On improved crack tip plastic zone estimates based on T ...webserver2.tecgraf.puc-rio.br/~lfm/papers/Sousa-FFEMS-2013.pdf · IMPROVED CRACK TIP PLASTIC ZONE ESTIMATES 27 Fig. 1 Mode

doi: 10.1111/j.1460-2695.2012.01684.x

On improved crack tip plastic zone estimates based on T-stress and oncomplete stress fields

R. A. SOUSA 1, J . T . P . CASTRO 2, A . A . O. LOPES 3 and L . F . MARTHA 1

1Department of Civil Engineering, 2Department of Mechanical Engineering, Pontifical Catholic University of Rio de Janeiro, PUC-Rio, Brazil3Petrosoft Design Development of Software LTDA, Rio de Janeiro, Brazil

Received in final form 28 February 2012

A B S T R A C T Cracked ductile structures yield locally to form a plastic zone (pz) around their crack tips,which size and shape controls their structural behaviour. Classical pz estimates are basedsolely on stress intensity factors (SIF), but their precision is limited to very low σ n/SY

nominal stress to yield strength ratios. T-stresses are frequently used to correct SIF-basedpz estimates, but both SIF and SIF plus T-stress pz estimates are based on truncated linearelastic (LE) stress fields that do not satisfy boundary conditions. Using Griffith’s platecomplete LE stress field to avoid such truncated pz estimates, the influence of its Williams’series terms on pz estimation is evaluated, showing that T-stress improvements are lim-ited to medium σ n/SY values. Then, corrections are proposed to introduce equilibriumrequirements into LE pz estimates. Finally, these improved estimates are compared withpz calculated numerically by an elastic–plastic finite element analysis.

Keywords crack tip plastic zone estimates; crack tip stress fields; elastic–plastic fracturemechanics; fracture toughness; T-stress.

N O M E N C L A T U R E a = crack sizeCTOD = crack tip opening displacement

E = Young’s modulusKI = stress intensity factor in mode IKt = stress concentration factor

pl − ε = plane strainpl − σ = plane stress

pz = plastic zoneSIF = stress intensity factorSY = yield strengthν = Poisson’s coefficient

σ n = nominal stressσ n/SY = nominal stress to yield strength ratio

I N T R O D U C T I O N

Most linear elastic fracture mechanics (LEFM) analysisand design routines are based entirely on stress inten-sity factors (SIF), a desirable and useful feature for manypractical applications. However, it is well known that SIFalone do not accurately model some crack problems. Forexample, the idealized linear elastic (LE) stress field σ ij ina Griffith’s plate loaded in mode I by a nominal stress σ n,

Correspondence author: Jaime T.P. Castro. E-mail: [email protected]

induced by its SIF KI = σ n√

πa , where 2a is the crack size,does not satisfy its boundary conditions far from the cracktips. Indeed, σ ij(r, θ ) = [KI /

√2πr] · fij(θ ) ⇒ σ θ (r → ∞,

θ = 0) = 0, instead of σ θ (r → ∞, θ = 0) = σ n as it shouldbe, where r is the distance from the crack tip, θ is theangle measured from the crack plane and fij(θ ) are Irwin’sθ functions. Moreover, SIF-based LE analyses obviouslycannot be used to describe stresses and strains inside theplastic zones pz(θ ) that always form around real crack tipseither. However, for engineering design purposes, pz(θ )are traditionally estimated from such simplified analyses,

c© 2012 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 36, 25–38 25

26 R. A . SOUSA et al.

assuming that they depend only on SIF values. Indeed,equating the Mises stress obtained from σ ij to the yieldstrength SY , the simplest mode I elastic–plastic (EP)boundary estimations in plane stress (pl − σ ) and in planestrain (pl − ε) are expressed by1:

pz(θ )pl−σ = (K 2

I /2π S2Y)

cos(θ/2)2[1 + 3sin(θ/2)2], (1)

pz(θ )pl−ε = (K 2

I /2π S2Y)

cos(θ/2)2[(1 − 2ν)2 + 3sin(θ/2)2],

(2)

in which ν is the Poisson’s coefficient.According to these classical estimates, the pz(θ ) size di-

rectly ahead of crack tips in pl − σ , which is the ref-erence used here to normalize pz(θ ) plots, should bepz(0)pl−σ = pz0 = (1/2π )(KI/SY ).2 However, stress fieldsbased solely on SIF are exact only at r → 0, just where theassumed LE behaviour makes no sense. Consequently,such traditional pz(θ ) estimates are reasonable only forvery low SIF values, associated with nominal stress toyield strength ratios σ n/SY much smaller than those usedin most practical design applications. Nevertheless, theerrors introduced by this crude approximation are fre-quently ignored in mechanical design and analysis, evenwhen dealing with crack problems associated with thehigher σ n/SY ratios found in practice. Ignoring such er-rors may cause mistaken decisions, for example evaluatingwhich component dimensions can be treated as pl − σ orpl − ε dominated for design purposes; estimating the sizesof overload-affected zones in fatigue crack propagationunder variable amplitude loads; analysing how much ma-terial should be removed when repairing a fatigue crack,etc. Moreover, before proceeding, it is important to em-phasize that idealized EP singular pz(θ ) estimates, such asthose generated by the Hutchinson, Rice, and Rosengren(HRR) field,2 do not solve this problem either, becausethey also do not satisfy stress boundary conditions farfrom the crack tips.

