(x, x)and lJti(x, x) represent the Kelvin traction and dis-placement fundamental solutions, respectively, at aboundary point x. The distance between the points xand x is denoted by r. The integrals in eqn (1) areregular, provided r # 0. As the distance r tends tozero, the fundamental solutions exhibit singularities;they are a strong singularity of order I./r in Tt i nd aweak singularity of order In l/r in [email protected] symbol fstands for the Cauchy principal-value integral, andthe coefficient cO(x) is given by 6,/2 for a smoothboundary at the point x (6, is the Kronecker delta).

The present paper is concerned with the appli-cation of the DBEM to the analysis of crack-growth in linear elastic fracture mechanics. The dualboundary integral equations are presented, the crackmodelling strategy defined and the stress intensityfactors evaluated by the f-integral technique. Anincremental crack-extension analysis is performed todetermine the crack path. For each increment of theanalysis, in which the crack extension is modelledwith new boundary elements, the DBEM is appliedfor the stress analysis and the J-integral techniqueis used for the stress intensity factors evaluation.The incremental analysis is based on a prediction-correction technique to define the direction of thecrack-extension increment. The maximum principalstress crack-growth direction criterion is applied topredict the tangent direction of the crack path andthen a correction is introduced to determine theactual direction of the increment of crack extension.Results of incremental crack-extension analysis arepresented for several geometries. Fatigue crack-growth is caused by cyclic loading: in the simplestcase, the loading cycle has a constant amplitude

In the absence of body forces and assumingcontinuity of both strains and tractions at x on asmooth boundary, the stress components aii aregiven by

(x3 -t s SiJx, x)uk(x) dT(x)r= f D, (x, x)tk04 df(x). C-4I-

In this equation, S,,(x, x) and D,,(x, x) containderivatives of T#(x, x) and U&x, x), respectively.The integrals in eqn (2) are regular, provided r 0.As the distance r tends to zero, S,, exhibits a hyper-singularity of the order l/r*, while D, exhibits astrong singularity of the order l/r. The symbol fstands for the Hadamard principal-vaIue integral.On a smooth boundary, the traction components, ti,are given by

itj(x) + q(x} s S x, x)+(x) dT(x)r= n,(x) f D x, x)fktx) W(x), (3)r


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39

nodeX- lement end pointA- Displacement equationB -Traction equation

Fig. 1. Crack modelling with discontinuous quadratic boundary elements.where ni denotes the i component of the unitoutward normal to the boundary, at the point x.Equations (1) and (3) constitute the basis of theDBEM, see Portela et al. [I I].

CRACK MODELLING STR TEGYFor the sake of efficiency, and to keep the

simplicity of the standard boundary element, thepresent formulation uses discontinuous quadraticelements for the crack modelling, as shown in Fig. 1,The general modelling strategy, developed by Portelaet al. [I I], can be summarized as follows:

the crack boundaries are modelled with discon-tinuous quadratic elements, as shown in Fig, i;continuous quadratic elements are used alongthe remaining boundary of the structure, exceptat an intersection between a crack and an edge,where discontinuous or ~mi-discontinuouselements are required on the edge in orderto avoid a common node at the intersection,as shown in Fig. 1.the displacement equation (1) is applied forcollocation on one of the crack surfaces;the traction equation (3) is applied forcollocation on the other crack surface;the displacement equation (1) is applied forcollocation on all non-crack boundaries.

This simple strategy is robust and allows the DBEMto effectively model general edge or embedded crackproblems; crack tips, crack-edge corners and crackkinks do not require special treatment, since they arenot located at nodal points where the collocation iscarried out. In general, cracking processes can extendthe cracks along curved paths. However, in practice,curved paths are usually modelled with flat incre-ments which lead to piece-wise flat cracks. For suchcracks, all the integrals in eqns (I) and (3) are mosteffectiveIy carried out by direct analytic integration,as shown by Portela et al. [II].

When the present crack modelling strategy isapplied in an incremental crack-extension analysis,such that each new crack-extension increment is

modeled with new discontinuous boundary elements,it becomes apparent that remeshing is not requiredin the DBEM. The new discontinuous boundaryelements of the crack-extension increment willgenerate new equations and up-date the ones alreadyexisting with new unknowns. In other words, newboundary elements will generate new rows and newcolumns in the matrix of the system of equations.Assuming that the crack extension is traction-free,the right-hand side of the system of equations is onlyextended for the positions corresponding to the newunknowns introduced. This procedure is illustratedschematically in Fig. 2. If the LU decompositionmethod is adopted for the solution of the system ofequations, a very e&Sent incremental anaIysis canbe carried out. For each increment of the analysis,only the new rows and new colunns need to beLU-decomposed. The existing rows and columns,already decomposed, are brought from the previousiteration into the current one.

