The interaction of doubly periodic cracks

Theoretical and Applied Fracture Mechanics 42 (2004) 249–294

www.elsevier.com/locate/tafmec

The interaction of doubly periodic cracks

G.S. Wang

Key Laboratory of Rock and Soil Mechanics, Institute of Rock and Soil Mechanics,

Chinese Academy of Sciences, Wuhan 430071, China

Available online 5 November 2004

Abstract

In this paper, an extremely accurate and efficient method for computing the interaction of a set of or multiple sets ofgeneral doubly periodic cracks has been presented on the basis of superposition principle, pseudo-traction method, andisolating analysis technique. A great number of typical examples are given in this paper. The stress intensity factors(SIF), the minimum strain energy density factors (SED) of crack tip and the critical stress (CRS) of crack growthare calculated with the accuracy of six significant digits for the rectangularly distributed periodic cracks and five signif-icant digits for the general doubly periodic cracks. The relation of the interaction effect of the double periodic crackswith the periods and the ratio of crack length to crack spacing is analyzed. Also in this paper, the key technique prob-lems for this method are discussed.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Interaction of doubly periodic cracks; Stress intensity factor; Minimum strain energy density factor; Pseudo-tractionmethod

1. Introduction

There are a great number of parallel and doublyperiodic cracks in the stratified rock mass andcomposite materials. Research on the interactionof doubly periodic cracks under the load actionis important to the safe application of the compos-ite materials and the stability analysis of the rockmass engineering structure. A great quantity of re-search work on the interaction of periodic cracks

0167-8442/$ - see front matter � 2004 Elsevier Ltd. All rights reservdoi:10.1016/j.tafmec.2004.09.003

E-mail address: [email protected]

has been performed. Because of the complex nat-ure of the interaction problem, the analytic solu-tion exists only under the condition of singleperiodic collinear cracks, which has been obtainedin [1,2]. And for nearly all the other periodic crackproblems, the analytical solution does not exist orcan not be obtained, and the numerical computa-tion methods have to be used. The numerical solu-tion of the single row of echelon cracks has beenobtained through the Laurent series expansionmethod in [3], and through the Chebyshev poly-nomial and Cauchy integral method in [4]. Themethod of pseudo-traction is introduced in [5]

ed.

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250 G.S. Wang / Theoretical and Applied Fracture Mechanics 42 (2004) 249–294

and it is used to calculate the interaction of multi-ple holes, the interaction of multiple cracks andgeneralized to an infinite row of equal circularholes and an infinite row of parallel cracks. Thestress intensity factor for two rows of overlappingperiodic equal-size collinear cracks is calculated byusing the singular integral equation method in [6].In [7], the interaction of multiple rows of periodiccracks is analyzed by using the superposition prin-ciple, pseudo-traction method and isolating analy-sis technique. The other reports related with theinteraction of single periodic cracks can be foundin [8–12]. And the reports related with the multiplecracks are still more in number and are not listedin details here.

For the problems of plane doubly periodiccracks, even though under the condition of rectan-gularly distributed periodic cracks, the analyticsolution can also not be obtained. The stress inten-sity factors for rectangular array of crack is calcu-lated by singular integral equation method in [13]and by the boundary collocation technique in[14]. A systematic method of estimating the overallproperty of solids with periodic cracks is presentedin [9]. The methods of dealing with the problems ofperiodically distributed Z-shape and U-shapecracks, of the doubly periodic array of curvilinearcracks, and of the double periodically distributedflat inclusions can be found in [15,16] and [17],respectively. The other reports related with theinteraction of doubly periodic cracks can be foundin [18–23].

The methods of dealing with the interaction ofdoubly periodic cracks could be roughly classifiedinto three types, that is, the boundary collocationmethod based on the unit cell, the integral equa-tion method based on the complex potential func-tion of Muskhelishivili, and the pseudo-tractionmethod based on the series expansion and super-position principle.

In the boundary collocation technique devel-oped by [14], the problem of infinite plate contain-ing rectangular distributed doubly periodic cracksis transformed into a problem of finite rectangularplate containing a center crack which is known asunit cell. The length and width of the rectangularblock are equal to the spacing and pitch of the rec-tangular distributed doubly periodic crack array.

Since the rectangularly distributed doubly periodiccrack array is axial symmetric about axis x andaxis y in local coordinate system of any crack,the boundary conditions on the periphery of therectangular unit cell are easy to be given as knownquantities. But for the general doubly periodiccracks, the axial symmetry does not exist, the unitcell is not rectangle but parallelogram, the bound-ary conditions on the periphery are impossible ordifficult to be given as known quantities. So theboundary collocation technique in [14] is notapplicable in solution of the interaction of the gen-eral doubly periodic cracks.

The integral equation method for dealing withthe problem of doubly periodic cracks [13,16,20,23] is mainly based on the mathematical theoryof plane elasticity of Muskhelishivili. In themethod, the complex potential functions with dou-bly periodic character are taken as the unknownfunctions, and the Cauchy-type singular integralequations are established based on the boundarycondition. But solving these integral equations isvery difficult. For the general doubly periodiccracks and multiple sets of doubly periodiccracks, the difficulty of solution is greater, needlessto say.

The pseudo-traction method and the stresssuperposition technique [5,24–27] are the most effi-cient methods for solving the interaction of cracks.It is applicable to research on the interaction ofmultiple cracks, periodic cracks as well as cracksand holes. The main steps of this method are asfollows. Firstly, the problem of an infinite platecontaining cracks under a far-field uniform stressis decomposed into a series of sub-problems. Theone of them is an infinite no-crack plate under afar-field uniform stress. And each of the othersub-problems is that an infinite plate contains acrack with pseudo-traction exerted on the crackface but without uniform stress field at the infinite.Secondly, for all sub-problems with a crack, thepseudo-traction exerted on crack faces are ex-panded into the form of series with unknown coef-ficients, and the corresponding stress functionsexpressed by complex variable function are ob-tained. Thirdly, the stress fields produced by allsub-problems are superposed in order to formtotal stress field. Making the total stress field to

G.S. Wang / Theoretical and Applied Fracture Mechanics 42 (2004) 249–294 251

meet the boundary conditions of all cracks, thealgebraic equation system for the unknown coeffi-cients is obtained. Finally, solving the equationsystem, the unknown coefficients are determinedand the total stress field and the stress intensityfactors of all crack tips can be calculated. Thedecomposition process of problem of multiplecracks is called as isolating analysis of cracks in[7], for the sake of making the description of theprocess more conveniently and directly.

In the process of solving the interaction of peri-odic cracks using the isolating analysis technique,pseudo-traction method and superposition princi-ple, the most key problem is the summation ofcomplex variable function series with infinite num-ber of terms. Any negligence may cause the unac-ceptable error, and lead to the instability ofequation system or distortion of solution. In [7],a famous summation equation of complex variablefunction series with infinite number of terms isintroduced, and the interaction of multiple rowsof periodic cracks is successfully solved with extre-mely high accuracy. In this paper, the techniquewill be extended to the research on the interactionof doubly periodic cracks and multiple sets of dou-bly periodic cracks.

2. General formulation

2.1. The isolating analysis of doubly periodic

cracks

Let us consider an infinite elastic plate contain-ing a set of doubly periodic cracks applied by a farfield uniform stress f rx ry sxy gT. The coordi-nate system is set up on any one crack and letx-axis be on the crack faces and y-axis verticalto the crack faces and through the center of crack.The complex z0 and z1 are defined as the main per-iod and the auxiliary period respectively, and theycan be set to any complex values but have differ-ent argument, that is, Im(z1/z0)5 0, otherwisethe points z0, z1 and the original point are in astraight line, and the problem will be reduced tothe singly periodic cracks. The first step of solvingthe double-periodic-cracks problem is the isolatinganalysis, as does in solving the single-periodic-

cracks problem in [7]. By using the superpositionprinciple and pseudo-traction method [5,24–27],the problem of an infinite elastic plate containingdoubly periodic cracks can be decomposed into aseries sub-problems. One of them is that a no-crackinfinite plate is under a far-field stress. And for theother sub-problems, each is that an infinite platecontains a crack with pseudo-traction acting onits surfaces but without uniform stress field at infin-ity. The pseudo-tractions on the face of an isolatedcrack are induced by all the other cracks andthe infinite uniform stress field. And they areunknowns. Since the crack distribution is periodic,and all cracks have the same length and orienta-tion, so all the isolated cracks have the same pseu-do-traction distribution and the same stress field.The stress field of an isolated crack, that is, singlecrack contained in an infinite elastic plate, withpseudo normal and tangential tractions appliedon its face, can be expressed by the Westergaardstress functions as follows (see [28,29]):

rxðzÞ¼ReZIðzÞ�yImZ 0IðzÞþ2ImZIIðzÞþyReZ 0

IIðzÞryðzÞ¼ReZIðzÞþyImZ 0

IðzÞ�yReZ 0IIðzÞ

sxyðzÞ¼�yReZ 0IðzÞþReZIIðzÞ�yImZ 0

IIðzÞð1Þ

where ZI(z) and ZII(z) are the Westergaard stressfunctions, respectively, and Z 0

IðzÞ and Z 0IIðzÞ are

their derivative functions. z is the complex varia-ble, z = x + iy, where i ¼

ffiffiffiffiffiffiffi�1

p. Based on [7],

ZI(z) and ZII(z) are given as following complexfunction series:

ZIðzÞZIIðzÞ

� �¼X1n¼0

An

Bn

� �GnðzÞ ð2Þ

where An and Bn are the unknown real coefficients,and

GnðzÞ ¼

za�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiza

� �2� 1

r" #nþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiza

� �2� 1

r ð3Þ

where a is the half length of the isolated crack.It must be pointed out that the Westergaard

stress function formulation missing constants inthe case of applied normal and shear stress at


infinity. But in the case of isolated crack withpseudo-tractions on their faces, the equation for-mulated by using Westergaard function methodis same as the result inferred by using Mushelishi-vili method. In [30], it is pointed out that Westerg-aard stress function is really a special of themethod of Muskhelishivili.

Let the stress fields induced by all isolatedcracks and the uniform far-field stress come intosuperposition. It is noted that though all the iso-lated cracks have the same stress field, the samefunction form Gk and the same efficient Ak andBk, but they take up different places in the planeconsidered. So the total stress field of doubly peri-odic cracked plate is obtained as follows:

rxðzÞ ¼ r1x þ

X1k¼0

fAk½Re/kðzÞ � ImwkðzÞ�

þ Bk½2Im/kðzÞ þRewkðzÞ�g

ryðzÞ ¼ r1y þ

X1k¼0

fAk½Re/kðzÞ þ ImwkðzÞ�

þ Bk½�RewkðzÞ�g

sxyðzÞ ¼ s1xy þX1k¼0

fAk½�RewkðzÞ�

þ Bk½Re/kðzÞ � ImwkðzÞ�g

ð4Þ

where

/kðzÞ¼Xþ1

l¼�1

Xþ1

r¼�1Gkðz�lz0�rz1Þ

wkðzÞ¼Xþ1

l¼�1

Xþ1

r¼�1Im ðz�lz0�rz1Þ

d

dzGkðz�lz0�rz1Þ

ð5ÞFor the problem of multiple sets of doubly peri-odic cracks, following procedure is adopted. Sup-pose that there are M sets of doubly periodiccracks in an infinite elastic plate. The cracks in dif-ferent sets have the different crack lengths and ori-entations. For the mth set of doubly periodic crackarray, the crack length is 2am, the orientation is hm,and the two periods are zm0 and zm1. The globalcoordinate system is expressed by z, and the localcoordinate system of the mth set is zm. Wherez = x + iy, zm = xm + iym, xm is along the face ofcenter crack of the mth set, and ym is vertical to

the face of the crack. In order to avoid overlappingor crossing each other of the cracks between thedifferent sets, following constrains are declared.The periods of each set can be different with refer-ence to the local coordinate system of itself, but tothe global coordinate system, the periods of everyset are of the same value, that is

z10eih1 ¼ z20eih2 ¼ � � � � � � ¼ zM0eihM ¼ z0

z11eih1 ¼ z21eih2 ¼ � � � � � � ¼ zM1eihM ¼ z1

ð6Þ

The problem of M sets of doubly periodic cracksin an infinite plate under far-field stress can bedecomposed into M + 1 sub-problems. One ofthem is that an infinite no-crack elastic plate is ap-plied by far-field stress. For the other M sub-prob-lems, each is that an infinite plate only containsone set of doubly periodic cracks with pseudo-traction exerted on their faces but without uniformstress at infinity. And each of the M sub-problemscan be further decomposed into infinite number ofsimpler sub-problems in which an infinite plateonly contains one crack with pseudo-traction ex-erted on its faces but without uniform stress atinfinity.

For any isolated crack of the mth set of doublyperiodic cracks, the Westergaard stress functionscan be expressed as follows:

ZðmÞI ðzmÞ

ZðmÞII ðzmÞ

( )¼X1k¼0

AðmÞk

BðmÞk

( )GðmÞ

k ðzmÞ ð7Þ

Let frðmÞðnÞ g express the stress column induced by the

mth set of isolated doubly periodic cracks in thelocal coordinate system zn, that is

rðmÞðnÞ

n o¼ rðmÞ

xnðzmÞ rðmÞ

ynðzmÞ sðmÞxnyn

ðzmÞn oT

ð8Þ

Then the transformation of stress column inducedby the mth set of doubly periodic cracks from zmlocal coordinate system to zn local coordinate sys-tem can be expressed as follows:

rðmÞðnÞ

n o¼ ½Rðhn � hmÞ� rðmÞ

ðmÞ

n oð9Þ


where [R(h)] is the following stress transformmatrix:

½RðhÞ� ¼cos2h sin2h sin 2h

sin2h cos2h � sin 2h

� 12sin 2h 1

2sin 2h cos 2h

264

375 ð10Þ

in which hn is the orientation of the nth set of dou-bly periodic cracks in the global coordinate sys-tem. Then the total stress field in the globalcoordinate system z and the local coordinate sys-tem zn are

rf g ¼ r1f g þXMm¼1

½Rð�hmÞ� rðmÞðmÞ

n oð11Þ

rðnÞ� �

¼ r1ðnÞ

n oþXMm¼1

½Rðhn � hmÞ� rðmÞðmÞ

n oð12Þ

where M is the number of sets of the doubly peri-odic cracks in the plate, {r1} is the far-field uni-form stress in the coordinate system z, and fr1

ðnÞgis the far-field uniform stress in the coordinate sys-tem zn.

