J R Rice - Path Independentt Integral - JAM68

7/28/2019 J R Rice - Path Independentt Integral - JAM68

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Journal of Applied Mechanics, vol. 35, pp. 379-386, 1968

J. R. RICE

APath Independent Integral and theApproximate Analysis of Strain Concentrationby Notches and Cracks

Assistant Professor of Engineering,Brown Univenity,Providence, R. I. A line integral is exhibited which has the same value for all paths surrounding the tip ofa notch in the two-dimensional strain field of an elasticor deformation-type elastic-plasticmaterial. Appropriate integration path' choices serye bOlh to relate Ihe integral to thenear tip e f o r m a t i o t ~ s and, in many cases. to permit its direct e:tlalutllion. This (l.t'eragedmeasure of the near tip field leads to approximate solutions for several strain-collcen/reltion problems. Contained perfectly plastic deformation near a crack tip is analy:.ed fortile plane-strain case with the aid of the slip-line theory. Near tip stresses are shown tobe significantly elet1ated by hydrostatic tension. and a strain singularity results varying;Ilt/erseiy u,ith distance from the tip ilt centered fan above and below the tip.Approximate estimates are given for the strain i n t e n s i t y ~ plastic zone size. and crack tipopening displacement, and the important role of large geometry changes in crack blunli1Jgis noted. A nother application leads to a general solution for crack tip separations inthe Barenblatt-Dugdale crack model. A proof follows on the equivalence IkeGrijfil.k energy balance and cohesive force theories ofelastic brittle fracture, and lu!jrdent1it9behavior is included in a model for plane"stress A final applicationfo approximate estimates of sHain (;Oncentrations at smooth-ended notch tips in elasticand elastic-plastic materials.

IntroductionCONSIDERABLE mathematical difficulties accompany

the determination of concentrated s train fields near notches andcracks, especially in nonlinear materials. An approximateanalysis of a variety of strain-ooncentration problems is carriedou t here through a method which bypasses this detailed solutionof boundary-value problems. Th e approach is first to identify aline integral which has the same value for all integration pathssurrounding a class of notch tips in two-dimensional deformationfields of linear or nonlinear elastic materials. The choice of anear tip path directly relates the integral to the locally concentrated strain field. But alternate choices (or the path often permi t a direct evaluation of the integral. This knowledge of anaveraged value for the locally concentrated strain field is thestarting point in the analysis of several notch and crack problemsdiscussed in subsequent sections.

All results are either approximate or exact in limiting cases.Th e approximations suffer from a. lack of means for estimatingerrors or two-sided bounds, although lower bounds.on strainmagnitudes may sometimes be established. Th e primary interes tin discussing nonlinear materials lies with eiasticplastic behaviorin metals, particularJy in relation to fracture. This behavior isbest modeled through incremental stress-strain relations. Butno success has been met in formulating a path integral forincremental plasticit y analogous to that presented here for elasticmaterials. Thus a udeformation" plasticity theory is employedand the phrase ftelastic-plastic material" when used here will beunderstood as denoting a nonlinear elastic material exhibiting awear Hookean response for stress states within a yield surfaceanq a nonlinear hardening response for those outside.

Contributed by the Applied Mechanics Division and presentedat the Applied Mechanics Conference. Providence, R. I. t June 12-14.1968, of TH E AMElUCAN SOCIETY OF MECII.ANIOAL ENGINEERS.Discussion of this paper should be addressed to the Editorial Dep"artment. ASME, Unit ed Engineering Center. 345 East 47th Street.New York. N. Y. 10011, and will be accepted u;ttU July 16. 1968.Discussion received alter the closing date will be returned. Manuscript rec!3ived by S ~ E Applied Mechanics Division, May 22, 1967;final draft, March 5, 1968. Paper No. 68-APM-31.

Journal of Applied Mecbanics

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F.... 1 .Flat surfaced notch In twMlmen.lonal d"ormatlon ftekl (aU . . . . . . . . . depend only on 11 and y). r Is anycurve surrounding the notch tip, r denot.. the curvednotch tip.

