Review of fatigue crack growth under non-proportional mixed-mode loading

Accepted Manuscript Review of fatigue crack growth under non-proportional mixed-mode loading P. Zerres, M. Vormwald PII: DOI: Reference:

S0142-1123(13)00099-6 http://dx.doi.org/10.1016/j.ijfatigue.2013.04.001 JIJF 3094

To appear in:

International Journal of Fatigue

Please cite this article as: Zerres, P., Vormwald, M., Review of fatigue crack growth under non-proportional mixedmode loading, International Journal of Fatigue (2013), doi: http://dx.doi.org/10.1016/j.ijfatigue.2013.04.001

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Review of fatigue crack growth under non-proportional mixed-mode loading P. Zerresa , M. Vormwalda a

Materials Mechanics Group, Technische Universität Darmstadt, Petersenstr. 12, D-64287 Darmstadt, Germany

Abstract Cyclic non-proportional loading is common experimental practise for investigations of large structures like vehicles. Numerical analysis of local nonproportional loading conditions is also a well established field of research and application. However, theoretical and practical support is rare for evaluating the growth of fatigue cracks under non-proportional cyclic loading conditions. At least seven influence factors - most of them not yet thoroughly understood - are listed and discussed in the paper: the mode-mixity, the material’s influence including its anisotropy if existent, the degree of cyclic plastic deformation and its direction ahead of the crack tip, the crack closure phenomenon, the related mean stress effect, the component’s geometry in general and especially the variable mode-mixity along a crack front. Two crack propagation mechanisms must be considered: (a) the tensile stress dominated, mode II minimising mechanism and (b) the shear stress dominated mechanism. Transition mode-mixities are observed. Some successful explanations of experimental findings have been published, however, a generally accepted and validated formulation of a crack driving force parameter has not yet been identified.

Preprint submitted to Journal of Fatigue

April 8, 2013

Keywords: Mixed mode, non-proportional loading, fatigue crack growth 1. Introduction Most of the engineering structures and components are subjected to fatigue load conditions, which are a combination of various load sequences originated from different sources. Only in rare cases, a correlation of these load sequences may be observed. For ground vehicles, for example, the excitation provided by the roadway surface is uncorrelated with load sequences from manoeuvring, be it curving or acceleration and braking. After an onset of fatigue damage – for metallic materials this generally means the initiation of a fatigue crack – the crack is cyclically loaded in a way such that at the crack front non-proportional mixed-mode situations will occur. In experimental investigations of the fatigue strength of such structures, it is common state of the art to reproduce the action of various load sequences in their realistic interconnection in a laboratory. Chassis and suspension test systems, for example, with twelve or more actuators simultaneously loading the structure are common practice in vehicle test laboratories. In numerical fatigue life assessments, methods for dealing with the initiation of fatigue cracks are available even for the complicated non-proportional cases of combined cyclic loading. The accuracy of the fatigue life estimates obtained by applying these methods and the associated software tools are still under thorough investigation. Nevertheless, the engineers responsible for the fatigue strength of the structures are supported by these helpful numerical tools. Already a remarkable number of systematic investigations of crack ini3

tiation lives under combined cyclic tension and torsion is available. Especially the 90◦ out-of-phase loading shows significantly different lives compared to the proportional loading. Fatigue lives under proportional and nonproportional loading (data mainly collected by Döring [1]) are reported in Table 1 (see also references [2]–[13]). A phase shift from in-phase loading to out-of phase loading leads to an increase in fatigue life if the test is performed in a stress or load controlled condition. According to Sonsino [14] this increase may by explained by a smaller local cyclic plastic deformation when load maxima do not coincide. In contrast to this, under strain controlled test, the fatigue life decreases under out-of-phase loading compared to in-phase loading. However, the material also plays an important role. The above statement holds for ductile materials whereas for semi-ductile materials a life reduction in strain-controlled condition could not be observed. The orientation of initiated cracks can be estimated by applying critical plane approaches. Numerous papers have been published on this topic and only some references are given here, [15, 16, 17, 18, 19, 20]. Approximation algorithms for calculating local non-proportional stress and strain sequences resulting from combined loading are available [21, 22]. If the same nonproportional load sequence continues to be applied to the structure with a short fatigue crack the mode changes are evolving continuously. The total life is the sum of the life to initiate a crack of technical size and subsequent fatigue crack growth. The previous comments on the crack initiation life are supposed to show that a lot of successful research has been done for non-proportional fatigue concerning the initiation stage. By contrast, investigations concerning the fatigue crack growth stage are still rare

