Constraint effects on crack-tip ﬁelds in elastic ... · Constraint effects on crack-tip ﬁelds in elastic-perfectly plastic materials X.K. Zhu, Yuh J. Chao * Department of Mechanical

Journal of the Mechanics and Physics of Solids49 (2001) 363–399

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Constraint effects on crack-tip fields in elastic-perfectly plastic materials

X.K. Zhu, Yuh J. Chao*

Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA

Received 8 September 1999; received in revised form 28 March 2000

Abstract

Asymptotic crack-tip stress fields accounting for constraint effects are developed for astationary plane strain crack under mode-I, mode-II or mixed-mode I/II loading. The mixed-mode loading is considered only within small-scale yielding. Materials are taken into accountin incompressible, elastic-perfectly plastic materials, and plastic deformation of materials obeysvon Mises yield criterion. This investigation is an extension of the solution obtained by Liand Hancock [Li, J., Hancock, J.W., 1999. Mode I and mixed mode fields with incompletecrack tip plasticity. International Journal of Solids and Structures 36 (5), 711–725] with specialattention on what constraint parameters existed in the elastic-plastic crack-tip fields.

Results indicate that the asymptotic crack-tip field is a 4-sector solution for mode-I cracksand a 6-sector solution for mixed-mode cracks, and is comprised of plastic sectors and elasticsector(s), and contain two undetermined parametersTp and Tπ which are hydrostatic stressesahead of the crack tip and on the crack flank, respectively. WhenTp andTπ vanish, the presentelastic-plastic crack-tip field reduces to the fully plastic Prandtl slip-line field. Comparisonshows that the asymptotic crack-tip stress fields can precisely match with elastic-plastic finiteelement results over all angles around a crack tip for various fracture specimens with constraintlevels from high to low. The magnitudes ofTp andTπ determine the level of crack-tip constraintin plastic sectors and in elastic sector, respectively, due to geometric and loading configurationsor mode mixity. Thus the parametersTp andTπ can be used as constraint parameters to effec-tively characterize the entire crack-tip field in elastic-perfectly plastic materials under the planestrain conditions. 2001 Published by Elsevier Science Ltd.

Keywords:Crack-tip field; C. Asymptotic analysis; Constraint effect; Plane strain; Elastic-perfectly plas-tic material

* Corresponding author. Fax:+1-803-777-0106.E-mail address:[email protected] (Y.J. Chao).

0022-5096/01/$ - see front matter 2001 Published by Elsevier Science Ltd.PII: S0022 -5096(00 )00030-2

364 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) 363–399

1. Introduction

Accurate characterization of asymptotic crack-tip fields is imperative in the con-straint analysis and choice of fracture parameters. There are many published workson elastic-plastic deformation of near-tip field and constraint effect at a crack tip forpower-law hardening materials. Since Hutchinson (1968a), Rice and Rosengren(1968) developed the HRR dominant singularity field for plane strain mode-I cracksbased on a single parameter ‘J-integral’ proposed by Rice (1968), theJ-integral-based single parameter fracture criterion is gradually applied to fracture assessmentof engineering structures. In practice, however, it is found that specimen geometryand loading configurations play a significant role in crack-tip fields. As such, theJ-integral fracture criterion and the HRR field only have limited application to realflawed structures. Recently, more attention has been paid to constraint analysis andmulti-parameter crack-tip fields. Typical representatives are the ‘two-parameter’crack-tip fields governed, respectively, byJ–T of Betegon and Hancock (1991),J–Q of O’Dowd and Shih (1991) andJ–A2 of Chao et al. (1994) through extension ofthe HRR field using higher-order terms for mode-I cracks. These parameters areidentified to be able to quantify the constraint effects on crack-tip fields for differentcracked specimens. Further comments can be found in Chao and Zhu (1998). Con-straint effects on mixed mode crack-tip fields in power-law hardening materials arepresented in Du et al. (1991) and Roy and Narasimhan (1997).

At complete yielding, analyses based on perfect plasticity could provide meaning-ful insights and reference values for low-hardening materials, and possibly for moder-ate-hardening materials. In fact, the earliest study of specimen geometry effect oncrack-tip fields is for perfectly plastic materials using the slip-line theory. Based onthe assumption that the plastic sectors entirely surround the crack tip, Prandtl (1920)first developed the well-known Prandtl slip-line field for a semi-infinite plane strainmode-I crack. Green and Hundy (1956), Green (1956), Ewing and Hill (1967) andEwing (1968) reported the slip-line fields for different crack depths of Single EdgedNotched plate under pure Bending (SENB) and Double Edged Crack Plate in tension(DECP). It is found that the Prandtl field can exist only in deeply cracked DECPspecimens. Hutchinson (1968b) presented the slip-line field for plane strain mode-II cracks and observed that the HHR field approaches the Prandtl field in the limitof non-hardening. McClintock (1971) introduced the slip-line field of Center CrackPlate in tension (CCP), and summarized the results of slip-line fields for differentnotch or crack specimens. He found that the Prandtl field and CCP slip-line fieldare two extreme cases, all other slip-line fields of cracked specimens are in betweenthe two extremes. As such, the stress and velocity slip-line fields around a crack tipare generally different for cracked specimens with different geometry or loadingconfigurations. Wu et al. (1990) reviewed the slip-line field solutions for severalconventional fracture specimens including both deep and shallow cracks. They con-cluded that the fully plastic crack-tip fields are similar to the Prandtl field for speci-mens with high triaxiality such as deeply cracked SENB and DECP, and Compact

365X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) 363–399

Tension (CT) specimens, but considerably different for specimens with low triaxial-ity such as CCP specimens.

Detailed investigation of constraint effects on near-tip fields in non-hardeningmaterials was presented first by Du and Hancock (1991). Based on the modifiedboundary layer formulation, they numerically examined the effects of elasticT-stresson the near-tip field of a plane strain mode-I crack under contained yielding con-ditions. They found that a positiveT-stress can cause plasticity to envelop the cracktip and yields the Prandtl field, while a negativeT-stress reduces the triaxiality ofthe stress state at the crack tip and forms an incomplete Prandtl field. Under large-scale yielding, Lee and Parks (1993) carried out the fully plastic analyses of singleedge cracked specimens subject to different combined tension and bending for asufficiently deep crack. Kim et al. (1996) and Zhu and Chao (2000) performeddetailed finite element analyses (FEA) for SENB, CCP and DECP specimens,respectively, to study the effects of crack depth and loading level on crack-tip con-straint for elastic-perfectly plastic materials under plane strain conditions. Bothresults indicate that only for deeply cracked SENB and DECP specimens, the valuesof crack-tip constraint remain ‘almost’ constant for all range of deformation levels,and the crack-tip fields are very close to the Prandtl field. Under other cases, withthe decrease of constraint levels, the hydrostatic stress ahead of the crack tipdecreased from the Prandtl field and an elastic sector occurred on the crack flanks.Zhu and Chao (1999) showed that the two-parameter asymptotic field such as theJ–A2 three-term solution can capture the essential features of elastic-plastic fieldsin the plastic sector ahead of the crack tip, but not in the elastic sector near thecrack flanks.

To explore the general structure of crack-tip fields, Ibragimov and Tarasyuk (1976)and Nemat-Nasser and Obata (1984) first discussed the possibility of near-tip fieldscontaining elastic sectors for plane strain mode-I stationary cracks in elastic-perfectlyplastic materials. More recently, Li and Hancock (1999) assembled crack-tip fieldswith slip-line plastic sectors and elastic sectors under contained yielding to matchwith the FEA results with the incomplete plasticity obtained by Du and Hancock(1991). The incomplete plasticity leads to a loss of crack-tip constraint and theappearance of an elastic sector on the crack flank.

For mixed-mode-I/II loading, using the theoretical methods of Hutchinson(1968a,b), Shih (1974) constructed an asymptotic crack-tip field under plane strainand small-scale yielding conditions based on the assumption that plasticity entirelysurrounds the crack tip at all angles. With the exception of near mode-II fields, theserequire a discontinuity of radial stress component in a sector trailing the crack front.In contrast, by use of FEA under the small-scale yielding, Dong and Pan (1990) andHancock et al. (1997) calculated mixed-mode near-tip fields. These near-tip fieldsdiffer from those constructed by Shih (1974) in that plasticity does not entirely sur-round the crack tip and stress components contain no discontinuities. And with theexception of fields close to mode-II cracks, an elastic sector appears on one crackflank. Dong and Pan (1990), Li and Hancock (1999) and Sham et al. (1999)assembled mixed mode crack-tip fields with slip-line plastic sectors and elastic sec-tors directly using the numerical border angles obtained in the FEA.