Because relatively high σ n/SY ratios are often used instructural designs nowadays, it is worthwhile to evaluatethe effect of σ n/SY on pz(θ ), instead of simply neglectingit. In fact, modern engineering structures are typically de-signed with yield safety factors 1.25 < φY < 3 ⇒ 0.33 <

σ n/SY < 0.8. However, it must be pointed out that suchhigh σ n/SY ratios do not preclude LE crack analysis, be-cause they can many times be associated with pz0 valuesthat are small compared with the residual ligament sizerl = w – a, where w is the cracked component width, par-ticularly in large structures. In this way, their stress fieldsremain predominantly LE and can be well analysed as so.Indeed, note that because those typical φY are associatedto 0.32 < pz0/a < 0.06, except for very deep cracks, theirpz0 size does in fact characterize a dominant LE stress field

in rl. Therefore, to explore how improved LE estimatescan be generated for pz(θ ) sizes and shapes may be aninteresting and profitable exercise, because more precisecalculations for them can only be obtained by non-trivialnumerical models that are far too expensive for most prac-tical applications.

For the Griffith plate, a crude engineering LE esti-mate for the σ n/SY effect can be made by forcing theσ yy stress component to obey the far field boundarycondition, σyy (x → ∞, y = 0) = σn, by adding a con-stant σ n stress to the Williams3 (or Irwin) stress field toobtain

σ (θ )KI +σnM,pl−σ = [

(κ · fx)2 + (κ · fy + σn)2 − (κ · fx)

× (κ · fy + σn) + 3(κ · fxy )2]1/2, (3)

in which κ = KI /√

2πr , and fx(θ ), fy(θ ) and fxy(θ ) are themode I θ functions associated with σ xx, σ yy and τ xy.

The somewhat heavy notation σ (θ )KI +σnM,pl−σ emphasizes

that Eq. (3) expresses the resulting Mises LE stress distri-bution around the crack tip in the limit 2D case of pl −σ conditions, considering the KI-induced stress field andthe additional σ n constant term (arbitrarily) added to itsσ yy component. A similar equation can be easily gener-ated for pl − ε. The corresponding simplistic estimatesfor the pz(θ ) border obtained by forcing σ M (θ ) = SY forthe Griffith’s plate, shown in Fig. 1, indicate that theσ n/SY ratio may indeed significantly affect pz(θ ) size andshape.

In other words, despite recognizing that adding a con-stant σ n term to the KI-induced σ yy stress component iscertainly not a sound approach, its simplistic pz(θ ) LE es-timates shown in Fig. 1 obey the far field boundary con-dition for the Griffith plate, thus have advantages overthe even more simplistic traditional estimates expressedby Eqs (1) and (2). Moreover, in this sense they pointout that their dependence on σ n/SY should be furtherexplored, as follows in this paper.

Successively refined LE estimates are thus generated inthe following sections. First, the complete LE stress fieldfor the Griffith’s plate is obtained from the equivalent In-glis plate and from its Westergaard stress function. Then,this stress field is used to estimate pz(θ ) for various σ n/SY

ratios. Next, the same complete LE stress field is obtainedby a Williams’ series expansion, to study the effect of itsnumber of terms on pz(θ ) estimation and to show that T-stress improvements are limited to medium σ n/SY ratios,not compatible with the loads used in high-performancestructures. In the sequence, four types of corrections areproposed to consider equilibrium effects neglected whenestimating pz(θ ) by forcing σ M (θ ) = SY . Finally, such im-proved LE estimates for pz(θ ) are compared with plastic

c© 2012 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 36, 25–38

IMPROVED CRACK TIP PLAST IC ZONE EST IMATES 27

Fig. 1 Mode I pz(θ ) roughly estimated for the Griffith’s plate by adding a constant σ yy = σ n stress to the KI-based LE stress field and thenfinding σ (θ )KI +σn

M = SY for (a) pl − σ and (b) pl − ε limit conditions, to evaluate the σ n/SY effect. Note the different scales used in pl − σ

and pl − ε plot.

zone boundaries obtained by a detailed EP analysis of theGriffith’s plate.

P L A S T I C Z O N E E S T I M A T E S B A S E D O NC O M P L E T E L E S T R E S S F I E L D S

Plastic zone boundaries estimated by the equivalentInglis plate

An improved estimate for the σ n/SY effect on pz(θ ) for theGriffith’s plate can be obtained from the LE stress field ofthe equivalent Inglis’ plate loaded in mode I, assuming ithas a sharp elliptical notch with major semi-axis a normalto σ n and minor semi-axis b � a. This way of modellingcrack problems is interesting, because no crack can havea zero radius notch tip under load, simply because noreal material can support singular stresses and strains. Byassuming x = c · cosh(α) · cos(β) and y = c · sinh(α) · sin(β),the sharp notch used to simulate the blunted crack canbe described in elliptical coordinates (α, β) by α = α0,where a = c · cosh(α0), b = c · sinh(α0) and c = a/cos(α0).The general LE stress field in Inglis’ plates is given bya series too long to be reproduced here.4 However, ifthe sharp notch in mode I has tiny, but finite, tip radiusρ = b2/a = CTOD/2 ∼= 2K 2