The total stress field should meet the wholeboundary conditions of all cracks. Practically, ifthe boundary conditions of the center crack ofeach set of doubly periodic cracks are met, thenthe boundary conditions of all cracks will be metdue to the periodicity of the functions. In the caseof the far-field stress being tension and shear, theboundary condition on the center crack faces ofthe nth set of doubly periodic cracks are as follows:

rynðxnÞ ¼ 0; sxnynðxnÞ ¼ 0

ð�an < xn < an; n ¼ 1; 2; . . . ;MÞ ð13Þ

In practical computation, the number of seriesterms in Eq. (7) is only able to be finite value N,so the boundary conditions are satisfied only inthe meaning of segmental averaging. To considerthe stress concentration in the zone around thecrack tip, the following collocation points areselected:

nj ¼xjnan

¼ cosj� ðN þ 1Þ

N þ 1p

� ;

j ¼ 0; 1; 2; . . . ;N þ 1 ð14Þ

Substituting the Eq. (12) into (13), and thenintegrating and averaging it on each segment, aftersome calculation and arrangement, the algebraicequation system about unknown coefficients AðmÞ

k

and BðmÞk (m = 1,2, . . .,M; k = 0,1,2, . . .,N) is

obtained:

XMm¼1

XNk¼0

½AðmÞk ðInm1kjþInm4kjÞþBðmÞ

k ð�Inm2kj� Inm3kjÞ�¼�p1n

XMm¼1

XNk¼0

½AðmÞk ðInm2kj�Inm3kjÞþBðmÞ

k ðInm1kj� Inm4kjÞ�¼�q1n

ðn¼1;2;...;M ; j¼0;1;2;...;NÞð15Þ

where

Inm1kj ¼1

jCjnjcosðhn � hmÞRe ½UðmÞ

k ðzmÞ�zm¼zjmBzm¼zjmA

Inm2kj ¼1

jCjnjsinðhn � hmÞRe ½UðmÞ


Inm3kj ¼1

jCjnjRe ½eiðhn�hmÞWðmÞ


Inm4kj ¼1

jCjnjIm ½eiðhn�hmÞWðmÞ


ð16Þ

jCjnj ¼ an cos

jþ 1� ðN þ 1ÞN þ 1

p

� �

� cosj� ðN þ 1Þ

N þ 1p

� �; j ¼ 0; 1; 2; . . . ;N

ð17Þ

UðmÞk ðzmÞ ¼

Xþ1

l¼�1

Xþ1

r¼�1G

ðmÞk ðzm � lzm0 � rzm1Þ

WðmÞk ðzmÞ ¼

Xþ1

l¼�1

Xþ1

r¼�1Im ðzm � lzm0 � rzm1Þ

� GðmÞk ðzm � lzm0 � rzm1Þ

ð18Þ

zjmA ¼ xjnAeiðhn�hmÞ þ ðzcn � zcmÞe�ihm ;

zjmB ¼ xjnBeiðhn�hmÞ þ ðzcn � zcmÞe�ihm

ð19Þ

Table 2Dimensionless SIF of the rectangularly distributed doubly periodic cracks under the action of far-field tension stress perpendicular tothe crack faces versus a/d, h/d and with lm and rm increasing (when N = 16)

hd

ad

rm, lm

2 4 6 8 10 12 14 16 18 20

0.5 0.10 0.983509 0.983511 0.983511 0.983511 0.983511 0.983511 0.983511 0.983511 0.983511 0.9835110.20 0.948297 0.948316 0.948317 0.948318 0.948318 0.948318 0.948318 0.948318 0.948319 0.9483190.30 0.921414 0.921494 0.921500 0.921503 0.921504 0.921505 0.921505 0.921505 0.921505 0.9215060.40 0.922314 0.922523 0.922542 0.922550 0.922553 0.922555 0.922556 0.922557 0.922558 0.9225580.50 0.961516 0.961954 0.962003 0.962021 0.962030 0.962035 0.962038 0.962040 0.962042 0.9620430.60 1.048921 1.049750 1.049865 1.049909 1.049931 1.049943 1.049950 1.049955 1.049958 1.0499610.70 1.203721 1.205295 1.205566 1.205669 1.205719 1.205747 1.205765 1.205776 1.205784 1.2057900.80 1.475527 1.478833 1.179504 1.479759 1.479883 1.479953 1.479995 1.480023 1.480043 1.4800570.90 2.067125 2.076117 2.078125 2.07889 2.079261 2.079468 2.079596 2.07968 2.079738 2.0797800.99 6.257672 6.336430 6.354610 6.361542 6.364904 6.366785 6.367943 6.368705 6.369234 6.369616

1.0 0.10 1.002829 1.002830 1.002830 1.002830 1.002830 1.002830 1.002830 1.002830 1.002830 1.0028300.20 1.012184 1.012188 1.012189 1.012189 1.012189 1.012189 1.012189 1.012190 1.012190 1.0121900.30 1.030585 1.030608 1.030613 1.030615 1.030616 1.030616 1.030617 1.030617 1.030617 1.0306170.40 1.062150 1.062225 1.062242 1.062249 1.062252 1.062254 1.062255 1.062256 1.062256 1.0622560.50 1.113012 1.113212 1.113258 1.113276 1.113284 1.113289 1.113292 1.113294 1.113295 1.1132960.60 1.193214 1.193687 1.193797 1.193839 1.193859 1.193870 1.193877 1.193882 1.193885 1.1938870.70 1.322816 1.323881 1.324127 1.324221 1.324266 1.324291 1.324307 1.324317 1.324325 1.3243300.80 1.554232 1.556676 1.557240 1.557455 1.557559 1.557617 1.557653 1.557677 1.557693 1.5577050.90 2.101015 2.107663 2.109199 2.109784 2.110068 2.110227 2.110325 2.110389 2.110434 2.1104660.99 6.293704 6.348034 6.360654 6.365472 6.36781 6.369119 6.369924 6.370455 6.370823 6.371088

Table 1Dimensionless SIF of the rectangularly distributed doubly periodic cracks under the action of far-field tension stress perpendicular tothe crack faces versus a/d, h/d and with N increasing (when lm = rm = 20)

hd

ad

N

2 4 6 8 10 12 14 16 30

0.5 0.10 0.983512 0.983511 0.983511 0.983511 0.983511 0.983511 0.983511 0.983511 0.9835110.20 0.948395 0.948318 0.948319 0.948319 0.948319 0.948319 0.948319 0.948319 0.9483190.30 0.922074 0.921495 0.921506 0.921506 0.921506 0.921506 0.921506 0.921506 0.9215060.40 0.924474 0.922516 0.922559 0.922558 0.922558 0.922558 0.922558 0.922558 0.9225580.50 0.966249 0.961959 0.962041 0.962043 0.962043 0.962043 0.962043 0.962043 0.9620430.60 1.057104 1.049910 1.049945 1.049962 1.049961 1.049961 1.049961 1.049961 1.0499610.70 1.216004 1.205966 1.205724 1.205792 1.205790 1.205790 1.205790 1.205790 1.2057900.80 1.491451 1.480420 1.479807 1.480053 1.480057 1.480057 1.480057 1.480057 1.4800570.90 2.070606 2.074251 2.077705 2.079439 2.079710 2.079764 2.079777 2.079780 2.0797810.99 5.730125 5.944739 6.106422 6.215002 6.283792 6.326778 6.353331 6.369616 6.394333

1.0 0.10 1.002830 1.002830 1.002830 1.002830 1.002830 1.002830 1.002830 1.002830 1.0028300.20 1.012191 1.012190 1.012190 1.012190 1.012190 1.012190 1.012190 1.012190 1.0121900.30 1.030632 1.030617 1.030617 1.030617 1.030617 1.030617 1.030617 1.030617 1.0306170.40 1.062325 1.062255 1.062256 1.062256 1.062256 1.062256 1.062256 1.062256 1.0622560.50 1.113482 1.113289 1.113296 1.113296 1.113296 1.113296 1.113296 1.113296 1.1132960.60 1.194168 1.193854 1.193887 1.193887 1.193887 1.193887 1.193887 1.193887 1.1938870.70 1.324075 1.324178 1.324324 1.324329 1.324330 1.324330 1.324330 1.324330 1.3243300.80 1.553224 1.556842 1.557626 1.557696 1.557704 1.557705 1.557705 1.557705 1.5577050.90 2.077202 2.102171 2.108774 2.110108 2.110389 2.11045 2.110463 2.110466 2.1104670.99 5.710076 5.943002 6.107918 6.216193 6.285138 6.328202 6.354786 6.371088 6.395824

KI ¼ KI=rffiffiffiffiffiffipa

p.


Table 3Dimensionless SIF, SED and CRS of the rectangularly distributed doubly periodic cracks under the action of far-field tension stress perpendicular to the crack faces atdifferent a/d, h/d (where KI ¼ KI=r

ffiffiffiffiffiffipa

p, Smin ¼ Smin=½ð1� 2mÞr2a=4l�, rcr ¼ rcr=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4lScr=ð1� 2mÞa

p, l is shear moduli. In computation, Poisson�s ratio is supposed as

m = 0.3, lm = rm = 20, N = 16 for a/d 5 0.9, N = 30 for a/d > 0.9)ad

h/d

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 1KI 0.1 0.966190 0.983511 0.992479 0.997420 1.000240 1.001876 1.002830 1.004064 1.004140 1.004145

0.2 0.899923 0.948319 0.976499 0.993090 1.002922 1.008749 1.012190 1.016686 1.016965 1.0169820.3 0.853130 0.921506 0.966635 0.995369 1.013217 1.024093 1.030617 1.039258 1.039798 1.0398300.4 0.847174 0.922558 0.977086 1.014133 1.038125 1.053122 1.062256 1.074509 1.075281 1.0753270.5 0.886239 0.962043 1.019070 1.059219 1.085898 1.102858 1.113296 1.127429 1.128325 1.1283790.6 0.976822 1.049961 1.104070 1.142158 1.167592 1.183845 1.193887 1.207541 1.208410 1.2084650.7 1.139723 1.205790 1.251552 1.282723 1.303247 1.316288 1.324330 1.335259 1.335958 1.3360050.8 1.430344 1.480057 1.511328 1.531555 1.544544 1.552701 1.557705 1.564494 1.564936 1.5649740.9 2.057467 2.079781 2.092600 2.100511 2.105480 2.108573 2.110467 2.113069 2.113265 2.1133070.99 6.393640 6.394333 6.394763 6.395093 6.395371 6.395613 6.395824 6.396513 6.396815 6.398004

Smin 0.1 0.933523 0.967294 0.985015 0.994847 1.000480 1.003755 1.005667 1.008145 1.008298 1.0083070.2 0.809862 0.899308 0.953551 0.986228 1.005852 1.017575 1.024528 1.033650 1.034218 1.0342520.3 0.727830 0.849173 0.934383 0.990760 1.026609 1.048766 1.062172 1.080056 1.081179 1.0812470.4 0.717703 0.851114 0.954697 1.028466 1.077704 1.109067 1.128389 1.154569 1.156229 1.1563280.5 0.785419 0.925526 1.038504 1.121945 1.179174 1.216296 1.239429 1.271096 1.273117 1.2732400.6 0.954181 1.102418 1.218972 1.304524 1.363271 1.401488 1.425367 1.458156 1.460256 1.4603870.7 1.298968 1.453929 1.566382 1.645379 1.698452 1.732614 1.753849 1.782917 1.784785 1.7849090.8 2.045884 2.190569 2.284113 2.345661 2.385615 2.410881 2.426446 2.447642 2.449025 2.4491430.9 4.233171 4.325490 4.378974 4.412145 4.433047 4.446082 4.454071 4.465060 4.465889 4.4660650.99 40.87863 40.88749 40.89300 40.89721 40.90077 40.90386 40.90657 40.91538 40.91924 40.93446

rcr 0.1 1.034993 1.016766 1.007578 1.002586 0.99976 0.998128 0.997178 0.995952 0.995877 0.9958720.2 1.111206 1.054498 1.024066 1.006958 0.997087 0.991327 0.987957 0.983588 0.983318 0.9833020.3 1.172155 1.085181 1.034517 1.004652 0.986955 0.976474 0.970292 0.962225 0.961725 0.9616960.4 1.180395 1.083942 1.023451 0.986064 0.963275 0.949557 0.941392 0.930658 0.929990 0.9299500.5 1.128364 1.039455 0.981287 0.944092 0.920897 0.906735 0.898233 0.886974 0.886270 0.8862270.6 1.023728 0.952416 0.905739 0.875536 0.856463 0.844705 0.837600 0.828129 0.827533 0.8274960.7 0.877406 0.829332 0.799008 0.779591 0.767314 0.759712 0.755099 0.748918 0.748526 0.7485000.8 0.699133 0.675650 0.661670 0.652931 0.647440 0.644039 0.641970 0.639184 0.639004 0.6389880.9 0.486035 0.480820 0.477874 0.476075 0.474951 0.474254 0.473829 0.473245 0.473201 0.4731920.99 0.156406 0.156389 0.156379 0.156371 0.156364 0.156358 0.156353 0.156336 0.156329 0.156299

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xjnA ¼ an cosj� N � 1

N þ 1p

�;

xjnB ¼ an cosjþ 1� N � 1

N þ 1p

� ð20Þ

where GðmÞk ðzmÞ is the integral function of GðmÞ

k ðzmÞ,and

Fig. 1. Dimensionless SED and CRS of the rectangularly distributed dperpendicular to the crack faces at different a/d, h/d. (The figure corrfactor. Bottom (b) Normalized critical stress.

p1nq1n

� �¼ sin2hn cos2hn � sin2hn

� 12sin2hn 1

2sin2hn cos2hn

" # r1x

r1y

s1xy

8><>:

9>=>;

ð21Þ

oubly periodic cracks under the action of far-field tension stressesponds to Table 3.). Top (a) Normalized strain energy density

Table 4Dimensionless SIF, SED and CRS of the rectangularly distributed doubly periodic cracks under the action of far-field shear stress parallel to the crack faces at differenta/d, h/d (where KII ¼ KII=s


p, Smin ¼ Smin=½ð2� 2m� m2Þs2a=12l�, scr ¼ scr=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12lScr=ð2� 2m� m2Þa

p, l is shear moduli. In computation, Poisson�s ratio is supposed as

m = 0.3, lm = rm = 20, N = 16 for a/d 5 0.9, N = 30 for a/d > 0.9)ad

h/d

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 1KII 0.1 1.024639 1.016024 1.011288 1.008493 1.006790 1.005744 1.005103 1.004210 1.004149 1.004145

0.2 1.088092 1.059459 1.042895 1.032860 1.026668 1.022841 1.020492 1.017221 1.016996 1.0169820.3 1.170442 1.120199 1.089585 1.070516 1.058596 1.051190 1.046637 1.040293 1.039858 1.0398300.4 1.257739 1.189377 1.146450 1.119299 1.102225 1.091602 1.085072 1.075989 1.075367 1.0753270.5 1.344678 1.263180 1.212065 1.179902 1.159793 1.147342 1.139716 1.129148 1.128425 1.1283790.6 1.432743 1.345051 1.291847 1.259236 1.239207 1.226946 1.219487 1.209211 1.208510 1.2084650.7 1.534896 1.452392 1.405238 1.377502 1.360903 1.350895 1.344861 1.336606 1.336043 1.3360050.8 1.702810 1.641846 1.60942 1.591183 1.580559 1.574252 1.570482 1.565354 1.565002 1.5649740.9 2.169589 2.143454 2.130369 2.123264 2.119208 2.116826 2.115409 2.113476 2.113332 2.1133070.99 6.398283 6.397788 6.397631 6.397581 6.397557 6.397531 6.397499 6.397308 6.397179 6.398001

