C20003 IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part C ...spiral.imperial.ac.uk/bitstream/10044/1/1187/1/Dini...hinterland where the stress state is given by the singular solution

The effect of a crack-tip radius on the validity of thesingular solution

D Dini and D A Hills*

Department of Engineering Science, University of Oxford, Oxford, UK

Abstract: The influence of the finite crack-root radius on the local stress field at the root of a crack isfound explicitly. This is then applied as an inner asymptotic solution, embedded with the conventionalcrack-tip singular solution, to quantify the possible influence of local rounding on the ability of thesingular solution to capture the characteristics of the crack-tip process zone. The scaling factoremployed is the conventional crack-tip stress intensity, and the example of a simple edge crack in atension field is used to illustrate the method.

Keywords: crack-tip radius, singular solution, local stress field, stress intensity

1 INTRODUCTION

Linear elastic fracture mechanics hinges on the encap-sulation of the crack-tip stress field by the stressintensity factor. The connection between the stressintensity factor and the strain energy release ratethrough the Irwin–Kolosov relation [1] demonstratesthat Griffith’s criterion is consistent with the stressintensity reaching a critical value. Of almost greaterimportance is the empirical correlation between therange of stress intensity experienced and the fatiguecrack growth rate [2]. Although the correlation itself isempirically established, the underlying connectionbetween the two hinges on the ‘process zone’, i.e. thezone of plasticity at the crack tip, where irreversibilitieslead to the exhaustion of ductility and hence to theextension of the crack, being characterized by the elasticsingular field. Thus, the key quantities influencing crackextension must be controlled by a substantial elastichinterland where the stress state is given by the singularsolution [3]. This is the basis for the proviso that ‘small-scale yielding’ should obtain in the crack-tip plasticregion. A paradox of crack-tip behaviour is that thelocal implied singularity means that there must alwaysbe some plasticity; and this plasticity is responsible bothfor crack growth (under repeated loading) and resis-tance to crack growth. Crack-tip plasticity means thatresidual stresses are always present after the first cycle ofload, at the crack tip, and these will both materially

influence the local stress state and cause the crack faceseither to remain apart (in the case of a stationary crack)or be pressed together (in the case of a propagatingcrack). Separately, the plastic deformation means thateven an initially atomically sharp crack tip will becomeradiused.

There have been many studies, at increasing levels ofsophistication, over recent years of the crack-tipplasticity problem [4], and most have focused on theform of the plastic zone and the nature of the residualstresses induced. This paper will not consider theseaspects but address, instead, solely the question of theinfluence of crack-tip rounding on the nature of the localstress state. In particular, the question arises as to howbig the radius at the end of a crack could be withoutinvalidating the domination of the fatigue process by thesingular stress field. The technique to be developed doesnot enquire how the crack-root end radius arose, and sois equally valid in assessing the effects of roundingpresent at the root of slots or ‘starter notches’, and isparticularly suited to judging whether EDM-formedslots are cracklike in their characteristics, as notch rootsformed this way are invariably almost semicircular inform.

The approach to be developed hinges on the use of aset of nested asymptotic elastic solutions, and auniversal solution for permitted root radius, butensuring singular response, will emerge. An exampleproblem will be considered, as shown in Fig. 1A. Thisconcerns the edge-breaking crack present on the surfaceof a half-plane and subject to uniform remote tension.Suppose that, firstly, the limit of small-scale yielding isto be determined. This could be found by taking thesemi-infinite crack solution (that corresponding to the

The MS was received on 8 October 2003 and was accepted after revisionfor publication on 27 February 2004.* Corresponding author: Department of Engineering Science, Universityof Oxford, Parks Road, Oxford OX1 3PJ, UK.

