Engineering Fracture Mechanics 86 (2012) 23–37

Contents lists available at SciVerse ScienceDirect

Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

2D weight function development using a complex Taylor seriesexpansion method

David Wagner, Harry Millwater ⇑Department of Mechanical Engineering, University of Texas at San Antonio, 1 UTSA Circle, TX 78249, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 September 2011Received in revised form 16 January 2012Accepted 1 February 2012

Keywords:Weight functionsStress intensity factorComplex variable sensitivity methodsComplex variable finite element

0013-7944/$ - see front matter � 2012 Elsevier Ltddoi:10.1016/j.engfracmech.2012.02.006

⇑ Corresponding author. Address: 1 UTSA Circle, EE-mail address: harry.millwater@utsa.edu (H. M

Weight functions are a critical component of a damage tolerance fracture control plan inthat they allow the stress intensity factor to be computed quickly from the stress alongthe uncracked crack line. The traditional method to compute weight functions is to use sev-eral (2–4) reference stress solutions or auxiliary conditions to develop the coefficients in aseries solution. While this method has been shown to provide good results in many scenar-ios, the truncated series provides a source of error that is difficult to quantify and themethod requires multiple high-quality reference solutions or other auxiliary information.In contrast, the WCTSE method presented here, provides a method to accurately and effi-ciently develop the weight function for an arbitrary geometry and loading scenario from asingle complex variable finite element solution without other reference solutions or auxil-iary information. The complex Taylor series expansion method is used within the finite ele-ment formulation to obtain the derivatives of the crack opening displacements withrespect to crack length directly from the finite element analysis. These derivatives allowthe direct evaluation of the weight function. The method requires a small perturbationof the crack length along the imaginary axis; the real coordinate mesh is unaltered. Sincethe real coordinate mesh is unaltered, standard finite element meshes and meshing algo-rithms can be used. Given that the error in the weight function is controlled by the accu-racy of the mesh, typical convergence tests can be used to obtain high confidence in theweight functions. Several numerical examples are computed and compared to other wellknown published weight function solutions or finite element (J integral) or boundary ele-ment solutions.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of a ‘‘weight’’ function was developed by Bueckner [1] and Rice [3]. Rice showed that the weight function,m(x, a), for mode I cracks in an isotropic material under a symmetric loading system can be computed from the equation:

mða; xÞ ¼ E0

2KA

@uAða; xÞ@a

ð1Þ

where E0 is an appropriate elastic modulus equal to E/(1 � m2) for plane strain and E for generalized plane stress, a representsthe crack length, u the crack opening displacement, KA the stress intensity factor for loading scenario A, and x is a coordinatealong the crack line. Given the solution for KA and ouA/oa for any mode I loading, the weight function can be computed andcan then be applied to determine the stress intensity factor for the same geometry but for any other mode I loading. That is,

. All rights reserved.

B 3.04.02, San Antonio, TX 78249, United States. Tel.: +1 210 458 4481; fax: +1 210 458 6504.illwater).

Nomenclature

a (m) crack lengthA crack opening displacement function coefficientC curve fit coefficientE (Pa) Young’s elastic modulusE’ (Pa) modulus for plane strain or plane stress conditionsh numerical differentiation step sizeH (m) model heightK (Pa m½) opening-mode stress intensity factorM weight function series coefficientn number of weight function termsQ (m) nodal perturbationR (m) hole radiusu (m) crack line opening displacementW (m) model widthx, y (m) crack face coordinateb weight function coefficientm Poisson’s ratior (Pa) crack face opening-mode stress or applied load

24 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

once the weight function is known for a geometry, the stress intensity factor, KB, for the same geometry but any other loadingcan be computed using the stress along the crack front from the uncracked geometry as

KBðaÞ ¼Z a

0rBðxÞmða; xÞdx ð2Þ

where rB(x) represents the crack front stress from loading scenario B, KB the stress intensity factor, and m(a, x) was deter-mined from the same geometry but a different loading scenario as given in Eq. (1).

Although Eq. (1) is exact, determining the partial derivative ou/oa is challenging in that most reference and numericalsolutions do not contain the functional form of the crack opening displacement. Therefore, computation of u(x) and subse-quently ou/oa must be determined by some other means.

Theoretically, ou/oa can be approximated as ou/oa � Du/Da using finite differencing. That is, the finite element solution issolved once using the crack size of interest to determine u(1)(x, a), then the crack size is perturbed by a small amount(a0 = a + da) and the finite element solution is resolved to compute u(2)(x, a + da). The partial derivative is then estimated as

@uðx; aÞ@a

¼ uð2Þðx; aþ daÞ � uð1Þðx; aÞda

ð3Þ

Computation of ou/oa from Eq. (3), while theoretically possible, rarely leads to accurate results since the numerical valuechosen for da is critical and not known a priori. da must be small in order to have an accurate estimate of the derivative;however, since this method requires the subtraction of two nearly equal numbers, machine round off error can have a sig-nificant effect. As a result, da cannot be too large or too small and formal application of finite differencing to determine thepartial derivative of the crack opening displacement with respect to crack length is hardly ever profitable.

A number of innovative methods have been developed in order to determine weight functions for common structuralconfigurations and approximate or circumvent the calculation of the crack opening displacement derivatives.

Parks and Kamenetzky use a virtual crack extension technique with the finite element method to compute the partialderivative of the crack opening displacement and subsequently the weight functions [22]. The computed weight functionfor a single edge crack was used to compute the stress intensity factor with good results. However, in this approach, a per-turbation of the stiffness matrix with respect to crack length is required and computed using finite differencing which maybe problematic. In addition, significant source code modifications are required.