I /π SY E ′, in which CTOD isthe crack tip opening displacement, E′ = E in pl − σ ,E ′ = E/(1 − ν2) in pl − ε, E is Young’s modulus and ν isPoisson’s coefficient, then its stress concentration factor

Kt = 1 + 2a/b is given by

Kt = 1 + 2ab

= 1 + 2√

aρ

= 1 + 2

√aπ E ′SY

2σnaπ

⇒ ab

√E ′

2σn

SY

σn=√

E ′φY

2σn. (4)

Using an a/b ratio for the notch shape that simulates ablunt crack with α0 = tanh−1(b/a), the LE stress field inthe Inglis plate can be calculated. Finally, considering thestress field describe in elliptical coordinates by σα , σβ , ταβ

and, in the pl − ε case, also by σ zz = ν(σα + σβ ), the resu-lting Mises stress can be used to estimate the Inglis’ plasticzones by numerically solving Eqs (5) and (6) for |θ | ≤ π :

σIngM,pl−σ =

√σ 2

α + σ 2β − σασβ + 3τ 2

αβ = SY , (5)

σIngM,pl−ε

=√

0.5[(σα − σβ )2 + (σα − σz)2 + (σz − σβ )2] + 3τ 2αβ

= SY . (6)

The results of these estimates are shown in Fig. 2.Therefore, the nominal stress to yield strength σ n/SY

ratio influence on Griffith’s plate pz(θ ) estimated fromthe exact LE stress field of its equivalent Inglis’ plate, al-though a little smaller than that estimated by the simplis-tic approximation used to generate Fig. 1 plots, is indeed



Fig. 2 Mises plastic zone boundaries pz(θ ) in (a) pl − σ and (b) pl − ε, calculated from the Inglis’ LE stress field for simulating a crackedGriffith’s plate loaded in mode I, modelling its blunt crack as a very sharp elliptical notch of tip radius ρ = CTOD/2.

significant. Hence, it should not be neglected in prac-tical applications. Note that to use Inglis stress field tosimulate the exact LE stress field for the Griffith’s plate,modelling its blunt crack as an elliptical sharp notch oftip radius ρ = CTOD/2, is a very reasonable hypothe-sis, because ideal cracks should open their tips by CTODunder load. Nevertheless, to avoid any doubts about thismodel soundness, it is interesting to use an alternativeapproach to confirm the importance of the σ n/SY ratioin pz(θ ) estimations, modelling the crack by the sameWestergaard stress function used by Irwin to calculatethe Griffith’s plate SIF KI . This is done in the followingsection.

Plastic zones estimated using Westergaard stressfunctions

The Westergaard stress functions Z(z) provide an alter-native way to rigorously estimate pz(θ ) from LE stressfields.5 However, as the EP boundary is not adjacent tothe crack tip, the complete stress field generated from Z(z)must be used in such a calculation. This can be demon-strated by revisiting Irwin’s classical solution for the Grif-fith’s plate loaded in mode I. Thus, if (x, y) and (r, θ ) areCartesian and polar coordinates centred at the crack tip,i = √−1, and z = r · exp(iθ ) = x + iy is a complex variable,Irwin’s solution is obtained from

Z(z) = (zσn)/√

z2 − a2 ⇒ Z′(z) = dZ(z)/dz

= (−a2σn)/(z2 − a2)3/2, (7)

⎧⎪⎨⎪⎩

σxx

σyy

σxy

⎫⎪⎬⎪⎭ =

⎧⎪⎪⎨⎪⎪⎩

Re(Z(z)) − yIm(Z′(z))

Re(Z(z)) + yIm(Z′(z))

−yRe(Z′(z))

⎫⎪⎪⎬⎪⎪⎭ . (8)

To solve the mode I problem from Z(z), a constant termσ n is added to σ xx to force σ xx(∞) = 0 in Griffith’s plate,an adequate mathematical trick, because a constant stressin the x direction does not affect the stress concentrationeffect introduced by the crack tip. However, the resultingσ yy stress is usually truncated to generate its well-knownSIF KI = σ

√πa . This is normally a highly desirable fea-

ture, but the stress field truncation required to obtain itis of no use for estimating pz(θ ), because it neglects theσ n/SY effect. The stress truncation is obtained by writing

σyy(θ = 0) = σn(x + a)/[(x + a)2 − a2]1/2

∼= σna/√

2ax = KI /√

2πr (if x << a), (9)

in which 2a is the crack length perpendicular to the nom-inal stress σ yy(r → ∞) = σ n.