Smin 0.1 1.049886 1.032304 1.022704 1.017059 1.013627 1.011520 1.010231 1.008438 1.008314 1.0083070.2 1.183944 1.122453 1.087629 1.066799 1.054046 1.046204 1.041405 1.034738 1.034280 1.0342520.3 1.369933 1.254846 1.187194 1.146004 1.120625 1.105001 1.095449 1.082210 1.081304 1.0812470.4 1.581907 1.414617 1.314347 1.252830 1.214899 1.191595 1.177382 1.157753 1.156413 1.1563280.5 1.808160 1.595623 1.469103 1.392169 1.345119 1.316393 1.298953 1.274976 1.273343 1.2732400.6 2.052753 1.809163 1.668869 1.585675 1.535635 1.505396 1.487149 1.462191 1.460496 1.4603870.7 2.355906 2.109443 1.974694 1.897511 1.852056 1.824918 1.808652 1.786515 1.785011 1.7849090.8 2.899563 2.695658 2.590232 2.531865 2.498167 2.478270 2.466414 2.450333 2.449230 2.4491430.9 4.707115 4.594393 4.538474 4.508252 4.491042 4.480952 4.474956 4.466782 4.466171 4.4660650.99 40.93802 40.93169 40.92969 40.92905 40.92873 40.92841 40.92800 40.92555 40.92390 40.93442

�scr 0.1 0.975953 0.984229 0.988838 0.991578 0.993255 0.994289 0.994923 0.995808 0.995869 0.9958720.2 0.919040 0.943878 0.958870 0.968186 0.974025 0.977669 0.979919 0.983071 0.983288 0.9833020.3 0.854378 0.892698 0.917781 0.934129 0.944648 0.951303 0.955441 0.961267 0.961670 0.9616960.4 0.795078 0.840777 0.872258 0.893417 0.907256 0.916085 0.921598 0.929377 0.929915 0.9299500.5 0.743672 0.791653 0.825038 0.847528 0.862223 0.871580 0.877411 0.885623 0.886191 0.8862270.6 0.697962 0.743466 0.774085 0.794132 0.806967 0.815032 0.820017 0.826985 0.827465 0.8274960.7 0.651510 0.688519 0.711623 0.725952 0.734806 0.740250 0.743571 0.748164 0.748479 0.7485000.8 0.587264 0.609071 0.621342 0.628463 0.632688 0.635222 0.636747 0.638833 0.638977 0.6389880.9 0.460917 0.466537 0.469402 0.470973 0.471874 0.472405 0.472722 0.473154 0.473186 0.4731920.99 0.156292 0.156304 0.156308 0.156309 0.156310 0.156310 0.156311 0.156316 0.156319 0.156299

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From the equation system (15), all unknown coef-ficients AðnÞ

k , BðnÞk (k = 0,1,2, . . .,N; n = 1,2, . . .,M)

can be determined.

3. The stress intensity factors (SIF) and strain

energy density factors (SED) of crack tip

After the coefficients AðmÞk ;BðmÞ

k (k = 0,1,2, . . .,N; m = 1,2, . . .,M) are solved, the total stress

Fig. 2. Dimensionless SED and CRS of the rectangularly distributedparallel to the crack faces at different a/d, h/d. (The figure correspondBottom (b) Normalized critical stress.

field can be easily evaluated by using the Eq.(11) or (12). Because the functions GðmÞ

k ðzmÞ(k = 0,1,2, . . .,N; m = 1,2, . . .,M) are singularat zm = ±am, the total stress field reflectsthe singularities of all crack tips. The stressintensity factors (SIF) of any crack tip can beevaluated by way of transforming the total stressfield in the local coordinate system of thecrack and on the basis of the equations asfollows:

doubly periodic cracks under the action of far-field shear: stresss to Table 4.) Top (a) Normalized strain energy density factor.

12l


KðnÞI;�an

KðnÞII;�an

( )¼ Lim

zn!�an

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pjzn � anj

p rynðznÞsxnynðznÞ

� �

ðn ¼ 1; 2; . . . ;MÞ ð22Þ

Substituting Eq. (12) into (22), noting that at thetip of the central crack for the nth set, only thestress part induced by the isolated crack itself(m = n, l = r = 0) has the contribution to the singu-larity of stress, and noting the equation as follows:

Limzn!�an

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p j zn � an j

pGðnÞ

k ðznÞ ¼ffiffiffiffiffiffiffipan

p ð�1Þk ð23Þ

the mode I and mode II stress intensity factors ofthe nth set cracks can be obtained

KðnÞI;�an

KðnÞII;�an

( )¼ ffiffiffiffiffiffiffi

panp XNn

k¼0

ð�1ÞkAðnÞk

BðnÞk

( );

n ¼ 1; 2; . . . ;M ð24Þ

The strain energy density factor (SED) for the tipof the nth set of cracks is (based on [31])

SðnÞ�an ¼ a11ðKðnÞ

I;�anÞ2 þ a12K

ðnÞI;�anK

ðnÞII;�an þ a22ðKðnÞ

II;�anÞ2

ð25Þwhere

a11 ¼1

16pl½ð3� 4v� cos hÞð1þ cos hÞ�

a12 ¼1

16pl2 sin h½cos h� ð1� 2vÞ�

a22 ¼1

16pl½4ð1� vÞð1� cos hÞ

þ ð1þ cos hÞð3 cos h� 1Þ�

Table 5Comparison of the computation results for the dimensionless SIF of rof far-field tension stress perpendicular to the crack faces with the re

ad

Method in this paper

hd¼ 0:5 1.0 1.5

0.10 0.983511 1.002830 1.004060.20 0.948319 1.012190 1.016680.30 0.921506 1.030617 1.039250.40 0.922558 1.062256 1.074500.50 0.962043 1.113296 1.127420.60 1.049961 1.193887 1.207540.70 1.205790 1.324330 1.335250.80 1.480057 1.557705 1.564490.90 2.079781 2.110467 2.113060.99 6.394333 6.395824 6.39651

where h is the angle taking the crack tip as originpoint and the crack face as polar axis.

The SED of crack tip varies with the angle h.Based on the assumption presented by [31], thecrack initiation will start in a radical directionalong which the strain energy density is the mini-mum. From the necessary and sufficient conditionsfor S to be a minimum, that is

oSoh

¼ 0;o2S

oh2> 0 at h ¼ hcr ð26Þ

the minimum of strain energy density factor Smin

and initiation angle hcr can be evaluated.If the minimum strain energy density factor

exceeds the threshold Scr, the cracks will extend.Where Scr is the critical strain energy density fac-tor, it is the material constant. Since the longerthe crack is, the greater the stress intensity factorand strain energy density factor will be, so Scr

corresponds to the limit stress field which thecracked body could bear. Defined rcr and scr asthe critical tension stress and critical shear stressof cracked body, respectively, it is not difficult tocalculate them if Scr is known. For the singlecrack problem, the minimum strain energy den-sity factor Smin can be given from Eq. (25) asfollows:

S0min ¼

ð1� 2mÞ4l

r2a for mode I

S0min ¼

ð2� 2m� m2Þs2a for mode II

ð27Þ

ectangularly distributed doubly periodic cracks under the actionsults computed by Isida (1981)

Isida (1981) [14]

hd¼ 0:5 1.0 1.5

4 0.983 1.003 1.0046 0.948 1.012 1.0178 0.922 1.031 1.0399 0.923 1.062 1.0749 0.962 1.113 1.1271 1.050 1.194 1.2079 1.206 1.324 1.334493

Table 6Dimensionless SIF, SED and CRS of doubly periodic cracks with one direction collinear applied by a far uniform tension stress field perpendicular to the crack faces atdifferent a/d, c0/d and h/d (KI ¼ KI=r


p, KII ¼ KII=r




p, the unit of hcr is degree, l is shear moduli.

Poisson�s ratio is supposed as m = 0.3. lm = rm = 20, N = 16 for a/d 5 0.9, N = 30 for a/d > 0.9)

hd

c0d

a/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99

0.5 0 KI 0.983511 0.948319 0.921506 0.922558 0.962043 1.049961 1.20579 1.480057 2.079781 6.394333KII 0 0 0 0 0 0 0 0 0 0Smin 0.967294 0.899308 0.849173 0.851114 0.925526 1.102418 1.453929 2.190569 4.32549 40.88749hcr 0 0 0 0 0 0 0 0 0 0rcr 1.016766 1.054498 1.085181 1.083942 1.039455 0.952416 0.829332 0.67565 0.48082 0.156388

0.1 KI 0.985137 0.953324 0.929424 0.932225 0.972528 1.060498 1.215342 1.487052 2.082802 6.394418KII �0.00602 �0.01994 �0.03436 �0.0449 �0.05022 �0.04983 �0.04296 �0.02935 �0.01194 �0.0004Smin 0.970573 0.909672 0.866335 0.873322 0.951164 1.129927 1.480975 2.213153 4.338368 40.88858hcr 0.350298 1.199567 2.12425 2.77422 2.977054 2.706199 2.030965 1.13195 0.328443 0.003572rcr 1.015047 1.048474 1.074378 1.070071 1.025351 0.940751 0.821724 0.672193 0.480106 0.156386






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1 KI 1.018103 1.072672 1.161698 1.273458 1.382694 1.467603 1.543869 1.690511 2.15877 6.396195KII 0 0 0 �1E�06 �4E�06 �0.00001 �2.4E�05 �5.4E�05 �0.00014 �0.00101Smin 1.036533 1.150626 1.349542 1.621696 1.911841 2.153859 2.383533 2.857829 4.660288 40.91132hcr 0 0.000004 0.00002 0.000065 0.000171 0.000405 0.000888 0.001824 0.003652 0.009047rcr 0.982219 0.932251 0.860809 0.785263 0.723226 0.681383 0.647723 0.591537 0.463227 0.156343

1.0 0.0 KI 1.00283 1.01219 1.030617 1.062256 1.113296 1.193887 1.32433 1.557705 2.110467 6.395824KII 0 0 0 0 0 0 0 0 0 0Smin 1.005667 1.024528 1.062172 1.128389 1.239429 1.425367 1.753849 2.426446 4.454071 40.90657hcr 0 0 0 0 0 0 0 0 0 0rcr 0.997178 0.987957 0.970292 0.941392 0.898233 0.8376 0.755099 0.64197 0.473829 0.156352




(continued on next page)

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Table 6 (continued)

hd

c0d

a/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99







1 KI 1.005438 1.021747 1.04913 1.088718 1.144002 1.223639 1.348139 1.572446 2.116028 6.396573KII 0 0 0 �1E�06 �3E�06 �7E�06 �1.5E�05 �3.5E�05 �9.4E�05 �0.00077Smin 1.010906 1.043967 1.100674 1.185307 1.30874 1.497292 1.817479 2.472586 4.477575 40.91614hcr 0 0.000003 0.000017 0.000057 0.000145 0.000322 0.000656 0.001279 0.00256 0.006872rcr 0.994591 0.978716 0.953171 0.918512 0.874125 0.817235 0.741763 0.635952 0.472584 0.156334


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then the critical stress (CRS) rcr, scr of single crackproblem can be expressed as

r0cr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4lScr

ð1� 2mÞa

s; s0cr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12lScr

ð2� 2m� m2Þa

sð28Þ

In this paper, the SIF, SED and CRS of doubleperiodic cracks are normalized by the SIF, SEDand CRS of single crack, respectively.

4. Numerical computation technique

The summation of infinite number of functionseries occurs in the calculation of Inmakj(a = 1,2,3,4), i.e.

UðmÞk ðzmÞ ¼

Xþ1

l¼�1

Xþ1

r¼�1G

ðmÞk ðzm � lzm0 � rzm1Þ

WðmÞk ðzmÞ ¼

Xþ1

l¼�1

Xþ1

r¼�1Imðzm � lzm0 � rzm1Þ

� GðmÞk ðzm � lzm0 � rzm1Þ

ð29Þ

Obviously, it is impossible to directly compute thesummation of two layers of infinite terms in prac-tice. But, if only finite terms are calculated, greaterror may be caused, especially when 2am/jzm0j > 1or 2am/jzm1j > 1. Even though the number of termstaken is very great, the accuracy is still not satisfac-tory. So it is important to develop a techniquewhich could provide satisfying accuracy with lesscomputer work needed. It is noted that if jzj islarge enough, the following approximation isreliable:

GkðzÞ ¼ � ak þ 1

a2z

� �kþ1

þ Oa2z

� �kþ3

GkðzÞ ¼ 2a2z

� �kþ2

þ Oa2z

� �kþ4ð30Þ

In the solution of single periodic cracks and multi-ple rows of periodic cracks [7] the following series(see [32]) has been used:

Xþ1

k¼�1

1

z� k¼ 1

zþXþ1

k¼1

2z

z2 � k2¼ pctgðpzÞ ð31Þ

By using the above equation, the following equa-tion can be obtained:

Xþ1

l¼�1

Xþ1

r¼�1

1

z� lz0 � rz1¼ 1

z0pctg p

zz0

�8>><>>:

þ 2pX1r¼1

sin 2pzz0

�

cos 2prz1z0

�� cos 2p

zz0

�9>>=>>; ð32Þ

In Eq. (32), though the problem is reduced to thesummation of a single layer of infinite terms, thedirect computation is also impossible, so the newtechnique is required.

The complex variable z can be on any point inthe whole plane, but its varying range can practi-cally be limited in a parallelogram due to the dou-bly periodicity of the distribution of cracks. Theparallelogram area takes the center of centralcrack as its center, and its four sides are parallelto the vector z0 and z1 in orientation and equalto jz0j and jz1j in length, respectively. So whenthe integer r is large enough, or in other words,when r is larger than a certain positive integerrm, following inequality is tenable:

cos 2pzz0

�� cos 2pr

z1z0

��

and j cosð2pr z1z0Þ j will exponentially increase with

the increasing of r. It is not difficult to prove thatif the integer r is large enough, then the approxi-mation as follows is accurate enough.

cos 2prz1z0

��

12e�i�2prz1z0 if Im

z1z0

�> 0

12ei�2prz1z0 if Im

z1z0

�< 0

8>>><>>>:

So it is easy to obtain the summation of doubleseriesXþ1

l¼�1

Xþ1

r¼�1

1

z� lz0 � rz1¼ 1

z0pctg p

zz0

�

þ 2pXrmr¼1

sin 2pzz0

�

cos 2prz1z0

�� cos 2p

zz0

�

þ 4p sin 2pzz0

�armþ1

1� a

!ð33Þ


where

a ¼ei�2pz1z0 if Im

z1z0

�> 0

e�i�2pz1z0 if Imz1z0

�< 0

8>>><>>>:

By way of the differential technique, followingequation can be obtained:

Fig. 3. Dimensionless SED and CRS of doubly periodic cracks withperpendicular to the crack faces at different a/d and h/d when h/d = 0.5energy density factor. Bottom (b) Normalized critical stress.