THE STRESS INTENSITY FACTORS EVALUATIONMany different techniques may be used in the

boundary element method for the evaluation of stress

nitial geometryIZZI 1st crack extensionIZZI 2nd crack extension@ I 3rd crack extension

Fig. 2. Schematic representation of the generation of thesystem of equations.


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240 A. Po~ra~ et al.intensity factors, as described by Aliabadi andRooke [8]. In this paper the Z-integral method waschosen to obtain the numerical results. Consider aCartesian reference system, with the origin at the tipof a traction-free crack, as shown in Fig. 3. Rice [13]introduced the path-inde~ndent J-integral, which inthe absence of body forces is given by

.J =sWn, - t,u,)dS,where Z is an arbitrary contour surrounding thecrack tip; W is the strain energy density; $ arethe traction components, given by aUrai, where n,are the components of the unit outward normalto the contour path. The relationship betweenthe J-integral and the stress intensity factors isgiven by

K: + K;,J=-, E' (5)where the constant E is the elasticity modulus equalto E for plane stress conditions and E = E/(1 - v2)for plane strain conditions. A simple procedure,based on the decomposition of the elastic field intosymmetric and antisymmetric mode components, canbe used to decouple the stress intensity factors of amixed-mode problem. The integral J is represented bythe sum of two integrals as follows:

J=JfJ, (6)where the superscripts indicate the pertinentdeformation mode. For this representation to bepossible, it is sufficient to introduce the followingdecomposition in the elastic fiefds

where a;, u; and eii, rri represent the elastic fieldat symmetric points P(x,,x,) and P(x,, -x2), asrepresented in Fig. 3. Equations (7) and (8) lead to thefollowing decompositions of the elastic field

ando+r;+cr; (9)

ui=u;+up. (IO)When eqns (9) and (10) are introduced into eqn (41,eqn (6) is obtained, with the J-integral com~nentsgiven by

J- - s (W , - t,fu,j) dZ, (11)rfor M = Z or M = ZZ. Finally, the followingrelationships hold

(12)The implementation of this procedure into theboundary element method is straightfo~ard.A circular contour path, around the crack tip, isdefined with a set of internal points, located atsymmetrical positions relative to the crack planeof the last increment, as shown in Fig. 3. The twocontour points on the crack surfaces are the first andthe last points of the path, respectively. At thesepoints, it is always verified that n, = - 1 and n2 = 0and thus, for a traction-free crack t2 = 0. The inte-gration along the contour path can be accomplishedwith the trapezoidal rule. For the sake of simplicity,only circular paths containing crack nodes, were

. - laworl Qotalsa) General contour b) Circular contour

Fig. 3. Coordinate reference system and contour path for J-integral.


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DBEM analysis of crack propagation 241considered in this paper. The accuracy of this tech-nique was already demonstrated with several bench-mark problems by Portela et al. [ll]. The J-integralis an effective method for the determination of stressintensity factors, because the interior elastic field canbe accurately determined along the contour path,with the boundary element method, since the exactvariation of the interior elastic field is built into thefundamental solution.

CRACK EXTENSION DIRECIION

Several criteria have been proposed to describe thelocal direction of mixed-mode crack growth. Amongthem, one of the most commonly used is based onthe maximum principal stress at the crack tip [14].The maximum principal stress criterion postulatesthat the growth of the crack will occur in a directionperpendicular to the maximum principal stress. Thus,the local crack-growth direction 0, is determined bythe condition that the local shear stress is zero, that is

K, sin 0, + K,,(3 cos 0, - 1) = 0, (13)where 0, is an angular coordinate centred at thecrack tip and measured from the crack axis aheadof the tip. As a continuous criterion, the maximum

principal stress does not take account of the discrete-ness of the crack extension modelling procedure. Inother words, the crack path is defined continuouslyby the trajectory of the maximum principal stresses,evaluated locally by eqn (13). Therefore, the incre-mental extension of a crack in a general mixed-mode deformation field, computed by eqn (13), isalways defined locally in the same direction, whateverlength of crack extension, Aa is considered. As aconsequence, uniqueness of the crack path cannot beassured with different sizes of the crack-extensionincrement. Therefore, in an incremental analysis,the tangent direction of the crack-path, predicted byeqn (13), must be corrected to give the direction of theactual crack-extension increment.