As Eq. (9) formally yields σ yy (θ = 0) = KI /√

2πr = 0only if r → ∞, this classical approximation is useful toobtain KI , as Irwin did. But it obviously cannot be usedto study the σ n/SY influence on pz(θ ). However, this taskcan be fulfilled by, first, calculating the complete stressfield generated from Z and Z′ to obtain the resultingMises (or Tresca, for that matter) stress and, then, byequating this stress to SY to obtain the required pz(θ ).This calculation results in Eq. (10), which is explicitlywritten here to demonstrate how the Westergaard stress



function can generate the complete LE stress field for theGriffith’s plate⎧⎨⎩⎡⎣Re

⎛⎝ (a + r cos(θ ) + ir sin(θ )) σn√

(a + r cos(θ ) + ir sin(θ ))2 − a2

⎞⎠

−yIm

(−a2σn[

(a + r cos(θ ) + ir sin(θ ))2 − a2]3/2

)− σn

]2

+⎡⎣Re


(a + r cos(θ ) + ir sin(θ ))2 − a2

⎞⎠

+ yIm

(−a2σn[

(a + r cos(θ ) + ir sin(θ ))2 − a2]3/2

)]2

−⎡⎣Re


(a + r cos(θ ) + ir sin(θ ))2 − a2

⎞⎠

−yIm

(−a2σn[

(a + r cos(θ ) + ir sin(θ ))2 − a2]3/2

)− σn

]

·⎡⎣Re


(a + r cos(θ ) + ir sin(θ ))2 − a2

⎞⎠

+yIm

(−a2σn[

(a + r cos(θ ) + ir sin(θ ))2 − a2]3/2

)]

+3

[−yRe

(−a2σn[

(a + r cos(θ ) + ir sin(θ ))2 − a2]3/2

)]2⎫⎬⎭

1/2

−SY = 0 (10)

The corresponding pz(θ ) estimations in pl − σ and pl −ε illustrated in Fig. 3 clearly show the importance of theσ n/SY effect, reinforcing the claim that that it should notbe neglected in practical applications.

Moreover, it should be noted that the plastic zoneboundaries estimated from complete Inglis and Wester-gaard LE stress fields, which satisfy all the boundary con-ditions for the Griffith’s plate loaded in mode I, namelyσ yy(|x| ≤ a, y = 0) = τ xy(|x| ≤ a, y = 0) = σ xx

∞ = τ xy∞ = 0,

σ yy∞ = σ n, visually coincide when the sharp ellipsis used in

the equivalent Inglis’ plate has its minor semi-axis (insteadof its tip radius) chosen as b = CTOD/2 = 2KI

2/πSY E′, asis shown in Fig. 4. As pzIng(θ ) and pzWtg(θ ) are obtainedfrom completely different equations, their near coinci-dence seem to not be fortuitous. Therefore, the largeσ n/SY effect estimated by these rigorous LE solutions re-ally should not be neglected in practice. It is worthwhileto emphasize this point again, because it is the plastic zonesize that validates most LEFM predictions used in designapplications.

Plastic zones estimated using Williams series andonly the T-stress correction

Based on photoelastic experimental results obtained byWells and Post5 for the stress field around crack tips, Ir-win proposed a long time ago to add a constant term tothe stress component σ xx parallel to the crack direction,naming it T-stress,6 to improve LEFM predictions. Lateron, Larsson and Carlsson,7 while investigating the lim-its recommended by ASTM for SIF use, noted that theaddition of the T-stress term could adjust pz(θ ) shapes es-timated by LE analysis, approximating them to the pz(θ )shapes obtained from EP finite elements (FE) numeri-cal analyses. Following that work, the T-stress has beenwidely explored in the literature to model some inter-esting problems. In this context, it is worth mentioningthe works of Rice,8 Leevers and Radon,9 Cardew et al.,10

Kfouri,11 Bilby et al.,12 Sham,13 Betegon and Hancock,14

Du and Hancock,15 O’Dowd and Shih,16 Nakamura andParks,17 Wang and Parks,18 Wang,19 Hancock et al.,20

Kim et al.,21 Ganti and Parks,22 Zhang et al.,23 Rameshet al.,24 Chen and Tian25 Kang and Beom,26 Smith et al.,27

Zhao et al.,28 Chen et al.,29 Karihaloo and Xiao,30 Tan andWang,31 and Su and Sun.32 Fett33 listed T-stress valuesfor several geometries. The T-stress addition results in amode I LE stress field given by⎧⎪⎪⎨⎪⎪⎩

σxx

σyy

σxy

⎫⎪⎪⎬⎪⎪⎭ = KI√

2πrcos(θ/2)

⎧⎪⎪⎨⎪⎪⎩

1 − sin(θ/2) sin(3θ/2)

1 + sin(θ/2) sin(3θ/2)

sin(θ/2) sin(3θ/2)

⎫⎪⎪⎬⎪⎪⎭

+

⎧⎪⎪⎨⎪⎪⎩

T

0

0

⎫⎪⎪⎬⎪⎪⎭ . (11)

However, because the KI + T-stress field cannot re-produce the σ yy

∞ = σ n boundary condition, despite itspopularity it must treated as a simplification for the com-plete LE stress field used above for analysing the Griffith’splate.

Fett showed that the T-stress is the Williams’ series con-stant or zero order term. Dumont and Lopes35 demon-strated that, to obtain KI values using Kelvin’s solutionin the Hybrid Boundary Element Method, the completeWilliams series should be used to generate a suitable stressfunction. This series is given by

= rλ+1

×{

c 1 sin [(λ + 1) (θ + π )] + c 2 cos [(λ + 1) (θ + π )]

+ c 3 sin [(λ − 1) (θ + π )] + c 4 cos [(λ − 1) (θ + π )]

},

(12)

where c1, c2, c3 and c4 are constants and λ is an exponentto be determined from the cracked component boundary



Fig. 3 Mises pz(θ ) for the Griffith’s plate loaded in mode I, estimated from its complete LE stress field calculated from its Westergaardstress function for (a) pl − σ and (b) pl − ε conditions.