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and isolated. In the following only the effect of non-proportional mixed mode loading in the fatigue crack growth stage will be reviewed. The theoretical and practical support is immediately lacking as soon as the growth of fatigue cracks under non-proportional cyclic loading conditions is a matter of concern. A great discrepancy exists between experimental and numerical feasibilities of performing a proof of structural durability under these conditions. The topic of non-proportional mixed-mode fatigue crack growth has, however, become a field of scientific interest. The intention of this paper is to provide a collection of references to already investigated cases, the experimental observations and the analytical and numerical models developed therein. A list of items influencing the fatigue crack growth is gathered. 2. Mixed-Mode Fracture Criteria An early hypothesis for mixed-mode fracture was published by Erdogan and Sih [23]. Their maximum tangential stress criterion postulates that a mixed-mode loaded crack extends in the direction perpendicular to the maximum tangential stress ahead of the crack tip. The stress involved is usually calculated for linear elastic conditions and only the near crack tip asymptotic singular stress field is exploited. Shih [24], however, extended the maximum tangential stress criterion to elastic-plastic analysis for strain hardening material. Sih [25] further proposed the strain energy density criterion according to which crack extension in the direction of the minimum strain energy density is assumed. Another energy-based approach was developed by Hussain et al. [26] who made the maximum energy release rate of a kinked crack 5

responsible for fracture propagation. All of the hypotheses listed so far predict very similar directions of a growing crack under mixed-mode I and II conditions. In the case that any of the aforementioned hypotheses is applied for fatigue crack growth analysis, the crack path is predicted such that the mode II loading at the crack tip is minimised. A completely different path is obtained by the maximum shear stress criterion [27, 28]. This criterion is especially useful in some cases when a crack subjected to mixed-mode I and II loading may remain or turn to propagate in a direction collinear with the plane of the maximum shear stress rather than the plane plane perpendicular to the maximum normal stress. Such a fatigue crack growth behaviour is observed, for example, during stage I of microstructurally short cracks as well as under enforced severe cyclic plastic deformation of notched axis-symmetric shafts under torsion. This initial list of most relevant mixed-mode fracture criteria is concluded with reference to extensive literature surveys of mixed-mode fatigue crack growth under proportional [14, 29, 30] and non-proportional loading [31]. 3. Observations of fatigue cracks growing under non-proportional loading Superimposing non-proportional mixed-mode conditions may be performed in innumerably different ways. The available test rig is the limiting feature for the choice of non-proportional load sequences. By using a uniaxial testing machine, the only possibility is to change the crack tip loading mode (or the mode-mixity) abruptly by changing the specimen’s fixing conditions. Investigations on mode I pre-cracked specimens which are subjected to a 6

mode-mixity, that is tan Φ = ∆KII /∆KI different from zero, may be seen as being investigations on non-proportional mixed-mode loading concerning the early growth after the mode-mixity change. 3.1. Abrupt change of the mode-mixity The crack growth rate for the non-proportional first cycle after a modemixity change is experimentally inaccessible. The investigations focus on the crack deflection angle from the direction of the pre-crack grown under pure mode I. Since the early investigation of Iida and Kobayashi [32] the majority of experimental results [29, 30] show a crack turning or kinking towards a path minimising ∆KII which may be well described by the maximum tensile stress criterion. However, Roberts and Kibler [33] found cases for which the maximum tensile stress criterion was not valid. For high mode-mixities (with the original Mode I pre-crack) coplanar fatigue crack growth was observed. In descriptions of this observations, the maximum shear stress criterion must be called. Besides the mode-mixity, this behaviour seems to be dependent on the material under investigation. No clear classification is available today with respect to which materials show preferred obedience to a mixed-mode criterion deviating from the popular maximum tensile stress criterion and its close relatives, strain energy density and energy release rate criterion. A third factor influencing the fatigue crack growth behaviour must be emphasised: the coplanar nearly maximum shear stress driven fatigue crack growth behaviour is observed preferably for higher stress intensity factor ranges. At the same time, this means that larger and more extended cyclic plastic deformations occur in the vicinity of the crack tip. Plastic deformations in metals – when observed on the microscopic scale – are dislocation motions 7

in planes with high shear stresses. It seems that these planes provide the opportunity for crack extension. Even under the conventional mode I fatigue crack growth situation, the micro mechanism of crack extension is explained by shear bands deviating from the mode I plane [34]. Eventually, at high mixed-mode stress intensity ranges, one shear plane remains dominating and the fatigue crack stays in this plane. For these cases, even crack growth rate data can be acquired. A mixed mode stress intensity factor range ∆KMM was introduced replacing the conventional stress intensity factor range in a crack growth rate equation: da = C (∆KMM )m . dn