Xþ1

l¼�1

Xþ1

r¼�1

1

ðz� lz0 � rz1Þkþ1

¼ ð�1Þk

k!dk

dzkXþ1

l¼�1

Xþ1

r¼�1

1

z� lz0 � rz1;

k ¼ 0; 1; 2; . . . ;N ð34Þ

one direction collinear applied by a far uniform tension field. (The figure corresponds to Table 6.) Top (a) Normalized strain


By means of the similar procedure, the series

summationPþ1

l¼�1Pþ1

r¼�1Im ðz�lz0�rz1Þðz�lz0�rz1Þkþ2 (k = 0,1,

2, . . .,N) also can be obtained. So the series func-

tions UðmÞk ðzmÞ and WðmÞ

k ðzmÞ can be expressed asfollows:

UðmÞk ðzmÞ ¼

Xlml¼�lm

Xrmr¼�rm

GðmÞk ðzm � lzm0 � rzm1Þ

� amkþ 1

am2

� �kþ1 ð�1Þk

k!dk

dzkmF ðmÞ

1 ðzmÞ(

�Xlml¼�lm

Xrmr¼�rm

1

ðzm � lzm0 � rzm1Þkþ1

)

WðmÞk ðzmÞ ¼

Xlml¼�lm

Xrmr¼�rm

Im ðzm � lzm0 � rzm1Þ

�GðmÞk ðzm � lzm0 � rzm1Þ

þ 2am2

� �kþ2 Im ðzm0Þzm0

ð�1Þk

k!dk

dzkmF ðmÞ

1 ðzmÞ(

þ Im ðzmÞ �zmzm0

Im ðzm0Þ �

�ð�1Þkþ1

ðkþ 1Þ!dkþ1

dzkþ1m

F ðmÞ1 ðzmÞ

þ Im ðzm1Þ �zm1zm0

Im ðzm0Þ �

� ð�1Þk

ðk þ 1Þ!dk

dzkmF ðmÞ

2 ðzmÞ

�Xlml¼�lm

Xrmr¼�rm

Im ðzm � lzm0 � rzm1Þðzm � lzm0 � rzm1Þkþ2

)

ð35Þ

where

F ðmÞ1 ðzmÞ ¼

1

zm0pctg p

zmzm0

�(

þ 2pXrmr¼1

sin 2pzmzm0

�

cos 2przm1zm0

�� cos 2p

zmzm0

�

þ 4p sin 2pzmzm0

�armþ1m

1� am

)

F ðmÞ2 ðzmÞ ¼

1

zm0

�2

4p2 sin 2pzmzm0

�(

�Xrmr¼1

r sin 2przm1zm0

�

cos 2przm1zm0

�� cos 2p

zmzm0

�� 2

þ8p2i sin 2pzmzm0

�

� armþ1m ½ðrm þ 1Þð1� amÞ þ am�

ð1� amÞ2

)ð36Þ

and

am ¼ei�2p

zm1zm0 if Im

zm1zm0

�> 0

e�i�2pzm1zm0 if Im

zm1zm0

�< 0

8>>><>>>:

By means of the above procedure, no difficulty willbe met in the computation of Inmakj (a = 1,2,3,4;n,m = 1,2, . . .,M; j,k = 0,1,2, . . .,N).

5. Numerical computing results

By means of this method, many typical exam-ples of doubly periodic cracks have been com-puted. It is found that this method has extremelyhigh accuracy and stability of solution.

5.1. The rectangularly distributed doubly periodic

cracks

The first example considered here is the rectan-gularly distributed doubly periodic cracks con-tained in an infinite elastic plate under the actionof the far-field stress. The two periods arez0 = 2d, z1 = 2hi, where i ¼

ffiffiffiffiffiffiffi�1

p. When the ap-

plied stress is tension stress perpendicular to thecrack faces, the dimensionless mode I stress inten-sity factors (SIF) of the rectangularly distributeddoubly periodic cracks computed by using themethod presented in this paper versus a/d, h/dand with N increasing (when lm = rm = 20) arelisted in Table 1. And the computing results ofthe dimensionless mode I SIF versus a/d, h/d andwith lm and rm increasing (when N = 16) are listed


in Table 2. It can be seen from the two tables thatthe computing accuracy converges rapidly with N

increasing, and as lm and rm increasing, the com-puting results also converge, but with a relativelyslow speed. When a/d = 0.9, N = 16, even thoughlm = rm = 2, the accuracy has reached to 0.6%;When a/d = 0.99, N = 16, lm = rm = 2, the erroris less than 1.8%. So if N is taken to be large

Fig. 4. Dimensionless SED and CRS of doubly periodic cracks withperpendicular to the crack faces at different a/d and c0/d when h/d =strain energy density factor. Bottom (b) Normalized critical stress.

enough, then lm = rm > 20 is sufficient. In thisexample, when a/d 5 0.9, N = 16, lm = rm = 20,the computing results can be within accuracy ofsix effective digits. The normalized SIF, SED andCRS of the rectangularly distributed doubly peri-odic cracks under the action of far-field tensionstress perpendicular to the crack faces at differenta/d, h/d are listed in Table 3 and the normalized

one direction collinear applied by a far uniform tension field1.0. (The figure corresponds to Table 6.) Top (a) Normalized

Table 7Dimensionless SIF, SED and CRS of doubly periodic cracks with one direction collinear applied by a far uniform tension stress field parallel to the crack faces atdifferent a/d, c0/d and h/d (KI ¼ KI=s


p, KII ¼ KII=s


p, Smin ¼ Smin=½ð2� 2m� m2Þs2a=12l�, �scr ¼ scr=


p, the unit of hcr is degree, l is shear

moduli. Poisson�s ratio is supposed as m = 0.3. lm = rm = 20, N = 16 for a/d 5 0.9, N = 30 for a/d > 0.9)

hd

c0d

a/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99

0.5 0.0 KI 0 0 0 0 0 0 0 0 0 0KII 1.016024 1.059459 1.120199 1.189377 1.26318 1.345051 1.452392 1.641846 2.143454 6.397826Smin 1.032304 1.122453 1.254846 1.414617 1.595623 1.809163 2.109443 2.695658 4.594393 40.93218hcr �82.3377 �82.3377 82.33774 �82.3377 82.33774 82.33774 82.33774 �82.3377 �82.3377 �82.3377scr 0.984229 0.943878 0.892698 0.840777 0.791653 0.743466 0.688519 0.609071 0.466537 0.156303

0.1 KI �0.00602 �0.0199 �0.03412 �0.04433 �0.04946 �0.0492 1 �0.04267 �0.02929 �0.01193 �0.00036KII 1.015 1.056377 1.115537 1.184006 1.257619 1.339512 1.447197 1.637831 2.141644 6.397774Smin 1.028407 1.10992 1.233926 1.387709 1.564926 1.776469 2.07729 2.668729 4.579032 40.93082hcr �82.4553 �82.7122 �82.9474 �83.0853 �83.1234 �83.0711 �82.9253 �82.6931 �82.4481 �82.3389scr 0.986092 0.949192 0.900234 0.848889 0.79938 0.750276 0.693827 0.612136 0.467319 0.156306

0.2 KI �0.01079 �0.03629 �0.06341 �0.08352 �0.09382 �0.09326 �0.08018 �0.05424 �0.02176 �0.00066KII 1.012242 1.047888 1.102398 1.168629 1.241585 1.323559 1.432392 1.626616 2.136692 6.397636Smin 1.02143 1.087714 1.197641 1.342242 1.513959 1.722043 2.022612 2.621761 4.551801 40.92847hcr �82.5492 �83.0289 �83.4921 �83.7771 �83.8611 �83.7564 �83.4606 �83.0029 �82.5398 �82.3398scr 0.989454 0.958832 0.91377 0.863147 0.812724 0.76204 0.703143 0.617594 0.468714 0.15631






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Table 7 (continued)

hd

c0d

a/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99




1 KI 0 0 0 �1E�06 �4E�06 �9E�06 �1.8E�05 �0.00004 �0.00011 �0.00088KII 0.997757 0.99243 0.989524 1.000661 1.042045 1.129173 1.278202 1.532816 2.102313 6.39728Smin 0.995519 0.984916 0.979158 1.001323 1.085857 1.275029 1.633794 2.349506 4.419654 40.92349hcr �82.3377 �82.3377 �82.3377 �82.3378 �82.3378 �82.3379 �82.338 �82.3383 �82.3387 �82.3405scr 1.002248 1.007628 1.010587 0.999339 0.959652 0.885605 0.782351 0.652397 0.47567 0.15632

1 0.0 KI 0 0 0 0 0 0 0 0 0 0KII 1.005103 1.020492 1.046637 1.085072 1.139716 1.219487 1.344861 1.570482 2.115409 6.397538Smin 1.01023! 1.041405 1.095449 1.177382 1.298953 1.487149 1.808652 2.466414 4.474956 40.92849hcr 82.33774 �82.3377 82.33774 �82.3377 82.33774 82.33774 �82.3377 �82.3377 82.33774 82.33775scr 0.994923 0.979919 0.955441 0.921598 0.877411 0.820017 0.743571 0.636747 0.472722 0.15631




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0.3 KI �0.00092 �0.00337 �0.00649 �0.00925 �0.01072 �0.01039 �0.00833 �0.00517 �0.00196 �0.00029KII 1.004699 1.019025 1.043827 1.081095 1.135125 1.215035 1.341277 1.568246 2.114568 6.397502Smin 1.00914 1.037385 1.087561 1.165816 1.28493 1.472585 1.795704 2.456965 4.470147 40.92746hcr �82.3559 �82.4031 �82.461 �82.5074 �82.525 �82.5072 �82.4608 �82.403 1 �82.3561 �82.3387scr 0.995461 0.981816 0.9589 0.926158 0.882186 0.824062 0.746247 0.63797 0.472976 0.156312







1.0 KI 0 0 0 �1E�06 �3E�06 �7E�06 �1.5E�05 �3.4E�05 �9.2E�05 �0.00076KII 1.00321 1.013559 1.033205 1.065875 1.117442 1.197907 1.327594 1.559798 2.1114 6.397215Smin 1.006431 1.027301 1.067513 1.136089 1.248676 1.43498 1.762501 2.432953 4.457953 40.92289hcr �82.3377 �82.3377 �82.3377 �82.3378 �82.3378 �82.3378 �82.338 �82.3382 �82.3386 �82.3401scr 0.9968 0.986623 0.967862 0.938197 0.894901 0.83479 0.753243 0.641111 0.473622 0.156321

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SED and CRS versus a/d, h/d are graphically dis-played in Fig. 1. In the case of the rectangularlydistributed doubly periodic cracks under the ac-tion of far-field shear stress parallel to the crackfaces, the normalized SIF, SED and CRS at differ-ent a/d, h/d are listed in Table 4 and the normal-

Fig. 5. Dimensionless SED and CRS of doubly periodic cracks with onto the crack faces at different a/d and c0/d when h/d = 0.5. (The figudensity factor. Bottom (b) Normalized critical stress.

ized SED and CRS versus a/d, h/d aregraphically displayed in Fig. 2. The comparisonof the computing results for the dimensionlessmode I SIF of rectangularly distributed doublyperiodic cracks when N = 16 for a/d 5 0.9,N = 30 for a/d > 0.9, lm = rm = 20 with the results

e direction collinear applied by a far uniform shear field parallelre corresponds to Table 7.) Top (a) Normalized strain energy


computed by [14] using the boundary collocationmethod are listed in Table 5.

5.2. Doubly periodic cracks with one direction

collinear

The second example is an infinite elastic platecontaining doubly periodic cracks with one direc-

Fig. 6. Dimensionless SED and CRS of doubly periodic cracks with onto the crack faces at different a/d and c0/d when h/d = 1.0. (The figudensity factor. Bottom (b) Normalized critical stress.

tion collinear where z0 = 2d, z1 = c0 + 2hi. Whenthe far-field stress is tension perpendicular to thecrack faces, the dimensionless SIF, SED andCRS of cracks at different a/d, c0/d and h/d(N = 16 for a/d 5 0.9, N = 30 for a/d > 0.9,lm = rm = 20) are listed in Table 6, and the normal-ized SED and CRS versus a/d, c0/d and h/d are dis-played in Figs. 3 and 4. It can be seen from the

e direction collinear applied by a far uniform shear field parallelre corresponds to Table 7.) Top (a) Normalized strain energy


table that KII = 0 only when c0/d = 0 or c0/d = 1.And in the other situations, that is, when0 < c0/d < 1, the mode II stress intensity factorsKII will not be zero. But the mode II stress inten-sity factors induced by mode I stress are far smal-ler than the mode I stress intensity factors.Similarly, if the far-field stress is shear parallel tothe cracks, the mode I stress intensity factors in-duced by mode II stress are also far smaller thanthe mode II stress intensity factors. In the case ofthe far-field stress being shear parallel to the crackfaces, the dimensionless SIF, SED and CRS of

Table 8Comparison of dimensionless SIF, SED and CRS of the axial-symperiodical collinear cracks under the far-field tension strSmin ¼ Smin=½ð1� 2mÞr2a=4l�, rcr ¼ rcr=


p, m = 0.3; N =

ad

Rectangularly distributed double periodiccracks (C0/d = 0)

Rhomcrack

hd¼ 0:5 1.0 1.5 2.0 h

d¼ 0

KI 0.1 0.983511 1.002830 1.004064 1.004140 1.010.2 0.948319 1.012190 1.016686 1.016965 1.070.3 0.921506 1.030617 1.039258 1.039798 1.160.4 0.922558 1.062256 1.074509 1.075281 1.270.5 0.962043 1.113296 1.127429 1.128325 1.380.6 1.049961 1.193887 1.207541 1.208410 1.460.7 1.205790 1.324330 1.335259 1.335958 1.540.8 1.480057 1.557705 1.564494 1.564936 1.690.9 2.079781 2.110467 2.113069 2.113265 2.150.99 6.394333 6.395824 6.396513 6.396815 6.39

Smin 0.1 0.967294 1.005667 1.008145 1.008298 1.030.2 0.899308 1.024528 1.03365 1.034218 1.150.3 0.849173 1.062172 1.080056 1.081179 1.340.4 0.851114 1.128389 1.154569 1.156229 1.620.5 0.925526 1.239429 1.271096 1.273117 1.910.6 1.102418 1.425367 1.458156 1.460256 2.150.7 1.453929 1.753849 1.782917 1.784785 2.380.8 2.190569 2.426446 2.447642 2.449025 2.850.9 4.325490 4.454071 4.465060 4.465889 4.660.99 40.88749 40.90657 40.91538 40.91924 40.90

rcr 0.1 1.016766 0.997178 0.995952 0.995877 0.980.2 1.054498 0.987957 0.983588 0.983318 0.930.3 1.085181 0.970292 0.962225 0.961725 0.860.4 1.083942 0.941392 0.930658 0.929990 0.780.5 1.039455 0.898233 0.886974 0.886270 0.720.6 0.952416 0.837600 0.828129 0.827533 0.680.7 0.829332 0.755099 0.748918 0.748526 0.640.8 0.675650 0.641970 0.639184 0.639004 0.590.9 0.480820 0.473829 0.473245 0.473201 0.460.99 0.156388 0.156352 0.156335 0.156328 0.15

cracks at different a/d, c0/d and h/d are listed inTable 7, and the normalized SED and CRS versusa/d, c0/d and h/d are displayed in Figs. 5 and 6.

