UNIVERSITY OF CALIFORNIA, SAN DIEGO

BZRELZY - DAVIS • UINE - LOS ANGELES • RIVERSIDE • SAN DLEGO - SAN FRANCISCO 0 SANTA BARBARA SANTA CRUZ

DEPARTMENT OF APPLIED MECHANICS AND LA JOLLA, CALIFORNIA 92093-0411ENGINEERING SCIENCES, MAIL CODE R-011

September 27, 1990

Ln Department of the NavyOffice of the chief of Naval Research

r% 800 North Quincy StreetN Arlington, VA 22217-5000 17' T .

Attention: M.M. Reischman E L" 'Director (Acting)OT1Mechanics Division OCT 111990

Reof--v - 00/l

Ser 1132/154 U

Dear Dr. Reischman,

Enclosed is the annual Performance Report for work performed under our project,"Mechanics of Interface Cracks", grant number N00014-90-J-1398.

We trust this complies with your request.

Sincerely yours,

Di._t:, I Dept. of AMESEncl

RA/ba

Introduction

Fracture of advanced materials, such as intermetallic alloys, ceramics, and compositematerials nearly always begins at interfaces of one sort or another. Fibrous fractures, forexample, initiate at the interfacews between second phase particles (or reinforcement fibers,whiskers, etc) and the ductile matrix; the conditions for this to occur are sensitve to bothmatrix constitutive properties as well as to the chemistry and structure of the interfacesbetween the two phases. Interfacial cleavage involves the initiation of micrto-cracks andtheir propagation along interfaces and transgranular cleavage often initiates at interfacialsites, e.g. twin boundaries in intermetallic compounds, especially those with non-cubiccrystal structures. Here again, the conditions required for this to occur axe sensitive tothe constitutive properties of the matrix materials as well as to the interfacial propertiesthemselves. Intermetallic alloys, in particular, are known t*o possess complicated thermaland strain rate dependencies that can lead to either briie or ductile behavior which inany event are very much dependent upon the loading conditions.

Our research over the past three years has been concerned with providing a theo-retical and comutational framework for analyzing the deformation of complex crystallinematerials, like intermetallic alloys, and in particular of the failure of interfaces withinthese materials and between these materials and other phases. In addition, we have beendeveloping experimental methods to study the failure of interfaces between metals andmetals and ceramics. Our experimental studies are to be directly correlated with ourtheoretical/computational work. Specifically, our aims are to

1) conduct computational, and, complimentary experimental studies of interface crackmechanics, including crack growth, at interfaces separating crystalline phases described byphysically based slip theories;

2) extend our physically based slip theories to specifically account for the dislocationmicromechanisms that are peculiar to more complex crystalline materials such as orderedintermetallic compounds;

3) conduct analytical and computational modeling of the deformation and failure oordered intermetallic alloys, and especially of the deformation and failure near interfaces;and

4) further develop our vacuum diffusion bonding methods to prepare metal/ceramic,alloy/alloy, and intermetallic/metal (ceramic) interfaces with controlled defects.

Our ongoing experimental work has been concerned with the Nb/Al2 03 interface. Wehave used vacuum diffusion bonding methods to prepare pre-cracked interfaces in thissystem and have been conducting experiments to measure the fractuare toughness. Wehave extended these studies to the interfaces of Ni 3 Al and Nb and TiAl/Nb for thepurposes of measuring interface toughness and also to correlate the interface toughness p t,with ductility and toughness of composites. We have also formulated a flow theory for

STIATEMEPINT "A" per D)r. LekouctisONR/Code 1132 Dist ,' cTELECON lu/10/90 cc

intermetallic alloy materials that accounts for deviations from Schmid's rule. We havepreviously shown that such deviations have important implications for the stability ofuniform flow.

During the next year, we plan to:

1) Continue our work of the past year on the analyses of the mechanics of cracks oninterfaces. Our intention is to perform computational analyses of the very near crack tipfields under a complete range of mixed mode loadings and of the full fields for crackedspecimen geometries of the type we are studying experimentally. We intend to explorethe effects of geometric constraints on the development of the crack tip fields, and inparticular, the influence of interface geometry on the buildup of hydrostatic stresses. Therole of hydrostatic stress in determining the failure modes of the interfaces, and the resultingductility of the microstructure in specimens containing the interface, will also be assessed.

2) Extend our computational studies of localized deformation to ordered intermetallicalloy systems, in particular Ni 3 Al and later TiAl alloys. As noted above, we have for-mulated a constitutive theory for such materials and have analyzed the behavior in singlecrystal Nis3 Al. We are now implementing the theory in our FEM codes to perform analysesof deformation processes ;n these materials.;

3) Residual stresses, due to incompatible thermal strains, are known to play an im-portant role in the development and propagation of flaws and cracks along interfaces. Wehave formulated a finite strain theory for thermal strains, and have implemented this the-ory in our finite-element models for composite behavior, and interface fracture. We intendto carry out simulations of crack growth along interfaces of bimateria couples such asNb/A12 03 for which the thermal strain mismatch is minimal and intermetallic compoundssuch as TiAl, Nb3 AI, and Ni 3 Al/Nb for which thermal mismatch is significant; and

4) We will continue our experiments on the fracture of metal/ceramic and metal/metalinterfaces prepared by diffusion bonding methods. We are currently conducting experi-mental studies of crack growth along Nb/A 203 interfaces which are prepared by vacuumdiffusion bonding. We will be exploring the use of the bend specimen to subject interfacecrack tips to a wide range of mixed mode loadings in addition to the compact tensionspecimen we developed in the initial phase of our project. We will extend these studies tointerfaces involving intermetallic compound/metal and intermetallilc compound/ceramiccouples.

Summary of Research

In the past three years, we have completed a series of analyses on 1) the structureof interface crack tip fields; 2) the full field analyses of the elastic-plastic fields associatedwith interface cracks and flaws; 3) the micomechanical and macromechanical process ofcrack and flaw propagation along interfaces; and 4) deformation at the tips of cracks lyingon interfaces with structure. We have also completed the formulation of a flow theory forordered intermetallic alloys. Our computations have been carried out for stationary cracksas well as for cracks propagating along interfaces where the phenomenology of propagationis governed by adhesive laws ascribed to the interface. Nine papers have been submitted

2

for publication. Four of these have appeared in print in 1990; one additional has beenaccepted for publication.

R.J. Asaro:

1) Cracktip Fields in Ductile Crystals (with J.RL Rice and D.E. Hawk), InternationalJournal of Fracture 42, 301 (1990).

R.J. Asaro and C.F. Shih:2) Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part I - Small Scale

Yielding, J. Appl Mechanics 55, 299 (1988).3) Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part II - Structure of

Small Scale Yielding Fields, J. Appl. Mechanics III, 763 (1990).

4) Elastic-Plastic Analysis of a Collinear Array of Cracks on a Bimaterial Interface, J.Material Science and Engineering A 107, 145 (1989).

R.J. Asaro, N.P. O'Dowd and C.F. Shih:5) Cracks on Bimaterial Interfaces: Plasticity Aspects, in proceedings of the Acta/Scripta

Metallurgica Conference on Bonding, Structure and Mechanical Properties ofMetal/Ceramic Interfaces, 1989.

6 Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part III - Large ScaleYielding, submitted to J. Appl Mechanics (1989).

7) Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part IV - Interfaces withStructure, submitted to J. Appl Mechanics (1990).

R.J. Asaro, N.P. O'Dowd, C.F. Shih and A.G. Varias:8) Failure of Bimaterial Interfaces, submitted for publication, Mat. Sci. and Eng. A

126, 65 (1990).9) Crack Growth on Bimaterial Interfaces, in proceedings of the Acta/Scripta Metallug-

ica Conference on Bonding, Structure, and Mechanical Properties of Metal/CeramicInterfaces (1990)

A tenth manuscript authored by R.J. Asaro and M. Dao, entitled The Mechanics ofPlastic Flow in Ni 3 AI is under preparation; the work associated with it was completedduring July/August 1990 at the University of California, San Diego.

Our initial studies under this project were concerned with the structure of plane straininterface crack tip fields. Specifically in Parts I and II of our four part series of articles (i.e.,refs. 2 and 3 listed above), the structure of the carck tip field that develop under conditionspertaining to small scale yielding was determined. In reference 4, the stress and strain fieldthat evolve under conditions producing contained yieldling and fully plastic conditionswre examined through a study of collinear interface crack geometries, these (primarily)numerical studies led to an understanding of the asymptotic structure of interface cracktip fields along with a correspondence of these fields to the well characterized small strain(HRR) fields in homogeneous media. In particular, it was found that the small scale

3

yielding fields of the interface crack are members of a family paremeterized by a plasticphase angle, C, defined as

where the parameters and L are defined vis-a-vis the linear elastic interface crack tip fieldas follows.

In terms of polar coordinates, r, 0, centered at the crack tip, the crack tip fields canbe organized in the form

0ij - ; ,n)

where n is a power law strainhardening index prescribed in connection with J2 deformation,or J2 flow theory. hij is a bounded function which., however, varies with F(= r/(J/o)).For -7-/6 < C < 7r/6 however, hi, varies slowly with F so that the near tip field conformsclosely with that of an HRR field. K is the complex stress intensity factor defined, withrespect to the elastic crack tip asymptotic field, as

crj= 1,... fRe{Kr'(1 &i,(O; e) + Im{Kr"I} &iy(O; c)]

E is a dimensionless bimaterial constant and L is a crack dimension. Our results showedthat for cases involving two elastic-plastic media that the fields, in both materials, areparts of a single asymptotic field; the intensity of the stresses are determined by the yieldstrength of the weaker material whereas the angular distribution of stress about the cracktip is set by the strainhardening characteristics of the more weakly hardening material.

In reference 4, analyses involving contained yielding and fully plastic states were carriedout. An interesting discovery was that the fields scaled with the value of the J integralwhen J was normalized by yield strength and crack length. In particular, a value J,prescribed as J/(uoeL) was found to scale the fields that developed between colineararrays under conditions that produced deformation states ranging from small-scale yielding- to contained yielding - to fully plastic behavior.

