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A multiaxial fatigue failure criterion considering the effects of the defects
International Journalof Fatigue
International Journal of Fatigue 29 (2007) 1996–2004
www.elsevier.com/locate/ijfatigue
A multiaxial fatigue failure criterion considering the effects of the defects E. Thieulot-Laure b
a,b,*
, S. Pommier a, S. Fre´chinet
b
a LMT, Ens-Cachan, 61, av. du pre´sident Wilson, 94235 Cachan, France SNECMA, Rond-point Rene´ Ravaud, Re´au, 77550 Moissy-Cramayel, France
Received 15 September 2006; received in revised form 2 January 2007; accepted 7 January 2007 Available online 31 January 2007
Abstract An endurance criterion is proposed which aims at capitalizing the advantages of both fracture mechanics and of energetic or stressbased approaches. As a matter of fact, fracture mechanics based approaches allow building probabilistic criterions by introducing the statistic distributions of defects sizes, positions and orientations. On the other hand, stress-based approaches allow building multiaxial failure and life criterions. However, as shown by the diagram of Kitagawa and Takahashi [Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks. In: ASM Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Metalspark, Ohio. 1976, pp. 627–631], none of these approaches applies over the whole range of possible defects dimensions. For large defects, fracture mechanics concepts apply and the fatigue limit is deduced from the threshold stress intensity factor and the defect size. There is usually a threshold below which the sensitivity of the fatigue limit to the defect size becomes negligible. In that domain, Linear Elastic Fracture Mechanics (LEFM) overestimates the fatigue limit and stress-based approaches are more pertinent. In the present paper, the LEFM framework is used, defects are assumed to be cracks. The size effect exhibited by Kitagawa and Takahashi [Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks. In: ASM Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Metalspark, Ohio. 1976, pp. 627–631] is accounted for through the T-stress parameter, which is the first non singular term in the asymptotic development of the stresses at crack tip. The proposed criterion is a non-propagation criterion based on a critical distortional elastic energy in the crack tip region. The use of an energetic criterion ensures the consistency with endurance criterion. The specificity of our approach lies in the calculation of the distortional energy using the displacement fields of LEFM which ensures the consistency with fracture mechanics. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Fatigue criterion; LEFM; T-stress; Probabilistic approaches
1. Introduction Conventional life prediction methodologies are based on concepts that don’t allow accounting for material, manufacturing or maintenances anomalies that can degrade the fatigue resistance of components. So as to consider these anomalies, the adoption of a probabilistic damage tolerance approach is useful. Within the framework of Linear Elastic Fracture Mechanics (LEFM), it becomes possible to consider random defects nucleation, size, orientation and location, random inspection schedules, and several *
Corresponding author. Tel.: +33 1 47 40 28 69. E-mail addresses: [email protected] (E. Thieulot-Laure), [email protected] (S. Pommier). 0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.01.017
other random variables. Such approaches were developed for instance to address hard alpha material anomalies in Titanium alloys [2]. However, LEFM based approaches do not apply for small defects sizes. The pioneering experiments of Kitagawa and Takahashi [1] have highlighted the effect of the defect size upon the fatigue strength of materials. These authors have established S N curves on specimens containing drilled holes. The fatigue limit was shown to be function of the typical dimension of the manufactured defects [1]. Above a threshold defect size, LEFM applies; the fatigue limit varies as the defect size to the power minus one half. Below that threshold, LEFM strongly overestimates the fatigue limit which is nearly constant. Stress based approaches are much more pertinent (Fig. 1) in the absence of large defects, but overestimate
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Nomenclature LEFM a a d m rf sf
Linear Elastic Fracture Mechanics crack length (m) parameter of the Dang Van’s criterion radius of the integration domain of the elastic distortional energy Poisson’s ratio fatigue limit in tension (Mpa) fatigue limit in torsion (Mpa)
the fatigue limit in their presence. These results have been reproduced by many authors [3] and for various metallic materials. Moreover, Nadot et al. [4,5] have also highlighted the importance of the defect shape and orientation upon the fatigue limit. As a consequence, methodological problems arise when the material contains a population of defects that extends itself over the transition domain, or when a competition between different populations of defects takes place. A component may contain for instance, small manufacturing defects, in the order of ten micrometers, with a large probability of occurrence and larger defects, in the order of one millimetre, created in service conditions with a very low probability of occurrence. The combination of failure probabilities issued from highly different calculations methodologies can be questionable. A unified model would be useful in the engineering practice. The aim of this paper is therefore to set up a fatigue failure criterion predicting the non-propagation domain for supposedly pre-existing defects in the material. The specifications of the model are as follows: it should tend continuously to a multiaxial endurance model for small defects and to a LEFM based model for large defects and include explicitly the various statistic distributions representative of the defects populations. The aim is to provide a probability of failure for a component subjected to complex loading conditions.
KI, KII mode I and mode II stress intensity factors (MPa.(m)1/2) KIth mode I threshold stress intensity factor (MPa (m)1/2) T T-stress (Mpa) w distortional part of the elastic energy density U distortional part of the elastic energy
2. Criterion In what follows, the material is supposed to contain defects – before any stresses are applied – and these defects are assumed to be cracks. The criterion is a non-propagation criterion based on a critical distortional elastic energy in the crack tip region. This choice arises from the hypothesis that fatigue crack growth is associated with plastic deformation. If the material obeys the Von Mises yield criterion, which is critical distortional elastic energy criterion, the crack tip region can also be assumed to obey a critical distortional elastic energy criterion. However, the distortional energy is calculated at a global scale for the whole crack tip region, using LEFM stress and strain fields. The LEFM fields used for the calculation of the distortional elastic energy include the T-stress which is the first non-singular term of the asymptotic development at crack tip. As an illustration, the expression of the asymptotic stress field at crack tip in mode I is reported in Eq. (1). KI h h 3h rxx ¼ pffiffiffiffiffiffiffi cos 1 sin sin þT 2 2 2 2pr KI h h 3h ð1Þ 1 þ sin sin ryy ¼ pffiffiffiffiffiffiffi cos 2 2 2 2pr KI h h 3h rxy ¼ pffiffiffiffiffiffiffi cos sin cos 2 2 2 2pr The displacements field at crack tip is the following under plane strain conditions along the crack front: rffiffiffiffiffiffi rffiffiffiffiffiffi KI r h K II r h ux ¼ cos ðj cos hÞ þ sin ð2 þ j þ cos hÞ 2 2 2l 2p 2l 2p T þ ðj þ 1Þr cos h 8l rffiffiffiffiffiffi rffiffiffiffiffiffi KI r h K II r h uy ¼ sin ðj cos hÞ þ cos ð2 j cos hÞ 2 2 2l 2p 2l 2p T ð3 jÞr sin h 8l uz ¼ 0 ð2Þ
Fig. 1. Schematic Kitagawa-Takahashi’s diagram, see [1].