The deviation from the HRR fields at a crack tip in power-law hardening materialsdue to specimen geometry and loading configuration is often attributed to the ‘lossof constraint’. Similarly, the deviation from the Prandtl fields at a crack tip in elastic-perfectly plastic materials can also be attributed to the effect of constraint. A keyissue in studying the constraint effect is to identify the proper parameter(s) that canquantify the constraint levels.

The present paper examines in detail the elastic-plastic crack-tip fields undermode-I, -II or mixed-mode-I/II loading. Our investigation is an extension of thesolution by Li and Hancock (1999) for incompressible, elastic-perfectly plasticmaterials under the plane strain conditions. Special attention is focused on whatconstraint parameters existed in the elastic-plastic crack-tip fields. Closed-formasymptotic solutions of crack-tip fields are developed. Results indicate that asymp-totic crack-tip fields are comprised of plastic sectors and elastic sector(s), containingtwo undetermined parametersTp, Tπ and no discontinuities of stresses. To verifythese asymptotic crack-tip fields, comparisons with numerical results are performedfor various fracture specimens with constraint levels from high to low. The level ofcrack-tip constraint quantified by the parametersTp and Tπ is addressed.

2. Governing equations and asymptotic analysis

Consider a stationary crack in an elastic-perfectly plastic material under planestrain conditions. Applied loading is accounted as mode-I, mode-II or mixed-mode-I/II loading, respectively. Since both FEA of Dong and Pan (1990) and Du andHancock (1991) showed that the effect of compressibility on the crack-tip stressdistributions is very small for elastic-perfectly plastic materials, the material con-sidered here is assumed to be incompressible, and deform plastically according tovon Mises yield criterion and the associated flow rule.

2.1. Equilibrium equations

Both Cartesian coordinates (x1, x2) and polar coordinates (r, q) (q=0 correspondingto the positivex1-direction) are introduced and centered at the crack tip. With refer-ence to the polar coordinates, the equilibrium equations can be written as

∂srr

∂r1

1r∂srq

∂q 1srr−sqq

r50 (1)

1r∂sqq∂q

1∂srq

∂r1

2srq

r50

For elastic-perfectly plastic materials, the numerical calculations of Dong and Pan(1990) indicated that the stresses near a crack tip are nonsingular within both theplastic and elastic sectors around the crack tip. Accordingly, all stress componentsnear the crack tip are only functions of the polar angleq, but not the distancer from


the crack tip (i.e.sij=sij(q), as r→0). Thus, the partial differential equations in (1)reduce to ordinary differential equations

dsrq

dq1srr2sqq50 (2)

dsqqdq

12srq50

2.2. Plane strain conditions

For the incompressible, elastic-perfectly plastic material, the elastic-plastic consti-tutive relations of materials are the Prandtl-Reuss equations

eij 51

2mSsij 213skkdijD1lsij (3)

where eij and sij are strain rate and stress rate components, respectively.m is theshear modulus,dij is the Kronecker delta,sij=sij2skk/3 are the deviatoric stresscomponents andl is the plastic flow factor.l;0 in elastic regions. Latin indicesi,j, k have a range of 1–3.

The plane strain conditions requiree33=0. From (3), we have

s33212(s111s22)5H0, in elastic region

C0e−2ml, in plastic region(4)

From the continuity of stress components at the border between an elastic regionand a plastic region, the integration constantC0 is equal to zero. For the incompress-ible, elastic-perfectly plastic material, therefore, the plane strain conditions are equiv-alent to

s33512(s111s22) or s335

12(srr1sqq) (5)

both in elastic regions and in plastic regions.

2.3. Yield criterion

Under the plane strain condition (5), the von Mises yield criterion can beexpressed as

14(srr2sqq)21s2

rq5k2 (6)

wherek is the yield strength in shear andk=s0/√3, s0 is the tensile yield stress.The yield criterion (6) is satisfied automatically if the in-plane stress components


can be expressed by using a stress functiony(q) (cf Zhu et al., 1997) in the follow-ing form

srr(q)=sm(q)−k cosy(q)

sqq(q)=sm(q)+k cosy(q)

srq(q)=k siny(q)

(7)

wheresm=(srr+sqq)/2 is a mean stress or hydrostatic stress. Comparing with (5), itholds thatsm=s33.

2.4. Asymptotic solution in plastic sector

Without loss of generality, it is assumed that a near-tip field is comprised of plasticsectors and elastic sectors around a crack tip. Substituting (7) into the equilibriumEq. (2), one obtains the governing field equations in plastic sectors as follows

cosy(q)Sdy(q)dq

−2D=0

dsm(q)dq

=k siny(q)Sdy(q)dq

−2D (8)

Solving the ordinary differential Eq. (8), we obtain two different sets of solutionswhich correspond to two different sets of plastic sectors at a crack tip.

(i) Plastic constant stress sector (sij=constant)

y(q)=2q+y0

sm(q)=C1

(9)

(ii) Plastic non-constant stress or fan sector (srr=sqq)

y(q)=p2+np

sm(q)=−2kq cos(np)+C2

(10)

The constantsy0, C1, C2 and the integern in (9) and (10) can be determined byboundary and continuity conditions, and then the stress components in plastic sectorsare given by (7).

2.5. Asymptotic solution in elastic sector

In an elastic sector, in addition to satisfying the equilibrium equations, the defor-mation field has to be compatible. Under the incompressible and plane strain con-ditions, the compatibility equation in terms of stress components gives


S ∂2

∂r211r

∂∂r

11r2

∂2

∂q2D(srr1sqq)50 (11a)

For the bounded stresses at a crack tip, Eq. (11a) becomes an ordinary differen-tial equation

d2

dq2(srr1sqq)50 (11b)

Solving (2) and (11b), one can obtain an asymptotic stress field in the elasticsector as follows

srr(q)=−A cos 2q−B sin 2q+2Cq+D

sqq(q)=A cos 2q+B sin 2q+2Cq+D

srq(q)=A sin 2q−B cos 2q−C

(12)

where the integration constantsA, B, C andD can be determined from the boundaryand continuity conditions.

2.6. Boundary conditions

The equilibrium considerations alone require that tractionσqq andsrq along theborder between any two sectors (either two plastic sectors or a plastic and an elasticsector) must be continuous, but radial stresssrr is not. To seek a fully continuoussolution, it is assumed thatsrr is also continuous along the border. And thus completecontinuity of all stress components is required, or it is expressed mathematically as

sab(q−i )5sab(q+

i ) (13)

where q−i and q+

i represent the angle just before and after the border-delimitationangleqi, respectively. The traction-free conditions on the crack flanks are

sqq(6p)50; srq(6p)50 (14)

3. Crack-tip stress fields under mode-I loading

For a mode-I opening crack, only the upper plane (0#q#p) needs to be con-sidered. The traction-free conditions on the crack flank and the deformation sym-metric condition require

srq(p)50, sqq(p)50, srq(0)50 (15)

Using boundary condition (15) and continuity condition (13), a fully plastic or elas-tic-plastic crack-tip stress field can be constructed.


3.1. Fully plastic crack-tip stress field

This section assumes that plasticity entirely surrounds the crack tip at all angles.In terms of the stress functiony(q) in (7), the boundary condition (15) becomes

siny(0)50, siny(p)50, sm(p)52k cosy(p) (16)

Solving the ordinary differential Eq. (8) with the boundary condition (16), two gen-eral solutions of this boundary-value problem are obtained. One of the general sol-utions is

y(q)552q+np, 0#q#

p4

p2+np,

p4

#q#3p4

2q−p+np,3p4

#q#p

(17a)

and

sm(q)55(1+p)k cos(np), 0#q#

p4

(1+3p2

−2q)k cos(np),p4

#q#3p4

k cos(np),3p4

#q#p

(17b)

This solution shows that the fully plastic stress field is comprised of three plasticsectors: a constant stress sector ahead of the crack tip, followed by a centered-fansector, and terminating with another constant stress sector adjacent to the crack sur-face.

The other general solution is

y(q)52q1np, sm(q)52k cos(np) (18)

The above equations hold for all angles in 0#q#p. This solution shows that the fullplastic stress field consists of a single uniform field around the entire crack tip.

Notice that both in (17) and (18), the parametern is an arbitrary integer, i.e.n=0, ±1, ±2,... Due to the periodic characteristics of the trigonometric function, oneobtains only four sets of plastic solutions for a plane strain mode-I crack when (17)and (18) is substituted into (7), respectively.