(1)

Several formulations for ∆KMM can be found in the literature. Based on a critical value for the displacements behind the crack tip, Tanaka [35] proposed a ∆KMM according to: ∆KMM = (∆KIm + q · ∆KIIm )1/m

(2)

Values of m = 2 together with q = 2 were suggested as well as m = 4 together with q = 4 or q = 8. Yan et al. [36] extended the maximum tangential stress criterion to fatigue crack growth and found the following expression for ∆KMM (with a lack of experimental verification): ∆KMM =

θ0 1 cos (∆KI (1 + cos θ0 ) − 3∆KII sin θ0 ) , 2 2

(3)

with θ0 being the crack growth direction obtained by the maximum tangential stress criterion. Richard et al. [30] proposed ∆KMM to be q ∆KI 1 ∆KI2 + 4 (1.155∆KII )2 . + ∆KMM = 2 2 8

(4)

They stated that the current crack growth rate can be obtained from the crack growth curve for mode I. In some cases, delayed deflection after a short distance of coplanar shear mode driven propagation was observed, see for example Gao et al. [37]. Opposite (or in addition) to what was said to the data from reference [33], the coplanar growth was observed in the near threshold region. In the discussion of the obtained results a fourth item with strong influence on the fatigue crack growth behaviour is outlined: the crack closure mechanism has to be considered. Besides plasticity induced crack closure the roughness induced crack closure becomes more important in high mode-mixity situations. An interlocking of the crack flanks due to roughness induced crack closure leads to a crack tip shielding of the mode II component. This phenomenon is especially observed for low stress intensity factor ranges. For large ranges, however, friction and wear of the crack flanks together with plastic deformations at the fracture surface may remove the roughness induced crack closure and with it the mode II shielding. Moreover, a decrease of the roughness induced crack closure also affects the mode I effective (closure-free) ranges because there is a mode I component associated to friction along crack flanks. As it is the case in mode I fatigue crack growth, crack closure and mean stress effect are closely related phenomena and therefore all effects should be discussed against the background of the effect of the mean stress or stress ratio, R = Kmin /Kmax . In a recent investigation by Highsmith [31], the abrupt change of modemixity was not only performed from pure mode I to proportionally combined mode II and mode I loading. Highsmith used mode I pre-cracked thin-walled

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tube specimens with the pre-crack oriented perpendicular to the specimen axis. With a testing equipment able to apply cyclic tension and torsion independently of each other, real long-ranging non-proportional load sequences can be applied. The material was the nickel-based super alloy Inconel 718. It was tested at room temperature and the maximum stress intensity factors at √ the pre-crack front were in the order of 10 to 25MPa m. Positive stress ratios were applied, R = 0.1 in most cases, with some results also for R = 0.6. In his accompanying study on proportional tension-torsion loading, he discovered the transition from the maximum tensile stress dominated crack deflection to the nearly coplanar maximum shear stress dominated crack growth occurring at a mode-mixity of 44◦ . The situation of two competing criteria may be visualised in an interaction diagram according to Figure 1. In non-proportional mixed-mode loading the mode-mixity varies during a cycle. Highsmith presents his results in diagrams showing the deflection angle as a function of the mode-mixity. Especially the results for constant torsion and cyclic tension, see Figure 2, show that none of the criteria mentioned so far is able to predict the correct deflection angle. The deflection angle cannot be explained by either the maximum tensile or the maximum shear stress criterion for any mode-mixity occurring during the cyclic loading. Highsmith also tested a few through-crack round specimens. The main difference between a thin-walled tube and the round shaft with a through(pre-)crack is that the mode-mixity varies considerably along the crack front in the latter case. This issue inserts a sixth factor of influence on the nonproportional (but also on the proportional) mixed-mode fatigue crack growth: at the different positions along the crack front, the crack may obey to different

