An energy based fatigue crack growth model

International Journal of Fatigue 25 (2003) 771–778 www.elsevier.com/locate/ijfatigue

An energy based fatigue crack growth model K.N. Pandey, S. Chand ∗ Mechanical Engineering Department, M. N. National Institute of Technology, Allahabad 211004, India Received 13 September 2002; received in revised form 8 November 2002; accepted 14 January 2003

Abstract A fatigue crack growth (FCG) model under constant amplitude loading has been developed considering energy balance during growth of the crack. Plastic energy dissipated during growth of a crack within a small area, known as process zone at the tip of the crack and area below cyclic stress–strain curve was used in the energy balance. The predictions of model are in good agreement with the experimental results.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Process zone; Fatigue crack growth; Cyclic plastic energy density

1. Introduction A crack growth law can be formulated with the help of the stress and strain field ahead of the crack tip and using a suitable failure criterion. Different failure criteria were used in the past incorporating critical stress, plastic/total strain ahead of crack [1], the magnitude of crack tip opening, damage accumulation ahead of the crack [2–5] and the energy criteria [6–20]. Among all these it is found that energy based criteria are more suitable than others [21].These criteria are mainly based on the dissipation of a critical level of energy within the material at the crack tip. One of the criteria was the area below the monotonic stress–strain curve, the specific energy Wc, as a failure criterion. This criterion was used by McEvily and Johnston [8] to modify the FCG model of Weertman [22].Gillemot [23,24] examined crack spreading under low cycle conditions and found that crack spreading appeared after the absorbed energy became equivalent to a characteristic value equal to the specific energy Wc. But as the local flow properties at the crack tip are likely to be more typical of the cyclic values than of the monotonic values, Chand and Garg [17] modified the above criterion by modifying the speCorresponding author. Tel.: +91-532-540212 (R); +91-532445103-04 Ext. 1110 (O); fax. +91-532-445101; +91-532-445102. E-mail addresses: [email protected] (S. Chand); [email protected] (S. Chand). ∗

cific energy term Wc by Rice’s superposition method [25] for cyclic loading and FCG model was developed adding some coupling material factors. The present study is aimed at developing a fatigue crack propagation model based on the above specific energy incorporating the cyclic deformation properties obtained from a low cycle fatigue (LCF) test. This concept has been used because the highly strained zone ahead of the crack is very much like a small LCF specimen. Also it is more advantageous to form a model mainly based on LCF properties since they are easier to obtain experimentally. The highly strained zone (process zone) very near to the crack is taken for the energy balance instead of taking the whole plastic zone with the premise that mainly it is the zone, where damage accumulates [21].

2. Fatigue crack growth model development Uni-axial elastic–plastic behaviour of materials under cyclic loading are often described by Ramberg-Osgood relation [21,26] as given by Eq. (1): ⌬e ⫽

冉 冊

⌬s ⌬s ⫹2 E 2k⬘

1/n⬘

(1)

where k⬘ is the cyclic strength coefficient and n⬘ is the cyclic strain hardening exponent. ⌬ε / 2 and ⌬s / 2 are the

0142-1123/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0142-1123(03)00049-5

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strain and stress amplitudes and E is the modulus of elasticity. In terms of number of reversal to failure 2Nf, ⌬s / 2 and ⌬ε / 2 are related as given by Eqs. (2a) and (2b)

Wc ⫽ Area OABCO ⫽ Area OFBEO ⫺ Area OABFO ⫺ Area CBEC

⌬s ⫽ s⬘f (2Nf)b 2

(2a)