3.1.1. First set of plastic solutionsAs n is taken as the even integers in (17), e.g.n=0, substitution of (17) into (7)

obtains the first set of fully plastic crack-tip stress field that is precisely the Prandtlfield under far-field tensile loading perpendicular to the crack line as follows


O Sector 1 (0#q#p/4)

srr=k(1+p−cos 2q)

sqq=k(1+p+cos 2q)

srq=k sin 2q

(19a)

O Sector 2 (π/4#q#3p/4)

srr=sqq=k(1+3p/2−2q)

srq=k(19b)

O Sector 3 (3π/4#θ#π)

srr=k(1+cos 2q)

sqq=k(1−cos 2q)

srq=−k sin 2q

(19c)

3.1.2. Second set of plastic solutionsAs n is taken as the odd integers in (17), e.g.n=21, substitution of (17) into (7)

obtains the stress fields that are (19) with a negative sign added, i.e. the negativePrandtl field. Thus, this second set of plastic solutions corresponds to a mode-I crackunder far-field compressive loading perpendicular to the crack line.

3.1.3. Third set of plastic solutionsAs n is taken as the even integers in (18), e.g.n=0, substitution of (18) into (7) has

srr=−k(1+cos 2q)

sqq=−k(1−cos 2q),

srq=k sin 2q

0#q#p (20)

Under the rectangular coordinates, (20) becomessxx(q)=22k, syy(q)=0, sxy(q)=0.Thus the uniform stress field in (20) appears to represent a crack under a far-fieldcompressive load,22k, parallel to the crack line.

3.1.4. Fourth set of plastic solutionsAs n is taken as the odd integers in (18), e.g.n=1, substitution of (18) into (7)

obtains the stress field that is (20) with a negative sign. Thus the fourth set of fullyplastic solutions corresponds to a crack subjected to a far-field tensile load, 2k, paral-lel to the crack line.

In conclusion, except for the positive or negative sign of the stress fields, there


Fig. 1. Angular distributions of fully plastic crack-tip stress fields for mode-I cracks (a) the Prandtlfield; (b) the uniform stress field subject to tension of far-field parallel to crack line.

are only two independent fully plastic crack-tip fields as shown in Fig. 1 (i.e. thePrandtl field (19) and the uniform stress field (20)). It is noted that both stress fieldsin Fig. 1 meet the conditionsqq(0)$srr(0) for mode-I tensile cracks. The followingsections will show that these two fully plastic solutions are the two extremes ofgeneral elastic-plastic fields. In other words, the elastic-plastic crack-tip fields for acommon specimen fall in between the two extreme cases shown in Fig. 1.

3.2. Elastic-plastic crack-tip stress field

Without loss of generality, it is assumed that the crack-tip field is comprised ofplastic and elastic sectors. Based on the FEA results available for the materials con-sidered, an elastic-plastic crack-tip stress field can be constructed by four sectors orthree sectors as shown in Fig. 2. It is referred to as a 4-sector solution or 3-sectorsolution in the present work. In the 4-sector solution, the crack-tip field consists of

Fig. 2. Structures of elastic-plastic crack-tip field for mode-I cracks (a) 4-sector solution; (b) 3-sector sol-ution.


three plastic sectors and an elastic sector: a plastic constant sector ahead of the cracktip over 0#q#q1, followed by a plastic fan sector inq1#q#q2 and another plasticconstant stress sector inq2#q#q3, terminating with an elastic sector adjacent to thecrack flank inq3#q#p. Here q1,q2,q3 are the border angles delimitating sectors.Using the continuity condition (13) and boundary condition (15), from (9), (10) and(12), one can obtain the stress function in plastic sectors

y(q)552q, 0#q#q1=

p4

p2, q1#q#q2

p2+2(q−q2), q2#q#q3

(21a)

and the hydrostatic stress around all angles

sm(q)55k(1+p)+Tp, 0#q#q1=

p4

kS1+3p2

−2qD+Tp, q1#q#q2

kS1+3p2

−2q2D+Tp, q2#q#q3

−kcos(q3−2q2)

sinq3

−2k(p−q)cos 2q2

1−cos 2q3

, q3#q#p

(21b)

whereTp is an undetermined constant related to the border anglesq2 andq3, and isgiven in (26). And thus, the stress components of the 4-sector solution are obtainedas follows

O Plastic constant stress sector 1 (0#q#p/4)

srr=k(1+p−cos 2q)+Tp

sqq=k(1+p+cos 2q)+Tp

srq=k sin 2q

(22a)

O Plastic fan sector 2 (p/4#q#q2)

srr=sqq=kS1+3p2

−2qD+Tp

srq=k

(22b)

O Plastic constant stress sector 3 (q2#q#q3)


srr=kS1+3p2

−2q2D+k sin 2(q−q2)+Tp

sqq=kS1+3p2

−2q2D−k sin 2(q−q2)+Tp

srq=k cos 2(q−q2)

(22c)

O Elastic sector 4 (q3#q#p)

srr=−kcos(q3−2q2)

sinq3

[1+cos 2q]−kcos 2q2

1−cos 2q3

[2(p−q)−sin 2q]

sqq=−kcos(q3−2q2)

sinq3[1−cos 2q]−k

cos 2q2

1−cos 2q3[2(p−q)+sin 2q]

srq=kcos(q3−2q2)

sinq3sin 2q−k

cos 2q2

1−cos 2q3[1−cos 2q]

(22d)

Comparing (22a) and (22b) with (19a) and (19b), it is found that the stress fieldof the 4-sector solution in the plastic sectors 1 and 2 is related to the Prandtl field by

sij 5sPrandtlij 1Tpdij (23)

wheresPrandtlij denotes the components of stresses in the Prandtl field. It is obvious

that the parameterTp in the solution of (22) is a hydrostatic stress and can be determ-ined atq=0 by

Tp5sappqq (0)2sPrandtl

qq (0) (24)

Similarly, we can define another hydrostatic stress parameterTp on the crack sur-face as

2Tp5sapprr (p)2sPrandtl

rr (p) (25)

wheresappqq (0) andsapp

rr (p) stand for the known stress components atq=0 andq=p,respectively, from the slip-line field or a FEA.

Using the two parametersTp and Tp, the delimitation anglesq2 andq3 in the 4-sector solution (22) can be determined by solving the following equation

Tpk

=−cos(q3−2q2)

sinq3−1

Tp

k=S2q2−

3p2

−1D−(p−q3)cos 2q2

sin2 q3−cos(q3−2q2)

sinq3

(26)


3.3. Discussions about the 4-sector solution

In terms of the yield condition (6), the radial stresssapprr (p) on the crack surface

has the range of |sapprr (p)|#2k. Accordingly, from (25) and (26), there exist several

special cases in the 4-sector solution as below:(a) Whensapp

rr (p)=2k, we have

Tp50, Tp50; q253p/4, q35p

Under this case, the elastic sector 4 disappears and the 4-sector solution (22) reducesto the fully plastic Prandtl field (19).

(b) Whensapprr (p)=22k, we have

Tp522k, Tp52(21p)k, q35q25q15p/4

Under this case, the plastic sectors 2 and 3 vanish and the elastic sector 4 becomesa plastic sector under constant pressure22k. And the 4-sector solution (22) reducesto the fully plastic uniform stress field (20).

(c) When 22k,sapprr (p),2k, an elastic sector 4 appears near the crack surface.

The angular spanf(=p2q3) of the elastic sector 4 increases with the decrease ofparametersTp andTp, thus stress componentssapp

rr (p) andsappqq (0). The Prandtl field

(19) and the plastic uniform stress field (20) are the two extremes of the generalelastic-plastic stress field (22). Ranges of all unknown parameters in (26) are

2(21p)k#TP#0, 22k#Tp#0; p/4#q2#3p/4, p/4#q3#p

With these ranges, from (21b), angular variation of the hydrostatic stresssm withdifferent values of the parametersTp and Tp can be illustrated in Fig. 3. It is seen

Fig. 3. Sketch of angular variation of the hydrostatic stresssm with different values of the parametersTp andTp for mode-I cracks.


that parametersTp and Tp can characterize constraint levels in the plastic sectorsahead of crack tip and in the elastic sector on the crack surface, respectively. Thetwo parameters are generally independent to each other and so both can be used asconstraint parameters.(d) When q3=q2, the 4-sector solution reduces to a 3-sector solution. From (26),we have

Tp=k(cotf−1)

Tp=kFp2−1+cotf−f

sin2 fG (27)

wheref=p2q2 is the angular span of the elastic sector 3 in 3-sector solution. Elimin-ate the unknownf from (27), the two constraint parametersTp andTp are related by

Tp5kFp21Tpk

2S11(Tp+k)2

k2 DarctanS kTp+kDG (28)

Notice that if the trigonometric function arctan[k/(Tp+k)] is less than zero, it mustbe substituted byp2arctan[k/(Tp+k)]. Under the condition (28), only one of the twoconstraint parametersTp andTp is independent, the 4-sector solution degenerates to3-sector solution as shown in Fig. 2(b). Eq. (28) is thus the existence condition ofthe 3-sector solution.