Path Independent J ." ..... . Consider a homogeneous body oflinear or nonlinear elastic material free of body forces and subjected to a two-dimensional deformation field (plane strain, generalized plane stress, antiplane strain) so that all stresses tTiJ depend only on two Cartesian coordinates Zl( - :c) and %t('= 1/).Suppose the body contains a notch of the type shown in Fig. 1,having Hat surfaces parallel to the z-axis and a rounded tip denoted by the arc r p A straight crack is a limiti ng case. Definethe strain-energy W by

W .= W(z,1/) = W(e):= 1:1 (1 )where = [Eil} is the infinitesimal strain tensor. Now considerthe integral J defined by

J - J: (Wdll - T . ~ : dB)' (2)Here r is a curve surrounding .the notch tip, the integral beingevaluated in a oontraclockwise sense starting from the lower Hatnotch surface an d continuing along the path r to the upper Hatsurface. T is the traction vector defined to the out;.ward normal along r, T, - tTunjJ U is the displacement vector,and d8 is an element of arc length along r. To prove path independent, consider any closed curve r* enclosing an area A in a

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two-dimensional deformation field free of body forces. An application of Green's theorem II] 1 givesi. WdY - T, :i is)- fA. [ ~ : - ~ J ( ~ i i :') xdy.

Differentiating the strain-energy densit.y,~ W ~ W ?JE iJ ~ E i ;- = - .- = ~ i ' -C>x I' j ()x ()x

{by equation (1)]

I [0 (em,) 0 em,)]= '2 ui i Ox ox; +?lx ()x:(OU')" i i Oxi Ox' (since" i = (T j i)() ( OU)= ox , . "i f Ox' since -- '} = 0( ()(T .. )ox ,

(Sa)

(3b)Th e area integral in equation (3a) vanishes identically, and thusf (WdY - T. OU cis) = 0JJr. ox for any closed curve r*. (3e)

Consider any two paths f l and surrounding the notch tip, asdoes r in Fig. 1. Traverse r l in the contraclockwise sense,continue along the upper flat notch surface to where f2 intersectsthe notch, traverse 1'2 in the clockwise sense, and then continuealong the lower fiat notch surface to the starting point where f lintersects the notch. This describes a closed contour so that theintegral of Wdy - T (ou/()x)cis vanishes. But T = 0 and dy = 0on the portions of path along the flat notch surfaces. Thus theintegral along r 1 contraclockwise and the integral along r t clockwise sum to zero. J has the same value when computed by integrating along either P l or f 2, and path independent is proven.We assume, of course, that the area between curves PI and fl isfree of singularities.Clearly, by taking r close to the notch tip we can make theintegral depend only On . he local field. In particular, the pathmay be shrunk to the tip r, (Fig. 1) of a smooth-ended notch andsince T = 0 there,

J = f WdyJr, (4)so tha.t J is an averaged measure of the strain on the notch tip.Th e limit is no t meaningful for a sharp crack. Nevertheless, sincean arbitrarily small curve r may then be chosen surrounding thetip, the integral may be made to depend only on the crack tipsingularity in the deformation field. The utility of the methodrests in the fact that alternate choices of integration paths oftenpermit a direct evaluation of J. These ar e discussed in the nextsection, along with an energy-rate interpretation of the integralgeneralizing work by Irwin [2) for linear behavior. Th e J integral is identical in form to a static component of the Clenergy-momentum tensor" introduced by Eshelby I3} to characterizegeneralized forces on dislocations and point defects in elasticfields.Evaluation of the J Integral

Two Special Conftguratlons. The J integral may be evaluatedalmost by inspection for the configurations shown in Fig. 2.These are not of great practical interest, but are useful in il -lustrating the relation to potential energy rates. In Fig. 2(a), asemi-infinite flat-surfaced notch in an infinite strip of height ~loads are applied by clamping the upper and lower surfaces of thestrip so that the displacement vector u is constant on each

1 Numbers in brackets designate References at end of paper.380 / J U N E 1 9 6 8

J [ J : r J : : ~ : : : : : / , : ~ : t : ~ ; : : tJr: J11CLAMPED BOUNDARIES, U IS CONSTANT

(0 )

x(b )

Fig. 2 Two .peclal conftguratlons for which the path Independent Integral J Is readily evaluated on the dashed-line paths r shown. Intlnitestrip. with seml-Intlnite notches. (0) Constant displacements imposedby clamping boundaries, an d (6) pure bending of beamlfke arms.clamped boundary. Take r to be the dashed curve shownwhich stretches ou t to x == 0 0 . There is no contribution to Jfrom the portion of r along the clamped boundaries since dy = 0and ()ujox = 0 there. Also at x = - 0 0 , lV = 0 and ()ujox = O.The entire contribution to J comes from the portion of r at x =+ X) 1 and since ) u / ~ == 0 there,