Now the two kinds of specially distributed ofdouble periodic cracks are considered. One is therectangularly distributed (c0/d = 0), and the otheris the rhomboidally distributed (c0/d = 1). Forthese two kinds of distribution, whole crack arrayis symmetric about the midperpendicular of anyone crack, so they are called as the axial-symmetricdouble periodic cracks. The dimensionless of SIF,SED and CRS of these axial-symmetric double

metrically distributed double periodical cracks with the singleess perpendicular to the crack faces (KI ¼ KI=r


p,

16 for a/d 5 0.9, N = 30 for a/d > 0.9)

boidally distributed double periodics (C0/d = 1)

Single periodiccollinear cracks

:5 1.0 1.5 2.0

8103 1.005438 1.004225 1.004149 1.0041452672 1.021747 1.017277 1.01699S 1.0169821698 1.049130 1.040403 1.039862 1.0398303458 1.088718 1.076146 1.075373 1.0753272692 1.144002 1.129329 1.128432 1.1283797599 1.223639 1.209384 1.208516 1.2084653860 1.348140 1.336739 1.336045 1.3360050492 1.572449 1.565423 1.564997 1.5649748724 2.116036 2.113459 2.113310 2.1133075867 6.396635 6.396939 6.397027 6.398004

6533 1.010906 1.008469 1.008316 1.0083070626 1.043967 1.034853 1.034285 1.0342529542 1.100674 1.082438 1.081314 1.0812471695 1.185307 1.158090 1.156427 1.1563281837 1.308740 1.275384 1.273360 1.2732403847 1.497293 1.462610 1.460510 1.4603873504 1.817482 1.786871 1.785017 1.7849097764 2.472595 2.450548 2.449215 2.4491430090 4.477607 4.466709 4.466079 4.466065712 40.91694 40.92083 40.92196 40.93446

2219 0.994591 0.995792 0.995868 0.9958722251 0.978716 0.983016 0.983286 0.9833020809 0.953171 0.961166 0.961666 0.9616965264 0.918511 0.929242 0.929910 0.9299503227 0.874124 0.885482 0.886185 0.8862271385 0.817234 0.826867 0.827461 0.8274967727 0.741763 0.748089 0.748478 0.7485001544 0.635951 0.638805 0.638979 0.6389883237 0.472582 0.473158 0.473191 0.4731926351 0.156332 0.156325 0.156323 0.156299


periodic cracks under the far-field tension stressand under the far-field shear stress are listed in Ta-bles 8 and 9, respectively. The interaction of dou-ble periodic cracks with one direction collinearcan be taken as the interaction of infinite numberof rows of single periodic collinear cracks. In orderto investigate the interaction effects between thesecollinear crack rows, the ratio of SIF for theaxial-symmetric double periodic cracks to theSIF for single periodic collinear cracks underthe same load condition are calculated and listedin Table 10. The table shows that when the far-

Table 9Comparison of dimensionless SIF, SED and CRS of the axial-symperiodical collinear cracks under the far-field tensionSmin ¼ Smin=½ð2� 2m� m2Þs2a=12l�, scr ¼ scr=


p,

ad

Rectangularly distributed double periodiccracks (C0/d = 0)

Rhomcrack

hd¼ 0:5 1.0 1.5 2.0 h

d¼ 0

KII 0.1 1.016024 1.005103 1.004210 1.004149 0.990.2 1.059459 1.020492 1.017221 1.016996 0.990.3 1.120199 1.046637 1.040293 1.039858 0.980.4 1.189377 1.085072 1.075989 1.075367 1.000.5 1.263180 1.139716 1.129148 1.128425 1.040.6 1.345051 1.219487 1.209211 1.208510 1.120.7 1.452392 1.344861 1.336606 1.336043 1.270.8 1.641846 1.570482 1.565354 1.565002 1.530.9 2.143454 2.115409 2.113476 2.113332 2.100.99 6.397826 6.397538 6.397346 6.397217 6.39

Smin 0.1 1.032304 1.010231 1.008438 1.008314 0.990.2 1.122453 1.041405 1.034738 1.034280 0.980.3 1.254846 1.095449 1.082210 1.081304 0.970.4 1.414617 1.177382 1.157753 1.156413 1.000.5 1.595623 1.298953 1.274976 1.273343 1.080.6 1.809163 1.487149 1.462191 1.460496 1.270.7 2.109443 1.808652 1.786515 1.785011 1.630.8 2.695658 2.466414 2.450333 2.449230 2.340.9 4.594393 4.474956 4.466782 4.466171 4.410.99 40.93218 40.92849 40.92604 40.92438 40.92

scr 0.1 0.984229 0.994923 0.995808 0.995869 1.000.2 0.943878 0.979919 0.983071 0.983288 1.000.3 0.892698 0.955441 0.961267 0.961670 1.010.4 0.840777 0.921598 0.929377 0.929915 0.990.5 0.791653 0.877411 0.885623 0.886191 0.950.6 0.743466 0.820017 0.826985 0.827465 0.880.7 0.688519 0.743571 0.748164 0.748479 0.780.8 0.609071 0.636747 0.638833 0.638977 0.650.9 0.466537 0.472722 0.473154 0.473186 0.470.99 0.156303 0.156310 0.156315 0.156318 0.15

field stress is tension, the interaction between thecollinear crack rows is weakening each other forthe rectangular distributed double periodic crackarray, and intensifying each other for the rhom-boidal distributed double periodic crack array.Contrary to this, when the far-field stress is shear,the interaction between the collinear crack rows isintensifying each other for the rectangular distrib-uted double periodic crack array, and weakeningeach other for the rhomboidal distributed doubleperiodic crack array. The Table 10 also shows thatthe intensity of interaction between collinear crack

metrically distributed double periodical cracks with the singlestress parallel to the crack faces (KII ¼ KII=s


p,

m = 0.3; N = 16 for a/d 5 0.9, N = 30 for a/d > 0.9)

boidally distributed double periodics (C0/d = 1)

Single periodiccollinear cracks

:5 1.0 1.5 2.0

7757 1.003210 1.00408 1.004141 1.0041452430 1.013559 1.016743 1.016967 1.0169829524 1.033205 1.039368 1.039803 1.0398300662 1.065875 1.074667 1.075288 1.0753272046 1.117443 1.127613 1.128334 1.1283799175 1.197908 1.207723 1.208421 1.2084658206 1.327596 1.335413 1.335970 1.3360052825 1.559801 1.56461 1.564953 1.5649742339 2.111410 2.11318 2.113302 2.1133077504 6.397293 6.397156 6.397108 6.398004

5519 1.006431 1.008176 1.008299 1.0083074916 1.027301 1.033766 1.034223 1.0342529158 1.067513 1.080285 1.081190 1.0812471324 1.136090 1.154908 1.156244 1.1563285860 1.248678 1.271512 1.273137 1.2732405036 1.434984 1.458595 1.460281 1.4603873811 1.762511 1.783328 1.784817 1.7849099553 2.432980 2.448006 2.449079 2.4491439830 4.458052 4.465529 4.466046 4.466065805 40.92536 40.92360 40.92300 40.93446

2248 0.996800 0.995937 0.995876 0.9958727628 0.986623 0.983533 0.983316 0.9833020587 0.967862 0.962123 0.961721 0.9616969339 0.938196 0.930521 0.929983 0.9299509651 0.894901 0.886829 0.886263 0.8862275602 0.834789 0.828004 0.827526 0.8274962346 0.753241 0.748832 0.748520 0.7485002390 0.641107 0.639137 0.638997 0.6389885661 0.473617 0.473221 0.473193 0.4731926311 0.156316 0.156319 0.156321 0.156299


rows rapidly decreases with the increasing of theditch between rows. When h/d P 2, the SIF ofthe double periodic cracks with one direction col-linear is nearly same as the SIF of single periodiccollinear cracks.

5.3. General doubly periodic cracks

The example considered here is an infinite elasticplate containing general doubly periodic cracksunder the uniform tensile stress perpendicular tothe crack faces. This problem can be transformedas multiple sets of doubly periodic cracks withsame periods. For example, the doubly periodiccracks with periods z0 = 2d + 0.5di andz1 = 0.5d + 2di can be transformed as the four setsof doubly periodic cracks with same periodsz0 = 7.5d, z1 = 0.5d + 2di. The center coordinatesof them are zc1 = 0, zc2 = 2d + 0.5di, zc3 = 4d + di,zc4 = 6d + 1.5di, respectively. The calculation re-sults indicate that the dimensionless SIF of foursets are nearly entirely the same, the divergence be-tween them is less than 10�7, which shows that the

Table 10Ratio of the SIF for axial-symmetrically distributed double periodic cthe same load condition

Loads mode ad

Rectangularly distributed double periodic carray (C0/d = 0)

h/d = 0.5 1.0 1.5 2.0

Tension 0.1 0.979451 0.998690 0.999920 0.990.2 0.932484 0.995288 0.999709 0.990.3 0.886208 0.991140 0.999449 0.990.4 0.857933 0.987845 0.999239 0.990.5 0.852588 0.986633 0.999158 0.990.6 0.868839 0.987938 0.999236 0.990.7 0.902534 0.991261 0.999442 0.990.8 0.945739 0.995356 0.999694 0.990.9 0.984136 0.998656 0.999887 0.990.99 0.999426 0.999659 0.999767 0.99

Shear 0.1 1.011830 1.000954 1.000065 1.000.2 1.041768 1.003452 1.000235 1.000.3 1.077291 1.006546 1.000445 1.000.4 1.106060 1.009063 1.000616 1.000.5 1.119464 1.010047 1.000682 1.000.6 1.113025 1.009121 1.000618 1.000.7 1.087116 1.006629 1.000450 1.000.8 1.049120 1.003520 1.000243 1.000.9 1.014265 1.000995 1.000080 1.000.99 0.999972 0.999927 0.999897 0.99

symmetry and stability of solution are very good.When N = 16, lm = rm = 20, the dimensionlessSIF, SED and CRS of cracks at different a/d,C0/d and e0/d are listed in Table 11 and the dimen-sionless SED and CRS versus a/d, C0/d and e0/dare displayed in Fig. 7, the critical initiation angleversus a/d, C0/d and e0/d are displayed in Fig. 8.The table and figures shows that, with the increasingof ratio a/d, the SIF and SED are more and morelarge and the CRS are more and more small. Andmeanwhile, the variation of the critical initiationdirection of crack is not obvious. In most of cases,with the increasing of ratio a/d, hcr is varied fromzero to negative, and then from negative to zero.

5.4. Doubly periodic cracks with slightly

misalignment

In this section, doubly periodic crack array withslightly misalignment under the uniform tensionstress is considered. The dimensionless SIF, SEDand CRS of crack tip at different h/d, e/d anda/d are listed in Table 12 and the SED and critical

rack array to the SIF for single periodic collinear cracks under

rack Rhomboidally distributed double periodic crackarray (C0/d = 1)

h/d = 0.5 1.0 1.5 2.0

9995 1.013900 1.001288 1.000080 1.0000059984 1.054761 1.004686 1.000291 1.0000169969 1.117200 1.008944 1.000551 1.0000319957 1.184252 1.012453 1.000761 1.0000439952 1.225379 1.013845 1.000842 1.0000479955 1.214433 1.012557 1.000761 1.0000429965 1.155580 1.009083 1.000549 1.0000309976 1.080205 1.004776 1.000287 1.0000159980 1.021491 1.001291 1.000072 1.0000029814 0.999666 0.999786 0.999834 0.999847

0004 0.993638 0.999069 0.999935 0.9999960014 0.975858 0.996634 0.999765 0.9999860026 0.951621 0.993629 0.999555 0.9999740037 0.930565 0.991210 0.999386 0.9999640041 0.923489 0.990308 0.999321 0.9999600038 0.934388 0.991265 0.999386 0.9999640028 0.956737 0.993706 0.999557 0.9999740018 0.979457 0.996695 0.999768 0.9999870011 0.994810 0.999102 0.999940 0.9999989877 0.999922 0.999889 0.999867 0.99986

Table 11Dimensionless SIF, SED and CRS of general doubly periodic cracks (z0 = 2d + e0i, z1 = C0 + 2di) under the uniform tension stress perpendicular to the crack facesversus a/d at different e0/d, C0/d (where KI ¼ KI=r


p; KII ¼ KII=r


p; Smin ¼ Smin=½ð1� 2mÞr2a=2l�; �rcr ¼ rcr=


p, the unit of hcr is degree; l is shear

moduli of material; in the computation of SED, Poisson�s ratio is supposed as m = 0.3)

e0d

C0

da/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 0.0 KI 1.004066 1.017451 1.043692 1.088696 1.161004 1.272201 1.436374 1.665527 1.956509 2.273246KII 0 0 0 0 0 0 0 0 0 0Smin 1.008148 1.035207 1.089293 1.185258 1.347931 1.618496 2.06317 2.773979 3.827927 5.167646hcr 0 0 0 0 0 0 0.000001 0.000001 0.000002 0.000002rcr 0.995951 0.982848 0.958137 0.91853 0.861323 0.786039 0.696198 0.600411 0.511114 0.439900

0.25 KI 1.005776 1.024332 1.059384 1.117288 1.207183 1.340510 1.528018 1.771146 2.047998 2.307125KII �0.00103 �0.00373 �0.00693 �0.00863 �0.00581 0.005686 0.029979 0.067120 0.106962 0.130922Smin 1.011588 1.049285 1.122397 1.248492 1.457361 1.797036 2.336747 3.146525 4.218573 5.3591S9hcr 0.058849 0.209249 0.374946 0.442837 0.275548 �0.24306 �1.12514 �2.17858 �3.01148 �3.27580rcr 0.994256 0.976233 0.943902 0.894967 0.828355 0.745970 0.654175 0.563747 0.486875 0.431967

0.5 KI 1.008363 1.034619 1.082279 1.157070 1.266087 1.415313 1.604883 1.823005 2.041081 2.216050KII �0.00122 �0.00409 �0.00611 �0.00271 0.012883 0.048042 0.105103 0.172576 0.22394 0.235126Smin 1.016800 1.070473 1.171408 1.338826 1.603330 2.008012 2.599072 3.386347 4.271924 5.027683hcr 0.069536 0.227179 0.323356 0.134481 �0.58313 �1.94981 �3.78989 �5.53746 �6.46339 �6.23931rcr 0.991704 0.966523 0.923945 0.864247 0.789748 0.705695 0.620284 0.543418 0.483825 0.445981