Grandt used Eq. (1) to develop the weight function for a single and double edged radial crack(s) emanating from a fastenerhole in an infinite plate ([13]). In order to determine the partial derivative of the crack opening displacement, a two-termapproximation of the crack opening displacement using conic sections due to Orange was utilized ([24]) with the coefficientsdetermined using displacement data from a finite element model. Once the coefficients in the conic section were deter-mined, the two-term crack opening displacement approximation was differentiated to determine the weight function.The weight function was subsequently applied to fastener holes under various types of loading. The results for a crack froma hole in an infinite plate under remote tension were compared against an exact solution by Bowie [4] and agreed within 7%.However, as pointed out by the author, these results should be used with caution for situations where cracking extendssignificantly across finite-width plates due to the two-term approximation of the crack opening displacement.

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 25

A similar but more general approach was proposed by Petroski and Achenbach [23] to model the crack opening displace-ments with a two-parameter equation of the form

uða; xÞ ¼ r0

E0ffiffiffi2p 4Fða=LÞa1=2ðx� aÞ1=2 þ Gða=LÞa�1=2ðx� aÞ3=2

n oð4Þ

where x is measured from the crack mouth and x = a is the crack tip. The first term in Eq. (4) is the near-tip field and thesecond term is the correction for larger distances from the crack tip. F is the known geometry correction factor and G isto be determined from the consistency equation

Z a

0½KðaÞ�2da ¼ E0

Z a

0rðxÞuða; xÞdx ð5Þ

The displacement assumption given in Eq. (4) is based on the observation that the crack opening displacement of an edgecrack can be approximated by a power series

u ¼X1i¼0

Aið1� x=aÞiþ1=2 ð6Þ

where Ai are numerical constants.Eq. (4) can be substituted into Eq. (5) and the term G(a/L) determined. Thus, only one reference analysis is needed to com-

pute the weight function.Good results were obtained for many geometries [12,21]; however, it is clear that a two-term approximation will have

limits. Subsequent analyses using the Petroski–Achenbach approach confirmed this limitation. Fett applied this approachto strip loaded edge cracks in a semi-infinite body where the width of the load on the crack line was varied from uniformto point loads. A weight function derived by Bueckner was used for comparison [2]. The results showed that the Petro-ski–Achenbach assumption became highly inaccurate for rapidly changing loading [8]; the crack opening displacement couldnot be described satisfactorily by a two-term series. Glinka and Shen have reported that this method has been shown to beinaccurate for very non-uniform stress fields [11].

Wu and Carlsson proposed a series solution for the crack opening displacement consisting of up to four terms [30]. Theseries expansions for the crack opening displacements are in terms of (1 � (x/a)2)j�1/2 for center cracks and (1 � (x/a))j�1/2 foredge cracks, where j = 1, 2, 3, 4. The four conditions used to determine the coefficients are: crack tip relation, self-consis-tency, zero curvature at the crack mouth, and the crack opening displacement at the crack mouth. If information is not avail-able for all four conditions, the number of terms is reduced to coincide with the information available.

Glinka and Shen propose a modified ‘‘curve-fit-like’’ approach that bypasses the calculation of the ou/oa term. They exam-ine two proposed truncated series solutions for the weight functions, namely series solutions that are integer powers andhalf-integer powers of (1 � x/a), i.e.,

mðx; aÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pða� xÞ

p ½1þM1ð1� x=aÞ þM2ð1� x=aÞ2 þ � � � þMnð1� x=aÞn� ð7Þ

mðx; aÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pða� xÞ

p ½1þM1ð1� x=aÞ1=2 þM2ð1� x=aÞ1 þ � � � þMnð1� x=aÞn=2� ð8Þ

For either of these series solutions, the requirement is to determine the coefficients Mi Glinka and Shen report good accu-racy, �1%, when fitting to analytical results for an edge crack in a semi-finite plate, edge crack in a finite plate through crackin an infinite plate, central through crack in a finite-width plate, double edge cracks in a finite-width plate, edge crack in acircular disk. For these problems, the results generally indicate that three terms, (1, M1, M2), were required for Eq. (7) and 4terms, (1, M1, M2, M3), for Eq. (8).

Fett has proposed a series expansion for the crack opening displacement of edge cracks using the form (1 � (x/a))j�1/2 [9].Reference solutions and the auxiliary conditions for edge cracks at the crack mouth o2uy/oa2 = 0, and o3uy/oa3 = 0 were usedto estimate the coefficients.

Series solutions of one form or another have been shown to be a successful approach with the current state-of-the-artusage of three or four terms. However, it is usually not known a priori how many terms are necessary for sufficient accuracyand the answer will be problem specific. For example, deep cracks or rapidly changing loading may require more terms.

In contrast to the previous approaches, a new methodology is proposed and demonstrated here called Weight functionComplex Taylor Series Expansion (WCTSE) that uses the complex Taylor series expansion (CTSE) complex variable sensitivitymethod embedded within the finite element method to determine the required partial derivative ou/oa at the nodes alongthe crack line, and, subsequently the weight function, directly from a single finite element analysis. Since the quality of theweight function is dependent upon the finite element mesh, the accuracy is under control of the analyst and standard meshconvergence studies can be conducted.

CTSE was originally described by Lyness and Moler [17] and Lyness [18] and was brought to the attention of the engineeringcommunity by Squire and Trapp [25].

