Mechanisms associated with transient fatigue crack growth under variable-amplitude loading: An experimental and numerical study

Engineering Fracture Mechanics Vol. 32, No. 4, pp. 613-638, 1989 0013-7944/89 $3.00 + 0.00 Printed in Great Britain. 0 1989 Pergamon Press pk.

MECHANISMS ASSOCIATED WITH TRANSIENT FATIGUE CRACK GROWTH UNDER VARIABLE-AMPLITUDE

LOADING: AN EXPERIMENTAL AND NUMERICAL STUDY

C. M. WARD-CLOSE,? A. F. BLOMS and R. 0. RITCHIQ

TMaterials and Structures Department, Royal Aircraft Establishment, Farnborough, Hants GU14 6TD, U.K. SStructures Department, The Aeronautical Research Institute of Sweden (FFA), P.O. Box 11021, S-161 11 Bromma, Sweden. §Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory and Department of Materials Science and Mineral Engineering, University

of California, Berkeley, CA 94720, U.S.A.

Abstract-An experimental and numerical study has been made of the mechanisms of fatigue crack growth and crack-closure behavior in an a//? titanium alloy Ti4A14Mo-2SnO.SSi (IMI 550), following both single and block tensile overloads. Closure immediately behind the crack front (near-tip closure) was found to be the main factor controlling load-interaction effects. Single tensile overloads were found to remove near-tip closure, and slightly reduce far-field closure along the crack length, resulting in an initial acceleration in fatigue crack growth rates. Subsequent delayed retardation of crack growth rates was accompanied by an increase in the near-rip closure load, due to the enlarged compressive residual stress in the overload plastic zone. High/low block overloads caused greater retardation than single overloads of the same magnitude, and this was attributed to changes in the degree of closure in the wake of the crack. Numerical predictions of such transient behavior, based on a modified Dugdale model, are found to be in close agreement with experimental results, both in terms of observed crack growth rates and crack opening displacements. Load-interaction effects were found to be most severe when the baseline stress intensity range (AK) was close to the fatigue threshold (AKrn), or, when the overload maximum stress intensity (K,,) approached the fracture toughness of the material. At low AK levels, the magnitude of the delay was sensitive to microstructure and found to be enhanced in coarse-grained P-heat-treated microstructures compared to standard fine-grained a //I microstructures. Based on these results, mechanistic sequences are suggested to explain the transient fatigue crack growth behavior following single and block tensile overload cycles.

1. INTRODUCTION

FOR MANY fatigue-critical structures, fatigue crack propagation under service conditions generally involves random or variable-amplitude, rather than constant-amplitude, loading conditions, such that the loading history can be a major factor in determining fatigue life. Such excursions in the loading sequence are known to induce a number of different load-interaction effects which can result in significant transient accelerations or retardations in growth rate, although an under- standing of the micro-mechanisms is still uncertain. The most studied of these effects appears to be that of the crack growth-rate retardation of mode I cracks following tensile overloads, where several mechanistic interpretations have been proposed (Fig. 1); these include arguments based on (a) residual stresses, where the strain generated by the tensile overload results, on unloading, in a large increase in residual compressive stress ahead of the crack[l]; (b) crack closure in the wake of the advancing crack arising from these residual stresses, which results in contact between the crack faces at non-zero loads and hence to a locally reduced effective stress-intensity range (AK) experienced at the crack tip[2,3]; (c) plastic blunting of the crack tip at the onset of the overload and subsequent resharpening[4,5]; (d) crack deflection, where a branching or deviation of the crack path at the overload can additionally cause a local reduction in the stress intensity at the crack tip[6,7]; (e) strain hardening from monotonic strain hardening of material in the overload crack-tip plastic zone following single overloads or cyclic strain hardening following block-overload sequences[8,9], and (f) crack-front geometry, where high-amplitude tensile overloads cause a marked bowing of the crack front, particularly in thin sections, which precedes slower crack growth at the edges of the specimen[lO].

Of these many mechanisms, residual compressive stresses and the resulting (plasticity-induced) crack closure provides a primary basis for rationalizing many load-interaction effects. Con-

613

https://www.researchgate.net/publication/222634738_Micromechanisms_of_Fatigue_Crack_Growth_Retardation_Following_Overloads?el=1_x_8&enrichId=rgreq-852dae01-a109-4846-b8ba-eab5fa3c686b&enrichSource=Y292ZXJQYWdlOzIzODE3Mzc5NztBUzoyMDE4ODA1NTk5ODQ2NDBAMTQyNTE0MzQ4NDg3Mw==

614 C. M. WARD-CLOSE et al.

On overload or during subsequent growth

Time

a Residual stress d Oef!ection

i

_,/

Effectively reduces AK

Material strain - hardened on overload

Time

b Crack closure e Strain hardening

Fatigue cycles

,

R Gro to overload

__zL_ Large overload may change crack front shape- effectively

C Plastic blunilng / re - sharpening reducmg AK for subsequent growth

f Pop-In

Fig. 1. Schematic illustration of various proposed mechanisms to account for retardation following a tensile overload.

sequently, several successful models for predicting crack growth rates in variable-amplitude fatigue are based on the crack-closure concept (e.g. refs[l l-141). Direct experimental evidence for this prominent role of closure, however, is limited and in many cases conflicting, particularly for behavior under predominantly plane-strain conditions. Early studies by Hertzberg and co- workers[3, 151 report highly abraded post-overload regions on fatigue fracture surfaces of aluminum alloys, which are a clear indication of crack closure. However, surface measurements by Davidson and Hudak[l6] following large tensile overloads in aluminum alloys show a transient increase in closure at the overload, whereas the compliance measurements of others[ 17-l 91 in steels and aluminum alloys show a decrease. Paris and Hermann[20], conversely, report two measures of closure following tensile overloads, termed upper and lower opening loads; the upper opening load increasing and the lower opening load showing an immediate decrease at the onset of the overload.