⌬e ⌬ee ⌬ep s⬘f ⫽ ⫹ ⫽ (2Nf)b ⫹ ef⬘(2Nf)c 2 2 2 E

(2b)

p

冕 冉冊

2s⬘f ⫹ 2e⬘f )⫺ Wc ⫽ 2s ( E ⬘ f

where ⌬e and ⌬e are elastic and plastic part of the strain amplitude ⌬e / 2, respectively. s⬘f , e⬘f , b and c are cyclic fatigue strength coefficient, fatigue ductility coefficient, fatigue strength exponent and fatigue ductility exponent, respectively. For 2Nf = 1, ⌬s=2s⬘f and ⌬e = 2s⬘f / E + 2e⬘f . One typical cyclic stress–strain curve is shown in Fig. 1. Point A is corresponding to twice of the cyclic yield stress and strain (s⬘y, e⬘y) and point B is the point corresponding to stress and strain ranges with 2Nf = 1, i.e. ⌬s = 2s⬘f and ⌬e =(2s⬘f / E+2e⬘f ). It is based on the premise that the maximum stress/plastic strain range sustainable by a fatigue element is fatigue strength, 2s⬘f and ductility limits of the material, 2e⬘f [27]. For an elasticplastic material 2s⬘f / E is small in comparison to 2e⬘f , and the maximum value of ⌬e is very nearly equal to 2e⬘f . It is conceptualised that energy absorbed till fracture Wc is the area below the cyclic stress strain curve (Fig. 1), e

denoted by OABCO. Line BC represents typical elastic unloading.

Wc ⫽ 4s⬘f e⬘f ⫺

4n⬘ 1 n⬘ ⫹ 1 k⬘

2s⬘f

0

1 2s⬘f ⌬e d(⌬s)⫺ (2s⬘f ) 2 E

1/n⬘

(s⬘f )

1+n⬘ n⬘

⫹

2s⬘2 f E

neglecting the last term, as it is very small in comparison to the first two terms, Wc is given by Eq. (3a). Wc ⫽ 4s⬘f e⬘f ⫺

冉冊

1/n⬘

4n⬘ s⬘f 1 ⫹ n⬘ k⬘

s⬘f.

(3a)

For a material, which follows Eq. (1), (s⬘f / k⬘)1 / n⬘ equals e⬘f and energy absorbed till fracture Wc is given by Eq. (3b). Wc ⫽

4 s⬘e⬘. 1 ⫹ n⬘ f f

(3b)

The difference between Eqs. (3a) and (3b) was calculated for the four materials namely 10 Ni steel, Man-ten steel, 2219-T851 aluminium alloy and 8630 steel from ref. [19], which is given in Table 1. The maximum difference of 4.2% is for 8630 steel. So Eq. (3b) has been used in the model development. Under small scale yielding conditions, the cyclic stress and plastic strain components of HRR (Hutchinson, Rice and Rosengren) crack tip singularity fields [19,28], ahead of crack tip is given by Eq. (4a), (4b) [19,29,32], for materials which obey the stress–strain relation given by Eq. (1).

冉 冉

⌬ sij ⫽ ⌬s⬘y ⌬epij ⫽

⌬K2I a⬘⌬s⬘2 y In⬘r

冊 冊

a⬘⌬s⬘y ⌬K2I E a⬘⌬s⬘2 y In⬘r

n⬘/(1+n⬘)

s˜ ij(q ; n⬘)

(4a)

e˜ ij(q ; n⬘)

(4b)

1/(1+n⬘)

where ⌬KI, ⌬s⬘ y, are the range of stress intensity factors under mode I loading and cyclic yield stress (⌬s⬘y苲2s⬘y), respectively. r and q are the radial and angular position, respectively, of any point P from the crack tip, as shown in Fig. 2 and a⬘ is given by Eq. (5).

Table 1 Showing the difference in the result of Eq. (3a) and (3b)

Fig. 1.

Cyclic stress–strain curve.