where l and j are the Lame´’s coefficients. The strain tensor is derived from the displacements field, and the stress
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tensor is obtained from the strain tensor by the Hooke’s law. Knowing the expressions of these two tensors, the distortional part of the elastic energy density w is defined by: wðr; hÞ ¼ Trðr0 e0 Þ
ð3Þ
where r’ and e’ are the deviatoric parts of the stress and strain tensors. The energy density in Eq. (3) is integrated over a domain within a distance d to the crack tip: Z r¼d Z h¼p 1 Trðr0 e0 Þrdr dh U ðK I ; K II ; T Þ ¼ ð4Þ 2 r¼0 h¼p U is therefore the distortional energy per unit length of the crack front in a domain delimited by a circle at a distance d from the crack tip. With the above mentioned assumptions, the non-propagation criterion takes the form: U ðK I ; K II ; T Þ < U c ) No growth
ð5Þ
where Uc is the critical distortional elastic energy in the crack tip region defined by the parameter d. Besides, if the mode I non-propagation threshold stress intensity factor KIth is known, the expression of Uc is given by: U c ¼ U ðK Ith ; 0; 0Þ
ð6Þ
As a consequence, the non-propagation criterion can now be rewritten as follows: U ðK I ; K II ; T Þ < U ðK Ith ; 0; 0Þ ) No growth
ð7Þ
A few mathematical developments and simplifications of Eq. (7) allow expressing the criterion directly as a function of KI, KII and T (Eq. (8)). 2 2 2 KI K II T KI T þ þ þ fm : <1 T th T th K Ith K IIth K Ith ) No growth
ð8Þ
From Eq. (8) we can see that the criterion representation in the (KI, KII) plane (with T = 0) and in the (KII, T) plane (with KI = 0) will be an ellipse and that in the (KI,T) plane (with KII = 0), due to the coupled term between KI and T, it will be an inclined ellipse. For the sake of simplicity, three terms have been introduced fm, KIIth and Tth, whose expressions are provided in Eqs. (9), (10) and (11). KIIth is the non-propagation threshold of KII obtained for KI and T equal to zero. Tth is the non-propagation threshold of T obtained for KI and KII equal to zero. And, when computing the non-propagation threshold of KI for KII and T equal to zero, we satisfactorily obtain KIth. 32ð1 10m þ 10m2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15p ð1 m þ m2 Þð7 16m þ 16m2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 16m þ 16m2 K IIth ¼ K Ith pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 16m þ 16m2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K Ith 1 7 16m þ 16m2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T th ¼ pffiffiffiffiffiffiffiffi 1 m þ m2 2pd 2 fm ¼
Fig. 2. Case of a through thickness crack in an infinite media (Eq. (12)). Evolution of the non propagation threshold KInp, as calculated using the criterion (Eq. (8)), normalized by the mode I threshold stress intensity factor KIth (long crack) versus the crack length, for various values of d.
ð9Þ
ð10Þ ð11Þ
It is important to underline that provided the Poisson’s ratio and the mode I threshold stress intensity factor are known, fm, and KIIth are also known. For instance, if m = 0.3: fm 0.45 and KIIth 0.5KIth. However, either the term Tth or the dimension d of the integration domain remains to be identified. In this criterion, the stress biaxiality is accounted for by the T-stress level. Let consider, for instance, a through thickness crack lying in the plane (x, z) and subjected to remote biaxial loads which amplitudes are Sx and Sy, the expressions of mode I stress intensity factor and T-stress are the following: pffiffiffiffiffiffi K I ¼ S y pa ð12Þ T ¼ Sx Sy It is obvious in Eq. (12), that the T-stress varies with the stress biaxiality. Besides, the T-stress allows also accounting for the defect’s size [6]. Under uniaxial loading conditions, for instance, the threshold for crack propagation denoted by Knp can be easily calculated as a function of the crack length using Eqs. (12) and (8). Its evolution is plotted in Fig. 2 for various values of the dimension d of the integration domain. 3. Identification The non-propagation criterion proposed in the previous section is a function of the mode I and mode II stress intensity factors and of the T-stress in each point of the crack front assuming a plane strain condition along the crack front. The required ‘‘material’’ parameters are the Poisson’s ratio, the mode I threshold stress intensity factor for long cracks and either the parameter Tth or the dimension d of the integration domain. In this section, it is explained how this supplementary parameter can be identified. In what follows, the stress quantities considered refers to the stress amplitudes, in the case of a proportional loading.
E. Thieulot-Laure et al. / International Journal of Fatigue 29 (2007) 1996–2004
3.1. Material The criterion proposed in Eq. (8) was identified for a 30NCD16 steel for which the requested data could be found in the literature [7], namely the mode I threshold stress intensity factor KIth, the fatigue limit rf in tension and the fatigue limit sf in torsion (Table 1). Banvillet et al. [7] have also identified the parameter a of the Dang Van [8] multiaxial endurance criterion for this material. Very schematically the Dang Van [8] endurance criterion takes the form: max max ½s þ ap < sf ) No fatigue failure t ~ n sf 1 where a ¼ 3 ð13Þ rf 2 where s is the current shear stress component in a plane defined by its normal ~ n, and where p is the current hydrostatic pressure. sf is the fatigue limit in pure alternate shear. a is a material parameter that is identified using the data reported in Table 1. The Dang Van’s criterion belongs to the family of the critical plane endurance criterions, it was developed in particular so as to better represent the experimental results for non-proportional loading paths and has been established from observations at the microscopic scale. For this criterion a critical plane ~ n and a critical moment t that maximize the criterion can be determined. With the assumption that below the fatigue limit the material is not defect free but may contain small defects below their non-propagation threshold, it becomes possible to compare the criterion in Eq. (8) with the Dang Van endurance criterion identified by Banvillet et al. [7] for the 30NCD16 steel. 3.2. Defect’s modelling So as to compare the criterion in Eq. (8) to the Dang Van criterion, an assumption should be done on the defects. Various assumptions can be done according to the problem. Let consider, for instance, a penny-shaped crack embedded in an infinite body lying in a plane defined by its normal ~ n. Let consider a point of the crack front defined by its normal ~ t. Let rn be the normal stress, s be the tangential stress, and rT be the projection of the stress tensor along the axe parallel to the crack plane. From handbooks [9,10] we have: pffiffiffiffiffiffi 8 2 > < K I ¼ p rn papffiffiffiffiffiffi 4 K II ¼ pð2mÞ s pa > : 1þ2m T ¼ rT 2 rn
8 r ¼ ðr~ nÞ ~ n > < n ~ nÞ t with s ¼ ðr~ > : tÞ ~ t rT ¼ ðr~
ð14Þ
Table 1 30NCD16 steel material data and the corresponding set of parameters of the Dang Van endurance criterion (after [7]) m
KIth (MPaÆm1/2)
rf (MPa)
sf (MPa)
a
0.29
7
560
428
3.08
1999