From (22), the stress field of the 3-sector solution is simplified as

O Plastic constant stress sector 1 (0#q#p/4) Stress components are the same asthose in (22a)

O Plastic fan sector 2 (p/4#q#q2) Stress components are the same as those in (22b)O Elastic sector 3 (q2#q#p)

srr=−k cotq2[1+cos 2q]+k(1−cot2q2)F(p−q)−12sin 2qG

sqq=−k cotq2[1−cos 2q]+k(1−cot2q2)F(p−q)+12sin 2qG

srq=k cotq2 sin 2q+k(1−cot2q2)sin2 q

(29)

whereq2=p2f is determined by (27), andp/4#q2#3p/4. It should be noted that the3-sector solution is invalid for the elastic angular span 0#f,p/4 in which the yieldcriterion is violated in the postulated elastic sector. Comparison shows that our 3-sector solution is the same as the solution developed by Li and Hancock (1999).

Fig. 4 shows that the angular spanf, thus the size of the elastic sector, indeedincreases with the decreasing of constraint parametersTp and Tp in the 3-sectorsolution. Whenf=p/4, Tp=Tp=0 andq2=3p/4, stress components of (29) in the elastic


Fig. 4. Variation of constraint parametersTp and Tp in the 3-sector solution with the elastic angularspanf=p2q2 for mode-I cracks.

sector reduce to (19c) in the plastic sector, and so the 3-sector solution reduces tothe fully plastic Prandtl field (19). Whenf=3p/4, Tp=2(2+p)k, Tp=22k andq2=p/4,(29) reduces to (20) in the plastic sector, and so the 3-sector solution reduces to thefully plastic uniform field (20).

4. Crack-tip stress fields under mode-II loading

For mode-II crack problems under the plane strain conditions, the equilibriumequations and yield criterion are the same as (2) and (6), respectively. If the stresscomponents are still represented by (7), then the governing equation of the plasticstress field remains the same as (8). The difference between mode-II and mode-Icracks is only the boundary condition. For a mode-II shear crack, from the skew-symmetric deformation along the remaining ligament and traction free on the crackface, the boundary conditions are

srr(0)50, sqq(0)50; srq(p)50, sqq(p)50 (30)

4.1. Fully plastic crack-tip stress field

Again, it is assumed that plasticity entirely surrounds the crack tip. Using thestress functiony and the hydrostatic stresssm, the boundary conditions (30) can berewritten as


cosy(0)50, siny(p)50; sm(0)50, sm(p)52k cosy(p) (31)

Solving the ordinary differential field Eq. (8) with the boundary conditions (31),the general solution of this boundary-value problem is obtained as

y(q)55p2+np, 0#q#q1

2(q−q1)+p2+np, q1#q#q1+

p2

3p2

+np, q1+p2

#q#3p4

2q+np,3p4

#q#p

(32a)

and

sm(q)55−2kq cos(np), 0#q#q1

−2kq1 cos(np), q1#q#q1+p2

2kq−4kSq1+p4Dcos(np), q1+

p2

#q#3p4

−4kSq1−p8Dcos(np),

3p4

#q#p

(32b)

where the parametern is an arbitrary integer, i.e.n=0, ±1, ±2,....q1 is the delimitation

angle between the first sector and the second sector andq1=14+p8. This solution shows

that the fully plastic crack-tip stress field is comprised of four plastic sectors: a fansector ahead of the crack tip, followed by a constant sector and another fan sector,and terminating with another constant sector adjacent to the crack surface.

Due to the periodic characteristics of the trigonometric functions, substituting (32)into (7), one obtains only two sets of plastic solutions for a plane strain mode-II crack.

4.1.1. First set of plastic solutionsAs n is taken as the even integers in (32), e.g.n=0, substitution of (32) into (7)

yields fully plastic crack-tip stress fields as follows

O Sector 1 (0#q#q1)

srr=sqq=−2kq

srq=k(33a)


O Sector 2 (01#q#q1+p/2)

srr=−2kq1+k sin 2(q−q1)

sqq=−2kq1−k sin 2(q−q1)

srq=k cos 2(q−q1)

(33b)

O Sector 3 (q1+p/2#q#3p/4)

srr=sqq=2kq−4kSq1+p4D

srq=k

(33c)

O Sector 4 (3p/4#q#p)

srr=−k−k cos 2q

sqq=−k+k cos 2q

srq=k sin 2q

(33d)

The angular variations of stress components in (33) are shown as in Fig. 5(a).Through simple comparison, one can find that the present plastic solution (33) isexactly the slip-line field of plane strain mode-II crack given by Hutchinson (1968b).Accordingly, this set of plastic solutions corresponds to a plane strain mode-II crackunder far-field clockwise shear forces.

4.1.2. Second set of plastic solutionsAs n is taken as the odd integers in (32), e.g.n=1, substitution of (32) into (7)

obtains the stress field (33) with a negative sign added or the negative stress fieldof (33) which corresponds to plane strain mode-II crack under counter-clockwiseapplied shear forces. Therefore, the crack-tip stress field (33) for plane strain mode-II crack is indeed unique.


It is possible that an elastic sector occurs on one crack surface in the mode-IIcrack-tip stress field under action of the far-field ‘T-stress’. As such, without loss ofgenerality, this section assumes that the crack-tip field is fully plastic on the lowerplane, but on the upper plane it is comprised of three plastic sectors and an elasticsector: a plastic fan sector ahead of the crack tip over 0#q#q1, followed by a plasticconstant sector inq1#q#q2 and another plastic fan stress sector inq2#q#q3, termin-ating with an elastic sector adjacent to the crack surface inq3#q#p. Hereq1,q2,q3


Fig. 5. Angular distributions of crack-tip stress fields for mode-II cracks (a) fully plastic stress field;(b) elastic-plastic stress field.

are the border angles delimitating sectors. Using the continuity condition (13) andboundary condition (30), from (9), (10) and (12), one can obtain the elastic-plasticstress field as follows

O Plastic fan sector 1 (0#q#q1)

srr=sqq=−2kq

srq=k(34a)

O Plastic constant stress sector 2 (q1#q#q1+p/2)

srr=−2kq1+k sin 2(q−q1)

sqq=−2kq1−k sin 2(q−q1)

srq=k cos 2(q−q1)

(34b)


O Plastic fan sector (q1+p/2#q#q3)

srr=sqq=2kq−4kSq1+p4D

srq=k

(34c)


srr=k cotq3[1+cos 2q]+k cotq3 cot 2q3[2(p−q)−sin 2q]

sqq=k cotq3[1−cos 2q]+k cotq3 cot 2q3[2(p−q)+sin 2q]

srq=−k cotq3 sin 2q+k cotq3 cot 2q3(1−cos 2q)

(34d)

whereq3$q2=q1+p/2. q3 andq1 are dependent on the radial stresssrr(p) and determ-ined by

5q3=arctanS 2ksrr(p)

Dq1=

14[q3−cotq3−(p−q3)cot2 q3]

(35)

Fig. 5(b) plots the stress distribution of the elastic-plastic crack-tip field (34) withsrr(p)=21.5k, q1=35° andq3=127°. This is equivalent to the results of mode-II cracksunder a positive far-field ‘T-stress’. Comparing (34) with (33) and Fig. 5(b) withFig. 5(a), one can find that the stress components ahead of the crack tip and on thelower plane remain invariant due to the occurrence of elastic sector on the uppercrack surface. Therefore, it can be concluded that constraint has no effect on thestress states ahead of the crack tip for a mode-II crack in elastic-perfectly plasticmaterials. A similar conclusion is obtained by Chao and Yang (1996) for power-lawhardening materials.

5. Crack-tip stress fields under mixed-mode-I/II loading

For mixed-mode-I/II crack problems, our attention is concentrated within thesmall-scale yielding (SSY). The SSY field is one field subjected to very small appliedload levels for all geometries, which corresponds to theT=0 field. FEA results ofDong and Pan (1990), Hancock et al. (1997) and Li and Hancock (1999) showedthat under the SSY conditions the mixed-mode crack-tip fields are

(a) the Prandtl slip-line field for pure mode-I loading,


(b) the Hutchinson slip-line field for pure mode-II loading,(c) fully plastic crack-tip fields for near mode-II loading,(d) elastic-plastic crack-tip fields for other loading cases.

Therefore, only case (d) above will be addressed in the following sections.


