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\ \ \ \
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200
1
1
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10000
number of blocks to failure Dots: experiments, curves 1 to 8: calculations: 1: hot spot stress approach applied with critical plane shear stress criterion, 2: hot spot stress approach applied with critical plane normal stress criterion, 3: old geometry - local stress approach applied with critical plane normal stress criterion, 4: old geometry - local stress approach applied with max. principal normal stress criterion, 5: old geometry - local stress approach applied with critical plane shear stress criterion, 6: new geometry - local stress approach applied with critical plane normal stress criterion, 7: new geometry - local stress approach applied with mx. principal normal stress criterion, 8: new geometry - local stress approach applied with critical plane shear stress criterion. Fig. 22. Comparison between calculated and experimental fatigue lives.
The hot spot stress approach in conjunction with the criterion critical plane - normal stress shows a slight trend to conservative calculations at lower load levels. The use of the constant amplitude ^-A^ curve within the hot spot stress approach supported by the criterion critical plane shear stress seems to be unsuitable to calculate fatigue lives for this weld detail. Lifetimes predicted with the local stress approach are generally very conservative here. The predicted fatigue lives for the new geometry are slightly shorter than the ones calculated for the old geometry. However, the advantage of the new geometry compared to the old geometry is that the weld root is no longer failure-critical. Good agreement exists between the two criteria critical plane normal stress and maximum principal stress. But this was to be expected in a predominantly locally uniaxial situation.
Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 39
Validation of calculated fatigue results Calculated fatigue lives or damage sums are influenced by several factors which are worth to be discussed in the following: As mentioned previously, the local stress approach is applied in practice in connection with the submodelling technique. Within the latter, the usual procedure is to take displacements (and rotations) from a coarse mesh as boundary conditions for the finely meshed submodel. Attention should be paid to the requirement that the stiffnesses of the models do not differ too much. For example, a very stiff submodel will yield much higher stresses under applied displacements than a compliant submodel would. In the current investigation we found differences in resulting bending and torsion moments in the tube in the order of 5%. The higher stresses occurred for the stiffer new design variant. This is the reason for the lower predicted lives for this design. Some inherent uncertainties are obviously linked with the submodelling technique itself. Calculated results also depend on the multiaxial fatigue criterion used. In general nonproportional loading cases, criteria based on conventional equivalent stresses (Tresca, von Mises) are inappropriate. In special cases with locally proportional stress situation, they might work to a limited extent as well. The hypothesis of the effective equivalent stress introduced by Sonsino and Klippers [21] showed good predictions for welded flange-tube joints from fine-grained steel FeE 460 under bending and torsion with constant and variable amplitudes. As mentioned above, these cases - dominating uniaxial stresses for example - have some practical relevance. In case of doubts on local stress states, critical plane criteria should be preferred: Critical plane - normal stress (mode I) or critical plane - shear stress (mode n+m). The first one should normally be used when normal stresses are dominant. The second criterion is appropriate in the case of dominating shear stresses. Scatter of fatigue lives is an unavoidable matter of fact. It should be taken into account when comparing calculated and experimentally determined lives. The number of tested components here is quite low; thus, even mean values of lives are subject to uncertainties. On the other hand, the baseline stress-life curves for prediction, Figs. 7 and 15, are only mean curves in a scatter band. Within the local stress approach, the ratio T for probabilities of survival of 10% to 90% (in stresses) is 1.5 for normal stresses and 1.39 for shear stresses, respectively [19, 20]. Thus, a factor of 2 to 3 in lives can easily arise from this fact and can be qualified as minor inaccuracies. At last, the real fatigue life mainly depends on the geometrical form and quality of a single weld. It is possible to model the real welded form or geometry from a drawing. In this report the geometry of already existing welds is modelled. However, the plane surface of the weld and the notch radius 1mm are idealisations. But geometry and stress distribution depend on each other. Therefore, the element with maximum damage sum is not the only one to be looked at. Other elements with similar damage sums should also be regarded as failure-critical as well. These failure-critical locations can be determined quite accurately using the local stress approach. Using the hot spot stress approach, only one critical location on the tube has been detected. This location is verified and one more location has been detected with the local stress approach. The corresponding finite element results for the older geometry verified the weld root as failure-critical. This is in good agreement to the experimental results. After modification of the tube to the new geometry the weld root is no longer failure-critical. Figure 23 shows a tested component with new geometry. A further effect is shown: The crack initiation does not start from the weld at all. The notch at the tapering of the tube far away from the weld undercut is failure-critical. Finally, this result prevented the presentation of experimental fatigue lives for the new design (for the weld undercut). On the one hand, it is possible to calculate lives for this location based on local stresses or strains and on cyclic
40
a SAVAIDIS ETAL
material data, but this is not the subject of the current paper. On the other hand the result reveals some aspects of the problems when dealing with life prediction in an industrial environment: As the tapered tube is known from experience to survive quite a number of truck lives further investigations on the component have minor priority. Nevertheless, gaining experience with life prediction techniques is the prerequisite to apply them in design process.
Tapering of tube Forged arm
Failure-critical location calculated with hot spot approach and local approach
Fig. 23. Experimental tested new geometry, crack initiation at tapering of tube
CONCLUSIONS The mechanical behaviour and fatigue life of a thin-walled tube joined to a forged component by fillet welding has been investigated. The component is loaded by nonproportional random sequences of bending and torsion as measured during operation. The stresses in the welded structure have been calculated using finite element analysis. The fatigue life has been determined theoretically, by means of the hot spot and local stresses in conjunction with the critical plane approach, and experimentally. Though the experimental data base is quite narrow, basic trends can be derived from the present investigation concerning the possible field of application and capability of the theoretical approaches calculating fatigue life of vehicle components. 1. A coarse finite element model of the component has been created in accordance with the n w guideline for application of the hot spot stress approach. Comparison of numerically and experimentally determined end-results (fatigue lifetimes) shows that the coarse model is suitable to determine lifetimes with the hot spot stress approach, if a reliable hot spot stress-life curve at constant amplitude loading is existing for the detail investigated. Because of the coarse mesh, the geometry of the weld is not modelled in detail. For instance, in this investigation only one failure-critical location has been detected. 2. If experimental results are missing, appropriate S-N curves from publications, e.g. the nW guideline or Eurocode can be used. In cases of multiaxial loading causing normal and shear stresses, attention must be paid to the slope of the S-N curve used, since various suggestions are reported in the literature. The major advantage of the hot spot stress approach is a relatively low expense to model and calculate.
Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 41
3. The coarse finite element model is unsuitable for usage in conjunction with the local stress approach. Therefore, a submodel of the failure-critical detail has been created here. Due to the finer mesh, the number of elements is increasing. The results obtained here show that weld geometry optimisation is only possible with the local stress approach. 4. In general, higher numerical expense does only make sense, if failure-critical locations should be investigated more exactly. In the majority of engineering applications, a finer mesh or more detailed model is often an unrealisable option due to technical and commercial reasons. 5. To calculate fatigue lives in accordance with the local stress approach, no experimental input data are required, when using the universal local a-N curve.
REFERENCES 1. Hobbacher, A. (1996). Recommendations for Fatigue Design of Welded Joints and Components. Document Xm-1539-96/XV-845-96, International Institute of Welding, Paris. 2. Radaj, D. (1990). Design and Analysis of Fatigue Resistant Welded Structures. Abingdon Publishing Cambridge. 3. Radaj, D. and Sonsino, C M . (1998). Fatigue Assessment of Welded Joints by Local Approaches. Woodhead Publishing Limited, Cambrigde. 4. Maddox, S.J. (1991). Fatigue strength of welded structures, 2""^ edition, ISBN 1 85573 0138, Woodhead Publishing. 5. Bo vet-Griff on, M., Ehrstrom, J.C., Courbiere, M., Bignonnet, A., Thomas, J.J., Puchois, J.P., Rethery, S. and Liennard, C. (2001). Fatigue assessment of welded automotive aluminium components using the hot spot approach. Proc. 8^ INALCO, 28-30.03.2001, Munich. 6. Fayard, J.L, Bignonnet, A. and Dang Van, K. (1996): Fatigue design criterion for welded structures. Fat. Fract. Engng. Mater. Struct. 19, 723. 7. Savaidis, G. and Vormwald, M. (2000). Hot-spot stress evaluation of fatigue in welded structural connections supported by finite element analysis. Int. J. Fatigue 11, 85. 8. Niemi, E.J. (1995). Recommendations Concerning Stress Determination for Fatigue Analysis of Welded Components. Document Xm-1458-92/XV-797-92, International Institute of Welding, Abingdon Publishing, Cambridge. 9. Niemi, E.J. (2001). Structural Stress Approach to Fatigue Analysis of Welded Components. Document XHI-1819-00/XV-1090-01/Xm-WG3-06-99. International Institute of Welding, Abingdon Publishing, Cambridge. 10. Vormwald, M., Purkert, G. and Schliebner, R. (2000). Lebensdauerbewertung einer NFGFahrerhausschwinge mit dem Programm FALANCS. Technical report, Weimar. 11. Sonsino, C M . (1994). Festigkeitsverhalten von SchweiBverbindungen unter kombinierten phasengleichen und phasenverschobenen mehrachsigen Beanspruchungen. Materialwissenschaft und Werkstofftechnik 25, 353. 12. Sonsino, C M . (1995). Multiaxial Fatigue of Welded Joints Under In-Phase and Out-ofPhase Local Strains and Stresses. Int. J. Fatigue 17, 55. 13. N.N.: FALANCS. Version 2.9j, LMS Durability Technologies GmbH, Kaiserslautern. 14. Fatemi, A. and Socie D.F. (1988). A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fat. Fract. Engng. Mater. Struct. 11, 149. 15. Fatemi, A. and Kurath, P. (1988). Multiaxial fatigue life predictions under the influence of mean stresses. J. Engng Mat. Techn. 110, 380. 16. Maddox, S.J. and Ramzjoo, G.R. (2001). Interim fatigue design recommendations for fillet welded joints under complex loading. Fat. Fract. Engng. Mater. Struct. 24, 329.
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17. Sonsino, C M . (1999). Overview of the State of the Art on Multiaxial Fatigue of Welds. ESIS, Publication 25, Elsevier. 18. Haibach, E. (1989). Betriebsfestigkeit. Verfahren und Daten zur Bauteilberechnung. VDIVerlag, Dtisseldorf. 19. Olivier, R. (2000). Experimentelle und numerische Untersuchungen zur Bemessung schwingbeanspruchter Schweifiverbindungen auf der Grundlage des ortlichen Konzeptes. Phd-thesis, TU-Darmstadt. 20. Olivier, R. and Amstutz, H. (2000). Fatigue strength of shear loaded welded joints according to the local concept. Materialwissenschaft und Werkstofftechnik 32. 287. 21. Sonsino, C M . and Kueppers M. (2001). Multiaxial fatigue of welded joints under constant and variable amplitude loadings. Fat. Fract. Engng. Mater. Struct. 24, 309.
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
43
MULTIAXIAL FATIGUE ASSESSMENT OF WELDED STRUCTURES BY LOCAL APPROACH
Florence LABESSE-JIED\ Bruno LEBRUN\ Eric PETITPAS^ and Jean-Louis ROBERT^ LERMES, Blaise Pascal University, lUT, B.P. 2235, 03101 Montlugon Cedex-France GIATIndustries, CRET/MOD, 7, Route de Guerry, 18023 Bourges-France
ABSTRACT The moving of vehicles on chaotic ground induces dynamic multiaxial loading on structures and mechanical components. As a consequence, early fatigue damage occurs especially in structural details such as notched areas and welded parts. A multiscale approach has been developed to design the structures against fatigue, starting from the dynamics of the vehicle and ending with the calculation of structural details using a local approach to assess the fatigue life. The methodology of the local approach developed is introduced. The evaluation of the prediction capability of this local approach is described. Finally, the application to the fatigue life assessment of welded elements is presented and compared to experimental results. The major parameters of the weld geometry that govern the material resistance against fatigue are pointed out. They concern geometrical features depending on the quality of the weld. Their influence on the weld durability is outlined and the way the proposed assessment method accounts for them in a quantitative manner is detailed. KEYWORDS Welding, fatigue life, local approach, multiaxial loading, damage cumulation
INTRODUCTION Welding is a technology commonly used nowadays in many applications of the mechanical industry because it allows the making of complex structures from simple components as bars or plates. Engineering designers and researchers have thus developed different evaluation techniques for ensuring fatigue resistance of such welded structures [1]. Structural details and welded joints are assessed from the fatigue point of view in design codes principally on the basis of the nominal stress range. Practically they are classified into many different classes depending on the geometry and the applied loading. A fatigue strength curve is attributed to each class, the denomination of which corresponds to the characteristic fatigue strength at 5.10^ cycles [2]. Most design codes allocated to fatigue assessment of welded structures refer to this principle. Figure 1 depicts the conventional Wohler S-N curves generally consulted as referenced standard quality fatigue properties. In the case where the nominal stress can not be easily defined within the fatigue-prone welded structure, the proof against fatigue uses either hot-spot or structural stresses [1,3]. The
44
E LABESSE-JIED ET AL
hot-spot stress is obtained by the hnear extrapolation of the stress distribution to the weld toe (Fig. 2). The location of this considered stress is due to the fact that it is generally the critical point of the weld, i.e. the crack initiation site. The structural stress is the measured or Finite Element calculated stress obtained at a given distance afar from the weld toe. Because of the stress concentration generated by the geometrical discontinuity at the weld toe and directly related to the transition radius, this structural stress is in general higher than the hot-spot one and leads to a more conservative dimensioning.
Constant amplitude fatigue limit
1e7
logN
Fig. 1. Wohler S-N curves for classified weld geometries (from [2])
a, : structural stress Chi • hot-spot s^ess
a^:
f:
maw mum notch stress CT, - f (weld geometry, loading mode)
,cr, structural stress •distribution (strain gauge or FE)
notch stress distnbuton (F£) Stress-ccncerrtration factor: K, =
/^' linear extrapotation
weld-toe 1
^ = f (tNckness, gauge length) • 1 to 2 mm
Fig. 2. Conventional design stresses in a weld joint (from [1,3])
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
45
The advantage of these approaches is the relatively easy procedure for a design purpose. However two main drawbacks can be retained. The effective maximum stress existing at the weld toe is not the referred stress used for the assessment of the weld despite the fact indeed that this value is the origin of the fatigue damage. The second difficulty consists in the uniaxial stress states used for evaluating the weld durability. As a matter of fact the notch effect constraints existing at the weld toe and the possibly multiaxial loading of the structure generally induce multiaxial stress states. In the case of rotating principal stress directions, a multiaxial fatigue criterion is the only way to properly account for the effective influence of such stresses in fatigue. PRESENTATION OF THE PROPOSED LOCAL APPROACH This approach is developed for the design of welded structures of all-terrain vehicles. When such a vehicle is moving on chaotic ground at high speed, its structure and mechanical components are subjected to vibrations capable of inducing fatigue cracks especially in notched areas and welded parts. This is the reason why many all-terrain vehicles built all over the world suffer from progressive cracking. Controlling the fatigue dimensioning of structures under complex loading becomes thus an essential condition in the design process. Furthermore, controlling the fatigue strength of welded parts constitutes an important technico-economic advantage. The aim of this section is to present the approach developed and its application in order to ascertain fatigue strength under complex loading. Since the fatigue phenomenon is very local, whereas the phenomenon generating it is vehicle-wide, the design approach must necessarily be multiscale. The approach developed results in 3 steps. Starting from the terrain, the first step consists in simulating or testing a vehicle when rolling, in order to make it possible to determine the internal loads. Figure 3 shows for instance the forces induced by the suspension on the chassis due to the moving of a 6-wheeled vehicle on a training ground at high speed. As a matter of fact, the ratio between forces coming from the vehicle's suspension shows that the loadings are not proportional. The structure is actually subjected to random non-proportional multiaxial loading. These loads are used, during the second step of the multiscale approach, for the overall finite element calculation of the structure (dynamic or quasi-static), resulting in the identification of the potential fatigue damage areas [4]. The third step of the multiscale approach consists in assessing the lifetime of the critical areas by the local approach.
5.E+05
^
l.E+05
-l.E+05 O.E+00
2.E+01
4.E+01
6.E4-01
8.E+01
l.E+02
Time (s)
Fig. 3. Force versus time in one suspension of a 6-wheeled vehicle
E LABESSE-JIED ET AL
46
The approach adopted for the local calculation of fatigue damage under multiaxial random loading is shown on the flowchart of Fig. 4. The calculation is performed on each weak point of the surface of the structure. The inputs are the nodal plane stresses coming from a dynamic calculation (stationary random or transient) or from a quasi-static calculation.
rii
time
NiR
Fig. 4. Calculation of the fatigue damage by the local approach The operational service stresses obtained through the latter analysis have to be corrected for the effects of stress concentration resulting from geometric singularity overlooked during the calculation. This may concern a fillet, a hole, or even the micro-geometry of the weld toe or the weld root. The correction is performed with a two-dimensional stress concentration factor matrix [K], the components Ky of which are identified from a local calculation of the structural detail. The local stress states are thus obtained from the nominal stress states as: [ci]local = [ K ] . [ a ] n
(1)
It is well recognised now that residual manufacturing stresses, combined with service stresses, play an important role from the fatigue point of view. These stresses are taken into account in the proposed approach in the form of initial stresses to which the service stresses are added. [c?]total - [Cljresidual + [^] local
(2)
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
47
Stress states histories may present peaks overshooting the material yield stress. The plasticity resulting from these stress peaks modifies the material stress-strain response because of stress hardening. Hence, plasticity modifies the damage kinetics. The plastic correction is made with the hypothesis that the total strain remains constant. The elastic-plastic model corresponding to the Chaboche non-linear isotropic and kinematic hardening model is used [5]. Eqs (3), (4) and (5) describe the yield surface and the kinematic and isotropic stress hardening rules of this model.
f ( s , X , R ) = J(s-X)-R(p) = [ | ( s - X ) : ( s - X ) ] 2 - R ( p ) = 0
(3)
R(p) = Ro+Q.[l-exp(-y.p)]
(4)
dX = -Ca.d8P-C.X.dp
(5)
where ? is the deviatoric stress tensor, X and R(p) are the kinematic and isotropic parts of the stress hardening respectively. R(p) is divided into the initial yield stress Ro and the isotropic hardening which maximum value is Q. Q < 0 corresponds to a softening of the material whereas Q > 0 corresponds to a material hardening. The multiaxial fatigue criterion used is based over the critical plane concept. Such a criterion defines a so-called damage indicator related to any material plane (or facet). This indicator is generally a fiinction of the shear and normal components acting onto this facet. The principle is to search the most damaged plane, i.e. the critical plane. The assumption is made that this material plane drives the fatigue behaviour of the material as it is the first plane to experience a fatigue crack initiation. Bannantine and Socie [6] showed that fatigue cracks initiate from free surface on only 2 sets of facets, 90° or 45° inclined from the normal to the free surface (Fig. 5).
Fig. 5. Expected crack initiation planes (from [6])
The fatigue criterion is thus applied to those 2 groups of facets in order to find out the one that is submitted to the highest amount of fatigue damage. On each facet, the shear stress history is calculated; the rainflow procedure applied to one projection of that shear stress makes it possible to identify and extract the cycles from the stress states histories. For each cycle, the criterion is constructed by combining the shear stress amplitude and the maximum hydrostatic stress encountered during the cycle, that is:
48
F LABESSE-JIED ET AL.
a | , = M A X [ T a ( t ) + |3.aH(t)]
(6)
cycle Ta(t) = T ( t ) - T
(7)
The damage associated with each cycle is calculated by using Miner's linear rule and the material fatigue strength curve expressed in the form of Basquin's law [7]. d^=S-^-S
!—r
i Nf
(8)
The lifetime is calculated by maximising the damage over all the examined planes. In other words, it means that the critical plane enforces its fatigue life to the material.
d = MAX[d"J
(9)
N - d
(10)
EVALUATION OF THE METHOD O N SPECIMENS The local approach is assessed on cylindrical and tubular shaft specimens from experiments reported in the round-robin program performed by the SAE in the 80's [8]. The shafts are submitted to proportional or non-proportional strain-controlled tension and torsion loading. The strain amplitudes are constant or variable. The random spectra come from measurements made on log skidder and agriculture tractor axles and recorded as Markov matrix. Fatigue tests are performed on shaft specimens with different deterministic or random amplitudes until complete fracture. Calculations are carried out according to the local approach flowchart (Fig. 4). Starting from the deterministic or random strain cycles, the stress states are calculated by using a cyclic constitutive law with non-linear kinematical and isotropic hardening. Then, the damage corresponding to the cycles extracted by the rainflow method is calculated and cumulated. The life corresponding to each strain sequence is then calculated by using the most damaged facet. The shaft is made up of C45 carbon steel. The mechanical properties of this steel are summarised in Table 1.