With such an assumption it becomes possible to formulate the non-propagation criterion in Eq. (8) as a function of the applied stress tensor r and of the dimension a of the defect. 3.3. Identification of the parameter d in the non-propagation criterion (Eq. (8)) The data in Table 1 are employed to determine the parameter d in the non-propagation criterion in Eq. (8). There are some restrictions on the value of the parameter d. As a matter of fact, the distortional elastic energy is calculated using LEFM fields which validity is restricted to the near crack tip region so as to keep the error of the asymptotic development reasonable. For instance, if the distance to the crack tip remains below a/10 – a being the crack size – the average error is around 10%. Let now assume that the endurance limit of a material is not determined for a defect-free material but for a material containing the smallest possible defects with a typical size denoted by arf. If we identify the value of d so that the non-propagation threshold corresponds to the fatigue limit, since d is the size of the integration domain, it should therefore be a fraction e of this typical size: d ¼ e arf
ð15Þ
The smaller e is, the more precise the LEFM fields are and the smaller the d parameter is. But d should also be large enough compared with the typical microstructural dimensions because of the continuum mechanics hypothesis used for establishing the LEFM fields. However, provided that these restrictions are respected, a large range of values of d can be chosen for which the condition that the non-propagation threshold corresponds to the fatigue limit is verified. As a matter of fact, with the assumptions on the defects geometry in Eq. (14), the criterion in Eq. (8) yields at the fatigue limit: 8 pffiffiffiffiffiffiffiffiffi 2 > < K I ¼ p rf parf with arf ¼ d=e K2 K II ¼ 0 ) d ¼ gðe; mÞ Ith2 > rs f : T ¼ 1þ2m rf 2 8 > < gðe ¼ 0:01; m ¼ 0:3Þ ¼ 0:008 with gðe ¼ 0:5; m ¼ 0:3Þ ¼ 0:04 ð16Þ > : gðe ¼ 0:1; m ¼ 0:3Þ ¼ 0:075 Therefore, the mode I threshold stress intensity factor and the fatigue limit in alternate tension if not sufficient to identify the criterion. The fatigue limit in shear is used to identify precisely the value of d for a given material. The response of the criterion under multiaxial loading conditions is examined so as to show graphically how to identify precisely the value of d. For this purpose, various values of e were fixed. For each value of e, the corresponding value of d that ensures that the non-propagation threshold match the fatigue limit in alternate tension is determined. Then the non-propagation criterion was plotted for a biaxial stress state in Fig. 3. The crack plane is
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chosen so as to maximize the criterion. The non-propagation threshold surface is plotted in the plane (r1, r2) r1 and r2 being the two components of the applied biaxial stress. For the sake of comparison the Dang Van criterion [8] identified for the 30NCD16 steel [7] was also plotted in the same graph (Fig. 3). It is clear on this graph that the
value of d controls the sensitivity of the criterion to the stress biaxiality. For a value of e equal to 0.1, the proposed criterion is very close to the Dang Van criterion. The value of d was identified so as to obtain the best possible agreement between our criterion and the Dang Van’s one (Fig. 4). The value identified for d and the corresponding values for arf and e are reported in Table 2. It is worth to underline that the value of d identified corresponds to a value of arf which is in the order of the grain size. Beside the matching value of e ensures that the LEFM fields are reasonably valid. In Fig. 4 is also reported the Von Mises criterion for a biaxial stress state. In bi-compression cases, our criterion is close to the Von Mises criterion , which is not the case for the Dang Van’s criterion. As a component won’t fail in bi-compression because of crack opening, but because of generalized plasticity, it is satisfactory to find the criterion agreeing with the Von Mises plasticity criterion in the bi-compression region. 4. Applications
Fig. 3. Graphical illustration of the identification of the parameter d using the fatigue limit in torsion and in tension.
In this section the response of the proposed model in various configurations is discussed. First of all, its ability of reproducing the size effect in the Kitagawa and Takahashi diagram is examined. Then the defect modelling is discussed, in particular its position with respect to the surface. Then its response under multiaxial loading conditions is also illustrated and finally it is shown how the criterion can be used to build up a probabilistic model. 4.1. Size effect in uniaxial tension In what follows, crack initiation is assumed to occur in the plane perpendicular to the maximal principal stress direction. In such a case, the criterion in Eq. (8) reduces to: 2 2 KI T KI T þ þ fm : ¼1 ð17Þ T th T th K I th K I th
Fig. 4. Comparison between the non-propagation criterion in Eq. (8) identified for the 30NCD16 steel with either the Dang Van’s endurance criterion or the Von Mises yield criterion.
Table 2 Set of parameters identified for the criterion in Eq. (8) and for a 30NCD16 steel, and the corresponding values of Tth, e and arf KIth (MPaÆm1/2)
d (lm)
e
arf (lm)
Tth (MPa)
7
7.16
0.14
49.15
1120
Let consider a penny-shaped crack with a variable radius a. Knowing the stress intensity factor and the Tstress for a penny-shaped crack (Eq. (14)) and using Eq. (17), the non-propagation threshold stress can be determined as a function of the crack length. The criterion was plotted in the Kitagawa and Takahashi diagram in Fig. 5. It reproduces qualitatively well the features made known by Kitagawa and Takahashi [1], namely an independency of the ‘‘no-growth’’ threshold stress to the crack length, when the defect size tends to zero, and a dependency similar to that of LEFM when the defect size is large. It is important, however, to underline that when the defect size tends to zero the stress tends to a constant value close to 1400 MPa, which is well above the fatigue limit of the material. In fact, the criterion was identified with the assumption that the material always contains defects, which dimension was ‘‘identified’’ to be equal to arf at the fatigue limit rf, therefore the part of the diagram below
E. Thieulot-Laure et al. / International Journal of Fatigue 29 (2007) 1996–2004
2001
Fig. 5. Non propagation threshold stress range as calculated using Eq. (17) in uniaxial tension and plotted in the Kitagawa and Takahashi [1] diagram.
arf is irrelevant in this particular case. This domain would correspond to a ‘‘perfect’’ material without defects, without grains etc. . . However, this doesn’t mean that the model does not predict a size effect for defects sizes above arf. In Fig. 6 is plotted for instance an enlarged plot of the criterion for defects sizes above arf. The difference between the proposed criterion and the prediction of LEFM is up to 190 MPa for defect sizes close to arf. which is far from being negligible. It was discussed in Section 3.3 (Fig. 3) that the fatigue limit determined in uniaxial tension is not sufficient to identify d . The fatigue limit in torsion was employed to identify the value of d. But, since the T-stress allows also accounting for the crack length (Fig. 2), one could expect to adjust the value of d so as to obtain the desired size effect on the ‘‘no-growth’’ threshold stress. In Fig. 7, the criterion is plotted for different value of d (or alternatively e). It is obvious in this graph that the size effect can not be
Fig. 7. Effect of d on the size effect in the Kitagawa–Takahashi diagram.