693

C20003 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part C: J. Mechanical Engineering Science

crack-tip stress intensity) (Fig. 1B), and noting thedomain in which agreement with the finite cracksolution is obtained to within a given degree of accuracy[5]. This would then limit the extent of the plastic zonethat could be accommodated as a necessary condition isfor the elastic hinterland to be characterized by the usualsingular field, to within a given tolerance. At the sametime, the stress intensity factor would emerge. The nextphase of the procedure is to establish a solution for thesemi-infinite rounded slot (Fig. 1C). It should be notedthat, remote from the notch root, the stress field for thisproblem approaches that of a semi-infinite sharp crack(i.e. the solutions to problems B and C diverge when r=rbecomes small, adjacent to the notch root). Theargument is made that, if the majority of the processzone experiences a strain field dominated by the elasticstrains generated by an exterior singular field, thepresence of deviation from this field near the innerboundary will have a small effect. The present authorsargue that the exhaustion of ductility within the processzone will, under these conditions, still be controlled by

the stress intensity factor. Note that this is despite thefact that at the notch root the strain state must bebounded while at the crack tip it must be singular. Thus,by noting the discrepancy between these two solutions,the lower extent of the process zone, given that it mustbe controlled by the singular field, may be found. Thispart of the method gives rise to a universal solution,which may be applied to any crack, without furthercalculation, once the stress intensity factor is known.Thus, the solution for a crack in a finite body but with aroot radius (here Fig. 1D) may be inferred withouthaving to solve the problem itself, within certainlimitations.

2 FORMULATION

The enabling solutions, here, are those relating to theauxiliary problems (Figs 1B and C), and these are nowgiven.

Fig. 1 (A) Edge crack in the example finite body, (B) semi-infinite edge crack, (C) rounded edge crack in theexample finite body and (D) semi-infinite rounded edge crack

D DINI AND D A HILLS694

Proc. Instn Mech. Engrs Vol. 218 Part C: J. Mechanical Engineering Science C20003 # IMechE 2004

3 SEMI-INFINITE CRACK

The solution for a semi-infinite crack suffering mode Iloading is well known, and may be derived directly fromWilliams’ solution for a semi-infinite wedge [6], andchoosing the internal wedge angle to be 2p rad, or byusing a series expansion on, for example, the Wester-gaard [7] solution for a finite crack. Either way, thestress state may be written in the form

si j½r, y� ¼KIffiffiffiffiffiffiffi2pr

p fi jðyÞ ð1Þ

where the spatial distribution is given by

fyyfrrfry

8<:

9=; ¼ 1

2

3

2cos

y2

� �5

2cos

y2

� �1

2sin

y2

� �

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ 1

2

cos3

2y

� �

� cos3

2y

� �

sin3

2y

� �

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

26666664

37777775

26666664

37777775ð2Þ

and KI is the stress intensity factor.

3.1 Semi-infinite rounded slot

This problem is depicted in Fig. 1C and is a special caseof the semi-infinite radiused-root V-notch recentlyanalysed very effectively by Fillippi et al. [8], whosework should be consulted for details. The present papermerely records the relevant results for the stress field, forthe special case where the notch sides are parallel,forming a ‘slot’, i.e. something geometrically equivalentto a crack, but with a tip radius. In polar coordinates,centred as shown in the figure [with r0 ¼ rðq� 1Þ=q andq ¼ ð2p� 2aÞ=p], the stress state is given by

si jðr, yÞ ¼KIffiffiffiffiffiffiffi2pr

p ½ fi jðyÞ þ gi jðr, yÞ� ð3Þ

where the functions gi jðr, yÞ, which may be thought of asthe perturbation of the singular field, are given by

gyygrrgry

8<:

9=; ¼ r

2r

cosy2

� �

� cosy2

� �

siny2

� �

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

0BBBBBB@

1CCCCCCA

26666664

37777775

ð4Þ

and r is the crack tip radius.

4 NESTING OF INNER AND OUTER

ASYMPTOTES

The two asymptotic solutions described above (prob-lems B and C) have been set out in a way that makes itclear that, for large values of r=r, the two fields musttend to the same stress distribution, if KI is the same foreach. Thus, once the value of the stress intensity factorhas been found for the outer asymptotic (the standardsemi-infinite crack solution), the same scaling factormay be used to determine the inner stress field when aradius is present. It is therefore possible to compare thetwo asymptotic fields for the general case. Figure 2displays, at two magnifications, the state of stress aheadof the ‘crack’ tip, and the contours of two quantities aregiven. One (shown in solid lines) is the location of theplastic front, or the process zone size, as a function ofthe normalized applied load ðKI=k

ffiffiffir

p Þ, where k is theyield stress in pure shear (only the half-plane y � 0 isshown for economy of space; the lower half-plane is, ofcourse, symmetrical). The von Mises yield criterion hasbeen assumed, and transverse plane strain, with Poisson’sratio set to 0.3. The second set of contours (broken lines)displays the fractional mismatch between the inner andouter asymptotic solutions as characterized by the seconddeviatoric stress invariant (von Mises parameter). Asexpected, the fractional difference increases as the ‘crack’tip is approached, but it does so in a very non-uniformway (in a polar sense), because the eigenfunctions fi jðyÞ,relating to the outer asymptote, and gi jðr, yÞ, relating tothe inner asymptote, are very different in form.

