An engineering model for propagation of small cracks in fatigue

Engineering Fracture Mechanics Vol. 56, No. 3, pp. 357-377, 1997

Pergamon

PII: S0013-7944(96)00057-4

Copyright © 1996 ElsevierScience Ltd Printed in Great Britain. All fights reserved 0013-7944/97 $17.00 + 0.00

A N E N G I N E E R I N G M O D E L F O R P R O P A G A T I O N OF SMALL CRACKS IN FATIGUE D. L. MCDOWELL George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, U.S.A. Abstract--The propagation of small cracks in fatigue has received considerable attention over the past decade. Microstructurally and even mechanically small cracks have been shown to consistently exhibit higher crack growth rates than predicted using standard threshold and Paris growth law concepts, based on linear elastic fracture mechanics (LEFM) applied to mechanically long cracks. This has been commonly attributed to several factors, including the influence of microstructure, the breakdown of LEFM parameters for representation of the crack tip field, and the transient development of plasticityinduced closure towards some steady-state value associated with long crack behavior. Other mechanisms related to microstructural effects such as roughness-induced closure and crack face bridging/interference are also potential contributors. Quantitative attempts to explain the fatigue propagation of small cracks in terms of plasticity-induced closure, along with adoption of an additional component of the driving force (e.g. crack tip opening displacement) to reflect the contribution of cyclic plastic strain have met with some success in correlation of the so-called "anomalous" propagation behavior, including crack deceleration and acceleration transients. However, these models rely on the adoption of highly idealized assumptions regarding the self-similarity of crack growth, neglect of local anisotropy and heterogeneity associated with microstructure, etc.; in spite of these compromises, they still involve a considerable degree of complexity. Here, we adopt the viewpoint that multiple, microstructure interactions and closure effects may simultaneously influence the propagation of small cracks; moreover, driving force parameters based on self-similar crack growth arguments of elastic-plastic fracture mechanics (EPFM) for mechanically long cracks, such as the cyclic J-integral or crack tip opening displacement, may apply in principle, but not rigorously, as the driving force for small cracks. As an engineering approach, we consider a recent extension of the multiaxial microcrack propagation model first proposed by McDowell and Berard[1,2] for the growth of microstructurally small and mechanically small fatigue cracks[3] in multiaxial fatigue. Integrated between initial and final crack lengths, the model is fully consistent with standard strain-life laws of fatigue crack initiation mechanics under various states of stress, [1,2] and therefore bridges the mechanics of classical initiation and LEFM/EPFM to some extent. The existence of a fatigue limit (nonpropagating crack limit) is neglected in this particular work. It is shown for uniaxial loading of both 1045 steel and Inconel 718 that the model is able to describe, to first order, the anomalous high propagation rates of small cracks and convergence with long crack d a / d N - A K data as the crack transitions from small to mechanically long scales. The limits of validity of engineering schemes based on decomposition of total fatigue life into "initiation" and propagation phases that rely on strainlife and long crack propagation laws are discussed[4]. Moreover, it is shown that the model essentially reflects a closure transient in the context of a cyclic J-integral approach, similar to the EPFM plasticity-induced closure modeling concepts set forth by Newman [5] and McClung et al. [6] for small cracks in fatigue. However, the present model implicitly reflects multiple forms of crack tip shielding effects, not just plasticity-induced closure. Finally, the model is shown to provide realistic treatment of cumulative damage in two level loading sequences, as reflected by comparison with the damage curve approach of Manson and Halford[7]. Copyright © 1996 Elsevier Science Ltd

NOMENCLATURE

#o, ,Sp

A7 Aymax

~m~x A~ AEe, A~P AJ AJ¢, AJp AO'n, ATn Aft

AK

aKj

constant in DCA approach biaxiality functions for fully elastic and plastic cases, respectively range of engineering shear strain maximum engineering shear strain range elastic part of Aymax plastic part of Aymax range of axial strain ranges of uniaxial elastic and plastic strain, respectively cyclic J-integral based on range of stress and strain fields elastic and plastic components of cyclic J-integral, respectively normal and shear stress ranges, respectively, acting on the plane of maximum shear strain range range of axial stress range of stress intensity factor equivalent AK based on range of elastic-plastic cyclic J-integral 357

358

Eft, yft /z p Of/, "t'f/ 0", G m 0"rain, 0"ma x 0"no 170 0"op 0", "C

~Pe, ~p a

b, bo C, c o

d

da/dN kd co, Cp E,G Je, jp jme, imp A, n

K', Ko'

M,m my

Nr l'l~, n o r

R

Rn

D. L. MCDOWELL fatigue ductility coefficients in tension and torsion, respectively constraint coefficient constraint parameter fatigue strength coefficients in pure tension and torsion, respectively stress and mean stress, respectively, in uniaxial case minimum and maximum uniaxial stress levels, respectively mean stress normal to plane of maximum shear strain range average flow stress measure for cyclically stable material opening stress level remote axial stress and shear stress driving force parameters in fully elastic and plastic microcrack propagation laws, respectively crack length fatigue strength exponents in tension and torsion, respectively fatigue ductility exponents in tension and torsion, respectively grain size cyclic crack growth rate transition crack length, where k is a scaling factor fully elastic and plastic microcrack propagation law coefficients, respectively Young's modulus and shear modulus, respectively biaxiality exponents for fully elastic and plastic forms for da/dN, respectively mean stress exponents for fully elastic and plastic forms for da/dN, respectively constants in fatigue crack propagation law for fully open cracks cyclic strength coefficients in pure tension and torsion, respectively exponents for fully elastic and plastic terms in daMN, respectively crack length exponent in da/dN law number of cycles to fatigue crack of prescribed length considered as "failure" cyclic strain hardening exponents in pure tension and torsion, respectively 0"min/O'max in uniaxial case the ratio A~max/Aymax the ratio Atrn/Azn.

INTRODUCTION THE PROBLEM o f the fatigue c r a c k g r o w t h o f small c r a c k s (from the o r d e r o f 1 # m to 5 0 0 1000/~m) has received m u c h recent a t t e n t i o n in the fracture mechanics c o m m u n i t y . C r a c k s are c o n s i d e r e d as small when all p e r t i n e n t d i m e n s i o n s are small c o m p a r e d to some characteristic length scale. In the case o f m i c r o s t r u c t u r a l l y small cracks, this length scale is on the o r d e r o f the d i m e n s i o n s o f m i c r o s t r u c t u r a l p e r i o d i c i t y (e.g. g r a i n diameter). F o r physically o r m e c h a n i c a l l y small cracks, it is typically on the o r d e r o f 5 - 1 0 times the m i c r o s t r u c t u r a l scale. M e c h a n i c a l l y long cracks are in excess o f p e r h a p s 10-20 times this dimension. A t t e m p t i n g to d e v e l o p a correlation between d a M N a n d A K as in the case o f m e c h a n i c a l l y long cracks, the so-called a n o m a lous b e h a v i o r o f m i c r o s t r u c t u r a l l y a n d p h y s i c a l l y small c r a c k s has been widely d e m o n s t r a t e d . In

• largecrack

/

S, i.~'x Xx ..xx xx~xX,q"

I

Fig. 1. Typical propagation behavior of small cracks. Note that da/dN is higher for a given AK than for long cracks, and the apparent scatter in da/dN is significant (from Ref.[6]).