Finite deformation analyses were carried out in references 5, 6, 7, 8 and in PartIV. Computational studies were performed using crystal plasticity theory as well as phe-nomenological J2 flow theory. Our finite strain calculations have included studies of thedeformation of cracks on interfaces with structure. An example of this is Fig. 1 whichshows the deformed meshes for cracks lying on an interface between two materials de-scribed by a rate dependent, finits strain J2 flow theory. The materials on either side ofthe interface differed by a factor of two in yield strength; their properties were taken to

4

match those of the dissimilar steels that we studied experimentally. The interface regionhad properties intermediate between these two phases.

Figure 1 indicates the strong influence of mixed mode loading in that our calculationswere carried out for three values 4 = -7r/6, 0, 7r/6. The crack tip openings, and associatedextents of plastic deformation, are greatest for the mixed mode sta6tes. In fact, for agiven value of the applied far field J, the crack tip opening was largest for the mixed modestate corresponding to C = -7r/6. Figure 2 shows contours of hydrostatic stress whichalso illustrate the effects of mixed mode loading. A particularly remarkable feature is theshift in hydrostatic stress with mode from the softer to the harder materials as the modeshifts from that corresponding to C = 0. These particular calculations were part of a setof model studies of our experiments concerned with dissimilar steel interfaces. One suchiinterface is shown in Fig. 3. Note that this interface is rather broad and is characterzpdby an interdiffusion zone of about 60 pm in thickness. Our observations showed that, as theinitial crack began to propagate, it quickly travelled to the boundary of the transition zoneand the softer steel. Given that the fracture mode was fibrous, this is, in fact, expectedsince our computations showed that the hydrostatic stresses are larger in the transitionzone toward the softer steel. Fibrous fractures, which are driven by void initiation andgrowth, are accelerated by hyd-ostatic tensile states of stress.

Recent Work

Our more recent work has been concerned with the formulation of a flow theoryfor ordered intermetallic alloys. Ali 3 Al was proposed as an initial example. One of theparticularly interesting features of the dislocation mechanics in such crystalline materials isthat they display strong departures from Schmid's rule of a critiacal resolved shear stress.Deviations from the Schmid rule of this type represent non-normality in the flow rule.Asaro and Rice have shown that there are indeed important implications of non-normalityof ths type regarding stable, uniform, plastic flow.

As part of this project, Dao and Asaro have formulated a flow rule for both rateindependent and rate dependent materials. For rate independent materials, the flow ruletakes the form (for each slip system)

1

where r" is the shear stress phrased on slip system coordinates as shown in Fig. 4. Thisfigure also indicates some typical micromechanical processes, involving dislocations, thatgive rise to the non- schmid effects. Q is the dyad

Q = sm+a : H :L - 1

where a is the matrix of non-Schmid factors, i.e.

/ &. 0 Cia0 0 cmm amz •

( a8 z amz Cezz

For rate dependent materials, the flow law instead takes the form

-y = function(Q : a; material state).

Dao and Asaro have analyzed the behavior of Ni 3Al within this framework and havecalculated, based on experimental data, values for their non-Schmid coefficients.

We are currently in the process of implementing our theory in one of our FEM codes.Our initial objectives will be to study the development of flow patterns in single andpolycr-ystalline microstructures of these materials and to then analyze the behavior ofinterfaces between Ni 3Al and Nb.

List of Participants and Status

S. Schoenfeld, Graduate Student

M. Dao, Graduate Student

Both of these students are in their second year.

Other Sponsored Research

"Fundamental Studies of Localized Plastic Deformation", NSF, 71,900.

Awards

RLJ. Asaro has been awarded the TMS Champion H. Mattewson Gold Metal for 1991for his research on micromechanics.

6

UCSD 90-5318

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UNIVERSITY COMMUNICATIONSUL= NE LID 5

Yedia Contact:

Warren R. Froelich, University Communications, 534-8564

Aug. 10, 1990

Robert J. Asaro Honored by Metallurgical Society

Robert J. Asaro, a professor in the Department of Applied

Mechanics and Engineering Sciences (AMES) at the University of

California, San Diego (UCSD), has been awarded the TMS Champion

H. Mathewson Gold Medal for 1991.

The Mathewson Gold Medal is presented to an author, or

authors, of a paper published in "Metallurgical Transactions,"

which is considered to be the most notable contribution to

metallurgical science during the period under review. The award

was established in 1950 to honor Dr. Champion H. Mathewson, a

noted scientist and teacher.

Formal presentation of the award will be made during the

annual dinner of the Minerals, Metals and Materials Society, to

be held Feb. 19, 1991, at the Radisson Suite Hotel in New

Orleans, La.

The Minerals, Metals and Materials Society is a member

society of the American Institute of Mining, Metallurgical and

Petroleum Engineers, Inc.

###S

S-038 • Univesityof California, SanDego * LaJol,CA • 9203-0036 •(619)534-3120

Ill t miioti ~I i'mial ,I Iriat till-, 421 30 1 121 1990.6t ,kmn, md 'I . R o,zai cd ,,. sn-litiar /)oi lurc.

Crack tip fields in ductile crystals

JIR. RI(iL.* DT. fl \\%:K* and RiJ. ASARO

Ili,!, Io I 4!1 1 ( ,ilib'iiii4i at SaIn I) ireo". to I. C. 1 _'iiQs 1 N I

ReCCI\ ed I \(ILtIi~l NSS. AOe -pled IS vAiiei'lt NNN5

\bs~lract. Re-.uh-. on lihe i-. , mploiinal-. oll LrIZI lip field-. InI CLo-lie-plai1 l-iiile a i- re pre-.eiited aiid,,,fle preIIInIIIII' e-Itit-, Of iII lelentI .OIliiois IoM criacked solid-. oF k t, pe ie C till)IIIA~ri In tle ed-.e--.t,,ticki. iiiwkllt\ Ii~ liie tlriill tibnile and dilli-plicliem er riatk- it -celt pl,lie I ce mi'd bet c r\-.ldl- .ilm~ls/ed\III~I thiiiNConii.'I-ll di-.pldeenilel idm la dsll iLIIPIIOl- tihe ML1\1c rcl~lci~dX- e jI nieil'ik ing i.oiiiiofeld,. at the ciack ipl

I .' ,Ii. l, L i li\ :iek the tiCNN.- 'idle I'. Ioiiid 1t) be loemilli\ i11001111 Ill C~ilil 01 d 1 dtil (11 Ailiiikii1 -. CCIIr. 11the icttIk tlp. hitll Itiii di'c111(ftiiuioui-.1I' l sec h~0_IOmindLIArie. \%'iI Me dl-.,i th 'r 'ic- WCh -.1 d dII -. (1COllIFImLme-.ill (lie .i-.pldscementi field. F-or the quai-i-shilhh grtmiingc crlck tIle t.le-.i -.lde k- Itilli\ ciinllinLOI-. 'r011 01neneil1-tip in1ctLlir eeilor 1til Ile\(. hilt no%% 'orlI ol, tile -.celor-. il'.el% ehmi-tmC tilAY1iMn h-011. minld rllOmidiiieU 10.

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lie peipIitdmeiIl~I 1 tile mim I>' 01i AF\ icIe-.ip plle-. NO0 thdt tile del CIIItillie-. Corre-.pond 1o a kiiikiuie mode ofheiar

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elment I101t1i111i01i0n -.uiitmible for MjritrMri delorniliat mol hid- beett u-.edI to ,.Okie lor tile plaiie Strdinl teniiot oh miI m> or-i~rdesiiiie >-.ihp;IIel Coltaiuliiiu. a reniter erdek \wih all iiiitili r-oundedCL tip. is,- -.kO\N effect-. dueC to

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1. Introduction

* This, pZIper slimmllarizes recent anal\ tical aid nmler-C1ic l mx SticationlS into thle nature of thenear-crack-tip stress and delfbrmation fields Inl dLuctile single crystals. DItCtilC crx stals def'orm1

*plasticalli. by thle motion of' dislocations onl a limitedl set of' slip 5\ stemis. A coniM11..11representation of' this plastic def'ornution consistent wkith the Schmid rulle., which states thatflow onl a si.stemi is actix ated whenl the shear stress resolved onl that system reaches a critical\alule. is used inl the aralvses to be presented. This fbOrmla~tion leas to a yield SLutfaIce inl

stres;s space Consisting of' planrar f',icets joi ned at vertices and to an ''associated- plasticstraining, relation.

General methods of' construIcting aI sI mptotic necar-tip fields for01 SuCh1 CryAls wU \ith either,stationar\ or quasi-staticall\ growi.ing cracks. hawe heen obtained inl the ideally plastic caseFor both anti-plane strain (miode Ill: [ll) and tensile planec strain ( modle 1: 121) cracks. Thieresults, ais illustrated for common crack orientationis inl fcc anld h cc crystals, lead to

302 J.R. Rice. D.E. thawk and R.J. ,saro

striking predictions of discontinuities at the crack tip. Full scale elastic-ideally plasticsolutions to the near-tip stress and deformation fields have been given for stationary cracksin the mode Ill study. These show that all flow is confined to planar plastic zones emanatingfrom tlh- crack tip, across which both displacement and stress are discontinuous. Asymptoticanalysis of dynamic crack growth. i.e., including inertia, has -een developed as well for thatmode [3] and reveals that an elastic-plastic shock discontinuity moves along with the tip. Inaddition, asymptotic fields of the HRR type have been developed for stationary cracks incrystals showing Taylor hardening, with a powci-law stress-strain relation at large strain, inmode 111 [4] and mode 1 [5].

The mode I asymptotic analysis [2]. based on ideal plasticity and a "small displacementgradient" formulation, shows that for material at yield the stress state is constant withinangular sectors whose houndaries are certain crystallographic directions on which discon-tinuities in either displacement (stationary crack) or velocity (quasi-statically growing crack)are possible. A direct comparison of the mode I analyses is made here with the numericalresults of Hawk and Asaro [6]. When comparing different types of crystals. or crackorientations within a given crystal, the structure of the dislocations necessary to produce thesame continuum field discontinuity is different. Furthermore, certain dislocations structuresmay induce rotation of the lattice relative to the material, thus changing the resolved shearstresses on slip systems and causing a geometric hardening or softening of the crystal. Thisis important particularly when large deformations are taken into account.