The objective of the current work was to compare numerical predictions with experimental studies on the transient fatigue crack growth rate behavior associated with single and block overloads under plane-strain conditions in an a//? titanium alloy. In addition, the intent was to determine the role of crack closure (in terms of magnitude and location) using surface strain measurement, metallographic sectioning and fractographic examination. Microstructural effects were investigated by examining two widely different grain sizes, as previous studies[21-251 on constant-amplitude fatigue crack growth in a- and a/B-titanium alloys have indicated that

Mechanisms associated with transient fatigue crack growth under variable-amplitude loading 615

coarser-grained microstructures often show lower crack growth rates due to more irregular crack paths.

2. EXPERIMENTAL PROCEDURES

The material used in this investigation was an c1 //?-type titanium alloy Ti-4Al+Mo-2Sn-OSSi (IMI 550), supplied in the form of 25-mm rolled plate. The alloy was tested in two heat-treated conditions, intended to produce widely different microstructures: the normal a //I solution-treated and aged condition (9OOC, air cooled, 24 h at 55O’C), which gave a microstructure consisting of approximately 55% primary a, and the remainder transformed /I, with a primary a grain size of approximately 14 pm, and a p-solution treated and aged condition (lOlOC, air cooled, 24 h at 500°C), which gave 100% transformed /I, with a prior /I grain size of approximately 240pm. Optical micrographs of the two microstructures are shown in Fig. 2; ambient temperature mechanical properties are presented in Table 1.

Fatigue crack growth testing was performed on 10 mm-thick compact C(T) test pieces (L-T orientation), at sinusoidal frequencies in the range 4-40 Hz at a baseline load ratio (R = K,i”/K,,,) of 0.1. Crack length was measured using a d.c. electrical potential system, which incorporated load shedding to maintain constant AK conditions as the crack advanced. Both single tensile overloads and block overloads were applied to constant baseline AK conditions. Single overloads were applied at 0.01 Hz. For block overloads, the high block was continued until the crack had extended a minimum of twice the distance required for the crack growth rate to reach a steady-state. Figure 3 illustrates the various load-cycle parameters associated with the overload experiments. All tests were performed under predominantly plane-strain conditions, with the computed maximum overload plastic zone always less than 1/50th of the test piece thickness.

The magnitude of the post-overload retardation was quantified in terms of delay cycles and delay distance, which are defined in Fig. 2(b). The delay cycles represent the increase in total fatigue life due to the overload, whereas the delay distance is the equivalent extent of crack growth at the baseline growth rate. The use of a delay distance to quantify overload retardation has more physical significance than a simple cycles calculation, and enables a direct comparison to be made between overloads applied at different baseline AK levels.

The magnitude of (far-field) crack closure was determined using the back-face strain unloading compliance technique, with the closure stress intensity K, defined at the point of first contact of the crack faces from changes in slope of the elastic unloading curve. On the assumption that crack growth could not take place when the crack closed, the local “crack driving force” actually experienced by the crack tip was estimated in terms of the effective stress intensity range,

AK,% = K,,, - Kc, (see Fig. 3). An offset elastic-displacement method was used to improve the sensivity of the closure-load

determinations, as illustrated in Fig. 4. The displacement signal was offset by an amount inversely proportional to the applied load, such that the linear (crack open) part of the load/displacement curve became constant with respect to load. The resultant load/displacement signal was then amplified to exaggerate deviations from linearity caused by crack closure. In this study, it was found that under certain circumstances two distinct stages of closure could be identified by this method, indicating changes in crack closure both in the immediate vicinity of the crack tip and globally along the crack length, as described below.

Table I. Ambient temperature mechanical properties of IMI 550

Microstructure

Young’s modulus

W @Pa)

0.2% Yield

strength Tensile strength (MPa)

Elong. (%I

Reduction in

area (“~1

9 /b-annealed 107 984 1130 15 18 86 b-annealed 107 874 1094 12 9 100

616 C. M. WARD-CLOSE er at.

Fractography was performed using both optical and scanning electron microscopy (SEMI. Crack-path profiles were examined by taking metallographic sections at the center of the specimen, impregnating the crack with epoxy for edge retention, and imaging using optical microscopy.

3. NUMERICAL PROCEDURES

Transient crack growth behavior was predicted using an analytical model based on the modified Dugdale approach proposed by Newman[ 121. The current model, however, was extended to include the concept of weight functions in order to facilitate the analysis of any two-dimensional geometry. Specific details and comparisons with elastic-plastic finite element analyses have been presented elsewhere~Z6~; the essential features and limitations are briei?y described below.

(4

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AKeff- effective AK8

f% - baseline R-ratio (K,i, / K,.,B 1

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Fig. 3. (a) Definition of parameters, and (b) curve of crack length, a, vs load cycles, baseline AK conditions for a single tensile overload. illustrating the definition of “delay

distance”.

N, under constant cycles” and “delay


/ compliance curve D

Load -)

Fig. 4. Illustration of conventional compliance curve, from back-face strain measurements, and its resultant offset-elastic-displacement curve, used to determine closure loads. Segment AB represents a fully

open crack, BC partial closure, and CD final closure.

Crack-surface displacements, influenced by crack-tip plasticity and the residual plastic deformation in the wake of an advancing crack tip, are used to calculate the contact stresses and hence the degree of crack closure. Upon reloading, the applied stress at which the crack surfaces become fully separated is directly related to the contact stresses and is here defined as the crack-opening stress. In the modified Dugdale model, the material’s constitutive relation is assumed to follow elastic-ideal plastic behavior, i.e. no consideration is given to strain hardening. The plastic zone size and the crack-surface displacements are obtained by superposition of two elastic problems (a cracked body subjected to remote loading and the cracked body subjected to a uniform stress, applied over a segment of the crack surface).

