On macroscopic and microscopic analyses for crack initiation

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On macroscopic and microscopic analyses for crack initiation

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On Macroscopic and Microscopic Analyses for Crack Initiation and Crack Growth Toughness in Ductile Alloys
Relationships between crack initiation and crack growth toughness are reviewed by examining the crack tip fields and microscopic (local) and macroscopic (continuum) fracture criteria for the onset and continued quasi-static extension of cracks in ductile materials. By comparison of the micromechanisms of crack initiation via transgranular cleavage and crack initiation and subsequent growth via microvoid coalescence, expressions are shown for the fracture toughness of materials in terms of microstructural parameters, including those deduced from fractographic measurements. In particular the distinction between the deformation fields directly ahead of stationary and nonstationary cracks are explored and used to explain why microstructure may have a more significant role in influencing the toughness of slowly growing, as opposed to initiating, cracks. Utilizing the exact asymptotic crack tip deformation fields recently presented by Rice and his co-workers for the nonstationary plane strain Mode I crack and evoking various microscopic failure criteria for such stable crack growth, a relationship between the tearing modulus TR and the nondimensionalized crack initiation fracture toughness Ji~ is described and shown to yield a good fit to experimental toughness data for a wide range of steels.

THE fracture toughness of a material is conventionally
assessed in terms of the critical value of some crack tip field characterizing parameter at the initiation of unstable crack growth. In plane strain, for example, under smallscale yielding (ssy) conditions, the critical value of the linear elastic stress intensity factor, K~c, is generally determined at the onset of crack extension, and can be referred to as the "toughness.''~ With appreciable nonlinearity in the load-displacement curve, however, the (crack initiation) toughness is measured in terms of the critical value of the J-integral, Jlc,2'3 or the crack tip opening displacement, ~ or 8i, 4 For ssy conditions, these parameters are explicitly related in terms of the flow stress, Co, and the elastic (Young's) modulus E, i.e.:

K~ 1

Jlc = -- -~ --~iO'o,


E ' c~

where E ' = E in plane stress and E/(1 - v 2) in plane strain, and c~ is a proportionality factor of order unity, dependent upon the yield strain (eo = co~E), the work hardening exponent (n), and whether plane stress or plane strain conditions are assumed. 5
Although in "brittle" structures, catastrophic failure or instability is effectively coincident with this onset of crack extension, in the presence of sufficient crack tip plasticity crack initiation is generally followed by a region of stable crack growth. Under elastic-plastic conditions (or plane stress, linear elastic conditions), such subcritical crack advance has been macroscopically characterized in terms of crack growth resistance curves, i.e., the JR(Aa) and 8R(Aa) R curves (Figure 1). 6,7'8 Crack growth toughness is now assessed in terms of the slope of the resistance curve, which

R. O. RITCHIE, Professor, is with the Lawrence Berkeley Laboratory and the Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720. A.W. THOMPSON is Professor, Department of Metallurgical Engineering and Materials Science, Carnegie-Mellon University, Pittsburgh, PA 15213.
Manuscript submitted December 31, 1983.


Blunting line




dj R

9 Experimental data

C ock
1 initiation -1~
Crack blunting ~_

Fig. 1--JR(Aa) resistance curve of J vs crack extension Aa, showing definition of Ji = J,c at initiation of crack growth where the blunting line intersects the resistance curve.

in the J approach can be evaluated in terms of the nondimensional tearing modulus (TR = E / c ~ " dJ/da), 7 or in
the CTOD approach in terms of the crack tip opening angle (CTOA = d~/da), 8'9 where:

E dJ E d6


TR - - - - - - =




c~ da Co da Yield Strain

Whereas crack initiation toughness values (i.e., Klc, J~c, etc.) are by far the most widely measured and quoted, it has been noted in high toughness ductile materials, for example, that stable ductile crack growth can occur at J values some 5 to 10 or more times the initial J~c value prior to instability,1~ e.g., Figure 2. Furthermore, microstructural influences on fracture resistance would appear to be enhanced in the crack growth regime, compared to initiation behavior (Figure 2). Evaluating the toughness of such materials with crack initiation parameters, such as J~c, would appear overly conservative, and accordingly there has been a recent trend, both for engineering fracture mechanics design and for metallurgical toughness assessment, to consider additionally crack growth parameters,

VOLUME 16A, FEBRUARY 1985--233




r A516-70

(ram) 1.0



' ~

-~1000 ,

" := "~ 2000


Co T

~ 1t





- 600
I -- 400 I I


.," ... -"

- 200








0.02 0.04 0.06 0.08 0.10

~(1 (in)

Fig. 2--Experimental data showingJR(Aa) resistancecurvesfor several
heats of A516Grade70 plaincarbonsteelplate(tr0 ~ 260 MPa). Sulfide and oxide nonmetallicinclusionshave been controlledby both conven-
tional techniques(CON)usingvacuumdegassingand calciumtreatments (CAT).Notehow modifyingthe inclusiondistributionhas a moresignificant effecton crackgrowthcomparedto crackinitiation(fromRef. 63).