J == W",h (5)where W " is the constant strain-energy density at x = + co.Now consider the similar configuration in Fig. 2(b), with loadsapplied by couples M per unit thickness on the beamlike arms soa state of pure bending (all stresses vanishing except (Tu) resultsat x = - ( X I . For the contour r shown by the dashed line, no contribution to J occurs at x = + X) as Wand Tvanish there, andno contribution occurs for portions of r along the upper andlower surfaces of the str ip as dy and Tvanish. Thus J is given bythe integral across the beam arms at z = - (X) and on this portion of r, dy = -cis, T" = 0, and Ts = - Tu . We end upintegrating

(60)

across the two beam arms, where n s the complementary energydensity. Thus, letting n6(M ) be the complementary energy perunit length of beam arm per unit thickness for a state of purebending under moment per unit thickness M,J.".2{MM). (ab)

Small Seal. Ylefdlngln Elastk-Pla.tk Materials. Consider a narrownotch or crack in a body loaded so 88 to induce a yielded zonenear the tip that is small in size compared to geometric dimensions such as notch length, unnotched specimen width, and so on.The situation envisioned has been termed Hsmall-scale yielding,"and a boundary-layer style formulation of the problem [4] isprofitably employed to discuss the limiting case, The essentialideas are illustrated with reference to Fig. 3. Loadings symmetrical about the narrow notch art> imagined to induce a deformation state of plane strain. First, consider the linear elastic solution of the problem when the notch is presumed to be a sharpcrack. Employing polar coordinates r, 8 with origin at the cracktipt the form of stresseS in the vicinity of the tip are known [2, 51to exhibit a characteristic inverse square-root dependence on r :

Ktui i = (27rT)1IJij(8)+ other terms which a.re bounded at the crack tip. (7a)

Here KI is the stress intensity factor and the se t of functionsf'i(8) are the same for all symmetrically loaded crack problems.For an isotropic material

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yx

PLASTIC --2OttAS r . . . IX)

(0) (b )

fig.3 (a) Small-scale yielding near a narrow notch or crack In an elastlcplastic material. (II) The actual configuration I I replaced by a .emiinfinite notch or crack In an infinite Itody; actual boundary conditions arereplaced by the requirement of an asymptotic approach to the Unearelaltie crack lip singularity streu fleld.

cos (8/2)[1 - sin (8/2) sin (38/2)1f l l l l(8) cos (8/2)[1 + sin (8/2) sin (30/2)} (7b)!ZII(J} = f llz(8) = sin (6/2) cos (0/2) cos (38/2).

Now suppose the material is elastic-plastic and the load level issufficiently small so that a yield zone forms near the tip which issmall compared to notch length and similar characteristic dimensions (small-scale yielding, Fig. 3Ca. One anticipates that theelastic singularity governs stresses at distances from the notchroot that are large compared to yield zone and root radius dimensions bu t still small compared to characteristic geometric dimensions such as notch length. The actual configuration in Fig. 3(a)is then replaced by the simpler semi-infinite notch in an infinitebody, Fig. 3(b), and a boundary-layer approach is employed replacing a c t u ~ . l boundary conditions in Fig. 3(a) with th e asymptotic boundary conditions

KICTii - (21rr)1/J ,j(8) asT-CO, (7c)

where KI is the stress intensity factor from the linear elasticcrack solution.Such boundary-layer solutions for cracks are mathematically

exact in the plastic region only to the first nonvanishing term of aTaylor expansion of complete solutions in the applied load. Butcomparison [4] with available complete solutions indicates theboundary-layer approach to be a highly accurate approximationup to substantial fractions (typically, one half) of ne t sectionyielding load levels. We now evaluate the integral J from theboundary-layer solution, taking r to be a large circle of radius rin Fig. 3(b):f+r [ ()u ]J = r - r (W(T, 8) cos 8 - T(r, 8 ) . ~ (r,8) dO. (Sa)By path independence we may le t r -+ co and since W is quadraticin strain in the elastic region, only the asymptotically approachedinverse square-root elastic-stress field contributes. Working ou tthe associated plane-strain deformation field, one finds

J = 1 - Vi K;E for small scale yielding,where E is Young's modulus and v Poisson's ratio.