The correct direction of the crack-extension incre-ment ensures that, for different analysis of the sameproblem with different crack-extension increments,a unique final crack path is achieved. The procedureapplied to define the direction of the n th crack-extension increment introduces a correction angle /3to the tangent direction erCnJ redicted by the maxi-mum principal stress criterion, as shown in Fig. 4.Using geometric relationships, this correction angle isgiven by /J = B,(,+ ,/2, in which e,,,,+ ,) is the directionof the next crack-extension increment, also computedwith the maximum principal stress criterion. This

n - current crack-extension incrementAa - increment sizei iteration numberBt - increment direction computed with the tangent criterionj3 - correction angle (p = Bttn+1j/2)P crack tip location

Fig. 4. Incremental crack-extension direction.


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predictor-corrector procedure can be applied whileeach correction is smaller than the previous one. Forthe current nth crack-extension increment, the ithiteration can be summarized as follows:

for the first iteration only, evaluate the crack-path tangent direction e:,,, with the maximumprincipal stress criterion, eqn (13);along the direction computed in the previousstep, extend the crack one increment Aa to Piand evaluate the new stress intensity factors;

* with the new stress intensity factors and themaximum principal stress criterion, eqn (13),evaluate the next crack-path direction 6:,, + ,,;define the correction angle /Ii = O;,,,+ ,/2,measured from the increment defined in thesecond step;correct the crack-extension increment, defined inthe second step, to its new direction given by8:&l = %(., + B;starting from the second step, repeat the abovesteps sequentially while l/P+1 < lpl.

When the size of the crack-extension increment,Aa tends to zero, the angle B,tR+f also tends to zeroand so does the correction angle, fl. This means thatin the limit, the direction of the incremental crack-extension tends to the direction of the tangent ofthe continuous crack path. The present procedure issimple, but more research may be needed to define anefhcient criterion for the prediction of the direction ofthe crack-extension increment.

INCREMENTAL ANALYSIS

The incremental crack-extension analysis assumesa piece-wise linear discretization of the unknowncrack path. For each increment of the crack exten-sion, the DBEM is applied to carry out a stressanalysis of the cracked structure and the J-integral isthe technique used for the evaluation of the stressintensity factors. The steps of this basic compu-tational cycle, repeatedly executed for any numberof crack-extension increments, are summarized asfollows:

. carry out a DBEM stress analysis of the crackedstructure;for each crack tip compute the stress intensityfactors with the j-integral technique;for each crack tip compute the direction of thecrack-extension increment;extend each crack one increment along thedirection computed in the previous step;repeat all the above steps sequentially until aspecified number of crack-extension incrementsis reached.

For the sake of simplicity, the increment of crackextension is discretized with a fixed number of newboundary elements. In order to avoid numericalproblems, concerned with the relative size of neigh-

bouring elements, the crack-increment length iskept between convenient limiting bounds, defined interms of the size of the crack-tip element. Apart fromthis constraint, the length of the crack-extensionincrement may be defined as the result of a compro-mise between accuracy and computational cost; thesmaller the crack increment the more accurate andexpensive is the analysis.

The results obtained from an incremental crack-extension analysis are a crack path diagram for thestructure and diagrams of the stress intensity factorvariation along each crack path. In a mixed-modeanalysis, an equivalent mode I stress intensity factorcan be defined [151. For the maximum principal stresscriterion, it is given by

Kr, = Ki co? $ - 3Kn cos2 + sin $ . (14)The fracture condition then follows from &, = K,,,in which 4, is the fracture toughness.

CRACK EXTENSION APPLICATIONS

Two applications of the incremental crack-extension analysis will be studied in the presentsection. The first application characterizes thebehaviour of a crack that initially deforms under puremode I or pure mode II. The second application

t iFig. 5. Square plate with an edge crack under pure mode Iand pure mode II.


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DBEM anaiysis of crack probation 243I

fj

srm crack path3 4.0 -6J8

ia

5.0 -~.0 -2.0

$J 1.0fi 0.0 r

-Mode I- Mode II

-l*O 1: 2 3 4 5 6 7 8 9 10Number of Crack-Extension Inoremsnta

Fig. 6. Incremental extension analysis of a crack initially under pure mode I.makes the analysis of a crack that initially deformsunder mixed mode.

For the first application, consider a square plateof height h, with an edge crack of length Q = h/3,subjected to either pure mode I or pure mode IIloading, as represented in Fig. 5. Results of theincremental crack-extension analysis of these prob-lems were obtained with the maximum principalstress criterion and a crack-extension increment equalto three times the length of the initial crack-tipelement. Figures 6 and 7 present these results in termsof crack paths and stress intensity factor diagrams,for each one of the problems.