Propagation of small cracks in fatigue

359 /

MFM ,-\high A~_ / ',

,, I

EPFM

-. , t

,, I

~....

.-~ '. !

[

b

.--\._,) / :

, ~

, ~'

:

/i /

\../

Z

~.~ d

low Ao (kd) Crack

Length, a

Fig. 2. Schematic of typical progression of crack growth rate behavior as a function of crack length and stress amplitude for microstructurally small, physically small and long cracks.

particular, the cyclic crack growth rate of small cracks exceeds that of long cracks at the same level of AK, as shown in Fig. 1. The apparent scatter of fatigue crack growth rate of small cracks at a given AK level is considerable. Furthermore, particularly at lower stress amplitudes, deceleration of crack growth is observed which is associated with a dip in the da/dN vs AK behavior. As small cracks propagate, their da/dN vs AK responses are typically observed to eventually merge with the long crack response as shown in Figs 1 and 2. Clearly, the concept of a threshold AK for fatigue crack growth based on long crack experiments is inapplicable to small cracks since they may grow vigorously at even lower AK levels. An important experimental observation is that the propagation behavior of microstructurally small and physically small cracks depends significantly on both the R-ratio and stress amplitude. Although small crack behavior is more subject to scatter due to dependence on microstructure, this observation explains that some of the apparent scatter in da/dN is really due to the non-uniqueness of the LEFM da/dN vs AK relation. Moreover, this scatter essentially remains even if standard EPFM parameters are used to account for effects of plasticity. It is generally observed that higher stress amplitudes lead to higher, more nearly constant values of da/dN at a given AK level for small cracks prior to merging with the long crack response. As the stress amplitude decreases, da/dN tends to depend more strongly on AK. At low stress amplitudes, the crack growth rate may decrease with AK and then accelerate prior to merging with the long crack data. At sufficiently low amplitudes, small cracks may become arrested, nonpropagating cracks. There are several prevalent explanations for the nonconformity of small/short crack behavior with that of mechanically long cracks. These include: •

• •

•

plasticity-induced closure transients; microstructural roughness-induced closure/bridging, crack deflection and interaction with microstructural features; violation of validity limits of LEFM or EPFM, and intensification of local driving forces relative to nominal applied stresses and strains due to heterogeneity of cyclic slip in the vicinity of the small crack(s); mixed mode growth effects for small cracks, even under apparent mode I remote loading.

Indeed, all of these factors may contribute significantly to the fatigue crack growth behavior of a given material, with one or more weighing more heavily at high stress amplitudes and others at low stress amplitudes, for a given R-ratio. It is a potentially very complex set of physical mechanisms to address with a detailed model. Consequently, as a practical matter only certain

360

D.L. MCDOWELL

of these aspects are considered by current models, with detailed treatment based on neglect of the other factors. For example, the treatment of plasticity-induced closure (see Refs [5, 6]) typically assumes validity of LEFM or EPFM concepts, even for microstructurally small cracks, while neglecting microstructural roughness-induced closure/bridging or interaction with microstructural features. Even with this simplification, the application of plasticity-induced closure models requires a considerable level of idealization and assumption. On the other hand, models which consider interaction with periodic microstructural barriers (see Refs [8-12]) typically do not consider closure or bridging effects, although they may recognize the lack of applicability of LEFM or EPFM for small cracks. In addition, models for the growth of small cracks have largely been confined to simple uniaxial (mode I) loading conditions; formal treatment of multiaxial loading conditions within the fracture mechanics methodology is indeed formidable in view of the plethora of mechanisms and local mixity of crack tip opening and sliding displacements, distinct from the remote loading history. The range of validity of LEFM or EPFM concepts diminishes even further under such general loading conditions. As perhaps is clear from the preceding discussion, it is extremely challenging, if indeed possible at all, to explicitly include all the pertinent mechanisms and modes of growth in a fatigue crack growth law for microcracks. Because of the complexity of the physics of the problem, practical application of "fracture mechanics-like" approaches for propagation of small cracks requires estimating the values of constants and parameters employed in the idealization by essentially fitting experimental data. From an entirely different viewpoint, fatigue crack "initiation" approaches (e.g. strain-life or stress-life) have been applied for many years to address the small crack problem. In these approaches, the fatigue life for a crack of preassigned dimension (e.g. 1 mm) is correlated with the remote or bulk cyclic stress and/or strain amplitude, along with the influence of mean stress. These approaches tend to be highly empirical, with the implicit incorporation of various mechanisms affecting propagation of small cracks. A major deficiency of initiation approaches is their lack of explicit treatment of evolution of microcracks; consequently, correlation of amplitude sequence effects requires adoption of some measure of cumulative damage, such as cycle fraction, which is not physically descriptive. Another deficiency is the lack of consistency of the fatigue crack initiation approach with LEFM or EPFM methodologies for long cracks. In particular, the precise definition of the crack length at which long crack mechanics apply has not reached consensus. This is in part due to the fact that the common definition of "initiation" in reality pertains to both nucleation and propagation of microcracks. A major strength of the fatigue crack initiation approach is its capability to correlate fatigue life and microcrack orientation under a wide range of multiaxial loading conditions with relatively simple parameters [1, 2, 13, 14]. In contrast, fracture mechanics approaches have been much less ambitious in their treatment of combined stress states for small cracks. Several recent investigations have attempted to bridge the concepts of fatigue crack initiation and propagation of small cracks. Recent studies[15-17] have employed the AJ-integral of EPFM to correlate the uniaxial fatigue propagation behavior of small cracks. For combined tension and torsion fatigue, Hoshide and Socie[16], Socie et al. [19] and Berard et a/.[20], have shown that the simple bulk stress and strain range parameters used in critical plane fatigue crack initiation laws serve to correlate the fatigue crack propagation rate; in these particular studies, the dependence of da/ dN on crack length is that traditionally associated with EPFM. It has been shown [12, 15, 18] that the growth of small cracks may be correlated with a crack length dependence which differs from that of conventional EPFM. McDowell and Berard [1, 2] extended the form of the AJ-integral to address small crack growth along critical planes in multiaxial fatigue, addressing both case A and case B cracking [21,22]. This approach is based on the assumption of cyclic crack opening and sliding displacements as driving force parameters for microcrack extension [16], and was able to provide excellent correlation with experimental trends for completely reversed low cycle fatigue (LCF) under various stress states ranging from torsion to uniaxial to internal/external pressure [2]. Familiar strain-life relations were adhered to for constant amplitude, completely reversed uniaxial and torsional loading conditions to establish a link with existing databases containing cyclic stressstrain and strain-life constants for various materials. Extensions were made to include mean stress effects in a manner consistent with well-established fatigue crack initiation approaches.