0.75 KI 1.011241 1.045887 1.106326 1.195061 1.312280 1.453753 1.611378 1.773919 1.922329 2.027692KII �4.3E�05 0.001162 0.007796 0.026923 0.066579 0.129848 0.206922 0.273779 0.303532 0.284494Smin 1.022609 1.093883 1.224087 1.429712 1.731486 2.149080 2.686756 3.304106 3.888598 4.281782hcr 0.002422 �0.06388 �0.40379 �1.29231 �2.92439 �5.21352 �7.64165 �9.33604 �9.57543 �8.40957rcr 0.988884 0.956125 0.903845 0.836326 0.759960 0.682140 0.610079 0.550140 0.507112 0.483268

1 KI 1.013331 1.053243 1.118553 1.204925 1.304548 1.410209 1.519369 1.629931 1.731102 1.805159KII 0.003166 0.014549 0.039251 0.083013 0.146688 0.221043 0.287955 0.327227 0.324342 0.278725Smin 1.026861 1.109771 1.254432 1.466453 1.747275 2.091196 2.481357 2.87937 3.216139 3.421648hcr �0.17961 �0.7918 �2.0159 �3.99118 �6.63318 �9.49786 11.7722 �12.5876 �11.6176 �9.34097rcr 0.986834 0.949256 0.892846 0.825783 0.756518 0.691517 0.634827 0.589320 0.557613 0.540608

0.75 0 KI 1.003896 1.016740 1.041989 1.085500 1.156066 1.266618 1.435443 1.683616 2.018541 2.401200KII 0.000560 0.002412 0.006083 0.012471 0.022692 0.037299 0.053755 0.060787 0.032896 �0.05711Smin 1.007807 1.033772 1.085819 1.178640 1.337584 1.607277 2.066634 2.842411 4.076809 5.772689hcr �0.0321 �0.1364 �0.33452 �0.65847 �1.12566 �1.69085 �2.15266 �2.07497 �0.93433 1.364521rcr 0.996119 0.983530 0.959669 0.921106 0.864648 0.788778 0.695614 0.593139 0.495268 0.416208

0.25 KI 1.005250 1.022056 1.053621 1.105459 1.185911 1.307114 1.485646 1.738616 2.064891 2.413360KII �0.0003 �0.00064 0.000619 0.006053 0.018859 0.041656 0.072976 0.099737 0.089914 0.006977Smin 1.010527 1.044600 1.110119 1.222118 1.407141 1.712232 2.218448 3.043888 4.280940 5.824410hcr 0.017044 0.035749 �0.03381 �0.31373 �0.91171 �1.83023 �2.83028 �3.31207 �2.50594 �0.16565rcr 0.994778 0.978419 0.949107 0.904573 0.843007 0.764221 0.671391 0.573173 0.483315 0.414356


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Table 11 (continued)

e0d

C0

da/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.5 KI 1.007245 1.029744 1.069831 1.13156 1.221111 1.347634 1.523844 1.761834 2.054203 2.346988KII �0.00031 �0.00038 0.002336 0.011844 0.032841 0.068423 0.114942 0.153372 0.14373 0.045305Smin 1.014542 1.060372 1.14455 1.280727 1.493404 1.826054 2.350096 3.153853 4.263541 5.512712hcr 0.017629 0.021045 �0.12556 �0.59986 �1.54403 �2.92655 �4.37923 �5.07612 �4.05478 �1.10697rcr 0.992807 0.971115 0.934722 0.883633 0.818298 0.740020 0.652315 0.563092 0.484300 0.425910

0.75 KI 1.009120 1.036783 1.083839 1.151687 1.242851 1.362681 1.521004 1.727954 1.975455 2.216164KII 0.000904 0.004633 0.014119 0.033590 0.066853 0.113906 0.165698 0.197505 0.167241 0.038290Smin 1.018325 1.074964 1.175130 1.328780 1.554162 1.884374 2.371443 3.068167 3.961641 4.914496hcr �0.05149 �0.25683 �0.74667 �1.67466 �3.10276 �4.86758 �6.41517 �6.74918 �4.93191 �0.99062rcr 0.990961 0.964502 0.922480 0.867508 0.802143 0.728478 0.649372 0.570901 0.502415 0.451087

1 KI 1.009802 1.038816 1.085870 1.149467 1.229587 1.330723 1.463374 1.638652 1.851985 2.07088KII 0.003417 0.014487 0.035209 0.067360 0.110313 0.158668 0.199471 0.207863 0.146197 �0.01554Smin 1.019725 1.079585 1.181747 1.330900 1.537637 1.823933 2.225209 2.776239 3.475123 4.289059hcr �0.19455 �0.79939 �1.86225 �3.38382 �5.23745 �7.05854 �8.14985 �7.54183 �4.58886 0.430095rcr 0.990281 0.962435 0.919X94 0.866817 0.806442 0.740450 0.670370 0.600166 0.536432 0.482858

0.5 0.0 KI 1.003470 1.014940 1.037587 1.076881 1.141539 1.246372 1.418994 1.711279 2.194202 2.848222KII 0.000811 0.003511 0.008959 0.018766 0.035500 0.06254 0.101788 0.142135 0.123082 �0.08170Smin 1.006953 1.030129 1.076757 1.160421 1.305789 1.561744 2.035502 2.971256 4.846665 8.126545hcr �0.04648 �0.19888 �0.49479 �0.99914 �1.78586 �2.89189 �4.15940 �4.83561 �3.23757 1.646669rcr 0.996542 0.985268 0.963698 0.928308 0.875112 0.800194 0.700913 0.580136 0.454233 0.35079

0.25 KI 1.004465 1.018757 1.045606 1.089867 1.159584 1.268956 1.445245 1.740252 2.223320 2.866654KII 0.000032 0.000735 0.003898 0.012423 0.030233 0.061794 0.109425 0.161087 0.152642 �0.04857Smin 1.008950 1.037866 1.093324 1.188138 1.346577 1.618353 2.114104 3.08338 4.992546 8.222717hcr �0.00184 �0.04152 �0.21429 �0.65327 �1.49679 �2.80559 �4.39622 �5.40992 �3.97697 0.971374rcr 0.995555 0.981588 0.956369 0.917416 0.861756 0.786074 0.687760 0.569490 0.447547 0.348732

0.50 KI 1.005980 1.024453 1.057141 1.107451 1.181843 1.293182 1.467990 1.757515 2.228219 2.845365KII �7E�06 0.000767 0.004642 0.015271 0.037229 0.075064 0.130092 0.187668 0.179083 �0.03239Smin 1.011995 1.049505 1.117592 1.226944 1.399697 1.684273 2.190814 3.163266 5.032886 8.098329hcr 0.000402 �0.04303 �0.25242 �0.79043 �1.80901 �3.35195 �5.17075 �6.28130 �4.67443 0.652455rcr 0.994056 0.976130 0.945928 0.902792 0.845246 0.770537 0.675612 0.562254 0.445750 0.351400


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0.25 0.0 KI 1.003022 1.013021 1.032753 1.066858 1.122642 1.213009 1.366116 1.660717 2.369447 4.049082KII 0.000586 0.00255 0.006583 0.014098 0.027793 0.052945 0.100408 0.191471 0.320897 �0.04346Smin 1.006053 1.026224 1.066670 1.138609 1.261965 1.477342 1.887639 2.835359 5.831040 16.39908hcr �0.03359 �0.4475 �0.36526 �0.75743 �1.42121 �2.51197 �4.26435 �6.81143 �8.09302 0.615173rcr 0.996987 0.98714 0.968244 0.937158 0.890177 0.822734 0.727848 0.593876 0.414121 0.246939

0.25 KI 1.003698 1.015556 1.037875 1.074698 1.132729 1.224411 1.377767 1.672158 2.382451 4.066460KII �0.00018 �0.00023 0.001344 0.006977 0.020313 0.047236 0.098767 0.196025 0.333237 �0.02560Smin 1.007410 1.031353 1.077188 1.155079 1.283951 1.503919 1.918917 2.877195 5.909676 16.53749hcr 0.010567 0.01308 �0.07446 �0.37202 �1.02823 �2.21808 �4.15665 �6.93282 �8.38137 0.360668rcr 0.996315 0.984683 0.963505 0.930453 0.882522 0.815432 0.721891 0.589543 0.411356 0.245904

0.5 KI 1.004873 1.019900 1.046427 1.087234 1.147832 1.239925 1.391587 1.683267 2.391804 4.072790KII �0.00035 �0.00076 0.000573 0.006559 0.021244 0.050673 0.105687 0.206966 0.347620 �0.01261Smin 1.00977 1.040197 1.095011 1.182169 1.318478 1.542866 1.960181 2.923708 5.974718 16.58796hcr 0.019915 0.043082 �0.03147 �0.34568 �1.06129 �2.35066 �4.41009 �7.29417 �8.73938 0.177336rcr 0.995150 0.980488 0.955632 0.919730 0.870891 0.805074 0.714253 0.584834 0.409111 0.245529



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initiation angle versus e/d and a/d at h/d = 1.0 aredisplayed in Fig. 9. It can be found that when the dkeeps constant, and e slightly increases from zero,though the crack space becomes slightly largerthan the collinear situation, the SIF and SED ofcrack tip are not decreased but increased. For

Fig. 7. Dimensionless SED and CRS of general doubly periodic crackperpendicular to the crack faces at different a/d, C0/d and e/d. (The figdensity factor. Bottom (b) Normalized critical stress.

the sake of convenient to comparison, the SIFand SED of single periodic crack array with mis-alignment are calculated as shown in Table 13.From the table it can be seen that if the ratio a/dkeeps constant, then the SIF and SED will firstincrease and then decrease with the increasing of

s (z0 = 2d + e0i,z = C0 + 2di.) under the uniform tension stressure corresponds to Table 11.) Top (a) Normalized strain energy


ratio e/d. Let (e/d)cr express the critical value of theratio e/d when the SIF and SED come to the max-imum, it is found that there is different (e/d)cr fordifferent a/d, and (e/d)cr will decrease with theincreasing of a/d. For example, (e/d)cr � 0.8 fora/d = 0.1; (e/d)cr � 0.2 for a/d = 0.9; (e/d)cr � 0for a/d = 0.99.

Fig. 8. Critical initiation angle of general doubly periodic cracksperpendicular to the crack faces at different a/d, C0/d and e0/d. (Theh0/d = 0.5.

5.5. Two sets of doubly periodic cracks with the

same crack length and perpendicular orientation

The example given here is an infinite elasticplate containing two sets of doubly periodic crackswith the same length and orientations perpendicu-lar to each other under the uniform equiaxial

(z = 2d + e0i, z = C0 + 2di.) under the uniform tension stressfigure corresponds to Table 11.) Top (a) h0/d = 1.0. Bottom (b)

Table 12Dimensionless SIF, SED and CRS of general doubly periodic cracks (z0 = 2d + ei, z1 = 2hi) under the uniform tension stress perpendicular to the crack faces versus h/d,e/d, and a/d (where KI ¼ KI=r


p; KII ¼ KII=r




p, the unit of hcr is degree; l is shear moduli of material; in

the computation of SED, Poisson�s ratio is supposed as m = 0.3)

hd

ed

a/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99



0.2 KI 0.983518 0.948358 0.921657 0.923068 0.963686 1.055196 1.222665 1.537743 2.293831 4.231390KII �1.5E�05 �3.7E�05 0.000036 0.000517 0.002394 0.008745 0.029056 0.091188 0.245187 0.018567Smin 0.967307 0.899384 0.849452 0.852055 0.928703 1.113602 1.496703 2.382291 5.388666 17.9054hcr 0.000890 0.002242 �0.002220 �0.03224 �0.14283 �0.47494 �1.36339 �3.42554 �6.28807 �0.25142rcr 1.016758 1.054454 1.085002 1.083343 1.037675 0.947622 0.817395 0.647892 0.430784 0.236324

0.3 KI 0.983515 0.948361 0.921723 0.923402 0.964944 1.059403 1.235650 1.571568 2.298163 3.593533KII �8.6E�05 �0.00032 �0.00060 �0.00065 0.000461 0.005564 0.022903 0.070991 0.139809 �0.06472Smin 0.967301 0.899388 0.849574 0.852673 0.931117 1.122400 1.527945 2.480523 5.323011 12.92238hcr 0.005012 0.019309 0.037566 0.040279 �0.02745 �0.30092 �1.06285 �2.60050 �3.51574 1.032747rcr 1.016761 1.054451 1.084924 1.082951 1.036329 0.943901 0.808995 0.634934 0.433432 0.278182

0.4 KI 0.983492 0.948286 0.921618 0.923403 0.965523 1.062030 1.244130 1.589208 2.268983 3.284119KII �0.00022 �0.00087 �0.00189 �0.00316 �0.00422 �0.00361 0.002862 0.022295 0.033951 �0.09329Smin 0.967256 0.899248 0.849388 0.852694 0.932272 1.127935 1.547877 2.526639 5.150734 10.80392hcr 0.012997 0.052836 0.118178 0.196579 0.251162 0.195302 �0.13228 �0.80419 �0.85776 1.630615rcr 1.016785 1.054533 1.085043 1.082937 1.035687 0.941582 0.803770 0.629113 0.440621 0.304235

1 0 KI 1.00283 1.01219 1.030617 1.062256 1.113296 1.193887 1.324330 1.557705 2.110466 6.371088KII 0 0 0 0 0 0 0 0 0 0Smin 1.005667 1.024528 1.062172 1.128389 1.239429 1.425367 1.753849 2.426446 4.454067 40.59077hcr 0 0 0 0 0 0 0 0 0 0rcr 0.997178 0.987957 0.970292 0.941392 0.898233 0.837600 0.755099 0.641970 0.473829 0.156959



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tension stress as shown in Fig. 10. When z0 = 2d,z1 = 2di, N = 16, lm = rm = 20, the dimensionlessstress intensity factors of crack tip versus a/d arelisted in Table 14 and displayed in Fig. 10. In the

Fig. 9. Dimensionless SED and critical initiation angle of general doubtension stress perpendicular to the crack faces at different h/d, e/dNormalized strain energy density factor. Bottom (b) Normalized criti

case of equiaxial tension, the two sets of doublyperiodic cracks have the same mode I stress inten-sity factors, and their mode II stress intensity fac-tors are equal to zero.

ly periodic cracks with slightly misalignment under the uniform, and a/d. (The figure corresponds to the Table 12.) Top (a)cal stress.