26 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

CTSE is a numerical differentiation method similar in spirit and concept to finite differencing but with significant advan-tages. CTSE uses the orthogonality of the real and imaginary axes of the complex plane to calculate first order derivativeswithout any differencing operations. CTSE requires the difference of two analyses but with a small perturbation along theimaginary axis. That is, variable X = x0 is perturbed to X = x0 + ih, where i denotes an imaginary number and h denotes thestep size. The formulae for the derivatives can be derived from the Taylor series representation of the function evaluatedat the complex sample point,

f ðx0 þ ihÞ ¼ f ðx0Þ þ f ð1Þðx0Þih1!þ f ð2Þðx0Þ

ðihÞ2

2!þ f ð3Þðx0Þ

ðihÞ3

3!þ H:O:T: ð9Þ

where f(1) denotes the first derivative, f(2) the second, etc. Taking the imaginary part of both sides of Eq. (9), solving for thefirst derivative and ignoring terms of O(h2) yields an estimate of the first derivative.

f ð1ÞðxoÞ �Imðf ðxo þ ihÞÞ

hð10Þ

Note that no difference operation is needed for the first derivative; compare with Eq. (3) for finite differencing. Thismeans that the step size h can be made arbitrarily small with no concern about round-off error. Note that the real portionof Eq. (9) is also accurate up to O(h2). Thus, use of a small h allows an accurate computation of the traditional real portion ofthe finite element analysis in addition to accurate first order derivatives.

When implemented within a finite element code, the sensitivity of the displacement field with respect to the crack lengthis computed by perturbing the nodal coordinates of the crack length along the imaginary axis, e.g., a + ih, where i denotes animaginary number and h the step size. Upon the establishment of complex nodal coordinates, the stiffness matrix, displace-ments and all derived quantities such as strains and stresses become complex valued. Per the CTSE method, the real portionof the displacement field contains the traditional estimate of the displacements, up to O(h2), and the imaginary portion of thedisplacements contain the derivative of the displacement with respect to the crack size times h. Therefore, the partial deriv-ative of the displacement with respect to crack length is recovered, up to O(h2), simply as

@uða; xÞ@a

� Im½uða; xÞ�h

ð11Þ

which is straightforward to evaluate at each node along the crack line.The power of CTSE is that no differencing operations are required and the step size h can be made as small as desired. It is

quite remarkable that good results have been obtained with very small step sizes of the order 10�150 and smaller. Since h issmall, the accuracy of the displacements and their derivatives are quite good since they are O(h2) accurate. In addition, sinceEq. (11) requires no differencing of near equal numbers and the results are stable over a large range of h, the selection of thestep size is immaterial.

The simplicity of the CTSE approach must be emphasized. There is no need of programming the virtual crack extensionmethod, no truncated series approximation of the crack opening displacements, no series fit required to the weight function(although this can be done for conciseness as shown below), no multiple reference solutions under different loadings re-quired, and no required auxiliary conditions to determine the coefficients. The only requirement is to implement complexnodal coordinates within a standard finite element code and perturb the crack length along the imaginary axis.

CTSE has been applied in several engineering fields but as yet is not widely known nor applied. In fluid dynamics, CTSEhas been used to find sensitivities for the solution of the Navier–Stokes equation [5]. Furthermore, researchers have appliedCTSE techniques to finite element methods in the field of aerodynamics and aero-structural analysis [19,6]. CTSE has alsobeen applied in the study of heat transfer [10], dynamic system optimization [15], pseudospectral [7], eigenvalue sensitivitymethods [29], finite element shape sensitivity analysis ([27]), and fatigue analysis ([28]).

In this research, the WCTSE method is developed and applied to generating weight functions for 2D geometries with pub-lished weight functions. As will be shown, the WCTSE weight functions are at least as good as and often better than pub-lished weight function solutions. It is our contention (as yet unproven) that WCTSE will provide superior resultscompared to all other methods due to its simplicity, generality and accuracy. The only limitation of the WCTSE method isthe quality of the finite element model. In addition, the WCTSE method provides the weight function from a single finiteelement analysis as a post processing operation.

2. Complex variable finite element formulation

A 2D complex variable finite element software (CFEM) was developed that is a standard formulation except for the inclu-sion of complex variable nodal coordinates; commercial finite element software does not allow for the nodal coordinates tobe complex. Internal computations are standard but complex, e.g., computation of a complex stiffness matrix, solution ofcomplex displacements, strains and stresses, etc. The code is linear elastic with eight noded isoparametric quadrilateral ele-ments. Singular elements at the crack tip are modeled with collapsed quarter point elements. The stress intensity factor usedfor comparison of WCTSE weight functions is determined using a domain-based J integral. Using the weakly-typed languageMatlab, the conversion of a standard finite element code to complex was straightforward and handled internally by Matlab

Fig. 1. Schematic of the effect of perturbing only the nodes along the crack line.

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 27

whenever nodal coordinates were complex. Other investigators have developed strategies for modifying strongly-typedcompiled languages [20]. An analytical example for a simple beam problem is presented in Voorhees et al. [27].

2.1. Perturbation of the crack length

When a 2D finite element mesh represents a solid body with a straight crack, there are many ways to perturb the nodes toincrease the crack length by a small imaginary amount ih. One method is to move all the nodes along the crack line in thedirection of crack extension proportional to the distance from the crack mouth so that the tip node moves ih and the nodes atthe mouth remain unperturbed. This method has the disadvantage of changing the relative locations of some of the affectedelements’ mid-side nodes, i.e., the mid-side nodes on the crack line, so they are no longer precisely at mid- or quarter-points,and therefore represent a change in both crack length and a change to the elements’ shape functions, resulting in less accu-rate results. A schematic of this effect is shown below in Fig. 1. As a result, calculation of ou/oa using this method is not suf-ficiently accurate, hence, this method is not recommended.

A second approach is to move only the node at the crack tip a small amount in the direction of the crack extension alongthe imaginary axis. However, this approach also changes the relative location of the side node and represents a change to theshape function in addition to a change of the crack length.

A third approach, recommended here, is to perturb all the elements within a user-defined group around the crack tip as arigid body in the imaginary domain. For example, in Fig. 2, all the nodes along the black line (ring 5) remain unperturbed,whereas the nodes within ring 4 are all perturbed as a rigid body. This perturbation is called Q5 to indicate the all the nodeswithin ring 5 are perturbed. Empirical evidence indicates that the results of perturbing a small domain such as Q5 is generallymore accurate than perturbing the crack tip node only, but either method provides accurate derivatives of the crack openingdisplacement. As described below, the derivatives within the element being perturbed are poor and should be either cor-rected or discarded.