Recently, full scale elastic-plastic solutions for mode I cracks in ductile crystals have beenobtained numerically by the detailed finite element analyses of Hawk and Asaro [6]. Someof their results are briefly summarized here for several different loading cases. These analysesmodel the constitutive behavior of the crystal with a visco-plastic formulation in the nearlyrate-independent limit. The numerical analysis of Asaro [6] models from small-scale togeneral yielding a center cracked panel with a blunted crack tip under uni-axial tension. Theslip systems of the crystal are idealized by a planar double-slip model. The effect of Taylortype hardening of the slip systems and large displacement gradients (e.g., lattice rotation)are included. In the small-scale-yielding analysis of Hawk [6], both a stationary and quasi-statically propagating perfectly sharp tensile crack are simulated. The crystal is modeled aselastic-ideally plastic using a complete description of the slip systems in a fcc crystal. Smalldisplacement gradients are assumed.

2. Constitutive law

The crystals considered can undergo both elastic and plastic deformation. The plasticdeformation is consistent with a continuum description of single crystals [7-10]. The totalstrain-rate is taken as the sum of the elastic strain-rate and plastic strain-rate

Under plane strain conditions, which are possible for the high-symmetry crack orientationsconsidered here, ., , !. o-, and a, are zero. The plastic deformation of the crystal occursby the motion of dislocations along certain preferential crystallographic planes. The move-ment of these dislocations causes a permanent dilationless straining of the crystal. A slip

(rack ip /fi('l in dtic/cih crsAlI.\ 303

system denoted by a is defined by two unit vectors giving the slip direction s' and the nornalto the slip plane n"'. The parameter 7'' describes the amount of shear strain on each slipsystem. These prelerential slip systems defined by s' and n'"' ary according to the crystalstructure (e.g., face centered cubic, fcc. or body centered cubic, bcc). It is possible to expressthe plastic strain rate as a sum over all the N slip systems of the crystal as

Here jip, is termed the Schmid f'actor and is determined from the slip direction s'' and thenormal to the slip plane n'" as

I . , + . ') } (3)

The resolved shear stress. r''. on a system is expressible in terms of these Schmid Iactors as

- a,l1', (4)

The plastic deformation of a crystal is said to obey the Schmid rule if the instantaneousshear response :' of any given system a depends on the current stress state only through thecombination in (4). In a time-independent pksticity lormulation, a necessary but notsulicient condition for slip to occur is that the resolved shear stress r reaches a critical valucg''. This criteria results in a yield surface in stress space which consists of planar facets thatjoin at vertices at which two or more systems are simtllaneously active Equation (2) thenassures that the flow rule is of an -associated" type. Rice [Il] showed that for rigid-plasticincompressible solids of this class in plane flow, the yield surface can be represented as acurve in the reduced stress space of (r11 - (7-)/2 and a,. Such a yield surface is schematic-Ally shown in Fig. I and for crystals as described here. it is polygonal. In tl castic-plastic

Or I - CY22

2

/1,'. 1. Schema Inic of ield Surlace For a ducti le cr. stal undergoi ng plane st raini ng.

304 J.R. Rice. D.E. Hawk and R.J. .- aro

material the same result applies for stress states in large sustained plastic strain. However.

in such a case limited deformation may occur on slip systems corresponding to segments of

the yield surface sensitive to the value of ((T, + (,.) but which cannot produce sustainedplane flow. We will term the sides of the polygon as "flats" and the point of junction of two

sides as a "vertex". The flow law of (2) being associative, the direction of plastic straining.

with components (4!P - i!(P{ ),2 and i.',, is normal to the yield surface on a "flat" and withinthe forward fan of normals at a "vertex".

It will also be convenient to represent the slip systems of the crystal as traces in theV, x -Vplane. For example, in Fig. 2a. a unit cube representing the lattice of an fc c crystal is shown.

The slip planes of a fcc crystal are the I I 1 1I1 planes indicated by the various shaded planesin the cube. The slip directions are directions which are the diagonals of the cubefaces. If the crack lies in the plane (0 I 0) and the crack tip lies along the diagonal of the cube

face as indicated, then the traces of the slip systems (i.e. intersection of the slip planes withthe x,, v, plane) are as indicated in Fig. 2b. It is then possible to represent the slip directions and slip plane normal n as the directions S and N respectively in the x1, x, plane as shownin the figure. Only those traces shown correspond to systems which can produce sustained

crack plane (010)

X 2 '0101tip along 11011-

i [1011

(a) X 1

(b)2[0101011

traces of slip planes

Fig. 2. (a) lace centered cubic slip systems, and orientation of the crack plane and crack tip for case illustratedbeloss. (hN Crack on (( 1)) plane grosing in [I0 01 direction. Traces are shown of those slip plane families whichcan accom1modate sustained plane straining.

-- Mo

('rack tip fields in ductile ir.itu.s 305

plane flow. The solid line traces correspond to individual 'I I I : planes. and the dashed linesto two such planes which intersect along a < 1 1 0> slip direction common for each and which.by equal coincident slip. produce plane flow.

The finite element analyses use a visco-plastic formulation where slip can occur on an\system so long as resolved shear stress on that system is non-zero. A simple power-lawrelation [12-14] can be used to describe the rate of' slip on each system as

* ~,sgn (r") (5

As the exponent m -* 0 the visco-plastic model approaches the rate-independent formu-lation. Use of a visco-plastic law in a finite element formulation, pioneered in the work ofPierce et al. [14]. eliminates certain problems of uniqueness and considerations of when apoint is at yield. The g" reflects the current level of strain hardening in the crystal. In [61,results are presented for both ideal plasticity i.e., g" is a constant. r,,, and cases where g'"is a function of the sum of the magnitudes of the slips (I = Y, ;"I), which coincides withTaylor hardening.

3. Asymptotic analysis

The asymptotic analyses (e.g., lim, ., a,1 (r. U. t)) of Rice [2] are summarized here for the caseof ideal plasticity in a "'small displacement gradient" formulation of the mode I problem.Based on the equations of equilibrium with a bounded crack tip stress state and with thefurther condition that the stress state is at yield relative to a slip plane trace in the directionS,, the following requirement is found:

(N,e, )(S,e, )(a,' + a", ) = 0 (6)

Greek indices range from I to 2. As indicated, Fig. 2b, the e, are the components of the radialunit vector in the v. x, plane; (e. e,) = (cos 0, sin 0). The terms S, and N, are thecomponents of the traces of the slip direction and slip plane normal as previously defined.The (T,, denote

= lim ;' (7)

Equation (6) implies a;, + (T, = 0 for all 0 except for four special values when e is alignedwith either N or S. Based on the form of the stresses consistent with equilibrium for materialat yield, this statement further implies that within sectors bounded by these four specialvalues of 0. all (7t, are zero: i.e. the stress state is independent of 0. For a tensile crack it isshown that. in order to meet boundary conditions, either (i) the stresses in certain angularsectors around the crack tip are not at yield or (ii) that the stresses change discontinuouslyat the special values of U as mentioned above.

306 JA Rice. D.E. Hawk and R.J . Asaro

3.1. StationarY crack

For the stationary crack, as r - 0. the yield condition can be met in all angular sectorsaround the crack tip, thus requiring discontinuous jumps in the stress state. The stressdiscontinuity must be from one point on the yield surface to another, and considering thenature of restrictions on such discontinuities. this path must be a straight line in the stressspace and the line must lie everywhere along the yield locus (and hence correspond to a flatsegment in Fig. 1) since otherwise elastic unloading would occur. The solution which satisfiesthe jump conditions for the stress discontinuity is one where the stress state changes fromvertex to vertex on the yield surface. The deformation fields, consistent with the above stressstate, must have a shear type of discontinuity in the displacement field along the samedirection, e. for which the stress is discontinuous. These correspond to concentrated shearparallel to the slip plane traces when e is aligned with S, and to kink-like shear perpendicularto the traces when e is aligned with N. The specifics of constructing such solutions may befound in [2]. Specific examples of these solutions will be given in the discussion of the finiteelement solutions of [6].

3.2. Growing crack

For the quasi-statically growing crack, Drugan and Rice [15] demonstrated for this type ofmaterial that discontinuities in stresses and displacements cannot exist. However, it ispossible to have velocity discontinuities. It is therefore necessary to have bordering on theplastic sectors angular sectors in which elastic unloading (and perhaps reloading) occur.Construction of the solution for a particular crystal orientation, again for which the detailsare given in [2], shows a complicated pattern of plastic sectors and elastic unloading andreloading sectors is necessary to model the crack tip fields for quasi-static growth. A specificexample will be presented for comparison with the finite element solutions of Hawk [6] irthe next section which shows that one of the physically active system for the stationary crackbecomes inactive in the growing case.

3.3. Note on role o'S and N

In the asymptotic analyses presented, based on the "small displacement gradient" formula-tion, the role of the slip direction S and slip plane normal N are interchangeable since theycome in through the symmetric form of (3). In certain cases [2] fc c crystals and bc c crystalsdiffer only by an interchange of S and N. Therefore solutions obtained for such fcc crystalsare also valid for bcc crystals as well (aside from a slight scaling). However, the structureof the actual dislocations necessary to produce the shear deformation is quite different whenS and N are reversed. For example, shown in Fig. 3a and 3b is shear parallel to the slip planesand shear perpendicular to the slip planes respectively. The motion indicated in Fig. 3a canbe generated by the emission of dislocations from the crack tip along the slip direction asshown in Fig. 3c. Notice however that in Figure 3b the slip planes must form a kink in orderto accommodate the deformation. Experiments in Fe-3% Si crystals [16] show resultsconsistent with kink-like shear. This kink requires dislocation dipoles as in Fig. 3d. Thesedipole loops, illustrated as pairs of dislocations, cannot be swept out from the crack tip and

('rack lip /ield in (hi,'iht, crtiNals 307

( a) (b) /

(C) ~(d) %

Fiz. 3. (a) Shear in band lying parallel to the active slip systems. (b) Shear by a kinking mode in a band lying

perpendicular to the active system. (c) Dislocations can be generated at the crack tip and swept out along slipplanes, or can be generated from internal sources, to produce the slip-plane-parallel shear band of (a). (d)Oislocations dipole loops must be nucleated from internal sources and expand as illustrated to produce the kinkingshear band of (h).

must therefore depend on the availability of internal sources to produce them. Requiring theavailability of internal sources, as opposed to allowing dislocations to be swept out from thecrack tip along slip planes, may influence whether the fracture of a particular crystal is brittleor ductile. It is seen also that the lattice between the pair of dislocations in Fig. 3d isapparently rotated. This can cause either a geometric hardening or softening of the materialbecause the resolved shear stress r"' on each slip system varies as the Schmid factor, i,.changes due to rotation of S and N directions. The effect of such lattice rotation will be seenlater in the large deformation results of Asaro [6].

