Effect of cyclic plastic strain on fatigue crack growth

International Journal of Fatigue xxx (2015) xxx–xxx

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Effect of cyclic plastic strain on fatigue crack growth Michael Vormwald ⇑ Materials Mechanics Group – Technische Universität Darmstadt, Franziska-Braun-Str. 3, D-642987 Darmstadt, Germany

a r t i c l e

i n f o

Article history: Received 2 February 2015 Received in revised form 1 June 2015 Accepted 13 June 2015 Available online xxxx Keywords: Fatigue crack growth Plastic strain Crack closure Variable amplitude loading Multiaxial loading

a b s t r a c t Because fatigue life calculations, which are based on the stress intensity factor, are restricted to certain limited applications, one must develop substitute procedures in more general settings. Various proposals for a crack driving force parameter in elastic–plastic fracture mechanics will be discussed. A preference in favour of the cyclic DJ-integral is elaborated, keeping its theoretical limitations in mind. Crack closure is also an important issue under large-scale yielding conditions. Experience cannot be extrapolated from the small-scale to the large-scale cyclic yielding regime or vice versa. The consequences for fatigue lives under variable amplitude and multiaxial loading are discussed. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The fatigue crack growth rate is determined by stresses and deformations at the crack front. According to the linear theory of elasticity, a singularity appears at crack fronts. In a first-order approximation crack growth rates are usually linked to the range of the corresponding stress intensity factor. However, plastic and disruptive processes at the crack tip are responsible for crack propagation. The applicability of a parameter derived from a linear theory to describe highly non-linear phenomena is limited. Various observable phenomena of fatigue crack growth cannot be explained without considering cyclic plasticity. The influence of mean stress on the fatigue crack growth rate is a well known example where plasticity-induced crack closure supplies an explanation. The crack closure argument also provides an explanation for the various phenomena of load history effects, which may accelerate or decelerate the growth rate depending on a variety of conditions. Experience concerning fatigue crack closure cannot be transferred from the small-scale to the large-scale cyclic yielding regime or vice versa. Multiaxial and mixed mode aspects in the large-scale cyclic yielding regime are a final topic of the present paper.

2. Crack driving force parameters The overview on the development of crack driving force parameters has been well documented by McClung et al. [1]. A first ⇑ Tel.: +49 6151163645; fax: +49 6151163038. E-mail address: [email protected]

obvious attempt to move from small to large scale yielding was to replace stress by strain quantities, see for example Boettner et al. [2], McEvily [3], or El-Haddad et al. [4]. At first, only the plastic strain range replaced the stress range in stress intensity factor formulas of a given geometry. Later the total strain range was used.

DK e ¼ ðDeel þ Depl ÞE

pffiffiffiffiffiffi paY

ð1Þ

This makes more sense because a large scale yielding parameter should smoothly approach the small scale yielding parameter for vanishing plastic deformations. However, a strain-based intensity factor DKe does not provide a measure of the strain singularity at the crack front. The most important advantage of this method is its pre-eminent ease of application. If applied as suggested, not even the growth rate constants have to be re-determined. They can be directly used as determined in the small scale yielding formulation and expressed in terms of the stress intensity factor. The theoretical shortcomings of DKe can be overcome by using the cyclic crack tip opening displacement Ddt as the crack driving force parameter. This was proposed by many researchers, see for example McEvily et al. [5], Tomkins [6], or Tanaka et al. [7]. This measure of cyclic deformation is taken as closely as possible to the location of material separation. It is generally assumed to provide a sound and unique correlation with the growth rate. However, determining it is a difficult task. McClung et al. [1] have already emphasised that knowledge of a proper driving force is of little value unless the driving force can either be calculated exactly with reasonable effort or estimated with sufficient accuracy. Simple extensions of the Dugdale [8] model led to approximation formulas, see Vormwald and Seeger [9]. An alternative way to

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Please cite this article in press as: Vormwald M. Effect of cyclic plastic strain on fatigue crack growth. Int J Fatigue (2015), http://dx.doi.org/10.1016/ j.ijfatigue.2015.06.014

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M. Vormwald / International Journal of Fatigue xxx (2015) xxx–xxx

Nomenclature a D E i J K K0 m md n n0

crack length damage sum Young’s modulus counter J-integral stress intensity factor cyclic hardening coefficient exponent in crack growth law constant counter for the number of applied load cycles cyclic hardening exponent

N R t u W x Y dt

e r

determine the cyclic crack tip opening displacement is to apply the strip yield model, Dill and Saff [10], Führing and Seeger [11], Newman [12]. The strip yield model is usually applied only to calculate the crack opening stresses. Schlitzer et al. [13] calculated fatigue crack growth based on the cyclic crack tip opening displacement extracted from strip yield model calculations. When going from large to small scale yielding conditions the parameter should asymptotically approach a bijective functional relationship with the stress intensity factor. For the crack tip opening displacement this function is provided by