The model is composed of three regions as depicted in Fig. 5: (i) a linear elastic region with a fictitious crack of length c + p, (ii) a plastic region of length p and (iii) a residual plastic deformation region along the crack surface. The physical length of the crack is c. Region (i) is treated as an elastic continuum and crack-surface displacements have been derived from the weight-function approach in ref.[26]. Regions (ii) and (iii) are composed of rigid-perfectly plastic constant stress bar elements with a flow stress q,, which is the average between the yield stress and the tensile stress. The bar elements are either intact (in the plastic zone) or broken (in the wake of the growing crack). The broken elements carry compressive loads only when they are in contact. The elements yield if the contact stress reaches -u,, . Those elements which are not in contact do not affect the calculation of crack-surface displacements.

The model predicts variable-amplitude behavior from a knowledge of the constitutive law and constant-amplitude fatigue crack growth properties (Fig. 6) in three consecutive steps. Firstly, a da/dN-A& relationship is derived from constant amplitude fatigue crack growth data by incorporating calculated values of crack closure. Secondly, closure levels are calculated as function of crack length for the variables-amplitude loading case. Here, crack extension is chosen to be a certain fraction of the maximum plastic zone size (rmaX); crack growth of the order of l-4% of r,,, yields convergent results[26]. Lastly, transient crack growth behavior is obtained by a cycle-by-cycle integration of the da /dN-AK,, relationship.

It should be noted that the Dugdale model was originally derived for plane-stress conditions[27]. To apply this model to the current predominantly plane-strain conditions, we introduce a constraint factor (a) to elevate the effective flow stress (a a,,), in accordance with the

C. M. WARD-CLOSE ef al.

a) b)

Fig. 5. Schematic illustration of the modified Dugdaie model showing crack-surface d~spla~ments and stress distributions along the crack line at (a) maximum stress and (b) minimum stress[lZ, 261. W is the

compressive plastic zone size.

approach of Newman[lZ]. A value of a = ,,h is assumed, consistent with Irwin’s plane-strain constraint factor. A better fit to the data could be obtained by using a as a fitting parameter, although this requires prior knowledge of the experimental results and thus detracts from the predictive capabilities of the model.

4. RESULTS

a. Single tensife ouer~ou~s

As first described by Schijve[ 1 J, the application of a single tensile overload results primarily in a delayed retardation in crack growth rates. The specific transient response is shown schematically in Fig. 7, and indicates several distinct stages, namely steady-state crack growth at baseline levels (A), crack growth during the overload cycle (B), accelerated crack growth (C-D), retarded crack growth (D-E), and crack arrest or return to steady-state conditions (E-F). Such behavior for the present alloy is shown in Fig. 8(a,b) for 100% overloads at baseline AK levels of 8 and 15 MPa&. respectively. Each solid curve represents the best fit to experimental data from up to 6 tests. Also shown are the numerical predictions (dotted lines) from the modified Dugdale model; agreement with experiment is clearly excellent.

As shown in Figs 7 and 8, the immediate effect of the tensile overload was to blunt plastically the nominally sharp fatigue crack, such that, at zero load, the crack was visibly open for some distance behind the crack tip (Fig. 9a). Additionally, the higher overloads (i.e., 100% overloads at baseline AK of 15 MPa&) produced an increment of crack growth by ductile tearing during the overload cycle itself (Fig. 10). In this figure, the magnitude of this growth increment is N 50 pm, some 50 times larger than that expected for constant-amplitude cycling. Following the overload cycle, there invariably was an acceleration in growth rates relative to baseline steady-state levels, followed by delayed retardation and an eventual return to steady-state pre-overload behavior. In Fig. 9 where the crack-path profile (taken at the center-thickness of the specimen) and corresponding SEM fractograph for a 100% overload in the a/j? microstructure at a baseline AK of


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Fig. 6. Constant amplitude fatigue crack propagation rates for IMI 550 at R = 0.1.

8 MPa,/& are shown, crack closure in the form of contact between the crack faces is clearly visible for the delayed portion of the crack (marked by the arrow at A), in contrast to the pre-overload portion which shows an enhanced crack-opening displacement due to blunting. Similar behavior is apparent with larger overloads (100% overload at baseline AK of 15 MPafi), although the post-overload crack path shows an irregular region of ductile fracture due to crack growth at the overload (marked at C), which is clearly subject to closure at zero load (B), followed by a region showing extensive fretting damage (D).

Experimental and numerical results for the single overload tests are summarized in Table 2. Comparing the two microstructural conditions, the extent of the retardation measured in terms of the delay cycles was clearly larger for the coarser-grained p-heat-treated material, and in all cases the distance over which crack growth rates were affected by the overloads were far in excess of computed maximum overload plastic zone sizes.

6. Block overloads

Results for the low-high and high-low block overloads are shown in Figs 11 and 12, respectively, for AK levels between 8 and 17 MPafi and between 15 and 32 MPafi. For the low-high sequences, the start of the overload block was accompanied by an acceleration in crack growth rates, followed by a gradual reduction to the new steady-state. The numerical model again provided a close prediction of the transient crack growth behavior. The acceleration was most

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Fig. 7. Typical crack growth rate curve showing delayed retardation following tensile overload. A = steady-state crack growth rate, B = crack during overload cycle, C-D = accelerated crack growth,

D-E = retarded crack growth and E-F = gradual return to steady-state.

pronounced for the low AK block-sequence (8-l 7 MPa,_,/-) m , and in the B-annealed microstructure (Fig. 11).