3.4. Some.fur ther analyvses of mode III cracks

The analyses disLassed so far have been for tensile cracks, which is the physically moreinteresting case. However they are approximate in that their validity is only in the limit asr - 0. The simpler nature of the equations for mode II (anti-plane strain) type loadingallows for more complete solutions [1. 3] which we briefly summarize here. Three suchsolutions are shown in Fig. 4 exclusively for the fcc crystal with the orientation of the crackthe same as in Fig. 2a with the crack along the (0 10) plane with crack tip along the[l 01] direction. Under mode III loading conditions the only non-zero displacement isu1(x., x,) which gives rise to only two non-zero stress components a and a,.. In this caseonly the solid-line sip plane traces in Fig. 2b can provide anti-plane straining. The yieldsurface can thus be represented in o. a, stress space and for this orientation is diamond

308 J.R. Rice. D.E. Hawk and R.J. Asaro

032

B /B 031 B"

(a) so A

X B 4 displacement

discontinuity

0 32

A

(b JB C C' B' _0J3-

Bc 3 A

elocity4- discontinuity

032

B \ B' J31 0(c, A

V,,.. elastic-plasticshock

Fg. 4. Summary of analyses for anti-plane shear of a cracked ideally plastic crystal: (a) Stationary crack (b)Quasi-static crack growth; (c) Dynamic crack growth asymptotic field (inertial effects included).

shaped. The assembled stress sectors for these solutions will be labeled with A. B, etc. withcorresponding points or trajectories labeled on the yield surface. As is often the case, similarfeatures arc seen to carry over from the mode II solution to the mode I solution.

The stationary case is shown in Fig. 4a with the corresponding yield surface and stresstrajectories as indicated. Only two sectors A, B in the upper half plane (a correspondingsector to B labeled B' is shown in the lower half plane) of the constant stress type. aspreviously discussed for plane strain in the asymptotic limit as r --* 0, are necessary toconstruct the full solution. Both of the corresponding stress points are on the yield surface

(ra, A tip /ichls in hct i/c cr .tl/s 309

as indicated. These sectors are separated by a line of discontinuity in both displacement andstress. The stress discontinuity can be thought of as a rapid transition (in a vanishing smallsector) from point A to point B as indicated by the arrow on thc yield surface. The completesolution of Rice and Nikolic [ II shu,',s that these sectors are actually elastic with all plasticitycollapsed into a shear zone along the line of the displacement discontinuity. The size of thisplastic zone is found exactly in their analysis.

The quasi-static crack growth case is shown in Fig. 4b. Three sectors A. B' and C' in theupper half plane (with corresponding sectors B and C in the lower half plane) are necessaryto assemble the solution. Sector A is of the constant stress type with the state of stress at theyield surface as indicated on the yield surface plot. As necessary from the asymptotic solutionthis sector is bounded by an elastic sector C'. Sector A is separated from C' by a velocitydiscontinuity (expected to extend over a finite size region based on the approximate analysisof Rice and Nikolic [1]) along which all the plastic flow takes place in the region forward ofthe crack tip. As shown by the lightly shaded region in Fig. 4b, as the crack grows thisvelocity discontinuity also leaves behind a wake of plastic deformation. Sector C'. asindicated by the corresponding path within the yield surface. unloads elastically and reloadsto yield by the time it reaches the other plastic sector B' creating a thin wedge of plasticdeformation along the crack face. shown in darker shading. The exact details of theconstruction of the angles for these sectors is presented in [11.

Finally, the case of dynamic crack growth where the effects of inertia are taken intoaccount is shown in Fig. 4c [3]. In this case the entire field consists of constant stress sectorsseparated by an elastic-plastic shock (possible only in single-crystal-like materials with flatsegments along their yield locus). Both velocity and stress are discontinuous across theshock. Remarkably, the strain accumulated in crossing the shock is finite at the crack tip.It is of the order of the strain at first yield divided by the elastic Mach number associatedwith the crack speed. The angle 0,. in Fig. 4c. is proportional to the Mach number at lowspeed and approaches (ir - 0,,) 2 at the sonic speed. There must be a complicated transitionbetween the near tip field with the inertia included, Fig. 4c, and that without Fig. 4b. Thisis an example of nonuniform asymptotic limits as one considers r --- 0 and /Liv2,G -- 0where o is the density, G is the shear modulus and v is the crack speed. E.g.. the quasi-staticgrowth case may be considered as letting L -- 0. or v -- 0. before letting r -- 0.

4. Comparison of asymptotic and numerical analyses for mode 1

4.1. imall-scale-yielding resu/ts

Under small-scale-yielding conditions, a stationary and quasi-statically propagating planestrain mode I crack in a fcc crystal has been investigated by the finite element analysis ofHawk [6]. The material is elastic-ideally plastic. modeled by a visco-plastic formulation asin (5). with g"' = r,, and in = 0.005. Small-scale-yielding conditions exist when the size of theplastic zone is much smaller than the region over which the elastic singular stress fielddominates. The dominance of the elastic singularity allows the crack tip region to be modeledas an infinite solid with a semi-infinite crack where the stresses approach those of the elasticsingularity as r -- Y .The finite element mesh of both analyses is shown in Fig. 5. Since themode I problem is symmetric, only the tipper half plane is modeled. The finite element mesh

310 J.R. Rice, D.E. Hawk and R.J. Asaro

C

I---- ............... k-

Fhg. 5. Finite element mesh for "small displacement gradient" analysis of small-scale yielding in an ideally plasticcrystal.

is shown in three sections A, B and C for clarity. Section A fits into the rectangular spaceof section B and section B in turn fits into the space in section C. The ratio of the size of thesmallest rectangular element of region A to the outermost radius of region C is approximately101. Each quadrilateral element is actually a so called cross triangle element [17] made upof four constant-strain triangles (formed from the diagonals of the element) which as a groupbehave well under incompressible conditions. The crack opens to the left with nodes aheadof the crack tip constrained from vertical movement by the symmetry boundary condition.Tractions corresponding to the elastic K field singularity are applied to the outer boundaryof section C. The results are primarily from section A and the region immediately surround-ing it. The details of the finite element method are presented in [6].

The analysis of the stationary crack in an f cc crystal was performed with the orientationof the crack the same as shown in Fig. 2a. The crack lies in the (0 1 0) plane with thecrack tip along the [10 01 direction. Elasticity was modeled as isotropic. The yield surface

Crack lip .iehs in ductilh cr.xval. 311

0 stress discontinuities

(311 022N

70.50

(a) (b) D \ AB C 54.7'

0.030

0.015

(C)

0.000

-0.015 0.000 0.015

X/(K/o) 2

Fig. 6. Stationary crack in ideally plastic crystal of Fig. 2: (a) Yield surface, (h Asymptotic structure of crack tipfield; (c Finite element results: grey zone with F > 0.01 (r,, G) essentially denotes plastic zone; black zone withF > 10 (r, G) shows zone of more concentrated plastic strain.

(appropriate to sustained flow) for this orientation is shown in Fig. 6a. The asymptoticsolution obtained by Rice [2] for the stationary crack case is shown in Fig. 6b. In thestationary case, the entire crack tip region may deform plastically. For the upper half plane,the solution consists of four constant stress sectors labeled A, B. C, and D separated by stressand displacement discontinuities as indicated in Fig. 6b. The stress state of each sector issimilarly labeled as points on the yield surface A, B, C and D in Fig. 6a. The discontinuitybetween A and B at 54.70 and the one between C and D at 125.3' correspond to the slipdirections for the solid traces in Fig. 2b. The discontinuity between sectors B and Ccorresponds to the normal of the slip plane indicated by the dashed traces in Fig. 2b. Eachdiscontinuity in stress represents a rapid transition from point to point on the yield surfaceas shown by the arrows.

Let the sum over the slip systems of the magnitude of slip. _, ;' be denoted as F. Thisquantity is an overall measure of the amount of plastic straining and is shown in Fig. 6c

312 J.R. Rice. D.E. Hawk and R.J. Asaro

normalized by the elastic strain at yield, r,G. The grey zone, formed by shading individualtriangular elements whose value of F/(ziG) is between 0.01 and 10, is representative of theoverall plastic zone shape. The black bands are those elements whose value of V is greaterthan 10. Therefore. the most intense straining is along the predicted discontinuities of theasymptotic solution. It is interesting to note that all the sectors are predicted to be stressedto yield as r -, 0, but only the interfaces between them arc proven to be deforming plasticallyin the asymptotics. The finite element solution indicates that sector D bounding the crackface is relatively free of plastic deformation while the others deform. Some plastic strain

0.030-

0.015

(a)

0.000 _

-0.015 0.000 0.015

X/(K/t 0 )2

0.030-

fi '' " I ''II-.-

_ ,, , ........ ..1 1I Ill ......

Ill Ill Ill Il I. . . . .., e,,,IIIIIIIIIIIlllltt~,*......IIIIII llI#lIIIII ~ t~g . . ..... .

1111111l1llllltl*........

(b) .........

IIIIII III I: ,, .... ....OAXO I I t 111 l ll ,i.... . .

0.000 . . .,. .... ;.... I ' ' I

-0.015 0.000 0.015

X/(K/To)2

Fig. 7. (a) Deformed mesh for stationary crack (factor of 50 amplification of displacements) note shear zonescoincident with directions shown in Fig. 6(b). (b) Line lengths proportional to displacements accumulated in aload increment; suggests that displacement discontinuities predicted by asymptotic analysis extend well into plasticzone.