DK d ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi md EDrY Ddt

ð2Þ

where md = 1 for plane stress conditions and md  2 for plane strain. The extension of the J-integral for application with cyclic loading according to Dowling and Begley [14], and Dowling [15] has by far drawn the greatest attention of researchers and engineers who want to model fatigue crack growth in the large scale yielding regime. It has also been the subject of extreme academic controversy. The definition of the cyclic DJ-integral is given by

  DJ ¼ J Drij ; Deij ; Dti ; Dui ¼

 Z  @ðDui; Þ DWdx2  Dt i ds @x1 C

ð3Þ

with

DW ¼ WðDrij ; Deij Þ ¼

Z

~ ij Þ Deij ðDr

~ ij dðD~eij Þ: Dr

ð4Þ

0

The prefix ‘‘D’’ of the variables for stress rij, strain eij, traction ti and displacement ui, designates the changes in these quantities. These changes must be evaluated from a reference state. This reference state of all field variables serves as a new origin for defining increment-field variables, the latter designated with the prefix. The stress and displacement state at the time of a load reversal is a natural and sound reference state. Then the increments D(  ) designate the changes from the respective reference values. At the instant of the next reversal the field variables with a prefix are identical with the conventional ranges. However, the ‘‘D’’-prefix does not represent changes in J and W; instead DJ and DW are functions of their arguments as defined in Eqs. (3) and (4). Wüthrich [16] proposed that a new variable, Z for DJ, should be used. The proposal was not widely accepted. McClung et al. [1] suggested referring to a ‘‘DJ-integral’’ rather than the ‘‘range of the J-integral’’; the latter is, strictly speaking, wrong. The path independence is the outstanding property of the J-integral and the DJ-integral. A path independent value is a measure of stresses and strains very near the crack front responsible for material separation processes, on the one hand, but, on the other hand, it can also be determined by far field values, uncontaminated by numerical deficiencies. The most frequent objections raised to the DJ-integral argued that since the J-integral was based on the

number of cycles to failure stress ratio stress vector displacement vector strain energy density coordinate geometry influence function crack tip opening displacement strain stress

theory of non-linear elasticity or (with limitation) deformation plasticity, it does not allow unloading. However, Wüthrich [16] proved the path independence if the DJ-integral is defined as given above. The use of the DJ-integral has some remaining theoretical limitations. A first type of limitation has to do with the material’s stress–strain behaviour. Yoon and Saxena [17] have pointed out that path independence is violated if the material is not completely cyclically stabilized. Also, strict compliance with path independence conditions cannot be achieved in the presence of temperature gradients and material behaviour dependent on temperature. Some remedies have been proposed, e.g. Blackburn [18], Kishimoto et al. [19], Atluri et al. [20]. These proposals for removing mathematical restrictions have recently found their way into engineering applications, see Bauerbach et al. [21]. A second type of limitation has to do with crack closure. There should be no stresses at the crack flanks; otherwise path independence is violated. The instant of a complete loss of contact would be a natural reference state. This state is recommended by McClung et al. [1]. However, even at this instant, the material at the various locations in the structure is at different stress– strain-positions on the ascending hysteresis branch. There will be no path independence during further loading. A valid DJ-integral can be calculated for a reversal from maximum load to crack closure load. This opens a route to deal with crack closure in connection with a DJ-based fatigue crack growth calculation. Choosing the instant of the maximum load as reference state, path independence is maintained during the descending reversal until crack face contact first occurs. Calculating the DJ-integral at the instant of first contact will provide a path independent effective DJ-integral, DJeff. 3. Crack closure under large-scale yielding conditions Measurements of the crack opening and crack closure levels under large-scale cyclic yielding conditions have been performed by many researchers, Dowling and Iyyer [22], Rie and Schubert [23], McClung and Sehitoglu [24,25], Vormwald and Seeger [9,26], DuQuesnay et al. [27], El-Zeghayar et al. [28,29], Pippan et al. [30]. Some results for the constructional steel S460 [9] are shown in Fig. 1. Short fatigue cracks were initiated by strain controlled fatigue loading of conventional cylindrical material specimens used for low-cycle fatigue testing. The deformations in the neighbourhood of the crack were measured using a strain microstrain-gauge with a grid length of 0.6 mm. While the crack is closed, its flanks can transfer stresses by contact. This leads to a nearly homogeneous uniaxial stress state. Local near-crack strains do not differ from global strains used in the control loop of the testing machine. Upon crack opening, the micro strain gauge gets into the stress and strain shadow of the crack. The local strains are smaller than the global strains. The deviation between the

