Theoretical analysis on the behaviour of short fatigue cracks

International Journal of Fatigue 24 (2002) 719–724 www.elsevier.com/locate/ijfatigue

Theoretical analysis on the behaviour of short fatigue cracks C.A. Rodopoulos *, E.R. de los Rios Structural Integrity Research Institute of the University of Sheffield (SIRIUS), Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, UK Received 6 June 2001; received in revised form 24 October 2001; accepted 25 October 2001

Abstract Based on the Navarro–Rios model, the boundaries of short crack growth are predicted. Theoretical analysis reveals that the extent of short crack growth regime is principally governed by the relation between the fatigue limit and the cyclic yield stress of the material. Materials with low values of sFL/scy show extensive short crack behaviour, while in material with high values of sFL/scy short crack behaviour is very limited. The results of the analysis are validated with experimental data.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Navarro–Rios model; Short fatigue cracks; Fatigue limit; Cyclic yield stress

1. Introduction In the last twenty years, a significant amount of research effort has been allocated to understand the complex behaviour of short fatigue cracks [1–3]. The growth rate of short fatigue cracks has been shown experimentally to be significantly faster than that of the long cracks under the same nominal stress intensity factor range ⌬KI [4–7]. This is because the fatigue damage, a term that represents the combined lengths of the crack and the crack tip plastic zone, of a short crack is seriously underestimated by classical linear elastic fracture mechanics (LEFM). In this respect short crack propagation is dominated by relatively large cyclic crack tip plasticity, which significantly alters the strength of the stress field ahead of the fatigue crack. Essentially, this implies a breakdown of the similitude concept generally assumed by fracture mechanics and represents a LEFM limitation. Several modifications and adaptations of LEFM have been reported as an attempt to predict the fatigue behaviour of short fatigue cracks. The most known are, the crack closure approach [8], the crack deflection approach [9] and the J-integral [10]. These models have been reasonably successful for short cracks several times * Corresponding author. Tel.: +44-114-222-7710; fax: +44-114222-7890. E-mail address: [email protected] (C.A. Rodopoulos).

longer than the microstructural dimension defined by the length of a grain [11]. Furthermore, most of the above models acknowledge the fact that different materials are likely to show different short crack growth tendencies. However, an analysis of the short crack growth problem, that is based on fundamental issues like the mechanical properties of the material, has never been attempted. In this work, the physical characteristics that define the growth of short and long cracks are analysed. The analysis reveals the boundaries where the growth of short cracks take place. Further analysis, based on the parameters that governed the above boundaries, acknowledges their dependency on the cyclic properties of the material.

2. Short crack growth boundaries The observations that grain boundaries retard or even arrest the growth of short fatigue cracks [6], prompted Navarro and de los Rios [12] to develop a micromechanical model, known as the NR model, which describes microstructural sensitive crack propagation. According to the model, the crack tip plastic displacement and hence the crack propagation rate, changes in value in an oscillatory manner every time crack tip plasticity is generated into a new grain. This oscillation represents the blocking and unblocking of the plastic zone, modelled as a persistent slip band (PSB), by the grain bound-

0142-1123/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 1 9 6 - 7

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Nomenclature D i mi n1 n2 ro a sarrest sFL scy strans

Grain size Half grain interval Grain orientation factor Normalised value of crack length Normalised value of fatigue damage Grainy boundary width Crack length Crack arrest stress Fatigue limit Cyclic yield stress Short to long crack transition stress

ary. The amplitude of the oscillation can be illustrated by plotting either the crack tip plastic displacement or the crack propagation rate against the crack length [7,11,12]. According to the NR model, the end of the oscillation represents the end of the microstructurally dependent crack growth (short crack). Briefly, the amplitude of the oscillation is reported to decrease with crack length as shown in Fig. 1. The end of the oscillation shown in Fig. 1 represents the point from where the crack tip plastic zone, and consequently the corresponding crack tip strain, can be predicted by LEFM. This comes to reinforce the belief of many researchers that the transition from a short crack behaviour to a long crack behaviour is defined when both cracks experience matching, LEFM based, propagation rates under the same nominal ⌬KI [13]. According to the NR model which is based on the continuous distribution of dislocations as proposed by Bilby et al. [14], the fatigue damage (crack and plastic zone) at any crack size can be represented by three zones

as shown in Fig. 2. The first zone represents the crack, the second the crack tip plasticity and the third a microstructural barrier (grain boundary). In terms of the material resistance to crack propagation, the crack is considered as stress free unless some closure stress, s1, is acting on the crack flanks; the stress at the plastic zone is equal to the resistance of the material to plastic deformation (cyclic yield stress), scy; and at the grain boundary, of width ro, the stress developed due to the constraint exerted on the plastic zone is s2. This stress represents a measure of the reaction stress developed on the barrier due to the blocking of the PSB. Based on the equilibrium of dislocations along the three zones, the constraint stress ahead of the slip band for an opening mode loading, s, is given by [11], s 2=