For mixed-mode crack-tip fields, it is observed from elastic-plastic FEA that theconstant stress sector 1 ahead of the crack tip under mode-I loading rotates clockwiseas the mode-II loading is applied, and one elastic sector on the upper crack flankexpands. It seems that this kind of mixed-mode crack-tip field is the perturbation ofborder angles in the mode-I crack-tip field. Under SSY, thus, it can be assumed thatthe elastic-plastic mixed-mode crack-tip field is constructed generally by 6-sectorsolution or 5-sector solution as shown in Fig. 6. These two solutions are the extensionof the 4-sector solution and the 3-sector solution of mode-I cracks. The 6-sectorsolution is comprised of an elastic sector overq5#q#p and five plastic sectors: thefirst constant stress sector in2p#q#q1, the first fan sector inq1#q#q2, the secondconstant stress sector inq2#q#q3, the second fan sector inq3#q#q4 and the thirdconstant sector inq4#q#q5. Hereq1,q2,…,q5 are the border angles delimitating sec-tors. The 5-sector solution is a special case of the 6-sector solution whenq5=q4.

Using the continuity condition (13) and boundary condition (14), from (9), (10)and (12), we obtain stress fields in the 6-sector solution as follows

O Plastic constant stress sector 1 (2p#q#q1)

srr=k(1+cos 2q)

sqq=k(1−cos 2q)

srq=−k sin 2q

(36a)

Fig. 6. Structures of elastic-plastic crack-tip field for mixed-mode-I/II cracks (a) 6-sector solution; (b)5-sector soution.


O Plastic fan sector 2 (q1#q#q2)

srr=sqq=kS1+3p2

+2qDsrq=−k

(36b)

O Plastic constant stress sector 3 (q2#q#q3)

srr=kS1+p2+2q3D+k sin 2(q−q3)

sqq=kS1+p2+2q3D−k sin 2(q−q3)

srq=k cos 2(q−q3)

(36c)

O Plastic fan stress sector 4 (q3#q#q4)

srr=sqq=kS1+p2+4q3−2qD

srq=k

(36d)

O Plastic constant sector 5 (q4#q#q5)

srr=kS1+p2+4q3−2q4D+k sin 2(q−q4)

sqq=kS1+p2+4q3−2q4D−k sin 2(q−q4)

srq=k cos 2(q−q4)

(36e)


srr=−kcos(q5−2q4)

sinq5[1+cos 2q]−k

cos 2q4

1−cos 2q5[2(p−q)−sin 2q]

sqq=−kcos(q5−2q4)

sinq5[1−cos 2q]−k

cos 2q4

1−cos 2q5[2(p−q)+sin 2q]

srq=kcos(q5−2q4)

sinq5

sin 2q−kcos 2q4

1−cos 2q5

[1−cos 2q]

(36f)

and the hydrostatic stress components over all angles are


sm5

k, −p#q#q1

kS1+3p2

+2qD, q1#q#q2

kS1+p2+2q3D, q2#q#q3

kS1+p2+4q3−2qD, q3#q#q4

−kcos(q5−2q4)

sinq5−2k(p−q)

cos 2q4

1−cos 2q5, q5#q#p

(37)

where the delimitation anglesq1=23p4

, q2=q32p2; q3, q4 and q5 can be determined

to solve the following equation system

Tpk

52cos(q5−2q4)

sinq521 (38a)

4Sq32p4D5S2q42

3p2

21D2(p2q5)cos 2q2

sin2 q5

2cos(q5−2q4)

sinq5

(38b)

Tp

k55S2q3−1−

p2D+sin 2q3; srq(0),k

4q3−p; srq(0)=k

(38c)

in which the parametersTp andTp is defined as hydrostatic stresses ahead of cracktip and on the crack flank, respectively, which is similar to the mode-I case. Theycan be determined atq=0 andq=p by

Tp5Hsappqq (0)−sPrandtl

qq (0); srq(0),k

sappqq (0)−(1+3p/2)k; srq(0)=k

(39)

and

2Tp5sapprr (p)2sPrandtl

rr (p) (40)

wheresappqq (0) andsapp

rr (p) stand for the known stress components atq=0 andq=p,respectively, from the slip-line field or FEA. Once the parametersTp and Tp aredetermined from (39) and (40),q3, q4 and q5 can be solved from (38). Then theelastic-plastic stress field over all angles are given by (36).


5.2. Discussions about the 6-sector solution

Due to |sapprr (p)|#2k for elastic-perfectly plastic materials, from (38) and (40),

several special solutions exist in the 6-sector solution of mixed-mode I/II cracks:(a) Whensapp

rr (p)=2k, we have

Tp50, Tp50; q252p4, q35

p4, q45

3p4

, q55p

Under this case, the elastic sector 6 disappears and the 6-sector solution (36) reducesto the fully plastic Prandtl field (19).

(b) Whensapprr (p)=22k, we have

Tp522k, Tp52(21p)k; q25212

2p2,q352

12, q45q55

p4

Under this case, the plastic constant stress sector 5 vanishes and the elastic sector6 becomes a plastic sector under constant pressure22k. And the 6-sector solution(36) degenerates to a fully plastic near mode-II stress field.

(c) When 22k,sapprr (p),2k, an elastic sector 6 appears near the crack surface.

The angular spanf(=p2q5) of the elastic sector 6 increases as the parametersTp

andTp, thus stress componentssapprr (p) andsapp

qq (0), decrease. Ranges of all unknownparameters in (36) are

2(21p)k#TP#0, 22k#Tp#0;

212

2p2

#q2#2p4,2

12

#q3#p4,p4

#q4#3p4

,p4

#q5#p

With these ranges, from (37), angular variation of the hydrostatic stresssm withdifferent values of the parametersTp and Tp can be illustrated in Fig. 7. It is seenagain that parametersTp andTp can characterize constraint levels in the plastic sec-tors ahead of crack tip and in the elastic sector on the crack surface, respectively.The Prandtl field (19) and the fully plastic near mode-II stress field are the twoextremes of the elastic-plastic stress field (36). Notice that the pure mode-II slip-line field cannot be obtained from (36) because the elastic-plastic solution (36) isbased on the extension from the pure mode-I solution [see Fig. 6(a)].

(d) Whenq5=q4, the constraint parametersTp and Tp depend on each other, andthe 6-sector solution above reduces to the 5-sector solution, as shown in Fig. 6(b),which is the same as the solution developed by Li and Hancock (1999). The stressfield of 5-sector solution can be obtained from (36) by lettingq5=q4.

5.3. Plasticity mixity for mixed-mode cracks

Based on the current solution and the well-accepted concept of mode mixity, wedetermine in this section under what mode mixity an elastic sector will occur on thecrack flank. The near-tip plasticity mixity parameter,Mp, is defined by Shih (1974)in terms of opening and shear stresses ahead of the crack tip as


Fig. 7. Sketch of angular variation of the hydrostatic stresssm with different values of the parametersTp andTp for mixed-mode-I/II cracks.

Mp52parctanFlim

r→0

sqq(r,q=0)srq(r,q=0)G (41)

with Mp=1 for the pure mode-I andMp=0 for the pure mode-II case.From the present elastic-plastic solution (42),Mp is related toq3 as

Mp552parctanF1+p/2+2q3+sin 2q3

cos 2q3G, 0#q3#

p4

2parctanF1+

p2+4q3G, −

12

#q3#0

(42)

The analysis above gives the range of the delimitation angleq3. Therefore from(42), the range ofMp for the elastic-plastic 6-sector solution (36) can be determ-ined as

(a)Whenq3=p4, Mp=1; the 6-sector solution reduces to the Prandtl slip-line field.

(b)When,q3=212,Mp=0.3302; the 6-sector solution reduces to the fully plastic near

mode-II stress field.

(c)When212

,q3,p4, 0.3302,Mp,1; an elastic sector occurs on the crack flank and


increases in size with the decrease ofMp value. The crack-tip field is determined

by the general 6-sector solution. Moreover, as 0#q3,p4, 0.7638#Mp,1, the con-

stant stress sector 3 crosses the lineq=0; and as212

,q3#0, 0.3302,Mp#0.7638,

the fan sector 4 crosses the lineq=0.(d)When 0,Mp,0.3302, the crack-tip field is for fully plastic near pure mode II

cracks.

6. Constraint analysis of crack-tip field with elastic sector

As displayed in previous sections, except for the pure or near mode-II cracks, thecrack-tip field is influenced by the constraint of specimen geometry and loadingconfigurations. As a result, the opening stress ahead of the crack tip decreases andan elastic sector occurs on the crack flank with decreasing crack-tip constraint. Thiskind of mechanics behavior and constraint level can be theoretically described bythe two hydrostatic stress parametersTp and Tp. To verify the present asymptoticcrack-tip fields, comparisons with numerical results are performed for various frac-ture specimens. Quantification of constraint levels using the parametersTp and Tpis also addressed in this section.