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criteria and therefore it may follow different deflection angles. The final fracture surface appears either strongly warped or it shows a step-like joining of cracks initially grown in different planes according to different criteria. Moreover, the out-of-plane constraints differs between the specimens. In the thin-walled tube a plane-stress situation dominates and in the through-crack round specimen a plane-strain situation prevails (especially in the centre of the specimen). If the starter crack is cut in a direction which d eviates from being perpendicular to the specimen axis, mode III non-proportional conditions may be investigated, too. Observations on this loading case are unaware to the present authors. Highsmith concludes that there is no single formulation at hand to predict crack direction for all cases. A best-practise approach is proposed, which is based on the stress intensity factors at an infinitesimally small kink crack’s tip, k1 and k2 : ϑ k1 = cos 2 k2 =

µ

ϑ 3 − KII sin ϑ KI cos 2 2

¶

2

ϑ 1 cos (KI sin ϑ + KII (3 cos ϑ − 1)) 2 2

(5) (6)

In the Eqs. (5) and (6), ϑ is the kink angle. Since crack deflection angles fall between the angles of maximum kink tip stress intensity factor, ki,max , and the maximum kink tip stress intensity factor range, ∆ki , a crack driving force combining the influence of both was suggested: 1−w ∆k¯i = ∆kiw · ki,max

(7)

A fitting parameter, w, appears in Eq. (7), which is allowed to take different values for the two cases, i = 1 (tensile stress dominated), and i = 2

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(shear stress dominated). A transition criterion similar to what is shown in Fig. 1 completes the approach. Highsmith formulated his concept against the background of his overview on published results. He took some advantage of previous work by Hourlier et al. [38, 39] who also used the kink angle’s stress intensity factors for predicting the deflection angle and, alternatively, a criterion based on the maximum crack growth rate calculated using da/dn-∆KI data. The reference to two more summary papers by Liu [40] and Bold [41] is given here, both focussing on rolling contact fatigue which is a special type of non-proportional mixed mode loading. Some of these papers are discussed in a later paragraph; the others are referenced here for an attempt towards – however, unreachable – completeness, see references [42] to [50]. Highsmith did not intend to compare crack growth rates when applying Eq. (7) in connection with a crack growth law for pure-mode loading. He only restricted his work to the description of deflection angles which, however, is an important pre-requisite for any fatigue crack growth modeling. In the following amendments to Highsmith’s overview are listed. 3.2. Effect of interposed mode II load cycles on mode I fatigue cracks It is well known that mode I overloading leads to a crack retardation or arrest [51, 52] under pure mode I loading. Investigating the effect of mode II overloads – as a special case of non-proportional loading – Nayeb-Hashemi and Taslim [53] found that in contrast to mode I overloads, mode II overloads give rise to short time crack growth acceleration with no retardation. In contrast to their observations, Gao and Upul [54], who applied ten overload cycles instead of only one, Srinivas and Vasudevan [55], Sander and Richard 12

[56] as well as Dahlin and Olsson [57, 58, 59] observed a retardation of the mode I crack after the mode II overload, as shown in Fig. 3. Based on the results shown in Fig.3, Dahlin and Olsson [57, 58, 59] stated that there should be a certain threshold value for the mode II overload, below which no decrease of the subsequent mode I crack growth rate occurs. They identified the main reason for the decrease in crack growth rate in crack closure. Due to the mode II displacement of the crack-surface, the crack surface roughness causes mismatch between the upper and lower crack faces. Increased crack closure stresses decreases effective ranges and fatigue crack growth rates. Changing the overload from purely mode II to mixed-mode, Sander and Richard [56] and Srinivas and Vasudevan [55] observed that, by fixing the size of the transient plastic zone, the crack growth rate retardation decreases with an increasing mode II portion. Dahlin and Olsson [60] extended their study to periodic mode II cycles during mode I loading. They stated that ∆KI , the R-ratio of the mode I loading, the magnitude of the mode II load cycles and the frequency of the mode II cycles, i.e. the number of mode II load cycles per mode I load cycles, are the main parameters concerning the influence of the sequential mode II cycles on the crack growth behaviour. They found two mechanisms resulting from the mode II load cycles: mode II-induced crack closure, which results in a reduction of the crack propagation rate, and a mechanism that increases the growth rate temporary at every mode II load. For a crack with high mode I R-ratios, the crack is open during the whole mode I load cycle. The decreasing effect of the mode II load on the crack growth rate vanishes. They also stated that the crack path deviation from the mode I crack path is only significant at high mode II frequencies,