2.2. Adjustment of CTSE solutions within perturbed elements

The imaginary parts of the displacement of all the nodes represent the sensitivity of displacement with respect to increas-ing crack length (ou/oa) as shown in Eq. (9). However, the imaginary displacements of the nodes that themselves have non-zero imaginary perturbations of their coordinates (x) also include the sensitivity of displacement with respect to a change inlocation (ou/ox), which are simply strains. As a result, the estimate of the derivative ou/oa becomes inaccurate for thesenodes. One option is to discard the results for these nodes. For example, if the finite element mesh uses 8-node quadrilateralsalong the crack line and the nodes within ring 5 are perturbed, the ou/oa results from the ten nodes nearest the crack tip willhave nonzero imaginary displacements applied to their nodal coordinates and these ou/oa values should be discarded.

Alternatively, these results can be corrected by subtracting these strains, ou/oa � Im[u]/h � Q ou/ox. However, the strains(ou/ox) computed from a finite element analysis are generally less accurate than the displacements, so the results for ou/oaare less accurate for the nodes with imaginary perturbations applied, where Q is nonzero.

3. WCTSE methodology

The general procedure summarizing WCTSE is as follows:

� Model the cracked geometry for a specific value of a/W where W is a characteristic length, typically a plate width.� Perturb the crack length along the imaginary axis as described above.

(a) (b)

Fig. 2. Schematic of rigid body perturbation of nodes around the crack tip in the imaginary domain. (a) unperturbed mesh and (b) perturbed mesh.

28 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

� Run the complex finite element code and extract the partial derivative,ou/oa, for each node along the crack line. Eithercorrect or ignore the results for the nodes within the perturbed domain since the quality of the derivatives is poor.� Using the results from the nodal locations along the crack line, fit the results to a convenient representation such as a

series expansion in integer powers or half-integer powers of 1 � x/a using least squares minimization. For symmetricproblems, an expansion in terms of (1 � x/a)2 may be used.� Repeat the above process for different crack lengths of a/W and determine the coefficients as a function of a/W to develop

m(a, x).� Assess the accuracy of the resulting weight function through comparisons with known stress intensity factor results, by

conducting a convergence study of the finite element model, and/or computing the stress intensity factor vs. the numberof terms in the series expansion.

From the finite element analysis, the weight function can be computed piecewise over each element. However, it is con-venient for tabularizing the weight function to fit the computed weight to well known series expansion such as proposed byWu and Carlsson [30], Glinka and Shen [11], Fett [9] and others. However, unlike previous researchers, there is no need toassume a three or four term expansion. Since there is a large number of nodal results available, least squares minimizationcan be used to determine the values for a large number of coefficients. This avoids predefining a small number of coefficientsto consider.

Combining the weight function definition, equation (1), with an appropriate series expansion such as Glinka and Shen’sequation (7) truncated to n terms, allows approximating the weight function by a simple least-squares polynomial fit to afunction of only the opening derivatives, material properties, and geometry. That is, an independent solution for KA is notrequired.

mða; xÞ ¼ E0

2KA

@uAða; xÞ@a

� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pða� xÞ

p ½1þM1ð1� x=aÞ þM2ð1� x=aÞ2 þ � � � þMnð1� x=aÞn� ð12Þ

Rearranging Eq. (12) yields

E0

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipða� xÞ

2

r@uAða; xÞ

@a� KA½1þM1ð1� x=aÞ þM2ð1� x=aÞ2 þ � � � þMnð1� x=aÞn� ð13Þ

Using the WCTSE approximation for the derivatives and applying a coordinate change of x0 = 1 � x/a, x = a(1 � x’) to theright hand side makes the polynomial form clear.

E0

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipða� xÞ

2

rIm½uAða; xÞ�

h� C0 þ C1x0 þ C2x02 þ � � � þ Cnx0n ð14Þ

Matching the fitted polynomial from Eq. (14) term-by-term to the right hand side of Eq. (13) yields the stress intensityfactor and the coefficients of the weight function

KA � C0; Mj �Cj

C0ð15Þ

As seen from Eqs. (14) and (15), the intercept from the least squares curve fitting process provides the stress intensityfactor. Therefore, a stress intensity factor solution is not required but can be obtained from this same finite element modeland used in place of C0 for the normalization, if desired.

This process can be used with the appropriate coordinate transform for other weight function forms such as Glinka andShen’s equation (8), or Wu and Carlsson’s form in terms of (1 � (x/a)2)j�1/2.

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 29

4. Numerical examples

The power of the WCTSE method is that the weight function for any 2D geometry of interest can be modeled and com-puted from a single finite element analysis for each normalized crack length of interest. For verification purposes, four stan-dard two-dimensional problems in fracture mechanics and a crack from a hole were analyzed and compared with exact orother published weight function solutions: (a) an infinite array of cracks, (b) a single edge crack in a finite-width panel, (c) adouble edge crack in a finite-width panel, (d) a center crack in a finite-width panel, and (e) a single edge crack from a hole ina finite-width panel.

The four standard two-dimensional problems may all be modeled by applying four different fixed-displacement (Dirich-let) boundary conditions to the same finite-width strip geometry as shown in Fig. 3.

The infinite strip geometry in the vertical direction was approximated by modeling a finite strip with a large aspect ratioof height to width of 20. The horizontal symmetry plane allows modeling only the top portion of the strip and ensures onlyopening-mode displacements along the crack line for all four problems, regardless of the load applied. Remote tension is theobvious choice for reference loading to determine the weight function, and the finite height of the single edge crack also al-lows remote bending to be applied counter-clockwise to open the crack.