The constraint effect is discussed both for the SSY case and for finite-sized speci-mens. For finite-sized specimens, it is generally acknowledged that a deeply crackedDECP or SENB specimen is the representative for high constraint specimengeometry. And a CCP specimen is the representative for low constraint specimengeometry. The constraints of all other commonly used specimens generally fall inbetween these two extreme cases. As such, our study for finite-sized specimensincludes these three specimens to cover a wide range of constraints. Particularemphasis is placed on how well the parametersTp and Tp can be used to quantifythe constraint level over the wide range.

6.1. Mode-I cracks under small-scale yielding

For mode-I cracks in elastic-perfectly plastic materials under small scale yielding,Shih and German (1981) and Zhu and Chao (1999, 2000) showed that since theapplied loading is small, the plastic zone at the crack tip is very small, but indeedencompasses the entire crack tip in practical structures. In this circumstance, theboundaries of a finite-sized specimen hardly affect the crack-tip fields. Their FEAresults for SENB, CCP and DECP specimens indicate that as the crack tip isapproached the tensile stress ahead of the crack tip attains the limiting value of2.97s0 given by the Prandtl field. For the far-field stressT=0, Dong and Pan (1990),Betegon and Hancock (1991) and Du and Hancock (1991) presented the crack-tipstress field which is close to the Prandtl field for the non-hardening materials by


using a boundary layer FEA approach. Consequently, it can be concluded that themode-I crack-tip field under SSY is almost identical to the Prandtl field, and so theconstraint parametersTp=Tp=0. In other words, the mode-I crack-tip field is nearlyunaffected by the constraints of specimen geometry and loading under SSY.

6.2. SENB specimens in pure bending

Consider a SENB specimen in pure bending with the specimen widthW, the crackdeptha and the ligament lengthb. For elastic-perfectly plastic materials, Kim et al.(1996) presented the numerical crack-tip fields for SENB specimens with variouscrack depths through detailed FEA. Using their numerical results (see Fig. 5 in Kimet al., 1996), we can determine the values of constraint parametersTp andTp from(24) and (25). Then the angular crack-tip stress distributions for a specific case canthen be obtained from (22).

Figs. 8–10 shows the angular distributions of the stress componentssrr, sqq andsrq for the SENB specimens witha/W=0.1, 0.2, 0.3 determined from the 3-sectorsolution, the 4-sector solution and from the FEA by Kim et al. (1996) at the limitload. Fig. 11 is the angular variation of the hydrostatic stress or mean stresssm forthe same specimen. Notice that the crack becomes ‘enough deep’, i.e.a/W$0.5, boththe FEA results and the 4-sector or 3-sector solutions approach to the Prandtl fieldas shown in Fig. 1(a). The results in Figs 8(b)–11(b) show that the 4-sector solutioncan match exactly with the numerical results both for the stress distributions and thedelimitation angles. However, the 3-sector solution can only match with the numeri-cal results within the plastic sector ahead of the crack tip [see Figs 8(a)–11(a)], butnot in elastic sectors. In the 3-sector solution the border angleqEP delimitating theelastic sector and the plastic sector is quite different from the FEA result. In general,

Fig. 8. Angular distributions of stresses in SENB specimens witha/W=0.1. Notice that the symboldenotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution(a) the numerical results and the 3-sector solution withTp=20.969s0 and q2<75°; (b) the numericalresults and the 4-sector solution withTp=20.969s0, Tp=20.827s0, q2<88.5° andq3<110.8°.


Fig. 9. Angular distributions of stresses in SENB specimens witha/W=0.2. Notice that the symboldenotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution(a) the numerical results and the 3-sector solution withTp=20.588s0 and q2<89.5°; (b) the numericalresults and the 4-sector solution withTp=20.588s0, Tp=20.877s0, q2<107.4° andq3<143°.

Fig. 10. Angular distributions of stresses in SENB specimens witha/W=0.3. Notice that the symboldenotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution(a) the numerical results and the 3-sector solution withTp=20.119s0 andq2<122.4°; (b) the numericalresults and the 4-sector solution withTp=20.119s0, Tp=20.352s0, q2<128.2° andq3<157.8°.

therefore, it is necessary to use the two independent parametersTp andTp to charac-terize the constraint effects on the crack-tip field both in the plastic sector ahead ofthe crack tip and in the elastic sector on the crack flank, respectively.

It is seen that the values of constraint parameterTp andTp increase with increasinga/W and approaches zero as coming closer to the limiting case, i.e. the Prandtl field.The values ofTp/s0 in the 4-sector solution are20.969, 20.588, 20.119, 0.0,respectively for SENB specimens corresponding toa/W=0.1, 0.2, 0.3, 0.5 (or 0.7),


Fig. 11. Angular distributions of hydrostatic stress of SENB specimens fora/W=0.1, 0.2, 0.3, 0.5 (or0.7). Notice that the symbol denotes the FEA solution from Kim et al. (1996) and the solid line denotesthe present asymptotic solution (a) the numerical results and the 3-sector solution; (b) the numericalresults and the 4-sector solution.

and the values ofTp/s0 are 20.827, 20.877, 20.352 and 0.0. These values areplotted in Fig. 12 and are curve-fitted to yield

Tp5H−7.715(a/W)2+7.0993(a/W)−1.6259 fora/W,0.5

0.0 for a/W$0.5(43a)

and

Fig. 12. Variation of the parametersTp and Tp with the crack deptha/W for SENB specimens at thelimit load.


Tp5H−130.5(a/W)3+107.05(a/W)2−23.48(a/W)+0.581 fora/W,0.5

0.0 for a/W$0.5(43b)

For a specific crack deptha/W of SENB specimens in elastic-perfectly plasticmaterials or low-hardening materials, we can estimate the values of the constraintparametersTp andTp from (43) or Fig. 12, and then predict the crack-tip stress fieldusing the 4-sector solution (26) without FEA calculations.

Figs. 11 and 12 and the two equations above indicate thatTp andTp values increasewith increasing constraint (e.g. largea/Wor high stress triaxiality as reflected by thehigh hydrostatic stress or mean stresssm at the crack tip). Therefore, a specimenhaving low (high)Tp andTp values implies that it is a low (high) constraint speci-men geometry.

6.3. DECP specimens in tension

A DECP specimen loaded by remote tension is considered, 2W, 2a, 2b are thespecimen width, the crack length and the ligament length, respectively. For perfectlyplastic materials, both the slip-line solution of Ewing and Hill (1967) and the FEAresults of Zhu and Chao (2000) show that for deeply cracked DECP specimens withabouta/W.0.9 the crack-tip field is the Prandtl field, and thus the constraint para-metersTp=Tp=0. When a/W,0.9, the crack-tip field is affected by the boundaryconstraint of the specimen.

Fig. 13 plots the angular distributions of the stress componentssrr, sqq andsrq

of the 3-sector solution and the 4-sector solution, respectively, for the DECP speci-mens witha/W=0.5 and compares them with the FEA results of Zhu and Chao (2000)at the limit load. In the 4-sector solution,Tp=21.092s0, Tp=20.856s0, q2<84°,q3<106°. In the 3-sector solution,Tp=21.092s0 andq2<71°. Again, the results here

Fig. 13. Angular distributions of stresses in DECP specimens witha/W=0.5. Notice that the symboldenotes the FEA solution from Zhu and Chao (2000) and the solid line denotes the present asymptoticsolution (a) the numerical results and the 3-sector solution withTp=21.092s0 andq2<71°; (b) the numeri-cal results and the 4-sector solution withTp=21.092s0, Tp=20.856s0, q2<84° andq3<106°.


show that the 4-sector solution can match exactly with the FEA results both for thestress distributions and the delimitation angles, whereas the 3-sector solution canonly match with the numerical results within the plastic sector ahead of the cracktip. In the 3-sector solution, the stresses in the elastic sector and the elastic-plasticborder angleqEP are quite different from the FEA results. Therefore, it shows againthat it is necessary to use the two independent parametersTp andTp to characterizethe constraint effects on the crack-tip field.