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as shown in Fig. 4. As a limiting case of the aforementioned loading Doquet and Pommier [61] studied sequentially applied mode I and mode II load cycles in ferriticpearlitic steel. They observed coplanar growth for mode-mixities Φ between 45◦ and 76◦ , Fig. 5. For a mode-mixity of 76◦ they found that the crack growth rate is the sum of each contribution. However, for higher modemixities they found a synergetic effect resulting in a faster growth compared to the simple sum of both contributions. This effect was also observed by Wong et al. [62, 63] for a rail steel. Doquet and Pommier explained the different bifurcation behaviour between their and Wong’s results, who observed deflection to tensile dominated crack propagation at mode-mixities below 63◦ , by the different loading conditions in both tests. 3.3. Effect of superimposed static component on fatigue crack growth Another often investigated special case of non-proportional loading is the superposition of a static and a cyclic load component. Here, the following kinds of non-proportional loading can be distinguished: 1. Cyclic mode I loading and static mode II load 2. Cyclic mode II loading and static mode I load 3. Cyclic proportional loading and static mode I or II load 4. Cyclic mode I or II loading and static Mixed-mode load Plank and Kuhn [64] studied the crack growth behaviour under load cases (1), (2) and (4) on different aluminum alloys using compact tension shear specimen [65]. Within their studies they distinguish two modes of propagation, as shown in Fig. 6: a mode I controlled deviation, named tensile mode, 14

and a coplanar mode II controlled crack growth, named shear mode. They stated that stable shear mode crack growth can only be observed under cyclic mode II with superposed static mode I, while for cyclic mode I with superposed static mode II, as well as for cyclic mixed-mode with superposed static mode I only, stable tensile mode I crack growth is observed. Moreover, once the crack has turned into tensile mode it will not reverse to shear mode any more. Based on their experimental findings, they stated that two loading parameters and one material-specific parameter are decisive for the kind of crack propagation. For the initiation of shear mode crack growth, the effective range of the mode II stress intensity factor ∆KII,eff must exceed a certain threshold value ∆KII,th,sm , which is material-specific. Furthermore, the ∆KII -value of the starter crack has to be larger than the mode I range ∆k1 at the infinitesimally short kink crack’s tip. They also found ∆KII,th,sm to be indirectly proportional to the average grain size. Concerning the crack growth rates they found that under cyclic mode II a static mode I load leads to a significant increase in crack growth rate, because of the reduction of crack surface contact. This result is confirmed by several other researchers, e.g. [66, 67, 68, 69]. Moreover, at identical cyclic stress intensity factors, the crack growth rate is higher, if the crack is growing in shear mode than in tensile mode, what would be in accordance with Eq. (2). The introduction of a static mode II component on cyclic mode I loading leads to a decreasing crack growth rate and therefore to longer fatigue lives. Plank and Kuhn [64] explained this behaviour by increasing friction between the two crack faces.

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3.4. Effect of phase shift on fatigue crack growth As stated in the introduction, the type of the loading, i.e. proportional or non-proportional loading, has a significant influence on the total live. A variety of such experimental results have been published by Brüning et al. [70, 71, 72] concerning the influence of the phase shift on the crack growth life solely. In these experiments thin-walled tubes, made of fine grained steel S460N and aluminium alloy AlMg4.5Mn, were tested under proportional and non-proportional (stress-controlled) tension and torsion loading. For both types of materials slightly longer fatigue crack growth lives were found for non-proportional loading compared to proportional loading, as shown in Figures 7 and 8 for a multiaxial ratio f (ratio of gross section nominal normal stress Sgr and gross section nominal shear stress Tgr ) equal to 1.6. The found observations correspond well with the observations made by Sonsino [14]. For the specimens investigated by Brüning et al. [70, 71, 72] the wall thickness increased considerably from the smallest net cross-section towards the clamping region due to shouldering. This mild notch effect enforced the fatigue crack to search its path in the region of the small cross section, thus increasing the mode II contribution by torsion compared to a tube specimen with constant wall thickness. A seventh influence parameter – the component’s geometry in general – has therefore been identified as affecting the non-proportional (and the proportional) mixed-mode fatigue crack growth. 4. Fatigue crack growth prediction In section 3.1 equations for the determination of the crack growth direction under non-proportional loading based on linear-elastic fracture me16