Fig. 8. Influence of the defect size in biaxial loading.
Fig. 6. Enlarged plot of the criterion in its domain of validity. Comparison with LEFM.
Fig. 9. Non-propagation criterion for internal and surface particles in a component subjected to a biaxial stress range (r1, r2).
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employed to identify d. Provided that the validity of LEFM fields is ensured (e is well below 1), there is almost no effect of d on the position of the criterion in the Kitagawa diagram. 4.2. Influence of the size of the defect under biaxial loading conditions Now the criterion identified using the fatigue limit in tension and in torsion (Table 2) is plotted in a (r1, r2)
plane for a biaxial stress range in Fig. 8. So as to illustrate the effect of the defect size, the criterion is plotted for a few crack’s dimensions i.e. 90, 500 lm and 5 mm. First of all, it can be noticed that the size of the defect has no effect on the criterion under a bi-compression state, which was expected. On the contrary, when one of the principal stress components is positive, the size of the defect has a large effect on the position of the criterion which tends to zero when the size of the defect increases (Fig. 8).
Fig. 10. (a) Non-propagation criterion in a Kitagawa–Takahashi [1] diagram and two choices of probability density functions of flaw sizes. Calculated failure stress range probability density associated with (b) the first flaw sizes distribution (c) the second flaw sizes distribution.
E. Thieulot-Laure et al. / International Journal of Fatigue 29 (2007) 1996–2004
4.3. Influence of the position of the defect Let consider a biaxial stress range (r1, r2) and a circular particle either embedded in the material or located just below the surface. When the particle is well inside the material, the defect can be modelled as a penny-shaped crack. When the particle is located just below the surface, the defect should tend quasi instantaneously to a semi-elliptical crack. For the sake of comparison the defect is modelled as a semi-circular crack at the surface (Fig. 9). The non-propagation threshold for both the embedded and the surface crack were plotted in the same diagram (Fig. 9). As could be expected, the borderline corresponding to the particle located at the surface is entirely contained within the borderline corresponding to the internal particle. A defect located at the surface of the material is more critical than a defect of equivalent size embedded in the same material.
2003
size and each defect orientation. Then the flaw size distribution (Eq. (18)) allows computing the survival probability corresponding to each direction and each value of ac. From these results, a failure probability for each stress range and corresponding to each direction can be computed. The cumulus of every probability in every direction is made in the framework of the weakest-link theory. This process is illustrated in Fig. 10. It is important to underline that for a given width of the flaw size distribution, the width of the distribution of the failure stress range varies with the location of the mean flaw size within the Kitagawa diagram. If the flaw size distribution is embedded within the domain where the non propagation threshold stress is insensitive to the defects size, the width of the distribution of the failure stress tends to zero. The opposite effect appears when the flaw size distribution is located within large defect domain of the Kitagawa diagram. 5. Conclusions
4.4. Application of the criterion within the framework of a probabilistic model The main objective of this research is to build a probabilistic model for fatigue failure. Using the proposed criterion it is now possible to introduce various random variables such as for instance the size of the defect, its position with respect to the surface, its orientation (a plane defined by its normal ~ nÞ etc. For this purpose, the weakest-link assumption is adopted. The probability that the component survives is therefore the product of the probabilities for each volume of material to survive. This second probability is itself the product of the probabilities that the volume of material survives for each possible defect orientation, for each step of the loading scheme etc. It is worth to underline that the weakest link theory makes easier the use of the criterion under non-proportional loading conditions with criticalplane endurance criterions. As a matter of fact, it is not necessary to determine the critical plane, each plane is examined and the survival probabilities are multiplied. Second, following Weibull, an exponential form is chosen for the defect distribution so as to transform the products of survival probabilities into the exponential of an integral. An example of application is given in the sequel. So as to illustrate the applicability of the model within a probabilistic framework the following assumptions were done. A uniaxial loading stress state is considered. The defects are all modelled as penny-shaped cracks. The defects are assumed to be randomly oriented, each normal ~ n is equally probable. A flaw size distribution is chosen: ac P ða P ac Þ ¼ 1 exp n exp ð18Þ a0 The probability density function of the flaw distribution is derived from the previous expression. The criterion in Eq. (8), allows computing the ‘‘nogrowth’’ threshold stress which corresponds to each defect
In this paper, a multiaxial fatigue criterion is proposed which allows accounting for the distributions of the defect sizes, orientations, shapes and positions. It is based on a critical distortional elastic energy calculated within the framework of the LEFM. However the first non-singular term of LEFM fields (i.e. the T-stress) is included so as to introduce a defect size effect and a sensitivity to the stress biaxiality or triaxiality. The material parameters in the criterion expression are the Poisson’s ratio m, the mode I threshold stress intensity factor KIth and alternatively the threshold T-stress Tth or the size of the integration domain d. It was discussed that the best way to identify d is to adjust it so that for small defects the criterion predicts both the fatigue limit in tension and in torsion. The corresponding defect size arf is a result of this identification process. Once the criterion is identified it can be plotted versus the defect size under uniaxial loading conditions. It was shown that the size effect made known by Kitagawa and Takahashi can be successfully reproduced using this criterion. Then it can also be plotted for various multiaxial loading conditions. It was shown that for small defects sizes it is very close to the Dang Van multiaxial endurance criterion. Finally, the use of this criterion within a probabilistic framework was briefly discussed. It is worth to underline that a probabilistic model makes easy the application of the criterion to non-proportional loading schemes. References [1] Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks. In: ASM Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Metalspark, Ohio. 1976, pp. 627–631. [2] Wu YT, Enright MP, Millwater HR. Probabilistic methods for design assessment of reliability with inspection. AIAA J 2002;40(5): 937–46.
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[3] McEvily AJ, Endo M, Ishihara S. The influence of biaxial stress on the fatigue behaviour of defect-containing steels. ICF11. 2005. [4] Billaudeau T, Nadot Y, Bezine G. Multiaxial fatigue limit for defective materials: mechanisms and experiments. Acta Mater 2004;52:3911–20. [5] Nadot Y, Billaudeau T. Multiaxial fatigue limit criterion for defective materials. Eng Fract Mech 2006;73:112–33. [6] Hamam R, Pommier S, Bumbieler F. Mode 1 fatigue crack growth under biaxial loading. Int J Fatigue 2005;27(10):1342–6.
[7] Banvillet A, Palin-Luc T, Lasserre S. A volumetric energy based high multiaxial fatigue criterion. Int J Fatigue 2003:755–69. [8] Dang Van K. Sur la re´sistance a´ la fatigue des me´taux. Sciences et Techniques de l’Armement 1973;47(3e´me fascicule): 641–722. [9] Murakami Y. Stress intensity factors handbook. Oxford: Pergamon Press; 1987. [10] Wang X. Elastic T-Stress solutions for penny-shaped cracks under tension and bending. Eng Fract Mech 2004.