Table 1. Monotonic and cycUc properties of C45 carbon steel (fi-om [8]) Gy (MPa) 280
K (MPa) 1185
n 0.23
a'y (MPa) 180
K' (MPa) 1258
n' 0.208
aV (MPa) 948
I'f (MPa) 505
b -0.092
Figure 6 gives the comparison between the experiments and calculated lives for all kinds of loading: deterministic uniaxial tension, proportional tension-torsion, non proportional tension-
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
49
torsion and random uniaxial tension for 3 different random spectrums. In the case of random loading, the life is expressed as the number of blocks up to crack initiation (Nf). The results are plotted with two straight lines indicating a deviation interval band between Nf/3 and 3 Nf. This comparison makes it possible for the prediction to be assessed within an interval between Nf/4 and 4 Nf. lE+06
lE+00 lE+00
lE+01
lE+02
lE+03
lE+04
lE+05
lE+06
Nf experimental Fig. 6. Comparison of predicted lives against experimental ones
The highest deviations between predictions and experiments are obtained for nonproportional loading and random amplitude, hi the case of non-proportional loading, the most important part of the error comes from the fatigue criterion used, hi fact, several authors [9,10] showed that, for non-proportional loading, the critical plane criterion taking account of the maximum of the shear stress amplitude encountered during one cycle give inaccurate results. The average of the shear stress on the facet over the cycle or the integration of shear stresses over all planes may give better results. This concept developed in some multiaxial fatigue criteria is the so-called integral approach. Considerations on how to integrate such a criterion for random loading are necessary, hi the case of random tension loading, the calculation is always optimistic, because the accelerating influence of previous small amplitude cycles onto the damage rate due to large amplitude cycles is not taken into account. To improve the prediction, a more physical damage parameter such as cumulative micro-plasticity on each facet or micro-cracks nucleation on each facet should be used.
APPLICATION OF THE LOCAL APPROACH TO WELDED COMPONENTS This section details all the steps of the local approach procedure which are successively run on in order to assess the fatigue lifetime of a welded mechanical assembly. Welded joints are particular structural details characterised by: - Stress concentrations induced by local shape of joint and manufacturing geometrical defects produced by the welding process (angular deviation and misalignment of the connected metal components).
50
E LABESSE-JIED ET AL
- Metallurgical transformation resulting in local changes of the mechanical static and fatigue properties, - Residual stresses induced by a non-homogeneous cooling of the welded components. The purpose of the procedure included in the local approach software is to take these peculiarities into account in the fatigue calculation. The approach is local as the fatigue life prediction is made from the local stress states in the most critical welded area. Studies concerning the influence of these three peculiarities showed that the stress concentration factor is the most important parameter to be considered. Figure 7 shows the stress concentration on the weld toe or on the weld root of a butt joint when it is subjected to axial loading. The stress concentration can easily exceed a factor equal to 2.
-EEEEiiEEEE}
(a) Fig. 7. Stress distributions within a butt-welded joint: (a) with angular distortion; (b) without angular distortion. Geometrical defects modify also the stress distributions and have consequently to be taken into account. Frequently geometrical defects are the angular distortion of a butt-welded joint and the misalignment of a cruciform welded joint as shown in Fig. 8. The importance of the angular distortion is illustrated on Fig. 7; as it shows that the location of the weakest point of the weld moves from the weld root (b) to the weld toe (a) when the angular distortion is taken into account for the stress calculation. The strong modifications regarding the stress distributions are explained by a significantly different change of the applied loading. For example, an angular distortion often generates a static bending moment when the welded sample is installed and clamped in the jaws of the fatigue testing machine. This bending moment is due to the fact that the sample is straightened because of jaws' alignment. This load is induced in fact by boundary conditions and is superimposed to the fatigue loading. The stress concentration is represented by the matrix [K] associated with the stress concentration located at the critical area. This critical zone may be the weld toe or the weld root. It depends in fact on the particular conditions as loading and local geometry of the case encountered. The partial penetration at the weld root can be interpreted as a geometrical defect too. The corresponding stress concentration factors are calculated from Finite Element Analysis of the real geometry of the weld subjected to the given particular loading. Figure 9 shows as an example the calculation results of the [K] stress concentration matrix.
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
(a)
51
(b)
Fig. 8. Welding geometrical defects: (a) angular distortion of a butt-welded joint; (b) misalignment defect of a cruciform welded joint Global typical model
Local typical model
' 1 0.5 0' [K] defined as [ajiocai = [K].[a]nom with [K] = 0.5 1.9 0 0 0 1 Fig. 9. Calculation of the stress concentration matrix associated with a butt weld for a given loading The constitutive cyclic law and the fatigue properties correspond to those of the critical zone of the welded joint, i.e. the weld toe or the weld root. The material properties are identified from specimens simulating the metallurgical states of these overheated areas. Such properties may be available in the technical literature, as for example within reference [11]. Another possible way to identify these properties is to use inverse simulation method from some fatigue results obtained for welded joints of the same material [12]. The local residual stresses existing at the weld toe or at the weld root could be calculated using coupled simulation of the thermal, metallurgical and mechanical phenomena during welding. An alternative method is to measure the local residual stresses using X-ray diffraction
52
F. LABESSE-JIED ET AL
or the incremental hole method. These residual stresses are then combined with the local stresses in order to determine the total operational stress states from which the fatigue life procedure is performed: W,„,^, =Mres.dual+[KlWnom
dD
The plasticity correction is performed when it is necessary with the cyclic plasticity parameters of the critical area. The multiaxial fatigue criterion function and the Miner's damage amount are calculated on each material facet. By this way the fatigue life on the facet maximising the damage is assessed.
EVALUATION OF THE METHOD ON WELDED COMPONENTS The proposed fatigue analysis method has been performed on welded structures. Figure 10 shows experimental data on butt-welded joints. Two series of tests corresponding to loading ratios equal to -1 and 0.5 respectively are plotted in the figure. These results have been obtained for the 6 mm thick 16MnNiCrMo5 high strength steel plates welded in 2 layers with the MAG process. The real geometry and angular distortion of the joint has been measured and then modelled by the Finite Element method. Figure 7a shows the stress states involved when the joint is clamped in the jaws of the fatigue testing machine and subjected to nominal axial loading. The weakest point of the joint is focussed on the weld toe. The local stress states are calculated by the Finite Element method using the actual geometry of the weld. For an axial cyclic loading they are obtained by combining the cyclic nominal stresses modified by the stress concentration matrix and the initial stress states due to the distorted welded specimen clamped in the jaws of the testing machine. The local mechanical properties of the overheated thermally affected area corresponding to the weld toe are presented in Table 2. These properties come from cyclic loading tests performed on specimens on which a heat treatment simulating the overheat of the thermally affected area has been realised and validated using the inverse method [12]. Table 2. Mechanical properties of the thermally affected material 'y
(MPa) 565
K' (MPa) 1150
n' 0.111
aV (MPa) 2080
-0.13
As the welded joints are subjected to high-temperature stress relieving, the residual stresses are disregarded. The fatigue strength of the joint under tensile cyclic loading corresponding to the ratios R = -1 and R = 0.5 respectively are calculated. They are plotted with straight lines in Figure 10. The comparison against experimental fatigue tests results shows a rather good agreement of predicted fatigue behaviour despite the fact that the scatter of these results is rather wide.
53
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
1000
•--
1
•
,,,^^
--| ! 1
•
CO
a.
•
V
•
X (0
\
E CO
—calculation R=0.5 •
calculation R=-1
^*i*^*
A
A A
1 A testR=-1 100 1E+03
1
A
test R=0.5
1E+04
1E+06
1E+05 cycles
j
1 1E+07
Fig. 10. Comparison between predicted and experimental lives of stress-relieved butt-welded steel
DEFINITION OF THE WELD QUALITY: THE GEOMETRICAL PARAMETERS The quality of welded joints is defined from a design point of view by geometrical parameters such as the width of the weld (denoted as Lpc and Lrc for the weld toe and the weld root respectively), the height of the joints (S), the transition radii, the axial misalignment and the angular distortion. Transition radii and geometrical defects (distortion and misalignment) constitute in fact the main parameters of the quality of welds. Figures 11 and 12 show the geometrical parameters of butt and fillet welded joints respectively.
Rl
Root
R2
Fig. 11. Geometrical and micro-geometrical parameters of butt welds
The mean values and standard deviations of the geometrical parameters measured on several butt welded joints in 7020 aluminium alloy of 20 mm thickness, welded using the multi-layer MIG process, are given in Table 3. The angular distortion and the axial misalignment measured are respectively 0.7 ± 0.9° and 0.5 mm.
54
F LABESSE-JIED ET AL
Table 3. Mean geometrical parameters measured on multi-pass MIG butt welds in 7020 aluminium alloy (thickness equal to 20 mm).
Mean value (mm) Standard deviation (mm)
Rl R2 12.6 4.3 9.1 4.5
Top bead Rmin Lpc 2.7 34 3.0 1.5
S 2.9 0.6
Rl 6.7 3.3
Root bead Rmin Lpc S 4.0 3.7 28.4 3.6 2.4 2.2 2.4 0.7 R2
R2
Fig. 12. Geometrical and micro-geometrical parameters of fillet welds The mean values and standard deviations of the geometrical parameters measured on several fillet welded joints in 7020 aluminium alloy of 30 mm or 20 mm thickness, welded using the muhi-layer MIG process, are given in Table 4. Table 4. Average geometrical parameters measured on multi-layer MIG fillet welds in 7020 aluminium alloy (thickness t and T equal to 20 mm and 30 mm respectively). Welds A & B Weld A Rl R2 Rl R2 2.9 2.8 2.6 2.9
Mean value (mm) 1.6 Standard deviation (mm)
1.8
1.2
1.2
Weld B Rl R2 1.7 1.3
Jl
J2
14.2
16.2
0.9
0.8
0.8
0.7
0.6
1.1
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
55
The precise measure of the characteristic geometry of these welded components makes it possible to accurately determine the local stress states existing within them. This is of major importance as these stresses are responsible for the local fatigue behaviour of the welds.
ANALYSIS OF THE EFFECT OF THE WELD QUALITY ON THE FATIGUE STRENGTH The local approach method was applied to welded joints in aluminium alloy to analyse the effect of the welds quality upon the fatigue strength. The examples presented here concern butt-welded joints in 7020 aluminium alloy of 20 mm thickness. The welds were obtained by using multi-pass MIG process with a 5183 filler metal. Figure 13 shows a macrograph of the weld.
Fig. 13. Aluminium alloy butt welds with the fatigue crack occuring in the vicinity of the weld toe
Material properties and geometrical parameters
Mechanical properties. Fatigue cracks always start at the weld toe or at the weld root. The material parameters of these zones must therefore be determined and used for the local approach. In order to determine these parameters, the metallurgical state at the toe of the bead is reproduced by an overheating thermal treatment on test samples. The material parameters that were obtained by this way are given in Table 5.
Table 5. Mechanical properties of heat affected zones of aluminium samples Elasticity parameters E V (MPa) 7.3-10^ 03
Ro (MPa) 200
Cyclic plasticity parameters Ca C Q b (MPa) (MPa) iF 150 ToO 50
Fatigue parameters a'f b p (MPa) 670 -0.095 2.3-10"^
56
F LABESSE-JIED FT AL
Mean residual stresses. The mean residual stresses were measured on the surface of the toe of the bead with means of X-Rays diffraction. They reach 25 MPa and 37 MPa in the transversal and longitudinal directions respectively.
Mean geometrical parameters. Experimental comparisons are made on the basis of the average geometrical parameters measured on welded test specimens. Table 6 summarises these geometrical parameters.
Table 6. Geometrical parameters of the aluminium alloy butt welds
Rmin Mean value (mm) 2.67 Standard deviation (mm) 1.55
Top bead Lpc S 34.03 2.94 3.00 0.59
Rmin 3.69 2.20
Root bead Lrc S 28.40 3.64 2.43 0.68
Moreover, the average axial misalignment of 0.5 mm and angular distortion of 0.7° are taken into account for the fatigue life assessment. The following sections describe the sensitivity of the butt welds, regarding their predicted fatigue life under constant amplitude loading, to residual stresses, angular distortion and transition radius.
Influence of residual stresses Residual stresses are known for their determinant role on the fatigue strength of materials, including of course welding joints. This influence depends on the local sensitivity of the material to hydrostatic stress. Fatigue properties, determined from results of fatigue tests performed on the overheated metallurgical state corresponding to the weld toe, indicate an important sensitivity to hydrostatic stress. This sensitivity can also be quantified by the ratio of fatigue strengths obtained under alternate tensile test and alternate torsion test respectively, for a given number of cycles. The smaller this ratio is, the more the material is sensitive to the residual stresses. A material insensitive to the hydrostatic stress effect has the following strength ratio:
^
(12)
=S
The present overheated material located at the weld toe provides the following fatigue strength ratio, at 10^ cycles: I"
\
-1.4
(13)
weld toe, N=10 cycles
It is important to note that this ratio is about 1.6 for high-strength low alloy steels hardened/tempered at high temperature. This indicates a lower sensitivity of these steels to residual stresses.
57
Multiaxial Fatigue Assessment of Welded Structures by Local Approach
The analysis of the influence of the residual stresses level on the fatigue strength of butt welded joints subjected to alternate cyclic load has been carried out by the proposed local approach. These calculations were done with the mean measured geometry of the welds. Figure 14 shows the influence of the residual stresses on the fatigue strength of the butt welds. The different curves are examined from the left side to the right side, i.e. in the increasing order of the fatigue strength.
20
-nil SIG_res
experimental results
-minimum SIG_res measured
-maximised SIG_res
-SIG_res shot peening
maximum SIG_res measured 01E+4
' ' '^ ' i 1E+5
1E+6
1E+7
1E+8
Number of cycles
Fig. 14. Influence of residual stresses on predicted alternate tensile fatigue strength (R = -1) of butt welds
The lowest S-N curve has been obtained with the maximised residual stresses, it is to say CTiong = G^trans = 140 MPa. The sccond S-N curve was plotted by considering the maximum measured values of residual stresses which correspond to: aiong = 70 MPa and Qtrans = 60 MPa. These two assessed S-N curves are found to be quite conservative. The minimum measured residual stresses (aiong = 1 5 MPa and Gtrans = -20 MPa) lead approximately to the same simulation results as the calculation carried out accounting for no residual stresses. These assumptions seem to be a little optimistic regarding to the experimental S-N curve. The last SN curve was obtained assuming that residual stresses are compressive and reach the maximum absolute value, i.e. aiong = CTtrans = -100 MPa. This assumption brings quite non conservative predictions. It is similar in fact to the effect of pre-stressed shot peening. As a conclusion, there is clearly a very considerable effect of residual stresses on the fatigue strength of aluminium alloy welded joints. The influence is beneficial in the case of compressive residual stresses and detrimental in the other case. A pre-stress shot peening type finishing treatment makes strongly increase the fatigue strength of welded joints.
58
F. LABESSE-JIED ET AL
Influence of angular distortion defects The effect of geometrical defects is more difficult to analyse and must be looked at on a caseby-case basis. Generally speaking, an axial misalignment or an angular distortion creates a supplementary bending moment, which induces some compressive stresses on one side of the joint and some tensile stresses on the other side. The effect of these static stresses on the fatigue strength of the welded joint depends on the loading and on the micro-geometry of the bead: - generally, when the geometrical defect induces stresses opposite to those induced by the service loading, some improvement in fatigue strength can be expected, - when the micro-geometry is symmetrical with respect to the mean fibre, the geometrical defect causes a reduction in the fatigue strength of the welded joint, - when the transition radii are smaller on the side on which the geometrical defect induces compressive stresses, then the defect makes improve the fatigue strength of the welded joint. On the contrary, i.e. with tensile static stresses applied on the smallest transition radii side, the fatigue strength of the welded joint is reduced. To sum up, the effect of geometrical defects may be positive or negative. Well-controlled defects could in principle contribute to improve the fatigue strength of welded joints. The effect of angular distortion is quantified by calculation of the fatigue strength of buttwelded joints subjected to alternate tensile loading. The direction of the modelled distortion, in relation to the geometry of the joint, corresponds to the direction that was measured, namely a closing of the welded joint on the root side. The S-N curves of Fig. 15 show the importance of angular distortion upon the predicted fatigue strength of welds. All the calculations are realised with the average geometry and the mean residual stresses. The highest fatigue strength is obtained with a zero angular distortion. The general trend is that the fatigue strength decreases as the angular distortion increases. Most of the experimental test results are close to the predicted fatigue strengths obtained with considering the average measured angular distortion.
100
l, 6sl26°).
L SUSMEL AND N. PETRONE
98
P26BT5 - Nf,2% = 64090 Cycles
P25BT5-Nf2% == 31000 Cycles
1^ HsV
V
n
1
\
'J li
20JI
mm
i
\
V
\
1-^
;
f —^-
P24BT4 - Nf,2% = 124460 Cycles
P23BT4 - Nf,2%= 132215 Cycles
P27BT6-Nf,2% = 232370 Cycles
P28BT7 - Nf,2% = 315795 Cycles
f/ 1
x'f
Fig. 11. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: in-phase bending/torsion tests (X=Txy,a/ax,a1 were positioned in the non-conservative zone, whereas fatigue points having X' + 3z^
1 2 3
5x^ -^y^ ^Ixy + Az^ 3JC^ + 3V^ + 2X); + 4Z^
x^ + y ^ - x y + 3z^
x^ + 5y^ + 2.ry + 42^
8(x + y)z
3x^ + 3v^ + 2x>' + 4z^
3x^ + 3y^ + 2xy + 4z^
1
Vac/^W 0.8
1
>*V-^o
1 0.6 i
130 test results
0.4 J j
•
steel (bending&torsion)
o
steel (tension&torsion)
0.2 J
•
Al-alloy
j •
^ *
•^LJ
ellipse equaliuii
0
1 — ' — '
0.2
0.4
0.6
0.8
^xac/^W Fig. 3. Fatigue limit under alternating normal and shear stresses
If one combines the elliptical equation with collected test results [25] to yield a standardised diag ram, Fig, 3, Eq. (19) then agrees with the test results. For the case of an alternating normal stress with a superposed static shear stress, the fatigue limit is decreased by the superposed static shear stress. Fig. 4. Up to a static shear stress Tj^^ which is lower than the yield strength RpQ 2, the influence of the superposed shear stress is correctly described by the SIH. Beyond this value, however, the influence of the superposed shear stress is overestimated. With r^y,^ > /?p0.2' severe plastic deformations occur; consequently, this case is defined by a static strength design, and is of no importance for practical applications.
154
J. LIU AND H. ZENNER
^xac/^V ^xym
•t
t-^ ^4-^
0.2
0.4
0.6
0.8
Fig. 4. Effect of the mean shear stress on the fatigue hmit for cycHc normal stress
1.2
1 1
* ^
^
1 '^xyac/'^W 0.8 'xya
0.6
H
0.4 J
•t
Ck35 (SIH)
0.2 J
•
] 0
^V o
onr^i-M!r»^ ('Oil i\ J U U I M 0 4 ( o i l 1)
-1.5
'
^va,T
equivalent stress amplitude
^vm' ^vm,cT> ^vm,T
equivalent mean stresses
a^ch
fatigue limit (double amplitude) for pulsating tension
(%
fatigue limit for alternating tension-compression
a-yfpa
normal stress amplitude in the intersection plane ytp
a^fp^
mean normal stress in the intersection plane y(p
T^ch
fatigue limit for pulsating torsion (double amplitude)
Tj^aD
fatigue limit for shear stress
%
fatigue limit for alternating torsion
Tyfpa
shear stress amplitude in the intersection plane y(p
Tyfp^
mean shear stress in the intersection plane yig>
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) Published by Elsevier Science Ltd. and ESIS.