chanics were reviewed. However, the applicability of these equations for the determination of the fatigue crack growth rate was not shown. In general, validated and accepted mixed-mode hypotheses for application with especially non-proportional crack tip conditions are missing. Specifying simply the sum of crack growth rates from each mode contribution as suggested by Doquet and Pommier [61] as well as Döring [1] is certainly not the final end of the development. A way out of purely empirical descriptions of experimental finding for special loading situations might be worth an investigation. A numerical study on the feasibilities of the ∆J-integral evaluating nonproportional cyclic loading conditions has been presented by Hertel et al. [73]. The integral is defined by Z ∆ε Z ij ∆J =

Z ∆σij d∆εij dy −

Γ

0

∆σij nj

∂ (∆ui ) ds ∂x

(8)

Γ

with (0)

(0)

(0)

∆σij = σij − σij ;∆εij = εij − εij ;∆ui = ui − ui

The upper index (0) denotes a reference state from which the stress σij , strain εij and displacement ui field variables must be evaluated. The integration path is called Γ and nj denotes the unit normal on the path. For elastic material the ∆J-integral would have the meaning of an energy release rate. A fatigue crack growth criterion based on the ∆J-integral therefore can be considered as being closely related to a maximum rate criterion as used by Hourlier at al. [38, 39]. All calculations have been carried out for a centre cracked specimen. A finite element model has been build up to deal with the mixed mode I and mode II normal and shear stress loading. A kinematic hardening plasticity model 17

of the Besseling type approximating the materials cyclic stress-strain curve was used. Crack closure was neglected. The ∆J-integral has been calculated by numerical integration for three different paths surrounding the crack tip. Nearly path-independent results were obtained. For non-proportional loading the load reversal points (this means the reference states) are automatically detected by the following algorithm. If the increase in ∆J during the integration of equation (8) becomes negative it is supposed that the last equilibrium state was a load reversal point and the reference variables denoted with the upper index (0) in equation (8) are redefined with stress, strain and displacement values of this last equilibrium state. Afterwards, the integration will be continued. The latter approach satisfies the monotonic increase of ∆J for every equilibrium state. In addition to ∆J the mode separated parts ∆JI and ∆JII are of special interest in a mixed mode analysis. According to Ishikawa et al. [74] JI and JII can be derived by splitting the stress, strain and displacement values of the crack tip field into a symmetric (mode I) and antisymmetric (mode II) part and then feeding this symmetric and antisymmetric field variables separately into equation (8). In case of proportional loading, Figure 9a, the reference states automatically determined by the proposed algorithm agree with the reversal points at minima and maxima of the external load. Due to plasticity, the separated values of ∆JI and ∆JII are larger than the values for pure mode I or mode II loading, but they sum up to the value of ∆J. For the 90◦ out-of-phase load case, Figure 9b, the predicted load reversal points for ∆JI and ∆JII are shifted relative to the extremes of the external loads σ and τ . This is a result

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of plasticity and the rounded σ-ε- and τ -γ-hysteresis loops. The result for ∆J is close to ∆JI because the ratio of the external loads leads to a mode I dominated crack tip field. ∆JI and ∆JII are different from pure mode I or mode II loading and cannot simply be added to get ∆J. In Figure 9c the external loads have different frequencies. The predicted reference states (reversal points) for separated mode I and mode II loading show the same frequency as the external loads but are shifted relative to the maxima and minima of σ and τ . The load case is again mode I dominated and the result of ∆J is close to ∆JI . With this algorithm it is possible to calculate the nearly path independent crack tip parameter ∆J, based on finite element calculations, even for non-proportional cyclic loading. Due to the practical path independence, the parameter should maintain its sound physical basis to describe fatigue crack growth even in non-proportional loading situations. However, further investigations on the definition of a load cycle and an appropriate mixed mode criterion for non-proportional loading including crack closure and direction of crack propagation have to be performed. 5. Conclusion The fatigue crack growth under non-proportional mixed-mode loading conditions at the crack tip or crack front is depending on many influence factors: • Mode mixity. A tendency is observable that increased mode mixity can lead to a shear dominated fatigue crack growth instead of a tensile stress dominated fatigue crack growth which is common experience. 19