Difference (%)

10 Ni steel

Man-ten steel

2219T851 Al

8630 steel

0.74

0.044

1.8

4.2

K.N. Pandey, S. Chand / International Journal of Fatigue 25 (2003) 771–778

773

region. Region II is the region between cyclic plastic zone and process zone (region I).This is the region where stress and strain ranges derived from HRR singularity field and given by Eq. (4a),(4b) are valid. Region III is the region between vast elastic region outside the monotonic plastic zone and cyclic plastic zone. Among these three regions ahead of the crack tip the process zone (region I) is the region where damage mainly accumulates [31–33]. Let the length of the process zone ahead of crack tip be d∗ (Fig. 2), which was taken as a constant by some workers [27,34], and variable and depending on ⌬K by others [30,31]. Taking it as a variable with agreement of the view of Ref. [31] that it is a parameter dependent on the level of driving force ⌬KI, d∗ is expressed by Eq. (9). Fig. 2.

a⬘ ⫽

d∗ ⫽

Three regions in front of a crack tip.

2E . ⬘ 1/n⬘ (2k ) ⌬s⬘y(n⬘⫺1)/n⬘

(5)

Further the s˜ ij (q ; n⬘) and e˜ ij (q ; n⬘) are non-dimensional angular distribution functions. Similarly In⬘ is the nondimensional parameter of exponent n⬘. From Eqs. (4a) and (4b) multiplication of equivalent stress and strain along the crack line (q = 0) is given by Eq. (6). ⌬seq⌬eeq ⫽

⌬K2I s˜ eq(0,n⬘)e˜ eq(0,n⬘) . EIn⬘r

(6)

冉 冊 ⬘

1⫺n (⌬s)eq(⌬ep)eq 1 ⫹ n⬘

(7)

substituting the value of ⌬seq⌬epeq from Eq. (6) into Eq. (7) ⌬Wp ⫽

冉 冊

1⫺n⬘ ⌬K2I s˜ eq(0 ; n⬘) e˜ eq(0 ; n⬘) . 1 ⫹ n⬘ E r In⬘

(8)

The Eq. (8) gives distribution of plastic strain energy density per cycle ahead of the crack tip with only unknown values of s˜ eq (0 ; n⬘), e˜ eq (0 ; n⬘), the angular distribution functions of equivalent stress and strain and In⬘ a non-dimensional parameter of exponent n⬘. The region near the crack tip is widely recognized as divided into three regions, Fig. 2. The region near the crack tip, known as the process zone [2,15,30]. Here, due to nonproportional plasticity and crack tip blunting, due to repeated loading and unloading, the stress/strain cannot be found from Eq. (4a),(4b). The product of stress and strain given by Eq. (6) has finite magnitude in this

(9)

where ⌬Kth is range of threshold stress intensity factor. The plastic energy dissipated in the process zone would be the integration of Eq. (8) over the process zone. The finite value of product of range of stress and plastic strain (⌬s⌬ep) in the process zone can be found by substituting d∗ in place of r in Eq. (6) [2,27,34]. Integrating Eq. (8) along the process zone ahead the crack tip to get the plastic energy, fp dissipated per cycle per unit growth, we get Eq. (10).

冕冉 冊 冉 冊 d∗

fp ⫽

0

Now, for the specified equivalent stress and strain range, the cyclic plastic strain energy density in the units of Joule per cycle per unit volume [19,29] is given by Eq. (7). ⌬Wp ⫽

⌬K2I ⫺⌬K2th pEs⬘y

fp ⫽

1⫺n⬘ ⌬K2I y(n⬘) dr 1 ⫹ n⬘ EIn⬘d∗

1⫺n⬘ ⌬K2I y 1 ⫹ n⬘ EIn⬘

(10) (11)

where y = {s˜ eq (0 ; n⬘) e˜ eq (0 ; n⬘)}. As load is increased from zero at the crack tip, it is blunted first and starts to open when the stress intensity factor reaches the threshold value Kth. Further loading makes the crack tip more blunt and when load reaches the maximum value, the crack moves by some distance. Fatigue crack growth takes place when crack is open, so the driving parameter of crack is Kmax–Kth instead of Kmax⫺Kmin, so substituting Kmax–Kth for ⌬KI in Eq. (11). fp ⫽

冉 冊

1⫺n⬘ y(Kmax⫺Kth)2 . 1 ⫹ n⬘ EIn⬘

(12)

Crack would grow by a length da in a cycle when energy absorbed in the same cycle Wc equals plastic energy dissipated in the process zone, i.e. energy absorbed per unit growth of crack equal to the plastic energy dissipated within the process zone per cycle. This concept is expressed by Eq. (13). Wcda ⫽ fp.