For the high-low sequences, the step-down in loads was found to have a similar effect to single tensile overloads, although for an equivalent reduction in K, the block overloads caused a greater reduction in crack growth rates (Fig. 12). In this case, the model predicts crack arrest. This can be attributed to the fact that the minimum growth rates following the overload event approach threshold levels, such that a small error in predictions of the effective stress-intensity range can lead to AKeK values less than the th~shold. Numerical and expe~me~tal results for the block-overload tests are listed in Table 3.

Figure 13(a) shows the center-section crack profile for a block overload sequence in the a/S-annealed material. In this test piece, even after the crack had extended N 10 mm beyond the block overload region, closure (crack surface contact) was still evident at zero load, both in the overload region (B) and in the post-overload region (A). The corresponding fractograph (Fig. 13b), shows the initial baseline fracture surface changing abruptly to the more ductile and irregular fracture surface at the onset of the low-high sequence, but a gradual return to the baseline fracture surface morphology when the K level is reduced again with the high-low sequence. Evidence of fretting damage (D) is apparent in the retarded region immediately following the high-low sequence.


Fig. 2. Optical micrographs of the titanium alloy microstructures, (a) u/b-annealed, (b) p-annealed.

622 C. M. WARD-CLOSE

Crack growth dir8ctlon -

et at.

K waveform (schematic)

Fig. 9. (a) Micrograph of crack-path profile taken at the center thickness of the specimen dter application of 100% overload in the z//I microstructure at baseline AK of 8 MPa,/%. Note post-overload crack closure at A, and enhanced crack-opening displacement of the pre-overload crack; (b) corresponding SEM

fractograph.

Mechanisms associated with transient fatigue crack growth under variable-amplitude loading

crack growth direction -

(a)


Fig. 10. (a) Micrograph of crack path profile taken at the center thickness of the specimen after application of 100% overload in the a/P microstructure at baseline AK of 15 MPafi. Note closure in overload region at B (test terminated after overload); (b) SEM fractograph of a corresponding area in another test piece (test continued after overload). Note ductile overload region C, and fretting damage

due to closure at D.

623

624 C. M. WARD-CLOSE & a!.

crack growth

direction

(a)


Fig. 13. (a) Center-section micrograph of fatigue crack at a baseline AK of I5 MPa,/h in the cf /~-annealed microstructure, after application of 100% low-high and high-fow block overload sequences. Note closure in the block-ov~rl~d region at B. and in post-overload region at A; (b) Corresponding SEM fractograph, showing the irregular and ductile-like fatigue fracture associated with high-block loading, and

fretting damage in the post-overload region (D).


-_- - -----___-~


Table 2. Summary of 100% single tensile overload results

Microstructure

Overload Exp. Baseline plastic Distance to delay Exp. Predicted

AK r_ zone size? steady-state distance delay delay (MPaJm 1 (mm) (mm) (-1 cycles cycles

a //T-annealed 8 0.03 15 0.09

/?-annealed 8 0.04 15 0.12

TComputed from robL - (1/3x) (Kot/a,)2.

0.12 0.22 35552 35490 0.40 0.88 9373 5430 0.27 0.52 301 I90 436210 0.40 0.77 16785 16483

c. Crack deJection

Crack deflection at approximately 45” to the main crack path was observed immediately following (low-high) block overloads in some, but not all, of the tests on a//?-annealed material (e.g. Fig. 13). Similarly, stepping-down after a 100% (high-low) overload block to a AK of 15 MPa& caused crack deflection at the surface in about half of the tests, but such deflection was not observed at the center of the specimen. Conversely, stepping-down to the same AK level from a 150% (high-low) overload block caused crack deflection both at the surface and in the interior. It was not possible to distinguish specific overload-related crack deflection in the &annealed microstructure, because of the already highly irregular and crystallographic nature of crack growth in this structure.

d. Crack closure

As noted above, center-section crack profiles revealed evidence of crack closure (crack-face contact at zero load) both for crack extension during the overload and for subsequent retarded post-overload crack growth, following both single and block overloads (Figs 9, 10 and 13). Despite such obvious metallographic indications of enhanced closure, standard global (far-field) & measurements with back-face strain compliance methods indicated only minimal changes before and after the overload event. In fact, large overloads were observed to lead to a small but immediate decrease in the closure load, i.e. for 150% overloads at a baseline AK of 15 MPa,/;;l in the a//I-annealed microstructure, the reduction in iu,, was between 8 and 17% (based on five observations).

However, on close examination of the unloading compliance curves, a second change in slope was frequently observed during the period of retardation following the overload (see Fig. 4). By monitoring this second closure point, approximately 130% increase in closure load was seen between the application of the overload and the point of minimum growth rates. Such measurements were considered to be associated with crack closure immediately behind the crack tip (near-tip closure), i.e. for post-overload crack extension into the overload plastic zone. This concept of dual closure levels is consistent with the observations of Nowack et a1.[28], and especially Paris and Hermann] who observed upper and lower opening loads following single tensile overload cycles. Although unexplained, their upper opening load increased at the overload, whereas their lower opening load showed a small but immediate decrease. Here we interpret the upper opening load in terms of near-tip closure principally for the increment of crack advance into the overload plastic zone, and the lower opening load in terms of far-field closure along the entire length on the crack.

Similar to experimental observations (Fig. 9), numerical predictions of the residual crack- opening displacements clearly show the blunting and enhanced opening of the crack at the overload, which reduces the far-field closure along the length of the crack (Fig. 14). Post-overload crack extension, conversely, involves very small opening displacements, which results in high levels of near-tip closure. Even after extensive post-overload crack growth, numerical predictions still show a large residual stretch of the crack at the point of the overload (similar to experiment- Fig. 91, which accounts for the dual levels of closure. It is worth noting here that conventional measurements of crack closure involving global compliance t~hniques will be relatively insensitive to near-tip closure and thus may detect little change in the closure level following the overload.