Propagation of small cracks in fatigue

361

This work was recently extended by McDowell and Poindexter [3] to include stress amplitude dependent propagation of microcracks via appropriate dependence of da/dN on crack length. A length scale was introduced to mark the transition from microstructurally small crack propagation to the physically small crack growth regime which is treated with a variation of EPFM. In this paper, this simple engineering model is applied to predict fatigue crack growth of small cracks subjected to uniaxial loading of smooth specimens for two structural alloys for various amplitudes ranging from LCF to high cycle fatigue (HCF) conditions and for several Rratios. The results are discussed in terms of crack closure and crack face interference concepts which are conventionally held to affect small crack propagation behavior. A discussion of the capability of the model to correlate cumulative damage under sequences of loading amplitude is also presented in relation to the damage curve approach of Manson and Halford[7]. SOME PRACTICAL CONSIDERATIONS FOR SMALL CRACK PROPAGATION Although of great practical relevance, we do not address here the propagation of small cracks from a notch root field, which is strongly influenced by plasticity-induced closure transients related to the notch root geometry and level of applied loading [23, 24]. The case considered here, propagation in a uniform applied stress field, is of perhaps more relevance to intrinsic crack growth resistance which is subject to influence of microstructure[25], and serves as a prerequisite to more fundamental consideration of notch root behavior of small cracks. The model which has been developed by McDowell and co-workers [1-3] assumes the preexistence of micron-scale small cracks associated with thermo-mechanical processing. In its present form, therefore, the model does not include crack nucleation processes associated with the generation and coalescence of excess vacancies along persistent slip bands [26-29], prevalent in ductile single crystals. In polycrystals, cracks may nucleate via fracture during processing[26] at intersecting slip bands (or twin) or by blockage of a slip band (or twin) by second phase particles. A second type of microcracking in polycrystals occurs along grain boundaries due to impurity embrittlement, or the presence of voids. Sometimes microcracking in polycrystals occurs at strong grain boundaries due to heterogeneous plastic deformation, governed by the degree of misorientation at the grain boundary, usually associated with a mixed mode of intercrystalline-transcrystalline fracture. If sub-grain scale small cracks cannot bypass strong barriers at the microstructural scale such as grain boundaries, then a fatigue limit results [8,9, 11, 12, 30]. Likewise, elastic shakedown or cessation of cyclic micro-plastic flow may occur due to heterogeneity of yielding among grains, also leading to a fatigue limit[31]. In this study, we will not consider either of these two possibilities of nonpropagating cracks. It is known that microcrack growth transitions from Stage I shear-dominated to Stage II normal stress-dominated character after attaining some length suitably in excess of the grain size [11, 12, 14]. The precise dependence of this transition phenomenon on grain size, slip planarity, and local microstructural detail is not known. There is also an apparent dependence of this transition on stress amplitude and stress state, with higher amplitudes promoting the transition for shorter cracks and uniaxial loading promoting the transition as compared to torsional fatigue. We distinguish here between a regime in which the microcrack length is on the scale of the spacing of strong microstructural barriers (microstructurally small cracks [9-11]), and one for which the crack length sufficiently exceeds the spacing of barriers. In the latter case, the process zone is increasingly well-confined to the crack tip and contains several microstructural barrier length scales (mechanically or physically small cracks [9, 11]). Under the former conditions, the crack is typically engulfed within the zone of intense cyclic plastic deformation (e.g. slip band) and it is inappropriate to assign crack length dependence following standard elastic-plastic fracture mechanics (EPFM) arguments for long cracks in homogeneous media. The crack growth process is instead governed by the balance of local driving and resisting forces associated with periodic microstructural barriers. The term microstructural fracture mechanics (MFM) has been used to describe the microstructure-dependent mechanics of crack growth in this regime[11, 12]. Upon reaching sufficient length, typically on the order of several grain diameters, physically small cracks begin to exert an influence on cyclic deformation and damage processes at their tips in accordance with EPFM crack opening and singularity concepts.

362

D . L . MCDOWELL

EPFM concepts become applicable when the scale of cyclic plasticity and the crack tip process zone size are small compared to the crack length and span over a sufficient number of microstructural barriers; at this point, linear elastic fracture mechanics (LEFM) may apply provided the crack tip plasticity is small scale. However, if the scale of crack tip plasticity is relatively small compared to the crack length but is still on the order of the microstructural barrier spacing, then microstructural influence may still be evident, manifesting a crack length dependence of the crack growth rate which varies with the stress (or strain) amplitude. The dependence on crack length under LCF conditions has often been reported to be similar to that of long cracks [9, 15-20, 32, 31, 33]. Relative to low amplitudes, such high amplitude loading conditions promote more extensive slip band formation and effective traverse of barriers in front of the advancing crack. As a result, an approach which is applicable to physically small cracks is similar in nature but not identical to standard EPFM which applies to long cracks with wellconfined crack tip process zones that enclose a statistically homogeneous array of microstructural obstacles. For this reason, we shall refer to this class of EPFM approaches as EPFM u. Furthermore, the applied stress state is known to have a profound influence on the character and distribution of slip among grains[34,35], which in turn affects the scale of the crack tip plasticity and the process zone relative to the microstructural barrier scale. An idealized schematic depicting the microcrack propagation through these various regimes is illustrated in Fig. 2, assuming pre-existing cracks. Note that the oscillatory character of fatigue crack growth is particularly pronounced for microstructurally small cracks and for low stress amplitudes. The EPFM u regime (a > kd, where a is the crack length, d is grain size, and k > 1) exhibits a relatively weaker dependence on microstructure, and the crack growth rate is observed to depend approximately linearly on crack length, i.e. da/dNoca [3, 11, 12, 15, 18]. Under LCF conditions, this linearity is very nearly consistent with EPFM concepts. Applicability of long crack EPFM/LEFM typically occurs at crack lengths on the order of several hundred to a thousand microns, depending on the microstructure, stress state and stress amplitude. The MFM and EPFMu regimes are characterized by a crack growth rate which exceeds that corresponding to LEFM parameters such as AK fit to experiments involving mechanically long cracks. The transition from Stage I to Stage II propagation of microcracks may be related to the transition from MFM to EPFMu regimes, although this has not been fully verified. MULTIAXIAL MICROCRACK PROPAGATION MODEL We make the heuristic assumption for purposes of an engineering model that the complexities of microstructural influence for small cracks below a transition crack length, kd, are taken into account in an average sense by suitable nonlinear dependence on crack length, amplitude and stress state. Moreover, we neglect here the specific possibility of small crack deceleration or arrest which may be associated with interactions with microstructural obstacles ahead of the crack, development of crack bridging, or transient development of plasticity-induced closure. Otherwise, effects of microstructural interactions are implicitly included via the nonlinear crack length dependence on the crack growth rate. Following McDowell and Poindexter [3], as a generalization of the McDowell-Berard model we introduce a microcrack propagation law of the form. da ( a ~ my d--N = DaN[ReC¢(*e)M + (1 - Re)Cp(*p) m] kd]

(1)

( A't'nA _~max) q'o = (1 + UR)(&Rn + 1) 2

(2)

O2P: (I + IzP)(flpRn + I)( 2 vn Aypax

(3)

where

and Re is a nondimensional number, 0 < Re < 1, representative of the relative degree of nominal elastic straining, i.e.

Propagation of small cracks in fatigue

363

Re = A~max

(4)

Aymax In eqs (2, 3), Aakk/2 p - 2A~n/2

Rn

(5)

where Rn = (A%/2)/(A*n/2); % and rn are the normal and shear stresses, respectively, on the plane of maximum range of shear strain. Parameter R, varies from zero for completely reversed torsional fatigue to unity for uniaxial or biaxial loading conditions. Parameters fie and tip introduce dependence of the crack opening and sliding displacements on stress ratios for fully elastic and plastic remote loading conditions, respectively; fie and tip reflect mixity and biaxiality of remote loading with respect to the crack plane. Secondary dependence of the microcrack propagation rate on the degree of biaxiality (or triaxiality) is introduced via constraint parameter p; [1,2] an increasing biaxiality ratio (ratio of in-plane surface principal stresses Atrl/Aa=) modifies the microcrack growth rate or, equivalently, fracture ductility in the integrated expression [1-3]. Note that p is nonzero only when multiple normal stress components are operative as in biaxial loading. Constant # controls the degree of this constraint effect. The first and second terms of eq. (1) correspond to fully elastic and fully plastic nominal conditions, respectively[I,2]. Coefficients Ce and Cp are given by

-(l+no~)m Ce = G M

Jnm+,RJmo "t'f' (O.n° +-~)A~"\Jme

Cp =(got)--m

'

yftno'

crjmp+l~Jmp vno *-n .