• Material. However, different materials show a different sensitivity to display shear dominated fatigue crack growth. • Cyclic plasticity. The cyclic plastic deformation occurs as shear stress driven dislocation motion. With increasing cyclic plastic deformation the shear dominated fatigue crack growth becomes more important • Crack closure. The cyclic plastic deformation is also the origin of the plasticity induced crack closure phenomenon. Especially in nonproportional mixed mode situations, the roughness and friction induced crack closure appears and interacts with the plasticity induced crack closure. This interaction gives rise to both accelerating and decelerating fatigue crack growth depending on the applied amplitudes. • Mean stress. As the mean stress dependence of fatigue crack growth rates may be explained by crack closure to a large extent, all phenomena of non-proportional mixed mode fatigue crack growth are expected to show a strong dependence on the applied mean stresses. • Geometry. The geometry of a structure may force a fatigue crack to grow in a (weak) cross section area although a (material dependent) mixed mode hypothesis might demand a different path. • Variable mode mixity along crack front. In three-dimensional structures the mode-mixity is generally changing along the crack front. The direction of the crack growth increment along the crack front, according to a criterion identified under plane stress or strain conditions, 20

might lead to incompatible crack surface geometries. Stepped and fissured cracks might evolve. These influence factors are severely interrelated. Therefore, a study of individual factors is hard to achieve. Some successful explanations of experimental findings have been published, however, a generally accepted and validated formulation of a crack driving force parameter has not yet been identified. Acknowledgement The authors express their gratitude to the Deutsche Forschungsgemeinschaft (DFG) for providing financial support under grant VO 729/13-1. References [1] R. Döring,

Zum Deformations- und Schädigungsverhalten met-

allischer Werkstoffe unter mehrachsig nichtproportionalen zyklischen Beanspruchungen, Institut für Stahlbau und Werkstoffmechanik, TU Darmstadt, Report 78, 2006, in German. [2] S.M. Tipton and D.V. Nelson, Advances in multiaxial fatigue life prediction for components with stress concentrations, Int. J. Fatigue 19(6) (1997) 503–516. [3] J. Hoffmeyer, R. Döring, T. Seeger, M. Vormwald, Defomation behaviour, short crack growth and fatigue lives under multiaxial nonproportional loading, Int. J. of Fatigue 28 (2006) 508–520.

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31

List of Figures 1

Schematic visualisation showing the competition of the maximum tensile and the maximum shear stress criterion. . . . . . 33

2

Loading scheme (a) and crack deflection angles over the range of mode-mixity for constant torsion with superimposed cyclic tension (b) [31]. Line end symbol geometries correspond to each other in figures (a) and (b). . . . . . . . . . . . . . . . . 33

3

Change in mode I crack growth rate due to a single mode II √ load cycle of three different magnitudes (KI = 20MPa m, R = 0.1) [59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4

Crack path of a specimen under cyclic mode I loading with gradually increasing mode II frequency [60]. . . . . . . . . . . 34

5

Crack path development in sequential mode I and mode II √ loading with KII = 20MPa m [61]. . . . . . . . . . . . . . . 35

6

Different modes of stable crack growth [64].

7

Crack growth lives of steel specimens for multiaxiality ratio f = 1.6 [71].

8

. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Crack growth lives of aluminium specimens for multiaxiality ratio f = 1.6 [72].

9

. . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . . . 37

∆J-integral for proportional and non-proportional cyclic loading [73]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

32

D K

II

M a x im u m s h e a r s tre s s c rite rio n

lin e s o f c o n s ta n t m o d e m ix ity , ta n F

= K

II

/K I

lin e s o f c o n s ta n t g ro w th ra te s M a x im u m te n s ile s tre s s c rite rio n

D K I

Figure 1: Schematic visualisation showing the competition of the maximum tensile and the maximum shear stress criterion.

K

M P a

II

m

8 0 °

E x p e rim e n ta l lo a d in g c o n d itio n s

6 0 ° 4 0 ° D e fle c tio n a n g le J

2 0

1 0

K 1 0

(a )

2 0

M P a

2 0 ° 0 °

m

2 0 °

4 0 °

-2 0 ° -4 0 ° -6 0 °

I

M a x im u m s h e a r s tre s s c rite rio n

-8 0 °

M a x im u m te n s ile s tre s s c rite rio n

6 0 °

M o d e m ix ity F 8 0 °

E x p e rim e n ta l re s u lts o p e n s y m b o ls : rig h t c ra c k c lo s e d s y m b o ls :le ft c ra c k

(b )

Figure 2: Loading scheme (a) and crack deflection angles over the range of mode-mixity for constant torsion with superimposed cyclic tension (b) [31]. Line end symbol geometries correspond to each other in figures (a) and (b).