('rack tip fic/dA ilhi /ctd/c cri-Stel.S 3131

ahlead of the crack Occurs on slip systems other than those of' Fig. 2b. i.e.. onl s\ stemns % hichcannot undergo sustained plane flow. The deformation of the finite element me1Sh1 aroun1d thlecrack tip is showNn in Fig. 7a % ith thle displacements magnified by a factor of 50. The mostintense deformation is inl those elements which lie along the discontinuity between sectors Aand B. The increment of displacemlent. scaled up for visibility, over a load increment is showniin Fig. 7b. Four distinct sectors exisi with thle motion in each sector firl\ uniform0-11. Thisindicates the majority of' deformiation is caused b\ the nmemlent of' nodes near thle predicteddiscontinuities.

A quasi-statically propagating crack in a fcc crystal is slimulated using at node releasetechnique the details of'kwhich are given in [6]. Thle crack propagates to thle right t Iirough21 9elements under constant load from an initial stationary position to the Centel- of section Ain Fig. 5. The same orientation of thle crack with respect to the crystal is Used Inl theCpropagating crack case as in the stationary crack case aboxe. The \ield surfatce f'or thisorientation is repeated in Fig. Xa for reference. As discussed In the prev[is section onl

0,~ Velocity discontinuities

/ elasticG - elastic B

Bk\ 70.5 ' A Plastic

(b) Plastic D 12.7- 54.70

(a)

0.060

S 0.030-

(C)

0.000

-0.030 0.000 0.030

X/(K/'r0 ) 2

: .s. Qimsi-,taticajfx gro~~ ing crack in ideally plastic cr,,stal or Fig. 2: (l) Yield Surtlace and stress lrajctors. NAs m plotic structure of near tip field. (c) Finite element results showing plaistic activtx . as measured b\ incrementor, F. during one finite element step or crack grow~th. (ire% zone corresp 'nds essentiall s\ it h plasticil\ 'Ictis /o11Cduring groskt black /one hats more concentrated plas:ic straining.

314 J.R. Rice. D.E. Hawk and R.J. Twvaro

asymptotic solutions, neither stress nor displacement dis.ontinuities may exist and if a sectoris plastic it must border an elastic sector. The assembly of sectors for this orientation [2] isshown in Fig. 8b. For the Lipper half plane. the solution again consists ol tour sectors labeledA. B. C. and D. Sector A is a constant stress plastic sector and it is separated from an elasticsector B by a discontinuitN in velocit\ at 54.7 . As indicated by the stress trajectory in Fi. 8a.sector B involkes an elastic unloading and reloading back to yield by the time a second\elocith discontinuity is reached at 125.3 ° . Sector C is another elastic unloading and reload-ig sector. Finally, sector L) is plastic.

The crack leaxes behind a wake of plastic deformation wAhich makes the quantit\F not as illustrative as it had been in the stationar\ crack case. The increment in F (r,, (= AF (z,, (i ) from just before the last node release to just after it. is shown in Fig. 8c. which

eliminates much of the accumulated plastic wake and shows only the regions k hich areplasticall. active during groth. Elements which sustained AF (,, G ) between 0.001 to I are.shaded gra and those with increments greater than I are in shaded black. There are verystrikinu similarities betwkeen the predicted asymptotic solution. Fig. 8b. and the tiniteelement solution. Fig. 8c. Sector A is plastic and is bounded by an intense band of plasticdeformation along the predicted velocit, discontinuity. Sector B is relatively free of plasticdel'ormation as predicted. At the second \elocit\ discontinuity at 125.30 where the pre-dicted stress trajector,, is tangent to the \ield surface an intense band of plastic defor-mation is encountered. Subsequently. an elastic sector roughly similar in angular extentto ector C is scen in the finite element solution. Finally. elements along the crackf'ace, are loading plasticall.. The deformed mesh after the crack growth has takenplace is shown in Fig. 9a. It is interesting to note that while the crack w\as stationar\ (as loadwas increased) the solutiun was the same as in Fig. 7a. as shown by the distinct kink in thecrack prolile at the crack's initial position. Howvever. once crack grow*th began and adisplacement discontinuit, %%as no longer allowed the crack profile wxas smooth. The incre-ment in nodal displacements. again scaled up for visibility. from just before the final crackgrow th increment to jut af'ter is shown in Fig. 9b. Two rather distinct bands ofdiscontinuitvare seen close to the crack tip corresponding to the lines of the velocit discontinuities. The,,mooth change of direction of the increments directl, above the crack clcarl\ indicates thatthe discontMuit 'seen at 90' in the stationary case is no longer present in the growing crackcase.

4.2. (Ctiter ( ra( k pancl rexult.%

A square. center cracked panel under plane strain conditions has been modcei from,mll-scale-. ielding to gceneral iclding in the finite element analyses of Asaro [6]. Threesections of the linite element mesh are shon in Fic. 10. Section A is imbedded in the densepart of section B M hich is in turn imbedded in the densest part of section C. Section C is thenextended out,%,ards in a similar pattern to slightly more than 20 times the size sho%%n. withthe outer boundaries of succcssive 'rincs" of elements forming t%%o adjacent sides of asuccession ol squares. S metr, rCducs th1e dilll y'I 1Jw, inIt l" (1e quarter of the speci-men. represented b\ section C as extended to a large square N ith sides of dinension b. Thetotal lencth of the crack is 2a w ith the ratio of a h equal to 0.01. The ratio of the si/c ofelements in section A to the size of the elements in the outermost mesh is approximatel\2 , 10 ". The crack tip is initiallk rounded as wxe can see from the mesh ;n section A. Stresses

)-at -A til fields ill ductl 3/ u si/ 315

0.060-

-0,030-

0.000-

-0.030 0.000 0.030

XI(K/T0 )2

0.060

S 0.030-

... .. . . .

....

..............

-0.03 0 0.000 0.030

XI(K/T0 )2

fwn. 1). [ Defo rmed inesh (I fdctor of MO am ,nplification) ater several steps of c rack K rokt h at constadn t lu r- lield,Iress intenslt K,. (h) Line lengths proportionali to displacmrent increments (Wring' cramck gross h oser one finiteclemlent step of crack growth. Suggests that %elocitN discontinuities predicmcd hN as,,nptotic zmtiil~sis C\tend %% ellinto the plastic /one.

corresponding to Uniform tension (;i' perpendicular to the craick are applied to the outerboundary.

[miite rotations of' both material and the crystal lattice are taken into account in this%kork. The computational procedures used in the finite clemnent calculations t'ollowN the initial" ork of' [ 141 and [181. The elements used are the same cross triangle type [17] discussed inconjunction with the small scale-vielding analysis. Thle plasticity is modeled by a visco-plasticformulation wAith the slip rate on each systemn as givyen by (5) with ti = 0.005. A Taylor type

316 J.R. Rice, D.E. Hawk anl R.J. A.aro

C

17

, , ,1 1 .-

Fig. 10. Finite element mesh for analysis of tensile load of a panel containing a center crack w ith initially roundedtip. Used for cr,stal models with double slip, as in insets of Figs. I Ila) and (b). Strain hardening and arbitrar\displacement gradients are included in the analysis.

of hardening of slip systems is included with g' a function of F = I,

g'" = g,[I + 0.8 tanh (11.1 F)] (8)

where g,/'G = 0.0026.The crystal is idealized by a planar double slip model for two orientations of the crack as

shown by the insets of Fig. II a and II b. The first orientation corresponds to that in Fig. 2b

Crack tip fiels" inilutlih, crystals 317

4,-35.3"

(a)

(b)

F 11 I. Contours of equivalent shear strain r near the tip, outermost is r 0.005, next is r 0.01 andinnermost is F = 0.02: (a) For 35.3' angle between slip planes and tensile directionm (b) For 54.70 angle.

for a f cc crystal, i.e., a (0 1 0) crack growing in the [ 10 11 direction, when we neglect the slipplane traces shown as the dashed lines. It also corresponds to a b cc crystal with the crackon the (1 0 1) plane growing in the [0 1 0] direction when we similarly neglect certain slip planetraces parallel to the crack. The angle between the x-, axis and the traces of the slip directionS, denoted by 4). is equal to 35.3' in this case. The second orientation has 4) = 54.7'. Itcorresponds. with neglect of certain systems as above, to a crack lying on the (1 0 1) planegrowing in the direction [0 10 Oin a fcc crystal. or to a crack on the (0 1 0) plane growing inthe [1 0 1] direction in a bcc crystal. See [2] for fuller discussion of yield surfaces in thesecases. These orientations with only the slip systems indicated correspond to a diamondshaped yield surface. e.g.. as in Figs. 6a and 8a but without the horizontal cut-offs. Anidentical diamond-shaped surface applies for the two cases shown in Fig. I I a and b. The fc ccrack orientation coinciding with Fig. I I a is sometimes observed in fatigue studies on ductileCu and Al crystals. whereas the b cc crack orientation noted to coincide with Fig. I l b is acommon cleavage crack orientation. e.g., in Fe-3% Si crystals [161. The term g,, in (8) is thecritical resolved shear stress in the vl. .%, plane of deformation for the double slip model. Itcorresponds to g, = (21J/,,/~z for the fcc interretations and to g,, = r, for bce [2].

318 J.R. Rice. D.E. Hawk and R.J. Asaro

Contours of constant plastic strain (as measured by the accumulated sum of the slips.F = 1, 1'' j) are shown in Figs. I Ia and I lb for the orientations o = 353o and

Crack ip fichA~ in dictile crv SIUl% 319

(0-35.3-

(b)

(C) 0.000

-2.0000

7-S.0000

-7. 5@00z

-1 .0 0

15.0000y~

20.0000

F~g 1. R taedorinttios f lipplnetraes(frm neof hetwoslp ystmsinthedobleslp ode) ea

th rc i:() o 53 nl etenpae n eniedrci hwsltl ~dneofltiertto

exep narblntngti. b)Fo 5.7 agl: ho s igifcat atic rtaio af i te inin sea zneo

Fi. (b. c Asofo 4.' nge sow cnous fcntat atic ottin er heti.4fNhchte4utrm s

corepod to ' lokwse

320 J.R. Rice. D.E. Hawk and R.J. Asaro

5. Conclusion

The analytical and numerical investigations into the near-crack-tip fields of ductile crystalshave been shown to be consistent. The results show that for ideally plastic crystals obeyingthe Schmid rule, the near-crack-tip deformation fields are characterized by discontinuities ineither displacement (stationary crack) or velocity (quasi-statically growing crack). Thesediscontinuities are either parallel or normal to crystallographic planes on which slip can takeplace (i.e. by the motion of dislocations) for the fcc and bcc orientations considered. Thestate of stress is constant within angular sectors bounded by these discontinuities for materialat yield.