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M. Vormwald / International Journal of Fatigue xxx (2015) xxx–xxx

500

Crack depth a = 1 mm

400

a = 0.5 mm

σref εref

Crack depth a = 1 mm a = 0.5 mm σref εref

200 no crack

no crack

100 0

σop σop

-200 -300 -400

σcl

-500 -0.6

-0.4

σcl

Steel S460 Global strain amplitude εa = 0.5 %

-0.2

0.0

0.2

εop = εcl

-100

εop = εcl

Global stress in MPa

300

0.4

0.6 -0.4

-0.2

0.0

Steel S460 Global strain amplitude εa = 0.2 %

0.2

0.4

Local strain near crack flanks in %

Local strain near crack flanks in %

Fig. 1. Crack opening and closure of semi-circular surface cracks of depth a in cylindrical specimens, Vormwald et al. [9].

global and local hysteresis loops allows for the determination of the crack opening and crack closure levels. It turns out that fatigue cracks open and close at nearly the same global strains, which is in accordance with results presented by others, e.g. [30]. The stabilized levels of crack opening stress that are nearly independent of crack length decrease with increasing applied stress, Fig. 2. An increase of applied stresses is accompanied by leaving the small scale yielding regime. The decrease of crack opening stress means that the crack driving force parameter increases more than linearly with the applied stress due to plasticity-influenced loss of crack closure. Newman’s [31] approximation formula, Eq. (5), for calculating crack opening stresses describes the trend correctly.





rop p rmax rmax þ 0:344R ¼ 0:535 cos rY 2 rY rY

ð5Þ

Fig. 2 shows a comparison of calculated and measured crack opening stresses. The general experimentally observed trend of decreasing crack opening stresses with increasing amplitudes is nicely mirrored by this approximation. Improved accuracy can be achieved by individually adjusting the yield stress, rY, used in Eq. (5). Here, the average (557 MPa) between ultimate tensile strength (682 MPa) and cyclic 0.2% cyclic plastic strain offset (432 MPa) is used.

under total strain control with amplitude ea = 0.5%. The stabilised hysteresis loop is shown in Fig. 3. The shape of the hysteresis loop branches can be described by

 1 ~ ~ n0 D~e 1 Dr 1 Dr ¼ þ 0 E 2 2 K 2

ð6Þ

where the material’s cyclic hardening coefficient and exponent are K0 = 1230 MPa and n0 = 0.126. Young’s modulus is E = 198 GPa. The ~ -coordinate sysloop’s reversal points are the origin for the D~e—Dr tem used in Eq. (6) and shown in Fig. 3. For the case shown, the global plastic strain amplitudes are in the same order of magnitude as the global elastic strain amplitudes. The growth of the fatigue cracks was measured by applying the replica technique. The fatigue test was interrupted regularly and surface replicas were taken before continuing the test. The collection of replicas was investigated under the microscope and surface crack lengths, 2a, were measured. An example of the appearance of the surface is shown in Fig. 4. The crack growth curve given in Table 1 and drawn in Fig. 5 is for the longest crack found in the specimen. da Table 1 includes the fatigue crack growth rates, a0i ¼ ðdn Þi , calculated using

a0i ¼



 da aiþ1  ai1 ¼ : dn i niþ1  ni1

ð7Þ

4. Short fatigue crack growth The growth of short fatigue cracks under large cyclic plastic strain conditions has been studied in more detail [32]. A cylindrical specimen machined from steel 42CrMo4V was cyclically loaded

600

Δε~

42CrMo4V

400

~ Δσ

0.2

stress in MPa

R = -1

σop / σY

0.0

-0.2

200

0

-200

S460N

-0.4

-400

Approximation, Newman, IJFract . 1984

-0.6 0.0

~ Δσ ~ Δε

-600 0.2

0.4

0.6

0.8

1.0

σmax / σY Fig. 2. Crack opening stress as a function of applied stress, Vormwald et al. [9].

-0.5

0.0

0.5

strain in % Fig. 3. Cyclically stabilised hysteresis loop, steel 42CrMo4V, ea = 0.5% [32].

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M. Vormwald / International Journal of Fatigue xxx (2015) xxx–xxx

crack tip

crack tip

inclusion

inclusion crack tip

crack tip

after 2750 cycles, surface crack length 2a = 36 μm

after 6000 cycles, surface crack length 2a = 225 μm

Fig. 4. Surface of specimens from steel 42CrMo4V inspected during a fatigue test with ea = 0.5% [32].