165
SEQUENCED AXIAL AND TORSIONAL CUMULATIVE FATIGUE: LOW AMPLITUDE FOLLOWED BY HIGH AMPLITUDE LOADING
Peter BONACUSE US Army Research Laboratory^ NASA Glenn Research Center, Brook Park, OH, USA and Sreeramesh KALLURI Ohio Aerospace Institute, NASA Glenn Research Center, Brook Park, OH, USA
ABSTRACT The experiments described herein were performed to determine whether damage imposed by axial loading interacts with damage imposed by torsional loading. This paper is a follow on to a study [1] that investigated effects of load-type sequencing on the cumulative fatigue behavior of a cobalt base superalloy, Haynes 188, at 538°C. Both the current and the previous study were used to test the applicability of cumulative fatigue damage models to conditions where damage is imposed by different loading modes. In the previous study, axial and torsional two load level cumulative fatigue experiments were conducted, in varied combinations, with the low-cycle fatigue (high amplitude loading) applied first. In present study, the low amplitude fatigue loading was applied initially. As in the previous study, four sequences (axial/axial, torsion/torsion, axial/torsion, and torsion/axial) of two load level cumulative fatigue experiments were performed. The amount of fatigue damage contributed by each of the imposed loads was estimated by both the Palmgren-Miner linear damage rule (LDR) and the non-linear, damage curve approach (DCA). Life predictions for the various cumulative loading combinations are compared with experimental results. Unlike the previous study where the DCA proved markedly superior, no clear advantage can be discerned for either of the cumulative fatigue damage models for the loading sequences performed. In addition, the cyclic deformation behavior under the various combinations of loading is presented. KEYWORDS Multiaxial, cumulative fatigue, axial loading, torsional loading, tubular specimens.
166
R BONACUSE AND S. KALLURI
INTRODUCTION Many multiaxial fatigue damage models are based on the premise that damage is a tensor, i.e., it has both magnitude and direction. These Sensorial' approaches imply that damage imposed by loading in one direction does not readily interact with subsequent damage imposed by loading in another direction. By subjecting thin-walled tubular specimens to fatigue in different loading directions, in sequence, this hypothesis can be tested. In this study, a block of lower amplitude loading, at a given fraction of the estimated life, preceded a second block of higher amplitude loading to failure. This work is a complement to a previous study [1] where higher amplitude cycles were imposed initially, followed by lower amplitude loading to failure. Various combinations of axial and torsional load-type and load-sequence interactions have been explored in both studies. The most common method of accounting for the fatigue damage accumulated in a material subject to variable amplitude loading is to estimate the fraction of the fatigue life expended for each cycle of a given amplitude and then sum up all the fractions for each of the load excursions. When this sum reaches unity, the material is said to have used up its available life. This model is commonly referred to as the linear damage or Palmgren-Miner rule (LDR) [2]. However, many studies have shown LDR to be inaccurate by as much as an order of magnitude for certain combinations of variable amplitude loading [3-6]. Halford [3] outlines the uniaxial loading combinations where the LDR is likely to break down and where alternative approaches may prove useful. In the present study, two cumulative fatigue damage models are assessed for their ability to predict the remaining cyclic life after prior loading at both different amplitudes and different loading directions. The first model is the LDR [2]. The second is the damage curve approach (DCA) of Manson and Halford [7]. In the previous study (high amplitude loading followed by low amplitude loading), the damage curve approach (DCA) was found to model the load interaction behavior remarkably well for all the cases investigated including the mixed load experiments (axial followed by torsional and torsional followed by axial). This seemed to indicate that the fatigue damage was most likely isotropic, at least when the loading sequence was high amplitude followed by low amplitude. A possible explanation for apparent isotropic nature of damage accumulation under fully reversed fatigue loading is that the cyclic hardening behavior of superalloys is, at least to some extent, isotropic in nature. This hardening should influence damage accumulation even when loading is imposed in another direction. The magnitude of plastic deformation is often used as a measure of the accumulated damage in a loading cycle. If the material is in a work hardened state from previous mechanical cycling it should be able to absorb a larger portion of ensuing deformations elastically, resulting in lower plastic strains. Thus, a smaller increment of damage would then be accrued in each subsequent cycle. This mechanism would be in competition with propagating cracks that may have initiated in the prior cycling; higher stresses due to work hardening would increase the crack propagation rate in the subsequent loading.
MATERL\L, SPECIMENS, AND TEST PARAMETERS Specimens used in this study were fabricated from hot rolled, solution annealed, Haynes 188 superalloy, 50.8 mm diameter bar stock (heat number: 1-1880-6-1714). This is the same heat of material used to perform the experiments described in Ref. [1]. The measured weight
Sequenced Axial and Torsional Cumulative Fatigue: ...
167
percent composition of the superalloy was: < 0.002 sulfur, 0.003 boron, < 0.005 phosphorus, 0.052 lanthanum, 0.09 carbon, 0.35 silicon, 0.8 manganese, 1.17 iron, 14.06 tungsten, 22.11 chromium, 22.66 nickel, with the balance made up of cobalt. All experiments were performed on thin walled tubes with nominal gage section dimensions of 26 mm outer diameter, 22 mm inner diameter, 41 mm straight section and 25 mm gage length. The interior surfaces of the tubes were honed in an attempt to preclude crack initiation on the inner surface of the specimen. Outer surfaces were polished with final polishing direction parallel to the specimen axis. Further details on the specimen geometry and machining specifications can be found in Ref. [8]. The baseline axial and torsional fatigue lives for this material, specimen geometry, and test temperature can be found in Ref. [1]. The specimens were heated to 538°C with an induction heating system. All specimens were subjected to sequential constant amplitude fatigue loading under strain control. A commercially available, water-cooled, biaxial, contacting extensometer with a 25 mm gage length, designed specifically for axial-torsion testing, was used. The loading actuator that was not being used for fatigue strain cycling (the torsional actuator during axial cycling or the axial actuator during torsional cycling) was maintained in load control at zero load. This procedure allowed strains to accumulate in the zero load direction. During the axial strain cycling segments, relatively small mean strains in the load controlled torsional axis were observed. However, the torsional strain cycling segments always showed increasing mean axial strains. When torsional strains were applied in the first segment, the magnitudes of these axial strains were recorded and then electronically set to zero prior to commencing the second loading segment. The specimen failure criterion programmed into the testing software was a 10% drop in the measured load in the strain controlled direction. Five experiments were terminated due to a controller interlock. Details on the testing system and test control procedures can be found in Ref. r 11.
TEST MATRIX The test matrix for this study is shown in Table 1. Seventeen different combinations of low amplitude followed by high amplitude, two load level experiments were performed. The loading sequences were axial followed by axial (axial/axial), torsion followed by torsion (torsion/torsion), axial followed by torsion (axial/torsion), and torsion followed by axial (torsion/axial), with at least four different, first load level life fractions imposed in each combination. A fifth life fraction was imposed in the torsion/torsion subset. One torsional experiment was repeated as a cursory check on the expected specimen-to-specimen variability in fatigue life. This summed to a total of 18 tests performed for this study. Table 1 also contains the stress range and mean stress at half-life for each load level, the number of cycles imposed, and the final crack orientation.
TABLE I: Test matrix and interaction fatigue data for Haynes 188 at 538°C
Specimen
AEI
First Load Level; v = 0.5 Hz A01 0ml A ~ I A Zml n~ (MPa) (MPa) (MPa) (MPa) (Cycles)
AxialIAxial HYII-103 0.0067 811 -9 ... ... HYII-116 0.0066 849 -6 ... ... HYII-119 0.0066 882 -10 ... ... ... HYII-114 0.0066 905 -9 ... Torsion/'Torsion ... 0.0120 515 HYII-117 ... ... ... 0.0120 536 HYLI-1 12 ... ... ... 0.01 19 554 HYII- 1 15 ... ... ... 0.0121 521 HYII- 109 ... ... ... 0.01 19 588 HYII-118 ... ... ... 0.0120 579 HYII- 104 ... ... Axial/Torsion HYII-I 10 0.0069 841 -10 ... ... ... HYII-111 0.0069 862 -8 ... ... HYII-108 0.0065 892 -8 ... HYII-105 0.0066 888 -9 ... ... Torsion/Axial ... 0.0121 498 HYII-120 ... ... ... HYII- 107 ... ... 0.0120 523 ... 0.01 19 540 HYII- 106 ... ... ... 0.0119 581 HYII-113 ... ... * Angle measured with respect to the specimen axis.
AE?
Second Load Level; v = 0.1 Hz A02 Gm2 Ay2 A Tm2 n2 Crack* (MPa) (MPa) (MPa) (MPa) (Cycles) Orientation
...
... ... ... ...
3926 7851 15702 23553
0.0203 0.0202 0.0203 0.0205
1254 1247 1244 1267
-14 -14 -1 1 -12
... ... ... ...
... ... ...
... ... ... ...
789 758 659 815
75" 80" 85" 90"
-1 0 1 0 0 -2
5857 11714 23427 23427 35141 40998
... ...
... ... ... ... ...
... ... ... ... ... ...
0.0345 0.0349 0.0346 0.0347 0.0347 0.0349
73 1 740 731 709 748 732
1 1 0 2 0 -2
1250 1100 816 1343 1467 1294
0" 0" 0" 0" 0" 0"
...
... ... ...
...
!m tu
0 0
5
sh
* 2b b
... ... ... ...
3926 7851 15702 23553
... ... ... ...
... ... ... ...
...
0 -1 0 -1
5857 11714 23427 35141
0.0201 0.0203 0.0200 0.0204
1400 1414 1432 1452
781 761 794 783
2 0
... ...
0.0348 0.0347 0.0344 0.0346
-15 -16 -20 -18
... ... ... ...
... ... ... ...
... ... ...
...
1
2
...
1189 1218 930 1253
0" 5" 0" 0"
560 494 459 427
90" 90" 75" 80"
C
?
Sequenced Axial and Torsional Cumulative Fatigue: ...
169
CUMULATIVE DAMAGE MODELS The results of the cumulative fatigue experiments performed for this study were compared with predictions of two load interaction models: the LDR (Eq. 1) [2], and the DCA (Eq. 2) [7].
n.
(2)
vNw The LDR assumes that damage accumulated during each load excursion can be simply added to the already accumulated damage in the material. Load sequence and changes in the properties of the material are not taken into account. The LDR has the distinct advantage of being straight forward to implement for virtually all loading histories, provided sufficient baseline fatigue data are available and an adequate cycle counting method is employed. The DCA attempts to model the observed non-linear interactions between two load-level experiments. The underlying assumption of the DCA is that high amplitude loading initiates cracking early in life whereas under lower amplitude loading measurable cracking does not occur until late in life. In the case of the experiments performed in this study, the implication is that the initially imposed lower amplitude loading might not initiate cracks. In the subsequent higher amplitude loading the material might then 'ignore' the previous cycling or even derive a benefit from it, allowing the sum of life fractions to be greater than unity.
RESULTS AND DISCUSSION The initially imposed lower amplitude cyclic loading (0.65% axial and L24% torsional nominal strain ranges) had sufficient plasticity to cause this solution-annealed material to isotropically harden. The magnitude of the hardening in the first load level was proportional to the number of imposed cycles. The cyclic hardening behavior for the lower amplitude, first load level, loading is presented in Fig. L The horizontal lines in each of these figures correspond to stress range for the last cycle of the lower amplitude loading. The vertical drop lines indicate the last cycle of the first load level for each experiment. There was some variation in the first cycle stress range for both the axial and torsional loading. In the first load levels, the average axial first cycle stress range (the left most data point in each of the curves in Fig.l (a) and (c)) was 646 MPa with a standard deviation of 15 MPa, while the torsional first cycle average stress range (the left most data point in each of the curves in Fig.l (b) and (d)) was 382 MPa with a standard deviation of 14 MPa. The most likely explanations for these variations include the natural variability in the material properties and discrepancies in the machining and/or measurement of the specimen gage section. A 0.1% error in the measurement of a gage section dimension (inner or outer radius), which is the approximate precision of the micrometers employed, would lead to a 0.2% error in the axial stress and a 0.4% error in the shear stress calculations. This dimensional measurement error would account for only about 10% of the variability observed. The specimen to specimen variation in the hardening rate, however, in all the axial and torsional experiments was small.
170
P. BONACUSE AND S. KALLURI
1800
1500 H
< CD D) C
= •O- "i = -y-- ^^ = = -^-^^ - • -
CO
Q.
^1
3926 7851 15702 23553
1200 H
CO
CL if) CO CD
900
1-
(D
"S x
a • W., S(To) =
(39)
b N, for W., \ r
/ >"•
tin
>^
\y B3v.
C3J ^:J^
NW
Loi
al
Aa2
o2
C4\
LGI
/
(b) (c) Fig. 1. In-phase and out-of-phase biaxial loading histories where stresses are at different frequencies (a) a = l , (b) a=2 and (c) a=3.
206
A. VARVANI-FARAHANI
In load histories B3 and B4, two axes of loading start with a phase delay of (t)=90°: the former history contains no mean stress, while the latter history contains a transverse mean stress value. Histories B3 and B4 have non-Hnear C-shaped load paths (see Fig. 1(b)). (c) Frequency ratio a=3: two axes are loaded with a frequency ratio of 3:1. Histories CI and C3 correspond to biaxial tests including no mean value, however, histories C2 and C4 contain a non-zero transverse mean stress value. Histories CI and C2 have non-linear Z-shaped load paths, while histories C3 and C4 have non-linear S-shaped load paths (see Fig. 1(c)).
FATIGUE DAMAGE MODEL AND ANALYSIS
Cyclic stress and strain analyses The stress and strain tensors for a thin-walled tubular specimen are given by Eq (1) and Eq (2), respectively:
^.,=
=
0
^ 22
0
0
V
^ii
0
(1)
an J
0
0
0
£22
0
0
0
£•33
(2) J
Figure 2 illustrates a thin-walled tubular specimen subjected to biaxial fatigue loading. The two controlled applied stresses are the principal stresses in the plane shown on the tubular specimen. A Ell
Wrrrn (a)
(b)
(0
Fig. 2. (a) Thin-walled tubular specimen subjected to biaxial loading, (b) 3D presentation of principal stress state, and (c) of principal strain state.
Critical Plane-Energy
Based Approach for Assessment of Biaxial Fatigue Damage where ...
207
In equation (1) Gjj is the stress tensor (both i and j are equal to 1,2,3), and the stresses Gu, 022, and 033 are principal stresses. Since the thin-walled tubular specimens are in plane stress condition, a22=0. Principal stresses can be calculated from applied stress amplitudes: f^ll = ^ / n l + L ^ l S l n(^)]
(3a)
(722 = 0
(3b) (3c) a )]
where Gmi and (Jmi are the longitudinal and the transverse mean stresses, respectively. The angle 6 is the angle during a\ cycles of stressing in a block loading history at which the Mohr's circles are the largest and has the maximum value of shear strain. Angle 9 varies from 0 to 2an. Angle (j) corresponds to the phase delay between loads on the longitudinal and transverse axes. In Eq (2) 8ij is the strain tensor (both i and j are equal to 1,2,3) where the strains 8ii, 822, and 833 are principal strains calculated using the elastic-plastic strain constitutive relation (Eq(4)). In Eq (4), the first square bracket presents the elastic and the second square presents the plastic components of strains: cP 1 -eq_
\ + v.
- ^ . - ^ ^ ^ k^'j\
2 '^
(4)
where the deviatoric stress Sjj is defined as the difference between the tensorial stress Ci^ and hydrostatic stress (-cr^^t)"
S =a ij
where 5ij=l
if i=j
5.j=0
if i^j
--a,,S ;/
^
kk
(5)
ij
E is the elastic modulus, Ve=0.3 is the elastic Poisson's ratio for the material used in this study and Okk is the summation of principal stresses. The stress amplitude of the hysteresis loop at half-life cycle was associated with the stabilized cyclic stress-strain loop. The cyclic stressstrain curve can be described with the same mathematical expression as for the monotonic stress-strain curve. The relationship between the equivalent cyclic plastic strain e^^ and the equivalent cyclic stress crgq obtained from uniaxial stabilized cyclic stress-strain data is: ^1227(£-P )/2.78
(6)
The coefficient and the exponent in Eq (6) are the cyclic strength coefficient, K*=1227 MPa, and the cyclic strain hardening exponent, n*=0.36, respectively. Eq (4) presents elastic-plastic deformation and correlates the tensorial cyclic stress and strain components for 3D state of stress and strain. A simple form of this equation for the
208
A. VARVANI-FARAHANI
uniaxial
loading
condition
is
the
well-known
Ramberg-Osgood
relation
(^11 = ^ n / ^ + P i i / ^ j " )' which is commonly used to present cyclic uniaxial stress-strain curve. The range of maximum shear stress and the corresponding normal stress range are calculated from the largest stress Mohr's circle during the first reversal (at the angle 61) and the second reversal (at the angle 62) of a cycle as: _^ll-^33
AW=P^^^ >^u±^]
1
f (Til-0-33
-P^^^
(7a)
_£iLl£B|
(7b)
where a n , O22 and 033 are the principal stress values calculated using Eq (3). Similarly, the range of maximum shear strain and the corresponding normal strain range on the critical plane at which Mohr's circles are the largest during the first reversal (at the angle 9i) and the second reversal (at the angle 62) of a cycle are calculated as:
A|ijii^Vf£LLl£3ll _f£lLl£31| ^
^
l^lilfB
Je\ ^
^
_£iil£3i|
(8a)
Jei
(8b)
where en, £22, and 833 are the principal strain values (£n>£22> £33) which are calculated using Eq (4).
Proposed fatigue parameter In this paper for the convenience of presentation, first the proposed parameter and its capability to take into account the effects of out-of-phase strain hardening and mean stress are discussed for the load histories with frequency ratio of a=l [17,23,24] and then the damage model is extended for other ratios of a=2 and a=3. The range of maximum shear stress AXmax and shear strain AI-^-'^^ I obtained from the largest stress and strain Mohr's circles at angles 9i and 62 during a cycle and the corresponding normal stress range AGn and the normal strain range ASn on that plane are the components of the proposed parameter in the present paper. Multiaxial fatigue energy-based models have been long discussed in terms of normal and shear energy weights. In Garud's approach [8] he found that an empirical weighting factor equal to C=0.5 in the shear energy part of his model (Eq 10) gave a good correlation of multiaxial fatigue results for 1% Cr Mo V steel for both in-phase and out-of-phase loading conditions: Afilo- + CA}Ar = /(Nf)
(9)
Critical Plane-Energy Based Approach for Assessment of Biaxial Fatigue Damage where ... where Nf is the number of cycles to failure, hi Eq (9) Ae and Aa are the range of normal strain and stress, and Ay and Ax are the range of shear strain and stress, respectively. Tipton [25] found that a good multiaxial fatigue life correlation was obtained for 1045 steel with a scaling factor C of 0.90. Andrews [26] found that a C factor of 0.30 yielded the best correlation of multiaxial life data for AISI 316 stainless steel. Chu et al. [15] weighted the shear energy part of their formulation by a factor of C=2 to obtain a good correlation of fatigue results. Liu's [14] and Glinka et al. [16] formulations provided an equal weight of normal and shear energies. The empirical factor (C) suggested by each of the above authors gave a good fatigue life correlation for a specific material which suggests that the empirical weighting factor C is material dependent. In the present study, the proposed model correlates multiaxial fatigue lives by normalizing the normal and shear energies using the axial and shear material fatigue properties, respectively, and hence the parameter uses no empirical weighting factor. Both normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively: pi-,(Aa„AfJ+
'
^AwA^i^jj=/(Nf)
(10a)
where dt and £f are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and ri and /f are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively.
Out-of-phase strain hardening Under out-of-phase loading, the principal stress and strain axes rotate during fatigue loading (e.g. see [13]) often causing additional cyclic hardening of materials. A change of loading direction allows more grains to undergo their most favorable orientation for slip, and leads to more active slip systems in producing dislocation interactions and dislocation tangles to form dislocation cells. Literactions strongly affect the hardening behavior and as the degree of outof-phase increases, the number of active slip systems increases. Socie et al. [27] performed inphase and 90° out-of-phase fatigue tests with the same shear strain range on 304 stainless steel. Even though both loading histories had the same shear strain range, cyclic stabilized stressstrain hysteresis loops in the 90° out-of-phase tests had stress ranges twice as large as those of the in-phase tests. They concluded that the higher magnitude of strain and stress ranges in the out-of-phase tests was due to the effect of an additional strain hardening in the material [28]. During out-of-phase straining the magnitude of the normal strain and stress ranges is larger than that for in-phase straining with the same applied shear strain ranges per cycle. The proposed parameter via its stress range term increases with the additional hardening caused by out-of-phase tests whereas critical plane models that include only strain terms do not change when there is strain path dependent hardening. To calculate the additional hardening for out-ofphase fatigue tests, these approaches may be modified by a proportionality factor like the one proposed by Kanazawa et al. [29].