such as dJ/da, CTOA, or the tearing modulus TR (for
reviews, see References 6 through 12). The purpose of this paper is to provide an interpretative
review of recent major advances in continuum mechanics relevant to the fracture toughness of engineering materials which rely on a consideration of salient microstructural dimensions. While perhaps lacking in detailed metallurgical formulation for the micromechanisms of fracture, which in few cases are known, the review seeks to integrate both microscopic (local) and macroscopic (continuum) viewpoints based on the current knowledge of the mechanics and microstructural aspects of crack tip processes. Specifically, the relationship between crack initiation and (quasi-static) crack growth toughness, at both macroscopic and microscopic levels, is examined and in each case the role of microstructure identified. Continuum and local models for the initiation and continued propagation of cracks, by both cleavage and hole growth mechanisms, are considered in terms of near-tip stress and deformation fields and macroscopic/microscopic fracture criteria for the quasistatic plane strain advance of stationary and nonstationary tensile cracks. Specifically, model expressions for crack initiation and crack growth toughness are presented which
indicate a relationship between Jic and TR, the latter repre-
senting an extension of the brief assessment of crack growth
toughness originally reported by Shih et al.t3

A. Crack Tip Fields for Stationary Cracks
The stress and deformation fields local to the near-tip region of stationary cracks subjected to tensile (Mode I)
234--VOLUME 16A, FEBRUARY 1985

opening are well known for both linear elastic and nonlinear elastic solids. Asymptotic continuum mechanics analyses of the local singular fields yield, for linear elastic conditions, a local stress distribution at distance r from the crack tip, in the limit of r ~ 0, of."14'15

o'ij(r, O) ~ K~.rfJ(O)



where K~ is the Mode I stress intensity factor, 0 the polar angle measured from the crack plane, and fj a dimensionless function of 0. For elastic-plastic (actually nonlinear elastic) conditions, asymptotic solutions by Hutchinson, Rice, and Rosengren (HRR) for the local stress, strain, and displacements ahead of a stationary tensile crack in a powerhardening (incompressible nonlinear elastic) solid, with a constitutive relationship of the form:

= ~,(~p)",


yield, in the limit of r ~ 0:16'17

~r,i(r, O) , ( j y/,+l

-~ ,

\-~ , l ,-----~/r 6%(0),


(j]l:.+, eli(r, O) ~ ,,~/



r ~ \-~, l,----~r/ t~i(0),


where J is the J-integral, ~8~j the equivalent stress at unit strain, 6-0(0), ~0(0), and tTi(0) are normalized stress, strain,
and displacement functions of 0, and I, is a numerical con-
stant, weakly dependent upon the strain hardening exponent, n, given within 2 pct by empirical relation: ~9'z~

I, = 10.3X/0.13 + n - 4.8 n.


Numerical values of the functions in Eq. [5] have recently been tabulated by Shih. 2~
Incorporating Rice and Johnson's 22 and McMeeking's 23
near-tip blunting solutions which consider large geometry changes at the crack tip, and Tracey's24 numerical power hardening solutions, the distribution of local tensile (opening) stress, O~y,directly ahead of a stationary Mode I
crack (i.e., at 0 = 0 when r = x) can be defined for various values of n and tro/E, as shown in Figure 3. The corre-
sponding near-tip equivalent plastic strain (~p) distribution,2z
which is essentially independent of strain hardening for the stationary crack, 23 is shown in Figure 4. Also plotted is the near-tip variation of stress state, defined as the ratio of hy-
drostatic to equivalent stress (~rm/-~).

B. Continuum (Macroscopic) Fracture Initiation Criteria

To define macroscopic fracture criteria for crack initiation, reference is made to the functional form of the local singular fields from the continuum asymptotic analyses (Eqs. [3] and [5]). For a brittle solid, the stress intensity factor/s can be considered as the (scalar) amplitude of the linear elastic singularity in Eq. [3]. Under conditions of small-scale yielding (ssy), where the plastic zone size (ry) at the crack tip, given approximately by: ~5

1 (K, 2

ry -~ ~ ~ 0 0 ] '



"1 n=0.2

4r ~ ' ~ " ~.. ~.


2o;, - /

...... ....