Consider the form of the discrepancy contoursqualitatively (see Fig. 2a). They show that, as y?+p,i.e. along the flanks of the crack, the stress fields arequite different, even though the discrepancy must tendto zero for large values of r=r. The region (approxi-mately) p=6 < y < 3p=4 shows a remarkably consistentcorrelation between the two solutions, with a maximumabsolute dicrepancy of about 5 per cent. As y? 0 (aheadof the crack tip), the discrepancy again increases inmagnitude, although the maximum value attained ismuch less than along the crack flanks.

Clearly, because of the considerable extent of themismatched region as y?+p, it is impossible to findany error contour circumscribed by a process zone frontcontour. It follows that it is impossible, no matter howsmall the ‘crack-tip radius’, for a regime to be foundrigorously when the process zone strains are everywherecontrolled by a singular elastic hinterland. It isproposed, therefore, to discount any attempt to matchin this neighbourhood, but instead ensure that aroundthe majority of the process zone front the strains arecontrolled by KI (in practice 04y9 3p=4 as shown inFig. 2a, where the shaded region highlights the domainoutside the chosen process zone front in which thediscrepancy exceeds 5 per cent). The KI dominatedhinterland encompasses both the region in which, for

THE EFFECT OF A CRACK-TIP RADIUS ON THE VALIDITY OF THE SINGULAR SOLUTION 695


any given load, the process zone radius is largest and theregion where the exhaustion of ductility leads toextension of the crack ðy&0Þ. In practice, the line y ¼0 therefore delineates the maximum mismatch incurredand the corresponding plastic front location (whosevalues are shown in Fig. 2) which may be controlled.

By comparing the location of the plastic front with therelative discrepancy in the stress state, as captured bythe singular solution, it is accepted that the minimumload needed to achieve this degree of accuracy may befound. Note that this figure actually gives the mismatchimplied by the elastic solutions at this point. In order to

Fig. 2 Discrepancy between solutions for problems C and B as implied by the von Mises parameter, i.e.

jffiffiffiffiffiffiJC2

q�

ffiffiffiffiffiffiJB2

qj=

ffiffiffiffiffiffiJC2

q,

together with process zone front locations, for a given normalized stress intensity factor, KI=kr0:5: (a)global view, (b) close-up



achieve this degree of absolute accuracy in the processzone, the elastic hinterland would have to extendbeyond the plastic front by a considerable amount.Nevertheless, this procedure provides a good measure ofthe accuracy to be expected.

It is emphasized that, because each of the character-istic fields under consideration scales with the stressintensity factor, the comparison made in Fig. 2 iscompletely general in its application. In order to use itfor a particular finite geometry, it is necessary only tofind the value of KI for a conventional sharp crack inthat geometry.

5 EXAMPLE PROBLEM: EDGE CRACK

As an example in the use of the procedure described,consider the problem of an edge crack under remoteuniform tension. This problem may be solved in anumber of ways, but one that not only enables the cracktip stress intensity factor but also the complete localstress field to be found readily is the distributeddislocation method, which is described in detail byNowell and Hills [9, 10] and only the briefest outline willbe given here.

The first step of the procedure is to determine the stateof stress, sdi j (with i, j ¼ x, y), induced by a singledislocation of magnitude by (the Burgers vector) placedat a point ðxd, 0Þ in a half-plane. Using results reportedin reference [11], the stress components at a particularpoint ðx, yÞ are given by

sdi jðx, yÞ ¼m

pðkþ 1Þ byðxdÞGyi jðx, xd, yÞ ð5Þ

where m is the shear modulus and k is Kolosov’s con-stant [equal to 3� 4n in plane strain or ð3� nÞ=ð1þ nÞin plane stress, where n is Poisson’s ratio] and thecoordinate set for the half-plane is as shown in Fig. 1.The functions Gyi j may be obtained from reference [9].