4.1. WCTSE example using an infinite array of collinear cracks

A detailed explanation of the WCTSE weight function development method for an infinite array of collinear cracks is pre-sented since this problem has an exact solution for the crack opening displacement, its derivatives, and the resulting weightfunction.

Fig. 3. Schematic of geometries for single edge crack, double edge crack, center crack, and infinite array of collinear cracks.

30 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

The exact stress intensity factor solution is [30].

Fig. 4.five dif

Kexact ¼ rffiffiffiffiffiffipap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Wpa

tanpa2W

rð16Þ

with a and W defined in Fig. 3.The exact weight function is [30].

mexactða; xÞ ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pa=Wp cos

px2W

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipa2W

tan pa2W

sin2 pa2W� sin2 px

2W

!vuut ð17Þ

Finite element models of an infinite array of collinear cracks were developed for a/W values of 0.1, 0.3, 0.5, 0.7, and 0.9consisting of between 67 to 181 nodes along the crack line, where the number of nodes depends upon the crack depth, bymodeling a single half crack and using symmetry as described in Fig. 3. Accurate stress intensity factors and crack line open-ing derivatives were calculated by using a finite element mesh with 18 semi-circumferential elements and 30 radial ele-ments around the crack tip. The smallest element was approximately 10�4 of the crack length. Symmetry was used suchthat only one-half of the strip was modeled as shown in Fig. 3. The loading was pure tension.

The partial derivative of the displacement field with respect to the crack length was computed using CTSE by perturbingthe crack tip and the nodes within the fourth contour around the crack tip along the imaginary axis, similar to the rigid-bodymotion illustrated in Fig. 2b, using a step size of 10�9 times the smallest element at the crack tip, e.g., h/a � 10�13.

Fig. 4a shows the crack opening displacement (black markers) plotted along the crack line computed using the CTSEmethod against the exact solution (solid line) [30] for five different crack lengths. Fig. 4b shows the partial derivative ofthe crack opening displacement with respect to crack length computed using CTSE (black markers) against the exact solution(solid line). These results confirm the high accuracy of ou/oa obtained using CTSE.

A comparison of the WCTSE weight function with an approximate 3-term weight function for the infinite array is avail-able, with the bi coefficients tabulated for selected crack lengths using an equation of the form [30]

mWuða; xÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pa=Wp Xn

i¼1

bi 1� xa

� �2� �i�3=2

ð18Þ

The WCTSE solution was directly obtained from a single complex variable finite element analysis for each normalizedcrack length. The WCTSE crack opening displacement derivatives were fit by a least-squares polynomial for direct compar-ison to the reference approximation using an equation of the form

10-5

10-4

10-3

10-2

0.1 0.3 0.5 0.7 0.9

duy/d

a

x/W

Crack Face Displacement Derivatives with Respect to Crack Length

CTSEExact

0

5e-06

1e-05

1.5e-05

2e-05

2.5e-05

3e-05

0.1 0.3 0.5 0.7 0.9

u y/W

x/W

Infinite Array Normal Crack Face Displacements at Selected Crack Lengths

CTSEExact

Crack opening displacement (top) and crack opening displacement partial derivatives with respect to crack length (bottom) along the crack line forferent crack lengths.

Table 1Weight

a/W

0.100.200.300.400.500.600.700.800.90

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 31

E0

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipða2 � x2Þ

a

rIm½uða; xÞ�

h� KA

Xn

i¼0

Ci 1� xa

� �2� �i

ð19Þ

Using Eq. (19), Fig. 5 shows a least squares fit of the left hand side versus 1 � (x/a)2 for a/W = 0.7 that was used to computethe Ci coefficients. The results from the regression analysis indicate that the exact stress intensity factor value (1.98122), i.e.,the intercept, is recovered accurately using only the crack opening displacement derivatives, that is, no reference solution KA

is required; the stress intensity factor is computed from the displacement derivative results.Using the relation bi+1 = 2Ci/KA i e [0, 1, 2, 3], the approximate weight function coefficients using the WCTSE results were

computed for direct comparison with a reference approximation [30]. The WCTSE results are in agreement with published bvalues at tabulated crack lengths, as shown in Table 1, and provide more accurate results at deeper crack lengths.

Fig. 6 compares the approximate weight functions to the exact solution for a/W = 0.7 in terms of the relative error in theweight function, i.e., Err = (m(a, x) �mexact)/mexact. Three weight functions are shown: the 3-term expansion by Wu (red line),a three term expansion computed by WCTSE (pink line), and a 7-term expansion (blue line). The WCTSE results were all ob-tained from the same single set of finite element analyses; higher order expansions could be computed from the sameresults.

As shown in Fig. 6, the 3-term approximation due to Wu is within 0.03% for a/W = 0.7. The 3-term WCTSE expansion iswithin 0.007% and the 7-term WCTSE expansion is within 0.004%.

Further numerical comparisons yield the following results. For a/W = 0.9, the max error of the WCTSE 3-term weight func-tion is about �0.75%, and the error in Wu’s 3-term weight function is about + 1.65%. The 7-term WCTSE weight function iswithin ±0.002% over the entire crack length. For all 17 crack lengths analyzed, from a/W = 0.1 to 0.9 at intervals of 0.05, the 7-term WCTSE weight functions are within ±0.01% of the exact weight function over the entire crack length.

4.2. Single edge crack in a finite-width strip

The behavior of an infinite strip cracked along its width can be approximated accurately by a finite rectangular platewhen the plate’s height-to-width ratio is much greater than its crack-length-to-width ratio (H/W P 20a). Accurate stress

y = 0.0447x3 - 0.2511x2 + 1.4917x + 1.9812 R² = 1

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fitte

d Fu

nctio

n: y

= K

m(a

,x)

Transformed Crack Face Ordinate, Crack Tip at Left: x=1-(x'/a)2

K m(a,x)

Poly. (K m(a,x))

Fig. 5. Least squares fitting of WCTSE results for infinite array of cracks (a/W = 0.7).

function coefficients at selected crack lengths.