Table 1 Fatigue crack growth curve, steel 42CrMo4V, ea = 0.5%. i n a in lm a0 in nm/cycle

1 1250 6.5 5

2 1500 7.5 6

3 1750 9.5 5

4 2000 10 1

5 2250 10 1

6 2500 10.5 16

7 2750 18 19

8 3000 20 5

9 3250 20.5 2

10 3500 21 5

11 3750 23 10

12 4000 26 13

i n a in lm a0 in nm/cycle

13 4250 29.5 28

14 4500 40 29

15 4750 44 27

16 5000 53.5 39

17 5250 63.5 51

18 5500 79 52

19 5750 89.5 67

20 6000 112.5 146

21 6250 162.5 146

22 6500 185.5 134

23 6750 229.5 107

24 7000 239

In Fig. 6 these calculated fatigue crack growth rates are plotted over the crack length. The rates are considerably decelerated at crack lengths of 10 lm and 20 lm. Although the reason for this behaviour has not been studied in detail, in the current investigation the hypothesis is that the deceleration is due to the influence of the microstructure on the fatigue crack growth rates. As the crack becomes longer the micro-structural influence decreases. The intention of the present paper is to discuss the dependency of the crack growth rates on the global cyclic plastic strain. Micro-structural issues, therefore, cannot be taken into account in detail – only the growth rates in average when the crack front passes over several grains can be related to the macroscopic quantities of continuum mechanics. For long cracks and within the range of applicability of linear elastic fracture mechanics the fatigue crack growth rate can be empirically correlated to the range of the crack-closure-free stress intensity factor, DKeff. For many steels a power-law type expression describes this behaviour quite accurately,

0.30

42CrMo4V

0.25

σa = 567 MPa εa = 0.5 %

0.20

a in mm

2a

0.15 0.10 0.05 0.00

0

1000

2000

3000

4000

5000

6000

7000

8000

n Fig. 5. Fatigue crack growth curve, steel 42CrMo4V, ea = 0.5% [32].

10

    da da DK eff m ¼ ; dn dn ref DK eff;ref

3

42CrMo4V

where a reference point is defined by taking values of pffiffiffiffiffi da mm ðdn Þref ¼ 105 cycle and DK eff;ref ¼ 9 MPa m. These values, together

σa = 567 MPa

da/dn in nm/cycle

10

εa = 0.5 %

2 2a

10

10

10

R = -1

1

da mm ⎛ ΔKε ⎞ = 10−5 ⎜ ⎟ dn cycle ⎝ 9MPa m ⎠ 0

ð8Þ

⎛ ⎞ da mm ⎜ ΔJ eff ⎟ = 10−5 ⎜ ⎟ dn cycle ⎜ 0.4 N ⎟ ⎝ mm ⎠ 3 ⎞ da mm ⎛ ΔK = 10−5 ⎜ ⎟ dn cycle ⎝ 9MPa m ⎠

3

1.5

-1

10

100

a in μm Fig. 6. Fatigue crack growth rates, steel 42CrMo4V, ea = 0.5%, experimental results and estimations based on various simplified approximations.

with the exponent m = 3, are used in the following. For nearly semi-circular surface cracks in structures in which all geometrical dimensions are much larger than the crack length and which are loaded by a homogeneous stress r (in the structure without crack), the stress intensity factor can be approximated pffiffiffiffiffiffi by K ¼ 0:7r pa. Eq. (8) is used to provide a first estimate of fatigue crack growth rates. It is assumed that the crack is open throughout the total applied stress–strain range. This assumption may hold for the very beginning of naturally initiated fatigue cracks when no plasticity induced crack closure could have developed. The comparison in Fig. 6 shows that, on average, the growth rates are underestimated. Next, the performance of the strain based intensity factor defined according to Eq. (1) is checked. Again, the crack is supposed to be open for the entire range. The corresponding curve in Fig. 6 shows that one can postulate an acceptable representation of the non-decelerated fatigue crack

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M. Vormwald / International Journal of Fatigue xxx (2015) xxx–xxx

ð9Þ

The following general relationship has been given by Dowling:

"

DJ ¼ 2pa

#  2f ðn0 Þ ðDrÞ2 f ðn0 Þ 1 0 þ 0 DrDepl n þ1 n þ1 2E

ð10Þ

where f(n0 ) has to be determined for the individual geometry. Many specific formulations have been proposed for f(n0 ). In considering the case of a semicircular surface crack including crack closure the following expression [9] was used adopting Dowling’s [38] approach,

"

DJeff

# ðDreff Þ2 1:02 ¼ a 1:24 þ pffiffiffiffi Dreff Depl;eff : E n0

ð11Þ

In contrast to many other applications, the effective ranges entering Eq. (11) are taken from the descending hysteresis loop branch. The upper reversal point is used as reference point. For this portion of a load cycle – and for the cyclically stabilized deformation behaviour – all material elements follow the same stress– strain curve. The crack is completely open and crack flanks are free of stresses. These conditions for a path-independent value of the DJ-integral are met. The effective ranges entering the approximation Eq. (11) are the difference of the reference point value and the value for the onset of crack closure, D(  )eff = (  )ref  (  )cl, see Fig. 1. The link to applications in the small scale yielding regime is provided by