7 ) Ko' ( crno+

(6)

'm'

where %0 is the mean stress normal to the plane of maximum shear strain range. These forms for Ce and Cp recover classical strain-life and cyclic stress-strain laws for completely reversed loading and include mean stress effects. This formulation is insensitive to torsional mean stress, and the sensitivity to mean normal stress is established by the exponents Jme and J,np [2]. Specifically, increasingly positive values of these exponents lead to reduced sensitivity to mean stress. For jme = Jmp = 0, we recover the Morrow mean stress correction of the strain-life equation which is applicable to hard metals [2, 36]. We will conservatively assume that jme = Jmp = 0 for all calculations in this paper, Note that eq. (1) yields essentially the same life under either completely reversed LCF or HCF conditions as that given by the earlier McDowell-Berard model [1,2] integrated between selected initial and final crack lengths. However, a change of crack length dependence of da/dN in eq. (1) at a = kd affects the nonlinearity of crack growth between given initial and final limits. The effective crack length exponent, m r, is of a different form for each of the two regimes of propagation, i.e. my = fftv(Re, Rn, Ayeax) for --~<1; a m v = l for ~a > 1

(7)

The parameters A~max and Re are used, in addition to Rn, as independent variables in the nonlinear crack growth relation for a < kd. This decomposition of crack length dependence into two regimes is based on the observed role of microstructure in the nonlinear MFM growth relation (a < kd), while the behavior of physically small cracks for a > kd is in agreement with the observation that da/dN oc a for cracks beyond several grain diameters in length[9-12, 15, 18]. For high strain amplitudes in the MFM regime, m~ ~ 0 as Re ~ 0, in accordance with the observations of essentially crack length-independent growth [14] in high strain LCF. The simple treatment of nonlinearity associated with the MFM regime afforded by eq. (7) is recognized as an approximation valid only in the absence of microcrack deceleration effects or arrest, in general. Assuming that standard uniaxial and torsional strain-life and cyclic stress-strain relations, i.e.

364

D.L. MCDOWELL

A~

of #

AY2-- rf'(2Nf)b°G + Yf'(2Nr)C°' 2 - E (2Nf) b + Ef'(2Nr) c Ar = Ko'

o, Aa ~-K'

(8) (9)

are obeyed for completely reversed uniaxial and torsional loading from a given initial to final crack length under predominantly HCF and LCF conditions, respectively, we assign

[4( rf' "]2 -- 1](Rri)J"

tip =

i_3

Kt@ft)(l+n,

) -

]

1

(10)

(11)

(Rn) j"

in addition to the forms for C, and Cp in eq. (6). Homogeneous dependence on Rn is assumed in eqs (10, 11), so that fie = tip = 0 for pure torsion. Exponents Je and jp may differ, in general [2]. Both uniaxial and torsional tests are required to characterize most of the constants of this approach (except for #), since no universal relationship is known to exist which uniquely relates fatigue under these two states of stress. No assumptions are made here regarding equivalent relationships between uniaxial and pure torsional behavior. Material constant # in eqs (2, 3) can only be assessed from fatigue experiments involving cyclic normal stresses on orthogonal planes, e.g. biaxial tension-compression. To recover eq. (8) for uniaxial and torsional fatigue, we write exponents m and M[2] as -1 -1 m = c(Rn)(1 + n'(Rn)) ' M = 2b(Rn-----)

(12)

where the Rn-dependent parameters may be defined by interpolation between torsional and uniaxial cases, i.e. b(Rn) = bo + Rn(b - bo), c(Rn) = Co + Rn(c - Co), n'(Rn) = no' + Rn(n' - no')

(13)

The coefficient DaN in eq. (1) is determined by integrating the expression for constant amplitude and stress state loading conditions between given initial and final crack lengths, i.e. a i ( kd'] m~"- k d f a r / a \ -my

DaN = 2Li t~--~)

da=2kdln

(af)k_d

+2

k, a i ]

m×- 1

(14)

where it is understood that m r in the second term on the right-hand-side corresponds to the regime a < kd. The decomposition into MFM (a < kd) and EPFM u (a > kd) regimes is clearly reflected by the two terms of eq. (14). Since m r is a function of stress amplitude and stress state, it is apparent that DaN depends on these as well. The material length scale kd affects the nonlinearity of cumulative damage, along with the expression for m r in eq. (7) and differs from the microstructural threshold discussed by Tanaka [9] and Miller [11, 12] for the microstructurally small crack regime, for which the grain size or strong barrier spacing defines the fatigue limit for nonpropagating cracks. In the absence of definitive data regarding possible dependence of the transition crack length, kd, on stress state and amplitude, we assume here that kd is a constant, with k typically in the range of 3-10. This appears to be consistent with the observed domain of microstructural influence on crack length dependence [11,12, 37]. Within this framework, the existence of a fatigue limit under HCF conditions could be addressed as a nonpropagating crack with length on the order of d[8-12, 32], although this is not undertaken here. The orientation distribution function of grains and shakedown analyses for microscopic cyclic plastic deformation are known to be relevant to the existence of a fatigue limit as well [31], in accordance with MFM concepts. This model does not explicitly address a

Propagation of small cracks in fatigue

365

transition from Stage I crystallographic growth to Stage II growth normal to the direction of the range of maximum principal stress. FATIGUE CRACK PROPAGATION OF 1045 STEEL AND INCONEL 718 In this section, we consider the propagation of small cracks in smooth specimens, through the microstructurally and mechanically small crack regimes, and up to the onset of applicability of long crack fracture mechanics. McDowell and Berard [1, 2] demonstrated correlation of fatigue life (1 mm crack) to within a factor of two on life for a wide range of multiaxial loading conditions for several structural alloys by separately integrating different growth laws for LCF and HCF, and then adding the elastic and plastic components of maximum shear strain to form a total strain-life equation. Equation (1) yields essentially the same results as this procedure. For purposes of examining the performance of the proposed microcrack propagation law in eq. (1), we consider only uniaxial loading. Accordingly, the parameters of the growth law in eq. (1) simplify to rf'

q~e = 2 ( ~ f , )

21q-v

At7 2

---E-- (--~-)

Ko'(yr') (l+n°') Aa AeP ~P = Kt(ef') (l+n') 2 2

(15)

(16)

and

1 Re =

3 Ae p

1 + ~---~-((1

Aee "x-1

+v)--~-)

(17)

where v is the elastic Poisson's ratio, Act is the stress range, and Ace and Aep are the elastic and plastic strain ranges, respectively; p = 0 and Rn = 1 for the uniaxial case. Note that the torsional fatigue constants appear in these equations as well. While approximations could be made for these torsional constants in terms of the uniaxial constants, we regard them here as fully independent.