(8b)

Primarily, we will later deal with one configuration, the narrownotch or crack of length 2a in a remotely uniform stress field CToo,Fig. 4. Here [2J

andfor small-scale yielding (9)

Journal Dt Applied Mechanics

For plane stress, the same result holds for J with 1 - ,,1 replacedby unity. Th e same computation may be carried ou t for moregeneral loadings. Lett ing KI, Ku, and KIll be elastic stress-intensity factors [2] for the opening, in-plane sliding, and antiplanesliding modes, respectively, of notch ti p deformation one readilyobtains

1 - 1'1 t t l V tJ ... ~ (KI + Kn) + K J I I(small-scale yielding) (10)

Interpretallon In Terms of Energy Comparisons for No tche. of NeighborIng Sin. Let A denote the cross section and r ' the boundingcurve of a two-dimensional elastic body. Th e potential energyper unit thickness is defined as

P = f Wdxdy - f Tuds,A' Jr' ( il)

where r ' s that portion of r ' on which tractions Tare prescribed.Let P(l} denote the potential energy of such a body containing aflat-Burfaced notch as in Fig. 1 with tip at z = l. We comparethis with the energy P( l + a l) of an identically loaded body whichis similar in every respect except that the notch is now at x = i+ al, the shape of the curved tip r, being the same in both cases.Then one may show that

J = _ lim pe l + a l) - P(l) = _ OP (12)az ()l'

the rate of decrease of potential energy with respect to notch size.The proof is lengthy and thus deferred for brevity to a section ofa forthcoming treatise chapter [6]. Equations just mentionedprovide a check. Fo r loading by imposed displacements onlyas in Fig. 2(a), the potential energy equals the strain energy sothat equation (5) results. Similarly, the potential energy equalsminus the complementary energy for loading by tractions only asin Fig. 2(b), 80 that equation (66) results. Equation (10) is thelinear elastic energy-release rate given by Irwin [21, and reflectsthe fact that a small nonlinear zone at a notch tip negligiblyaffects the overall compliance of a notched body.

In view of the energy-rate interpretation of J and its alternaterelation to the near tip deformation fieldJ the present work provides a generalization of the connection between crack-tip stressintensity factors and energy fates noted by Irwin for linear materials. Further, J. W. Hutchinson has noted in a private communication that an energy-rate line integral proposed by Sanders [71for linear elasticity may be rearranged so as to coincide with the Jintegral form. Th e connection between energy rat es and locallyconcentrated strains on a sIDooth-ended notch tip, as in equations(4) and (12), has been noted first by Thomas [8] and later byRice and Drucker [9] and Bowie and Neal [lOJ. Since subsequent results on strain concentrations will be given in t erms of J,means for its determination in cases other than those representedby equations (5)-(10) are useful. The energy-rate interpretationis pertinent here. In particular, the compliance testing methodof elastic fracture mechanics [2} is directly extendable throughequati on (12) to nonlinear materials. Also, highly approximateanalyses may be employed since only overall compliance changesenter the determination of J. For example, the Dugdale model

fig. 4 Narrow notch or crack of length 20 In Infinite body;uniform remote ,frets Ua>

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discussed next or simple an iplane strain calculations 111] maybe employed to esti mate the deviation of J from its linear elasticvalue in problems dealing with large-scale plastic yielding near anotch. Once having determined J (approximately), the modelmay be ignored and methods of the next sections employed todiscuss local stra in concentrations. Such estimates of J aregiven in reference [6J for the two models just noted . As anticipated, deviations from the linear elastic value show little sensitivity to the particular model employed.Perfectly Plastic Plane-Strain Yielding al aCrack TipA13 a first application we consider the plane-strain problem ofcontained plastic deformation near a crack up in anonhardeningmaterial. Behavior is idealized as linear elastic until the principal in-plane shear stress reaches a yield value T y, at which un-restricted further deformation may occur with no increase inshear stress. This conveniently permits utilization of the slipline theory [12] in plastic regions. The idealization is rigorouslycorrect for an isotropic nonhardening material exhibiting elasticas well as plastic incompressibility. Elastically compressiblematerials approach a constant in-plane shear stress in plasticregions when plastic stra ins are somewhat in excess of initial yieldvalues, as anticipated in the near crack tip region. Constanc y ofthe J integral requires a displacement gradient singularity at thecrack t.ip since stresses are bounded and the path may be shrunkto zero length. Thus we construct a near tip stress state satisfying the yield condition and varying only with the polar angle O.