Modified stress distribution due to bluntin 9 for o'o/E = 0 , 0 0 2 5 (offer Rice and Johnson)
SGC solution of power hardeninq (after Rice and Rosengren and Hutchinson)


element results (after Rice and Tracey

and Trocey )

0 2B



O.Ol (K/(ro)z







I 0.02 (K/0-o)2 ~ x





Fig. 3--Distribution of local tensile stress o-~ as a function of distance x directly ahead of a crack tip in plane strain based on HRR small geometry changes (SGC) asymptotic solution for a power hardening solid (from Refs. 16 and 17) and the corresponding finite element solutions (from Refs. 24 and 37), modified for an initial yield strain o'~/E of 0,0025 by the finite geometry solution of Rice and Johnson which allows for progressive crack tip blunting (from Ref. 22). The abscissa is normalized with respect to both (K/cro) z and 6, the CTOD, whereas the ordinate is normalized with respect to the flow stress cr,j.



1.5 -


l.O --

\~ / FULLY '

o.,~ ,













Fig. 4--Distribution of local equivalent plastic strain ~p as a function of distance x, normalized with respect to 8, the CTOD, directly ahead of a crack tip in plane strain, showing the corresponding variation of stress-state (cr~/5). Solutions based on finite geometry blunting solutions of Rice and Johnson (from Ref. 22) and McMeeking (from Ref. 23) for both smallscale yielding and fully plastic conditions.

is small compared to the length of the crack (a) and uncracked ligament (b), provided this asymptotic field "dominates" the local crack tip vicinity over dimensions large compared to the scale of microstructural deformation and fracture events involved, Kt can be utilized as a single, configuration-independent parameter which uniquely and autonomously characterizes the local stress and strain field ahead of a linear elastic crack. In such circumstances it can thus be utilized to correlate microscopically with crack ex-


tension. For the monotonic loading of plane strain stationary cracks, the onset of brittle fracture is thus macroscopically defined at Kt = K~c, where Kic is the Mode I plane strain fracture toughness. 1.15
In the presence of more extensive plasticity where ssy conditions no longer apply (i.e., typically for ry < 1/15 (a, b)), J, taken as the scalar amplitude factor of the HRR singularity (Eq. [5]), can be utilized in somewhat analogous fashion. Provided this field can be considered to dominate over the relevant crack tip dimensions, J uniquely and autonomously characterizes the local stresses and strains ahead of a stationary crack in a power hardening solid, and the corresponding macroscopic failure criterion for the onset of crack extension in a ductile solid becomes J = Jic. 2'3
It should be noted here that the HRR singularity and the J-integral are strictly defined for a nonlinear elastic solid, where stress is proportional to current strain, rather than for the more realistic elastic-incrementally plastic solid, where stress is proportional to strain increment (Figure 5). Provided the crack remains stationary and is subjected only to a monotonically increasing load, plastic loading will not depart radically from proportionality and this approach is appropriate. However, for growing cracks where regions of elastic unloading and nonproportional plastic flow will be embedded in the J-dominated field, behavior is not properly modeled by nonlinear elasticity, and this poses certain restrictions to the J characterization for large-scale yielding (cf. Reference 12). Moreover, the uniqueness of the crack tip fields implied by the HRR singularity is relevant only in the presence of some strain hardening, since the crack tip fields for rigid/perfectly plastic bodies under full yielding conditions are very dependent on geometry. As noted by McClintock,25 the plane strain slip-line field for a fully yielded edge-cracked plate in bending (essentially the Prandtl field) has a fundamentally different near-tip stress and strain field compared to the center cracked plate in tension (Figure 6). The former Prandtl field develops high triaxiality and normal stress ahead of the tip, with r -~ singular shear strains in the fan above and below, whereas in the latter case only modest triaxiality occurs ahead of the tip, but intense shear strains develop on planes at 45 deg to the crack. Rationalizing such nonunique fully plastic solutions with the concept of a unique HRR field, and Jl,. being a single valued configuration-independent measurement of toughness, requires that some strain hardening must exist. This implies that, unlike Kt characterization, the specimen size

5-y 5- / j





Fig. 5--Idealized constitutive behavior, of equivalent stress ~ as a function of equivalent plastic strain ~p, for (a) nonlinear elastic material conforming to deformation plasticity theory, and (b) incrementally-plastic material conforming to flow theory of plasticity.

VOLUME 16A, FEBRUARY 1985--235



Fig. 6--Fully plastic plane strain slip-line fields for rigid/perfectly plastic solids for (a) deep edge-cracked bend and deep double-edge-cracked tension plates (Prandtl field), and (b) center-cracked tension plate, k = shear yield stress = o'o/V'3.

limitations for single parameter J characterization must depend upon geometry. Finite strain, finite element calculations by McMeeking and Parks26 suggests that these critical size limitations, in terms of the uncracked ligament dimension b, vary from

b > 25 J/cro, for the edge-cracked bend specimen [8]


b > 200 J/o%,

for the center-cracked tension specimen

for materials of moderately low strain hardening (n = 0.1), where o'0 is the flow stress, usually defined as the mean of the yield and ultimate tensile strengths.