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Table 3. Summary of 100% block overload results

Microstructure

Low High Accel.

(MkKfi) (M:&) $1

Low-High High-Low

Delay AUXl. Predicted dist. Delay cyclest cyclest (mm) cyclesj

a //l-annealed 8 16.9 0.12 936 1062 0.98 244887 15 31.7 0.02 17 406 0.60 11960

p-annealed 8 16.9 0.25 4984 2300 0.40 400239 IS 31.7 0.29 451 611 0.83 27866

TExpressed as high-block cycles. SExpressed as low-block cycles.

These variations in experimentally measured crack closure and associated growth-rate behavior following 100 and 150% single tensile overloads in the a/j?-annealed microstructure are shown in Fig. 15. It can be seen that the extent of delayed retardation is consistent with an increase in near-tip crack closure (i.e. a decrease in AK&/AK), whereas after the overload cycle the far-field closure remains essentially unchanged.

5. DISCUSSION

The present study has shown that the magnitude and location of crack closure are principal factors governing load-interaction effects on fatigue crack growth in a//? titanium alloys under predominantly plane-strain conditions. Based on observations of larger crack-growth delays in the coarser-grained p-annealed microstructure, where the crack paths are more deflected and the fracture surfaces consequently far rougher, the primary mechanisms of closure are reasoned to be associated with the wedging of crack-face asperities (roughness induced closure[29-311) and more importantly with closure induced by cyclic plasticity[2] in the crack wake.

Work on the modelling of crack-face contact due to wedging during fatigue crack growth[32,33] suggests that contact close to the crack tip has a much greater influence on the effective near-tip stress intensity (AK,,) than contact remote from the tip. This notion is clearly supported by the numerical calculations of plasticity induced closure and the current measurements on near-tip and far-field closure during the overload sequences, which suggest that changes in crack closure in the immediate vicinity (typically within -500 pm) of the crack tip are primarily responsible for the transient crack-growth rate behavior associated with overloads. Specific mechanisms are now examined.

a. Single tensile overloads

Crack growth at the overload. At high AK levels, measurable increments of ductile crack growth, larger than anticipated from constant-amplitude fatigue data, occur during the single overload cycle. The enhanced magnitude of such crack extension at the overload may be a consequence of several factors. These include the smaller extent of residual compressive stresses ahead of the crack and the corresponding lower crack-closure (and opening) loads at the baseline AK, which cause the overload cycle to generate a larger effective AK, and hence a larger growth increment, than it would if it were part of continuous constant-amplitude cycling. Fractographic evidence (Fig. 11) and compliance measurements (Fig. 15) suggest that the ductile nature of such overload-induced crack growth results in irregular, non-matching fracture surfaces, which promote increased closure loads once cycling is returned to the lower baseline level (Fig. 11). Such closure on the fracture surfaces formed during large-amplitude overload cycles (shown at B in Fig. 10a) may partly negate the tendency for initial acceleration of growth rates on return to baseline cycling (see below). Indeed, the largest post-overload accelerations were observed at the lowest baseline AK levels, where crack growth during the overload cycle was very small and subsequent closure in the immediate post-overload region less apparent.

Initial acceleration. Following the overload cycle, crack-growth rates remained initially at up to two orders of magnitude faster than the steady-state baseline rate. Similar accelerated growth


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The residual crack surface displacement at minimum crack growth rate

The residual crack surface displacement for crack growth return to baseline crack growth rate

-2.00 -iI 0.87 0 .m 0.99 1 .oo 1 .Ol 1.02

CRRCK SURFACE X/A E+OO

Fig. 14. Numerical predictions of the residual crack-opening displacements during the overload event (for 100% single overload at baseline AK of 15 MPa$) in a/B-annealed microstructure. Note how the overload causes crack-tip blunting; subsequent post-overload crack growth occurs with vastly

decreased crack-opening displacement.

rates immediately following overloads have been previously reported in steels[ 13,341 and aluminum alloys[35]. In this study, metallographic sections (Figs 9 and 10) and numerical predictions (Fig. 14) show that the crack was blunted by the overload and remained open at zero load for some distance behind the crack front. Acceleration, therefore, is caused by the removal or reduction of crack closure along the crack length by the overload. This is supported by observations of larger crack opening displacements along the crack following the overload (Figs 9a, 10a and 14), and by the back-face strain-gauge results which indicate a reduction in far-field closure at the overload


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(Fig. 15b). In this sense, the immediate post-overload crack-growth response may be likened to short-crack behavior. Many workers (e.g. ref.[36]) have shown that growth rates of “short” cracks (typically less than 500 pm long) may be initially high, due primarily to the minimal development of closure along their limited wake; with continued extension, however, closure levels build-up and the growth rates decay or approach those of long cracks.

Delayed retardation. Delayed retardation, as modelled numerically and observed in all overload experiments, was reasoned to be caused primarily by the enhanced residual compressive stress field in the overload plastic zone and the consequent increase in crack closure generated as the crack grew into this zone. The overload cycle produces a zone of plastically-deformed material ahead of the crack front, accommodated by crack-tip opening and shear at the surface, which on return to minimum load results in an enlarged region of compressive residual stresses ahead of the tip. These stresses effectively reduce the load ratio and promote plasticity induced closure in post-overload region as crack enters the zone. The effective stress-intensity range experienced at the tip is thus progressively lowered with a consequent reduction in growth rates. In the present study, a large increase in near-tip crack closure was detected at this stage, for both 100% and 150% overloads in the a/P-treated material (Fig. 15).