International Journal of Fatigue 29 (2007) 1996–2004
www.elsevier.com/locate/ijfatigue
A multiaxial fatigue failure criterion considering the effects of the defects E. Thieulot-Laure b
a,b,*
, S. Pommier a, S. Fre´chinet
b
a LMT, Ens-Cachan, 61, av. du pre´sident Wilson, 94235 Cachan, France SNECMA, Rond-point Rene´ Ravaud, Re´au, 77550 Moissy-Cramayel, France
Received 15 September 2006; received in revised form 2 January 2007; accepted 7 January 2007 Available online 31 January 2007
Abstract An endurance criterion is proposed which aims at capitalizing the advantages of both fracture mechanics and of energetic or stressbased approaches. As a matter of fact, fracture mechanics based approaches allow building probabilistic criterions by introducing the statistic distributions of defects sizes, positions and orientations. On the other hand, stress-based approaches allow building multiaxial failure and life criterions. However, as shown by the diagram of Kitagawa and Takahashi [Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks. In: ASM Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Metalspark, Ohio. 1976, pp. 627–631], none of these approaches applies over the whole range of possible defects dimensions. For large defects, fracture mechanics concepts apply and the fatigue limit is deduced from the threshold stress intensity factor and the defect size. There is usually a threshold below which the sensitivity of the fatigue limit to the defect size becomes negligible. In that domain, Linear Elastic Fracture Mechanics (LEFM) overestimates the fatigue limit and stress-based approaches are more pertinent. In the present paper, the LEFM framework is used, defects are assumed to be cracks. The size effect exhibited by Kitagawa and Takahashi [Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks. In: ASM Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Metalspark, Ohio. 1976, pp. 627–631] is accounted for through the T-stress parameter, which is the first non singular term in the asymptotic development of the stresses at crack tip. The proposed criterion is a non-propagation criterion based on a critical distortional elastic energy in the crack tip region. The use of an energetic criterion ensures the consistency with endurance criterion. The specificity of our approach lies in the calculation of the distortional energy using the displacement fields of LEFM which ensures the consistency with fracture mechanics. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Fatigue criterion; LEFM; T-stress; Probabilistic approaches
1. Introduction Conventional life prediction methodologies are based on concepts that don’t allow accounting for material, manufacturing or maintenances anomalies that can degrade the fatigue resistance of components. So as to consider these anomalies, the adoption of a probabilistic damage tolerance approach is useful. Within the framework of Linear Elastic Fracture Mechanics (LEFM), it becomes possible to consider random defects nucleation, size, orientation and location, random inspection schedules, and several *
Corresponding author. Tel.: +33 1 47 40 28 69. E-mail addresses: [email protected] (E. Thieulot-Laure), [email protected] (S. Pommier). 0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.01.017
other random variables. Such approaches were developed for instance to address hard alpha material anomalies in Titanium alloys [2]. However, LEFM based approaches do not apply for small defects sizes. The pioneering experiments of Kitagawa and Takahashi [1] have highlighted the effect of the defect size upon the fatigue strength of materials. These authors have established S N curves on specimens containing drilled holes. The fatigue limit was shown to be function of the typical dimension of the manufactured defects [1]. Above a threshold defect size, LEFM applies; the fatigue limit varies as the defect size to the power minus one half. Below that threshold, LEFM strongly overestimates the fatigue limit which is nearly constant. Stress based approaches are much more pertinent (Fig. 1) in the absence of large defects, but overestimate
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Nomenclature LEFM a a d m rf sf
Linear Elastic Fracture Mechanics crack length (m) parameter of the Dang Van’s criterion radius of the integration domain of the elastic distortional energy Poisson’s ratio fatigue limit in tension (Mpa) fatigue limit in torsion (Mpa)
the fatigue limit in their presence. These results have been reproduced by many authors [3] and for various metallic materials. Moreover, Nadot et al. [4,5] have also highlighted the importance of the defect shape and orientation upon the fatigue limit. As a consequence, methodological problems arise when the material contains a population of defects that extends itself over the transition domain, or when a competition between different populations of defects takes place. A component may contain for instance, small manufacturing defects, in the order of ten micrometers, with a large probability of occurrence and larger defects, in the order of one millimetre, created in service conditions with a very low probability of occurrence. The combination of failure probabilities issued from highly different calculations methodologies can be questionable. A unified model would be useful in the engineering practice. The aim of this paper is therefore to set up a fatigue failure criterion predicting the non-propagation domain for supposedly pre-existing defects in the material. The specifications of the model are as follows: it should tend continuously to a multiaxial endurance model for small defects and to a LEFM based model for large defects and include explicitly the various statistic distributions representative of the defects populations. The aim is to provide a probability of failure for a component subjected to complex loading conditions.
KI, KII mode I and mode II stress intensity factors (MPa.(m)1/2) KIth mode I threshold stress intensity factor (MPa (m)1/2) T T-stress (Mpa) w distortional part of the elastic energy density U distortional part of the elastic energy
2. Criterion In what follows, the material is supposed to contain defects – before any stresses are applied – and these defects are assumed to be cracks. The criterion is a non-propagation criterion based on a critical distortional elastic energy in the crack tip region. This choice arises from the hypothesis that fatigue crack growth is associated with plastic deformation. If the material obeys the Von Mises yield criterion, which is critical distortional elastic energy criterion, the crack tip region can also be assumed to obey a critical distortional elastic energy criterion. However, the distortional energy is calculated at a global scale for the whole crack tip region, using LEFM stress and strain fields. The LEFM fields used for the calculation of the distortional elastic energy include the T-stress which is the first non-singular term of the asymptotic development at crack tip. As an illustration, the expression of the asymptotic stress field at crack tip in mode I is reported in Eq. (1). KI h h 3h rxx ¼ pffiffiffiffiffiffiffi cos 1 sin sin þT 2 2 2 2pr KI h h 3h ð1Þ 1 þ sin sin ryy ¼ pffiffiffiffiffiffiffi cos 2 2 2 2pr KI h h 3h rxy ¼ pffiffiffiffiffiffiffi cos sin cos 2 2 2 2pr The displacements field at crack tip is the following under plane strain conditions along the crack front: rffiffiffiffiffiffi rffiffiffiffiffiffi KI r h K II r h ux ¼ cos ðj cos hÞ þ sin ð2 þ j þ cos hÞ 2 2 2l 2p 2l 2p T þ ðj þ 1Þr cos h 8l rffiffiffiffiffiffi rffiffiffiffiffiffi KI r h K II r h uy ¼ sin ðj cos hÞ þ cos ð2 j cos hÞ 2 2 2l 2p 2l 2p T ð3 jÞr sin h 8l uz ¼ 0 ð2Þ
Fig. 1. Schematic Kitagawa-Takahashi’s diagram, see [1].