C. Local (Microscopic) Fracture Initiation Criteria
Since both macroscopic criteria, based on Kt or J, result from the asymptotic continuum mechanics characterization, realistic evaluation of toughness using Kic or J~ does not necessitate any microscopic understanding of the fracture events involved. However, in the interest of a full comprehension of a fracture process and specifically to define which microstructural features contribute to a material's toughness, it is often beneficial to construct microscopic models for specific fracture mechanisms. Such models are generally referred to as "micromechanisms". Unlike the continuum approach, this requires a microscopic model for

the particular fracture mode, which incorporates a local failure criterion and consideration of salient microstructural features, as well as detailed knowledge of both the asymptotic and very-near tip stress and deformation fields. Physical fracture processes, and consequently the local failure criterion and characteristic microstructural dimensions, vary substantially, however, with fracture mode, as Figure 7 illustrates for the four classical fracture morphologies, i.e., microvoid coalescence, quasi-cleavage, intergranular, and transgranular cleavage.
In view of the specificity of such models to particular fracture mechanisms for particular microstructures, a complete microscopic/macroscopic characterization of toughness has been achieved only in a few simplified cases. For example, for slip-initiated transgranular cleavage fracture (Figure 7(d)) in ferritic steels, Ritchie, Knott, and Rice (RKR) 27have shown that the onset of brittle crack extension at KI = K~cis consistent with a critical stress model in which the local tensile opening stress (~ryy)directly ahead of the crack must exceed a local fracture stress (o-?)* over a micro-

*Extensive studies on cleavage fracture in mild steels indicate that ere*is essentially independent of temperature below the ductile/brittle transition (see Reference 4).

structurally significant characteristic distance (x = l*), as depicted in Figure 8(a). Using the HRR field in Eq. [5] to define the crack tip stress field, the RKR model for the cleavage fracture toughness implies: 27'28'29

K,,, oc [(o-i*)t'+")n/(o-o)(a-n)n]l*'n '


where the proportionality factor is simply a function of In in the HRR solution, which can be inferred from tabulations in Reference 21.
In mild steels, with ferrite/carbide microstructures, the characteristic distance was found to be on the order of the spacing of the void initiating grain boundary carbides, i.e., typically - two grain diameters (dg), 27 although different size scales have been found when the analysis is applied to other materials.* The model has been found to be particu-

*In addition, recent modeling studies by Evans 3~ of cleavage in mild steel, using weakest link statistical considerations of the size distribution of cracked carbides, have interpreted the characteristic distance as the carbide location with the highest elemental failure probability pertinent to crack advance.

larly successful both in quantitatively predicting cleavage fracture toughness values in a wide range of microstructures and furthermore in rationalizing the influence on K~cof such variables as temperature,27'29'3~strain rate, 29.31,32neutron irradiation, 29,32 warm prestressing, 33 and so forth. Somewhat
similar microscopic models involving a critical stress criteflon have been suggested for other fracture modes, including intergranular cracking (Figure 7(c)) in temper embrittled steels 34'35 and hydrogen-assisted fracture. 36
For initiation of ductile fracture by microvoid coalescence (Figure 7(a)), McClintock, 9 Rice and Johnson, 22 and Rice and Tracey37 considered the criterion that the critical crack tip opening displacement must exceed half the mean voidinitiating particle spacing (i.e., 2t~ i ~ lo* ~ dp), based on the notion that, in nonhardening materials, this would take place when the void sites are first enveloped by the intense strain region at the crack tip (i.e., at distance x - 23 from the tip). This model implies that:

3i = ~c ~ (0.5 to 2)dp,



Fig. 7--Classical fracture morphologies showing (a) microvoid coalescence, (b) quasi-cleavage, (c) intergranular cracking, and (d) transgranular cleavage. Fractographs (a) and (c) obtained using scanning electron microscopy whereas (b) and (d) are from transmission electron microscopy replicas.


Jlc -- o-ol*,


although it is unusual to find the fracture toughness to increase directly with increasing strength.
This problem is overcome by the approach of McClintock,38 Mackenzie et a1.,39 and others29'4~who have alter-
natively utilized a stress-modified critical strain criterion. Here, at J = J[c, the local equivalent plastic strain ~p must exceed a critical fracture strain or ductility ~'(trm/~), specific to the relevant stress state, over a characteristic distance l* comparable with the mean spacing (dp) of the void initiating particles, as shown schematically in Figure 8(b). Following the approach of Ritchie et al. ,29the near-tip strain distribution ~e from Figure 4 is considered in terms of distance (r = x) directly ahead of the crack, normalized with respect to the crack tip opening displacement &

ep ~

~ Cl ,


where c~ is of order unity. The crack initiation criterion of -ep exceeding ~ ' ( o - J ~ ) over x --- I* ~ dp at J = Jl~ now implies a ductile fracture toughness of: 29



~ esl~,



J i c - tr0e~lo* ,



Kic -- N/~I~E' ~ N/E'tro-eTl * .