40.0

$ 30.0i

i: 2o.oBrn 10.01

B63 0.0

-10.0

The second application, where the initial crackis under mixed-mode deformation, is the cruciformplate represented in Fig. 8. The plate has square armsof length h which are loaded with uniform tractions6 and 6, applied at the ends of each arm, respect-ively. At one of the interior corners, a crack of lengtha = h/4 was introduced in the direction of the diago-nally opposite comer of the plate. In the presentapplication, incremental crack extension analyseswere carried out for four different cases in whicht,=Oand


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DBEM analysis of crack propagation 245increment equal to three times the length of the initialcrack-tip element, are presented in Fig. 9.

FATIGUE CRACK GROWTH

In general, fatigue crack-growth is driven byvariable amplitude loading. In the simplest case, theloading cycles have a constant amplitude and maybe described by a constant amplitude load with aconstant amplitude stress ratio. The aim of this typeof analysis is to obtain residual-strength and fatigue-life diagrams for each crack. From these results, boththe maximum permissible size of each crack and thesafe operational life of the structure can be evaluatedin a damage tolerance assessment. In the presentwork, analysis of fatigue crack growth is introducedas a post-processing procedure on the results of thecrack-extension analysis.

The residual-strength diagram shows the variationin the maximum load that the cracked structure cansustain, that is, the load that causes fracture instabil-ity, as the crack length varies. Thus, for a given cracklength a, the residual strength, Q, is given by thecritical load which is defined as

KI,uc=G=, (15)

where Y is the geometry factor. For the same cracklength, the geometry factor is constant and hence,can be defined in terms of the applied reference stress,0, and the corresponding mode I equivalent stressintensity factor, K,q s

Y=$=--$L. (16)From this equation, the following relationshipbetween the residual strength and the appliedreference stress holds

K,,ac=-dK

(17)ieqAt each step of an incremental analysis, the residual

strength is represented conveniently in a normalizedform, derived from eqn (17) as(18)

where cc. and KIq.epresent the residual strength andthe corresponding mode I equivalent stress intensityfactor, respectively, computed at the initial cracklength.

The fatiguelife diagram shows the variation inthe number of loading cycles, required to extend thecrack, as a function of the crack length. It wascomputed from the generalized Paris modeldefined as

where a is the crack length, N is the number of loadcycles, C and are material dependent constants andAKeff is the range of the effective stress intensityfactor. The model of Tanaka [16], in which AK,* isdefined by

AK&= AKf+2AKf,, (20)is usually applied in mixed-mode analyses, sinceit was verified experimentally. Thus, althoughAK, = AKics, when the maximum principal stresscriterion is used, the model of Tanaka wasapplied in the present work. In eqn (20), the stressintensity factor range of the individual modes isgiven by AK = Km,,K,,,,K,,,,(l -R) in whichK = Knin IKm,, = cmin omax s the stress amplitude ratioof the loading cycle. The number of loading cyclesrequired to extend the crack a given increment isevaluated by integration of eqn (19) with thetrapezoidal rule, applied for each increment of theanalysis.

Results of fatigue crack-growth analysis wereobtained for the cruciform plate represented in Fig. 8,for the case in which 6 = &. The residual strengthdiagram is presented in Fig. 10. The fatigue-lifediagram, obtained for the case in which the stressratio = 0.66666, C = 4.624 x lo-* and m = 3.3, ispresented in Fig. 11. With these diagrams and aknowledge of the fracture toughness of the material,a complete damage tolerance analysis can be carriedout.

DISCUSSION AND CONCLUSIONS

In this paper, the dual boundary element methodis applied to the incremental linear elastic analysis ofcrack-extension problems. For each increment of thecrack extension, a stress analysis of the structure iscarried out and the stress intensity factors are evalu-ated with the J-integral technique. This basic compu-tational cycle is repeated for each crack extension inthe direction determined by the maximum principalstress criterion.

The dual boundary element method incorporatestwo independent boundary integral equations: oneis the displacement boundary integral equation andthe other is the traction boundary integral equation.When the displacment equation is applied for col-location on one of the crack surfaces and the tractionequation is applied for collocation on the other,general mixed-mode crack problems can be solvedin a single-region formulation. This feature consti-tutes a practical advantage of the dual boundaryelement method over the finite element and multi-region boundary element methods, because remesh-ing, extensively used in these methods, is no longerrequired when the crack-extension increment ismodelled with new elements.

The new boundary elements, introduced for thediscretization of each crack extension increment,


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DBEM analysis of crack propagation 247principal stress criterion and then introduces acorrection to this direction with information derivedfrom one step ahead of the current crack-extensionincrement.The results of incremental crack-extension analysesof several structures are presented. fatigue crack-growth analysis is also presented for the simplest caseof constant amplitude loading cycles.

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