The asymptotic methods developed [1, 2] provide a general way of constructing such fieldsaround the crack once the yield surface has been determined for the particular crystal andorientation of the crack. In the case of a mode Ill crack [1. 31 a more complete analysis ispossible of stationary cracks (exact full analysis) and quasi-static crack growth (asymptoticplus approximate analysis) and even dynamic crack growth (asymptotic). The generalfeatures seen in the mode Ill cases carry over to the physically more interesting mode I planestrain asymptotic analyses [2].

The numerical investigation of Hawk [6] by detailed finite element analyses providedconfirmation that the features seen in the mode I asymptotic analyses (stationary andquasi-static growth) are valid over a finite size region. The finite element analysis of Asaro[6] of a center crack with initially rounded tip in a tension panel from small-scale yieldingto general yielding, including hardening and full account of arbitrary displacementgradients, shows the effects of lattice rotation and the limits of small-scale yielding. However,the overall feature of concentrated deformation along the predicted discontinuities isretained throughout.

Finally, the analyses summarized here may provide some insight into why certaincrystals undergo ductile fracture while others are brittle. One possible factor is the struc-ture of the dislocations necessary to produce the predicted slip patterns which in onecase involves a shear band parallel to the active slip system and in another involvesa band perpendicular to the active system that deforms by kinking shear. In the parallelshear case it is possible for these dislocations to be generated at the crack tip and sweptout along the slip planes while in the kinking case it is necessary to have present, and toactivate, internal sources to generate the necessary dislocation dipoles. The finite strainpattern associated with large ductile opening at the crack tip is also significantly different inthe two cases.

Acknowledgements

J.R. Rice and D.E. Hawk acknowledge the support of the Office of Naval Research undercontract N00014-85-K-0045 to Harvard University. The computations of D.E. Hawk werecarried out at the John von Neumann Center, Princeton N.J., under grant NAC-519supported by the U.S. National Science Foundation. R.J. Asaro also acknowledges thesupport of the Office of Naval Research under contract N00014-88-K-01 19 to BrownUniversity. His computations were done during a June 1987 visit to Sandia NationalLaboratory, Liveri.iore, Calif.

(rm k ip l h ill d/uile 1cr-t.ubl 3 21

References

I ,J.R . Rice anid R. Nik oltit toritrnof /Uc. i( etiltm Niand PhlI 'it ,t of Solids 33 ( 985) 59 -622.2. 1 R Rice. AItOinit.N o atc ~hrja/t 6 (9S7) 317 335.3. R . N ik olic andl J. R . Rice. .%lIct(hii tif Aftitcritat 7 098,' 163 173.4. J .R . Rice anld M . SaICed\,l 'I. .1itlot-nattt i/fi' t, t ehii N an / P/i i t/ 5 of iii 36 ( 19"'X) $9 2 4.i. M. Saeedx a 'a a nil J-R . Rice. it itoil tf te AA hia \~ti al tn,/ P/ti N ttf .So/it/t (19'S9) ill pre '.6. 1D. E. I!a K anil R .J. Asaro. \\tork in progre,;ss, onl finit e eletment a ha I \ Si orcracWked Cleaqti Ic LIt ii C r A I

(19M).7-.1 l. lsr anil C.F. larn. Prttccctding\ t tl i/ic, R taln St tti jt t/ ttii/ttti :X 102 (1913 14 4.8. GA. Tia), or anil C . H1amn. Pittcvm o tt/h/e Rtt aol ttciti tt oft Lnd/ttn A 11)8 1 925;)2$9. G. 1.Fa\IoW. i .St'ft/ici Tjmttluicn to 00th .Initiittrt )-if i lo it. \1actiillan. Nc\\ Yttrk ( 1918) 21$ 2214.

10). R .J. Asa to, in .4 i/root-c ill Apptied't 11(i/talic.st J. W . ti I tChiltt Mt ledi. .. 23) 1 19S.3 I - I115.11. J. R . Rice. Joturnal,/ it /c 11ch/i,s and /i It ll t/ .Sotjld 2 1 ( 19731) 63 74.

I 2..i.W. Hi utei inson. Proccui't tt/g, o ic Rtt it Sttt iwo of t Lono ttittt ( 301 976) 101- 1 I 7-113. J . Pan anil J. R. Rice. hiitrio iat/.tio oil o/ S "tiid aitid Sfiteturtc 1 9 (19S 31 97.1-987.14. 1). Pierce. RI. Asaro. aid A. Needleman. A(cf ta It'iali//i a 31 (1 9-3) 195 1 1976.1iS. WIJ. 1)rUL'a anid J. R. Rice. in Afcc/ianii.\ (if Materrial Bchorittr. (i.I. D\ tirak anid RIT. Shiielid 4 es. ). 1lse\ icr,

A\msterdam (19X4) 59.73.1 6. AS. Tetelna ma iad \\.D1. Rober --on.to a lcui//ttreita I I ( 1963) 41 S -42_ (1 7. J.C. N a utecaal. I). I. Parks, a ndl I. R Rice. ( Comtitr ciitid ti/. i .lidt/icc/ a/ nii stiil Etinhct'ritii 4 ( I1974)

153- 177.

M$ R .J. A. iro and A. Needlenian. Ati,' Alcit,/hot.ito 33 (19,15) 921.95119. .1. R. R ice. in loi~t, Cra( k Prttitoiioi. Fl NTA TP 4 /5 ( 1967) 247-109.2)). J .R . Rice .It onal tf t/ /c Mchc/Iuoi all toO/ S ic t/ .Sotlids 2(1974) 1 7- 20.

Reprinted from December 1989, Vol. 11, Journal of Applied Mechanics

Elastic-Plastic Analysis of Crackson Bimaterial Interfaces: Part II-Structure of Small-Scale YieldingFieldsIn Part I we found that although the near tip fields of cracks on bimaterial interfacesdo not have a separable form of the HRR type, they appear to be nearly separablein an annular zone within the plastic zone. Furthermore, the fields bear strongsimilarities to mixed mode HRR fields for homogeneous medium. Based on our

C. F. Shih numerical results, we have been able to identify a clear mathematical structure. Wefound that the small-scale yielding crack tip fields are members of a family param-eterized by a near tip phase angle t, and that the fields nearly scale with the value

R. J. Asaro of the J-integral. In Part II, the original derivation of the mathematical structureof the small-scale yielding fields is elaborated upon. The issue of crack face contact

Division of Engineering, is addressed and the phenomenology is described in terms of the phase parameterBrown University, t. Crack tip plastic deformation results in an open crack for a range of which is

Providence, R. I. 02912 nearly symmetric about the state corresponding to pure remote tension. Plane-strain

plastic zones and crack tip fields for the complete range of t are presented. Overdistances comparable to the size of the dominant plastic zone, the stress levels thatcan be achieved are limited by the yield stress of the weaker (lower yield strength)material. On the other hand, the stresses well within the plastic zone are governedby the strain-hardening behavior of the more plastically compliant (lower strain-hardening) material. We observe that the extent of the annular zone where the fieldsare nearly separable (i.e., of the HRR form) is dependent on the remote loadcombinations and the material combination. When the tractions on the interfaceare predominantly tensile, there are no indications of crack face contact over anylength scale of physical relevance. Instead, the crack tip opens smoothly and cracktip fields as well as the crack opening displacement are scaled by the J-integral. Thepaper concludes with a discussion on the range of load combinations which couldbe applied to two fracture test specimen geometries to obtain valid fracture toughnessdata.

1 IntroductionIn Part I of this article (Shih and Asaro, 1988), numerical facilitated making the connection between the interface crack

solutions were presented for the elastic-plastic fields of a crack solutions, the existing framework for nonlinear fracture me-on the interface between a nonlinear power-law hardening chanics, and specific solutions for crack tip fields in homo-material and a rigid substrate. Specifically, the problem of an geneous media (e.g., Hutchinson, 1983). In one sense the workinfinite crack embedded in an infinite bimaterial body (see Fig. complimented analyses such as that of Knowles and Sternberg1) subject to combinations of remote tension and shear was (1983) who studied the behavior of an interface crack betweenanalyzed under loading conditions that caused small-scale two neo-Hookean sheets. In these analytic asymptotic solu-yielding. The calculations were performed for a material de- tions, as well as the numerical solutions of Part I, it was shownscribed by a small strain, isotropic J2 deformation theory. This that pathological features of the linear elastic solutions, such

as the near tip oscillations in stresses and displacements are

Contributed by the Applied Mechanics Division of THhi AMERICAN SOCIETY strongly mitigated by nonlinear kinematics or material behav-OF Ma cHAmncL ENornamns for publication in the JOURNAL OF APPLIED MIE- ior. The linear elastic asymptotic solutions, which providedc nSth the point of departure for our numerical analyses, were dis-

Discussion on this paper should be addressed to the Editorial Department, cussed in Part I.ASME. United Engineering Center, 345 East 47th Street. New York, N. Y. The numerical analyses described in Part I also provided10017. and will be accepted until two months after final publication of the paperitself in the JOURNAL OF APPLIED MEcHAmcs. Manuscript received by ASME clarification of the structure of the small-scale yielding fields.Applied Mechanics Division, April 25, 1988; final revision, February 1, 1989. In particular, it was found that the interface crack small-scale

Journal of Applied Mechanics DECEMBER 1989, Vol. 56/763

has reexamined elastic fracture mechanics concepts for inter-'y . ,face cracks.