Unlike the critical CTOD criterion (Eq. [lib]), the stressmodified critical strain criterion (Eq. [13]) now implies that J~c for ductile fracture is proportional to strength times ductility, which is more physically realistic and permits rationalization of the toughness-strength relation for cases where microstructural changes which increase strength also cause a more rapid reduction in the critical fracture strain. Furthermore, in terms of a critical plastic zone size for Mode I fracture initiation, ryl, it implies that:

ryi ~- --l*--,


7T E o

where e0 is the yield strain ( t r o / E ) , and ot in Eq. [1] is taken as 0.5.
There is no conceptual difficulty with the term eT, but defining it as a material constant has some difficulties in

VOLUME 16A, FEBRUARY 1985--237

5 -~ " r

.,, 3 - ~ ,,',









1 0.02







I _-._(~O~'_mdp/ O 'j)e(pl l_l ~-. e f -" - over x~ ~ '

0 i




x / 8 ~--


8 i

Fig. 8 -- Schematic idealization of microscopic fracture criteria pertaining to (i) critical stress-controlled model for cleavage fracture (RKR) and (ii) critical stress-modified critical strain-controlled model for microvoid coalescence.

practice. It cannot, for example, necessarily be equated to either the tensile or plane strain ductilities as conventionally measured. Analysis by Rice and Tracey37 for the rate of void expansion in the triaxial stress field ahead of a crack tip in a nonhardening material, in terms of the void radius Rp, suggests:

dRp = 0.28d~p exp(1.5urm/~).



For an array of void initiating particles of diameter De and mean spacing dp, setting the initial void radius to Dp/2 and
integrating Eq. [15] to the point of ductile fracture initiation gives an expression for the fracture strain, ~l*, as




e~ ~ 0.28 exp(1.5o'm/~)'

An earlier analysis by McClintock38 for a strain hardening material (of exponent n) containing cylindrical holes similarly suggests:

238--VOLUME 16A, FEBRUARY 1985


ln(dplDp) (1 - n)


e7 sinh[(1 - n)(O'a + cr~)/(2~/X/3)]'

where O'a and cr~ are the transverse stress components. Although both analyses consider the fracture strain to be
limited by the simple impingement of the growing voids and thus tend to overestimate ~' by ignoring prior coalescence due to shear banding by strain localization, they correctly suggest a dependence of eI* on stress state (o-m/~), strain
hardening (n), and purity (dJDp). For example, a large effect of stress state (i.e., triaxiality) on fracture strain is
predicted such that from Eq. [17], eI* would be expected to be reduced by an order of magnitude by going from an unnotched plane strain condition to that ahead of a sharp crack. Increased strain hardening, however, can enhance ~I*, particularly at high triaxiality, but the benefits of in-
creased purity (i.e., increased hole spacing dp) are pronounced only at low DJdp ratios due to the logarithmic
terms in Eqs. [16] and [17]. For example, reducing the volume fraction f, of inclusions from 0.001 to 0.000001 would increase ~ ' only by a factor of 2. 20
More recently, a local means of evaluating ~I* has been suggested41 through use of the fracture surface micro-
roughness M, defined42as h/W in Figure 9(a) for microvoid
coalescence, and analogously43 for other locally-ductile
fracture modes as quasi-cleavage (Figure 7(b)), the tearing topography surface (TTS) and ductile intergranular, as shown in Figure 9(b). The basis for this approach is the recognition4] that the ratio of void height h to the diameter
Dp of the initiating particle is a measure of the local fracture
strain, such that:

-e~" = ln(h/Dp),


or, in terms4] of M and volume fractionfp of void-initiating particles:

e~ = -~- In



Thus, Eq. [13b] would be written as:
~r0 In[M2-~;~I*

The success of these microscopic models for crack initiation toughness can be appreciated in Figure 10 where the RKR critical stress model for cleavage (Eq. [ 10]) and stressmodified critical strain model for ductile fracture (Eq. [ 13])








Fig. 9--Definition of fracture surface roughness, M = h/w, for
(a) microvoid coalescence and (b) other locally ductile fracture modes, such as quasi-cleavage.






240 - SA533B-I (HSST 02) - IbA




;E~IX 10"2s "1

dQ- 2S/~m






9: !i:: : : i ! i : : i i "
9 :





- , 1-T TO 11-T CT & WOL (L-T)

z 3: 120








; o


:::i: :

:9 ~ @

o :" ,, ...... ":: "'

~o* = lO0#'m= 4dg

l-- 80 -




,, : : : ~ : oi a:i: ~* ::~: : :

75/~m," 3dg


9: ~::~. . . . .::::"

Jto~ 50~m " 2dg

' 4O




..... "