where l and j are the Lame´’s coefficients. The strain tensor is derived from the displacements field, and the stress
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tensor is obtained from the strain tensor by the Hooke’s law. Knowing the expressions of these two tensors, the distortional part of the elastic energy density w is defined by: wðr; hÞ ¼ Trðr0 e0 Þ
ð3Þ
where r’ and e’ are the deviatoric parts of the stress and strain tensors. The energy density in Eq. (3) is integrated over a domain within a distance d to the crack tip: Z r¼d Z h¼p 1 Trðr0 e0 Þrdr dh U ðK I ; K II ; T Þ ¼ ð4Þ 2 r¼0 h¼p U is therefore the distortional energy per unit length of the crack front in a domain delimited by a circle at a distance d from the crack tip. With the above mentioned assumptions, the non-propagation criterion takes the form: U ðK I ; K II ; T Þ < U c ) No growth
ð5Þ
where Uc is the critical distortional elastic energy in the crack tip region defined by the parameter d. Besides, if the mode I non-propagation threshold stress intensity factor KIth is known, the expression of Uc is given by: U c ¼ U ðK Ith ; 0; 0Þ
ð6Þ
As a consequence, the non-propagation criterion can now be rewritten as follows: U ðK I ; K II ; T Þ < U ðK Ith ; 0; 0Þ ) No growth
ð7Þ
A few mathematical developments and simplifications of Eq. (7) allow expressing the criterion directly as a function of KI, KII and T (Eq. (8)). 2 2 2 KI K II T KI T þ þ þ fm : <1 T th T th K Ith K IIth K Ith ) No growth
ð8Þ
From Eq. (8) we can see that the criterion representation in the (KI, KII) plane (with T = 0) and in the (KII, T) plane (with KI = 0) will be an ellipse and that in the (KI,T) plane (with KII = 0), due to the coupled term between KI and T, it will be an inclined ellipse. For the sake of simplicity, three terms have been introduced fm, KIIth and Tth, whose expressions are provided in Eqs. (9), (10) and (11). KIIth is the non-propagation threshold of KII obtained for KI and T equal to zero. Tth is the non-propagation threshold of T obtained for KI and KII equal to zero. And, when computing the non-propagation threshold of KI for KII and T equal to zero, we satisfactorily obtain KIth. 32ð1 10m þ 10m2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15p ð1 m þ m2 Þð7 16m þ 16m2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 16m þ 16m2 K IIth ¼ K Ith pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 16m þ 16m2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K Ith 1 7 16m þ 16m2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T th ¼ pffiffiffiffiffiffiffiffi 1 m þ m2 2pd 2 fm ¼
Fig. 2. Case of a through thickness crack in an infinite media (Eq. (12)). Evolution of the non propagation threshold KInp, as calculated using the criterion (Eq. (8)), normalized by the mode I threshold stress intensity factor KIth (long crack) versus the crack length, for various values of d.
ð9Þ
ð10Þ ð11Þ
It is important to underline that provided the Poisson’s ratio and the mode I threshold stress intensity factor are known, fm, and KIIth are also known. For instance, if m = 0.3: fm 0.45 and KIIth 0.5KIth. However, either the term Tth or the dimension d of the integration domain remains to be identified. In this criterion, the stress biaxiality is accounted for by the T-stress level. Let consider, for instance, a through thickness crack lying in the plane (x, z) and subjected to remote biaxial loads which amplitudes are Sx and Sy, the expressions of mode I stress intensity factor and T-stress are the following: pffiffiffiffiffiffi K I ¼ S y pa ð12Þ T ¼ Sx Sy It is obvious in Eq. (12), that the T-stress varies with the stress biaxiality. Besides, the T-stress allows also accounting for the defect’s size [6]. Under uniaxial loading conditions, for instance, the threshold for crack propagation denoted by Knp can be easily calculated as a function of the crack length using Eqs. (12) and (8). Its evolution is plotted in Fig. 2 for various values of the dimension d of the integration domain. 3. Identification The non-propagation criterion proposed in the previous section is a function of the mode I and mode II stress intensity factors and of the T-stress in each point of the crack front assuming a plane strain condition along the crack front. The required ‘‘material’’ parameters are the Poisson’s ratio, the mode I threshold stress intensity factor for long cracks and either the parameter Tth or the dimension d of the integration domain. In this section, it is explained how this supplementary parameter can be identified. In what follows, the stress quantities considered refers to the stress amplitudes, in the case of a proportional loading.
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3.1. Material The criterion proposed in Eq. (8) was identified for a 30NCD16 steel for which the requested data could be found in the literature [7], namely the mode I threshold stress intensity factor KIth, the fatigue limit rf in tension and the fatigue limit sf in torsion (Table 1). Banvillet et al. [7] have also identified the parameter a of the Dang Van [8] multiaxial endurance criterion for this material. Very schematically the Dang Van [8] endurance criterion takes the form: max max ½s þ ap < sf ) No fatigue failure t ~ n sf 1 where a ¼ 3 ð13Þ rf 2 where s is the current shear stress component in a plane defined by its normal ~ n, and where p is the current hydrostatic pressure. sf is the fatigue limit in pure alternate shear. a is a material parameter that is identified using the data reported in Table 1. The Dang Van’s criterion belongs to the family of the critical plane endurance criterions, it was developed in particular so as to better represent the experimental results for non-proportional loading paths and has been established from observations at the microscopic scale. For this criterion a critical plane ~ n and a critical moment t that maximize the criterion can be determined. With the assumption that below the fatigue limit the material is not defect free but may contain small defects below their non-propagation threshold, it becomes possible to compare the criterion in Eq. (8) with the Dang Van endurance criterion identified by Banvillet et al. [7] for the 30NCD16 steel. 3.2. Defect’s modelling So as to compare the criterion in Eq. (8) to the Dang Van criterion, an assumption should be done on the defects. Various assumptions can be done according to the problem. Let consider, for instance, a penny-shaped crack embedded in an infinite body lying in a plane defined by its normal ~ n. Let consider a point of the crack front defined by its normal ~ t. Let rn be the normal stress, s be the tangential stress, and rT be the projection of the stress tensor along the axe parallel to the crack plane. From handbooks [9,10] we have: pffiffiffiffiffiffi 8 2 > < K I ¼ p rn papffiffiffiffiffiffi 4 K II ¼ pð2mÞ s pa > : 1þ2m T ¼ rT 2 rn
8 r ¼ ðr~ nÞ ~ n > < n ~ nÞ t with s ¼ ðr~ > : tÞ ~ t rT ¼ ðr~
ð14Þ
Table 1 30NCD16 steel material data and the corresponding set of parameters of the Dang Van endurance criterion (after [7]) m
KIth (MPaÆm1/2)
rf (MPa)
sf (MPa)
a
0.29
7
560
428
3.08
1999