The slip lines of this field are shown in Fig. 5. Traction-freeboundary conditions require yield in uniaxial tension along thecrack line and the 45-deg slip lines carry this s t r ~ state into theisosceles right triangle A:

(T il l = (T:l1I == 0 (region A) (13a)Any slip line of region A finding its way to the x-axis in front ofthe crack must swing through an angle of 11"/2. The accompanying hydrostatic stress elevation [12} and 45-deg slip lines determine the constan t stress state(T n = 1rTy, (T IfI = (2 + 1r)TYI

(TZII = 0 (region B) (13b)in the diamond-shaped region ahead of the crack. A centered fanmust join two such regions of constant stress, and in the fan(Trr = (f88 = (1 +3;) Ty - 28T y ,

Urf = Ty (region C) (13c)This stress field is familiar in the limit analysis of double-edgenotched thick plates [13]. I t is emphasized that the boundaryof the slip-line field in Fig. 5 is no t intended to represent theelastic-plastic boundary.Large strains can occur only when slip lines focus, as in thecentered fans, bu t no t in constant stress regions (A and B) unlessstrains are uniformly large along the straight slip lines and thus

y

CRACK

f ig. 5 perfectly plaltlc plan . traJn IUpailn. ftelcl at a crack tipJ constanl,tre . retlonl A. an d B lolned by cenl.red fan C382 / JU N E 1 9 6 8

ou t toward the elastic-plastic boundary (where strains must besmall), One then anticipates strains on the order of initial yieldvalues in the constan t stress regions. Only the strain singularityin the centered fan enters the results toward which we are heading. Assume elastic incompressibility for the moment, and thatprincipal stress and strain directions coincide. Then EN' = E8& = 0in the fan and displacements are thus representable in the formu" = f'(O), U/I = -f(O) + g(r},

where g(O) = 1(1r 4) = 0 (14)These equations apply to velocities (12] rather than displacementsin a proper incremental theory. Th e nonvanishing strain component in the fan is

1 C>u" C>u/l U/I'Yr' = -;: be + Or - -;:1== - [1'(0) +1(0) + rg'(r) - g(T)] (14b)r

Now consider the path independent integral J (which wasdiscovered as an extension of work directed toward establishingthe strength of the 'Y rf singularity in this problem). Taking r asa circle of radius r centered at the crack tip and employing polarcomponents,J = r J: . {w cos 8 'Yy (l7a)

where 'Y is the principal resolved in-plane shear strain. ThusR(B)W -Ty'Yy-r as r- 0 in C. (l7b)

While it was simplest to perform calculations for an incompressible material, one may show that the asymptotic form for 'Y .. inequation (16a) as well as all subsequent formulas in this sectionapply also to an elastically compressible material.The average value of R(O) is now determined in terms of J bylettingT- 0 in equation ( 1 5 ~ , resUlting in

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J == 2Ty"'(y I3w-/4 R(6) f cos 0 + (1 + 3; - 26) sin 0] dO.w/4 L (18)

Another useful form results by solving for displacements as r - 0in the fan. Converting equations (14a) to Cartesian componentsand employing equation (16b), one findsuti -.. "'(y fe R{O) sin OdO, - "'(y I e R(fJ) cos OdDJ /4 w/4as r -. . 0 in C. (19a)

An integration by parts in equation (18) then leads tof 3r/ 4J = 2Ty 'U1I(t})(3 + ctn O)dO1(/4 (19b)where ull (8) is the near t ip vertical displacement in the fan.Plastic Zone Exlent and Crack Opening Displacement. We definethe "crack opening displacement" as the total separation distance between upper and lower crack surfaces at the tip due tothe strain singularity of the fan:

0, = 2 u ~ ( 3 1 1 " /4). (20)A lower bound on 0 is established from equation (19b) since ulI(O)is monotonic by equation (19a):

f 31(14J < (3 + ctn t O)dO7/4J (1!"(1 - J l i ) u ~ a 11 I . Id' ):. {I, > = E 1 sma -sea eyle mg(2 + 1r}Ty (2 + 'Jr)Ty(21)

The latter form employs the small-scale yielding relation betweenJ and the stress-intensity factor approp riate to the crack of length2a in a remotely uniform stress field. Asimilar bound is obtainedfor the maximum value Rmax of the function R(O), and thus approximately for the maximum extent of the yielded region.From equation (18),J ,:; 2TY'YyR_.1.::/4 [ cos 8 (1 + 3; - 28) sin 8] d8