6.4. CCP specimens in tension

In this section we consider a CCP specimen loaded by remote tension. 2W, 2a,2b are the width of specimen, the crack length and the ligament length, respectively.For perfectly plastic materials, McClintock (1971) gave the slip-line solution of thisspecimen as

syy(q)=2k<1.1547s0

sxx(q), sxy(q)=0, 0#q#45° and 0#x#b (44)

For elastic-perfectly plastic materials, our FEA results (Zhu and Chao, 2000)reveal that the stresses ahead of the crack tip gradually decrease from those in thePrandtl field (20) toward the CCP slip-line field (44) with the increasing loading. Atthe limit load, the numerical distributions of angular stresses over 0°#q#45° canapproach to, but cannot attain to, the slip-line field solution (44) near the crack tipbecause of a non-zero moment difference acting on the remaining ligament. Fig. 14graphs the FEA results of the crack-tip stress fields for the CCP specimen with

Fig. 14. Angular distributions of stresses in CCP specimens witha/W=0.5. Notice that the symboldenotes the FEA results from Zhu and Chao (2000) and the solid line denotes the 3-sector asymptoticsolution withTp=21.577s0 andq2<61°.


a/W=0.5 from Zhu and Chao (2000) and the 3-sector solution withTp=21.577s0

andq2<61°. Comparison shows that the numerical results and the 3-sector asymp-totic solution are in good agreement. The 4-sector solution is almost identical to the3-sector solution because the Eq. (28), the existence condition of 3-sector solution,holds approximately here. Notice that when using the 4-sector solution, the unknownparametersTp=21.577s0, Tp=20.903s0, q2<62°, q3<65°. Under this special case,therefore, one can use one parameterTp to characterize the constraint effect on theelastic-plastic crack-tip field.

6.5. Mixed-mode-I/II cracks under small-scale yielding

Under SSY, Dong and Pan (1990), Hancock et al. (1997) and Li and Hancock(1999) carried out the FEA calculations for mixed-mode cracks in elastic-perfectlyplastic materials using the elastic stressT=0 and various elasticity mixityMe. Theirnumerical crack-tip fields contain one elastic sector on the crack flank except fornear mode-II cracks. Fig. 15 depicts the stress angular distributions of the FEA resultsfrom Dong and Pan (1990) and the 6-sector asymptotic solutions for mixed-mode-I/II cracks with the elasticity mixityMe=0.84 andMe=0.54. Fig. 16 shows the stressangular distributions of the FEA results from Li and Hancock (1999) and the 6-sector solutions for mixed-mode-I/II cracks with the elasticity mixityMe=0.71 andMe=0.50. The comparison from the four graphs shows that the present 6-sectorasymptotic solution can precisely match with the FEA results over all angles for themixed-mode cracks. It should be noted that if the 5-sector solution is used, the elastic-plastic border angleqEP and the solution in the elastic sector cannot match with theFEA results. For example, in the Fig. 7 of Li and Hancock (1999), the differenceof qEP from the 5-sector solution and from FEA is more than 10°.

When the elasticity mixityMe=0.84, 0.71, 0.54 and 0.50, from (41) or (42), thecorresponding plasticity mixity are obtained asMp=0.917, 0.818, 0.727 and 0.688.All of plasticity mixity fall into the range 0.3302,Mp,1 and thus an elastic sectoroccurs on the crack flank as predicted in the Section 5.3. For the first two cases, theplasticity mixities are within 0.7638#Mp,1, so the constant stress sector 3 of the6-sector solution crosses the ligament lineq=0 as shown in Fig. 15(a) and Fig. 16(a).While for the two later cases, the plasticity mixities are within 0.3302,Mp#0.7638,so the fan sector 4 crosses the ligament line as shown in Fig. 15(b) and Fig. 16(b).For these four mode mixities, the elastic angular spanφ(=π2θ5) are 67°, 90°, 106°,93°, and the constraint parametersTp=20.429s0, 21.093s0, 22.038s0, 22.218s0;22.218s0; Tp=20.677s0, 20.907s0, 21.017s0, 21.087s0, respectively. It is seenthat with the decrease of the mode mixity or increase of mode-II loading, the sizeof elastic sector increases and the constraint parametersTp andTp decrease. There-fore, The 6-sector solution with the two parametersTp andTp can properly charac-terize the crack-tip field of mixed-mode I/II cracks in elastic-perfectly plasticmaterials.


Fig. 15. Angular distributions of stresses for mixed-mode-I/II cracks. Notice that the symbol denotesthe FEA results from Dong and Pan (1990) and the solid line denotes the 6-sector asymptotic solution.(a) Me=0.84. The parameters of the 6-sector solution areTp=20.429s0, Tp=20.677s0, q2=262°, q3=28°,q4=97°, q5=113°; (b) Me=0.54. The parameters of the 6-sector solution areTp=22.038s0, Tp=21.017s0,q2=296°, q3=26°, q4=58°, q5=74°.

7. Conclusions

Crack-tip stress fields accounting for constraint effects are examined for a station-ary plane strain crack under mode-I, mode-II or mixed-mode-I/II loading. It isassumed that materials are incompressible, elastic-perfectly plastic materials, andplastic deformation obeys the Mises yield criterion. Special attention is focused onwhat constraint parameters existed in the elastic-plastic crack-tip fields. Closed-fromasymptotic solutions are developed both for the fully plastic crack-tip fields and forthe elastic-plastic crack-tip fields. The results are summarized as follows:

1. When the plasticity entirely surrounds the crack tip, the fully plastic crack-tipfield is unique for mode-II cracks, but non-unique for mode-I cracks. For mode-


Fig. 16. Angular distributions of stresses for mixed-mode-I/II cracks. Notice that the symbol denotesthe FEA results from Li and Hancock (1999) and the solid line denotes the 6-sector asymptotic solution(a) Me=0.71. The parameters of the 6-sector solution areTp=21.093s0, Tp=20.907s0, q2=280°, q3=10°,q4=73°, q5=90°; (b) Me=0.50. The parameters of the 6-sector solution areTp=22.218s0, Tp=21.087s0,q2=2100°, q3=210°, q4=58°, q5=87°.


I cracks, the fully plastic crack-tip stress fields are the Prandtl slip-line field anda uniform stress field. For mode-II cracks, the fully plastic crack-tip stress fieldis the Hutchinson slip-line field. The non-uniqueness of the crack-tip field undermode-I loading is caused by the constraint effect on the crack tip.

2. For mode-I cracks, due to constraint effect, two elastic sectors occur on the upperand down crack flanks and the crack tip is surrounded by plastic and elastic sec-tors. In general, (i) the asymptotic crack-tip field is the 4-sector solution withthree plastic sectors and one elastic sector, and (ii) there are two independentparametersTp and Tp that cannot be determined in the asymptotic analysis butcan be determined from the far-field solution such as the FEA results. The twoparameters are hydrostatic stresses ahead of the crack tip and on the crack flanks,respectively. And thus they can characterize the constraint effects on the stressstates in the plastic sectors ahead of the crack tip and in the elastic sector nearthe crack flank due to geometric and loading. When the parametersTp and Tpare dependent, the general 4-sector solution degenerates to the 3-sector solution.Furthermore, whenTp and Tp vanish, the asymptotic crack-tip fields degenerateto the fully plastic Prandtl slip-line field.

3. For mode-II cracks, the stress states ahead of the crack tip are invariant althoughan elastic sector can occur on one crack frank under the far-fieldT-stress. Thisshows that the constraint does not influence the plastic stress field ahead of thecrack tip for elastic-perfectly plastic materials, which is in accord with the con-clusion of Chao and Yang (1996) for power-law hardening materials.

4. For mixed-mode-I/II cracks under SSY withT=0, the constraint due to modemixity decreases the crack opening stress and one elastic sector appears on thecrack flank except for near mode-II cracks. The elastic-plastic crack-tip field formixed-mode cracks is the extension of that for pure mode-I cracks as the mode-II loading increases. In general, the asymptotic crack-tip field is the 6-sector sol-ution with two independent hydrostatic stress parametersTp and Tp. When theparametersTp andTp are dependent, the 6-sector solution reduces to the 5-sectorsolution. Furthermore, whenTp and Tp are equal to zero, the asymptotic fielddegenerate to the fully plastic Prandtl slip-line field. For mixed-mode cracks undervarying elasticT-stress, the constraint effect on the crack-tip field need furtherinvestigations.

5. The present 3-sector solution for mode-I cracks and the 5-sector solution formixed-mode cracks are the same as the solutions obtained by Li and Hancock(1999). However, the 3-sector solution and the 5-sector solution are only the spe-cial cases of the present 4-sector solution and the 6-sector solution, respectively,and are invalid the elastic angular span 0#f,p/4 because the yield criterion isviolated in the postulated elastic sector. On the other hand, the 4-sector solutionand the 6-sector solution are valid for all elastic angular spans, and so are themore general and reasonable elastic-plastic crack-tip fields. Our investigationtherefore is a reasonable extension of the solution of Li and Hancock (1999) forincompressible, elastic-perfectly plastic materials within the plane strain con-ditions.