With the post-overload decrease in growth rates, effective stress-intensity ranges approach levels comparable with near-threshold conditions. Accordingly, fracture processes appropriate to near-threshold behavior, such as crystallographic crack growth, crack branching and enhanced roughness induced closure, may become apparent. As these mechanisms act to reduce further the effective AK and prolong the retardation[7], the distance over which the crack growth rate is affected by the overload in plane strain becomes far larger than the size of the overload plastic zone (Table 2); this is in contrast to observed behavior in predominantly plane-stress conditions[3].

The transient growth-rate response and associated retardation phenomenon due to the application of a single tensile overload can thus be seen as the result of the following mechanistic sequence (Fig. 16).

(a) The overload cycle-rack-tip blunting and associated increased crack-opening displacements removes near-tip crack closure and reduces (far-field) closure along the length of the crack. At higher AK levels, the overload results in an increment of ductile crack growth.

(b) Crack growth-rate acceleration-on return to baseline cycling, if the ductile crack advance at the overload is small, the absence of near-tip crack closure leads to an initial crack growth rate one or two orders of magnitude higher than would be normal for steady-state cycling conditions. With larger overload cycles, the removal of closure behind the pre-overload crack tip may be offset by closure generated by the ductile crack-growth increment formed by the overload cycle, in which case the acceleration may be reduced or be absent.

(c) Delayed retardation-as the fatigue crack extends into the overload plastic zone, it encounters an enlarged zone of residual compressive stresses which act to promote plasticity induced crack closure in the wake of the crack tip; the effective stress-intensity range is thus reduced, and fatigue crack growth rates are retarded. Mechanisms such as crack deflection and consequent enhanced roughness induced closure may prolong the retarded region over distances far larger than the overload plastic zone.

Fatigue crack profiles associated with single tensile overloads, together with corresponding fatigue crack growth behavior, are illustrated schematically in Fig. 16 for low K and high K overloads.

b. Block overloads

The observed transient growth-rate response associated with block overloads can also be rationalized in terms of the role of residual stress and near-tip crack closure. With a low-high block overload, the observed initial acceleration in growth rates, above the steady-state level for the high block, has been attributed to transient changes in plastic zone size[3]. The smaller reversed plastic zone, “inherited” from the low block, essentially permits more crack-tip plasticity during the first cycle of the high block, resulting in an initially increased crack growth rate before the steady-state plastic zone becomes established. However, in the present study, the accelerated growth rates persisted, in some cases, for several thousand cycles (Fig. 1 l), implying that the removal of near-tip crack closure by the first cycle of the overload block was more significant (see Fig. 13). In this case,


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the fatigue crack experiences an increasing effect of closure with growth at the higher AK, leading to a gradual reduction in growth rates until steady-state conditions are attained. The initial acceleration was targest for low-high block overloads from a baseline of 8 MPa,/%, consistent with the effect being caused by a reduction in &, , which would have a proportionaily greater effect on AK,, the lower the baseline AK.

The transient growth-rate response associated with high-low block overloads was similar in effect to that of single tensile overloads. However, whereas with a single overload the retardation is preceded by an initial acceleration caused by the removal of near-tip closure from the overload cycle, with the step-down from the high to low block, near-tip closure has been re-estab~ish~ during growth at the higher block. Consequently, the retardation following the high-low block overload event should be immediate and more severe than for an equivalent single overload. Such behavior was observed in both microstructures (compare Figs 8 and 12), particularly at low baseline AK levels, where the increase in K,, has the larger influence on AK&. Figure 17 illustrates schematically the proposed mechanistic sequence.

c. Microstructural efects

The coarse-grained P-annealed microstructure showed longer periods of retardation following both single and block tensile overloads compared to the standard fine-grained M: /P-annealed microstructure, consistent with previous comparisons using the simulated-flight loading spectrum


FALSTAFF[24]. However, the superior retardation response of the B-annealed microstructure was more prominent at lower baseline AK levels. This appears to be partly associated with the far larger effective grain size of this structure, which results in some crack deflection and crack-path meandering, which in turn promotes roughness induced crack closure. As the latter shielding mechanism involves wedging of the crack via fracture-surface asperities[29-3 11, it is most effective where the size of the wedges is comparable with crack-tip-opening displacements, i.e. at low AK levels.

d. Influence of baseline AK

The influence of baseline AK level on the magnitude of the delay produced by 100% single tensile overloads in the two titanium microstructures is shown in Fig. 18, both in terms of delay cycles and delay distance; results[37] on an aerospace steel BSS99 are included for comparison. Similar to results reported by Vecchio et a1.[38,39] and others[40,41] on aluminum alloys, the degree of retardation is increased, both with increasing baseline AK toward instability and with decreasing AK toward the threshold. Such behavior reflects competition between the various mechanisms contributing to the retardation effect. At high AK levels, the extent of the crack-tip residual stress field, and hence the magnitude of plasticity induced closure, is clearly related to the size of the plastic zone, provided the zone does not become too large such that the constraint of the surrounding material is lost. As this appears to be the dominant mechanism, the magnitude of the retardation will increase with increasing AK. Conversely, at low AK levels where the crack-tip-opening displacements are small, the dominant closure mechanisms are also associated with wedging processes[28-33, 361 which are promoted by decreasing AK levels. Moreover, changes in Kc, will have a proportionally larger effect the lower the AK. Consequently, plane-strain load-interaction effects will tend to increase with decreasing AK. Since wedge-related closure mechanisms are primarily microstructure-related, differences between the post-overload behavior of the a //I- and B-annealed structures are likely to be most apparent in this regime, consistent with the present results.