Crack growth rate predictions for small cracks in smooth uniaxial specimens Microcrack propagation under predominately LCF conditions has been treated within the framework of elastic-plastic fracture mechanics by a number of researchers [9, 15-17, 19, 20, 32, 33, 38]. The cyclic J-integral, AJ, has been frequently applied to correlate microcrack propagation rate using the power-law relation da d--N = C j ( A J )ms

(18)

where Cj and ms are constants. The AJ-integral is formally defined[39--41] as

AJ =

I p

AWdy-

OAui . Ati--~x ClS

(19)

where A W = Sha~aeo., hti = Aaj,Dj and nj is the outward unit normal vector to contour F. In eq. (19), F is a contour taken counterclockwise from the lower crack surface to the upper surface and Arr•, Ae0 and Au; are the stress, strain and displacement ranges for cyclic loading. For uniaxial loading of smooth specimens with small semi-circular surface cracks, AJ is given by Dowling[42] as [" AO"2 A J = AJe + AJp = 2 / r r 2 /+- ~L-

n' 2 f ( n ' ) AaAep] a = A K + AJp l +n' J E

(20)

where Y is a geometric correction factor (= 1.12(2/n) for a semi-circular surface flaw), n' is the cyclic strain hardening exponent, and f(n') = (1 + n')(3.85(1 -n')/(n') 1/2 + nn')/(2n). Plane stress

366

D.L.

MCDOWELL

conditions are assumed. This form has been used extensively for correlation of fatigue crack growth behavior of small cracks. An equivalent stress intensity factor range, AKs, may be defined based on the AJ-integral according to AKj = ~

= x/E(AJe + AJp) = AaqC~-d

71(

4.25E ( Aa "~(l/n')-I'~ 1.6 + K'(1 + n') \~--7 / ]

(21)

As distinguished from AK = YAa(rca)i/z, AKs includes a contribution from plasticity which may be very significant in the case of small cracks subjected to moderate to high remote stress amplitudes. In this section, we consider plots of predicted da/dN from the microcrack propagation model for both 1045 steel and Inconel 718 vs AK or AKj, without explicit regard for closure or bridging effects. Such effects will be considered in the next section. The calculated values of AK or AKj in this section are based on only the fraction of the stress range (coefficient in eq. (21)) which is tensile, since closure effects are not considered. The purpose of these calculations is to compare the predicted propagation behavior of small cracks as a function of stress amplitude and R-ratio with long crack data on the basis of conventional fracture mechanics parameters, irrespective of the breakdown of applicability of such parameters. The model is formulated to achieve proper correlation of fatigue life based on propagation to a preselected crack length (1 mm in this case), so this is not this issue; however, it is of interest to consider the nature of the predicted trajectories in the da/dN vs AK or AKs space for a < 1 mm. McDowell and Berard [2] previously employed an integrated form of this model to correlate the fatigue life, Nr, to a 1 mm crack for 1045 steel and Ni-base superalloy IN 718 under various stress states and amplitudes, including both proportional and nonproportional straining. The constants of the standard relations in eqs (8, 9) for uniaxial and torsional fatigue of both materials are presented in Table 1; the uniaxial constants for Inconel 718 are taken from the work of Waill[43]. For the uniaxial case considered here, the value of/~ is irrelevant. To determine the constants in the expression for m r for the uniaxial case, consider the data of Socie [14] for completely reversed, constant amplitude loading in tension-compression ( R = 6rnin/frnax1). Figure 3 presents the relative number of cycles for nucleation and growth to a 0.1 mm surface crack, and the number of cycles of growth to a 1 mm crack for both 1045 steel and Inconel 718. Note that the curve representing fraction of life to a 0.1 mm crack is termed "nucleation" in Fig. 3, but actually reflects microcrack propagation to this length. Here, we shall presume that this curve predominantly reflects microcrack growth from some initial size of the order of several/~m. For 1045 steel, the transition crack length kd and the constants q, r, s and t in the relation a

mr =q(1

a

+rRn)R~s-tR") for ~-d 1

(22)

were determined by integrating eq. (1) between given initial and final crack lengths and comparing the life to a crack length of 0.1 mm with that of the curve in Fig. 3. The initial crack length was assumed as 10 #m [38], somewhat less than d for this material, while the final crack length was taken as 1 ram, in accordance with the failure definition in Fig. 3. For 1045 steel, the constants were determined as kd = 0.25 ram, q = 2, r = 1, s = 7 and t = 4. For Inconel 718, the initial crack length was assumed as 20/~m, while the final crack length was taken as 1 ram, in acT a b l e 1. S t r a i n - l i f e a n d c y c l i c s t r e s s - s t r a i n c o n s t a n t s 1045 Steel Uniaxial c = - 0.474 ~/ = 0 . 2 3 0 n' = 0 . 2 3 9 K'= 1289MPa b = -0.111 a/ = 914MPa

I N 718 Torsional

Uniaxial

Co = - 0.413 7( = 0.436 no' = 0 . 1 9 6 Ko' = 642MPa bo = - 0 . 0 8 3 zl = 541MPa

c = - 0.82 ~( = 2 . 6 7 n' = 0 . 0 7 K' = 1530MPa b = -0.06 a( = 1640MPa

Torsional Co = ~( = no' = Ko ' = bo = T( =

- 0.872 6.85 0.0721 850MPa -0.095 1088MPa

367

Propagation of small cracks in fatigue 1045 Steel, Tension

=---Region A

=1"

B

-

08

_L.rl_ C

/

%

~"~'~ Tensile

_06

Z

Z(M

y

0.1 mm

Sheor Crock G r o w t h /

Crock Nucleotion

O2 102

I

I

I

I

I

Nf (cycles)

I

IN-718, Tension =

. I. B -r - " " ' ~ , . Tensile Crock Growth

RegionA

~0

_0.6 Z Z

0.4

~

o,,~r ~ru~ ~

0.1 m m

Crock Nucleation

02 %

2

,

@

,

1o4

,

lo5 NI (cycles)

,

lo6

Fig. 3. Data of Socic for completely reversed uniaxial fatigue for number of cycles to 0.1 mm (lower curve) and !mm cracks (N/Nr = 1) for 1045 steel (top) and [ncond 718 (bottom)[]4].

cordance with the failure definition in Fig. 3. For Inconel 718, the form of the microcrack length exponent was taken as my = q(1 - rRn)(1

a a A~max~//(s-t'%)for~--d <1; m r = 1 for k-d -> 1

(23)

Constants were selected as kd = 0.5mm, q = 80, r = 0.75, s = 12 and t = 8.5. The Macauley brackets in eq. (23) are defined as < F > = 0 if F < 0; < F > = F if F >0; constant Ay0 defines a threshold value of Ay~ax above which the microcrack growth rate is independent of crack length (i.e. m r = 0). A value of A?0 = 0.016 was selected to fit the LCF regime evident in microcrack iso-length curves for a = 0.1 mm in Fig. 3. A good correlation of the amplitude dependence indicated by this R = - 1 data is achieved as shown in Fig. 4 for both materials. Having determined the parameters of the microcrack propagation law, it is of interest to consider how the crack growth rate in the resulting relation compares with relevant data for both small and long cracks. Since the fitting procedure considered only one intermediate point (0.1 mm crack) for completely reversed loading to establish the nonlinearity of crack propagation with applied stress amplitude, computed crack growth rates from the model for other crack lengths and R-ratios may be regarded as predictions. It is useful to recall that the preceding crack growth relation in eq. (1) was derived based on (i) accordance with critical plane multiaxial fatigue concepts; (ii) accordance with known strain-life equation parameters; (iii) a simple formulation for mean stress effects [2] consistent with the Morrow approach [36] for hard materials; and (iv) a simple two-regime treatment of microstructurally small and mechanically small microcrack propagation. It is emphasized that experimental data expressing da/dN as a

368

D.L. MCDOWELL 1.0 0.8

z Z

0.6 0.4 0.2 -0"002

I

I

I

I

I

103

104

105

106

107

Nf

1.0 --

0.8 --

z Z

0.6 -0.4 -0.2 -0"002

I

I

I

I

I

103

104

105

106

107

Nf Fig. 4. Model correlation o f data in Fig. 3:0.1 m m (curves) and 1 m m cracks (top) and Inconel 718 (bottom).