Mean stress correction Under multiaxial fatigue loading, mean tensile and compressive stresses have a substantial effect on fatigue life. Sines [30] showed compressive mean stresses are beneficial to the fatigue life while tensile mean stresses are detrimental. He also showed that a mean axial tensile stress
209
210
A. VARVANI-FARAHANI
superimposed on torsional loading has a significant effect on the fatigue life. In 1942 Smith [31] reported experimental results for twenty-seven different materials from which it was concluded that mean shear stresses have very little effect on fatigue life and endurance limit. Sines [30] reported his findings and Smith's results by plotting mean stress normalized by monotonic yield stress versus the amplitude of alternating stress normalized by fatigue limit (R=-l) values. The relation is linear as long as the maximum stress during a cycle does not exceed the yield stress of the material [28]. Concerning the effect of mean strain on fatigue life, Bergmann et al. [32] found almost no effect in the low-cycle fatigue region and very little effect in the high-cycle fatigue region. Mean stress effects are included into fatigue parameters in different ways [28]. One approach was applied earlier by Fatemi and Socie [33] to incorporate mean stress using the maximum value of normal stress during a cycle to modify the damage parameter. Considering the effect of mean axial stress, a mean stress correction factor 1 + ^
inserted into Eq (10a)
showed a good correlation of multiaxial fatigue data containing mean stress values for both inphase and out-of-phase straining conditions. This correction is based on the mean normal stress, &!!, applied to the critical plane. To take into account the effect of mean axial stress on the proposed parameter, Eq (10a) is rewritten as:
1+ ^ 1 ^^[^a,^e,)+vrty
^Ar,,, A Ym ^ ^ JJ= /(N,)
(10b)
where the normal mean stress cC acting on the critical plane is given by:
^n"=ik"+^n™")
(11)
where cf^"" and cC" are the maximum and minimum normal stresses, respectively, which are calculated from the stress Mohr's circles. The mean normal stress correction factor can be applied for both <j^ >0 and <j^ 0, the tensile mean normal stress, increases the fatigue damage and cj^ -(T
Fig. 16: Mohr's circle Fatigue Limit - - - - -
-Stress Amplitude
Safety Factor
I 0.'
40
50
60
70
80
Polar Angle [deg]
Fig. 17: Safety factor against fatigue limit in dependence on the polar angle 6
Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX
Fatigue Limit
Stress Amplitude
Safety Factor
8 £
0,8
H
i
0,6
10
20
30
40
50
60
70
Polar Angle [deg]
Fig. 18: Safety factor against fatigue limit in dependence on the polar angle 6
- • — Bending Test - • — Torsion Test ~4—Comb. 0° Test "-•—Comb. 90° Test
1000
10000
- -s - -^ --L- -e -
100000
Bending Femfat Torsion Femfat Comb. 0° Femfat Comb. 90° Femfat
1000000
10000000
Number of cycles
Fig. 19: S/N-curves for tempering steel with additional corrections for in phase loading
237
238
C. GAIER AND H. DANNBAUER
The result quality for combined in phase loading has been moved in the right direction, but still it has not reached the test result. Additional improvements are not possible anymore, because a further increase of the fatigue limit at ^ = 0° would also change the result for pure bending. But the result for bending should not be affected, because it has been already adapted to the test result.
SUMMARY AND OUTLOOK The critical plane criterion is proved to deliver good results for brittle materials, but it is shown that it has some drawbacks when applying it to ductile ones in combination with out of phase loading. A method, where a new material parameter has been introduced, is proposed to overcome these problems. This method can be applied also for stochastic load combinations. Additionally a method has been introduced for the assessment of both normal and shear stress for stochastic nonproportional loadings. Advantages of the proposed "critical plane critical component" method are: - hidependency from coordinate system - Normal and shear stress and all possible combinations are taken into account. - rainflow-counting can be applied for lifetime evaluation without any problems. - Applicable for both brittle and ductile materials by specifying cyclic data in dependence on the polar angle 0 (S/N-curves, Haigh-diagrams, etc.) - The interpolation function between tension/compression and torsional load can be adapted to test resuhs. - Generally applicable for biaxial and triaxial stress (or strain) states - Physical interpretation possible as „critical component" of stress vector Disadvantages are: - High computational effort - Sometimes additional corrections are necessary for out-of-phase loading (e.g. for tempering steel). Alternatively, integral methods can be applied instead of a maximum search, to obtain an averaged value for the lifetime. This has been already proved for infinite lifetime evaluation (safety against fatigue limit) of hard metals, e.g. by Zenner et al. [9-10] and Papadopoulos et al. [12]. For lifetime prediction till crack initiation, further investigations are necessary. Future efforts take aim to extend the proposed method by integral solution algorithms. The methods, as presented here, are based merely on stresses and therefore applicable only to the high cycle fatigue domain (HCF) for lifetimes > -5000. Nevertheless, in principle these methods could be extended for low cycle fatigue (LCF) too by exchanging stresses for strains and introducing suitable damage parameters. Such strain based fatigue evaluations need to be combined with usually extensive non-linear elastic-plastic stress-strain analyses. Experiences in this direction are still outstanding.
Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX
239
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12.
13. 14. 15.
16.
17.
Eichlseder W. (1989), Rechnerische Lebensdaueranalyse von Nutzfahrzeugkomponenten mit der Finite Elemente Methode, Dissertation, University of Technology Graz, Austria. Eichlseder W. and linger B. (1994), Prediction of the Fatigue Life with the Finite Element Method, SAE Paper 940245. Unger B., Eichlseder W. and Raab G. (1996), Numerical Simulation of Fatigue Life - Is it more than a prelude to tests?, Proc. Fatigue'96, Berlin. Steinwender G., Gaier C. and Unger B. (1998), Improving the Life Time of Dynamically Loaded Components by Fatigue Simulation, SAE Paper 982220, pp. 465-470. Gaier C., Unger B. and Vogler J. (1999), Theory and Application of FEMFAT - a FEPostprocessing Tool for Fatigue Analysis, Proc. Fatigue'99, Beijing, pp. 821-826. Gaier C , Steinwender G. and Dannbauer H. (2000), FEMFAT-MAX: A FE-Postprocessor for Fatigue Analysis of Multiaxially Loaded Components, NAFEMS-Seminar Fatigue Analysis, Wiesbaden, Germany. German FKM Guiding Rule (1998), Rechnerischer Festigkeitsnachweis fuer Maschinenbauteile, VDMA Verlag Frankfurt/Main, 3"^^ edition. Nokleby J. O. (1981), Fatigue under Multiaxial Stress Conditions, Rep. MD-81001, Div. Mach. Elem., The Norway Inst. Technol., Trondheim,. Zenner H., Heidenreich R. and Richter I. (1985), Fatigue Strength under Nonsynchronous Multiaxial Stresses, Z. Werkstofftech. 16, Germany, pp. 101-112. Sanetra C. and Zenner H. (1991), Betriebsfestigkeit bei mehrachsiger Beanspruchung unter Biegung und Torsion, Konstruktion 43, Springer-Verlag, Germany, pp. 23-29. Chu C. C , Conle F. A. and Huebner A. (1996), An Integrated Uniaxial and Multiaxial Fatigue Life Prediction Method, VDI Berichte Nr. 1283, Germany, pp. 337-348. Papadopoulos I. V., DavoH P., Gorla C , Filippini M. and Bemasconi A. (1997), A comparative study of multiaxial high-cycle fatigue criteria for metals. Int. J. Fatigue, Vol. 19, No. 3, pp. 219-235. Socie D. F. and Marquis G. B. (2000), Multiaxial Fatigue, SAE, Warrendale, U.S.A. Matsuishi M., and Endo T. (1968), Fatigue of Metals Subjected to Varying Stress, Proc. Kyushi Branch JSME, pp. 37-^0. Simbuerger A. (1975), Festigkeitsverhalten zaeher Werkstoffe bei einer mehrachsigen phasenverschobenen Schwingbeanspruchung mit koerperfesten und veraenderlichen Hauptspannungsrichtungen, Fraunhofer Institute Structural Durability (LBF), Darmstadt, Germany, Report Nr. FB-121. Sonsino C. M. and Grubisic V. (1985), Mechanik von Schwingbruechen an gegossenen und gesinterten Konstruktionswerkstoffen unter mehrachsiger Beanspruchung, Konstruktion 37, Springer-Verlag, Germany, pp. 261-269. Sonsino C. M. (2001), Influence of load deformation-controlled multiaxial tests on fatigue life to crack initiation. Int. J. Fatigue 23, Elsevier, pp. 159-167.
Appendix : NOMENCLATURE d
Unit direction vector
dc
Unit direction vector of the critical stress component
dM
Degree of multiaxiality
240
C GAIER AND H. DANNBAUER
Kf
Fatigue notch factor
Kt
Stress concentration factor
n
Unit normal vector of a plane
He
Unit normal vector of the critical plane
A^;,
Normal stress component of S„
Nn,d
Normal stress component of S^,^
R
Stress ratio
SFe
Safety factor against fatigue limit for direction d inclined by 6 to the plane's normal n
SM
Sensitivity of multiaxiality
S„
Stress vector in the plane specified by its unit normal vector n
Sn,d
Stress component in direction d of stress vector acting on plane n
t
Time
Tn
Shear stress component of S„
Tn,d
Shear stress component of S„,^
6
Polar angle spanned by n and d
^
Normal stress
^
Maximum principal stress
^
Minimum principal stress
^
Alternating tension/compression fatigue limit
^
Tension/compression fatigue limit including mean stress influence
(J Q
Fatigue limit for direction d inclined by 6 to the plane's normal n
^
Mean tension/compression stress
^
Compession ultimate strength
^
Tension ultimate strength ^
Shear stress
^
Alternating shear fatigue limit
^
Shear fatigue limit including mean stress influence
Tr
Mean shear stress
'^uts
Shear ultimate strength
4. DEFECTS, NOTCHES, CRACK GROWTH
This Page Intentionally Left Blank
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
243
THE MULTIAXIAL FATIGUE STRENGTH OF SPECIMENS CONTAINING SMALL DEFECTS
Masahiro ENDO Department of Mechanical Engineering, Fukuoka University, Jonan-ku, Fukuoka 814-0180, Japan
ABSTRACT A criterion for multiaxial fatigue strength of a specimen containing a small defect is proposed. Based upon the criterion and the ^area parameter model, a unified method for the prediction of the fatigue limit of defect-containing specimens is presented. In making this prediction, no fatigue testing is necessary. To validate the prediction procedure, combined axial and torsional loading fatigue tests were carried out using smooth specimens as well as specimens containing holes of diameters ranging from 40 to 500 fim which acted as artificial defects. These tests were conducted under in-phase loading condition at i? = -1. The materials investigated were annealed 0.37 % carbon steel, quenched and tempered Cr-Mo steel, high strength brass and nodular cast irons. When the fatigue strength was influenced by a defect, the fatigue limit was determined by the threshold condition for propagation of a mode I crack emanating from the defect. The proposed method was used to analyze the behavior of the materials, and good agreement was found between predicted and experimental results. The relation between a smooth specimen and a specimen containing a defect is also discussed with respect to a critical size of defect below which the defect is not detrimental. KEYWORDS Multiaxial loading, fatigue thresholds, small defects, small cracks, -[area parameter model, steels, brass, cast irons.
INTRODUCTION Over a number of years a great deal of effort has been expended in the attempt to establish reliable predictive methods for the determination of the fatigue strength under both uniaxial and multiaxial loading conditions. However, prior to the 1970's, the methods proposed did not provide a useful mean for the analysis of materials which contained either non-metallic inclusions or small flaws that are usually encountered in engineering applications. This was in part because most of the proposed methods were applicable only to two-dimensional cracks or
244
M ENDO
notches of simple shapes, whereas actual inclusions are often of a three-dimensional irregular shape. In addition, whereas the large crack problem has attracted attention in fatigue studies since the birth of fracture mechanics, the behavior of small cracks could not be analyzed in a similar way, for their behavior has been found to be anomalous with respect to large cracks, as pointed out by Kitagawa and Takahashi [1]. These authors, in the first quantitative characterization of the fatigue threshold behavior of small cracks, showed that the value of AK\\x decreased with decreasing crack size. This finding led the development of many subsequent studies on small or short cracks. Since their initial work a number of models and predictive methods for the determination of the fatigue strength of defect-containing components have been proposed, although most of these have dealt only with uniaxial fatigue. These models have been reviewed in detail by Murakami and Endo [2]. Research has shown [3,4] that the fatigue strength of metal specimens containing small defects above a critical size is essentially determined by the fatigue threshold for a small crack emanating from the defect. Based upon this consideration, Murakami and Endo [4] used linear elastic fracture mechanics (LEFM) to propose a geometrical parameter, 4area , which quantifies the effect of a small defect. Using this parameter they succeeded in deriving a simple equation [5] for predicting the fatigue strength of metals containing small defects. Subsequently, this model, referred to as the 4area parameter model, has been successfully employed in the analysis of a number of uniaxial fatigue problems which dealt with small defects and inhomogeneities [6,7]. However, in many applications, engineering components are often subjected to multiaxial cyclic loading involving combinations of bending and torsion. A number of studies have been concerned with this topic [8-13], but with the exception of pure torsional fatigue very few studies have been directed at the study of the behavior of small flaws under multiaxial fatigue loading conditions despite the importance of small flaws in design considerations. Nisitani and Kawano [14] performed rotating bending and reversed torsion fatigue tests on 0.36 % carbon steel specimens which contained defect-like holes of diameters ranging from 0.3 to 2 mm. They reported that the ratio of torsional fatigue limit to bending fatigue limit, ^ = r^ / cr^, was about 0.75 and attributed the result to the ratio of stress concentrations at the hole edge at fatigue limits; that is, 3 cr^ under bending and 4 r^ under torsion. (Here r^ and a^ are the fatigue strengths of specimens containing small flaws in reversed torsion and tension, respectively.) Mitchell [15] also predicted ^ = 0.75 for specimens having a hole in the similar way. Endo and Murakami [16] drilled superficial holes which simulated defects ranging from 40 to 500 |im in diameter in 0.46 % carbon steel specimens to investigate the effects of small defects on the fatigue strength in reversed torsion and rotating bending fatigue tests. Based upon the observation of cracking pattern at the holes, they correlated the fatigue strength under torsion with that under bending by comparing the stress intensity factors (SIFs) of a mode I crack emanating from a two-dimensional hole. They predicted ^ = -0.8 for specimens containing a surface hole. In that study, they also observed that there was a critical diameter of a hole below which the defect was not detrimental to the fatigue strength, and that the critical size under reversed torsion was much larger than under rotating bending. In recent papers [17-19], the further application of the -Jarea parameter to multiaxial fatigue problems has been made. Combined axial-torsional fatigue tests were carried out using annealed 0.37 % carbon steel specimens containing a small hole or a very shallow notch [17]. It was concluded that the fatigue strength was related to the threshold condition for propagation of a mode I crack emanating from a defect, and an empirical method for the prediction of the fatigue limit of a specimen containing a small defect was proposed [17]. Murakami and Takahashi [18] analyzed the fatigue threshold behavior of a small surface crack in a torsional
The Multiaxial Fatigue Strength of Specimens Containing Small Defects
IAS
shear stress state and extended the use of the ^area parameter to mixed-mode threshold problems. In addition, Nadot et al. [19] have discussed the extension of Dang Van's multiaxial fatigue criterion [20] to the defect problem by using the -Jarea parameter. Beretta and Murakami [21,22] used numerical analysis to calculate the stress intensity factor (SIF) for a three-dimensional mode I crack emanating from a drilled hole or a hemispherical pit under a biaxial stress state. By comparing with the previous experimental data [17], they concluded that the value of SIF at the tip of a crack emanating from a defect determined the fatigue strength of a specimen which contained a small defect above the critical size subjected to combined stresses. The present author [23] subsequently proposed a new criterion for fatigue failure which was also based upon the SIF. This criterion was expressed in the form of an equation which, by including within the criterion the -Jarea parameter model, provided a unified method for predicting the fatigue strength of a metal specimen containing a small defect. The applicability of the method was investigated with an annealed steel [23] and nodular cast irons [23,24]. The essence of this approach will be presented in the present paper. The principal objective of this study is to determine the generality of the author's predictive method [23] with additional experimental newly obtained data. In the present study the relation between the fatigue strengths of smooth specimens and specimens containing defects in multiaxial fatigue will also be discussed.
BACKGROUND FOR THE PREDICTION OF THE MULTIAXIAL FATIGUE STRENGTH
The ^area parameter model Murakami and Endo [4] have shown that the maximum value of the SIF, K^^^^^, at the crack front of a variety of geometrically different types of surface cracks can be determined within an accuracy of 10% as a function of ^area , where the area is the area of a defect or a crack projected onto the plane normal to the maximum tensile stress, see Fig. 1. The expression for ^imaxas a function of area (Poisson's ratio of 0.3) is:
area
Maximum tensile stress direction
Fig. 1. Definition of area.
246
M ENDO
^imax =0.6500-0 V W ^
(1)
where CTQ is the remote applied stress. Thereafter Murakami and Endo [5] employed the Vickers hardness value as the representative material parameter and showed that the threshold level for small surface cracks or defects could be expressed by the following equation for uniaxial loading at the stress ratio, R, of-1: A^,, =l>3x\0-\HV
+ \2Q){4area)"'
(2)
In addition, they found that the fatigue limit could also be expressed as a function of 4area by: ^ 1 . 4 3 ( / / F + 120)
where AAr,^, the threshold SIF range, is in MPa Vm , CTW, the fatigue limit stress amplitude, is in MPa, HV, the Vickers hardness, is in kgf/mm^ and ^[area is in |im. Equations (2) and (3) were derived on the basis of LEFM considerations. More recently a justification for the exponents of 1/3 in Eq.(2) and -1/6 in Eq.(3) was provided by McEvily et al [25] in a modified LEFM analysis which also considered the role of crack closure in the wake of a newly formed crack. The prediction error involved in the use of Eqs (2) and (3) is generally less than 10 percent for values of V area less than 1000 jim, and for a wide range of HV [5,6]. Murakami and co-workers [26-28] further extended Eq.(3) to include the location of the defect, i.e., whether it was at the surface or sub-surface, and also to include the effect of mean stress. The generalized expression they developed for the fatigue strength is: _ C ( / / F +120) \-R (si area) 1/6
(4)
where the value of C depends on the location of the defect being 1.43 at the surface, 1.41 at a subsurface layer just below the free surface, and 1.56 for an interior defect. The value of the exponent a was related to the Vickers hardness by a= 0.226 + HV x 10"^. Equations (2)-(4) are useful for practical applications in that they require no fatigue testing in making predictions. The yfarea parameter model has been applied to deal with many uniaxial fatigue problems including the effects of small holes, small cracks, surface scratches, surface finish, non-metallic inclusions, corrosion pits, carbides in tool steels, second-phases in aluminum alloys, graphite nodules and casting defects in cast irons, inhomogeneities in super clean bearing steels, gigacycle fatigue, etc. They are summarized in detail in the literature [6,7].
Criterion for multiaxial fatigue failure of defect-containing specimens The author has previously shown [17,23,24] that, in in-phase combined axial and torsional fatigue loading tests, the fatigue limit for specimens containing small defects is determined by the threshold condition for propagation of a small crack emanating from a defect. The materials investigated were an annealed 0.37 % carbon steel [17,23] and nodular cast irons [23,24].
The Multiaxial Fatigue Strength of Specimens Containing Small Defects
247
Figure 2 shows typical examples of non-propagating cracks observed in those materials at the fatigue limit. The direction of a non-propagating crack is approximately normal to the principal stress, cTi, regardless of combined stress ratio, TIG. Under a stress slightly higher than the fatigue limit, a crack which propagated in a direction normal to a\ resulted in the failure of the specimen. Based upon such observations, the fatigue limit problem for specimens containing small surface defects when subjected to combined stress loading was considered to be equivalent to a fatigue threshold problem for a small mode I crack emanating from defects in the biaxial stress field of the maximum principal stress, ai, normal to the crack and the minimum principal stress, 02, parallel to the crack. Consider an axi-symmetric surface defect containing a mode I crack under the remote biaxial stresses, cjy and a^, as shown in Fig. 3. The mode I SIF, K\, at the crack tip is given by the
50 jim Axial direction
50jim , - ^ , i (a) 0.37% carbon steel; a ^ /? = 100 \xm, cTa 175 MPa, ra=87.5MPa[17].