Rather than installing discrete dislocations, a con-tinuous distribution of the Burgers vector density, By, isemployed along the line of the crack, so that the stressdue to this distribution, together with the farfield stress,s0, is given by

si jðx, yÞ ¼ s0 þm

pðkþ 1Þ

ðb0

ByGyi jðx, xd, yÞ dxd ð6Þ

Since the crack faces are traction free, the followingCauchy singular integral equation, applied along the liney ¼ 0, is obtained:

s0 þm

pðkþ 1Þ

ðb0

ByðxdÞGyyyðx, xd, 0Þ dxd ¼ 0 ð7Þ

where Gyyyðx, xd, 0Þ can be rewritten in the simplified

form

Gyyyðx, xd, 0Þ

¼ 21

x� xd� 1

xþ xd� 2xd

ðxþ xdÞ2þ 4x2d

ðxþ xdÞ3

" #ð8Þ

This equation may not be inverted in closed form, andso a numerical procedure, using Gauss–Jacobi quad-rature [12], is employed. This yields n simultaneouslinear algebraic equations, which may be solved usingstandard techniques. The procedure is as follows:

1. Normalize the range of integration to + 1 by anappropriate change in variables.

2. Consider the behaviour of the unknown function atthe ends of the crack. For a crack of size b whoseorigin is at the surface, such as that shown in Fig. 1A,the unknown function ByðxdÞ is singular at the cracktip, as x? b, and bounded at the crack mouth, asx? 0.

3. Represent the unknown function as a product of anappropriate weight function, w, which incorporatesthe expected behaviour at the end-points and abounded function f. Since the singularity order isexpected to be r� 0:5, the appropriate choice of weightfunction is

wðuÞ ¼ffiffiffiffiffiffiffiffiffiffiffi1þ u

1� u

rð9Þ

where u is the normalized variable, bounded at u ¼� 1 and singular at u ¼ 1.

4. Gauss–Jacobi integration may now be used to reducethe integral in equation (7) to a set of simultaneouslinear equations in order to obtain the values of thebounded function f at N integration points (N wasset to 50 for good accuracy). The discretized form ofequation (7) is

s0ðvkÞ þm

pðkþ 1ÞXNi¼1

2ð1þ uiÞ2N þ 1

Gyyyðui, vk, 0ÞfðuiÞ

¼ 0 ð10Þ

where

ui ¼ cos2i � 1

2N þ 1p

� �, i ¼ 1, 2, . . . , n ð11Þ

vk ¼ cos2k

2N þ 1p

� �, k ¼ 1, 2, . . . , n ð12Þ

The solution of equation system (10) gives n values off and hence ByðuÞ, from which the crack tip stressintensity may be found by Krenk interpolation. Itsvalue is that obtained by Kaya and Erdogan [13] andis

KI

s0ffiffiffiffiffiffipb

p ¼ 1:12152 ð13Þ



Once the values of the dislocation density are known,it is a routine matter to obtain the complete stressfield by evaluation of equation (6) which nowrepresents a simple integral. Note that, away fromthe crack itself, the integral is regular, and hence thechoice of the field coordinates is free and is notconstrained to those implied by the numericalquadrature scheme.

6 SMALL-SCALE YIELDING

The limit of small-scale yielding represents the largestload that may be applied while maintaining a gooddescription of the crack-tip plastic zone by the stressintensity factor (in contrast to the minimum loadcalculated in section 4). Several measures of the qualityof the singular field could be employed, but the one thathas greatest relevance in the present context would seemto be the plasticity parameter itself. Also in contrast tothe lower bound to the load, this result is, of course,geometry specific, and the results presented here are forthe problem in Fig. 1A alone.

Accordingly, Fig. 3 gives plots incorporating two setsof contours. The first is the fractional mismatch betweenthe von Mises parameter (the second invariant of thestress deviator), as measured by the full stress field andby the singular term only. The second set of contoursshows the position of the plastic front for various valuesof applied load ðs0=kÞ. Thus, for example, if adiscrepancy in the characterization of the surroundingelastic crack tip stress field of + 5 per cent is to bepermitted, the maximum load that may be applied, forthis particular geometry, is s0=k ¼ 0:036, while, if adiscrepancy of + 30 per cent is acceptable, themaximum load that may be applied goes up to s0=k ¼0:945 (see Fig. 3a).