(N/Nf =

1) for 1045 steel

function of AK or some other fracture mechanics parameter such as AKj (i.e. AJ) were not used in determining constants or establishing the form of the microcrack propagation relation. As such, no universal relation is assumed between da/dN and AK or AKj. Hence, we may consider the predicted da/dN from eq. (1) vs calculated values of AKs based on eq. (21) at each point in light of typical experimental data to examine the performance of the simple model in eq. (1). Figures 5-7 show plots of crack growth rate vs AKj for 1045 steel at room temperature in air for three different R-ratios for various applied remote stress amplitudes, Aa/2, and peak 10

-2

10-3

10 -3

Omax/Oo =0.8

-

/

// 10

/

0.6

10-4

10

_..___~//" ,, 0.4 ~ ."

-5

~, .~

=

10-5

,~E 10.6 Z

10-6 J

Omax /

/ /

B

Z

-4

i R-- 0, long cracks

,~

//// l

!

cracks

10__7

10-7 !

10-8

10-9

0.2

lO-8

1045 steel R=-l I

I

I

I I I III

I

10 AKj

(MPa

I

......

I

100 -

m 1/2)

Fig. 5, Predicted da/dN vs A K j for small cracks in 1045 steel at several applied stress amplitudes and R = - 1. Long crack data appears as dashed lines[19].

10_ 9

0.2 i

1045 steel R=0.1 i

~

, ,,I

,

10

,

J

,

, ,

~] 100

AK I (MPa - m I/2) Fig. 6. Predicted da/dN vs AKj for small cracks in 1045 steel at several applied stress amplitudes and R = 0.1. Long crack data for R = 0 appears as a dashed line[19].

Propagation of small cracks

in fatigue

369

s t r e s s e s , O'max, ranging from 20 to 100% of the quantity a0 = (ay' + au')/2 = 516 MPa; here, t;0 is the average of the 0.2% offset-defined cyclic yield strength, ay', and au' = IC(n')"'/(1 + n'), where tru' is a measure of ultimate strength for the cyclically stable material. Accordingly, high ama× values indicate considerable levels of cyclic plasticity, particularly for R = - l; the lives for R = - 1 range from 550 cycles for amax/aO = 0.8, to 1.67 x l0 s cycles for amax/aO = 0.2. In contrast, the lives for R = 0.1 range from 5120 cycles for amax/aO = 1.0, to 1.4 x l0 ll cycles for O'max/O"0 = 0.2. The lives for R = 0.5 range from 3.6x 105 cycles for O'max/O"0 = 1.0, to 2.18 x 10 l° cycles for O'max/fr0 = 0.4. The calculations for lower Aa/2 levels with lives in excess of 108 cycles may not be entirely realistic since the crack may actually decelerate or even arrest under these conditions; they are considered here to study behavior of the microcrack propagation model under a wide range of amplitudes without consideration of such limits. Long crack da/dN vs AK data are presented as well for comparison[19] in these figures. It is noted that AK and AKj differ little for low stress amplitudes and/or high R-ratios which promote predominately elastic nominal conditions (typical long crack conditions). Several observations are noteworthy. The predicted small crack fatigue crack growth trajectories share key characteristics of experimentally observed trajectories. First, the threshold AK for small cracks is below that obtained from long crack experiments. Second, the microcrack propagation rate generally exceeds, at a given AK level, that of the long crack data. Third, at sufficient crack length (on the order of 1 mm), the predicted small crack growth curves essentially merge into the long crack data, particularly for the R = - 1 case. Fourth, crack growth rates for higher applied stress amplitudes exceed those for lower amplitudes at the same level of AK, often significantly. There is no single trajectory, or even a common shape or trend among the various stress levels. Fifth, high stress amplitudes are associated with a nearly constant da/dN prior to approaching the long crack curve, owing to the relative insensitivity to crack length in the LCF regime of the microcrack propagation relation. Since the present form of the model does not consider the details of crack deceleration due to blockage by microstructural features [8-12] or plasticityinduced closure effects for small cracks growing away from notches or heterogeneities [5], dips in the da/dN vs AKI curves which are often observed experimentally are not predicted. However, to first order the predicted trends are in good agreement with typical observations. Predicted fatigue crack growth rate curves are presented for Inconel 718 at room temperature in air in Figs 8 and 9 for R = - 1 and R = 0 and several peak stresses. For this high strength material, a0 = 1088 MPa. Similar observations are made as those for the case of 1045 steel. For this material, the lives for R = - 1 range from 930 cycles for trmax/tr0 = 0.97 to

10-3 10-4

10-2 I

10-5L

/

0.6

//// ///

/

/////

Omax/O°= ~ . 8

/..'/

~ 10-4 ~

+

R = 0.5, long cracks

10-7

10-5

0.83

F / /// 4 / "~

//~.,--'// //

0.74

'~ 0.4

10-7

10-8 10-9

Omax/Oo = 0.97

10-3

1045 steel

l

10 AKj (MPa -

10-8

100 m1/2)

Fig. 7. Predicted da/dN vs AKj for small cracks in 1045 steel at several applied stress amplitudes and R = 0.5. Long crack data for R = 0.5 appears as a dashed line[19].

10-9

0.y/ 0.5 / /

IN 718 R = -1 10

AKj (MPa -

100 m t/2)

Fig. 8. Predicted da/dN vs AKj for small cracks in lncon¢l 718 at several applied stress amplitudes and R = - 1. Long crack data for R = - 1 appears as a dashed line[19].

370

D.L. MCDOWELL 10-2 / /

10-3 z O

10-4 10-5 Z

10-6 10-7 0.92 10-8

/

/

/ /0.74

IN 718 R=0 R=0

10-9 10

100

AKj (MPa - m 1/2) Fig. 9. Predicted

da/dN vs

AKj for small cracks in Inconel 718 at several applied stress amplitudes and R = 0. Long crack data for R = 0 appears as a dashed line[19].