Table 13Dimensionless SIF, SED and CRS of single periodic cracks with slightly misalignment under the uniform tension stress perpendicular to the crack faces versus e/d anda/d

ed

ad

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99

KI 0 1.004145 1.016982 1.039830 1.075327 1.128379 1.208464 1.336004 1.564972 2.113302 6.3970400.1 1.004176 1.017117 1.040180 1.076090 1.129969 1.211882 1.344183 1.589709 2.239406 5.6690090.2 1.004265 1.017502 1.041167 1.078223 1.134344 1.220993 1.3646 1 9 1.641747 2.359993 4.2450970.3 1.004399 1.018075 1.042622 1.081305 1.140448 1.232931 1.388106 1.683744 2.341078 3.4902380.4 1.004558 1.018750 1.044302 1.084751 1.146907 1.244345 1.406117 1.697786 2.246625 3.0053350.5 1.004719 1.0)9425 1.045936 1.087944 1.152402 1.252534 1.414061 1.684901 2.127903 2.6592120.6 1.004860 1.020002 1.047272 1.090354 1.155940 1.255970 1.411161 1.652928 2.007465 2.3958950.7 1.004961 1.020399 1.048113 1.091605 1.156961 1.254250 1.398889 1.609611 1.893880 2.1870520.8 1.005008 1.020557 1.048330 1.091502 1.155318 1.247770 1.379548 1.560689 1.789936 2.0165900.9 1.004992 1.020447 1.047871 1.090015 1.151179 1.237352 1.355442 1.509962 1.696095 1.8745741 1.004911 1.020061 1.046745 1.087241 1.144902 1.223968 1.328526 1.459795 1.611916 1.754463

KII 0 0 0 0 0 0 0 0 0 0 00.1 0.000411 0.001726 0.004215 0.008470 0.015726 0.028766 0.054823 0.117661 0.328719 0.4031030.2 0.000786 0.003290 0.008002 0.015969 0.029305 0.052494 0.095923 0.185494 0.358236 0.0939260.3 0.001092 0.004552 0.010998 0.021701 0.039081 0.067745 0.116245 0.195570 0.262114 �0.027410.4 0.001304 0.005411 0.012953 0.025166 0.044189 0.073393 0.116537 0.167918 0.157189 �0.098750.5 0.001410 0.005813 0.013750 0.026199 0.044582 0.070313 0.102024 0.123645 0.069767 �0.147730.6 0.001406 0.005751 0.013407 0.02494 0.040847 0.060526 0.078760 0.075829 0.000996 �0.183580.7 0.001300 0.005264 0.012047 0.021744 0.033930 0.046360 0.051686 0.030840 �0.05243 �0.210450.8 0.001106 0.004419 0.009866 0.017087 0.024881 0.029920 0.024169 �0.00879 �0.09382 �0.230600.9 0.000843 0.003301 0.007091 0.011472 0.014683 0.012852 �0.00177 �0.04243 �0.12577 �0.245471 0.000532 0.002001 0.003953 0.005364 0.004146 �0.00369 �0.02508 �0.07027 �0.15025 �0.25605

Smin 0 1.008307 1.034252 1.081247 1.156328 1.273239 1.460386 1.784907 2.449137 4.466045 40.922120.1 1.008370 1.034533 1.082011 1.158122 1.277356 1.470415 1.813209 2.556513 5.242008 32.482080.2 1.008550 1.035332 1.084165 1.163108 1.288560 1.496675 1.881688 2.767974 5.839027 18.039590.3 1.008820 1.036521 1.087318 1.170222 1.303865 1.529856 1.955452 2.915710 5.625711 12.183360.4 1.009140 1.037914 1.090923 1.178030 1.319543 1.559820 2.005923 2.942098 5.099698 9.0527510.5 1.009464 1.039298 1.094384 1.185081 1.332252 1.579330 2.021629 2.871291 4.538308 7.1177120.6 1.009747 1.040474 1.097162 1.190193 1.339740 1.585235 2.004537 2.744376 4.029917 5.8117130.7 1.009950 1.041272 1.098850 1.192606 1.341003 1.577707 1.962565 2.592867 3.592622 4.8768660.8 1.010044 1.041579 1.099204 1.191998 1.336076 1.558831 1.904393 2.435916 3.222547 4.1788830.9 1.010011 1.041335 1.098140 1.188413 1.325671 1.531392 1.837229 2.283809 2.910257 3.6409411 1.009847 1.040534 1.095708 1.182153 1.310837 1.498127 1.766317 2.141484 2.646036 3.215933

hcr 0 0 0 0 0 0 0 0 0 0 00.1 �0.02357 �0.09756 �0.23293 �0.45103 �0.79776 �1.36179 �2.34590 �4.29499 �8.83495 �4.122240.2 �0.04502 �0.18587 �0.44043 �0.84901 �1.48310 �2.47394 �4.07398 �6.66706 �9.16681 �1.269160.3 �0.06251 �0.25700 �0.60456 �1.15098 �1.96847 �3.17037 �4.87682 �6.86521 �6.60323 0.450010

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283

(continued on next page)Table 13 (continued)

ed

ad

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99

0.4 �0.07467 �0.30526 �0.71091 �1.33090 �2.21520 �3.40682 �4.82485 �5.79648 �4.05464 1.8874280.5 �0.08071 �0.32672 �0.75353 �1.38240 �2.22432 �3.24007 �4.18416 �4.25751 �1.88309 3.2058350.6 �0.08049 �0.32310 �0.73377 �1.31213 �2.03011 �2.77607 �3.22107 �2.64141 �0.02854 4.4503030.7 �0.07441 �0.29652 �0.65878 �1.14237 �1.68388 �2.12454 �2.12370 �1.09872 1.589032 5.6326420.8 �0.06330 �0.24889 �0.53932 �0.89748 �1.23528 �1.37575 �1.00452 0.322877 3.022348 6.7523940.9 �0.04824 �0.18596 �0.38777 �0.60315 �0.73106 �0.59525 0.075186 1.613365 4.303167 7.8038831 �0.03046 �0.11281 �0.21710 �0.28271 �0.20816 0.173331 1.082505 2.773014 5.449292 8.779272

rcr 0 0.995872 0.983302 0.961696 0.929950 0.886227 0.827497 0.748501 0.638989 0.473193 0.1563220.1 0.995841 0.983168 0.961356 0.929229 0.884798 0.824670 0.742636 0.625426 0.436768 0.1754600.2 0.995752 0.982789 0.960400 0.927236 0.880943 0.817403 0.728998 0.601062 0.413837 0.2354430.3 0.995619 0.982225 0.959007 0.924413 0.875757 0.808490 0.715116 0.585636 0.421610 0.2864950.4 0.995461 0.981566 0.957421 0.921344 0.870539 0.800687 0.706062 0.583004 0.442821 0.3323610.5 0.995301 0.980912 0.955906 0.918599 0.866377 0.795726 0.703314 0.590149 0.469411 0.3748260.6 0.995162 0.980357 0.954695 0.916624 0.863952 0.794243 0.706306 0.603640 0.498141 0.4148090.7 0.995062 0.979981 0.953962 0.915696 0.863545 0.796135 0.713819 0.621026 0.527587 0.4528240.8 0.995016 0.979837 0.953808 0.915930 0.865136 0.800941 0.724639 0.640721 0.557058 0.4891810.9 0.995032 0.979952 0.954270 0.917311 0.868525 0.808085 0.737766 0.661714 0.586185 0.5240751 0.995113 0.980329 0.955328 0.919736 0.873425 0.817007 0.752429 0.683349 0.614755 0.557630


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5.6. Comparison in SIF, SED and CRS between

several kinds of typical distributed double periodic

cracks

In this section, several kinds of typically dis-tributed double periodic cracks are put togetherinto consideration in order to compare with

Fig. 10. Dimensionless SED and CRS of two sets of doubly periodiorientation under the uniform equiaxed tension stress. Top (a) Normalstress.

each other. The SIF, SED and CRS ofthem under far-field tension stress and shearstress are calculated. The results for tensionmode are listed in Table 15 and graphicallyshown in Fig. 11. And the results for shearmode are listed in Table 16 and shown in Fig.12.

c cracks with the same length and perpendicular-to-each-otherized strain energy density factor. Bottom (b) Normalized critical

Table 14Dimensionless SIF, SED and CRS of two sets of doubly periodic cracks with the same length and perpendicular-to-each-otherorientation under the uniform equiaxed tension stress (KI ¼ KI=r


p, KII ¼ KII=r


p, Smin ¼ Smin=½ð1� 2mÞr2a=4l�,

rcr ¼ rcr=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4lScr=ð1� 2mÞa

p, m = 0.3)

ad

KI KII Smin hcr(�) rcr

0.1 0.995512 0 0.991044 0 1.0045080.2 0.984268 0 0.968783 0 1.0159840.3 0.972081 0 0.944942 0 1.0287200.4 0.966594 0 0.934303 0.000001 1.0345610.5 0.976150 0 0.952868 0.000001 1.0244330.6 1.010700 0 1.021515 0.000001 0.9894130.7 1.086844 0 1.181230 0.000002 0.9200950.8 1.246341 0 1.553366 0.000003 0.8023490.9 1.659669 0 2.754500 0.000006 0.6025300.99 4.985688 �3E�06 24.85708 0.000039 0.200574


It can be seen that when the far-field stress istension mode, the �rcr–a=d curve for rectangularlydistributed double periodic cracks is always onthe upper of the �rcr–a=d curve for single periodiccollinear cracks, and the �rcr–a=d curve for rhom-boidally distributed double periodic cracks is al-ways under the lower of the �rcr–a=d curve forsingle periodic collinear cracks. When h/d = 1,whether rectangularly distributed or rhomboidallydistributed double periodic cracks, the �rcr–a=dcurve is very close to the �rcr–a=d curve for singleperiodic collinear cracks. It means that whenh/d = 1, the interaction effect between the collinearcrack rows has become very weak. Whenh/d = 0.5, it is obvious that the �rcr–a=d curvesfor rectangularly distributed or rhomboidallydistributed double periodic cracks are close tothe curve for single periodic collinear cracks ata/d ! 0 and a/d ! 1, but relatively distant fromthe curve for single periodic collinear cracks whena/d is located in the middle of interval (0,1).

Now let us look the �rcr–a=d curve for rectangu-larly distributed double periodic cracks when h/d = 0.5 and under the normal tension stress. Asa/d increases start from zero, �rcr is not decreasingbut increasing from value 1.0. When a/d = 0.35,�rcr reaches to the maximum 1.0896. And then asthe increasing of a/d, �rcr start decease, when a/dis large than 0.55, the normalized critical stress isless than 1.0. The reason of this phenomenon isthe stress shield effect of the tiered cracks againstthe normal tension stress. The interaction ofcracks in a collinear crack row is intensifying each

other, and the interaction between the tiered col-linear crack rows is weakening each other. Whena/d is less than 0.55, the former is less than thelater, and when a/d is large than 0.55, the formerwill exceed the later.

For the �rcr–a=d curve of rhomboidally distrib-uted double periodic cracks when h/d = 0.5 andunder the normal tension stress, the interaction be-tween the staggered collinear crack rows is intensi-fying each other, so the normalized critical stress isalways less than 1.0.

In the case of far field stress being shear stress,the �rcr–a=d curve for rectangularly distributed dou-ble periodic cracks is always under the lower of the�rcr–a=d curve for single periodic collinear cracks,and the �rcr–a=d curve for rhomboidally distributeddouble periodic cracks is always on the upper of the�rcr–a=d curve for single periodic collinear cracks. Itindicates that the interaction between the tieredcollinear crack rows is intensifying each other,and the interaction between the staggered collinearcrack rows is weakening each other.

For two sets of rectangularly distributed doubleperiodic cracks, whether under the equiaxial ten-sion stress or under the shear stress, the normal-ized critical stress is largely higher than that ofsingle periodic collinear cracks under the samemode of load. It is also largely higher than thatof rectangularly distributed double periodic crackswhen a/d = 1.0 and under the tension stress. Thisis because that the arrangement of crack array ishomogeneity for X-axes direction and Y-axesdirection.


6. Discussion on the periods of doubly periodic

cracks and the reliability of this method

It is necessary for us to investigate whether theEq. (31) could be directly generalized to the sum-mation of dual complex series or not. Since thesolution of the rectangularly distributed periodiccracks has been obtained by [14] through the

Table 15Comparison of normalized SIF, SED and CRS for different arrangecrack faces (KI ¼ KI=r



pad

Different arrangement of crack array

a b c d

KI 0.1 0.995512 1.002830 0.983511 1.00540.2 0.984268 1.012190 0.948319 1.02170.3 0.972081 1.030617 0.921506 1.04910.4 0.966594 1.062256 0.922558 1.08870.5 0.976150 1.113296 0.962043 1.14400.6 1.010700 1.193887 1.049961 1.22360.7 1.086844 1.324330 1.205790 1.34810.8 1.246341 1.557705 1.480057 1.57240.9 1.659669 2.110467 2.079781 2.11600.99 4.985389 6.395786 6.394294 6.3965

Smin 0.1 0.991044 1.005667 0.967294 1.01090.2 0.968783 1.024528 0.899308 1.04390.3 0.944942 1.062172 0.849173 1.10060.4 0.934303 1.128389 0.851114 1.18530.5 0.952868 1.239429 0.925526 1.30870.6 1.021515 1.425367 1.102418 1.49720.7 1.181230 1.753849 1.453929 1.81740.8 1.553366 2.426446 2.190569 2.47250.9 2.754500 4.454071 4.325490 4.47760.99 24.854105 40.906081 40.887001 40.9164

rcr 0.1 1.004508 0.997178 1.016766 0.99450.2 1.015984 0.987957 1.154498 0.97870.3 1.028720 0.970292 1.085181 0.95310.4 1.034561 0.941392 1.083942 0.91850.5 1.024433 0.898233 1.039455 0.87410.6 0.989413 0.837600 0.952416 0.81720.7 0.920095 0.755099 0.829332 0.74170.8 0.802349 0.641970 0.675650 0.63590.9 0.602530 0.473829 0.180820 0.47250.99 0.200586 0.156353 0.156389 0.1563

a Two sets of rectangularly distributed double periodic cracks w(under the equiaxed tension).

b Single set of rectangularly distributed double periodic cracks, z0c Single set of rectangularly distributed double periodic cracks, z0d Single set of rhomboitally distributed double periodic cracks, z0e Single set of rhomboitally distributed double periodic cracks, z0f General double periodic cracks, z0 = 2d + di, z1 = 2di.g General double periodic cracks, z0 = 2d + 0.5di, z1 = 0.5d + 2di.h One row of periodic collinear cracks, z0 = 2d.

boundary collocation method, which can be usedto compare the calculation results, so let the inves-tigation start from such a special case. After agreat deal of calculating work it is found that whenthe main period is selected as the period in the col-linear direction, the calculation results for thedimensionless stress intensity factors versus h/dand a/d are all agree with the results by [14]. This

ments of crack array under the tension stress perpendicular toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4lScr=ð1� 2mÞa)

e f g h

38 1.018103 1.004066 1.00598 1.00414547 1.072672 1.017451 1.024453 1.01698230 1.161698 1.043692 1.057141 1.03983018 1.273458 1.088696 1.107451 1.07532702 1.382692 1.161006 1.181843 1.12837939 1.467599 1.272207 1.293182 1.20846540 1.543860 1.436387 1.467990 1.33600549 1.690492 1.665558 1.757515 1.56497436 2.158724 1.956579 2.228219 2.11330797 6.395829 2.242347 2.782943 6.398001