A crack of total length L lying on the interface between twosemi-infinite slabs of isotropic elastic solids with differing ma-

® terial moduli is shown in Fig. 1. The shear moduli and Poisson'sratios are ul, A2, and "i, v2, and r and 0 are polar coordinatescentered at the crack tip. There are three independent non-

material i dimensional material parameters but, as has been shown byDundurs (1969), the solution for this class of problems depends

r0 on only two parameters. In plane strain they aree

--- I x 1 = [All- P2)-,1420 - V)]/[lA1l - V2)+/A2(I - VIA] (1)

E ; mr[ia(l 2 2v 2) -1 2(1 -2l)]/[,tI(l - V2) +/A2(1 - V1). (2)b 7,v2 moteriol 2

04 These parameters vanish for identical materials across the in-terface and change sign when the materials are interchanged.

At small distances from the crack tip, the in-plane stressesGO have the singular form

T -Yyy

Fi.ICako itaeilItraeaj =Re L r ii'40;() (3)

where i = N/1, Q is the complex stress intensity factor,yielding fields are members of a family parameterized by a I (r/LL)el = 1, and 6;E) is the universal complex dimen-near-tip phase angle, t, defined as = 46 + E In (QQ/o2L). sionless angular function which depends on the bimaterial con-Q is a complex stress intensity factor defined by the linearelastic asymptotic solution, 0 is its phase angle, L is a char-acteristic crack dimension, ao is the yield stress of the weaker 1 In (I - 12(material, and e is a elastic bimaterial constant. We found that 2 = \l 1 -" (4)the dependence of the crack tip fields on radial distance isnearly of the HRR form for asymptotic fields in homogeneous In (3), L is a characteristic dimension of the crack geometry.media (Hutchinson, 1968; Rice and Rosengren, 1968) and that As an example, L is identified as the crack length of the ge-the small-scale yielding fields nearly scale with the value of the ometry depicted in Fig. 1. It is convenient to write Q asJ-integral (Rice, 1968). This structure is reviewed in Section 2 Q= Ql + iQ2 = I Q I eiO (5)of this paper, where IQI and 4) are the amplitude and phase of Q. Thus,

In the present paper we describe the small-scale yielding tractions on the bond line arefields in more detail and also demonstrate how the pheno-menology of crack face contact can be described in terms of tQ (mi_ e,o+ In(r/L)) (6)E. In the absence of plastic deformation, the range of for t= (7y iy)T~o (6)

which the crack is open is skewed about the state correspondingto pure remote tension. In contrast, crack tip plastic defor- and displacement jumps across crack faces take the formmation results in an open crack range, which is more nearly Au=Auy+iAu = (uy+iux)o-=- (uy+iu _ _symmetrical about the state corresponding to pure remote ten-sion. We make contact with a dimensional analysis by Rice IQI r ln(,/L ((1988) and a subsequent study by Zywicz and Parks (1988), - 8A (7)which contain results which are consistent with our general = ( 742) - Ir(framework. framework.where O = tan - l2e, and for plane strain A = [(I - pl)/,ul

The plan of the paper is as follows. In the next section thestructure of the small-scale yielding fields is described. Nu- + 0 - P2 )/ 1121/(4cosh

2,).

merical methods and the finite element boundary problems are Different normalizations of the singular crack tip fields whichdescribed in Section 3. Results of plane-strain crack tip fields result in stress intensity factors that differ by phase angles andfor two material combinations involving an elastic-plastic solid scaling constants involving e have appeared in the literaturebonded onto an elastic substrate and, secondly, to an elastic- on interface cracks. Rice (1988) introduced the stress intensityplastic material with different material properties, are pre- factor K which is related to Q bysented in Section 4. The range of validity of linear elasticityand small-scale yielding solutions as they pertain to two frac- K=QL-l, IKI = IQI, KK=QQ and ,p=4-e In L (8)ture test specimens are discussed in Section 5. where p is the phase of K. It is clear from (3) that K or QL- i2 Structure of Fiells Under Small-Scale Yielding uniquely characterizes the crack tip fields. In other words, K

2.1 Linear Elasticity Solutions. Solutions to specific prob- fully accounts for the effects of both load and geometry onlems of cracks lying along bimaterial interfaces of isotropic the crack tip field whereas Q and a characteristic length L arerequired for the same purpose. The merits of various defini-media have been given by Cherepanov (1962), England (1965), tions of stress intensity factors and possible approaches forErdogan (i963), and Rice and Silt (1965). More recently, Ting recording and using fracture toughness data have been dis-(1986) has prese.ted a framework for determining the degree cussed by Rice (1988).of singularity and the nature of the asymptotic fields for the For the geometry depicted in Fig. I and stressed by remotegeneral interfacial crack between two elastic anisotropic ma-terials. Park and Earnme (1986), Hutchinson, Mear, and Rice traction T(T = I TIe = a- +i), Q = (I + i2)TNi -2 (Rice(1987), and Suo and Hutchinson (1988) have obtained solutions and Sih, 1965). The connection between the phase of Q andfor several elastic interfacial crack problems, and Rice (1988) T is 4 = 0 + tan- 2f. Thus, changes in crack length at fixed

7641Vol. 56, DECEMBER 1989 Transactions of the ASME

( IT I is free to vary) do not change the phase of Q. For later Directing our attention to the dependence on o.j on Q, Luse, we note that , = - ElnL + d/+ tan - '2E for the geometry of and distance from the crack tip we writeFig. 1.

A relation which will be of use later in the analysis is the or2°o (-1 00fIenergy release rate for the crack advancing along the interface 'ii = U04 - + n(Malyshev and Salganik, 1965; Willis, 1971)

where i0 + c In(r/L) is the phase of Q(r/L)". To reveal theg = AQQ = AKK (9) structure offj, it is advantageous to express the third argument

where A is the parameter defined for (7). as the sum of a constant phase angle and a variable phaseThe linear elasticity solution for the displacement jumps (7) angle which depends on distance r. To this end, the third

predicts that overlapping of crack faces always occurs, though argument is modified by incorporating a dependence on thethe zone of overlapping crack faces is very small compared to first argument, viz.,L for load states in the range - 45 deg 5 _5 45 deg, i.e., [ L / \ ] = , InQ, > I Q21. To redress this physically objectionable behavior, 0 + E In[ =(f+) (ininvestigators have proposed various models and approaches. L\L o2 LComninou (1977a, b) and Comninou and Schmueser (1978)reformulated the linear elasticity boundary value problem to + (m- 1)f I (Q.) (12)allow a zone of contact to develop at the crack tip. Solutionsfor a range of remote load combinations were obtained.Achenbach et al. (1979) introduced a Dugdale-Barenblatt strip We should point out that the phase can depend on In r in ayield zone at the crack tip which eliminated crack face over- manner which is more complex than that suggested by the lastlapping as well as stress singularities altogether. Ortiz and term in (12). In Part I we introduced a near-tip phase param-Blume (1988) have obtained an inner solution, based on a zone eter, Z, defined byof decohesion (and sliding) at the crack tip, which does not/lead to interpenetrating crack faces in the region of dominance = =+f In -iQ- (13)of this inner solution. Along similar lines of inquiry, Needle- \Lav/man (1987) has implemented a decohesion model in a finite varies linearly with 0, depends weakly on QQ and L, andelement study of (rigid) inclusion debonding which takes ac- does not involve r. Making use of (13), the right-hand side ofcount of finite geometry changes. (12) can be written as

We take the view that while the elasticity solutions are invalidon the scale of the contact zone, they still provide an accurate + phase/-r ()14)description of the fields in an annular region surrounding the SLKQQ ) 3 (14)contact zone, as long as the size of the contact zone and plasticzone (if any) is small compared to the zone of dominance of where E is a dimensionless combination of material parametersthe elastic fields given by (3). In this sense, crack face contact yet to be determined. We use the latter result to restate (11)on a size scale which is much smaller than the crack length (or asthe relevant crack dimension) can be treated as a small-scale /ra C/ro~\ ' )nonlinearity. Along this line of argument, Atkinson (1982) had o%=a0fiJ1--, 0, phase $.-=Q ; . (15)used (3) as the outer solution and matched asymptotic expan- QQ /sions to derive an analytic contact zone solution for the For the small-scale yielding analysis, it is convenient to phraseComninou (1977a) model. We will return to the issue of crack the remote load in terms of I KI (or KK/a ) and its phase ,P.face contact following the discussion on the structure of the Use (8) in (13) to obtainsmall-scale yielding fields. /KKX

2.2 Small-Scale Yielding Formulation. In the small-scale =p+e In-). (16)yielding formulation, the actual crack problem is replaced bya semi-infinite crack in an infinite media with the asymptotic The characteristic crack dimension, L, does not appear ex-boundary condition that at large r the field approaches that plicitly in the equation (16) for , but has been absorbed ingiven by (3). The original derivation of the results to be dis- the phase of K. Now, the general result in (15) has the alter-cussed here is contained in an earlier publication (Shih and native representation:Asaro, 1988). It is elaborated upon here and contact is made I ra.2with subsequent studies (and alternative approaches) which aij=Ooj -- , 0, phase ON ; t (17)lead to results which are in agreement with our main conclu- KKsions. Since is the phase angle of a complex quantity, f.j has a

Let ao, and 02 be the yield strengths of materials I and 2. periodicity of 21r with re3pect to the argument , i.e.,It is convenient to designate the yield strength of the weaker f/,( ...... =f/ ...... ; t + mir) m = 2,4,6. (18)solid by a0, i.e., 0o = min (a01, 02). Within the boundary layerformulation, the stresses depend on QL - ' and Oo, and on Furthermore, due to the linearity of the equilibrium and strain-dimensionless material parameters, e.g., 90o/ao, /1, 02, etc. By displacement equationsdimensional analysis, fjA ...... ; Z)= -f/,( ...... ;+mr) m= 1,3,5. (19)

r O (r) Thus, t serves as the phase parameter of the fields in the small-au= aafij L-,0phsQ ' L) scale yielding formulation just as p (or ', - eInL) is the phase

angle of the linear elastic singular fields. Using a differentdimensionless material parameters . (10) approach, Zywicz and Parks (1988) introduced a phase pa-

rameter ' which differs inconsequentially from (16) by a con-Here, f, is a dimensionless function of the dimensionless ar- stant involving E.guments and it may be noted that Q0 has dimensions of square The mathematical structure of the fields expressed by (15),of stress times length. Henceforth we will assume that the and (17)-(19) was derived without regard to contact betweenimplicit dependence of f4 on material parameters is under- the crack faces. In fact, displacement jumps across the crackstood. faces must have the form,