4.06 x 10 7 cycles for trmax/6 0 = 0.5. The lives for R = 0 range from 4.03 x 105 cycles for O'max/ a0 = 1.0 to 2.76 x 108 cycles for trmax/tr0 = 0.74. Comparison of predicted crack growth behavior with experimental data[19] for Inconel 718 at room temperature appears in Fig. 10 for R = - 1 at controlled strain amplitudes of 0.5 and 1%, revealing reasonable agreement, notwithstanding the experimental scatter, for these relatively high amplitude loading conditions; the comparison is made in terms of AK here since this is how the data were reported. The predicted crack growth curves for Inconel 718 in general display exceedance of long crack AK levels, for given values of da/dN, prior to transition to long crack behavior. This is possibly due to neglect of crack face interference, crack deflection and closure effects in calculating AKj. Note that the data in Fig. 10, evaluated in the same manner, also cross the long crack data, in agreement with the predictions. In all of these plots, the point where the slope of da/dN vs AKj plot changes slope is associated with the transition crack length, kd. Clearly, this occurs before or in the vicinity of the point of convergence with the long crack data. Also, the linearity of da/dN with crack length in the mechanically small (EPFM~,) regime is evident just after the transition. The bilinear log-log behavior of the small crack growth offered by this model is recognized as an idealization of reality, since actual experimental data exhibits a smooth curvature for the trajectory. Since the kd value appears to be related to the transition from Stage I to Stage II propagation as indicated by the data of Socie[14], it is clear that the microstructure-dependent regime plays a key role in the propagation of small cracks by establishing the slope of da/dN for a < kd. Interestingly, the small crack trajectories reveal a trend of very nearly merging with the long crack data for a crack length of 1 mm for R = - 1 loading. Since this result has not been imposed upon the model, it indicates that the common assumption that a 1 mm crack length is present at "initiation" is perhaps reasonable for this case. For R > 0, the curves do not yet merge into the long crack data for R = 0 at a crack length of 1 mm. This suggests that some criterion other than a fixed crack length may correspond to "initiation", the point where long crack fracture mechanics is first applied. Since the present microcrack propagation law is operationally analogous to long crack fracture mechanics, a tentative recommendation would be to simply carry out the small crack propagation analysis until the long crack data are reached for the given R-ratio, and then shift to the long crack case. This essentially provides another nonarbitrary definition of initiation crack length [44, 45], valid for unnotched or notched specimens. It should also be pointed out that the predicted crack growth rate curves are sensitive to the mean stress formulation. In this study, Jmc = Jmp = 0 was assumed for simplicity, leading to maximum sensitivity to mean stress. For the rather ductile 1045 steel, the predicted crack

Propagation of small cracks in fatigue 10 -~

. . . . . . .

371

I

IN 718 R = -1 10 -2

t)

E E

As/2

10 -`3

= 0.01

/-

,/

smoli crocks ,/'i

/

10 -4

Z "[3 E9 7D

10 -5 R=-I R=0 Long cracks 10 -6

*

L

i

J

,

t

i l i

i

i

,

,

i

10 (MPo-m

AK

,

,

,

i

,

,

i,

~...L

I O0

1/2)

I

Inconel 718

~2

/

¢v

Long CrOCkS

#

°

J

A

,

, ill

:

t

lO

1.00

Fig. 10. Comparison of predicted (top) and experimental[19] (bottom) da/dN vs AK for small cracks in Inconel 718 under completely reversed strain-controlled conditions. Long crack data for R = - I and 0 appears as dashed lines[19].

growth rates for R = 0.1 or 0.5 are likely higher than in reality due to this assumption; of course, Jme and jmp may be selected to fit available data at high R ratios. It is interesting to note that tensile mean stress tends to elevate da/dN, whereas compressive mean stress decreases da/ dN, as might be expected•

Closure and shielding effects on retardation of small crack growth rate Closure effects are often invoked to explain the complex behavior of small cracks such as anomalously high growth rates and transients in the slope of the da/dN vs AK curve [5, 6]. It has also been clearly demonstrated that interaction with microstructural barriers may suppress the crack growth rate[8-12]. Since these have not been explicitly considered in developing the present microcrack propagation model, it is informative to calculate the magnitude of such retardation effects; in this case, a//sources of crack tip shielding such as plasticity-induced closure, microstructure-roughness induced bridging, etc. are embedded in the relation, along with an averaged treatment of interaction of the advancing crack front with microstructural barriers. Let us assume the existence of a da/dN vs an effective AKefr = AKj curve, defined in the absence

372

D.L. MCDOWELL

of closure effects (see [5]), where AKj is based on the entire applied stress range, Aa. We may postulate the da/dN relation for this curve as da dN = A(AKerf)"

(24)

Constants A and n pertain to this fully open, unobstructed case. Small cracks often start out essentially fully open and then develop closure as the crack extends [5,46]. Following Elber's assertion [47], we define AKcfr as

A K e f f = (O'max - -

4.25E (Aa~(l/n')-l~ 1.6 -~ K'(1 + n') \ ~ - S ] ]

/1( tYop)~r~-d

(25)

where the effective stress intensity factor incorporates only the stress range over which the crack is open; Cropis the opening stress level. By inserting that value of da/dN from eq. (1) into eq. (24) and solving for AKefr, we may then employ eq. (25) to determine the opening stress, aop, at each point in the history which is implied by the growth law eq. (1). For 1045 steel, the constant n = 3.5 was selected to conform with long crack data for R = - 1 and R = 0. The coefficient A = 1.29 × 10-8 m m / c y c - (MPa m]/2)-35 was selected to provide levels of well-developed closure as the crack length approaches 1 mm for the R = - 1 or R = 0 case which conform to those often observed experimentally or numerically (i.e. O'op/O'ma x = 0.3-0.4). The shifted curve for fully open cracks is shown relative to the long crack data in Fig. 11. Figure 12 shows plots of the calculated aop/amax as a function of number of cycles to failure for 1045 steel for R = - 1 and R = 0.1. Considering these plots, a relatively strong apparent closure transient is evident, particularly for increasing stress amplitude. At high stress amplitudes, the microstructurally small cracks are subjected to essentially fully open conditions (aop/amax = -- 1), and eventually develop closure levels typical of long cracks under the remote loading conditions (aop/ amax approximately 0.3-0.4). These results are very similar to those discussed by Newman [5] in the context of a plasticity-induced closure model, as are the stress amplitude dependent trajectories of da/dN vs AKj reported in the previous section for each R-ratio. At such high amplitudes, it is probable that plasticity-induced closure mechanisms dominate, and this predicted behavior is very much in accordance with such concepts [5]. At low stress amplitudes, such as R = 0.1 and amax/aO= 0.4, the opening stress starts out fairly high and actually drops as the fatigue crack grows, perhaps indicative of the influence of microstructure roughness-induced crack tip shielding in this case during the early stages of growth. As pointed out by Suresh [48], low strain amplitudes promote a predominantly mode II crystallographic growth and a higher 10 -2 10 -3

/

10 -4

,// 7/

10-5

AKeff

,/I /

curve ~

Z

10-6

i t"

[

10 -7

i

lO -8

10 -9

R=O long cracks I

I

*

1045 steel

~ I ILl

I

10

~

~

L * J J*J

100

AKj (MPa - m I/2) Fig. 11. Assumed

da/dN vs AKj curve for fully open conditions (solid line) and long crack data (dashed lines) for 1045 steel.

Propagation of small cracks in fatigue

373

0f

0.8

0.6 0.4

0.2 -E

0.0 --0.2 -0.4

/

-0.6

°4

°max'° R

l

-0.8 -1.0

0.0

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

N/Nf

0.8

0.4

°I

0.6 0.4 0.2

E 0.0 -0.2 -0.4 -0.6 -0.8

-1.0 0.0

I 0.2

0.4

0.6

0.8

1.0

N/Nf Fig. 12. C a l c u l a t e d O'op/O'max as a function of number of cycles for 1045 steel for R = - 1 (top) and R = 0.1 (bottom).

degree of microstructural roughness along the crack faces, leading to enhanced crack tip shielding effects. This apparent transition to dominance of crack face interference mechanisms is a natural consequence of the formulation, and has not previously been manifested in predictive fashion within a given framework. Typically, even the incorporation of only plasticity-induced closure effects in a growth law for small cracks is a very cumbersome undertaking, involving numerous assumptions and idealizations.