06 1.036533 1.008148 1.011995 1.00830767 1.150626 1.035207 1.049505 1.03425274 1.349542 1.089294 1.117592 1.08124707 1.621695 1.185260 1.226944 1.15632840 1.911837 1.347936 1.399697 1.27324093 2.153847 1.618511 1.684273 1.46038782 2.383504 2.063208 2.190814 1.78490995 2.857764 2.774082 3.163266 2.44914307 4.660090 3.828202 5.032887 4.46606552 40.906633 5.028121 7.744770 40.934416

91 0.982219 0.995951 0.994056 0.99587216 0.932251 0.982848 0.976130 0.98330271 0.860809 0.958137 0.945928 0.96169611 0.785264 0.918530 0.902792 0.92995024 0.723227 0.861322 0.845246 0.88622734 0.681385 0.786036 0.770537 0.82749663 0.647727 0.69691 0.675612 0.74850051 0.591544 0.600400 0.562253 0.63898882 0.463237 0.511096 0.445750 0.47319233 0.156352 0.445961 0.359332 0.156299

ith perpendicular-to-each-other orientations, z0 = 2d, z1 = 2di

= 2d, z1 = 2hi. C0/d = 0, h/d = 1.= 2d, z1 = 2hi, C0/d = 0, h/d = 0.5.= 2d, z1 = C0 + 2hi, C0/d = 0, h/d = 1.= 2d, z1 = C0 + 2hi, C0/d = 0, h/d = 0.5.


can be seen from the former Table 3. So the Eq.(32) is absolutely reliable to the problem of doublyperiodic cracks with one direction collinear if themain period is set on the collinear direction. Forthe problems of more general doubly periodiccracks, some can be transformed into the problemof multiple parallel doubly periodic cracks with

Fig. 11. Comparison of normalized SED and CRS for different arrangcrack faces (For the two sets of double periodic cracks, it is under theTop (a) Normalized strain energy density factor. Bottom (b) Normal

one direction collinear, as is done in example inSection 5.3. When the main period z0 is not of purereal, the Eq. (32) is not applicable, in general.But when jIm z0/Re z0j 6 0.1, the error of theresults based on the Eq. (36), comparing withthe results based on the transformation proce-dure, are at least less than 0.4% in the range of

ements of crack array under the tension stress perpendicular toequiaxed tension stress). (The figure corresponds to Table 15.)ized critical stress.


j2a/Re z0j = 0.1–0.99. The comparison of the di-rect computing procedure and the procedure oftransforming into multiple sets of doubly periodiccracks with one direction collinear is listed inTable 17. Besides, in the Eq. (32), the exchangea-bility of z0 and z1 does not exist. For example,the calculation results for the dimensionless stress

Table 16Comparison of normalized SIF, SED and CRS for different arrangem(KII ¼ KII=s


p, Smin ¼ Smin=½ð2� 2v� v2Þs2a=4l�, scr ¼ scr=

ffiffiffiffiffiffiffiffiffiffiffiffi4lScr=

pad

Different arrangement of crack array

a b c d

KII 0.1 0.997751 1.005103 1.016024 1.00320.2 0.992101 1.020492 1.059459 1.01350.3 0.986168 1.046637 1.120199 1.03320.4 0.984811 1.085072 1.189377 1.06580.5 0.994732 1.139716 1.263180 1.11740.6 1.025870 1.219487 1.345051 1.19790.7 1.096284 1.344861 1.452392 1.32750.8 1.250104 1.570482 1.641846 1.55980.9 1.660160 2.115409 2.143454 2.11140.99 4.986178 6.397499 6.397788 6.3972

Smin 0.1 0.995508 1.010231 1.032304 1.00640.2 0.984264 1.041405 1.122453 0.27300.3 0.972528 1.095449 1.254846 1.06750.4 0.969853 1.177382 1.414617 1.13600.5 0.989492 1.298953 1.595623 1.24860.6 1.052409 1.487149 1.809163 1.43490.7 1.201839 1.808652 2.109443 1.76250.8 1.562761 2.466414 2.695658 2.43290.9 2.756131 4.474956 4.594393 4.45800.99 24.86197 40.92800 40.93169 40.9248

scr 0.1 1.002254 0.994923 0.984229 0.99680.2 1.007962 0.979919 0.943878 0.98660.3 1.014026 0.955441 0.892698 0.96780.4 1.015423 0.921598 0.840777 0.93810.5 1.005296 0.877411 0.791653 0.89490.6 0.974783 0.820017 0.743466 0.83470.7 0.912172 0.743571 0.688519 0.75320.8 0.799933 0.636747 0.609071 0.64110.9 0.602352 0.472722 0.466537 0.47360.99 0.200554 0.156311 0.156304 0.1563

a Two sets of rectangularly distributed double periodic cracks w(under the equiaxed tension).

b Single set of rectangularly distributed double periodic cracks, z0c Single set of rectangularly distributed double periodic cracks, z0d Single set of rhomboitally distributed double periodic cracks, z0e Single set of rhomboitally distributed double periodic cracks, z0f General double periodic cracks, z0 = 2d + di, z1 = 2di.g General double periodic cracks, z0 = 2d + 0.5di, z1 = 0.5d + 2di.h One row of periodic collinear cracks, z0 = 2d.

intensity factors when z0 = 2hi, z1 = 2d are far dif-ferent from the results when z0 = 2d, z1 = 2hi. Thisfurther shows that main period z0 cannot be takenas the complex value with large intersection angleto crack faces.

It must be pointed out that the two periods ofthe doubly periodic cracks have many kinds of

ents of crack array under the shear stress parallel to crack facesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2� 2m� m2Þa)

e f g h

10 0.997757 1.003094 1.002445 1.00414559 0.992430 1.011696 1.009820 1.01698205 0.989524 1.023807 1.022270 1.03983075 1.000662 1.036239 1.040107 1.07532743 1.042046 1.044970 1.063845 1.12837908 1.129175 1.046111 1.093676 1.20846596 1.278206 1.038066 1.127147 1.33600501 1.532825 1.024870 1.153989 1.56497410 2.102339 1.018254 1.156525 2.11330655 6.397465 1.030911 1.148208 6.397994

31 0.995519 1.006197 1.004893 1.0083071 0.984916 1.023528 1.019504 1.03425213 0.979158 1.048181 1.043635 1.08124790 1.001324 1.073791 1.077303 1.15632878 1.085860 1.091963 1.121245 1.27324084 1.275036 1.094348 1.176657 1.46038711 1.633811 1.077580 1.241076 1.78490980 2.349553 1.050358 1.296317 2.44914252 4.419830 1.036842 1.303150 4.4660647 40.92756 1.062777 1.283416 40.93433

00 1.002248 0.996916 0.997562 0.99587223 1.007628 0.988440 0.990388 0.98330262 1.010587 0.976747 0.978871 0.96169696 0.999339 0.965028 0.963454 0.92995001 0.959651 0.956965 0.944386 0.88622789 0.885602 0.955922 0.921881 0.82749641 0.782346 0.963330 0.897637 0.74850007 0.652390 0.975734 0.878303 0.63898817 0.475661 0.982073 0.875997 0.47319217 0.156312 0.970016 0.882706 0.156299

ith perpendicular-to-each-other orientations, z0 = 2d, z1 = 2di

= 2d, z1 = 2hi, C0/d = 0, h/d = 1.= 2d, z1 = 2hi, C0/d = 0, h/d = 0.5.= 2d, z1 = C0 + 2hi, C0/d = 0, h/d = 1.= 2d, z1 = C0 + 2hi, C0/d = 0, h/d = 0.5.


combination. All these combinations of z0 and z1can form absolutely the same doubly periodic crackarray. For example, the rectangularly distributeddoubly periodic crack array can be composed ofz0 = 2d, z1 = 2hi; or of z0 = 2d, z1 = 2d + 2hi; orof z0 = 2d, z1 = 4d + 2hi; or of other periods asshown in Fig. 13. As stated previously, the mainperiod z0 must be of pure real value, or though it

Fig. 12. Comparison of normalized SED and CRS for different arranfaces. (The figure corresponds to Table 16.) Top (a) Normalized stra

is of complex value but the intersection angle tothe crack face must be small. After z0 is given, theauxiliary period can be taken as different complexvalues if by combination of it together with themain period the same doubly periodic cracks canbe composed. For example, when z0 = 2d,z1 = 2d + 2hi, the calculation results are the sameas the results when z0 = 2d, z1 = 2hi. This can be

gements of crack array under the shear stress parallel to crackin energy density factor. Bottom (b) Normalized critical stress.

Table 17Comparison of the direct computing procedure and the procedure of transforming as multiple sets of doubly periodic cracks with onedirection collinear: the dimensionless stress intensity factors of a general doubly periodic cracks under a uniform tension perpendicularto the crack faces

ad

Regard as single set of generaldoubly periodic cracks

Regard as ten sets of doubly periodiccracks with one direction collinear

Relative error (%)

Main period: z0 = 2d + 0.2di Main period: z0 = 19.8dAuxiliary period: z1 = 0.2d + 2di Auxiliary period: z1 = 0.2d + 2diCenter coordinate: zc = 0 Center coordinate of each set:

zcn = (n � 1) * (2d + 0.2di), n = l,2, . . ., 10

0.1 1.003391 1.003393 0.00020.2 1.014364 1.014370 0.00060.3 1.035302 1.035317 0.00140.4 1.070288 1.070319 0.00290.5 1.125878 1.125940 0.00550.6 1.213852 1.213986 0.01100.7 1.360288 1.360616 0.02410.8 1.640858 1.641856 0.06080.9 2.362923 2.367115 0.17740.99 4.255634 4.272398 0.3939

Fig. 13. Rectangular distributed double periodical cracks composed by different periods.


Table 18Dimensionless stress intensity factors of rectangular distributed doubly periodic cracks based on the different periods (N = 16 fora/d 5 0.9, N = 30 for a/d > 0.9, lm = rm = 20)

ad

Periods

z0 = 2d z0 = 2d z0 = 2d z0 = 2dz1 = 2di z1 =2d + 2di z1 = 4d + 2di z1 = 6d + 2di

0.1 1.002830 1.002830 1.002830 1.0028300.2 1.012190 1.012190 1.012190 1.0121900.3 1.030617 1.030618 1.030618 1.0306180.4 1.062256 1.062259 1.062258 1.0622580.5 1.113296 1.113304 1.113301 1.1133010.6 1.193887 1.193905 1.193898 1.1938980.7 1.324330 1.324369 1.324353 1.3243530.8 1.557705 1.557794 1.557757 1.5577570.9 2.110466 2.110709 2.110608 2.1106080.99 6.395824 6.397831 6.396997 6.396995


proved mathematically. But the auxiliary periodstill needs to be given a certain limit because ofthe following fact. In the computing process ofthe interaction of doubly periodic cracks, the inte-ger lm and rm must be given, then a finite area(�lm lm)z0 · (�rm rm)z1 are defined. In thisarea, the interaction effects of each crack on thecenter crack are directly computed based on the ex-act function equation (3), and out of the area, theinteraction effects of each crack on the center crackare computed based on the approximation equa-tion (30). When the area is rectangular, the calcu-lated results are most accurate. When z1 is notperpendicular to z0 and their intersection angle isrelatively small, then the area is an askew parallel-ogram. In the calculation of the effect of each crackon the center crack, such an askew parallelogramarea will lead to the unwished-for consequence.That is to say, for those cracks, which are distantto the center crack, their effects are computed bythe exact function equation (3). And on the con-trary, for those cracks, which are relatively closeto the center crack, their effects are computed bythe approximation equation (30) instead of the ex-act function equation (3). This will cause the errorof calculating results. The dimensionless stressintensity factors of rectangularly distributed dou-bly periodic cracks based on the different periodsare shown in Table 18. So the selection of the aux-iliary period z1 should make the parallelogramwhich is composed of z0 and z1 become a rectangleor as close to a rectangle as possible.

7. Conclusion

The theoretic analysis and a great quantity ofcomputing results show that:

(1) The method presented in this paper is a highlyaccurate and efficient method, which can beused to solve the problems of general doublyperiodic cracks and multiple sets of generaldoubly periodic cracks. It has the extensiveprospects of application.

(2) The key of this method is the summation com-putation of the double series with infinitenumber of terms. Since the Eq. (32) is usedin the method, the high accuracy solutioncan be obtained. And any required accuracycan be given by increasing N, the order num-ber of Chebyshev polynomials, and lm, rm,the number of series terms which are requireddirectly computing. The computing resultsshow that increase of N can make the resultsrapidly converge to the accurate solution,and the increase of lm, rm also can enhancethe accuracy of results, but the convergencespeed is relatively slow. Generally, whenN = 16, lm = rm = 20, the accuracy hasreached to 5–6 significant digits even thoughthe crack spacing is very small.

(3) The same doubly periodic crack array can becomposed of many kinds of period, but toobtain the right and accurate computingresults for the interaction of doubly periodic


cracks only one kind of combination of z0 andz1 can be selected. That is, the main period z0should be the period that the intersectionangle of period line to the crack face is theminimum in the all possible periodsconstituting the considered doubly periodiccrack array. And the auxiliary period z1should be the period which makes the parallel-ogram composed of z0 and z1 become a rectan-gle or become as close to a rectangle aspossible.

(4) The investigation for the validity of the Eq.(32) shows that the computation results ofthe stress intensity factors for the doubly peri-odic cracks in one direction collinear are abso-lutely reliable and accurate if the main periodz0 is set to be the period in collinear direction.For the more general doubly periodic cracks,in which the collinear condition does notexist, if jIm z0/Re z0j 6 0.1, then the errorcaused by using the Eq. (32) will be less than0.4% even though when j2a/Re z0j = 0.99; ifjIm z0/Re z0j > 0.1, the problem can be trans-formed as multiple sets of parallel doubly peri-odic cracks in which the main period z0 isof pure real value or jIm z0/Re z0j is lessthan 0.1.

(5) Whether the interaction between the periodiccollinear crack rows is weakening or strength-ening each other depends on the relative stag-gered shift of the two rows (i.e. C0/d) and themode of load applied on cracks. when the far-field stress is tension, the interaction betweenthe periodic collinear crack rows is weakeningeach other for the rectangular distributed dou-ble periodic crack array and intensifying eachother for the rhomboidal distributed doubleperiodic crack array. Contrary to this,when the far-field stress is shear, the interac-tion between the collinear crack rows is inten-sifying each other for the rectangulardistributed double periodic crack array, andweakening each other for the rhomboidal dis-tributed double periodic crack array. Theintensity of interaction between the periodiccollinear crack rows rapidly decreases withthe increasing of relative ditch between rows(i.e. h/d).

Acknowledgment

This study is supported by Chinese NationalNatural Sciences Foundation (No: 40172097).

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The interaction of doubly periodic cracks