;^3i.r Axial direction (b) FCD700 nodular cast iron; smooth specimen, cJa = ^a = 160 MPa [24].
Fig. 2. Small non-propagating cracks emanating from a defect observed at fatigue limit under combined loading.
+
(3=
Fig. 3. A three-dimensional defect leading to a crack subjected to biaxial stress.
248
M ENDO
following superposition.
where FIA and FIB are the correction factors for the cases A and B in Fig. 3, respectively, and c is the representative crack length. It is hypothesized that the threshold SIF range under biaxial stress, A^th,bi, is equal to that under uniaxial stress, AArth,uni, or: A/:,,., = A/:„„„
(6)
This criterion has previously been used by Endo and Murakami [16] in the correlation of the pure torsional fatigue limit, w, the biaxial fatigue limit, with the rotating bending fatigue limit, ow; the uniaxial fatigue limit, for specimens having a small hole at the surface. Based upon this criterion, Beretta and Murakami [21,22] predicted that ^ , the ratio of the fatigue limit in torsion to that in tension, i.e., T^la^, for a mode I crack emanating from a three-dimensional surface defect under cyclic biaxial stressing should have a value between 0.83 and 0.87. They found that the predicted value of (j> agreed well with previously reported experimental results for various steel and cast iron specimens which contained small artificial defects. For fully reversed loading; /.e., i? = -1, AA\h,bi and A/Cth,uni were expressed using Eq.(5) as A^th,bi = ^lA (2cT,) V ^ + F,B (2cT2 ) V ^
(7)
A^.H,u„i=^.A(2cTjV^
(8)
where G\ and 02 are the maximum and minimum principal stress amplitudes resulting from the combined stress at fatigue limit, respectively, and ow is the threshold stress amplitude for a mode I crack under tension-compression cyclic loading; that is, the uniaxial fatigue limit of a specimen containing the same sized defect under /? = -1 loading. When the crack length, c, under uniaxial loading is equal to that under biaxial loading, Eq.(6) is reduced to G\ + kOi = CTw
(9)
where k = FIB/FIA, and represents the effect of stress biaxiality. If the torsional fatigue limit is designated by TW, since a\ = -02 = rw, then ^ = TJCTW = 1/(1 - k). Equation (9) as well as Eq.(6) provides a criterion for fatigue failure of specimens containing small defects when subjected to multi-axial loading. For round-bar specimens subjected to combined axial and torsional loading, Eq.(9) can be expressed as (\/(Pf(Tja^f
+ (\/(/>- l)(c7a/ow)' + (2 . l/^)(cTa/aw) = 1
(10)
where ok and Ta are the normal and shear stress amplitudes, respectively, at the fatigue limit under combined loading. Equation (10) is identical in form to Gough and Pollard's "ellipse arc" relationship [29], which has been used to fit the experimental data for brittle cast irons and specimens with a large notch [29,30]. The ellipse arc is empirical, and as such it requires fatigue tests for the determination of CTW and rw. In contrast, in the case of small defects, aw can
The Multiaxial Fatigue Strength of Specimens Containing Small Defects
249
be predicted using Eq.(3) without the need for a fatigue test. In addition, the value of (j) can be estimated by stress analysis, as verified by Beretta and Murakami [21,22]. If the average value ^calculated by Beretta and Murakami is used, i.e., 0.85, Eq.(lO) becomes: 1.38(ra/crw)^ + 0.176(cra/crw)^ + 0.824(cra/crw) = 1
(11)
This expression is considered to be applicable to specimens containing a round defect on or near the surface where plane stress condition is satisfied. The use of this expression will be demonstrated in the following.
MATERIALS AND EXPERIMENTAL PROCEDURE The materials investigated in this study are: steels, nodular cast irons and a high strength brass. Although experimental data for an annealed steel and the cast irons have previously reported elsewhere [23,24], they will be included below for purposes of discussion. The chemical compositions of the various metals are listed in Table 1. The 0.37 % carbon steel (JIS S35C) was annealed at 860°C for 1 hour. The Cr-Mo steel (JIS SCM435) was heat-treated by quenching from 860°C followed by tempering at 550°C . The two nodular cast irons (JIS FCD400 and FCD700) have different matrix structures; ferritic for FCD400 and almost pearlitic for FCD700. The cast irons and the brass were used in the as-received condition. Mechanical properties are given in Table 2, and microstructures are shown in Fig. 4.
Table 1. Chemical composition (wt.%)
S35C steel SCM435 steel FCD400 cast iron FCD700 cast iron High strength brass
C Si Mn 0.37 0.21 0.65 0.36 0.30 0.77 3.72 2.14 0.32 3.77 2.99 0.44 Cu Pb 59.1 0.0030
P 0.019 0.027 0.008 0.023
S 0.017 0.015 0.018 0.11 Fe 0.0032
Ni Cr Cu 0.13 0.06 0.14 0.02 0.02 1.06 0.04 0.47 Mn Zn 0.021 bal.
Mo 0.18 -
Mg
0.038 0.058 Al 0.0047
Table 2. Mechanical properties
Annealed S35C steel Quenched/tempered SCM435 steel FCD400 cast iron FCD700 cast iron High strength brass
Tensile strength MPa 586 1030 418 734 467
Elongation % (Gage length: 80 mm) 25 14 25 8.0 42
Vickers hardness HV (kgf/mm^) 160 380 190(ferrite) 330 (pearlite) 110
250
M ENDO
lOO^im Transverse section Longitudinal section (a) Annealed S35C steel
Transverse section Longitudinal section (b) Quenched/tempered SCM435 steel
100 ^im
100 pm
Transverse section Longitudinal section (c) High strength brass
100 ^im
FCD400 FCD700 (d) Nodular cast irons
Fig. 4. Microstructures. Figure 5 shows the geometries of the smooth specimens. They have a uniform cross section which was either 8.5 or 10 mm in diameter, and 19 mm in length. After surface finish with an emery paper of grade #1000, about 30 jam of a thickness was removed by electro-polishing in order to remove the work-hardened layer. During electro-polishing the presence of inclusions or graphite nodules at the surface led to the development of undesired pits which were larger in size than the defect or nodule. These undesired pits were removed by polishing the surfaces of smooth specimens with alumina paste after electro-polishing. A small hole shown in Fig. 6 was then drilled into the surface of a number of specimens. Such specimens will be referred to as hole-containing specimens in this paper. The diameter d of the holes ranged from 40 to 500 jim. The depth h was equal to the diameter d\ so that the defects were geometrically similar. The hole-containing Cr-Mo steel specimens were electro-polished after drilling of a hole of 80 \mi in diameter in order to make a larger pit having a 4area of 83 |im. The other smooth and holecontaining specimens were slightly electro-polished to remove a surface layer of 1-2 |im in thickness before the fatigue tests. For steels, it was confirmed by an X-ray stress analyzer that the residual stresses on the specimen surface were almost zero. Uniaxial load tests were carried out using either a servo-hydraulic uniaxial fatigue testing machine with an operating speed of 50 Hz or a rotating bending testing machine with 57 Hz. Another servo-hydraulic axial/torsional fatigue testing machine which ran at 30-45 Hz was used for combined load and pure torsional fatigue tests. Pure torsional fatigue tests of FCD400 cast iron were conducted with a Shimadzu TBIO-B testing machine of the uniform moment type at 33 Hz with the specimen shown in Fig. 5c. All tests were performed under in-phase ftilly
251
The Multiaxial Fatigue Strength of Specimens Containing Small Defects
4 strain gauges attached
& ^
,_
\ ^ / in>^
1
20
20
50
(a) For combined axial/torsional load test or reversed torsion test 40
(c) For reversed torsion test
^/Hii J8!. 80
50 210
80
(d) For rotating bending test
Fig. 5. Shapes and dimensions of smooth specimens.
cy= /? = 40 ~ 5 0 0 / i m
'area=y/d{h-d/4V3)
Fig. 6. Hole geometries.
reversed (R = -1) loading and a sinusoidal waveform. The combined stress ratios of shear to normal stress amplitude, r/a, were chosen to be 0, 1/2, 1,2 and oo. For the tension-compression tests, in order to eliminate bending stresses each specimen was equipped with four strain gauges to facilitate proper alignment in the fixtures. The nominal stresses were defmed as (J = 4P /(TTD'^ ) for tension-compression
(12)
a = 32A/^ /(TTD^ ) for rotating bending
(13)
T = 16M, l{nD^) for reversed torsion
(14)
where a is the normal stress amplitude, r is the torsional shear stress amplitude, P is the axial load amplitude, Mb is the bending moment amplitude, M is the torsional moment amplitude and D is the specimen diameter. The fatigue limits under combined stress are defmed as the combination of the maximum nominal stresses, Ta and cTa, under which a specimen endured 10
252
M ENDO
cycles for a fixed value of r/a. The minimum increment in stress level in determining the fatigue limit was 5 MPa for greater value of crand r, except for Cr-Mo steels where 10 MPa was used.
RESULTS AND DISCUSSION
Behavior of small cracks at the threshold level Figures 7 and 8 show the non-propagating cracks emanating from a hole at the fatigue limit of hole-containing Cr-Mo steel and brass specimens. As seen in these figures, the direction of non-
±o. -•—I
Axial direction
50 ^m (a) Tension-compression; ow = 340 MPa.
Axial direction
50 \xm (b) Combined loading; da = ra = 200 MPa.
±o. Axial direction
mg^^^mmM:-^'mmm^ - B I B B (c) Pure torsion; TW = 320 MPa Fig. 7. Small non-propagating cracks emanating from hole at fatigue limit of quenched/tempered SCM435 steel; 4area = 83 |im (c/= 90 |im).
The Multiaxial Fatigue Strength of Specimens Containing Small Defects
253
^^^^^^^^^^^MM
Axial direction
200 {im (a) Combined loading; cja = Ta = 70 MPa
p^lillil^ipiiiWIK^ iiiiiiplii»%.
^11
'13
Fig. 1. Stress state at a notch tip (notation). Non-proportional load histories require the application of the theory of incremental plasticity. The Hooke law and the Prandtl-Reuss flow rule are the most frequently used models. For an isotropic body, the two models together give the basic incremental stress-strain relationship.
"
E
•' E
" '
2 a'
"
(3)
The multiaxial incremental stress-strain relation (3) is obtained from the uniaxial stress-strain curve by relating the equivalent plastic strain increment to the equivalent stress increment such that:
dKcr" )
(4)
The function, ZQ^= f(aeq), is identical to the plastic strain - stress relationship obtained experimentally from an uni-axial cyclic tension-compression test.
268
A. BUCZYNSKIAND G. GLINKA
THE NOTCH TIP STRESS-STRAIN RELATIONS The load or any other parameter representing the load is usually given in the form of the nominal or reference stress being proportional to the remote applied load and the stress concentration factor Kt. However, the use of the stress concentration factor, Kt, is not sufficient in the case of multiaxial stress states near the notch tip because it supplies information about only one stress component. Therefore the fictitious "linear elastic" stresses which would exist near the notch tip in the absence of plasticity are used in the method discussed below. In the case of notched bodies in plane stress or plane strain state the relationship between the load and the elastic-plastic notch tip strains and stresses in the localized plastic zone is often approximated by the Neuber rule [1] or the Equivalent Strain Energy Density (ESED) equation [2]. It was shown later [3, 4] that both methods can be extended for multiaxial proportional and non-proportional modes of loading. Similar methods were also proposed by Hoffman and Seeger [ 5] and Barkey et al. [6 ]. All methods consist of two parts namely the constitutive equations and the relationships linking the fictitious linear elastic stress-strain state (aij^,8ij^) at the notch tip with the actual elastic-plastic stress-strain response (aij^8ij^), as shown in Fig. 2.
Fig. 2. Stress states in geometrically identical elastic and elastic-plastic bodies subjected to identical boundary conditions. The Neuber rule [2,3] for proportional loading, where the Hencky stress-strain relationships are applicable, can be written for the uni-axial and multiaxial stress state in the form of equation (5a) and (5b) respectively. 0
(17)
The essential elements of the plasticity model can be presented in such a case graphically in a two-dimensional principal stress space. The load path dependency effects are modeled by prescribing a rule for the translation of ellipses in the a2^-a3^ plane. The translation of these ellipses is assumed to be caused by the sought stress increment, represented in the principal stress space as a vector. The ellipses can be translated with respect to each other over distances dependent on the magnitude of the stress/load increment. The ellipses move within the boundaries of each other, but they do not intersect. If an ellipse comes in contact with another, they move together as one rigid body. However, it has been found that the ellipses in the original Mroz model may sometimes intersect each other, which is not permitted. Therefore, Garud proposed [10] an improved translation rule that prevents any intersections of plasticity surfaces. The principle idea of the Garud translation rule is illustrated in Fig. 6. a) The line of action of the stress increment, Aa^, is extended to intersect the next larger nonactive surface, fj, at point B2.
An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...
275
b) Point B2 is connected to the center, O2, of the surface f2. c) A line is extended through the center of the smaller active surface , Oi, parallel to the line O2B2 to find point Bi on surface fi. d) The conjugate points Bi and B2 are connected by the line B1B2. e) Surface fi is translated from point Oi to point Oi' such that vector OiOi' is parallel to line B1B2. The translation is complete when the end of the vector defined by the stress increment, Aa, lies on the translated surface fi'.
Fig. 6. Geometrical illustration of the translation rule in the Garud incremental plasticity model. The mathematics reflecting these operations can be found in the original paper of Mroz [9] or Garud [10] or in any recent textbook on the theory of plasticity. The Mroz and Garud models are relatively simple but they are not very efficient numerically, especially in the case of long load histories with a large number of small increments. If the computation time is of some concern the model based on infinite number of plasticity surfaces proposed by Chu [11] can be used in lengthy fatigue analyses. The cyclic plasticity models enable the relationship, ASeq^'-Aagq^, to be established providing the actual plastic modulus for given stress/load increment, AGJ. In other words the plasticity model determines which piece of the stress-strain curve (Fig. 5) is to be utilized during given stress/load increment. Two or more tangent ellipses translate together as rigid bodies and the largest moving ellipse indicates which linear piece of the constitutive relationship should be used for a given
276
A. BUCZYNSKI AND G. GLINKA
Stress increment. The slope of the current linear segment of the stress-strain curve defines the plastic modulus, Aaeq/Aseq'', necessary for the determination of the parameter, dX, in the constitutive equation (10). The plasticity models are described in most publications, as algorithms for calculating strain increments that result from given series of stress increments or vice versa. This is called as the stress or strain controlled input. In the case of the notch analysis neither stresses nor strains are directly inputted into the plasticity model. The input is given in the form of the total deviatoric strain energy density increments and both the deviatoric strain and stress increments are to be found simultaneously by solving the equation set (12). Therefore, the plasticity model is needed only to indicate which work-hardening surface is to be active during current load increment, which subsequently determines the instantaneous value of the parameter dX. In order to find the actual elastic-plastic deviatoric stress and strain increment ASy^ and Ae/ from the equation set (12), the value of parameter dX is determined first according to the current configuration of plasticity surfaces. After calculating all stress increments, ASjj^, and subsequently, Aay^, the plasticity surfaces are translated as shown in Fig. 6. The process is repeated for each subsequent increment of the "elastic" input, Aay^. The Mroz [9] and Garud [10] models were chosen here as an illustration. Obviously, any other plasticity model can be associated with the incremental stress-strain notch analysis proposed above. The Garud plasticity model was employed in the analysis discussed below. COMPARISON OF CALCULATED ELASTIC-PLASTIC NOTCH TIP STRAINS AND STRESSES WITH ELASTIC-PLASTIC FINITE ELEMENT DATA The component used for the numerical validation was a cylindrical specimen (Fig. 7) with circumferential notch subjected to simultaneous tensile and torsion loading. The basic dimensions of the cylindrical component were p = 3 mm, R = 70 mm and t = 3 mm resulting in the torsional and tensile stress concentration factor K^ = 1.82 and Kta = 2.80 respectively. The ratio of the notch tip hoop to axial stress under tensile loading was ^^^^Icsii = 0.2179.
R=70 mm, t=3mm, p=3 mm
Kt =2.80, Kt =1.82, a,,/a„=0.2179
Fig. 7. Cylindrical specimen with circumferential notch subjected to tension-torsion loading The stress concentration factors for the axial and torsional loads were defined as:
An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...
K^
— K, =
and
32
277
(18)
The nominal stresses in the net cross section were determined as:
4R-tf
and
T„ =-
2T
4R-tf
(19)
The elastic-plastic finite element stress and strain stress results were obtained using the ABAQUS finite element package with built-in plasticity model. The torque T induced the 'linear elastic' shear stress G23^ at the notch tip and the axial load F induced the normal axial stress component a22^ and the normal hoop stress ass^. The fictitious elastic normal stress components maintained constant ratio throughout the entire loading history, i.e. a337a22^=const. The increments of the hypothetical "elastic" stress components Aa23^, Aa22^ and AGBS^ and associated strains were used as the input into the equation set (12). Two linear segments as shown in Fig. 8 were used to approximate the material stress-strain curve. The material properties were E = 200000 MPa, Ei= 4142.50 MPa, v = 0.3 and ay = 200 MPa.
E=200000 MPa Ei=4142.5MPa v=0.3
Fig. 8. Material elastic-plastic stress-strain curve The maximum applied load levels F and T were chosen to be higher than it would be required to induce yielding at the notch tip if each load was applied separately. The axial-shear notch tip elastic stress paths, a22^-a23^ applied to the notched component were those shown in Fig.9a and 9b. There were 10 full loading cycles applied in the case of the Stress Path #1 and 6 full cycles in the case of the Load Path #2. The FEM calculations for the Stress Path #1 required 72 hours CPU time on a Personal Computer with 800 MHZ processor. The finite element mesh of the notched component is shown in Fig. 10. The calculated from eq. (12)
A. BUCZYNSKI AND G. GLINKA
278
and the FEM determined axial and shear strain components, 822^ and 823^, and the axial and shear stress components, a22^ and a23^, respectively are shown in Figs. 11a - 12b. The resuhs corresponding to the Stress Path #1 (Fig. 9a) are shown in Figs. 11a and l i b . Note, that the calculated stresses and strains and the finite element data are identical in the elastic range. Load path #1
Axial elastic notch tip stress, MPa
Fig. 9a. Non-proportional load/stress path #1 Load path #2
25
50
75
100
125
150
175
200
-25 H -50 -75 J Axial elastic notch tip stress, MPa
Fig. 9b. Non-proportional load/stress path #2
225
250
275
300
An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...
279
Fig. 10. Finite element model of the notched component Load Path #1
0.002
Axial strain
Fig. 1 la. Evolution of the elastic-plastic axial strain and axial stress at the notch tip induced by the Stress Path #1
280
A. BUCZYNSKIAND G. GLINKA
Q.
-0.0012
^
0.0008
0.0012
-120 Shear strain
Fig. 1 lb. Evolution of the elastic-plastic shear strain and shear stress at the notch tip induced by the Stress Path #1 Load path #2
,>|?002 y /
0.0025
0.003
-FEM I -ENERGY!
Axial strain
Fig. 12a. Evolution of the elastic-plastic axial strain and axial stress at the notch tip induced by the Stress Path #2
An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic .
281
Load Path #2
0.0025
-120 Shear strain
Fig. 12b. Evolution of the elastic-plastic shear strain and shear stress at the notch tip induced by the Stress Path #1 Just above the onset of yielding at the notch tip, the strains predicted using the proposed model and the finite element data gradually deviate form the linear elastic behavior. Both sets of results reveal some ratcheting of the axial strain (Fig. 1 la); however it appears that the proposed model stabilizes quicker than the FEM data based on the ABAQUS software package plasticity model. The data obtained for the Stress Path #2 are shown in Figs. 12a and 12b. The proposed model overestimates the notch tip strains in comparison with the FEM data similarly to the classical uniaxial Neuber rule. However, this overestimation is not significant and it could be acceptable in many practical applications.