~'0" " 350p, m
9.o" " 300 p.m

- 160

- 120 z 3: r

O I..ua

- 80


Iu ,< m

- 40




















Fig. 10--Comparison of experimentally measured fracture toughness KI,. data for crack initiation in SA533B-1 nuclear pressure vessel steel (~o - 500 MPa) with predicted values based on RKR critical stress model for cleavage on the lower shelf (Eq. [10]). and on the stress-modified critical strain model for microvoid coalescence on the upper shelf (Eq. [13]), after Ref. 29.

are utilized to predict the respective lower and upper shelf toughness in ASTM A533B-I nuclear pressure vessel steel. 29Whereas the characteristic distance (l*) for cleavage fracture scales approximately with 2 to 4 times the grain size (essentially the bainite packet size), for ductile fracture 1" was found to be approximately five to six times the average major inclusion* spacing (dp).
*The alloy contained around 0.12 vol pct of manganese sulfide and aluminum oxide inclusions, roughly 5 to 10/xm in diameter.
A. Crack Tip Fields for Nonstationary Cracks
Neglecting large-scale crack tip geometry changes, the plane strain near-tip stress state for the stationary tensile crack described above can be represented by the Prandlt slip-line field (Figure ll(a)). This applies for a monotonically loaded crack under conditions of well contained yielding and at large-scale and general yielding in certain highly constrained configurations. For the nonstationary tensile crack, however, where applied load continuously varies with crack length, a, there are small differences in the crack tip stress field (Figure 12). Exact asymptotic analysis by Drugan, Rice, and Sham,44 and earlier analyses by Slepyan,45 Gao, 46 and Rice and co=workers 44'4v'48for



@ Y






v I


Fig. 11 - - P l a n e strain slip-line representation of the crack tip stress states of the Prandtl field for (a) stationary crack, and (b) modified with an elastic loading sector behind the tip for a nonstationary crack (after Refs. 44 and 48).

VOLUME 16A, FEBRUARY 1985--239

3 . . . . . . ~__YY= (2ยง k k(H-'n') k

Prondtl Field Exact Growing Crack Solution,
u :0.3




I -'1 ~176 I





180 8 (de(J)

Fig. 12--Comparison of local stresses o-~ ahead of the crack tip in the plane strain as a function of angle 0 for (a) stationary crack based on Prandtl field of Fig. 1l(a), and (b) nonstationary crack based on exact solution for v = 0.3 of the field shown in Fig. ll(b), which contains an elastic unloading sector, after Ref. 48. Note how the stress distribution is unchanged by the growing crack, except for 0 >~ 110~

quasi-static plane strain Mode I crack advance in an elastic-perfectly plastic solid have shown that stresses are unchanged from the Prandtl field for the stationary crack (i.e., numerical solutions within -+1 pet) except behind the tip in the neighborhood of 0 = 135 deg where differences of the order of 10 pet result from the presence of a wedge of elastic unloading between approximately 0 = 112 to 162 deg (Figure ll(b)).
The important point, however, about this crack tip field is that the strain distribution is quite different in that, at a fixed Kt, the strain at a given distance from the crack tip in the plastic zone of a stationary crack is larger than in the case of a nonstationary crack.44-s2This follows from the distinctly nonproportional straining of material elements above and below the crack plane for a growing crack, compared to the predominately proportional plastic straining of material elements near a stationary crack tip. As an elastic-plastic solid is more resistant to nonproportional strain histories, stable crack growth can result.5z As shown in Figure 4, the strains decay as 1/r ahead of a stationary crack in an elasticperfectly plastic solid, whereas for a nonstationary crack, the strain singularity is weaker, decaying as a function of In(l/r).
Asymptotic analyses of the strain fields for a growing Mode III crack were first reported over a decade ago by Chitaley and McClintock49 for elastic-perfectly plastic solids, and later by Hutchinson and co-workers5]'52for linear and power hardening solids.
For an elastic-perfectly plastic solid, the Mode III solutions for the shear strain 7p distance r ahead of the tip are

given in terms of the plastic zone size ry and the shear yield strain 70 = k / G as: 49

(ry], for the stationary crack [21] 7o \ r /

~ = 1 + In 70

+ -~- In2 ,

for the nonstation-

ary crack


where the plastic zones for stationary and growing cracks are assumed to be of equal size* and given approximately in

*Numerical calculations for Mode 148suggest that the plastic zone for the
growing crack (rr extends roughly 15 to 30 pet beyond the stationary crack (ry). This difference has been estimated49 to be smaller for Mode III.

terms of the stress intensity Kni as:

1 (Km] 2 ' ry = "~ \ T / ~- ry .