J (211"(1 - JI) (tT .. )1:. Rmu ::-- 0(2 + 1r}Ty"'(v = 0{2 + 1r ) 2T y a,small-scale yielding) (22)

The path integral leads to no upper bounds on deformationmeasures, bu t rough approximations can be obtained by choosinga reasonable form for the functions R(O) or containing anunknown constant, and employing the J to determinethe constant. Experience with nonhardening solutions in theantiplane strain case [14J suggest thatf while and 'U,,(8)will have a dependence on 0 in th e small-scale yieldingrange, the 8 dependence will vary considerably with applied stresslevel and detailed specimen shape in the large-scale yieldingrange. Thus no simple approximation will be uniformly valid,and we here choose approximations appropriate to a small plasticregion confined to the vicinity of the crack tip. Anticipating thatthe plastic region will then be approximately symmetrica l about.the midline of the fan, 0 = 11"/2, from equations (l9a) and (20)we will have 1[.,,(0) '{;:::S ~ , / 4 plus a function of 0 that is antisymmetric about 0 = 'Jr/2 . This antisymmetric term contributesnothing to the integral in equation (19b), and

J ~ (3 + ctn2 O)d8Jr /42 7/ 42J 211"(1 - pt)u!a l ' din

:. {I t '{;:::S (2 \- = ( \- E ' small-sea. e yte g.+ Jr}1 Y 2 + 11")1 Y (23)Journal of Applied Methanics

Were a slip-line construction similar to Fig. 5 made in the antiplane strain case, a centered fan of shear lines would result aheadof the crack for 101


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Barenblatt-Dugdale Crack ModelSharp cracks lead to strain singularities, The Griffith theory

[161 of elastic brittle fracture ignores this perhaps unrealisticprediction of conditions at the and employs an energy balanceto se t the potential energy decrease rate due to crack extensionequal to the energy of the newly created crack surfaces. Analternate approach due to Barenblatt [171 removes the singularity by considering a cohesive zone ahead of the r a c k ~ postulating that the influence of atomic or molecular att ract ions i s representable as a restraining stress acting on the separating surfaces,Fig. 7(a}. Th e restraining stress 0'"(0) may be viewed as a function of separation distance 0 as in Fig. 7(0), Although mathematical difficulties apparently prohibited Barenblatt from detailed solutions for specific restraining stress functions J we heretake the view that a q versus 0 curve falling off to zero is givenso that a fracture criterion requires no special assumptions no tinherent to the model. A mathematically similar bu t conceptually different model was proposed by Dugdale [181 to discussplane-stress yielding in sheets. There the influence of yieldingwas represented approximately by envisioning a longer crackextending into the region with yield level stresses opposing it sopening. For reasons that are no t yet clear, some metals actuallyreveal a narrow slitlike plastic zone [19] ahead of the crack ofheight approximately equal to sheet thickness when the zone islong compared to the thickness dimension. Except for a perturbation near the tip, yielding then consists of slip on 45-deg planesthrough the thickness so that plastic strain is essentially theseparation distance divided by the thickness. Rest raining stressestypical of plane-stress yielding are shown in Fig. 7(0}.We may evaluate our integral J by employing path independence to shrink. the contour r down to the lower and uppersurfaces of the cohesive zone as in Fig. (7)4. Then, since dy = 0onr, 1: ()uJ=- T-dBr ax

- q(o) dxi dOcohea ZODe dx== - - O'"(o)dO d3;J: d }cohea zone dx 0raj

== Jo q(o)dO (25)where 0, is the separation distance at the crack tip. Thus, if thecrack configuration is one of many for which J is known, we areable to solve for the crack opening displacement directly from theforce-displacement ourve.