CONCLUSIONS A method for calculating elastic-plastic strains and stresses near notches induced by multiaxial loading paths has been proposed. The method has been formulated using the equivalence of the total distortional strain energy density. The generalized equations of the total equivalent strain energy density yielded a conservative solution for the notch tip strains and stresses in the case of cyclic non-proportional loading paths. The method has been verified by comparison with finite element data obtained for identical notched components subjected to non-proportional loading paths. The notch tip strains and stresses calculated for cyclic load paths can be used for the estimation of fatigue lives for multiaxial cyclic loading histories. The calculated from equation (12) and the FEM determined axial and shear strain components, Z2i and 823^, and the axial and shear stress components, a22^and a23^, respectively are shown in Figures 11 and 12.
282
A. BUCZYNSKI AND G. GLINKA
REFERENCES 1. Neuber, H., " Theory of Stress Concentration of Shear Strained Prismatic Bodies with Arbitrary Non Linear Stress-Strain Law", ASME Journal ofApplied Mechanics, vol. 28, 1961, pp. 544-550. 2. Molski, K. and Glinka, G., " A Method of ElasticPlastic Stress and Strain Calculation at a Notch Root", Material Science and Engineering, vol. 50, 1981, pp. 93-100. 3. Moftakhar, A., Buczynski, A. and Glinka, G., "Calculation of Elasto-Plastic Strains and Stresses in Notches under Multiaxial Loading", International Journal of Fracture, vol. 70, 1995, pp. 357-373. 4. Singh, M.N.K., "Notch Tip Stress-Strain Analysis in Bodies Subjected to NonProportional Cyclic Loads", Ph.D. Dissertation, Dept. Mech. Eng., University of Waterloo, Ontario, Canada, 1998. 5. Seeger, T. and Hoffman, M., "The Use of Hencky's Equations for the Estimation of Multiaxial Elastic-Plastic Notch Stresses and Strains", Report No. FB-3/1986, Technische Hochschule Darmstadt, Darmstadt, 1986. 6. Barkey, M.E., Socie, D.F. and Hsia, K.J., "A Yield Surface Approach to the Estimation of Notch Strains for Proportional and Non-proportional Cyclic Loading", ASME Journal of Engineering Materials and Technology, vol. 116, 1994, pp. 173. 7. Moftakhar, A. A., "Calculation of Time Independent and Time-Dependent Strains and Stresses in Notches", Ph. D. Dissertation, University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, Canada, 1994. 8. Chu, C. -C, "Incremental Multiaxial Neuber Correction for Fatigue Analysis", International Congress and Exposition, Detroit, 1995, SAE Technical Paper No.950705, Warrendale, 1995. 9. Mroz, Z., "On the Description of Anisotropic Workhardening", Journal of Mechanics and Physics of Solids, vol 15, 1967, pp. 163-175. 10. Garud, Y. S., "A New Approach to the Evaluation of Fatigue under Multiaxial Loading, Journal of Engineering Materials and Technology, ASME, vol. 103, 1981, pp. 118-125. 11. Chu, C. -C, "A Three-Dimensional Model of Anisotropic Hardening in Metals and Its Application to the Analysis of Sheet Metal Forming", Journal of Mechanics and Physics of Solids, vol.32, 1984, pp. 197-212. 12. Barkey, M. E., "Calculation of Notch Strains under Multiaxial Nominal Loading", Ph. D. Dissertation, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, 1993.
Appendix: N O M E N C L A T U R E E
- modulus of elasticity
El
- plastic modulus
ey^
- actual elastic-plastic deviatoric strains at the notch tip
Cij^
- hypothetical elastic devaitoric strains at the notch tip
FEM
- finite element eethod
G
- shear modulus of elasticity
An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ...
KtCT
- Stress concentration factor due to axial load
KtT
- stress concentration factor due to torsional load
k, n
- load increment number
5ij
- Kronecker delta, Sy = 1 for i = j and Sy = 0 for i T^ j
AsyP
- plastic strain increments
Asy^
- elastic strain increments
A8y^
- actual elastic-plastic strain increments
Aseq^^
- equivalent plastic strain increment
Aay^
- pseudo-elastic stress components
Aay^
- actual stress components
Aaeq^ - actual equivalent stress increment Sy^
- deviatoric stresses of the elastic input
Sy^
- actual deviatoric stresses
Seq''^
- actual equivalent plastic strain
8y^
- actual elasto-plastic notch-tip strains
sy^
- elastic notch tip strain components
Sn
- nominal strain
V
- Poisson's ratio
Qeq^
- actual equivalent stress at the notch tip
ay^
- actual stress tensor components in the notch tip
Gy^
- notch tip stress tensor components of the elastic input
GY
- parameter of the material stress-strain curve (yield strength)
P
- axial load
T
- torque
R
- radius of the cylindrical specimen
283
This Page Intentionally Left Blank
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
285
THE BACKGROUND OF FATIGUE LIMIT RATIO OF TORSIONAL FATIGUE TO ROTATING BENDING FATIGUE IN ISOTROPIC MATERL4LS AND MATEIUALS WITH CLEAR-BANDED STRUCTURE
Takayuki FUKUDA^ and Hironobu NTSITANI^ Department of Mechanical Engineering, Sasebo National College of Technology, Sasebo, 857-1193, Japan Department of Mechanical Engineering, Kyushu Sangyo University Fukuoka, 813-8503, Japan
ABSTRACT The fatigue limit ratios of torsional to rotating bending fatigue (zjoy^) have been determined in several steels. This ratio was found to be equal to about 0.55 in an annealed rolled carbon steel (C=0.2, 0.45%) and about 0.65 in an annealed cast carbon steel, in a diffusion annealed carbon steel (C=0.45%) and in a quenched & tempered carbon steel (C=0.45%). The difference between the two values is due to the microstructure. The annealed rolled carbon steels delivered as round bars have a clear-banded structure of ferrite and pearlite. The other three materials have no banded structure. Namely, the former materials are regarded as anisotropic, while the latter ones are considered as isotropic materials. The clear-banded structure in the axial direction greatly affects the torsional fatigue limit, but it does not affect the rotating bending fatigue limit. This is due to the fact that, in the carbon steel with a clear-banded structure, large local strain concentration occurs within the ferrite in torsional fatigue but not in rotating bending fatigue. Because of this fact, the torsional fatigue limit in a banded structure relatively decreases and, therefore, the value of fatigue ratio TW/CTW becomes small. KEYWORDS Fatigue limit ratio. Torsional fatigue. Rotating bending fatigue. Isotropic material. Banded structure, Carbon steel.
INTRODUCTION A large number of studies have been made on the fatigue behavior under combined stresses [1-8]. In particular, Nishihara [2], Gough [3] and Findley [4] have studied the fatigue strength under torsional and rotating bending combined stresses. They proposed methods according to which the fatigue limit under general combined stresses is calculated from individual fatigue limits Qwandiw, where aw is the rotating bending fatigue limit and TW is the torsional fatigue limit. Since the fatigue limit under general combined stresses is usually discussed according to the
286
T. FUKUDA AND H. NISITANI
individual fatigue limits, it is important to understand the physical meaning of the fatigue limit ratio, zj cjw, of torsional to rotating bending fatigue. Most of the results of the above studies have been discussed using macroscopic criteria for plastic yielding. For example, it is said that the fatigue limit ratios zj Oy^oi many ductile materials follow the maximum distortion energy criterion (Mises' criterion; TS/(TS=0.58, where Is, as are the yield strengths measured in shear and tensile loading, respectively)[9]. In many experimental results, however, the ratio of fatigue limits ( zjo^) does not always follow this macroscopic criterion of yielding. In carbon steels, which are ductile materials, the ratio varies from 0.55 to 0.7 depending on the microstructure. According to the tensile and torsional static tests of two kinds of carbon steels in which one is a fully annealed rolled carbon steel having a clear-banded structure and another is an annealed cast carbon steel having no banded structure, the clear-banded structure greatly affects the local strain concentration in torsion, but it hardly affects that in tension [10]. Taking into consideration these facts, rotating bending and torsional fatigue tests have been carried out on plain specimens of several kinds of carbon steels which have clear-banded structure or not, and the fatigue limit ratio ( rjaw) values are herein discussed based on whether the specimen has a clear-banded structure or not. FATIGUE LIMIT RATIO OF BENDING AND TORSIONAL FATIGUE [DATA MAINLY OBTAINED IN THE PAST] Classification of fatigue limit Table 1 shows experimental data, mainly obtained in the past, of fatigue limit ratio of torsional to bending fatigue for various kinds of materials. The value of fatigue limit ratio zj aw varies between 0.5 and 1 depending on the material. As is well known, fatigue process depends on crack initiation and crack propagation. Crack initiation is controlled by the maximum shear stress ( Tmax) while crack propagation is mainly related to the maximum tensile stress ( a max). They are related to the microstructure or properties of material in a complex way. Because of these facts, there are three different cases concerning the dependence of fatigue limit. (a) The fatigue limit is controlled by the maximum tensile stress, amax(b) The fatigue limit is controlled by the maximum shear stress, imax(c) The fatigue limit is controlled by both amax and Tmax In the case where the fatigue limit is controlled by the maximum tensile stress only (defective material) In a defective material, defects, for example inclusions or small holes, are regarded as a kind of crack. Thus, in this case, crack initiation can be ignored and only crack propagation can be considered. Gray cast iron or nodular cast iron can be treated in this way, since flake or spheroidal graphite is regarded as a defect. In this case, the main factor deciding the fatigue limit is the maximum tensile stress (amax). In torsion, the principal stress ( ai) is equal to the principal shear stress( r i), as is shown in Fig. 1. Therefore, if the fatigue limit is controlled by the maximum tensile stress (amax) only.
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...
287
Table 1. Fatigue limit ratio between bending and torsional fatigue Material
Mechanical properties
Heat treatment gs
Op
' s
^
Fatigue limit HB,HV
a«
r «•
0^
239
138
0.58
265
147
0.56
Carbon steel
Annealed
L292
521
-
45.1
Carbon steel n
Annealed
L228
376
72.9
72.9
Normalized
L353
705
32.7
32.7
194
294
196
0.67
NiCr steel
Nothing
U482
704
59.2
59.2
237
392
255
0.65
Thermal refined
U879
962
44.9
44.9
295
530
304
0.57
"
HB103
Cast iron
Nothing
-
205
127
103
0.81
70/30 brass
Nothing
U337
409
70.4
70.4
122
131
54
0.41
Carbon steel
Normalized
U252
424
178
70.0
HB127
264
149
0.56
356
638
255
58.2
195
327
204
0.62
Spheroidized
283
470
178
67.0
144
271
154
0.57
Normalized
834
248
347
237
0.68
518
-
18.0
850t:oil que., 7001:air cooled
-
72.5
165
337
202
0.6
"
SSO'Coil que., 610'Cair cooled
578
711
422
67.5
237
438
263
0.6
CrVa steel
850t:oil que., 7001: air cooled
670
740
468
66.0
229
423
254
0.6
NiCr steel
830'Coil que., 620'C water cooled
748
882
574
65.0
282
532
347
0.65
"
" " " Ni steel
"
216
SSO'Coil que., two step cooled
743
883
-
60.5
278
502
319
0.64
NiCrMo steel
SSO'Coil que., 600'Cair cooled
1224
898
60.0
394
651
337
0.52
NiCr steel
8201: and 2001: air cooled
1642
479
798
445
0.56
Nothing
-
52.0
Silal cast iron
-
0
322
237
216
0.91
216
1.3
182
249
208
0.84
68.0
-
501
302
0.6
581
327
0.56
658
384
0.58 0.55
1 Nicrosolal "
Nothing
226
CrMo steel
900t:oil que., 580t:oil que.
844
939
NiCrMo steel
830'Coil que., 575X^011 que.
949
1087
CrMoVa steel
900T:oil que., 600t:oil que.
-
1376
NiCr steel
830t:oil que., 200'Cair cooled
970
NiCrMo steel
845'Coil que., 677'Ccooled
L876
70/30 brass
Annealed
113
WT80C
Thermal refined
761
809
Nothing
240
477
1 Nodular cast iron
54.0
1368
-
64.0
-
657
364
776
452
-
HRC25
462
286
0.62
320
-
73.1
83.4
73.5
0.88
461
275
0.6
10.6
-
196
186
0.95
CrMo steel
850'Cque., 600'Ctempering
U923
1041
" "
850X3que., 450X3tempering
1243
1373
850T3que., 300X3 tempering
1500
1756
850t3oil que., 630X3tempering
883
974
I NiCrMo steel Carbon steel n
845X1 water que., 6(K)X^tempering
U564
748
825X^ water que., 6U0X)tempering
700
860
Cr Steel
855X^oil que., 550X1 water cooled
935
1051
855X;oiI que., 6(X)X^water cooled
838
969
855X;oil que., 550X^water cooled
1046
1118
1081
1139
"
CrMo steel
66.4
61.3
Hv331
589
368
0.62
51.1
434
699
397
0.57
41.8
535
791
456
0.58
63.9
314
539
351
0.65
Nishihara,T(2)
Gough,H.J.(3)
Firth,P.H.(5)(6)
Findley,W.N.(7)
Matake,T(8)
Nishijima,S. (11)(12)
68.0
Hv245
415
284
0.69
283
458
318
0.69
61.0
328
582
401
0.69
NRIM fatigue
62.0
299
522
358
0.69
data sheet
59.0
362
586
405
0.69
(12)
62.0
372
610
416
0.68
60.0
339
85513oil que., 60()X1 water cooled
975
983
549
382
0.7
Carbon steel
Annealed
324
570
1170
51.3
235
140
0.59
Nodular cast iron
Nothing
343
596
-
3.7
206
186
0.9
Age-hardened
250
309
479
48.0
131
65
0.5
I Al-alloy
Nishihara,T(l)
58.0
-
" "
"
-
61.0
Reference
-
(13) Nisitani,H.(14)
as, Is- Yield stress(L:Lower, U:Upper)(MPa), o^\ Ultimate tensile strength (MPa), ^ : Reduction of area (%), HB,HV,HR : Hardness, Gw, T w: Fatigue limit (MPa)
(15)
288
T. FUKUDA AND H. NISITANI
f Fig.l. Principal stress and principal shear stress in torsion
the relation between both fatigue limits of bending and torsion becomes (Jw='^w in torsion. Then, the fatigue limit ratio of torsional fatigue to bending fatigue is obtained by the following equation: ^w/C^w=l
(1)
This equation corresponds to the maximum tensile stress criterion, which is related to the macroscopic criterion of yielding ( rs/as = 1). A defective material, which contains sharp defects like gray cast iron, mostly follows the Eq.(l). However, in the artificial defects of small circular holes or nodular cast irons, it is necessary to consider the stress concentration of a circular hole. Since the maximum stress due to the stress concentration of a circular hole is equal to approximately 3 a in bending and 4 r in torsion ( o max= "^ max), the ratio of fatigue limits is reduced to the following equation: L:^/a^ = 0.75
(2)
Nisitani presented some experimental results for an annealed rolled carbon steel bar with a small hole [13]. In the case of a single hole specimen, r J a ^ w a s equal to approximately 0.75, whereas it was equal to about 0.9 in the case of the connected hole specimen. It was also shown that Zj o^ was equal to approximately 0.9 (not 0.75) for a nodular cast iron. This means that spheroidal graphite is not always independently distributed and is regarded as a kind of connected hole. As is mentioned above, the fatigue limit of a defective material including cast iron is mainly controlled by the maximum tensile stress, and the value of fatigue limit ratio ij a^ follows Eq.(3), depending on the type or distribution of defects, etc. z^,/0^ = 0.75 ^ 1
(3)
In the case where the fatigue limit is controlled by the maximum shear stress only (non-defective material) If the cracks once initiated do not stop propagating, the fatigue limit is determined by whether a crack initiates or not. According to the successive observations by Nisitani, age-hardened Al-alloy corresponds to this case [15]. In this material a crack initiates in a fairly small region
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...
( ' ^ l / x m ) like a point, and gradually grows as the number of cycles N increases. Then the crack does not stop propagating due to the work softening effect [16], which finally causes the specimen to break. This phenomenon is quite different from the case of an annealed carbon steel, where fatigue process can clearly be divided into crack initiation and crack propagation [16], Figure 2 shows schematic illustrations of the crack initiation process of the age-hardened Al-alloy and annealed metals [17]. In Fig.2(a), the starting region of fatigue cracking of an age-hardened Al-alloy is much smaller than a grain size and extends gradually toward the grain boundary. In this material, it is difficult to distinguish the small slip band from a microcrack, or the initiation process from the propagation process. On the other hand, in Fig.2(b), the crack initiation process of the an annealed low carbon steel is entirely different from the crack propagation process. Until the initiation of a crack, fatigue damage is accumulated gradually in the same region (shaded area in Fig.2(b)), dimensions of which are closely related to the grain size, and then the region (a grain boundary or a slip band) turns into a crack as a whole. Since the factor controlling crack initiation is the maximum shear stress, the fatigue limit in this case is determined by only the maximum shear stress. The value of imax for the fatigue
Surface
r_[ }-•'N/Ni=0
l Fracture surface
(a)
Age-hardened Al-alloy , Surface
N/Ni=0
1 Fracture surface
(b)
Annealed low carbon steel, a -brass, Al-alloy
Fig.2. Schematic illustrations of fatigue crack initiation process [17]
289
290
T. FUKUDA AND H. NISITANI
limit in bending is equal to that in torsion. Thus considering o^.J 'rniax=2 in bending and o^.^J rniax=l in torsion, the fatigue limit ratio of torsional fatigue to bending fatigue is obtained by the following equation: r , , / a ^ = 05
(4)
This equation corresponds to the maximum shear stress criterion, which is related to the macroscopic criterion of yielding ( zJo^={)5).
In the case where the fatigue limit is controlled by both Of^ax cL^d tmax In annealed metals, the fatigue process can be divided into two different processes, i.e., an initiation stage and a propagation stage, as is shown in Fig.2 (b) [18,19]. (a) Process ( I ): By slip repetitions, the slip band or the grain boundary (or the part near the grain boundary), which is going to become a crack, is disrupted as a whole and is gradually turned into a free surface. Then a crack of the size of a crystal grain initiates. (b) Process (11): The crack initiated in process ( I ) increases in length and depth, and finally causes the specimen to break. As is mentioned above, since process ( I ) is controlled by the maximum shear stress and process ( A ) is controlled by the maximum tensile stress, the factor deciding the fatigue limit is not simple to be determined. That is, the fatigue limit in both bending and torsion is determined by the limting stress for propagation of a non-propagating microcrack. Considering a max/ '^max=2 in bending and a^ax/ '^max=l in torsion, the crack once initiated in bending is easier to propagate than that in torsion. Consequently, since the fatigue limit in bending becomes small compared to that in torsion, the value of fatigue limit ratio is more than 0.5, and the following equation holds on the basis of our present data: r^/a^ = 055^0.7
(5)
The fatigue limit ratios have frequently been discussed based on the maximum distortion energy criterion (Mises' criterion, rs/as = 038), which is related to the macroscopic criterion of yielding [9,20]. In this case, since the fatigue limit depends on the microstructure and the material properties, and is controlled by both crack initiation and crack propagation, the factor deciding the fatigue limit is complex to be determined. Therefore, it is difficult to apply the macroscopic criterion of yielding for deciding the fatigue limit. In the present study, through the successive observations of fatigue processes and the grid line method, the physical background of the value of r J a^ in carbon steel will be made clear.