6. CONCLUSIONS

Based on an experimental and numerical study of the transient crack growth rate response of mode I fatigue cracks subjected to 100 and 150% single and block tensile overloads (at baseline AK levels of 8 and 15 MPa,,/%) in a //?- and P-annealed microstructures of IMI 550 titanium alloy, the following conclusions can be made:

1. The close agreement between experimental results and numerical predictions of the modified Dugdale model imply that the transient fatigue crack growth rate behavior following tensile overloads is primarily a function of changes in the magnitude and distribution of plasticity induced crack closure.

2. The application of a tensile overload was seen to cause a significant change in the magnitude of closure in the immediate vicinity of the crack tip (termed near-tip closure), but to have far less effect on closure along the length of the crack remote from the tip (termed far-field closure).

3. On the application of a single tensile overload, crack-tip blunting was found to produce an increase in crack-opening displacement, resulting in the removal of near-tip closure and a slight reduction in far-field closure, which caused an initial transient acceleration in crack growth rates.

4. Subsequent delayed retardation in crack growth rates was accompanied by a significant increase in near-tip crack closure as the crack penetrated the enlarged compressive residual stress field, formed ahead of the crack tip during the overload cycle. As the retardation results specifically from plasticity induced closure in the wake of the crack tip, the distance affected by the retardation is generally larger than the size of the overload plastic zone. This may be further enhanced at low AK levels where additional mechanisms of closure are prevalent.

5. Low-high block loading sequences caused an initial transient increase in crack growth rates. This was attributed primarily to the temporary removal of near-tip crack closure by the initial cycles of the higher block.

6. High-low block loading sequences caused a retardation in crack growth rates, similar to that following a single overload. However, the retardation was immediate and more severe than

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for corresponding single overloads. This was considered to result from plasticity induced closure in the wake of high-block fatigue crack, coupled with additional closure developed as the crack penetrated the compressive residual stress field of the overload plastic zone at the lower AK level.

7. The magnitude of the load-interaction effects was found to be somewhat larger in the coarser-grained p-annealed microstructure than in a standard fine-grained a //S-annealed microstructure, especially at low baseline AK levels where additional wedge mechanisms of crack closure tend to be promoted in coarse microstructures.

8. The magnitude of the post-overload retardation effect was found to be enhanced with both increasing and decreasing baseline stress-intensity range, respectively consistent with an increase in plasticity induced closure at high AK and in closure from asperity wedging at low AK.

Acknow~e~gefflenrs-this work was supported by the Director, Ofice of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SFOOO98, and by the Swedish Board for Technical Development. Support for one of the authors (C.M.W-C.) was also provided by the Ministry of Defence, U.K., in the form of a leave of absence from the Royal Aircraft Establishment to spend a year in Berkeley. Particular thanks ate due to G. S. Wang for assistance with the numerical calculations, and to Madeleine Penton for help in preparing the manuscript.

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[2] W. Elber, The significance of fatigue crack closure, in Damage Toferance in Aircraft Structures, ASTM STP 486, pp. 230-242. American Society for Testing and Materials, Philadelphia, PA (1971).

[3] E. F. J. von Euw, R. W. Hertzberg and R. Roberts, Delay effects in fatigue crack propagation, in Stress Analysts and Growth of Cracks, ASTM STP 513, pp. 79-90. American Society for Testing and Materials, Philadelphia, PA (1972).

[4] R. H. Christensen, Metal Fatigue. MacGraw-Hill, New York (1959). [.5] C. M. Hudson and H. F. Hardrath, Effects of changing stress amplitude on the rate of fatigue crack propagation of

two aluminum alloys. NASA Technical Note D-960, National Aeronautical and Space Administration, Washington, DC. (1961).

[6] J. Lankford and D. L. Davidson, The effect of overloads upon fatigue crack tip opening displacement and crack tip opening/closing loads in aluminum alloys, in Advances in Fracture Research (Edited by D. Francois), Vol. 2, pp. 899-906. Pergamon Press, Oxford, U.K. (1982).

[7] S. Sutesh, Micromechanisms of fatigue crack growth retardation following overloads. Engng Fracture Mech. 18, 577-593 (1983).

[8] R. E. Jones, Fatigue crack growth retardation after single cycle peak overload in Ti-6Al-4V titanium alloy. Engng Fracture Meek. 5, 585604 (1973).

[9] J. F. Knott and A. C. Pickard, Effects of overloads on fatigue crack propagation: aluminum alloys. Metal Sci. 11, 399-404 (1977).

[IO] P. J. E. Forsyth, Causes of mixed fatigue-tensile-crack growth and significance of microscopic crack behaviour. Metals Technol. 5, 351-357 (1978).

[l 11 H. D. Dill and C. R. Saff, Spectrum crack growth prediction method based on crack surface displacement and contact analysis, in Fatigue Crack Growth Under Spectrum Loads, ASTM STP 595, pp. 306-3 19. American Society for Testing and Materials, Philadelphia, PA (1976).

[12] J. C. Newman, Jr, A crack closure mode1 for predicting fatigue crack growth under aircraft spectrum loading, in Methods and Models for Predicting Fatigue Crack Growth under Random Loading, ASTM STP 748, pp. 53-84. American Society for Testing and Materials, Philadelphia, PA (1981).

1131 A. F. Blom and D. K. Holm, Load interaction effects on fatigue crack propagation in steels of varying strength, in Role qfFracture Mechanics in Modern Technology (Edited by G. C. Sih and H. Nisitani), pp~ 235-249. North-Holland, Elsevier Science Publishers B.V., Amsterdam (1987).

[I41 L. N. McCartney, A theoretical explanation of the delay effect of overloads on fatigue crack propagation, Inr. J. Fracture 14, 213-232 (1978).