Equivalence of OrfHlth and CoM,Iv. Force Theorl... Now le t uscompare to Griffith theory of elastic brittle fracture with thefracture prediction from the Barenblatt-type cohesive forcemodel. Letting o be the separation distance in Fig. 7(b) whenthe atoms at the crack tip can be considered pulled out of range oftheir neighbors, the value of J which will just cause crack extension (or, if .crack extension is considered as reversible, willmaintain equilibrium at the current crack length) is then

J - J:' 17(8)


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for the opening displacement and length of the plastic zone. ATaylor development of the first of these checks equation (27c)at low applied stress levels. Further, since J = we may Usethis model to estimate the deviation of J from its linear elasticvalue in the large-scale yielding range. Taking the results for8, and Rmax as representative of plane stress, as observations suggest [18, 19] for materials in which Dugdale zone6 actually Corm,we may compare small-scale yielding behavior in plane stresswith plane strain as approximated by equations (23) and (24b).Thus, for v ::: 0.3, one finds a plane-strain opening displacementwhich is 61 percent of the plane-stress value for a Mises material(Uy = Vary ) and 70 percent for a Tresca material (O"y = 2ry).Th e plane-st.rain value of Rml1X is 55 percent t.he plane-stress va luefor a Mises material and 73 percent for a Tresca material. Anycompensation for stress biaxiality in raising the Dugdale yieldstress in a Mises material above the uniaxial yield would tend toreduce the differences between the Mises and Tresca comparisons.Etching observations [22J suggest a plane-strain to plane-stresszone size rat io in the neighborhood of one half.

Strain Concentration at Smooth Ended Notch TipsConsider a flat-surfaced notch with a smooth tip. Letting 4>

th e tangent angle at a point on the tip, 88 in Fig. 8, and r,(4)) theradius of curvature, equation (4) becomes

1. f +r/2J = Wdy = . W[E(4))}r,() cos #4>r, - r /2 {29}Here E(4)) is the surface extensional strain at the point with tan-gent angle 4>, and estimates of the maximum strain will requirean approximate choice for the functional dependence on 4>. Alower bound is immediate since W[E(4))] W(EmuJ:

(30)

where 2h is the distance between the flat notch surfaces. For alinear elastic material

1 EE'lW = -2 O"E = ( ) for plane strain21 - ViEE'l2 for plan e stress. (31)

Now consider a notch in the configuration in Fig. 4, which is sufficiently narrow so that the sharp crack value of J is appropriate.Then for either plane stress or plane stra in one has for the maximum concentrated stress

regardless of the detailed shape of the curved notch tip. An interesting unsolved problem is the det-ermination of the optimumproportioning of a notch tip. The lower bound would actually beobt&ined if the surface stress were constant at every point on thecurved tip, and the problem is to determine the shape of a tipleading to constant stress (i f such a shape exists).For approximations to the strain concentration through equation (29), we recall a feature of the linear elastic problem of anellipsoidal inclusion in an infinite matrix subjected. to a uniformremote stress state. All strain components are spatially const antwithin the inclusion [23]. The same result applies to an ellipsoidal void, in that surface displacements are compatible withthe homogeneous deformation of an imagined inclusion havingzero elastic moduli. Thus , for an elliptical hole in a linea r elasticplate, loaded symmetrically so as to cause no shear or rotation ofthe imagined inclusion, surface strains are given through th eusual tensor transformation as

E(4)) = Emu COS' 4> + Emin sin' 4>. (33a)Jaufnal of Applied Mechanics

Here the 4> notation is &8 in Fig. 8, E .. .x is the extensional stra inat the semimajor axis (q, - 0, 1F'), and Emili at the semi minor axis(4) :: s 7f'/2, 31F'/2). Thus, presuming this same i n t e r p r e ~ t a t i o n ofthe surCace st rains to be approximately valid in other eases andthat Eml n is small compared to Emu,f + r / 2J f:::S W(Emu COSI 4>)r(4)) COS 4xUP- r / 2 (33b)

For linear elastic behavior and the narrow notch in Fig. 4 witha semicircular tip, r,( Is tangent angle and r,(4)) (s radius of curvatureJUNE 1968 I 385


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AcknowledgmentSupport of this research by the Advanced Research ProjectsAgency, under Contract SD-86 with Brown University, is gratefully acknowledged.

References1 Phillips, H. B. t Vector AnalllN, Wiley, 1959.2 Irwin, G. R., "Fracture Mechanics." Structural Meehania(Proceedings of Firat Nat1al SympoBium,) Pergamon Press, 1960.3 Eshelby, J. D., "The Continuum Theory of Lattice Defects,"