6. Comparison shows that the 4-sector solution for mode-I cracks or the 6-sector


solution for mixed-mode cracks can precisely match with FEA results over allangles around a crack tip for SENB, DECP, CCP specimens or mixed-mode cracksunder SSY. The magnitudes ofTp and Tp can indeed determine the levels ofconstraint in the plastic sectors ahead of the crack tip and in the elastic sectornear the crack surface, respectively. Therefore, the parametersTp and Tp can beused as constraint parameters to effectively characterize the entire crack-tip fieldwith constraint levels from high to low in elastic-perfectly plastic materials underthe plane strain conditions.

As reported in Section 3 and summarized in item (1) above, there exist two fullyplastic crack-tip stress fields for a semi-infinite mode-I crack in an infinite elastic-perfectly plastic body. One is the Prandtl field under a tensile load perpendicular tothe crack line, the other is a uniform stress field under a compressive load parallelto the crack line. Both stress fields are subject to the uniaxial loading and meet thetraction-free condition, the deformation symmetric condition and the crack-openingcondition for tensile mode-I cracks. For fracture specimens or real flawed structures,however, the cracked body is finite, and not infinite in size. Even if the crackedbody is subject to an applied pure mode-I load, a biaxial stress state is generated inthe local field near the crack tip due to the deformation constraint of finite-sizedboundaries, except for the SSY case. Under the ‘biaxial’ stresses, the crack-tip fieldis no longer the Prandtl field but an elastic-plastic stress field determined by thepresent 4-sector solution. For fracture specimens under a large load or the limit load,therefore, the crack-tip field generally consists both plastic and elastic sectors. Welike to conclude the paper by pointing out that this result is analogous to the con-straint effect in elastic materials (Chao et al., 2000). In the elastic case, a uniformstress field parallel to the crack line, i.e. theT-stress, always exists in the neighbor-hood of the crack tip for finite size specimens even when the far-field loading ispure mode-I. And, thisT-stress does indeed affect the critical fracturing load ofthe specimen.

Acknowledgements

The authors sincerely appreciate the support of this work by NSF through Cooper-ative Agreement No. 9630167 and NASA grant number NCC5-174.

References

Betegon, C., Hancock, J.W., 1991. Two-parameter characterization of elastic-plastic crack-tip fields. Jour-nal of Applied Mechanics 58 (1), 104–110.

Chao, Y.J., Yang, S., 1996. Higher order crack tip fields and its implication for fracture of solids undermode II conditions. Engineering Fracture Mechanics 55 (5), 777–794.

Chao, Y.J., Zhu, X.K., 1998. J–A2 characterization of crack-tip fields: extent of J–A2 dominance and sizerequirements. International Journal of Fracture 89 (3), 285–307.


Chao, Y.J., Yang, S., Sutton, M.A., 1994. On the fracture of solids characterized by one or two parameters:theory and practice. Journal of the Mechanics and Physics of Solids 42 (4), 629–647.

Chao, Y.J., Liu, S., Broviak, B.J., 2000. Brittle fracture: constraint effect and crack curving under mode-I conditions, Proceedings of Pressure Vessel and Piping Conference, American Society of MechanicalEngineers, July 2000, Seattle, Washington.

Du, Z.Z., Betegon, C., Hancock, J.W., 1991. J dominance in mixed mode loading. International Journalof Fracture 52 (3), 191–206.

Du, Z.Z., Hancock, J.W., 1991. The effect of non-singular stresses on crack-tip constraint. Journal of theMechanics and Physics of Solids 39 (4), 555–567.

Dong, P., Pan, J., 1990. Plane-strain mixed-mode near near-tip fields in elastic perfectly plastic solidsunder small-scale yielding conditions. International Journal of Fracture 45 (4), 243–262.

Ewing, D.J.F., Hill, R., 1967. The plastic constraint of V-notched tension bars. Journal of the Mechanicsand Physics of Solids 15, 115–121.

Ewing, D.J.F., 1968. Calculations on the bending of rigid/plastic notched bars. Journal of the Mechanicsand Physics of Solids 16, 205–213.

Green, A.P., 1956. The plastic yielding of shallow notched bars due to bending. Journal of the Mechanicsand Physics of Solids 4, 259–268.

Green, A.P., Hundy, B.B., 1956. Initial plastic yielding in notch bending tests. Journal of the Mechanicsand Physics of Solids 4, 128–144.

Hancock, J.W., Nekkal, A., Kartensen, A.D., 1997. Constraint effects in mixed mode loading. In: Karih-aloo, B.L., May, Y.W., Ripley, M.I., Ritchie, R.O. (Eds.), Theoretical and Computational FractureMechanics, Proceedings of the Ninth International Conference on Fracture, vol. 4. Elsevier, pp.2015–2022.

Hutchinson, J.W., 1968a. Singular behavior at the end of a tensile crack in a hardening material. Journalof the Mechanics and Physics of Solids 16 (1), 13–31.

Hutchinson, J.W., 1968b. Plastic stress and strain fields at a crack tip. Journal of the Mechanics andPhysics of Solids 16 (2), 337–347.

Ibragimov, V.A., Tarasyuk, N.E., 1976. Asymptotic behavior of stress state near crack tip in elastic-ideally plastic medium. Izv. AN SSSR. Mekhanika Tverdogo Tela 11, 184–185.

Kim, Y.-J., Lin, G., Cornec, A., Schwalbe, K.-H., 1996. Fully plastic crack-tip fields for plane strainshallow-cracked specimens under pure bending. International Journal of Fracture 78 (1), 21–34.

Lee, H., Parks, D.M., 1993. Fully plastic analyses of plane strain single-edge-cracked specimens subjectto combined tension and bending. International Journal of Fracture 63 (4), 329–349.

Li, J., Hancock, J.W., 1999. Mode I and mixed mode fields with incomplete crack tip plasticity. Inter-national Journal of Solids and Structures 36 (5), 711–725.

McClintock, F.A., 1971. Plasticity aspects of fracture. In:. Liebowitz, H. (Ed.), Fracture: An AdvancedTreatise, vol. 3. Academic Press, New York, pp. 47–225.

Nemat-Nasser, S., Obata, M., 1984. On stress field near a stationary crack tip. Mechanics of Materials3, 235–243.

O’Dowd, N.P., Shih, C.F., 1991. Family of crack-tip fields characterized by a triaxiality parameter — I.Structure of fields. Journal of the Mechanics and Physics of Solids 39 (8), 989–1015.

Prandtl, L., 1920. Ueber die Haerte plastischer Koerper. Goettinger Nachr., math.-phys. Kl. 74–85.Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by

notches and cracks. Journal of Applied Mechanics 35 (2), 379–386.Rice, J.R., Rosengren, G.F., 1968. Plane strain deformation near a crack tip in a power-law hardening

material. Journal of the Mechanics and Physics of Solids 16 (1), 1–12.Roy, Y.A., Narasimhan, R., 1997. J-dominance in mixed mode ductile fracture specimens. International

Journal of Fracture 88 (3), 259–279.Sham, T.L., Li, J., Hancock, J.W., 1999. A family of plane strain crack tip stress fields for interface

cracks in strength-mismatched elastic-perfectly plastic solids. Journal of the Mechanics and Physicsof Solids 47 (9), 1963–2010.

Shih, C.F., 1974. Small scale yielding analysis of mixed mode plane strain crack problems. In: FractureAnalysis, ASTM STP 560. American Society of Testing and Materials, Philadelphia, pp. 187–219.


Shih, C.F., German, M.D., 1981. Requirements for one parameter characterization of crack tip fields bythe HRR singularity. International Journal of Fracture 17, 27–43.

Wu, S.-X., Mai, Y.-W., Cotterell, B., 1990. Plastich-factor (hp) of fracture specimens with deep andshallow cracks. International Journal of Fracture 45 (1), 1–18.

Zhu, X.K., Chao, Y.J., 1999. Characterization of constraint of fully plastic crack-tip fields in non-harden-ing materials by the three-term solution. International Journal of Solids and Structures 36 (29),4497–4517.

Zhu, X.K., Chao, Y.J., 2000. Fully plastic crack-tip fields for CCP and DECP specimens under tensionin non-hardening materials. International Journal of Solids and Structures 37 (4), 577–588.

Zhu, X.K., Hwang, K.C., Liu, G.T., 1997. Continuous near-tip fields for a dynamic crack propagating ina power-law elastic-plastic material. International Journal of Fracture 85 (3), 201–209.

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Constraint effects on crack-tip ﬁelds in elastic ... · Constraint effects on crack-tip ﬁelds in elastic-perfectly plastic materials X.K. Zhu, Yuh J. Chao * Department of Mechanical