DK eff ¼

pffiffiffiffiffiffiffiffiffiffiffiffi EDJ eff :

ð12Þ

Applying Eq. (12) also in the large scale yielding regime gives a plasticity-corrected crack driving force in the dimension of a stress intensity factor. This crack driving force can be used in combination with Eq. (8) for calculating fatigue crack growth rates. The result of this fatigue crack growth rate estimate is included in Fig. 6. The DJeff-based fatigue crack growth rate estimation also underestimates the experimentally determined average rates. However, this effect improves for increasing crack lengths. The conclusion that follows from this is that the plasticity induced crack closure develops gradually in reality, whereas in the estimate it is assumed to be already fully established from the first cycle on. It is obvious that so far the plasticity-induced effect on crack closure is well-considered only when crack lengths have increased and a stabilized closure level has adjusted. For future improvements in the accuracy of fatigue crack growth rate estimates, including the effect of plastic strain, one should focus on Eq. (11). In addition, Eq. (5) should be revisited. The influence of the microstructure should be characterized through multi-scale modelling of the fatigue process.

In the small scale yielding regime, plasticity induced crack closure mostly explains the mean stress effect, see for example Vormwald [39]. Moreover, the variety of load sequence effects leading to crack growth acceleration or retardation can be described by taking the sequence dependent crack closure into account, Führing and Seeger [11], Newman [12]. For example, a tensile overload usually causes delayed fatigue crack growth retardation. In cyclic loading with large plastic strain amplitudes the plasticity influenced crack closure behaviour differs considerably from what is known for the small scale yielding regime. In Fig. 7, the local strain measured in the stress shadow of a fatigue crack is plotted against the applied nominal gross section stress. Results for a simple variable amplitude load sequence are shown; cycles with nominal strain amplitude of 0.2% are interrupted by a large cycle of 0.5% strain amplitude every 26 cycles. The experiment was planned in such a way that the uncracked specimen undergoes the same hysteresis loops as shown in Fig. 1 for constant amplitude loading. Now, however, they are grouped together in a two-level test. The kinks in the hysteresis loop of the large cycle again indicate crack opening and closure. The crack opens and closes again at nearly the same strain values. Crack closure is governed by the large cycle. Therefore, the crack is completely open during the small cycles. Without a periodic overload by a large cycle a crack would be partly closed during a cycle with 0.2% strain amplitude, Fig. 1. Due to larger effective ranges, the crack grows faster during the smaller cycles than it would under constant amplitude loading. Therefore, in the large strain yielding regime overloads usually accelerate fatigue crack growth. The fact that crack opening and crack closure are observed at nearly identical global strains is even more important in variable amplitude fatigue life calculation than the accuracy of an approximation equation for crack opening stresses such as Eq. (5). The value of the global crack opening and closure strain, eop = ecl, is highly sensitive to the load sequence. Overloads cause its decrease and therefore an increase in the effective ranges. In the present example, during the n = 26 load cycles with strain amplitude ea = 0.2% the crack is completely open. In a constant amplitude test with the same strain amplitude and stress ratio R = rmin/rmin = 1 the crack was closed during a considerable part of the cycle. The consequences of this effect on fatigue life can be assessed based on a simple linear damage accumulation rule. From the constant amplitude lives of N = 7.7  103 cycles for ea = 0.5% and N = 2  105

500

Crack depth a = 1 mm

400

a = 0.5 mm

300 200 100

no crack

0 -100

σop

Steel S460 Two-level test 1 cycle with εa = 0.5 % and 26 cycles with εa = 0.2 %, sequence of 27 cycles repeated until technical crack initiation

-200 -300 -400 -500 -0.6

εop = εcl

DJ ¼ DJel þ DJpl

5. Variable amplitude loading

Global stress in MPa

growth rates in the early stage. As the crack grows the fatigue crack growth rates become more and more overestimated. The reason may be the development of plasticity induced crack closure, on the one hand, or a systematic overestimation of the crack driving force by Eq. (1), on the other hand, or even both. The calculation of the cyclic DJ-integral according to Eq. (3) based on a numerical integration of results of finite element calculation is cumbersome, especially taking into account that various crack lengths have to be dealt with. Despite this shortcoming, efforts in this area have been reported, e.g. [21,33–36]. In order to support the application of the approach, approximation formulas for the cyclic DJ-integral have already been derived together with the first proposal of this concept [14,15,37,38], initially without considering crack closure.