Although much work has been focused on the corresponding Mode I situation,44-48'5~ exact asymptotic solutions for
the growing plane strain tensile crack have only recently been presented by Rice, Drugan, and Sham for nonhardening solids.44 The latter solution, based on the flow theory of plasticity, shows that the opening displacement between the upper and lower crack surfaces 6 very near the crack tip can be written as: 44'48

6 __ ardJ+ o'o da

flr~Oln E




where the proportionality factors a and fl are defined numerically44 as ~-0.6 and 5.642 (for u = 0.3), respectively, e is the natural logarithm base = 2.718, and r~ is identified as approximately the maximum extent of the plastic zone size, given in Mode I by:

, seJ


ry = o'~ (0.11 0.13) o'0


The equivalent plastic shear strain distribution at small distances r directly above and below the advancing Mode I crack tip is given, in the limit of r --->0, as:48

m dJ+ 1"88(2--u)~176


yp - trod a

where the parameters m and L are undetermined by the asymptotic analysis, although L can be identified with the extent of the plastic zone size r~.53
The form of the expressions for opening displacements and shear strains yp ahead of a growing Mode I crack (Eqs. [24] and [26], respectively) both show a first term which represents the effect of proportional plastic strain increments due to crack-tip blunting of the stationary crack while the second term represents the effect of additional nonproportional plastic strain increments caused by the advance of the crack, as illustrated schematically in Figure 13.

B. Continuum (Macroscopic) Fracture Criteria
As noted above, the near tip vicinity of a growing tensile crack involves regions of elastic unloading and nonproportional plastic loading (Figure 13), both of which are inadequately described by the deformation theory of plasticity

240--VOLUME 16A, FEBRUARY 1985


C. Local (Microscopic) Fracture Criteria
A critical strain-based microscopic criterion for ductile crack growth was first proposed by McClintock and Irwin~5 for Mode III crack extension under elastic-perfectly plastic conditions and involved the attainment of a critical shear strain ~/~' over some characteristic radial distance r = I* into the plastic zone. Applying this local criterion for yp > y~ over distance r = l* both for crack initiation, using the Mode III plastic shear strain distribution for the stationary crack (Eq. [21]), and for crack growth, using the corresponding distribution for the nonstationary crack (Eq. [22]), yields estimates for the critical plastic zone sizes at initiation and instability, respectively, i.e.:

ryi = l* Y ~ , (initiation)




ryc = I* exp

1 , (instability) [33]

where Y0 is the shear yield strain. With the assumption that the critical fracture strains and distances are identical for initiation and growth, restated in terms of K[II or J (using Eq. [23]), this implies that stable crack growth would occur with Km or J increasing from initiation values (Ki, Ji) to steady-state values (Ks, when such terminology is appropriate, and J,s) where dJ/da --~ O, such that:

Ki/ - Ji y ? e x p 2

- 1 - 1 . [34]

Equation [34] implies that the potential for stable crack growth increases dramatically as y]' becomes large compared to the yield strain yo, although subsequent analyses52 for hardening solids have shown this potential to decrease with increases in strain hardening.
The concept of a critical strain being attained over some characteristic dimension directly ahead of a growing crack is not as amenable for the nonstationary Mode I case since the regions of intense strain are directly above and below the crack plane. Accordingly, Rice and his co-workers 44'47"48'5~
have proposed several alternative local failure criteria for initiation and continued growth of plane strain tensile cracks, all involving the notion of a geometrically-similar crack profile very near the tip. Prior to the development of the exact asymptotic analyses for the growing Mode I crack tip fields (Eqs. [24] through [26]), this was formulated as a constant crack tip opening angle (CTOA = dr/da) for crack growth, 9'5~as shown schematically in Figure 14. The crack tip displacement at the advancing crack tip 6p remains constant, whereas the crack tip displacement 6 at the original crack tip is increased by the amount of opening (Sp) to advance the ductile crack one particle spacing l* ~ dp for each increment of crack growth. With reference to Figure 14, the constant crack opening angle ~b is given by: 56'57

CTOA = 4~ = arctan 2 - " 2 d J = arctan -~-d6/da .
[35] Although for a rigid-plastic solid crack advance can occur with a finite CTOA, elastic-plastic analyses result in a







Fig. 14--Idealization of stable crack growth by microvoid coalescence showing (a) blunted crack tip, (b) crack growth to next inclusion based on constant CTOA (qS)or on critical CTOD (~p) distance (l* - d p ) behind the crack tip, (c) morphology of resulting fracture surface relevant to the definition of fracture surface microroughness (M = h/w), and (d) fractographic section (after Ref. 4) through ductile crack growth via coalescence of voids in free-cutting mild steel.

crack face profile with a vertical tangent immediately at the crack tip (i.e., as r ~ 0), thus making the CTOA impossible to define numerically. 44 Accordingly Rice and
Sorensen restated the crack growth criterion in more general fashion by requiring that a critical opening displacement 6p be maintained at a small distance l* behind the crack tip. 47With reference to Eq. [24], the local criterion of 6 = 6p at r = l* yields:

6 p _ ~ dJ + /3 In ( re~ ~)


l* cro da

By comparing Figures 9 and 14, it is apparent that the left side of Eq. [36], the ratio of local microscopic parameters, 6e/l*, can be identified with the fracture surface microroughness, M = h/w, for microvoid coalescence and possibly other modes. This point is further discussed in the following section. Rice and co-workers,48 however, have

242--VOLUME 16A, FEBRUARY 1985


Region of elastic

Region of nearly -proportional ~ loading, J-field

0 \kN\



non-ProPorflonol ~


plostic loeding ~

R=rodius of HRR field
Fig. 13--Schematic representationof the near-tip conditionsfor a nonstationary crack relevantto the definitionof J-controlled growth (after Ref. 10).

upon which J is based. ~~Following the deformation theory
analysis of Hutchinson and Paris, 9 which utilizes the incre-
mental form of the HRR singularity (Eq. [3]), i.e.:

deq(r'O)--~(~-~,r)~J"+J{ +l 1 J gu + dah'~r '~]

where O
h~ _ n +1~1 cos0 g~ + sin0 -~go(O) [27]

regions of elastic unloading, comparable with the scale of crack advance Aa, and nonproportional loading are assumed to be embedded within the HRR J-controlled singularity field of radius R (Figure 13). Their argument for J-controlled crack extension relies on the fact that these regions remain small compared to the radius of the HRR field, such that the singularity field can be said to be controlling. For the region of elastic unloading to be small, the increment of crack extension (Aa) must be small compared with the radius of the HRR field (R), whereas for the region of nonproportionality to be small, J must increase rapidly with crack extension. With reference to the form of Eq. [27], where, similar to Eq. [26], the first term corresponds to proportional load increments and the second to nonproportional load increments, the latter condition is achieved when the proportional straining (first) term dominates, i.e., when:

dJ J



> >r- ,



Aa ~ R,


Based on this nonlinear elastic analysis of crack growth9 and numerical computations by Shih and co-workers,8 the two requirements in Eq. [28] can be embodied into a single condition for J to uniquely characterize the near tip field of

the growth crack. Thus, in terms of the uncracked ligament b, J~c and the slope of the JR (Aa) resistance curve, J-controlled growth is feasible when:

w -=

-> 1,


where to must exceed 10 for Prandtl field geometries (e.g.,
deep-cracked single-edge-notch bend) and - 100 for centercracked tension geometries (for n -~ 0.1).
A similar criterion can be applied for the asymptotic crack tip fields for elastic-ideally plastic crack growth based on flow theory (Eq. [26]). For J-dominated crack extension, the first term in Eq. [26], representing proportional strain increments, must dominate the second term, representing nonproportional strain increments, such that:*

*A similar criterion based on crack tip opening displacements implies that the crack tip opening angle d r / d a must be large compared to the yield strain tro/E. ~


To provide a continuum measure of crack growth tough-
ness, the deformation theory analysis of Hutchinson and
Paris9 is applied to define the conditions for J-controlled
growth (Eq. [28]), and macroscopic toughness is then
assessed in terms of the tearing modulus, TR, representing the nondimensional slope of the JR(Aa) curve, TR = (E/o'~)dJ Ida. Crack instability is achieved when the tearing force, T =-(E/o'~)OJ/Oa exceeds TR. Analogous proce-
dures8 based on crack tip opening displacement have also
been suggested in terms of the nondimensional crack tip
opening angle, defined as dr/da normalized with respect to the yield strain oo/E' (Eq. [2]).
However, practically speaking, the deformation theory J
approach for macroscopic crack growth toughness9 is often
severely restricted by the limitation of Eq. [29]. For ex-
ample, ~zfor a 25 mm thick 1-T compact specimen in plane
strain, deformation J-controlled growth is only a reality for
the first 1.5 to 2.0 mm of crack extension (i.e., where
Aa < 0.06b), 8 whereas for a similar sized center-cracked
tension specimen, it is valid merely for the initial 0.5 mm or
so of a 25 mm ligament (i.e., where Aa < 0.016b). 8 This
means that for further crack extension, the shape of the
JR(Aa) resistance curve, and hence TR, for a given material
will differ with specimen geometry and with varying liga-
ment size in a given geometry. To overcome this problem,
Ernst54 has recently proposed a modified J parameter, JM,
based in part on the flow theory solution for the non-
stationary crack tip field (Eqs. [24] through [26]), in which:

JM = J - -[ amp' da,


J.ao Oa apt

where Jpt is the plastic portion of the deformation theory J,
evaluated at a fixed plastic load point displacement ~pl over
the extent of crack extension (a - a0). Use of this modified
J-integral, Jg, and associated modified tearing moduli, has
been shown to extend greatly the validity of J-controlled
growth, even to situations where oJ < 0 and Aa ~ 0.3b which normally would grossly violate the deformation theory requirement of Eq. [29]. 54


VOLUME 16A, FEBRUARY 1985--241
CrackStrainTermsRiceCrack Growth