MATERIALS AND EXPERIMENTAL METHODS The materials used are rolled carbon steel (S20C, S45C) and cast carbon steel (SC450). The chemical composition is shown in Table 2. Three kinds of specimens were prepared from a rolled round carbon steel bar (S45C) by heat treatment: the first kind is annealed (S45C), the second kind is diffusion annealed (S45C-DA) and the third one is quenched and tempered (S45C-H). Conditions of heat treatment and mechanical properties are shown in Table 3. The microstructures are shown in Table 4 in the following. In the longitudinal section, the banded
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in . 291
Structures of ferrite and pearlite areas are recognized in annealed rolled carbon steels (S20C, S45C), and not in cast carbon steel (SC450) nor in S45C-H or S45C-DA material. Namely, the former two are regarded as anisotropic materials, and the latter three are regarded as isotropic materials (see Table 4). In both tensile and torsional static tests, two types of specimens (S45C, SC450) were used. The diameter of the specimen was 8 mm, and all specimens were electropolished to the depth of about 20/xm to remove the surface layer. For the observation of the specimen surface, plastic replicas were taken from the surface after applying definite macroscopic strains. The surface states were taken from the replicas to negative films with the optical microscope, then digitized by using a film scanner and stored in the computer's memory. The local strain was measured using the image-processing method [21]. The macroscopic overall strains for the tensile tests are £g= 3, 6, 9, 12 %, and the macroscopic shear strains for the torsional test are Tg = 4.5, 9, 13.5, 18 %. The local strain for tensile test corresponds to the normal logarithmic strain which is calculated from the elongation of an axial grid line, and the local strain for torsional test corresponds to the shear strain which is calculated from the change of an intersectional angle of two grid lines. The fine grid lines were also drawn on the surface of the specimen with a diamond point needle [10]. The width of the lines was made less than 1 /i m and the depth of the lines less than 0.5 U m. The mesh size of the grid is 20 X 20 Mm. In the rotating bending and torsional fatigue tests, we used all the specimens shown in Table 3. The diameter of each specimen was 8 mm for SC450, S45C-DA and S45C-H, and 5 mm for
Table 2. Chemical composition
(%)
Material
C
Si
Mn
P
S
Cu
Ni
Cr
S20C
0.21
0.21
0.47
0.014
0.017
0.21
0.06
0.09
S45C
0.44
0.20
0.69
0.009
0.009
0.007
0.04
0.005
SC450
0.21
0.37
0.70
0.008
0.005
Table 3. Heat treatment and mechanical properties
Type Anisotropy
Isotropy
Material
Heat treatment
Osl
^B
S20C
890°C Ihr -^ F.C.
276
469
864
58.7
S45C
845°C Ihr -^ F.C.
335
569
1061
57.2
SC450
920°C Ihr ^ F.C. 1200°C 6hr -^ F.C. 845°C 0.5hr -> A.C. 845°C Ihr -> F.C. 845°C 0.5hr ^ A.C. 845°C Ihr -^ W.C. 600°C Ihr ^ W.C.
271
462
793
56.2
328
560
1050
55.5
528
751
1571
52.2
S45C-DA S45C-H
Osl '• Lower yield stress MPa o^:Ultimate tensile strength MPa Oj'.Tvuc fracture strength MPa ^ :Reduction of area %
Oj
^
292
T. FUKUDA AND H. NISITANI
S20C and S45C. After turning, the specimens were annealed in vacuum at 650°C for one hour, and were then electropolished to the depth of about 20 /i m to remove the surface layer. EXPERIMENTAL RESULTS AND DISCUSSION Local normal strain in tensile static test Figures 3 and 4 show the changes of surface state in the tensile tests of S45C and SC450 materials. The axial direction is the loading direction. It can be seen that square grids are deformed into nearly rectangular grids due to the axial elongation by tension. The degrees of their deformations are almost the same, since circumferential grid lines are nearly straight. Namely, there is not a large difference between the deformations of ferrite and pearlite bands. Consequently, the strain concentration in tension is small.
(a)
fg = 0 %
(b)
£g=12%
Fig.3. The changes of surface state in the tensile test of S45C
Axial direction
(a)
fg = 0 %
(b)
£g=12%
Fig.4. The changes of surface state in the tensile test of SC450
60 /i m
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...
50
t Axial direction I I I I I I I I I S45C(Tension)
I II
X CO
I I I M I I I I I 1 2345678910 Axial line number n (a) S45C
t Axial direction I I I I I II SC450(Tension) •
O ' ' 0 0 0 / ^ I I I I I I I I ? I 1 2345678910 Axial line number n (b) SC450
Fig.5. Local strains in tensile test ( £ g = 12%)
Figure 5 shows local strains at £g=12%, which are calculated from the elongation of one hundred axial grid lines (initial length of around 20/im). The ordinate (Tmax=l-5 £ ) is the maximum local shear strain in tension. The abscissa is the location number for grid lines. That is, the abscissa is the axial line number n shown in Fig.3 and 4. The local strains of ten axial grid lines are plotted in Fig.5. Each horizontal line indicates the average of one hundred values for maximum local shear strains, which approximately coincides with the applied macroscopic strain (Tgmax=l-5X 12=18%). The local strains are not uniform. That is, there exist various local strains, which are from several percent to some thirty percent, although the applied macroscopic shear strain is 18%. As can be seen in Fig.5, local strains in the tensile test are not uniform but the strain concentration due to the banded structure is small.
Local shear strain in torsional static test Figures 6 and 7 show the changes of surface state in the torsional test of both materials. It is recognized that square grids are deformed into nearly lozenge-shaped grids due to the distortion caused by torsion. Deformation of each grid is not uniform and circumferential lines are not straight. Especially in S45C steel, there is a large difference between the deformations of ferrite band and those of pearlite band. That is, as in the case of deformation of an elastic body sandwiched between two rigid bodies, the strain concentrates within the ferrite band sandwiched between two pearlite bands. Figure 8 shows the electron micrographs of the surface state of the specimens having the fine grid lines drawn with a diamond needle. In SG450 steel, circumferential lines are almost straight. In S45C steel, however, those are not straight and deformation of each grid is not uniform. There is a large difference between the deformation in ferrite band (gray portion) and pearlite band (white portion). Figure 9 shows one hundred values of the local shear strain measured at an overall shear strain of Tg=18%, which are calculated by the same method as that used in the tensile static test. The local strains in torsion are not uniform in both materials, and there exist various values of local strains, which are from several percent to some forty percent, although the
293
T. FUKUDA AND H. NISITANI
294
(a)
7g=0%
(b)
yg=18%
Fig.6. The changes of surface state in the torsional test of S45C
(a)
7g = 0 %
(b)
7g=18%
Fig.7. The changes of surface state in the torsional test of SC450
(a)
S45C
(b)
SC450
Fig.8. The local deformation in the static torsion test (SEM, y g=18%)
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in ...
T Axial direction
t Axial direction
1
2345678910 Axial line number n (a) S45C
295
1
2345678910 Axial line number n (b) SC450
Fig.9. Local strains in torsional test ( T g = 18%)
applied macroscopic strain is 18%. The variation of local strains in torsion is larger than that in tension. In S45C steel, local strains are unevenly distributed. That is, most of them on each axial grid line have a tendency to be distributed only above and/or below the average line. When the axial grid line exists on the pearlite band, which is harder than ferrite, most of local strains on the line are distributed below the average line, and when the axial grid line exists on the ferrite band, most of local strains on the line are distributed above the average line. This means that the strain concentrates within the ferrite in S45C steel which has a clear banded structure.
Local deformation and crack initiation in torsional fatigue of carbon steel with clear-handed structure Cracks initiated near grain boundaries. Figure 10 shows the changes of surface state in the torsional fatigue test of S45C steel. This is an example of the crack initiated near a grain boundary. Figure 10(a) shows optical micrographs (X 200). In the early stage of stress cycling ('^O.OlNf ,Nf: fatigue life), the long and narrow shadow, which will later become a crack, already appears near the grain boundary. Its darkness increases with increasing number of cycles N without increase in its size, and it develops later into a long crack. As is stated above, this fatigue process can be divided into two stages, crack initiation and crack propagation. Figure 10(b) shows the same region observed by scanning electron microscopy. At first, the region of the ferrite crystal grain which later becomes a crack (the long and narrow region that is white in color) slips. Then the width of the slip band and the extent of disruption increase with increasing N. Due to the large deformation on one side of the line, shown by the arrow in the figure, the deformation by slip is quite large in spite of high-cycle fatigue. Judging from the deformation, it seems that the region becomes more active due to work softening after work hardening by slip. On the other hand, the region near the pearlite side is scarcely deformed. Therefore, because of the compensation of deformation, large strains are concentrated near the grain boundary between pearlite and ferrite. Consequently, the cracks apparently appear near the grain boundary.
296
T. FUKUDA AND H. NISITANI
From the above observation, it can be said that this type of crack initiation in the region near the grain boundary is due to the nonconformity of deformation between peariite and ferrite. Cracks initiated in ferrite crystal grains. Figure 11 shows an example of a crack initiated in the ferrite crystal grain of the same specimen. In the optical micrographs [Fig. 11(a)], the entire region of stratiform ferrite areas sandwiched between peariite colonies with a width of about 10 M m is greatly damaged and dark. According to the observation of electron micrographs [Fig. 11(b)], only the ferrite zones are greatly damaged, and the first crack appears in the most damaged crystal grain. This is also confirmed from the fact that the central line in the ferrite is greatly and continuously deformed on one side. Judging from the inclination of the line, it seems that the shear strain of the region amounts to several hundred %. However, the macroscopic shear strain corresponding to nominal stress is about 0.2% (y ). As in the case of deformation of an elastic body sandwiched between rigid bodies, the strain is concentrated in the layer of ferrite bands sandwiched
\ Axial direction (a) N=0 (N/Nf)
N=0.5 X 10^ (0.011)
Optical micrograph N=2X10^ (0.044)
N=3X10^ (0.066)
N=9X10^ (0.198)
(b) Electron micrograph Fig. 10. Crack near a grain boundary in torsional fatigue (S45C)
1 1 20/im N=22X10' (0.484)
10/i m
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in .
297
between pearlite colonies. As can be deduced from the above observations, this type of crack initiation is due to the fact that the entire region of ferrite suffers large amounts of concentrated cyclic strain caused by the nonconformity in the deformation between the nondeformed pearlite and deformed ferrite areas. It seems that the slipped portion in the ferrite is softened after some stress repetitions, and then the slip concentrated there initiates cracks. The cracks are not necessarily initiated near the grain boundary. In the area where the cracks are initiated, several hundred % local shear strain is confirmed, which accumulated on one side. The circumstance is similar to the case as is stated above. For the reasons mentioned above, almost all the cracks were initiated and propagated first in the axial direction in this material. The reason for this is that this material has a clear-banded structure of ferrite and pearlite, that is, the ferrite and pearlite grains of the specimens are distributed in layers parallel to the axial direction, as is observed in Table 4. The ferrite zones sandwiched between pearlite colonies become active as a whole. Consequently, the crack is
(a) Optical micrograph N=0 (N/Nf)
N=0.5 X lO'* (0.011)
N=2X10 (0.044)
N=3 X lO'* (0.066)
20 Aim
N=9X10^ (0.198)
N=22X10^ (0.484)
(b) Electron micrograph Fig. 11. Crack in the ferrite crystal grain in torsional fatigue (S45C)
\Qjim
298
T. FUKUDA AND H. NISITANI
easily initiated in the ferrite grain in torsional fatigue of carbon steel which has a clear-banded structure.
Fatigue test The S-N curves of all specimens are shown in Fig. 12. Table 4 shows the fatigue limits a^and Tw and the fatigue limit ratios of torsional fatigue to rotating bending fatigue (^Jo^) of all materials. In anisotropic materials which have a clear-banded structure (S20C, S45C), the ratios are 0.55^0.57, and in isotropic materials which have no banded-structure (SC450, S45C-DA, S45C-H), the ratios are 0.65^0.68. As is mentioned above, in the torsional fatigue of carbon steel, which has a clear-banded structure, the large local strain concentration occurs in the ferrite. On the other hand, in the rotating bending fatigue of the same steel, the local strain concentration hardly occurs in the ferrite band. This produces a relative reduction of the torsional fatigue limit. Namely, the fatigue limit ratio ('Ojo,^) depends on whether the specimen has a clear-banded structure or not, and does not follow the macroscopic criterion for plastic yielding (Mises' criterion; TJO^=0.5S). The fatigue behavior of a specimen is related to the local microstructural details of the specimen. On the other hand, the macroscopic yielding is related to the average properties of the specimen. This is the reason why the criterion for macroscopic yielding is not applicable to fatigue loading under combined stresses in general. In discussing the results of fatigue tests under combined stresses (bending and torsion), it is important to consider whether the material used has a banded structure or not.
Classification of fatigue limit ratio Tj o^ The fatigue limit ratio zj a^ varies between 0.5 and 1 depending on the material as is shown in Table 1. The fatigue process consists of crack initiation and crack propagation. Therefore, by considering crack initiation, crack propagation, microstructure and type of defect, the value of fatigue limit ratio can be classified as is shown in Table 5.
CONCLUSIONS (1) The local deformations are not uniformly distributed in both static tension and static torsion, and therefore most of local strains are not the same as the applied macroscopic strain. (2) A clear-banded structure greatly affects the local strain in static torsion but it hardly affects that in static tension. That is, large shear strain concentration in torsion occurs in the ferrite band sandwiched between pearlite bands but the mean normal strain in tension in a ferrite band is nearly equal to that of a pearlite band. (3) Because of the above conclusion, the fatigue cracks of carbon steel with a clear-banded structure are initiated more easily in torsional fatigue than in bending fatigue. That is the reason why the fatigue limit ratio of the steel is close to 0.58 (Mises' criterion). (4) The fatigue limit ratio ( x j a^) depends on whether the specimen has a clear-banded structure or not, and does not follow the macroscopic criterion of yielding. In discussing the results of fatigue tests under combined stresses (bending and torsion), it is important to consider whether the material used has a banded structure or not.
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in . 299 500
ouu
S20C
(0 0.
§. 4 0 0 | -
S45C
400
^^"•QBending
Bending
300
300 h
MPa a^=235
0
(&
0 •o
1 200 h
"5. E
Q.
E CO
MPa =245
200 • U Torsion MPa 1 -Cw=135
CO CO
0)
CO
0)
100 10'
1
inn 10^
10^
10^
10
lO''
10°
1 10'
10"
N u m b e r of cycles N
N u m b e r of cycles N
(a) S20C
(b) S45C 500
500 ^
1
10^
S45C-DA
SC450
400 h
^
400 Bending
300
300 h
MPa 0^=240
Bending 0
0
1 200 h
1 200
Q.
Q.
E
MPa| rTv,=i55
E
CO
CO
Torsion
U) CO 0
0) 100"— 10^
CO
10"
10"
10^
lO''
100 10'
Number of cycles N
J-
_L
10"^
10°
(d) S45C-DA
(c) SC450 500 ^^Vi
Bending
r-
MPa
S. 400 h 300 h
"
MPa T^=255
^ Torsion
i
10'
Number of cycles N
•fc
200
Q.
E CO
S45C-H
CO 100 10^
L_ lO''
_L
J_
lO''
10'
N u m b e r of cycles N
(e) S45C-H Fig. 12. S-N curves
10°
10'
300
T. FUKUDA AND K NISITANI
Table 4. Fatigue limits
REFERENCES 1. Nishihara, T. and Kawamoto, M. (1940), 'The Fatigue Test of Steel under Combined Bending and Torsion", Trans. Jpn. Soc. Mech. Eng., Vol.6, No.24, 1. 2. Nishihara, T. and Kawamoto, M. (1941), "The Strength of Metals under Combined Alternating Bending and Torsion", Trans. Jpn. Soc. Mech. Eng., Vol.7, No.29, 85-95. 3. Gough, H.J. (1949), "Engineering Steel under Combined Cyclic and Static Stresses", Proc. Inst. Mech. Eng., Vol. 160-4, 417.
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in .
301
Table 5. Classification of fatigue limit ratios i J o, Characteristics of material Fatigue Crack limit does initiates not depend from on defects Fatigue Shape limit of depends on defect defects
4.
Point region
Work softening
'^ w/ ^ w
0.5
Example material Age-hardened Al-alloy
Banded structure
0.55--0.6
Rolled carbon steel
Isotropic
0.65-0.7
Cast carbon steel
Single hole
0.75
Steel with a hole
Connected hole
0.9
Nodular cast iron
Finite region
Crack-like defect
1
Gray cast iron
Findley, W.N. (1957), "Fatigue of Metals Under Combination of Stresses", Trans. ASME, Vol.79-6,1337. 5. Firth, P.H. (1956), "Fatigue of Wrought High-Tensile Alloy Steels", Int. Conf Fatigue, Inst. Mech. Eng., 462-499. 6. McDiarmid, D.L (1991), "A General Criterion for High Cycle Multiaxial Fatigue Failure", Fatigue Fract. Eng. Mater. Struct., Vol. 14-4, 429-453. 7. Findley, W.N. (1956), "Theory for Combined Bending and Torsion Fatigue with Data for SAE4340 Steel",/«r. Conf Fatigue Metal, Inst. Mech. Eng., 150-157. 8. Matake, T. (1976), "A Consideration for Fatigue Limit under Combined Stresses", Trans. Jpn. Soc. Mech. Eng., Vol.42, No.359, 1947-1953. 9. Peterson, R.E. (1956), "Torsion and Tension Relations for Slip and Fatigue", Colloquium on Fatigue, International Union of Theoretical and Applied Mechanics, 186-195. 10. Nisitani, H. and Fukuda, T. (1994), "Non-Uniformity of Local Strain Concentration in Static Deformation of Plain Specimens of Rolled Round Carbon Steel Bars", Proc.4^^ ISOPE, 222-227. 11. JSME, (19S2)JSMEData Book, Fatigue of Material 1, 16-73. 12. Nisijima. S. (1977), National Research Institute for Metals Fatigue Data Sheets, Vol.19, 119, 227. 13. Nisitani, H. and Kawano, K. (1972), "Correlation between the Fatigue Limit of a Material with Defects and Its Non-Propagating Crack", Bull. JSME, Vol.15, No.82, 433-438. 14. Nisitani, H. and Murakami, Y. (1973), "Part of Spheroidal Graphite of Nodular Iron Casting under Bending or Torsional Fatigue", Research of Machine, Vol.25, No.4, 543-546. 15. Nisitani, H. and Goto, T. (1976), "Notch Sensitivity in Fatigue of an Al-Alloy", Trans. Jpn. Soc. Mech. Eng., Vol.42, No.361, 2666-2672. 16. Forsyth, P.J.E. (1963), "Fatigue Damage and Crack Growth in Aluminium Alloys", Acta Metallurggica, Vol.11, 703-715. 17. Nisitani, H. (1985), "Behavior of Small Cracks in Fatigue and Relating Phenomena", Current Research on Fatigue Cracks, Jpn. Soc. Mat. Sci., 1-22. 18. Nisitani, H. and Takao, K. (1974), "Successive Observations of Fatigue Process in Carbon Steel, 7:3-Brass and Al-Alloy by Electron Microscope", Trans. Jpn. Soc. Mech. Eng., Vol.40, No.340, 3254-3266. 19. Nisitani, H. (1968), "Fatigue under Two-Step Loading in Electropolished Specimen of
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T. FUKUDA AND H. NISITANI
S25C Steel", Trans. Jpn, Soc. Mech. Eng., Vol.34, No.258, 220-223. 20. Gough, H.J. and Pollard, H.V., (1935), "The Strength of Metals under Combined Alternating Stresses", Proc. Inst. Mech. Eng., Vol.131, 3-103. 21. Fukuda, T., Kawasue, K. and Nisitani, H. (1999), "Quantitative Measurement of Local Deformation using an Image Correlation Technique", Proc. Eighth International Conference on the Mechanical Behaviour of Materials, Vol.2, 471-476.
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
303
INFLUENCE OF DEFECTS ON FATIGUE LIFE OF ALUMINIUM PRESSURE DIECASTINGS
Fernando Jorge LINO\ Rui Jorge NETO^ Alfredo OLIVEIRA^ and Fernando Manuel Femandes de OLIVEIRA* Faculdade de Engenhaha, Universidade do Porto, Departamento de Engenharia Mecdnica e Gestdo Industrial Rua Dr. Roberto Frias, 4200-465 Porto, Portugal ^INEGI, Instituto de Engenharia Mecdnica e Gestdo Industrial, Rua do Barroco, 174-214, 4465-591 Lega do Balio, Porto, Portugal
ABSTRACT Fatigue life of aluminium pressure diecastings is strongly dependent on the microporosity level of the parts. Even in a very controlled production process, it is almost impossible to obtain aluminium parts w^ithout micropores, which means that a considerable amount of parts are rejected, in accordance to internal companies criteria. Although these criteria are based on standards, they change from company to company, and depend on the type of the parts and the amount, size and location of the micropores. Many times, parts that could have a good fatigue life are rejected based on these criteria. The aim of the present work is to study the influence of the microporosity level and size on the fatigue life of aluminium pressure diecastings. Two different lots of samples, removed from aluminium components (considered unacceptable and acceptable) were tested using the staircase fatigue test. All the fractured parts were analysed macro and microscopically and the images obtained were digitalized in order to classify the size and amount of the micropores. The results obtained were compared with the fatigue life curves, in order to evaluate the influence of the microporosity on the fatigue life of aluminium components. KEYWORDS Fatigue life, manufacturing defects, aluminium, staircase fatigue test, diecasting.