σcl -0.4

-0.2

0.0

0.2

0.4

0.6

Local strain near crack flanks in % Fig. 7. Crack opening strains in a variable amplitude test [9].

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M. Vormwald / International Journal of Fatigue xxx (2015) xxx–xxx

cycles for ea = 0.2% a two-level test life can be calculated. The damage sum for the sequence is D = 1/7700 + 26/200,000  2.6  104. The sequence of one high amplitude cycle and 26 low amplitude cycles must be repeated 1/2.6  104  3850 times leading to 104,000 cycles for the predicted life. However, the experimentally determined life of only 26,500 cycles is overestimated by a factor of nearly four. According to Eq. (11) the crack driving force of the small cycles (ea = 0.2%) is larger by approximately a factor of 3.2 in the variable amplitude case discussed, here (two-level loading), compared to the constant amplitude case. Consequently, the fatigue crack growth rates were larger by a factor of approximately 3.21.5  6 where the exponent m = 1.5 was used. This factor of 6 can be interpreted in terms of a reduction factor of failure lives for calculating damage sum contributions. The damage sum, taking the crack closure caused sequence effect into account, becomes D  1/7700 + 6  26/200,000  9.1  104 leading to 30,000 cycles for the predicted life. This is in very good accordance with the experimental observation. More complex load histories than the simple two-level test have been examined [9,40]. Generally, the growth rate acceleration sequence effects dominate in a variable amplitude loading scheme with large plastic strain ranges. Usually this situation occurs in the vicinity of notch roots where fatigue cracks initiate. Under large cyclic plastic strain loading conditions the local stress ratio in the vicinity of notch roots inevitably tends to R = 1 even if the load ratio differs considerably from 1. On the one hand, the contracting plastic strains can only occur in connection with compressive local stresses, while, on the other hand, the mean stress relaxation effect causes remaining mean stresses to vanish quickly. Therefore, the investigation of the crack opening and growth behaviour is of vital importance for the case of R = 1. Under stress control test conditions, the case of stable mean stresses could be investigated as well. Severe ratcheting is expected for unnotched material specimens in combination with the occurrence of a different failure mechanism than pure fatigue. Corresponding results have so far not been reported in the literature. However, for notched coupons variable amplitude test and prediction results are reported for load ratios much different from R = 1 in Ref. [41]. For Gaussian-type pseudo random sequences with load ratio R = 0.1 and for the Minitwist sequence (R = 0.23) again large accelerating sequence effects have been observed in the stage of technical crack in initiation, i.e. in the short crack growth regime. Fatigue lives for initiating a crack of technical size are typically shorter than predicted with algorithms, ignoring this sequence effect. Under small scale yielding condition a retardation phenomenon dominates. The lower reversal point loading is no longer able to plastically compress crack wake material. Moreover, plastically elongated material (while located in the crack’s plastic zone) causes increasing crack opening stresses in the crack wake. The transition from accelerated to decelerated fatigue crack growth has barely been investigated.

6. Multiaxial loading Under uniaxial fatigue loading, some of the observed phenomena can be better understood when observing the fatigue crack growth. The sequence effects are so far the most important example. Such investigations have revealed that macroscopically observed results may be traced back to the effect of cyclic plastic strains. In addition to the fatigue damage accumulation problem, there are two more basic questions inspiring ongoing research activities: multiaxial fatigue and transferability of fatigue data. As the examination of the effect of cyclic plastic strains on variable

amplitude fatigue lives was rewarded with some success, attention was directed to the multiaxial fatigue problem, too. The first two modelling challenges remain the same as for uniaxial loading: the definition of a crack driving force and the determination of effective ranges. The complexity is further increased by the necessity to identify the direction of fatigue crack growth and to specify a relevant mixed-mode hypothesis. The issue of the mixed-mode may be distinguished in proportional and non-proportional cases. The general problem of large scale cyclic yielding fatigue crack growth under non-proportional combined loading with local non-proportional mixed mode situations is far from being solved, either theoretically or practically. It is worth noting that this general case is also still unsolved for small scale yielding conditions. The research community, however, has tackled some simpler special cases with increasing success. Research began with investigations of loading cases with fixed principal axes: Biaxial and proportional loading [42–55]. The common feature of most of these modelling approaches is that the driving force parameters are composed of two factors: a function of the crack length and the hardening exponent together with the ranges of the far-field loading. Again, the far-field loading is expressed in terms of the stresses and strains in the uncracked material state. The crack length independent factor is termed the damage parameter in conventional fatigue assessment approaches according to the local strain approach. The application-relevant crack closure effects have been taken into account by McClung [50] and Savaidis and Seeger [53,54]. In a straight forward way, Savaidis et al. extended Eq. (5) by including the biaxiality ratio K = r2/r1 where r1 and r2 are the principal stresses. Their approximation was based on results obtained by McClung [50] who carried out finite element calculations with the node release technique. He investigated various biaxiality ratios. McClung et al. later [1] revisited these investigations and reported the general trends found for R = 0 and R = 1 to be consistent with experimental data, McClung and Sehitoglu [24], Brown and Miller [42], Hoshide et al. [47]. The non-proportional case – short semi-elliptical surface cracks, large scale yielding, crack closure, and recently variable amplitudes – has been treated by Döring et al. [56], Vormwald and Döring [57], Hertel and Vormwald [58,59]. Their model follows the concept of a critical plane approach. The fatigue crack growth simulations must be performed for a variety of potential planes. The plane with the shortest life is decisive. Fig. 8 shows that even under a severely non-proportional strain sequence the fatigue crack growth curve can be modelled in a satisfactory way. The model assumptions are that the crack grows in a