Fig. 4 Mises pz(θ ) estimated from the complete Westergaard stress field for (a) pl − σ and (b) pl − ε are visually identical to Inglis estimatesif a sharp elliptical notch with b = CTOD/2 = 2KI

2/π SY E′ instead of ρ = CTOD/2 is used to model the crack.

conditions. Eq. (12) can be compactly rewritten as =rλ+1 F (θ, λ) and, to be a stress function, it must satisfy thefollowing expression

⎧⎪⎪⎨⎪⎪⎩

σrr

σθθ

σrθ

⎫⎪⎪⎬⎪⎪⎭ =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1r2

∂2

∂θ2 + 1r

∂

∂r∂2

∂r2

1r2

∂

∂θ+ 1

r∂2

∂r∂θ

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

⇒

⎧⎪⎪⎨⎪⎪⎩

σrr

σθθ

σrθ

⎫⎪⎪⎬⎪⎪⎭ = rλ−1

⎧⎪⎪⎨⎪⎪⎩

F ′′(θ ) + (λ + 1) F (θ )

λ (λ + 1) F (θ )

−λF ′(θ )

⎫⎪⎪⎬⎪⎪⎭ , (13)

where F ′(θ ) is the derivative of F with respect to θ .With the crack faces in the negative x-axis direction,their free surfaces require σ θθ (±π ) = σ rθ (±π ) = 0, then



F(±π ) = F′(±π ) = 0 if λ �= 0. Consequently, c2 = −c4,c1 = (1 − λ)/(1 + λ) − c3 and, because negative n would beassociated with an infinite strain energy at the crack tip,sin(2πλ) = 0 ⇒ λ = n/2, n = 0, 1, 2, 3, . . . . The resultingLE stress field given by Eq. (13) can be rewritten in termsof σ xx, σ yy and σ xy, as preferred in the LEFM literature,using proper fixx, fiyy and fixy θ-functions for each i-th termof the Williams’ series

⎧⎪⎨⎪⎩

σxx

σyy

σxy

⎫⎪⎬⎪⎭ = KI√

2πrcos(θ/2)

⎧⎪⎪⎨⎪⎪⎩

1 − sin(θ/2)(3θ/2)

1 + sin(θ/2)sin(3θ/2)

sin(θ/2)sin(3θ/2)

⎫⎪⎪⎬⎪⎪⎭

+

⎧⎪⎪⎨⎪⎪⎩

T

0

0

⎫⎪⎪⎬⎪⎪⎭ + κ2r1/2

⎧⎪⎪⎨⎪⎪⎩

f2xx(θ )

f2yy (θ )

f2xy (θ )

⎫⎪⎪⎬⎪⎪⎭ + κ3r3/2

×

⎧⎪⎪⎨⎪⎪⎩

f3xx(θ )

f3yy (θ )

f3xy (θ )

⎫⎪⎪⎬⎪⎪⎭ + · · · (14)

Similar equations can be obtained for pure mode II andfor mixed mode I-Mode II loading conditions but, becausethey are not required to demonstrate the σ n/SY effect,there is no need to reproduce them here. After propertransformations, the first term of the stress field generatedfrom the Williams series given by Eq. (14) reproduces theKI field and the second term (in fact, the constant or zeroorder term, correspondent to λ = 0) gives the T-stresscontribution expressed in Eq. (11). Its higher order terms(corresponding to λ = 2, 3, . . .) can be used to fit completeLE fields of any cracked planar component, calculated orexperimentally obtained by any means. For the Griffith’splate, the Williams series coefficients can be adjusted toits analytical solution, obtained, for example from Eq. (8).For more complex geometries, the higher order Williams’coefficients can be fitted to numerically calculated LEstress fields, generating in this way analytical expressionsfor them. This task may be achieved using for example theleast squares method, after defining the following vectorsand matrices:

(1) [F] matrix, which stores in its rows the n first Williams’Fn(θ ) terms evaluated at the points obtained from thereference stress function (for the Griffith’s plate analysedhere, calculated from the complete stress field generatedby its Westergaard stress function);

(2) {c} vector, containing the coefficients of the n firstWilliams’ terms;

(3) {sn} vector, containing the reference stresses (analyti-cally calculated here, but which may be numerically cal-

culated or else experimentally measured in more complexcases).

According to the least squares method, the vector {c}must be such that

({c}T[F]T − {sn}T)([F]{c} − {sn})= min ⇒ {c} = ([F]T[F])−1[F]T{sn}.

(15)

For low loads, that induce fairly small σ n/SY ratios, sayσ n/SY < 0.1, Figs. 2 and 3 show that just the first Williams’term (the KI value) is enough to reproduce reasonablywell pz(θ ) boundaries estimated from the complete LEstress field of the Griffith plate. For medium loads, sayfor σ n/SY < 0.4, this behaviour is achieved by LE stressfields generated by using two terms in Williams series,namely KI and the T-stress, as is shown in Fig. 5.