(13)

Substituting the value of Wc and fp from Eqs. (3b) and

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K.N. Pandey, S. Chand / International Journal of Fatigue 25 (2003) 771–778

(12), respectively, in Eq. (13), the expression for da is obtained which is given by Eq. (14). da ⫽

冉 冊

fp 1⫺n⬘ da ⫽ ⫽ dN Wc 1 ⫹ n⬘

y(Kmax⫺Kth)2 4 EIn⬘ s⬘e⬘ (1 ⫹ n⬘) f f

冉

冊

(14)

or da (1⫺n⬘)y ⫽ (K ⫺K )2. dn 4EIn’s⬘f e⬘f max th

(15)

The above crack relation is a particular case for R = 0 where R = Kmin / Kmax. For R = 0, Kmax = ⌬K, Kth = ⌬Kth. ⌬Kth is a material constant but it is sensitive to stress ratio R. A relation between ⌬Kth and R is given below based on experimental results [17,35–37]. ⌬Kth ⫽ ⌬Ktho(1⫺R)γ

(16)

where ⌬Ktho is the range of threshold stress intensity factor for the stress ratio R = 0, and g is a material constant which varies from zero to unity [35,38]. For most of the materials g comes out to be 0.71 [17]. Substituting the value of Kmax and Kth for a general stress ratio R, the fatigue crack growth relation is expressed as given by Eq. (17). da (1⫺n⬘)y (⌬K⫺⌬Ktho(1⫺R)γ)2. ⫽ dn 4EIn⬘s⬘f e⬘f

(17)

It is clear from Eq. (17) that with an increase in stress ratio R, ⌬Kth decreases and, ⌬K increases, increasing the fatigue crack growth rate but the influence is more pronounce in stage I (near threshold region of da/dN vs ⌬K plot) where ⌬K and ⌬Kth are comparable than in other regions of the plot. This is consistent with the experimental results [17,39–43]. The fatigue crack growth relation derived here requires mechanical and fatigue properties E, s⬘f , e⬘f and n⬘. The FCG model expressed by Eq. (17) has same form as derived by Smith [44], Zheng [45], and Bolotin [46] as given by Eq. (18). da ⫽ C(⌬K⫺⌬Kth)2 dN

(18)

where C is a material constant. Comparing the present model with the models which were based on fatigue properties [26,27,30–32,34], the present model has an edge. Although the present model is based on the analysis of the process zone, the FCG relation doesn’t need the value of d∗, making the analysis easier. The present model doesn’t need material coupling factors and any other data from experimental crack growth plots. 3. Results and discussion The crack growth model developed is compared with the experimental results available in the literature for the

materials given in Table 2 in which the mechanical and fatigue properties of the above materials are also listed. The values of In⬘ and y were calculated numerically and listed in Table 2. A computer program was developed for this purpose which gives the values of In⬘ and y for any value of q and n⬘ for both plane stress and strain cases. From the calculated values of y in Table 2, it is reasonable to take an average value of 0.95 for y. A vast amount of mechanical and fatigue properties required in the present model for different other materials can be found in the ASM handbook for fatigue and fracture [47]. For the materials for which results of fatigue crack growth rate are compared with the present model, the value of ⌬Kth available in the literature and here given in Table 2, were directly used instead of using Eq. (16). Comparison of da/dN vs ⌬K experimental data with model given by Eq. (17) is given in Figs. (3–9). From the comparison following observations are made:

1. The model compares well in both regions I (the threshold region) and II (the intermediate region of the da/dN vs ⌬K). 2. In the critical region predictions of model are somewhat conservative. 3. Comparison is done for materials with wide range of strength, whereas cyclic yield strength of 10 Ni steel is 1106 MPa, it is 331 and 334 MPa for Man-ten steel and 2219-T851 aluminium alloy, respectively. For all these ranges the predictions of the model are quite satisfactory. The only exception is C–Mn steel. The reason for this and the small difference in other materials, may be due to the deviation nature of stress–strain behaviour, which is represented here as the Ramboorg-Osgood type. Due to this assumption there is some difference in the value of experimental and numerical e⬘f . Also the effect of mean stress is not considered in the present model, which may be included as in other crack growth models based on LCF properties. This can be done by including local s∗m, introduced by Ellyin [21,27,32]. 4. Figs. 6 and 8 compare the crack growth results for different stress ratios R. Comparison is quite well with the experimental results.

Comparison is also done with the theoretical model of Ellyin [27,32] with HRR stress singularity in the same Figs. (3–9) and it compares quite well for both R = 0 and higher stress ratios as well. Although, both the present and Ellyin’s model are based on low cycle fatigue properties and concerned with the highly strained area, the process zone near the crack tip, the approach of the present model is different than from Kujawaski and Ellyin [27] and Ellyin [32] models. Ellyin’s work

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775

Table 2 Mechanical and fatigue properties E (GPa)

s⬘y (MPa)

s⬘f (MPa)

e⬘f

n⬘

k⬘ (MPa)

⌬Kth(MPa m1/2) R 0

10 Ni steel [48,50] Man-Ten steel [27] 2219 T 851 Al [50] 8630 steel [49] C–Mn steel [49] 4340 steel [26] E 36 steel [26]

0.1

0.5

In⬘

y

207

1106

2019

0.54

0.109

2177

–

5

–

3.02

0.94735

206

331

917

0.26

0.20

1200

–

15

–

3.4

0.94479

71

334

613

0.35

0.121

710

–

30

–

3.067

0.95152

207 208 209 206

661 372 724 350

1936 868 1713 1194

0.42 0.15 0.83 0.6

0.195 0.141 0.146 0.21

2267 896 1761 1255

13 13 7.8 5

– – – –

10 9 – –

3.082 3.163 3.184 3.446

0.94794 0.94726 0.94812 0.94411

Fig. 3. Comparison of the predicted crack growth rate with experimental data [48,50] and Ellyin’s model [27].

was similar to the work of Antolovich [30], Glinka [2,34] in which fatigue crack growth was presumed as a ratio of process zone size d∗ and the number of cycles, ⌬N required to propagate the crack through this distance. The process zone was considered as a factor of microstructure. Antolovich [30] had taken this distance as a fraction of cyclic plastic zone, whereas Glinka [2,34] took its value corresponding to the threshold stress intensity factor. Bolotin [46] had also taken it as a fraction

Fig. 4. Comparison of the predicted crack growth rate with experimental data [27] and Ellyin’s model [27].

of the cyclic plastic zone for fatigue crack propagation in regime I of the Paris diagram of crack growth rate versus stress intensity factor ⌬K. Citing the problem of experimental determination of d∗, Li [31] observed that the process zone should be representative of the crack driving force ⌬K and found d∗ as given by Eq. (9).The above discussion shows that the models become different by using various approaches to determine d∗. In

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Fig. 5. Comparison of the predicted crack growth rate with experimental data [50] and Ellyin’s model [27].

Fig. 7. Comparison of the predicted crack growth rate with experimental data [49] and Ellyin’s model [27].

Fig. 6. Comparison of the predicted crack growth rate with experimental data [49] and Ellyin’s model [27].

Fig. 8. Comparison of the predicted crack growth rate with experimental data [26] and Ellyin’s model [27].

K.N. Pandey, S. Chand / International Journal of Fatigue 25 (2003) 771–778

[11] [12] [13]

[14]

[15]

[16]

[17] [18] [19] [20] [21] Fig. 9. Comparison of the predicted crack growth rate with experimental data [26] and Ellyin’s model [27].

[22] [23]

present energy based FCG model the efforts have been made to eliminate d∗.

[24]

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[25]

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