0.4

S460N γa = 0.346 % 0.3

εa = 0.2 %

a in mm

6

0.2

crack 1 crack 2 crack 3

short crack model

0.1

0

crack 4 crack 5 crack 6

0

5000

n

10000

15000

Fig. 8. Comparison of experimentally determined and calculated crack growth curves [56].

Please cite this article in press as: Vormwald M. Effect of cyclic plastic strain on fatigue crack growth. Int J Fatigue (2015), http://dx.doi.org/10.1016/ j.ijfatigue.2015.06.014

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Fig. 9. Finite element based calculation of cyclic DJ-integrals for non-proportional loading [60].

(critical) plane without kink or path curvature, that effective crack driving force ranges (the cyclic DJ-integral was used) can be approximated by empirical formulas, and that the cyclic opening modes I and II lead to an individual crack increment that can simply be added. These assumptions and simplifications are quite restrictive and require further study. So far, the studies of fatigue crack growth under multiaxial elastic–plastic strain ranges have led only to arguments supporting critical plane approaches. However, an improved modelling accuracy compared to critical plane based results has not as yet been achieved. Recent investigations [58,59] have shown that variable amplitude and multiaxial loading can be addressed simultaneously in a model for calculating fatigue crack growth in fields of cyclic plastic strains. The sequence effects identified under uniaxial loading also act under multiaxial proportional and non-proportional loading. They can be taken into account and lead to realistic estimates of fatigue lives. For non-proportional loading the theoretical background of the cyclic DJ-integral is widely lost. Despite this drawback, the usefulness of the critical plane multiaxial damage parameters is beyond doubt. A numerical study on the performance of DJ-integral that evaluates non-proportional cyclic loading conditions has been presented by Hertel et al. [60]. The calculations have been carried out for a centre cracked plate specimen subjected to combined tensile and shear loading, Fig. 9. A finite element model has been created and the crack tip has been modelled with degenerated quadrilateral elements in order to achieve realistic crack opening displacements. The crack is assumed to be stationary. Crack closure is not taken into consideration; under compressive loading, overlapping crack flanks are allowed. The DJ-integral has been calculated by numerical integration of Eq. (3) for three different paths encircling the crack tip. Three circular paths with different radii are applied in order to assess the path-independence of the results. In the case of non-proportional loading the load reversal points, in terms of DJ, are detected by the following algorithm. If the change in DJ after the integration of Eq. (3) emerges as negative, it is presumed that the last equilibrium state was a load reversal point. In consequence, the reference variables are redefined with stress, strain and displacement values of the last equilibrium state. Thereupon the integration is continued. This procedure provides the monotonic increase of DJ for every load step. In addition to the total DJ, the mode related components DJI and DJII are of special interest in a mixed mode I and II analysis. The