Reference values [Wu] WCTSE values

b1(a) b2(a) b3(a) b4(a) b1(a) b2(a) b3(a) b4(a)

0 2.0000 0.0166 0.0000 0.0000 2.0000 0.0168 0.0004 �0.00040 2.0000 0.0683 �0.0005 0.0000 2.0000 0.0682 �0.0002 �0.00020 2.0000 0.1613 �0.0027 0.0001 2.0000 0.1612 �0.0024 �0.00040 2.0000 0.3086 �0.0101 0.0005 2.0000 0.3086 �0.0096 0.00000 2.0000 0.5355 �0.0316 0.0027 2.0000 0.5354 �0.0308 0.00200 2.0000 0.8949 �0.0921 0.0128 2.0000 0.8938 �0.0886 0.01060 2.0000 1.5135 �0.2756 0.0595 2.0000 1.5056 �0.2532 0.04500 2.0000 2.7845 �0.9523 0.3035 2.0000 2.7256 �0.7880 0.19740 2.0000 6.7888 �4.9892 2.2501 2.0000 6.1264 �3.1676 1.0744

Fig. 6. Relative error of approximate weight functions for infinite array of cracks for 3 [Wu], 3 [WCTSE], and 7 [WCTSE] terms.

Fig. 7. Close up of CFEM mesh for a single edge crack in finite-width strip.

32 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

intensity factors and crack line opening derivatives were calculated by using a finite element mesh with 18 circumferentialelements and 30 radial elements around the crack tip. The smallest element was approximately 10�4 of the crack length.Symmetry was used such that only one-half of the strip was modeled as shown in Fig. 3. The loading was pure tension.To preserve element squareness, the radial elements decreased in size at a fixed ratio, decreasing geometrically from the out-ermost contour to the crack tip. A close up of the finite element mesh near the crack tip is shown in Fig. 7. The J-integralresult over the domain within the sixteenth circular contour from the crack tip provides the reference stress intensity factorKRef = K16.

CFEM was used to analyze a set of single-edge-cracked models under remote tension with crack lengths from 0.1 to 0.9 ofthe width using fine meshes and the weight functions were determined using WCTSE. After the WCTSE weight function wasdetermined using remote tension, the stress intensity factors were computed for both tension and bending applied loads.Subsequently, other reference solutions (non-weight function) and other published weight function solutions were also usedto compute K under tension and bending for comparison. 16-point Gauss–Jacobi integration quadrature was used in all casesfor the WCTSE results. In addition, finite element J integral and boundary element methods were used to determine K forboth the tension and bending loading cases.

Fig. 8 compares the stress intensity factor computed using a 15-term WCTSE-developed weight function against othernon-weight function solutions (other weight function solutions are compared in Fig. 9). The reference load for WCTSEwas pure tension. All results are referenced against the J-integral result obtained from the finite element model. The tensionreference solutions are from [30,26]. The boundary element solution is from [16]. The application of the weight functions tobending is a more strident test than tension and a more realistic example of how the weight function is likely to be used inpractice. The WCTSE results are clearly very good over the full range of crack lengths of 0.1–0.9 of a/W with differences with-in 0.004% of the reference value.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

Single Edge Crack in a Finite−width StripWCTSE Weight Functions and Reference Stress Intensity Factors (H/W=20a)

Tension Ref.[Wu] Tension WCTSE 15−term Tension BEMTension Ref.[Tada] Bending WCTSE 15−term Bending BEM

Crack Length (a/W)

Com

paris

on to

Dom

ain

Inte

gral

(K/K

16)

Fig. 8. Comparison of stress intensity factors for tension and bending against non-weight function reference solutions for a single edge crack in a finitewidth strip.

Single Edge Crack in a Finite−width StripWCTSE Weight Functions and Published Weight Functions (H/W=20a)

Tension WCTSE 15−term Tension [Wu] Tension [Glinka91] Tension [Kaya]Bending WCTSE 15−term Bending [Wu] Bending [Glinka91] Bending [Kaya]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Crack Length (a/W)

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

Com

paris

on to

Dom

ain

Inte

gral

(K/K

16)

Fig. 9. Comparison of stress intensity factors for tension and bending against other published weight function solutions for a single edge crack in a finitewidth strip.

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 33

Fig. 9 compares the stress intensity factor computed using the 15-term WCTSE weight function against the results fromintegrating other published weight function solutions. All results are referenced against the J Integral result. The WCTSE re-sults are clearly very good over the full range of crack lengths of 0.1–0.9 of a/W and are more accurate than the results fromthe other weight functions. Wu’s stress intensity factor computed for bending shows a difference of about 2% at a/W = 0.9.Kaya (green line) [14] and Glinka (orange symbol) [11] results show a difference from the J-integral result that varies be-tween 1% and 3%. The WCTSE weight function results are all within 0.004% of the J-integral results. The WCTSE result isthe only bending result that consistently compares well with J integral results.

4.3. Double edge crack in a finite-width strip

The weight function results for a double edge crack in a finite-width strip was determined using the same finite elementmesh as for the single edge crack through the addition of a vertical symmetry boundary as shown in Fig. 3.

CFEM was used to analyze a set of double-edge-cracked models under remote tension with crack lengths from 0.1 to 0.9of the width using fine meshes and the weight functions were determined using WCTSE. After the WCTSE weight functionwas determined using remote tension, the stress intensity factors were computed for the same tension load case.

0.98

0.99

1.00

1.01

1.02

1.03

1.04

Double Edge CrackTension Stress Intensity Factor (H/W=20a)

Reference SIF [Wu] WCTSE 11−term Weight Fn. [Wu] Weight Fn. [Glinka91]Weight Fn. [Tada] Weight Fn. [Glinka96] BEM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Crack Length (a/W)

Com

paris

on to

Dom

ain

Inte

gral

(K/K

16)

Fig. 10. Comparison of stress intensity factor results for a double edge crack.