LCF-HCF sequence effects in microcrack propagation Fatigue life prediction under variable loading histories is of great practical importance. Accordingly, it is also of interest to assess how the microcrack propagation model in eq. (1) performs under sequences of amplitudes under uniaxial loading. It is instructive to compare the results of the model with the damage curve approach (DCA) for sequence effects in crack initiation developed by Manson and co-workers [7], since this model has been shown to correlate a rather exhaustive set of data for sequences of completely reversed high-low and low-high amplitude loading for a fairly wide range of ductile alloys. As such, we may regard it as descriptive of common experimental trends, even quantitatively in many cases. Detailed experimental data for such histories are not available for the two alloys considered here. Uniaxial sequence data have

D.L. MCDOWELL

374 1.o -

0.8

t~

Z

0.6 Nf2/Nfl = lO

0.4 0.2

I

o.o

0.2

0.4

0.6

0.8

1.0

0.8

1.0

NI/Nfl

1.o -

0.8

r,i

Z

~

DCAcurves

0.6 0.4 ~

~/Nfl

= l0

0.2

0.0

Fig. 13.

0.2

0.4 0.6 N~/Nfl

Comparison between microcrack propagation model (top) and the D C A model (bottom) for completely reversed high-low amplitude sequences for 1045 steel.

been reported for 1045 steel by Hua and Socie [49], but were obtained under predominantly LCF test conditions for which sequence effects are relatively weak. According to the DCA, the cumulative damage for two level loading sequences can be expressed as

, N22= 1 _ (\ . _NI ,]

°

(26)

where ~ is a constant (approximately 0.4) and nt, n2 are the number of cycles applied with amplitudes 1 and 2, respectively, while N1 and N2 are the corresponding cycles to fatigue crack initiation (Nf) for these two levels considered individually. Figure 13 shows the comparison between the results of eq. (1) and the DCA model in eq. (26) for completely reversed uniaxial high-low stress amplitude sequences for 1045 steel. The diagonal in this plot would correspond to a linear cumulative damage rule (i.e. Miner's law). The lives to a 1 mm crack for each amplitude range from 104 cycles for the highest stress amplitude (299 MPa) to 107 cycles for the lowest amplitude (140 MPa). Clearly, the prediction of experimentally observed trends of high-low amplitude sequence effects is qualitatively correct, with Y~ (nj/Nj) < 1 for j = 2. Low-high sequences would yield identical results but reflected

Propagation of small cracks in fatigue

375

about the diagonal in this plot, i.e. E(nj/Nj)> 1 for j = 2. The dependence of exponent m r in eq. (1) on strain amplitude for a < kd essentially engenders the nonlinear cumulative damage aspect of the present model. It should be noted that another value of • might be necessary to fit the experimental behavior of 1045, so this is an issue of uncertainty in a quantitative comparison. Furthermore, although the DCA approach is independent of the amplitudes of loading in the sequence, in reality, the degree of nonlinearity of the cumulative damage curves depends on these amplitudes. Hence, the curves in Fig. 13 based on the microcrack propagation model will shift somewhat if a higher or lower series of stress amplitudes are employed. Earlier application of the fully multiaxial form of the microcrack propagation law in eq. (1) considered prediction of torsional-axial sequences of loading[3]. As pointed out elsewhere [12, 50, 51], torsional cracks engaged in Stage I growth are effective as Stage II cracks in subsequent uniaxial loading. Hence, strong sequence effects with E (N/Nf)j < 1 are experimentally observed. On the basis of these results, it appears that the same physical processes which lead to the microcrack iso-length curves in Fig. 3 are to first order manifested in sequence effects. In other words, load sequence effects in the growth of small cracks are primarily due to the amplitude dependence of the crack tip shielding mechanisms which develop for a microstructuraUy sensitive growing crack. Overload retardation effects might play some role, but are likely not as influential as for long cracks which obey certain requirements of similitude and small scale plasticity. CONCLUSIONS This work provides insight into concepts which seek to unify the traditionally distinct analysis tools of fatigue crack initiation and propagation. We consider a simplified microcrack propagation model based on consistency with the traditional strain-life curve for fatigue crack "initiation" of a crack length of 1 mm. Microstructural influence is addressed by assigning a nonlinear crack length dependence below a given material length scale, kd, associated approximately with the extent of Stage I growth. The results of integrating the model under various uniaxial loading conditions for both 1045 steel and Inconel 718 suggest that important, first order characteristics of small crack propagation behavior are captured. Rather than adopting a construction based purely on a closure methodology using conventional fracture mechanics, the present engineering model acknowledges the breakdown of similitude and singularity concepts for small cracks and implicitly reflects contributions from all potential sources of crack tip shielding, crack deflection, etc. Closure effects inferred by the model are shown to be of realistic nature, with a transient of plasticity-induced closure indicated at higher stress amplitudes. Interestingly, microstructure-induced shielding effects are suggested at low stress amplitudes. The approach perhaps represents an advance of fatigue crack "initiation" mechanics since it attributes amplitude dependent nonlinearity to the evolution of small cracks and can therefore more realistically represent sequence effects. In addition, it quantitatively establishes a basis for bridging small crack "initiation" analyses and subsequent long crack LEFM analysis. Apart from the constants related to exponent m r in the growth law, all other constants are determined from conventional fatigue tests on smooth specimens. Many of these constants already exist in databases of stress-life and strain-life relations. In contrast to analogous damage mechanics approaches which address small crack fatigue growth using a damage evolution equation [52], the present approach (i) addresses both the microstructurally small and mechanically small crack regimes, and (ii) is a complete critical plane formulation which has been shown to be applicable to a wide range of multiaxial stress states [2]. By virtue of the latter, the very complex problem of mixed mode propagation of small cracks is addressed by this formulation[3, 53]. It is not even clear how to proceed with modification of the standard LEFM/EPFM fracture mechanics-based methodology with small crack closure modifications to address such mixed mode loading conditions, notwithstanding potential violations of the limits of its theoretical applicability. A principal shortcoming of the present form of the model is the lack of treatment of deceleration effects which occur predominantly at low stress amplitudes for cracks of length less than about 1-2 grain diameters. These effects arise chiefly from blockage by microstructural barriers [48], crack face microstructural roughness effects [48], differences in crystallographic orientation among grains and associated cyclic shakedown behavior [31], and plasticity-induced closure

376

D.L. MCDOWELL

transients which are coupled with microstructural features [5]. These features may contribute to an apparent fatigue limit in practice which is neglected here. A related topic of future development is the introduction of another length scale and associated loading- and microstructuredependent conditions which define a nonpropagating small crack limit at low amplitudes [9], along with consideration of distribution of microplasticity among grains to establish a shakedown fatigue limit [31]. Also, consideration of possible amplitude and stress state dependence of transition crack length kd is warranted; detailed analyses of transgranular crack growth are necessary to further resolve this issue. Acknowledgements--The author is grateful for the support of the Office of Naval Research MURI program in Integrated Diagnostics (ONR N00149510539) at the Georgia Institute of Technology.

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