Solid State PhyBics, Vol. 3, Academic Press, 1956.4 Rice, J. R., "The Mechanics of Crack Ti p Deformation andExtension by Fatigue," Fatigue Crack Growth, ASTM Spec. Tech.Pub!. 415, 1967.5 Williams, M. L., "On the Stress Distribution at th e Base of 8.Stationary Crack," JOURN.-'.L of ApPLIED MECH ...NIC8, Vol. 24, No. I,TRANS. ABME, Vol. 79, Mar. 1957, pp. 109-114.6 Rice, J. R., "Mathematical Analysis in th e Mechanics of Fracture," to appear in Treatise on Fracture, Vol. 2, ed., Liebowitz. H. tAcademic Press.7 Sanders. J. L., HO n the Griffith-Irwin Fracture Theory,"JOURNAL OF ApPLIED h!ECHANICS, Vol. 27, No.2, TRANS. MME,Vol. 82, Series E. June 1960, pp. 352-353.8 Thomas. A. G. , "Rupture of Rubber, II , Th e Strain Concentration at an Incision," Journal of Polymer Scient:e, Vol. 18, 1955.9 Rice, J. R., and Drucker, D. C., "Energy Changes in StressedBodies Due to Void an d Crack Growth," Internatwnal10urnal ofFracture Mechanics. Vol. 3, No. I, 1967.10 Bowie, O. L., an d Neal, D. M., "The Effective Crack Lengthof an Edge Notch in a Semi-Infinite Sheet Under Tension," to appearin International Journo.l oj Fracture Meclw.nics.11 Rice, J. R.. "Stresses Du e to a Sharp Notch in a Work-Hardening Elastic-Plastic Material Loaded by Longitudinal Shear,nJOURNAL OF ApPLIED MECHANICS. Vol. 34, No.2, TRANS. ASME, Vol.89, Series E, June 1967, pp. 287-298.

12 Hill. R. t TM Mathematical Theorll of Pla&ticitll. ClarendonPress, Oxford. 1960.13 Lee. E. H. t "Plastic Flow in a V-Notched Bar Pulled in Ten.sion:' JOURNA.L 01 ' APPLIED MECHANICS. Vol. 19. No.8. TRANI.ABME, Vol. 74 Sept. 1952, pp. 331-336.14 Rice. J. R., "Contained Plastio Deformation Near Cracks andNotches Under Longitudinal Shear." International Journal 0/ FrtM>lture Mechaniu, Vol. 2, No.2, 1966. .,15 Wang, A. J' t "Plastic Flow in a Deeply Notched Bar With &.Semi-Circular Root." (Juarterlll oj Applied M ~ . Vol. 11,1954.16 Grifiith, A. A" "The Phenomena of Rupture and Flow inSolids." Philosophical Transactiom oj the Rol/al Society, London. SeriesAI Vol. 221,1921.17 Barenblatt, G. I., "Mathematical Theory of EquilibriumCracks in Brittle Fracture," Advances in Applied Mechanics, Vol. 7,Academic Press, 1962.18 Dugdale, D., "Yielding of Steel Sheets Containing Slits,"JtJu:mal of he ,,;\fechamcs alld Physics of Solids. Vol. 8. 1960.19 Hahn, G. T. , an d Rosenfield. A. R . "Local Yielding and Extension of a Crack Under Plane Stress," Acta .rVetallU'l'aW. Vol. 13.No.3,1965.20 Goodier, J. N., and Kanninen, M. F., "Crack Propagation in aContinuum Model With Nonlinear Atomic Separation Laws," Stanford University. Division of Engineering Mechanics. Technical Report 165, July 1966.21 Willis. J. R., uA Comparison of th e Fracture Criteria ofGriffith and Barenblatt," JfJttrnal oj the Mechanics and Phllaic8 oj&tida. Vol. 15, 1967.22 Hahn, G. T., and Rosenfield, A. R. , "Experimental Determination of Plastio Constraint Ahead of a Sharp Crack Under Plane StrainConditions," Ship Structure Committee Report. SSC-180, Dec. 1966.23 Eshelby, J. D. , "The Determination of th e Elastic Field of anEllipsoidal Inclusion an d Related Problems," Proe&ding8 of theRoyal &ciety, London, Series:A. Vol. 241, 1957.24 Neuber, H. "Theory of Stress Concentration for ShearStrained Priamatical Bodies With Arbitrary Nonlinear Stress-StrainLaws," JOURNAL OF ApPLIED MECHANICS, Vol. 28, No.4, TRANS.ASME, Vo l 83, Series E, 1961, Dec. 1961, pp . 544-550.

Reprinted from the June 1968Journal of Applied .iUechanics

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J R Rice - Path Independentt Integral - JAM68