7 BOUNDS ON THE APPLIED LOAD

This paper is concerned with the validity of theassumption that the crack tip process zone is character-ized by the singular term, with and without localrounding. The maximum corresponds to the classicsmall-scale yielding requirement, and is, to repeat, ageometry dependent phenomenon. Using the results justfound in the preceding section, this requirement may besummarized in the plot shown in Fig. 4, where thehorizontal lines denote this limit. It is again emphasizedthat the term ‘5 per cent discrepancy’ means that there isa mismatch in the elastic fields of this value, ascharacterized by the second deviatoric invariant, whentheir plastic front, as implied by the elastic solution, lies

at this point. Thus, there is inaccuracy both in theestimated position of the plasticity front (which does notallow for stress redistribution) and because a significantregion of elastic hinterland must also be controlled bythe stress intensity solution to achieve the accuracycited. Nevertheless, the procedure described provides acompletely consistent procedure.

The second set of lines in the figure gives the lowerbound to the permitted applied load and corresponds tothe minimum load that can be tolerated when thesingular sharp semi-infinite crack stress field and semi-infinite rounded crack tip field agree. These were derivedby using the general results of matching displayed inFig. 2 and applying the calibration for the stressintensity factor, KI, for this particular geometry[equation (13)]. The permitted load range for a givendegree of accuracy decreases as the crack tip radiusincreases. For example, if a maximum discrepancy of5 per cent is required, the load and dimensionless rootradius must lie in the shaded region of Fig. 4. Nosolution to this degree of accuracy is possible if r=bexceeds 0.000 052 or s0=k exceeds about 0.036. If amismatch of 7 per cent is acceptable, then r=b mustcertainly be less than 0.000 242 and s0=k cannot exceedabout 0.074. These requirements are certainly stringent;with conversion to dimensional quantities, it suggeststhat, for the root radius to have less than a 7 per centeffect on the singular field, a 10mm edge crack wouldneed to have an end radius of less than 2:4 mm. Equally,the small-scale yielding assumption imposed at the samelevel of accuracy would imply a maximum remote loadof 7 per cent of the yield stress in pure shear (see Fig. 4a).A less ambitious choice in terms of acceptable mismatch(e.g. 20 per cent) will instead lead to less stringentrequirements in terms of both r=b and s0=k, as shown inFig. 4b.

8 CONCLUSION

The solution for a semi-infinite radiused-root crack hasbeen found and used as an inner asymptote to embedwithin the conventional crack-tip singular solution. Thisenables the lower bound load for a given degree ofaccuracy in the characterization of the process zone tobe found. These results are displayed in Fig. 2, andapply regardless of the geometry of the crack. As anexample of their application, they have been used toreconcile the edge crack in tension. This underlines thestringent requirements on load range and crack-tipradius imposed by the small-scale yielding and lowerbounds, and suggests that a wider tolerance of mismatchat the process zone front may be acceptable. Figure 5therefore summarizes the requirements imposed in Fig. 2



(for a general crack) for mismatch by up to 40 per cent.The point is emphasized that the results are independentof the geometry in which the crack lies—the geometryenters the solution only in the way in which it

determines the stress intensity factor. The maximumtolerable root radius, or the maximum amount ofblunting that may be accepted, has therefore beendetermined.

Fig. 3 Discrepancy between solutions A and B together with process zone front locations, for solution A, fora given remote load ðs0=kÞ starting from the crack tip: (a) global view, (b) close-up



Fig. 4 Range of the permissible normalized load ðs0=kÞ, as a function of r=b, to achieve characterization ofthe process zone by the singular solution B for the case of an edge crack under uniform remotetension, to within a given discrepancy: (a) 5 and 7 per cent; (b) 20 and 40 per cent



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Fig. 5 Lower bound load in the characterization of the process zone



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C20003 IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part C ...spiral.imperial.ac.uk/bitstream/10044/1/1187/1/Dini...hinterland where the stress state is given by the singular solution