However, when the value of σ n/SY is increased, the ad-dition of the T-stress only is no more sufficient to repro-duce the pz(θ ) estimated from the complete LE stressfield of the Griffith’s plate. This fact is illustrated inFig. 6, which plots pz(θ ) estimates for σ n/SY = 0.8. SixWilliams terms plus the T-stress are necessary to repro-duce pzWil(θ ) = pzWtg(θ ) in pl − σ , whereas five Williamsterms plus the T-stress are needed in pl − ε in this case.The coefficients of these terms are given by

σ Wil6txx = 0.56569√

rcos

(θ

2

)[1 − sin

(θ

2

)sin

(3θ

2

)]− 0.421

√r

× cos(

θ

2

)[cos

(θ

2

)2

− 2

]− 0.0768 r3/2

× cos(

θ

2

)[7 cos

(θ

2

)2

− 6

]+ 0.0168 r5/2

× cos(

θ

2

)[36 cos

(θ

2

)4

− 45 cos(

θ

2

)2

+ 10

]

− 0.00264 r7/2 cos(

θ

2

)[−14 + 176 cos

(θ

2

)6

− 308 cos(

θ

2

)4

+ 147 cos(

θ

2

)2]

+ 0.0002539 r9/2 cos(

θ

2

)[1368 cos

(θ

2

)4

− 345 cos(

θ

2

)2

+ 18 + 832 cos(

θ

2

)8

− 1872 cos(

θ

2

)6]

− σn (16)



Fig. 5 Mises pz(θ ) estimated for the Griffith’s plate loaded in mode I from Williams’ series with two terms, which induce a truncated KI +T-stress LE stress field, visually reproduce reasonably well pzWtg(θ ) (estimated from the plate complete LE stress field), both (a) in pl − σ

and (b) in pl − ε, for a medium nominal stress to yield strength ratio σ n/SY ≤ 0.4.

Fig. 6 Mises pz(θ ) estimated in (a) pl − σ and (b) in pl − ε for the Griffith’s plate in mode I from the Williams series needed up to six terms+ T-stress to reproduce pzWtg(θ ) for σ n/SY = 0.8.



σ Wil6tyy = 0.56569√

rcos

(θ

2

)[1 + sin

(θ

2

)sin

(3θ

2

)]+ 0.421

√r

× cos(

θ

2

)3

− 0.0768 r3/2 cos(

θ

2

)3

− 0.0168 r5/2

× cos(

θ

2

)3[

4 cos(

θ

2

)2

− 5

]+ 0.00264 r7/2

× cos(

θ

2

)3[

35 + 48 cos(

θ

2

)4

− 84 cos(

θ

2

)2]

− 0.0002539 r9/2 cos(

θ

2

)3[

504 cos(

θ

2

)2

− 105

+ 320 cos(

θ

2

)6

− 720 cos(

θ

2

)4]

(17)

σ Wil6txy = 0.56569√

rcos

(θ

2

)sin

(θ

2

)sin

(3θ

2

)

− 0.421√

r sin(

θ

2

)cos

(θ

2

)2

+ 0.23 r3/2 sin(

θ

2

)cos

(θ

2

)2

− 0.0843 r5/2

× sin(

θ

2

)cos

(θ

2

)2[

4 cos(

θ

2

)2

− 3

]

+ 0.01848 r7/2 sin(

θ

2

)cos

(θ

2

)2

×[

16 cos(

θ

2

)4

− 20 cos(

θ

2

)2

+ 5

]

− 0.002285 r9/2 sin(

θ

2

)cos

(θ

2

)2

×[

64 cos(

θ

2

)6

− 112 cos(

θ

2

)4

+ 56 cos(

θ

2

)2

− 7

].

(18)

However, for σ n/SY = 0.8 the Mises pz(θ ) estimated forthe Griffith’s plate loaded in mode I from the Williamsseries need at least 6 terms + T-stress to visually repro-duce pzWtg(θ ) in (a) pl − σ and 5 terms + T-stress to doso in (b) pl − ε, see Fig. 6.

Although the complete field generated, for example fromthe Westergaard stress function is the correct LE solu-tion for the Griffith’s plate, it is important to rememberthat the stress distribution inside pz(θ ) cannot be con-sistent with its (assumed) singular LE stress field. In-deed, the LE hypothesis inevitably leads to underesti-mated pz(θ ) boundaries. As a first approximation, thesestresses can be limited to the yield strength SY , neglect-ing strain-hardening effects inside pz(θ ). In this way, fouralternative models for compensating the stress trunca-tion inside the plastic zones by forcing the plate to satisfy

equilibrium requirements are discussed in the followingsection.

L E P L A S T I C Z O N E S E S T I M A T E S M O D I F I E DT O S A T I S F Y E Q U I L I B R I U M C O N D I T I O N S

The first correction to force a LE pz estimate to satisfyequilibrium conditions was proposed a long time ago7:analysing Griffith’s plate, Irwin corrected the plastic zonesize in the direction parallel to the crack plane, shifting theoriginally singular LE σ yy(x) stress component generatedby KI to compensate for its yield-induced truncation in-side pz(x). If strain hardening is neglected, it is well knownthat such an approach doubles the pz(x) size ahead of thecrack tip. Lacking a sound analytical solution for this EPstress analysis problem, some proposals to estimate theequilibrium requirement influence on the entire size andshape of the EP frontier pz(θ ) ahead of the crack tip aredescribed below.