integrals DJI and DJII can be derived by decomposing the stress, strain and displacement values of the crack tip field into a symmetric (mode I) and antisymmetric (mode II) component. The symmetric and antimetric field variables are separately fed into Eq. (3) obtaining thus the values of the mode related integrals DJI and DJII. In the case of non-proportional loading with different frequencies, Fig. 9, the DJ-integral remains practically path-independent. The load reversal points in terms of DJI and DJII show the same frequencies as the external loads but are displaced relative to the maxima and minima of the external loads expressed by r and s. Using the described algorithm, it is possible to calculate the practically path-independent crack tip parameter DJ based on finite element calculations even for non-proportional cyclic loading. The calculated DJ values should maintain their sound physical basis to describe fatigue crack growth even under non-proportional loading conditions. A superposition of solutions taken from pure mode I and pure mode II loading does not lead to reasonable results for a mixed mode load case. Suitable approximative solutions for DJ, DJI and DJII should be developed because elastic–plastic finite element calculations for cyclic loading are complicated and time-consuming. Such approximative solutions would have to take the interaction between mode I and mode II loading into account. Further investigations should be carried out on the definition of a load cycle and on an appropriate mixed mode crack growth criterion including crack closure applicable to non-proportional loading. 7. Conclusions In the case of large plastic strain amplitudes, fatigue crack growth simulations have to apply a crack driving force parameter of elastic–plastic fracture mechanics. The crack driving force scales with a power of the crack length if the crack grows in a homogeneous stress and strain field of an uncracked structure and if the crack length is still small compared with any geometrical item of the structure. Exponents are either 1 for Dd- or DJ-based parameters or 0.5 for DK-based parameters. This exponent is of marginal importance as it has to be combined with the exponent of the (empirical) fatigue crack growth rate power-law. Considering crack closure is crucial for modelling realistic crack growth rates for both long and short cracks. The plasticity-induced crack closure, however, is different in the small and large scale

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yielding regime. Under small scale yielding conditions, crack opening stresses increase with the applied stress ranges. The crack opening stresses depend only on the stress ratio. If the applied stress and strain ranges exceed a critical value the crack opening stresses and strains decrease with increasing ranges. This is due to the fact that the wake of the crack is exposed to plastic reduction under compressive stresses. Resulting large effective stress and strain ranges cause high crack growth rates. It seems to be worth emphasising that the crack closure stress is considerably lower than the crack opening stress. In a rough approximation it can be assumed that the cracks open and close at the same total strain. This behaviour is of great importance when evaluating fatigue lives under variable amplitude loading. In variable amplitude loading the sequence dependent crack opening and closure levels usually lead to fatigue crack growth acceleration for cycles with smaller amplitudes following cycles with large amplitudes. It is expected that progress in improving the simulation accuracy of fatigue life predictions can be achieved only by taking cyclic plastic deformation into account in a reasonable way. Under multiaxial fatigue the cracks initiate and grow principally in the same way as under uniaxial fatigue. Looking at a short fatigue crack under the microscope, it cannot be determined whether it has initiated and grown under uniaxial or multiaxial stresses. There is no alternative failure mechanism; therefore this is a unique way of modelling the process to be specified. In such trials, however, additional complexity enters into the model in the multiaxial case: mixed-mode hypothesis including the crack growth direction and rate. Current models require many assumptions and are so far not able to improve the prediction accuracy compared to critical plane based models for addressing multiaxial fatigue. The combined application for variable amplitude multiaxial fatigue has shown initial promising results. As the mechanism of metal fatigue in essence is fatigue crack growth, models ignoring this fact will scarcely be able to improve the understanding of the process or the accuracy of fatigue life predictions. Acknowledgements The author expresses his sincere gratitude to the German Federal Ministry of Economics and Energy (Bundesministerium für Wirtschaft und Energie, BMWi) for financial support via AiF (Arbeitsgemeinschaft industrieller Forschungsvereinigungen ‘‘Otto von Guericke‘‘ e.V.) of the research project ‘‘Betriebsfestigkeit von Hochdruckbauteilen mit kleinen Schwingspielen großer Häufigkeit‘‘ under the IGF project no. 16023 B. References [1] McClung RC, Chell GG, Lee Y-D, Russel DA, Orient GE. Development of a practical methodology for elastic–plastic and fully plastic fatigue crack growth. Report NASA/CR-1999-209428, Marshall Space Flight Center, NASA Center for Aerospace Information, Linthicum Heights, MD; 1999. [2] Boettner RC, Laird C, McEvily AJ. Crack nucleation and growth in high strain low cycle fatigue. Trans Metall Soc AIME 1965;233:379–87. [3] McEvily AJ. Fatigue crack growth and strain intensity factor. In: Proc air force conf on fatigue and fracture of aircraft structures and materials, Miami Beach, AFFDL-TR-70-144. p. 451–8. [4] El Haddad MH, Smith KH, Topper TH. A strain based intensity factor solution for short fatigue cracks initiating from notches. In: Fracture mechanics, ASTM STP 677; 1979. p. 274–89. [5] McEvily AJ, Beukelmann D, Tanaka K. On large scale plasticity effects in fatigue crack propagation. In: Proc symp mech behav mater, vol. 1, Kyoto, Japan, SMSJ; 1974. p. 269–81. [6] Tomkins B. The development of fatigue crack propagation models for engineering applications at elevated temperatures. ASME J Eng Mater Technol 1975;97:289–97. [7] Tanaka K, Hoshide T, Sakai N. Mechanics of fatigue crack propagation by cracktip plastic blunting. Eng Fract Mech 1984;19:805–25. [8] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100–8.

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