Journal of Applied Mechanics DECEMBER 1989, Vol. 561765

KK ro3 U31 will be taken to be the lower hardening material and its elasticAu, = -0 9

g KK , phase\ / );, (20) and plastic properties are designated by P, E, C1, 0, (0, and n.Let E, and E12 designate the tangent moduli of material Iwhere ,0 is a reference yield strain and the dimensionless func- and 2, respectively. As r - 0, and assuming that the stressestion gi has the periodic structure expressed by (18) and (19). are singular, EI/E,2 - 0. This suggests that, as r - 0, theIn this paper our interest is restricted to the range of where material system behaves increasingly like that of a plasticallythe crack faces are not in contact. The definition of an open deforming material which is bonded to a rigid substrate. Wecrack will be introduced shortly and the validity of the proposed will now argue that the form of the asymptotic fields is gov-fields will be established in Section 4 in terms of the range of erned by the strain-hardening characteristics of material 1.. For the present, it is useful to examine the mathematical We assume that the response of the plastically deforming

structure of the fields under the fictitious assumption that crack material is described by a J deformation theory of plasticity.faces may overlap. Path independence of the J-integral (Rice, 1968) is easily dem-

The effective stress a ( = 3s~sj/2, where sj is the deviatoric onstrated for the interface crack once it is recognized that thestress), has the form contribution to the J-integral from the upper interface, x, =

rc a: 0', is negated by the contribution from the lower interface,=f 0, phase . (21) x 2 = 0-. Now, take a circular contour centered at the tip andoe=oofe( , h) ) use path independence to shrink the contour onto the crack

The elastic-plastic boundary in the weaker material is the locus tip. Since J is nonzero, as it must be for a deformation theoryof points where o equals co. Substituting these values in (21) solid (in this case the J-integral is precisely the energy releaseand rearranging leads immediately to the following results for rate), we conclude that the strain energy density W - 0(l/r)the plastic zone (Shih and Asaro, 1988): as r - 0. This requires the product of stress and strain to have

KK a lr singularity. If it is further assumed that the fields haverp(O) = Ko R(O;Z) . (22) a product dependence on the first argument of fij (17), then

singular solutions, if they exist, would have the formHere, R(O; Z) is a dimensionless angular function which de- - (_ p((n + 1)pends on Z and on dimensionless material parameters. Since % =

O (!0k0, phase IP I ; 0 (25)0 e is quadratic in the stress components which have periodicityexpressed by (18) and (19), the angular function R has a pe- where i = r/(KK/o), and the implicit dependence ofh onriodicity of 7r with respect to , material parameters is understood.

1,2,3.............(23) The dimensionless function h, has a periodic dependence;=R(O; + ) = 1,2,(with period 27r) on , and the arguments leading to (25) require

Rice (1988) has derived the relation, that Ri be bounded. Though hij is a bounded function, it canKK ~ nevertheless oscillate rapi', as f - 0. Under small-scale yield-

rp = -a2 R (O-c ln(L/rp) ) (24) ing, J g (Rice, 1968) and noting the relation between q and0o KK (9), the singular fields can be arranged in the form

by a similar dimensional analysis. In (24), rp is a characteristic / , /("+ I)dimension of the plastic zone; for example, rp can be taken to a0, = 0 ) hifO, phase I R 'Y ; ) • (26)be the maximum extent of the plastic zone. Rice's implicitequation for rpcan be made explicit by replacing rp in the where hi, = (t/AE)"1 fi '1hj and A has been defined in con-argument by KK/a, whereupon a result similar to (22) is re- nection with (7). It may be noted that (26) is consistent withcovered, the product form of the linear elastic fields (3) involving a

The function R(O; Z) can be determined directly from plots bounded hij which depends on the phase I (r/L)" 1.of small-scale yielding plastic zones. This has been carried out It is instructive to examine the dependence of hoj on r forfor a range of phase angles t in Part I (Shih and Asaro, 1988) two special cases. Suppose that both materials are elastic andand for several material systems in this paper. Alternatively, d 0. At fixed 0, the stresses have the form oij Ct r-l/2hq(F;an estimate of R(O; t) can be obtained by approximatingf. in t, n). Comparing this form with (3), we conclude that the(21) by using the linear elasticity fields in (3). The latter pro- variation of h,, with r is bounded but rapidly oscillatory as rcedure has been adopted by Zywicz and Parks (1988) to es- - 0. Now suppose that one material is rigid-perfectly plastic.timate plastic zone sizes and shapes. Then, hi, is again bounded and the term multiplying it,

r I1(n+ 1) (n - c), is also bounded.As a third example, we take materials I and 2 to be strain-

2.3 Plausihle Form of Asymptotic Fields. The size of the hardening materials with n, > n2. Then by (26). the stressesdominant plastic zone (which develops in the weaker material) above the bond line (0 = 0') are of the form ai -is controlled by the yield strength of the weaker material 00 r 1', + '1h,j (F; n,). For the moment, suppose that the stresses(see (22)). Over length scales which are comparable to the below the bond line (0 = 0 -)have the form , - r-1102,+ hlhdominant plastic zone, the stress levels that can be achieved (F; n2). Continuity of traction across the bond line requiresin both materials are set by 0. On the other hand, we will that r - "l + 'lh2,4F; n,) = r 1 " ' 'h#; n2) = O(rX). For n,argue that the form of the asymptotic crack tip fields is gov- 4 n2, the equality can be met only if h2, is singular as r - 0,erned by the hardening characteristics of the material with the which contradicts the original assumption that h, is bounded.least hardening capacity. In fact, if the stresses are assumed to have a power dependence

We assume that each material of the material pair is char- on r such that traction is continuous across the bond line, i.e.,acterized by a Ramberg-Osgood stress-strain relation where a,, - r1h, for both materials 1 and 2, then continuity of dis-the uniaxial plastic strain is related to the stress by e"/o = placements across the bund l're cannot be satisfied. Similarly,a(o/oo)n where f0 = po/E. The plastic properties of material if the displacements are continuous across the bond line andI (top material) are designated by a,, 0o, and n, and those are of the form u, - roai, for materials I and 2, then continuityfor material 2 are designated by a 2 , a02, and n2. The reference of traction cannot be satisfied. However, if we assume that asstrains are defined by f0l = ao/E and Eo2 = o0 2/E,. We r - 0 the material system behaves like that of a plastically-designate the larger of n, and n2 by n, i.e., n = max(n,, n,). deforming material bonded to a rigid substrate, then (26) provesTo avoid ambiguity in the discussions to follow, material I to be an admissible form.

7661 Vol. 56, DECEMBER 1989 Transactions of the ASME

02 Q IQ~ei# q -Re ;(

OPEN CRACKREGION

Rb

0, Rb -R

(a)x2(y)

Fig. 2 Open crack domain in 0 plane

It is convenient to write (26) in the form

o0=o 0 h 9(O. P; ,n) (27) croc- -_Io-5Rb__ x(2)

where ho is a bounded function with respect to f and (b)SQ K) Fig. 3 (a) Small-scale yielding formulation and (b) finite element mesh

4)+f In = + E In =0+, In of upper half of crack tip region and crack tip conventions

(28) Table 1

The form of the fields (27) has been corroborated by the nu-merical solutions of Section 4. These full field solutions also 0O(deg) r/L O(deg)show that for load states in the range - r/6 _s t _s ir/6, hj 00 -2 9.9is a slowly varying function of P over all physically-relevant 10 - 4 13.2length scales. Thus, the near tip fields nearly scale with 10-6 19.8j,

1(n 1), since h* is only weakly dependent on J through the 0.050 5.71 10-2 13.2

10- 1 19.8phase parameter t. Moreover, these near tip fields of an in- 10- 4 26.4terface crack display strong similarities to mixed mode HRR 10-6 39.6fields. 0.100 11.3 10-2 26.4

When the plane ahead of the crack is stressed by tension l0- 1 39.610-4 52.8and shear, the HRR singularity (Hutchinson, 1968; Rice and 10-6 79.2

Rosengren, 1968) for homogeneous media has the form (Shih, 0.175 19.3 10-2 46.21974) 10-1 69.3

10- 4 92.3( / = o - (n + 1) &j#; (29) 10-6 139.0

Here, AMP is the mixity of the plastic singular fields defined bythe relative magnitudes of oo and a,@ along the radial line 0 For the case = 0. (31) states that the crack is open as long= 0 as r - 0. The angular functions i0 for several values of as the Mode I stress intensity factor is positive. Let Qi = -EAP have been tabulated by Symington et al. (1988). Under ln(rIL) and recall that 0 = tan- 2c. Then, by combining (30)small-scale yielding, the effects of load and geometry on the and (31), the open crack range in terms of 0 iscrack tip field of homogeneous media are characterized by J - ir/2 + Q) + Os _< ir/2 + 0. (32)and A. The structure of the near tip fields (27) suggests that Figure 2 depicts the open crack region in the Q plane. ValuesJ and are the characterization parameters of the near tip of 0 and 0 for four values of E and four values of rciL arefields of cracks on bimaterial interfaces, tabulated in Table I. Since 0 = 4, + 0 for the geometry in

2.4 Range of Z for an Open Crack. According to the linear Fig. I, the range of phase angles of the remote load for anelasticity solution, overlapping of crack faces will occur at open right crack tip issome distance for all values of 4) if E * 0. Nevertheless, for - ir/2 +12_5 _ 7r/2. * (33)a range of 0 the zone of interpenetration is confined to a The characteristic length in the small-scale yielding for-distance from the crack tip that is smaller than physically mulation is KK/o 2 (or QQ/ 2). Therefore, crack face contactrelevant length scales. For this reason, we define a crack to be ud Usopen ifunder small-scale yielding must be phrased in terms of theopen if normalized distance P (- r/(Kk/a2 = r/(QQ/o2)). The form

Au 2 >0