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

Center CrackTension Stress Intensity Factor (H/W=20a)

Reference SIF [And.] WCTSE 13−term Weight Fn. [Wu] Weight Fn. [Glinka91] BEM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Crack Length (a/W)

Com

paris

on to

Dom

ain

Inte

gral

(K/K

16)

Fig. 11. Comparison of stress intensity factor results for a center crack in a finite width panel.

34 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

Subsequently, other reference weight functions were also used to compute K for comparison and a J-integral and boundaryelement models were also solved. 16-point Gauss–Jacobi integration quadrature was used for all weight functions. The re-sults are referenced against the J-integral results. Fig. 10 clearly shows the accuracy of the WCTSE results and its superioraccuracy against other published solutions. The stress intensity factor computed using the WCTSE 11-term weight functionwas within 0.002% of the reference value.

4.4. Center crack in a finite-width strip

The weight function results for a center crack in a finite-width strip were determined using the same finite element meshas for the single edge crack through the application of the appropriate symmetry boundary conditions as shown in Fig. 3.

CFEM was used to analyze a set of center-cracked models under remote tension with crack lengths from 0.1 to 0.9 of thewidth using fine meshes and the weight functions were determined using WCTSE. After the WCTSE weight function wasdetermined using remote tension, the stress intensity factors were computed for the same tension load case. Subsequently,other reference weight functions were also used to compute K for comparison and J-integral and boundary element modelswere also solved. 16-point Gauss–Jacobi integration quadrature was used for all weight functions. The results are referenced

0.97

0.98

0.99

1.00

1.01

1.02

1.03

Hole with a Single CrackStress Intensity Factor (R/W=0.2, H/W=2)

Tension Ref.[Wu] Tension WCTSE Tension [Wu] Tension BEMBending WCTSE Bending [Wu] Bending BEM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Crack Length (a/B)

Com

paris

on to

Dom

ain

Inte

gral

(K/K

16)

Fig. 12. Comparison of stress intensity factor results for a single edge crack from a hole, R/W = 0.2.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550.97

0.98

0.99

1.00

1.01

1.02

1.03

Hole with a Single CrackStress Inensity Factor (R/W=0.5, H/W=2)

Tension Ref.[Wu] Tension WCTSE Tension [Wu] Tension BEMBending WCTSE Bending [Wu] Bending BEM

Crack Length (a/B)

Com

paris

on to

Dom

ain

Inte

gral

(K/K

16)

Fig. 13. Comparison of stress intensity factor results for a single edge crack from a hole, R/W = 0.5.

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 35

against J integral results. Fig. 11 clearly shows the accuracy of the WCTSE results and its superior accuracy against other pub-lished solutions. The 13-term WCTSE stress intensity factor was within 0.002% of the J-integral results.

4.5. Single radial crack from a hole in a finite-width strip

A crack emanating from a hole is an important structural configuration for fatigue. The WCTSE method was used to de-velop the weight function for a single edge crack from a hole and compared with other solutions. The weight function wasdeveloped under tension perpendicular to the crack then applied under both tension and bending loading.

Fig. 12 compares the stress intensity factors calculated using a 15-term WCTSE against those from published stress inten-sity factor and weight function formulas [30] for R/W = 0.2 (where R denotes the hole radius) and Fig. 13 shows the results for

36 D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37

R/W = 0.5. The J-integral result from finite element analysis is used as the reference. As seen from the figures, the WCTSEresults are clearly more accurate than other published solutions for this example. For R/W = 0.2, the WCTSE computed stressintensity factor results are within 0.06% for tension and 0.04% or bending of the J-integral results. For R/W = 0.5, the WCTSEcomputed stress intensity factor results are within 0.15% for tension and bending of the J-integral results.

5. Conclusions

Knowledge of weight function solutions for cracked geometries of interest is a staple of fatigue and lifing analysis of struc-tures. Weight functions allow the calculation of the stress intensity factor given the crack-line stresses from the uncrackedmodel.

In this research, a new method (WCTSE) for computing the weight function for an arbitrary 2D geometry (e.g. throughcracks) using the complex Taylor series expansion (CTSE) method was implemented and verified that has significant advan-tages over existing methods. In the WCTSE approach, the complex Taylor series expansion method was used to accuratelyand efficiently compute the partial derivative of the crack opening displacement with respect to the crack length using acomplex variable finite element code. In WCTSE, the crack length is given a small perturbation along the imaginary axisthrough the application of complex nodal coordinates (the crack length becomes complex) resulting in complex displace-ments, strains, stresses, etc. As a result, the partial derivatives of the crack opening displacement with respect to the cracklength are contained in the imaginary component of the displacements and easily and accurately determined. The partialderivatives are then used to compute the weight functions using only a single complex finite element solution for any geom-etry. These results can then be fit to any series expansion of interest for convenience and it is simple to consider any numberof terms in the polynomial expansion up to the number of nodes along the crack line; thus, it is straightforward to assess thebenefit of additional terms.

The new method was verified for various geometric cases: an infinite array of collinear cracks (analytical solution exists),and finite geometry cases of an edge crack, a single- and a double-edge crack, a central crack, and a radial crack emanatingfrom an open hole. In each of these cases, the stress intensity factor computed using the WCTSE method showed results moreaccurate than existing published weight functions when applied to tension and bending as compared to analytical or J-inte-gral results. It is anticipated that the WCTSE weight functions will show increased superiority under more severe loadingconditions.