[ 1 S] R. W. Hertzberg, Deformation and Fracture Mechanics ofEngineer~ng materials, Second edn. Wiley, New York (1983). [16] D. L. Davidson and S. J. Hudak, Alterations in crack tip deformation during variable-amplitude fatigue crack growth,

in Fracture Mechanics 18th Symposium, ASTM STP 945 (Edited by R. P. Read and D. T. Reed), pp. 934-954. American Society for Testing and Materials, Philadelphia, PA (1987).

[17] T. C. Lindley and C. E. Richards, The relevance of crack closure to fatigue crack propagation. Mater. Sci. Engng 14, 28l--293 (1974).

[18] J. T. P. de Castro and D. M. Parks, Decrease in closure and delay of fatigue crack growth in plane strain. Scripta Met. 16, 1443-1446 (1982).

[19] D. J. Alexander and J. F. Knott, Fatigue crack retardation in aluminium alloys, in Fatigue 87, Proc, 3rd Int. Conf. on Farigue and Fatigue Thresholds (Edited by R. 0. Ritchie and E. A. Starke), Vol. 1, pp. 395406. EMAS Ltd. Warley, U.K. (1987).

[20] P. C. Paris and L. Hermann, Twenty years of reflection on questions involving fatigue crack growth, Part 11: Some observations on crack closure, in Fatigue Threshofds, Proc. 1st Int. Co@ on Fatigue Thresholds (Edited By J. Backlund, A. F. Blom, and C. J. Beevers), Vol. 1, pp. 1 l-33. EMAS Ltd, Warley, U.K. (1982).

[2l] C. M. Ward-Close and R. 0. Ritchie, On the role of crack closure mechanisms in influencing fatigue crack growth following tensile overloads in titanium alloys: Near-threshold vs higher AK behavior, in Mechanisms ofFatigue Crack


P61

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[291

tz;

1321

I331

[341

[351

[361

[371

[381

[391

1401

[411

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(231 G. R. Yoder, L. A. Cooley and T. W. Crooker, 50-fold difference in region--II fatigue crack propagation resistance of titanium alloys: A arain-size effect. J. Enann Mater. Technol. 101. 8690 (1979). C. M. Ward-Cl&e, Simulated flight fatigue%-an a//? titanium alloy. Inr. J.‘Fa&ue 6, 139-146 (1984). J. E. Allison and J. C. Williams, The role of crack closure in rationalizing the variations in fatigue crack growth in titanium alloys, in Tifanium Science and Technology (Edited by G. Liitjering, U. Zicker and W. Bunk), Vol. 4, pp. 2243-2250. DGM Publishers, Oberursel, West Germany (1984). G. S. Wang and A. F. Blom, A modified Dugdale-Barenblatt model for fatigue crack growth predictions under general load conditions. FFA TN 1987-79, The Aeronautical Research Institute of Sweden, Bromma, Sweden, 1987. Submitted for publication. D. S. Dugdale, Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100-104 (1960). H. Nowack, K. H. Trautmann, K. Schulte and G. Liitjering, in Fracfure Mechanics, ASTM STP 677 (Edited by C. W. Smith). pp. 3653. American Society for Testing and Materials, Philadelphia, PA (1979). N. Walker and C. J. Beevers, A fatigue crack closure mechanism in titanium. Fatigue Engng mater. Strucr. 1, 135-148 (1979). K. Minakawa and A. J. McEvily, On crack closure in the near-threshold region. Scripta Met. 6, 633636 (1981). S. Suresh and R. 0. Ritchie, A geometric model for fatigue crack closure induced by fracture surface roughness. Mer. Truns. A 13A, 1627-1631 (1982). S. Suresh and R. 0. Ritchie, Some considerations on the modelling of oxide-induced fatigue crack closure using solutions for a rigid wedge inside a linear elastic crack. Scripra Met. 17, 5755580 (1983). C. J. Beevers, K. Bell, R. L. Carbon and E. A. Starke, A model for fatigue crack closure Engng Fracture Mech. 19, 93-100 (1984). C. Robin, M. Louah and P. Pluvinage, Influence of an overload on the fatigue crack growth in steels. Fatigue Engng mater. Struct. 6, l-13 (1983). D. M. Corby and P. F. Packman, On the influence of single and multiple peak overloads on fatigue crack propagation in 7075-T6511 aluminum. Engng Fracture Mech. 5, 479497 (1973). R. 0. Ritchie and W. Yu, Short crack effects in fatique: A consequence of crack tip shielding, in Small Fufigue Cracks (Edited by R. 0. Ritchie and J. Lankford), pp. 1677180. TMS-AIME, Warrendale, PA (1986). C. M. Ward-Close and R. 0. Ritchie, unpublished work, Lawrence Berkeley Laboratory, University of California, Berkeley, CA (1985). R. S. Vecchio, R. W. Hertzberg and R. Jaccard, Overload induced fatigue crack growth rate attenuation behavior in aluminum and steel alloys. Scripta Met. 17, 343-346 (1983). R. S. Vecchio, R. W. Hertzberg and R. Jaccard, On overload induced fatigue crack propagation behavior in alumium and steel alloy. Fatigue Engng mater. struct. 7, 181-194 (1984). G. R. Chanani, Effect of thickness on retardation behavior of 7075 and 2024 aluminum alloys, in Flaw Growth und Fracture ASTM STP 631, 365-387. American Society for Testing and Materials, Philadelphia, PA (1977). K. T. Venkateswara Rao and R. 0. Ritchie, Mechanisms for the retardation of fatigue cracks following single tensile overloads: Behavior in alumium-lithium alloys. Acta Met. 36, in review (1988).

Closure, ASTM STP 982 (Edited by J. C. Newman and W. Elber). American Society for Testing and Materials, Philadelphia, PA (1988).

(Received 16 February 1988)

Mechanisms associated with transient fatigue crack growth under variable-amplitude loading: An experimental and numerical study

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