With such an assumption it becomes possible to formulate the non-propagation criterion in Eq. (8) as a function of the applied stress tensor r and of the dimension a of the defect. 3.3. Identification of the parameter d in the non-propagation criterion (Eq. (8)) The data in Table 1 are employed to determine the parameter d in the non-propagation criterion in Eq. (8). There are some restrictions on the value of the parameter d. As a matter of fact, the distortional elastic energy is calculated using LEFM fields which validity is restricted to the near crack tip region so as to keep the error of the asymptotic development reasonable. For instance, if the distance to the crack tip remains below a/10 – a being the crack size – the average error is around 10%. Let now assume that the endurance limit of a material is not determined for a defect-free material but for a material containing the smallest possible defects with a typical size denoted by arf. If we identify the value of d so that the non-propagation threshold corresponds to the fatigue limit, since d is the size of the integration domain, it should therefore be a fraction e of this typical size: d ¼ e arf
ð15Þ
The smaller e is, the more precise the LEFM fields are and the smaller the d parameter is. But d should also be large enough compared with the typical microstructural dimensions because of the continuum mechanics hypothesis used for establishing the LEFM fields. However, provided that these restrictions are respected, a large range of values of d can be chosen for which the condition that the non-propagation threshold corresponds to the fatigue limit is verified. As a matter of fact, with the assumptions on the defects geometry in Eq. (14), the criterion in Eq. (8) yields at the fatigue limit: 8 pffiffiffiffiffiffiffiffiffi 2 > < K I ¼ p rf parf with arf ¼ d=e K2 K II ¼ 0 ) d ¼ gðe; mÞ Ith2 > rs f : T ¼ 1þ2m rf 2 8 > < gðe ¼ 0:01; m ¼ 0:3Þ ¼ 0:008 with gðe ¼ 0:5; m ¼ 0:3Þ ¼ 0:04 ð16Þ > : gðe ¼ 0:1; m ¼ 0:3Þ ¼ 0:075 Therefore, the mode I threshold stress intensity factor and the fatigue limit in alternate tension if not sufficient to identify the criterion. The fatigue limit in shear is used to identify precisely the value of d for a given material. The response of the criterion under multiaxial loading conditions is examined so as to show graphically how to identify precisely the value of d. For this purpose, various values of e were fixed. For each value of e, the corresponding value of d that ensures that the non-propagation threshold match the fatigue limit in alternate tension is determined. Then the non-propagation criterion was plotted for a biaxial stress state in Fig. 3. The crack plane is
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chosen so as to maximize the criterion. The non-propagation threshold surface is plotted in the plane (r1, r2) r1 and r2 being the two components of the applied biaxial stress. For the sake of comparison the Dang Van criterion [8] identified for the 30NCD16 steel [7] was also plotted in the same graph (Fig. 3). It is clear on this graph that the
value of d controls the sensitivity of the criterion to the stress biaxiality. For a value of e equal to 0.1, the proposed criterion is very close to the Dang Van criterion. The value of d was identified so as to obtain the best possible agreement between our criterion and the Dang Van’s one (Fig. 4). The value identified for d and the corresponding values for arf and e are reported in Table 2. It is worth to underline that the value of d identified corresponds to a value of arf which is in the order of the grain size. Beside the matching value of e ensures that the LEFM fields are reasonably valid. In Fig. 4 is also reported the Von Mises criterion for a biaxial stress state. In bi-compression cases, our criterion is close to the Von Mises criterion , which is not the case for the Dang Van’s criterion. As a component won’t fail in bi-compression because of crack opening, but because of generalized plasticity, it is satisfactory to find the criterion agreeing with the Von Mises plasticity criterion in the bi-compression region. 4. Applications
Fig. 3. Graphical illustration of the identification of the parameter d using the fatigue limit in torsion and in tension.
In this section the response of the proposed model in various configurations is discussed. First of all, its ability of reproducing the size effect in the Kitagawa and Takahashi diagram is examined. Then the defect modelling is discussed, in particular its position with respect to the surface. Then its response under multiaxial loading conditions is also illustrated and finally it is shown how the criterion can be used to build up a probabilistic model. 4.1. Size effect in uniaxial tension In what follows, crack initiation is assumed to occur in the plane perpendicular to the maximal principal stress direction. In such a case, the criterion in Eq. (8) reduces to: 2 2 KI T KI T þ þ fm : ¼1 ð17Þ T th T th K I th K I th
Fig. 4. Comparison between the non-propagation criterion in Eq. (8) identified for the 30NCD16 steel with either the Dang Van’s endurance criterion or the Von Mises yield criterion.
Table 2 Set of parameters identified for the criterion in Eq. (8) and for a 30NCD16 steel, and the corresponding values of Tth, e and arf KIth (MPaÆm1/2)
d (lm)
e
arf (lm)
Tth (MPa)
7
7.16
0.14
49.15
1120
Let consider a penny-shaped crack with a variable radius a. Knowing the stress intensity factor and the Tstress for a penny-shaped crack (Eq. (14)) and using Eq. (17), the non-propagation threshold stress can be determined as a function of the crack length. The criterion was plotted in the Kitagawa and Takahashi diagram in Fig. 5. It reproduces qualitatively well the features made known by Kitagawa and Takahashi [1], namely an independency of the ‘‘no-growth’’ threshold stress to the crack length, when the defect size tends to zero, and a dependency similar to that of LEFM when the defect size is large. It is important, however, to underline that when the defect size tends to zero the stress tends to a constant value close to 1400 MPa, which is well above the fatigue limit of the material. In fact, the criterion was identified with the assumption that the material always contains defects, which dimension was ‘‘identified’’ to be equal to arf at the fatigue limit rf, therefore the part of the diagram below
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Fig. 5. Non propagation threshold stress range as calculated using Eq. (17) in uniaxial tension and plotted in the Kitagawa and Takahashi [1] diagram.
arf is irrelevant in this particular case. This domain would correspond to a ‘‘perfect’’ material without defects, without grains etc. . . However, this doesn’t mean that the model does not predict a size effect for defects sizes above arf. In Fig. 6 is plotted for instance an enlarged plot of the criterion for defects sizes above arf. The difference between the proposed criterion and the prediction of LEFM is up to 190 MPa for defect sizes close to arf. which is far from being negligible. It was discussed in Section 3.3 (Fig. 3) that the fatigue limit determined in uniaxial tension is not sufficient to identify d . The fatigue limit in torsion was employed to identify the value of d. But, since the T-stress allows also accounting for the crack length (Fig. 2), one could expect to adjust the value of d so as to obtain the desired size effect on the ‘‘no-growth’’ threshold stress. In Fig. 7, the criterion is plotted for different value of d (or alternatively e). It is obvious in this graph that the size effect can not be
Fig. 7. Effect of d on the size effect in the Kitagawa–Takahashi diagram.