冋

册

1 p (syc−s1)sin−1n1−syc sin−1n2+ s −1 cos n2 2

a iD/2 n1= , n2= iD/2+ro iD/2+ro

(1)

The parameters n1 and n2 represent in a dimensionless form (see Fig. 2) the crack length and the fatigue damage size, respectively. The parameter syc represents the cyclic yield stress of the material. In [15] it was suggested that short crack growth is bounded between the regime for crack arrest (non-propagating cracks) and the regime for microstructurally dependent crack growth (short crack). In [15,16], the condition for crack arrest is reported as, sFL−s1 ⫹s1 sarrest⫽mi 冑i

(2)

where mi is the grain orientation factor given in [16] as, mi⫽1⫹0.5 ln(i) where 1ⱕmiⱕ3.07 Fig. 1. Oscillation of the crack propagation rate as a function of crack length.

(3)

where i is as defined in the caption of Fig. 2. In the case that s1=0, Eq. (2) is rewritten as,

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Fig. 2. Schematic representation of the three zones that constitute the fatigue damage. The parameter i=2a/D represents the normalised crack length in half grain intervals (i=1, 3, 5, . . .). The parameter iD/2 represents the extent of the fatigue damage and D is the grain size.

sarrest⫽mi

sFL

(4)

冑i

strans⫽

冉

冪2冉1− i 冊⫺s 冪2冉1− i 冊⫹2s 冊

2 s p cy

i−4

i−4

1

p

1

(7)

or in the case of s1=0 as, It is worth noting that mi increases with crack length from m1=1 until mi reaches the saturated Taylor value of 3.07 (truly polycrystalline behaviour). In [15], the transition between microstructurally dependent growth (short crack) and microstructurally independent growth (long crack) was assumed to take place when a crack generates PSB on two successive grains without growth. This stems from the assumption proposed by Yoder and others [17,18], that the transition from structure sensitive to structure insensitive crack growth, takes place when the plastic zone becomes larger than the grain size, D. In our case, to simplify matters and be conservative, the size of the plastic zone at transition is assumed equal to 2D. Using a simplified version of Eq. (1), according to the relationships cos⫺1(x)=(2(1⫺x))1/2, sin⫺1(x)=p/2⫺cos⫺1(x), the above assumption is interpreted as, strans⫽

冉

2 s 冑2(1−n2)⫹scy冑2(1−n1)⫺scy冑2(1−n2) p 2

(5)

冊

p ⫺s1冑2(1−n1)⫹ s1 2

where the parameters n1 and n2 are becoming, n1⫽

i−4 and i

n2⬇1

(6)

Using the boundary conditions set in Eq. (6), the transition from short to long crack propagation can be written as,

strans⫽

冢 冣

2 c s p y

冪i

8

(8)

The parameter strans is the minimum applied stress capable of generating a crack tip plastic zone of size 2D ahead of a crack of length a. It should be noted that the width of the grain boundary is neglected in Eq. (2) and Eq. (5) as it is small compared with the length of the fatigue damage. Close examination of the equations denoting crack arrest and short crack growth makes clear that the fatigue limit is the material property which controls the conditions for crack arrest, while it is the cyclic yield stress which controls the conditions for short–long crack transition. The effect of these two material parameters on the position of the boundaries that defined the short crack propagation zone is shown in Fig. 3. In this figure, the short crack propagation boundary is plotted for several ratios sFL/scy (assuming a constant sFL). The cut-off values of sarrest and strans found on the applied stress axis at i=1 represent the fatigue limit and the cyclic yield (multiplied by a value of Eq. (8) for i=1), respectively. Obviously, this situation is not realistic since changes in the cyclic yield stress will also change the fatigue limit. However, Fig. 3 was constructed to illustrate (a point

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boundaries of short crack growth over a wide range of crack lengths and loading conditions. Consequently cracks several times larger than the microstructural dimension can still exhibit short crack growth characteristics. 앫 Materials with intermediate values of w, 0.3⬍w⬍0.7, exhibit boundaries of short crack growth extending over a limited range of crack lengths and loading conditions. 앫 Materials with high values of w, 0.7⬍w⬍1, exhibit minimum or no short crack behaviour.