The advantages to the new approach are its accuracy and conceptual simplicity. The accuracy of the weight functions isonly limited by the accuracy of the utilized finite element mesh. Only a single finite element analysis is required rather thanthree or more reference solutions as currently needed. The method is simple in concept in that besides making the finiteelement program complex variable, no other significant modifications are needed. However, this will entail a systemic alter-ation of finite element software to be complex variable based. It is expected that this new approach will lead to the devel-opment of ‘‘personalized weight functions’’ for any particular problem of interest rather than relying on approximate weightfunctions in the literature.

Acknowledgements

This work was funded in part by the DoD High Performance Computing Modernization Program (HPCMP) User Produc-tivity Enhancement, Technology Transfer and Training (PETTT) initiative (Pre-planned Project PP-CSM-KY02-004-P3) and theNational Science Foundation (HRD-0932339) through the CREST Center for Simulation, Visualization and Real TimeComputing.

References

[1] Bueckner HF. A novel principle for the computation of stress intensity factors. Zeitschrift fur Angewandte Mathematik und Mechanik 1970;50:529–46;Rice J. Some remarks on elastic crack-tip stress field. Int J Solids Struct 1972;8:751–8.

[2] Buckner H. Field singularities and related integral representations. In: Sih GC, editor. Mechanics of fracture I – Methods of analysis and solution of crackproblems. Noordhoff, Leyden; 1979.

[3] Rice JR. Some remarks on elastic crack-tip stress field. Int J Solids Struct 1972;8:751–8.[4] Bowie L. Analysis of an infinite plate containing radial cracks originating from the boundary of an internal circular hole. J Math Phys 1956;35:60–71.[5] Anderson WK, Newman JC, Whitfield DL, Nielsen EJ. Sensitivity Analysis for Navier–Stokes Equations on Unstructured Meshes Using Complex

Variables. AIAA Journal 2001;39(1):56–63.[6] Burg COE, Newman JC. Computationally efficient, numerically exact design space derivatives via the complex Taylor’s series expansion method.

Comput Fluids 2003;32(3):373–83.[7] Cervino LI, Bewley TR. On the extension of the complex-step derivative technique to pseudospectral algorithms. J Comput Phys 2003;187:544–9.[8] Fett T. Limitations of the Petroski–Achenbach procedure demonstrated for a simple load case. Engng Fract Mech 1988;29:713–6.[9] Fett T. Direct determination of weight functions from reference loading cases and geometrical conditions. Engng Fract Mech 1992;42(3):435–44.

[10] Gao XW, He MC. A new inverse analysis for multi-region heat conduction BEM using complex-variable-differentiation method. Engng Anal BoundaryElem 2005;29(8):788–95.

[11] Glinka G, Shen G. Universal features of weight functions for cracks in mode I. Engng Fract Mech 1991;40(6):1135–46.[12] Gorner F, Mattheck C, Morawietz P. Limitations of the Petroski–Achenbach crack opening displacement approximation for the calculation of weight

functions. Engng Fract Mech 1985;22:269–77.[13] Grandt Jr AF. Stress intensity factors for some through-cracked fastener holes. Int J Fract 1975;11(2):283–94.[14] Kaya AC, Erdogan F. Stress intensity factors and COD in an orthotropic strip. Int J Fract 1980;16:171–90.

D. Wagner, H. Millwater / Engineering Fracture Mechanics 86 (2012) 23–37 37

[15] Kim J, Bates DG, Postlethwaite I. Nonlinear robust performance analysis using complex-step gradient approximation. Automatica 2006;42(1):177–82.[16] Li S, Mear ME, Xiao L. Symmetric weak-form integral equation method for three-dimensional fracture analysis. Comput Methods Appl Mech Engng

1998;151:435–59.[17] Lyness JN, Moler CB. Numerical differentiation of analytic functions. SIAM J Numer Anal 1967;4(2):202–10.[18] Lyness JN. Differentiation formulas for analytic functions. Math Comput 1968;22(102):352–62.[19] Newman JC, Whitfield DL, Anderson WK. Step-size independent approach for multidisciplinary sensitivity analysis. J Aircraft 2003;40(3):566–73.[20] Martins JRRA, Sturdza P, Alonso JJ. The complex-step derivative approximation. ACM Trans Math Software 2003;29:245–63.[21] Niu X, Glinka G. On the limitations of the Petroski–Achenbach crack opening displacement approximation for the calculation of weight function-do

they really exist? Engng Fract Mech 1987;26:701–6.[22] Parks DM, Kamenetzky EM. Weight functions from virtual crack extension. Int J Numer Meth Engng 1979;14:693–1706.[23] Petroski HY, Achenbach FD. Computation of the weight function from a stress intensity factor. Engng Fract Mech 1978;10:257–66.[24] Orange TW. Crack shapes and stress intensity factors for edge-cracked specimens. In: Stress analysis and growth of cracks, proceedings of the 1971

national symposium on fracture mechanics, Part I, ASTM STP 513, American Society for Testing and Materials; 1972. p. 71–8.[25] Squire W, Trapp G. Using complex variables to estimate derivatives of real functions. SIAM Rev 1998;40(1):110–2.[26] Tada H, Paris PC, Irwin GR. Stress analysis of cracks handbook. 3rd ed. ASME; 2000.[27] Voorhees A, Millwater HR, Bagley RL. Complex variable methods for shape sensitivity of finite element models. Finite Elem Anal Des 2011;47:1146–56.

doi:10.1016/j.finel.2011.05.003.[28] Voorhees A, Millwater HR, Bagley R, Golden P. Fatigue Sensitivity Analysis Using Complex Variable Methods. Int J Fatigue 2012. doi:10.1016/

j.ijfatigue.2012.01.016.[29] Wang BP, Apte AP. Complex variable method for eigensolution sensitivity analysis. AIAA J 2006;44:2958–61.[30] Wu X-R, Carlsson AJ. Weight Functions and Stress Intensity Factor Solutions. Pergamon Press; 1991.