Fig. 8. Influence of the defect size in biaxial loading.
Fig. 6. Enlarged plot of the criterion in its domain of validity. Comparison with LEFM.
Fig. 9. Non-propagation criterion for internal and surface particles in a component subjected to a biaxial stress range (r1, r2).
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employed to identify d. Provided that the validity of LEFM fields is ensured (e is well below 1), there is almost no effect of d on the position of the criterion in the Kitagawa diagram. 4.2. Influence of the size of the defect under biaxial loading conditions Now the criterion identified using the fatigue limit in tension and in torsion (Table 2) is plotted in a (r1, r2)
plane for a biaxial stress range in Fig. 8. So as to illustrate the effect of the defect size, the criterion is plotted for a few crack’s dimensions i.e. 90, 500 lm and 5 mm. First of all, it can be noticed that the size of the defect has no effect on the criterion under a bi-compression state, which was expected. On the contrary, when one of the principal stress components is positive, the size of the defect has a large effect on the position of the criterion which tends to zero when the size of the defect increases (Fig. 8).
Fig. 10. (a) Non-propagation criterion in a Kitagawa–Takahashi [1] diagram and two choices of probability density functions of flaw sizes. Calculated failure stress range probability density associated with (b) the first flaw sizes distribution (c) the second flaw sizes distribution.
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4.3. Influence of the position of the defect Let consider a biaxial stress range (r1, r2) and a circular particle either embedded in the material or located just below the surface. When the particle is well inside the material, the defect can be modelled as a penny-shaped crack. When the particle is located just below the surface, the defect should tend quasi instantaneously to a semi-elliptical crack. For the sake of comparison the defect is modelled as a semi-circular crack at the surface (Fig. 9). The non-propagation threshold for both the embedded and the surface crack were plotted in the same diagram (Fig. 9). As could be expected, the borderline corresponding to the particle located at the surface is entirely contained within the borderline corresponding to the internal particle. A defect located at the surface of the material is more critical than a defect of equivalent size embedded in the same material.
2003
size and each defect orientation. Then the flaw size distribution (Eq. (18)) allows computing the survival probability corresponding to each direction and each value of ac. From these results, a failure probability for each stress range and corresponding to each direction can be computed. The cumulus of every probability in every direction is made in the framework of the weakest-link theory. This process is illustrated in Fig. 10. It is important to underline that for a given width of the flaw size distribution, the width of the distribution of the failure stress range varies with the location of the mean flaw size within the Kitagawa diagram. If the flaw size distribution is embedded within the domain where the non propagation threshold stress is insensitive to the defects size, the width of the distribution of the failure stress tends to zero. The opposite effect appears when the flaw size distribution is located within large defect domain of the Kitagawa diagram. 5. Conclusions
4.4. Application of the criterion within the framework of a probabilistic model The main objective of this research is to build a probabilistic model for fatigue failure. Using the proposed criterion it is now possible to introduce various random variables such as for instance the size of the defect, its position with respect to the surface, its orientation (a plane defined by its normal ~ nÞ etc. For this purpose, the weakest-link assumption is adopted. The probability that the component survives is therefore the product of the probabilities for each volume of material to survive. This second probability is itself the product of the probabilities that the volume of material survives for each possible defect orientation, for each step of the loading scheme etc. It is worth to underline that the weakest link theory makes easier the use of the criterion under non-proportional loading conditions with criticalplane endurance criterions. As a matter of fact, it is not necessary to determine the critical plane, each plane is examined and the survival probabilities are multiplied. Second, following Weibull, an exponential form is chosen for the defect distribution so as to transform the products of survival probabilities into the exponential of an integral. An example of application is given in the sequel. So as to illustrate the applicability of the model within a probabilistic framework the following assumptions were done. A uniaxial loading stress state is considered. The defects are all modelled as penny-shaped cracks. The defects are assumed to be randomly oriented, each normal ~ n is equally probable. A flaw size distribution is chosen: ac P ða P ac Þ ¼ 1 exp n exp ð18Þ a0 The probability density function of the flaw distribution is derived from the previous expression. The criterion in Eq. (8), allows computing the ‘‘nogrowth’’ threshold stress which corresponds to each defect
In this paper, a multiaxial fatigue criterion is proposed which allows accounting for the distributions of the defect sizes, orientations, shapes and positions. It is based on a critical distortional elastic energy calculated within the framework of the LEFM. However the first non-singular term of LEFM fields (i.e. the T-stress) is included so as to introduce a defect size effect and a sensitivity to the stress biaxiality or triaxiality. The material parameters in the criterion expression are the Poisson’s ratio m, the mode I threshold stress intensity factor KIth and alternatively the threshold T-stress Tth or the size of the integration domain d. It was discussed that the best way to identify d is to adjust it so that for small defects the criterion predicts both the fatigue limit in tension and in torsion. The corresponding defect size arf is a result of this identification process. Once the criterion is identified it can be plotted versus the defect size under uniaxial loading conditions. It was shown that the size effect made known by Kitagawa and Takahashi can be successfully reproduced using this criterion. Then it can also be plotted for various multiaxial loading conditions. It was shown that for small defects sizes it is very close to the Dang Van multiaxial endurance criterion. Finally, the use of this criterion within a probabilistic framework was briefly discussed. It is worth to underline that a probabilistic model makes easy the application of the criterion to non-proportional loading schemes. References [1] Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks. In: ASM Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Metalspark, Ohio. 1976, pp. 627–631. [2] Wu YT, Enright MP, Millwater HR. Probabilistic methods for design assessment of reliability with inspection. AIAA J 2002;40(5): 937–46.
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[3] McEvily AJ, Endo M, Ishihara S. The influence of biaxial stress on the fatigue behaviour of defect-containing steels. ICF11. 2005. [4] Billaudeau T, Nadot Y, Bezine G. Multiaxial fatigue limit for defective materials: mechanisms and experiments. Acta Mater 2004;52:3911–20. [5] Nadot Y, Billaudeau T. Multiaxial fatigue limit criterion for defective materials. Eng Fract Mech 2006;73:112–33. [6] Hamam R, Pommier S, Bumbieler F. Mode 1 fatigue crack growth under biaxial loading. Int J Fatigue 2005;27(10):1342–6.
[7] Banvillet A, Palin-Luc T, Lasserre S. A volumetric energy based high multiaxial fatigue criterion. Int J Fatigue 2003:755–69. [8] Dang Van K. Sur la re´sistance a´ la fatigue des me´taux. Sciences et Techniques de l’Armement 1973;47(3e´me fascicule): 641–722. [9] Murakami Y. Stress intensity factors handbook. Oxford: Pergamon Press; 1987. [10] Wang X. Elastic T-Stress solutions for penny-shaped cracks under tension and bending. Eng Fract Mech 2004.