3. Limitation of short crack growth It is clear, from the above discussion, that material with intermediate or high values of w will exhibit limited short crack growth behaviour. One limit point within this transition is achieved when the crack arrest curve intercepts the transition curve. This interception can be determined by equating Eq. (4) and Eq. (8), mi

sFL

冪2冉1− i 冊

2 ⫽ scy p 冑i

i−4

(9)

From Eq. (9) and Eq. (3) the interception point is given as, ln(iint) 32 ⫹ln(iint)⫹1⫺ 2 2⫽0 4 pw

Fig. 3. The boundaries of short crack growth for several values of sFL/scy. The short crack regime is enclosed between sarrest and strans.

which is analysed later) that different materials exhibit different degrees of short crack behaviour. It is clear from Fig. 3 that the boundary of short crack growth is sensitive to the ratio w=sFL/scy and hence three distinct conditions can be identified: 앫 Materials with low values of w, 0.1⬍w⬍0.3, exhibit

(10)

where iint represents the number of half grains affected by the fatigue process at the transition, i.e. iint=2aint/D. Physically, point iint represents the maximum length that a short crack can achieve before starting to propagate as a long crack. In Table 1, the interception point for several materials is predicted using Eq. (10). Figure 4 shows the relationship between w and iint for selected materials. From Fig. 4 it is clear that materials with values of w⬍0.3 (such as the selected aluminium alloys) are most likely to exhibit wide regions of short crack growth. Materials with w⬎0.5, such as the selected titanium alloys and the M300 steel, exhibit intermediate size regions. Finally, material with w⬎0.9, such as the AISI 4340, exhibit small or negligible short crack growth regions. The above findings are in good agreement with experimental work published in [22–24].

4. Discussion The so-called anomalous behaviour of short fatigue crack propagation reflects the breakdown of fatigue analysis based on continuum mechanics (LEFM). This is caused due to the relatively large-scale yielding con-

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Table 1 Cyclic properties of selected engineering alloys and predicted iint at a stress ratio of R=⫺1 Material

Conditions

sFL (MPa)

scy (MPa)

w

iint

1100 2024 7075 6061 5456 AISI 4340 M300 Ti-5Al-2.5Sn Ti-6Al-4V

As received T4 T6 T651 H311 Q+T (243HB) Aged Annealed Annealed

34 [21] 150 [20] 130 [20] 97 [13] 165 [21] 480 [13] 800 [21] 400 [20] 480 [20]

55 [19] 448 [19] 517 [19] 296 [19] 352 [19] 517 [19] 1379 [19] 717 [19] 827 [19]

0.61 0.33 0.25 0.32 0.46 0.92 0.58 0.55 0.58

ⱕ49 ⱕ7466 ⬁ ⱕ10503 ⱕ341 ⱕ7 ⱕ67 ⱕ94 ⱕ67

within every grain. The emitted dislocations, assisted by the crack tip stress, pile up at the grain boundary were two possibilities arise. If the stress intensity ahead of the pile-up is incapable of operating a source in the next grain, the crack arrests. Conversely, if the pile-up has sufficient strength that allows the operation of a new dislocation source in a neighbouring grain, crack tip plasticity propagates from one grain to the next. It should be noted that the above analysis reflects the behaviour of the selected materials at a stress ratio (minimum to maximum stress) of ⫺1. Higher values of the stress ratio will affect, at different degrees, both the fatigue limit and the cyclic yield stress and consequently the short crack growth regime. Fig. 4. The predicted iint versus w ratio for the selected materials presented in Table 1.

ditions characterising the crack tip of the short crack and, due to the blocking of crack tip plasticity by microstructural barriers. In this work, the fatigue limit and the cyclic yield stress are acknowledged as the two mechanical properties that control the physical events that define the boundaries of short crack growth. The first dictates the propagation of crack tip plasticity beyond the limits of a single grain, while the second, controls the conditions for transition from microstructurally dependent to microstructurally independent crack growth. Herein, the response of different materials to the above two physical events is introduced by the use of the parameter w. The examination reveals that materials with low values of w can exhibit short crack growth behaviour over a wide area of crack lengths and stress levels. Alternatively, materials with high values of w are more likely to exhibit limited short crack growth. From Fig. 4 it is clear that different materials are expected to exhibit different regions of short crack growth. Here, the conditions that characterise crack arrest and transition from short to long crack growth are based on the assumption that dislocation sources operate

5. Conclusions From the analysis presented in this paper the following conclusions can be drawn: 1. The fatigue limit and the cyclic yield stress control the boundaries of short crack growth. 2. Coarser grained materials are likely to exhibit a broad region of short crack growth. Fine grained materials exhibit a narrow region of short crack growth. 3. Eq. (10) provides a practical quantification of the susceptibility of the material to short crack growth behaviour.

Acknowledgements The authors are indebted to the Royal Academy of Engineering, Airbus UK and to the EPSRC for their continuous support. References [1] Pearson S. Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks. Engng Fract Mech 1975;7:235–47.

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