33

K

C ra c k g ro w th ra te d iffe re n c e

m m /c y c le K

0

II

= 2 0 M P a .m

0 .5

3 0 M P a .m

0 .5

4 0 M P a .m

0 .5

-2 0

II

o n e (o r fe w ) m o d e II o v e rlo a d s

K c y c lic m o d e I

D K -4 0

2 0

4 0

6 0

C ra c k le n g th

I

= 2 0 M P a .m

I

0 .5

m m

Figure 3: Change in mode I crack growth rate due to a single mode II load cycle of three √ different magnitudes (KI = 20MPa m, R = 0.1) [59].

D ire c tio n o f K

II

lo a d in g

C ra c k

Figure 4: Crack path of a specimen under cyclic mode I loading with gradually increasing mode II frequency [60].

34

K

II

s e q u e n tia l m o d e I a n d m o d e II K

M o d e I p re c ra c k N o tc h

D K I

I

= D K

D K I= 0 .7 5 D K

D K I= 0 .5 D K II

II

D K I= 0 .2 5 D K

II

II

2 0 0 m m

Figure 5: Crack path development in sequential mode I and mode II loading with KII = √ 20MPa m [61].

T e n s ile m o d e g ro w th

M o d e I c o n tro lle d S ta tic m o d e I J

p re c ra c k

C y c lic m o d e II M o d e II c o n tro lle d S ta tic m o d e I

S h e a r m o d e g ro w th p re c ra c k

C y c lic m o d e II

Figure 6: Different modes of stable crack growth [64].

35

120 prop.

110

regr. prop.

45° 90°

S

gr

[MPa]

100 f=S

gr

90

/T

gr

=

1.6

80 70

60 4 10

5

10

5

2*10

Nc

gr

Figure 7: Crack growth lives of steel specimens for multiaxiality ratio f = 1.6 [71].

36

60 50 40

S

gr

[MPa]

30

20 prop. regr. prop.

45° 90° 10 4 10

f=S

/T

gr

=1.6

gr

10

Nc

5

6

10

gr

Figure 8: Crack growth lives of aluminium specimens for multiaxiality ratio f = 1.6 [72].

37

a )

Ö 3 t s

p ro p o rtio n a l b )

Ö 3 t s

9 0 o

p h a s e s h ift

c )

Ö 3 t s

fre q u e n c y ra tio 1 :2

Figure 9: ∆J-integral for proportional and non-proportional cyclic loading [73].

38

List of Tables 1

Comparison of fatigue lives under proportional and sinusoidal non-proportional 90◦ out-of-phase loading (collected mainly by Döring [1]).

. . . . . . . . . . . . . . . . . . . . . . . . . . 40

39

Table 1: Comparison of fatigue lives under proportional and sinusoidal non-proportional 90◦ out-of-phase loading (collected mainly by Döring [1]).

N90◦ /N0◦

N90◦ /N0◦

Material

Reference

under stress control under strain control C39

0.1

-

Tipton and Nelson [2]

S460N

10–20

0.21–0.28

Hoffmeyer [3]

S460N

2–3

0.3

Sonsino [4]

CK45

0.4

-

Sonsino [5]

SAE1045

5

-

Pan et al. [6]

25 CrMo4

0.6

-

Grün et al. [7]

30 CrNiMo8

1.1

0.4

Sonsino[4]

X6CrNiNb18-10 -

0.54–1.4

Hoffmeyer[3]

X6CrNiTi18-10

3.8

0.5

Hug [8]

X10CrNiTi8-9

1.1

0.5

Sonsino [4]

AlMg4.5Mn

1

0.77–0.9

Hoffmeyer [3]

Al6061 T6

-

0.2

Xia and Ellyin [9]

AlMgSi1

-

0.5

Ahmadi et al. [10]

AZ91

-

3–4

Renner [11]

Ti6Al4V

-

0.1

Nakamura et al. [12]

SAE1045

-

0.17–2.81

Fatemi and Socie [13]

40

Highlights Literature on fatigue crack growth under non-proportional loading is reviewed. Seven factors influencing fatigue crack growth under these conditions are identified. The current best practise modelling approaches are presented.