Correction proposed by Rodriguez et al.36

This correction compensates for the σ yy(r, θ ) componenttruncation by assuming that

pz(θ )Wtg+eqσyyM =

∫ pz(θ )WtgM

0σyy (r, θ )dr

σyy

(pz(θ )Wtg

M , θ) . (19)

This equation may be seen as an attempt to generalizeIrwin’s classical idea to compensate for the lack of equilib-rium induced by the yield-induced stress truncation insidethe entire pz(θ ) region. It basically forces the equilibriumof the net vertical forces that cannot exist within the plasticzone because σ yy(r, θ ) is limited by the yield stress insideit, if strain-hardening is neglected.4 Besides generalisingthe equilibrium requirement of vertical forces along anyθ-direction, the most important difference between Eq.(19) and Irwin’s model is that the former is based on thecomplete Westergaard stress function (thus the notationWtg + eqσ yy used to describe it), whereas the latter con-siders a truncated stress field based solely on the SIF.

Correction based on a constant pz(θ) increment ineach θ-direction

The constant pz(θ ) increment along each radius definedby a θ-direction is obtained by

pz(θ )Wtg+eqκ

M = pz(θ )WtgM + κ, (20)

in which κ = pz(θ = 0)Wtg+eqκ

M − pz (θ = 0)WtgM . This κ

constant has the same equilibrium rationale as the previ-ous correction, but it is based on a yield-induced stress



truncation compensation inside pz along the crack direc-tion θ = 0. For other radial directions, the same lengthcorrection is adopted, inspired by the idea of a constantT-stress correction.

Correction based on the equivalent Mises stress

This correction compensates yield-induced stress trunca-tion inside pz(θ ) by forcing

pz(θ )Wtg+eqMM =

∫ pz(θ )WtgM

0σM(r, θ )dr

SY. (21)

Because Eq. (19) only takes into account the effect of theσ yy stress component for θ �= 0, the equilibrium correctionof yield-induced stress truncation inside pz(θ ) by the Misesstress σ M (r, θ ) may be seen as a reasonable alternative,because it is affected by all stress components. Besides, itcan be extended to any type of loading.

Correction using the vertical traction component

This correction uses the vertical traction ty component tocompensate for the yield-induced stress truncation insidepz(θ ):

pz(θ )Wtg+eqTrM =

∫ pz(θ )WtgM

0ty (r, θ )dr

ty(

pz(θ )WtgM , θ

) , (22)

where ty is determined by{tx(r, θ )

ty (r, θ )

}=[

σx(r, θ ) τxy (r, θ )

τxy (r, θ ) σy (r, θ )

]{cos(θ )

sin(θ )

}. (23)

Once again, this correction only produces an exact equi-librium for θ = 0. However, by considering the verticaltraction component, the equilibrium may be seen as re-sulting from a free body diagram obtained by sectioningthe model along any θ-direction.

Figs. 7 and 8 compare the equilibrium correction al-ternatives described earlier, by showing the (slight) dif-ference between their pzM (θ ) estimates for plane stress(Fig. 7) and plane strain (Fig. 8). These figures also depictplastic zones obtained by truncated SIF, SIF plus T-stressand complete LE stress fields, which do not satisfy equi-librium requirements.

Note that the KI +T-stress pzM (θ ) are significantly dif-ferent from the pzM (θ ) corrected for equilibrium. Notealso that all pzM (θ ) estimates made until now were ob-tained from LE analyses. However, no matter how elab-orated, these approaches can have only a qualitativecharacter, because the EP pzM (θ ) estimation problemis intrinsically incremental. Therefore, the next section

shows pzM (θ ) estimates obtained from an incrementalnon-linear analysis using the Finite Element Method,to compare them with the LE estimates developedearlier.

P L A S T I C Z O N E S E S T I M A T E S F R O M E P F E MA N A L Y S I S

All estimates for pz(θ ) boundaries presented so far wereobtained from truncated LE stress fields consideringσ M (θ ) = SY , or at most have estimated equilibrium ef-fects induced by such a truncation. However, the pz(θ ) es-timation problem cannot be modelled well by such globalapproaches. Indeed, each small increment in the stresslevel causes yielding in the vicinity of the crack tip andredistributes the local stresses. Consequently, the result-ing plastic zone grows incrementally while the load isapplied to the structure. Therefore, the present sectionanalyses the Griffith’s plate properly considering this EPprocess to estimate pzEP

M (θ ) by incremental EP FE anal-yses. These analyses neglect strain-hardening effects, tobe consistent with the previous LE estimates. Their pa-rameters, for example the maximum number of equilib-rium iterations (eqint = 2000) and the load increments(incload = 0.1 of full load), were adjusted according toANSYS37 documentation. Fig. 9 compares the calculatedpzEP

M (θ ) with the equilibrium-corrected LE pzM (θ ) esti-mates that use σ yy component and σ Mises, and with thepzM (θ ) obtained from truncated SIF and SIF plus T-stressfields for pl − σ . Fig. 10 makes a similar comparison forpl − ε. Four values of σ n/SY are assumed: 0.4, 0.6, 0.7 and0.8.

Note how the shapes of LE plastic zone estimates arequite different from those calculated incrementally by aproper EP analyses. This is an indication that not evencomplete LE fields can generate satisfactory pz(θ ) esti-mates for some design purposes under high σ n/SY ratios.If precise plastic zone evaluations are needed in such cases,it may be indispensable to calculate them by proper EPnumerical models. However, as such models are complexand expensive, simplified LE estimates will probably re-main unavoidable for the foreseeable future. Moreover,because EP models are not foolproof, experimental ver-ification may be needed to guarantee their predictions,in particular when strain-hardening effects cannot be ne-glected.

LE estimates based on truncated stress fields gener-ated by the KI + T approximation lead to reasonablevalues for the maxima pzM (θ ) sizes in pl − σ in thisGriffith plate, but severely over-estimated the pzM (θ )boundaries calculated by the EP model in pl − ε forhigh σ n/SY loads. Equilibrium-corrected LE estimatesbased on the complete LE stress field are less bad in



Fig. 7 Equilibrium-corrected pzM (θ ) and pzMKI+T(θ ) estimated from the Griffith’s plate loaded in mode I in pl − σ , for (a) σ n/SY = 0.2 and

(b) σ n/SY = 0.8.

Fig. 8 Equilibrium-corrected pzM (θ ) and pzMKI+T(θ ) estimated from the Griffith’s plate loaded in mode I in pl − ε, for (a) σ n/SY = 0.2 and

(b) σ n/SY = 0.8.

this case. However, such behaviour cannot be extrapo-lated for other cases. Anyway, it can be certainly statedthat pz(θ ) estimates based solely on KI should not be as-sumed safe for modelling purposes, let alone for designapplications.

C O N C L U S I O N S

To evaluate the accuracy of plastic zones size and shapeestimates frequently used for mechanical design and struc-tural integrity evaluations of cracked components, es-timates for pz(θ ) ahead of Griffith’s plate crack tips



Fig. 9 Comparison of pzEPM (θ ) calculated by an EP incremental FE analyses for the Griffith’s plate loaded in mode I in pl − σ with pzM (θ )

estimated from complete LE stress fields corrected for equilibrium using the σMises component (which gives estimates similar to equilibriumcorrections based on σ yy and on ty) and with truncated LE estimates based only on the T-stress, pzKI +T

M (θ ).

obtained from five LE stress fields are compared to thatdetermined for its EP stress field calculated by FE. Thefirst LE field is based solely on its SIF. The second is ob-tained from its SIF plus its T-stress, which is its Williams’series constant or zero order term. The third is basedon the Griffith’s plate complete LE stress field generatedby its Westergaard stress function, the correct analyticalsolution for this stress analysis problem. The fourth isbased on the complete solution for the LE stress field ofan elliptical sharp notch in the equivalent Inglis’ plate,that is, the notch that simulates the opened Griffith crackby an elliptical hole with major axis equal to the cracklength and minor semi-axis equal to b = 2KI

2/πSY E′. Thelast LE stress field, based on the complete Williams se-

ries expansion for the Griffith’s plate, is used to eval-uate the number of its terms needed to fit well pz(θ )estimates generated by the complete Westergaard stressfield.

The accuracy of classical pz(θ ) estimates based solely onSIF are limited to very low σ n/SY ratios, and estimatesbased on SIF plus T-stress, frequently assumed to be anappropriate solution for this problem, do not solve it ei-ther. These approximate LE stress fields do not satisfyboundary conditions, in particular the nominal stressesfar from the crack tip. On the other hand, the completestress field generated from the Westergaard stress func-tion satisfies those conditions, because it is the correctLE solution for the Griffith’s plate. Its estimates for pz(θ )



Fig. 10 Comparison of pzEPM (θ ) calculated by EP incremental FE analyses for the Griffith’s plate loaded in mode I in pl − ε with pzM (θ )

estimated from complete LE stress fields corrected for equilibrium using the σMises component (which gives estimates similar to equilibriumcorrections based on σ yy and on ty) and with truncated LE estimates based only on the T-stress, pzKI +T

M (θ ).

can be reproduced by solving the equivalent Inglis’ plate.The Williams series can also reproduce such estimates,if a proper number of terms is used, which depends onthe σ n/SY ratio. This approach demonstrates that pz(θ )estimates based on SIF plus T-stress do not reproducethe estimates based on LE complete stress field for highσ n/SY values.

Nevertheless, none of the pz estimates based on LE stressfields satisfy equilibrium conditions, because their singu-lar (or very high, in the Inglis case) predicted stresses at thecrack tip must be limited inside the plastic zone. To copewith this problem, four alternative models for compen-sating the stress truncation neglecting strain-hardening

effects inside the plastic zone are considered, to force theplate to satisfy equilibrium conditions. Such equilibrium-corrected LE estimates have been compared with pz(θ )incrementally calculated using an EP FE code, also ne-glecting strain hardening to maintain compatibility withthe LE estimates. Whereas the shape of equilibrium-corrected pz(θ ) LE estimates are quite different fromthe EP ones, their maximum size for high σ n/SY lev-els is reasonably well reproduced both in pl − σ and pl− ε, contrary to what happens with estimates based ontruncated LE stress fields generated from SIF plus T-stress, which severely overestimate the plastic zones in pl− ε in this Griffith’s plate case.



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