INTRODUCTION One of today's greatest challenges in the foundry industry is the production of complex and structurally sound parts. At the forefront of this challenge is the porosity and inclusions size and levels, and how and where they develop. The presence, even in small levels, of these types of defects in pressure diecastings can lead to a significant reduction in the tensile strength, ductility, pressure tightness and fatigue life, affecting the life and the integrity of the cast parts [1].
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FJ.LINOETAL
Three factors can lead to the presence of porosity: shrinkage, coupled with a lack of interdendritic feeding during mushy zone solidification, evolution of hydrogen gas bubbles due to a sudden decrease in hydrogen solubility during solidification, and collapsed air [1, 2]. Aluminium diecasting alloys present very interesting properties, namely: good machinability, low weight, low transformation cost with the possibility of obtaining complex shapes, and, moreover, they are recyclable. However, they are very prone to present casting defects. Although the recent developments in pressure diecasting industry (use of low injection velocities, special feeding [2, 3], vacuum and "true isostatic pressure" [1]) and the use of simulation processes contributed to the improvement of the aluminium cast parts quality (possibility of structural parts production), it is almost impossible to avoid the presence of defects in the parts that are supplied to the customers [4-6]. Considering this, foundry companies follow international standard criteria (for example, ASTM Standard E 505 [7]) and also develop internal standards to classify the parts as acceptable or unacceptable. Frequently, a location or size of one defect is not critical in one part, but is unacceptable in other type of parts. This is especially important in structural parts, where the loading type during service can conduct to fatigue initiation. Pressure aluminium diecasting alloys have extremely low ductility (1-3%) [4, 8], which means that once initiated, cracks easily propagate until the failure of the component [9]. In the light of the above, it is obvious the interest, emphasized by current studies [1, 5, 10], in evaluating the effect of the presence of defects on fatigue life reduction of pressure diecastings. For example, some authors have been trying to determine the relationship between casting conditions and the amount of porosity in a casting. The majority of the models capable of providing a qualitative description of the level of microporosity fail to give accurate values because the prediction of microporosity requires a detailed understanding of pore nucleation and growth in the melt [1]. In the aluminium pressure diecasting industry, the main type of defects susceptible of being observed are oxidized surfaces, foreign material inclusions (oxide), gas cavities (hydrogen and gas porosity), macro shrinkage (cavities and sponginess), and microshrinkage (feathery, sponge, intercrystalline or interdendritic) [6, 11]. Gas pores, shrinkage pores and gas-shrinkage pores represent the main types of porosity that are detected in diecastings. In gas pores, liquid aluminium reacts with water vapour in the atmosphere to produce aluminium oxide and hydrogen gas. Gas porosity arises during the solidification, due to the difference in solubility of hydrogen gas in liquid and solid aluminium. If a casting is poorly fed during solidification, shrinkage will cause a hydrostatic stress in the liquid metal. This stress increases until a pore forms with the aid of a nucleus. Gas-shrinkage pores results from the fact that gas evolution and shrinkage occur in the same volume of liquid metal at the same time [1]. In this paper, the effect of the presence of porosity on fatigue life reduction of aluminium pressure diecastings is quantified. Such an effect will then be used in a software, under development, which will be able to predict fatigue life of high strength castings [12]. One of the advantageous of this study is the fact that all the fatigue tests are conducted in samples removed from real components or in real components and not in samples pressure diecasted separately in the metallic moulds. These last samples frequently have a section thickness that does not represent the usual parts thickness obtained in pressure diecasting.
Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings
305
EXPERIMENTAL ANALYSIS OF DEFECTS To study the influence of the presence of defects on fatigue life of aluminium pressure diecastings, two lots of 50 parts each, considered acceptable and unacceptable, were supplied by a foundry company (SONAFI, Porto, Portugal). The selection was made in accordance to the foundry internal criteria for the part selected, and using adequate quality control techniques (visual control, X-Rays, etc.). Figure 1 presents the component selected, with the fatigue test sample placed in the position from where it was removed.
Fig. 1. Brake pedal in an aluminium diecasting alloy AS9U3 (NF A57-703) [8] with the fatigue test sample. Figure 2 shows an image of the X-Rays control performed in one pedal. As one can see, different defects sizes can be detected in this control.
Fig. 2. X-Rays control in a brake pedal. The central part of the pedal shows pores with different sizes and geometries. The fatigue test sample was removed from region A.
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FJ. LINOETAL
All the parts were pressure diecasted in the AS9U3 (NF A57-703) [8] aluminium alloy. The typical microstructure of this alloy consists essentially in a aluminium dendrites, aluminiumsilicon eutectic cells, and intermetallics AbCu, Al7Cu2Fe and AlFeSiMg [6, 11] (see Fig. 3). Table 1 presents the alloy chemical composition specified by the Standard NF A57-703 and the medium values measured in the components using a spark spectrometer. This table shows that there are no significant differences between the two compositions.
Fig. 3. Typical microstructure of the pressure diecasting AS9U3 alloy (etched with HF 0.5%). This microstructure is composed by a aluminium dendrites, aluminium-silicon eutectic cells, and intermetallics A^Cu, AlyCuiFe and AlFeSiMg.
Table 1. Chemical composition (%) of the aluminium alloy AS9U3 (NF A57-703) [8].
NF A 57-703
Fe
Si
Cu
Zn
Mg
Mn
Ni
Pb
Sn
Ti
1.3
7.5-10
2.5-4.0
A-1.2
0.3
0.5
0.5
0.2
0.2
0.2
0.2
0.23
0.05
0.15
0.04
0.02
B-1.3 Measured in
0.78
9.24
2.99
0.81
the components
Figure 4 presents the cross section of an optical micrograph showing interdendritic (region A) and gas cavities (region B) regions, and also a macrograph of a transversal section of an unacceptable pedal, where these defects can also be observed. As one can see, pressure diecastings present a significant amount of micropores. This fact is responsible for considerable research in the foundry area, in order to produce aluminium diecastings with better characteristics [1,5, 13-15].
Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings
307
Fig. 4. Porosity in pressure diecastings: a) optical micrograph indicating a region A of interdendritic cavities and a region B of gas cavities, and b) macrograph of a polished transversal section of an unacceptable pedal showing small and large pores.
Table 2 presents the main properties of the aluminium alloy AS9U3 (NF A57-702/703) in the non heat-treated condition [8]. Table 2. Properties of the AS9U3 (NF A57-703) pressure diecasting alloy [8]. Alloy Properties Gr (MPa)
Elongation (%) Density (g/cc)
Value 200 0.5-1.5 2.8
Image analysis was performed with software called PAQUI (developed by the Centre of Materials of University of Porto - CEMUP, Porto, Portugal) in an Olympus optical microscope. The analyses were done in 10 brake pedals (5 acceptable and 5 unacceptable), in the transversal section of the fatigue test samples region. A routine to perform a specific analysis was defined in order to count and measure the defects (considered spherical) in the samples section. 50 to 80fieldswere characterised in each sample, using an optical microscope with a 5x magnification. Very small defects, with diameter 200
Acceptable
Unacceptable
Fig. 6. Mean number of defects detected in each field analysed. The defects are divided in five area (|Lim) classes for acceptable and unacceptable samples. The mean reletive defects area obtained in acceptable parts was 1.9%, while for unacceptable ones was 2.9%. This is a very small difference between the two lots, which can be explained by the fact that frequently a single defect is the reason for a part rejection.
FATIGUE TESTS One fatigue sample was removed from each pedal by EDM and machined for the following dimensions, 90 mm length, 20 mm width and 3 mm thickness. The geometry of the sample was defined according to the ASTM Standard E 466 [16] and is represented in Fig. 7.
Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings
309
Fig. 7. Fatigue test sample based on the ASTM Standard E 466 [16]. Two tensile tests were performed in both sets of samples in order to determine the main mechanical properties of the AS9U3 aluminium alloy. The geometry of the sample for the tensile test was modified in accordance with Standard E 8M-89b [17]. A constant cross section of 10 mm and 3 mm thickness was adopted. The mechanical properties obtained were very similar for acceptable and unacceptable samples. Considering this, the values indicated on Tab. 3 represent the medium value obtained from the 4 tensile tests performed in both acceptable and unacceptable samples. These values are similar with the ones specified by the Standard (NF A57-703) [8] (see Tab. 2), which can be the result of the natural defects presence in pressure diecastings. Density values, obtained from three samples of each set, are also lower due to the presence of micropores in the components and due to the slight difference in the alloy chemical composition. This table also includes the hardness values, which are equal for both sets of samples.
Table 3. Measured properties of the aluminium alloy AS9U3 (NF A57-703). Main Properties Gr (MPa) ao,2 (MPa) E (GPa) Elongation (%) Density (g/cc) Acceptable/Unacceptable Hardness (HB)
Value 204 148 64 1 2.70/2.72 93
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FJ. LINOETAL
Fatigue tests were performed in a 250 kN Servo-Hydraulic MTS machine (see Fig. 8), and using the staircase method (DIN EN ISO 9964 [14]). The loading ratio aMin/c^Max selected for the tests was 0.1, and tests started with 50 % of the aluminium alloy tensile strength. Figure 9 presents one pedal and some broken samples.
Fig. 8. Servo-hydraulic MTS machine for the fatigue tests.
Fig. 9. Brake pedal and broken fatigue aluminium samples. To start the staircase fatigue test [16, 18], it was estimated that the fatigue limit was 50% of the tensile strength. The number of levels used for each set of 20 samples was 7. Tables 4 and 5 indicate the values employed in the test and the resuhs obtained, respectively. Surprisingly, the unacceptable samples only have a small decrease in the fatigue limit;
311
Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings
however, they present a slightly higher standard deviation. Figures 10 and 11 present the Woehler curves obtained (in accordance with the ASTM Standard E 468 [19]) using these fatigue limits and the best fit of fatigue tests with three different levels of alternating stress, using 5 samples for each level. Table 4. Staircase fatigue test parameters. Staircase Method Value Estimated fatigue limit 102 50 % Tensile Strength (MPa) First applied range of stress (MPa) 103 Frequency of the test (Hz) 20 Number of samples 20 7 Number of levels 0.1 CTMin/c^Max Table 5. Fatigue test results. Data Fatigue limit (MPa) Standard deviation
Acceptable | Unacceptable 87 91 5 7
Curve S-N
a>
ii)!.iiiiiiiii5mii;i*«»i N I .
1
1'limlii
i^ilfn
M iiii.'iiii'».M..«.M
n'i!i.i.i!ij'|illllni.ViMifi..MiiiiilM»
f i Mj'tiim
ll{l..^..„
80
^—^—«i
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;..,. v . . . .)?•».. ^'>...>.-,:«..\;*,M . ,:....„-,.,'.;:,7:'a.^,au,u
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^
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60
:rR--sS"fe^iS'tEF;
i: 40
(/)
20
100
» ^ ^ f ^ ^ l * - - , . . f 7-?*5 Aij''. ^'
1
/
'1
^
!
r^
(b)
(c)
'^ max > 65 MPa Fig. 14. Intensity maps of the maximum resolved shear stress in copper: (a) uniaxial extension 8yy=0.1 %, (b) uniaxial extension exx=0.1 %, (c) non-proportional uniaxial extension at first 8yy=0.1 % and then 8xx=0.1 %. An element is considered as damaged (in black) if once during the fatigue cycle Tmax exceeds 65 MPa.
Discussion The problem of the distribution of stress and strain in the polycrystal was modelled here using a thin sheet. The first reason for this choice was to limit the number of elements, in order to perform statistic analyses within realistic times and with a number of grains in the model
338
S. POMMIER
sufficient to let a load percolation network appear. The second reason is that micro-crack nucleation occurs mostly at the surfaces of the samples. However, in a massive sample, grains located under the surface, may contribute to homogenize the stress state within the polycrystal. It was shown before [15], with a rough model of a massive sample, that the scatter is lower in the bulk than at the surfaces. Nevertheless, the scatter at the surfaces is very similar in the case of a thin sheet and in the case of a grain located at the surface of a massive sample. This point would need further investigation, since the difference between the stress and strain heterogeneity at the surfaces and in the bulk of a sample could contribute, with environmental effects, to explain the preferential nucleation of micro-cracks at the surfaces.
CONCLUSIONS Probabilistic approaches are developed to describe scale effects and scatter in fatigue. When fatigue cracks are nucleated on defects, the origin of the scatter is clear. When defects are nondamaging, micro-cracks are usually nucleated by cyclic slip in "weak" grains. In this case, the origins of the scatter in fatigue lives remain unclear. Under bulk elastic conditions, a "weak" grain is a grain within which the maximum resolved shear stress on the slip systems is the highest, which happens when two conditions are satisfied: the Schmid factor of the grain is high and the stress applied on the "weak" grain is high. The object of the paper was to discuss the second condition. Since the elastic behaviour of the grains is anisotropic, the stress and strain distribution is heterogeneous in a polycrystal. The spatial distribution of this heterogeneity was studied using experiments and finite element analyses. The spatial distribution of strain at the surface of a sample of TA6V titanium alloy was observed using the photostress method. Though the material is fully elastic, fine inclined lines appeared at the surface of the sample, where the strain is higher than the mean one in the sample. However the direction of the principal strain remains mostly coincident with the load axis. This experiment showed that there is a scale associated with the strain heterogeneity in the TA6V titanium alloy, which is larger than the grain size, approaching 10 grains. In order to reveal a scale for the spatial distribution of strain in a different material, thin sheet of OFHC polycrystalline copper have been subjected to a cyclic creep test. After failure, fine inclined lines forming a regular pattern are observed at the surface of the sample, revealing a scale for the heterogeneity of strain larger than one millimetre. Finite elements calculations were performed, in order to understand the above-mentioned effects. A polycrystalline thin sheet was modelled by FEM analysis. These computations showed that a load percolation network, analogous to that observed in a granular material, is formed through the polycrystal. The load is transferred through heavily loaded links whose direction is coincident with the principal stress directions of the equivalent homogeneous problem. This network possesses an intrinsic scale larger than the grain size. The probability of a given value of the maximum principal stress within a grain was calculated using the FEM. One grain located at the centre of the thin sheet was set to have a fixed orientation, while the crystalline orientations of the other grains in the model were selected randomly before each calculation. The variability of the maximum principal stress under uniaxial loading conditions depends on the elastic anisotropy of the grain. For a given crystal orientation, the maximum principal stress vary up to +/- 35 % for zinc and copper and
Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?
339
of +/- 24 % for iron. This variability is very high as compared with the width of the distribution of the Schmid factor in FCC crystal. The spatial distribution of the maximum resolved shear stress on slips systems was calculated by the FEM, and compared with the distribution of the Tresca equivalent stress on the one hand and with the distribution of the Schmid factor in the model on the other hand. It can be concluded from these calculations that the importance for fatigue crack nucleation of this load percolation network depends on the elastic anisotropy of the material on the one hand, and on the number of primary slip systems on the other hand. If the number of primary slip systems and the elastic anisotropy of grains are high, the nucleation process should be dominated by the self-organization of the stress and strain heterogeneity within the polycrystal. On the contrary, if the number of primary slip system and the elastic anisotropy of grains are low, the nucleation process should be dominated by the crystalline orientation of grains. When the distribution of Xmax is dominated by the load percolation effect, some effects of that network may arise in multiaxial fatigue. It was shown, that with a similar mean value of the maximum resolved shear stress on slip systems in a polycrystal, two different loading conditions are not equivalent in terms of the nucleation of micro-cracks, since the maximum bounds for Xmax can be different. For example, these calculations show that with a similar mean value , the maximum bound for Xmax is higher in torsion as compared with tension. The grains with the highest value of imax are located around the intersections between the heavily loaded links associated with each principal direction. These grains are sparse but overstressed. The role in fatigue of the self-organized spatial distribution of stress and strain in the polycrystal, which is described in this paper, should also be important for crack coalescence during subsequent crack growth, since it controls the number of damaged grains per unit surface and their mutual distance.
REFERENCES 1.
Sines, G. and Ohgi, G. (1981). Fatigue criteria under combined stresses or strains. Journal of Engineering Materials and Technology 103, 82-90. 2. Dang Van, K. (1993). Macro-micro approach in high-cycle multiaxial fatigue. In: Advances in Multiaxial Fatigue, ASTM STP 1191, pp. 120-130, Mc Dowell, D.L. and Ellis, R. (Eds.), ASTM, Philadelphia. 3. Murakami, Y., Toriyama and T.,Coudert, E.M. (1994). Instructions for a New Method of Inclusion Rating and Correlations with the Fatigue Limit. Journal of Testing & Evaluation 22,318-326. 4. Hild, F., Billardon, R. and Beranger, A.S. (1996). Fatigue failure maps of heterogeneous materials. Mechanics of Materials 22, 11-21. 5. Beretta, S. (2001). Analysis of multiaxial fatigue criteria for materials containing defects. In: ICB/MF&F, pp. 755-762, de Freitas, M., (Eds.), ESIS, Lisboa. 6. Murakami, Y. and Endo, M. (1994). Effects of defects, inclusions and inhomogeneities on fatigue strength. Int. J. Fatigue 16, 163-182. 7. Susmel, L. and Petrone, N. (2001). Fatigue life prediction for 6082-T6 cylindrical specimens subjected to in-phase and out of phase bending/torsion loadings. In: ICB/MF&F, pp. 125-142, de Freitas, M., (Eds.), ESIS, Lisboa. 8. Guyon, E. and Troadec, J.P., (1994). Du sac de billes au tas de sable, Odile Jacob (Eds), Paris.
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9. 10. 11.
12.
13.
14. 15.
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Savage, S.B. (1997). Problems in the static and dynamics of granular materials. In: Powder and grains, pp. 185-194, Behringer, R.P. and Jenkins, J.T. (Eds.), Balkema, Rotterdam. Dantu, P. (1968). Etude statistique des forces intergranulaires dans un milieu pulverulent. Geotechnique. 18. 50-55 Radjai, P., Wolf, D.E., Roux, S., Jean, M. and Moreau, J.J. (1997). Force networks in dense granular media. In: Powder and grains, pp. 211-214, Behringer, R.P. and Jenkins, J.T. (Eds.), Rotterdam. Roux, J.N. (1997). Contact disorder and nonlinear elasticity of granular packings: A simple model. In: Powder and grains, pp. 215-218, Behringer, R.P. and Jenkins, J.T. (Eds.), Balkema, Rotterdam. Le Biavant, K., Pommier, S. and Prioul, C. (1999), Ghost structure effect on fatigue crack initiation and growth in a Ti-6A1-4V alloy. In : Titane 99:Science and technology, pp 481487, Goryin, I.V. and Ushkov, S.S. (Eds), Saint Petersburg, Russia. Le Biavant, K., Pommier, S. and Prioul, C. (2002). Local texture and fatigue crack initiation in a Ti-6A1-4V Titanium alloy. Fat. Fract. Engng. Mater. Struct 25, 527-545. Pommier, S. (2002). "Arching" effect in elastic polycrystals. Fat. Fract. Engng. Mater. 5rrMcr. 25, 331-348.
Appendix : NOMENCLATURE OFHC FEM (1,2,3) and (x,y,z)
Oxygen Free High Conductivity Copper Finite element method Coordinate systems attached to the grain and to the model
(j^^
Stress tensor as calculated using the FEM in (1,2,3)
SF
Schmid factor
Tmax
Maximum resolved shear stress on the slip systems of a crystal
cr^
Tresca equivalent stress
G^
Principal stress
/g\
where S(Q) is given by S{0) = a,,-Kj^2a,,-Kj
K, +a,,-Kl
^a,,-Kli
(9)
and a^^ = 16/r/i ^12 =
( 3 - 4 v - c o s ^ ) ( l + cos^) sinO' {cosO-1 + 4v)
%7VJU
a^2 =^—•[4(l-v).(l-cos)] 16;r// 1 4;r// in which |i stands for the shear modulus of elasticity and v is the Poisson ratio. S/cos(/> represents the amplitude of intensity of the strain energy density field and it varies with the angle ^ and 0. It is apparent that the minimum of S/cos
--^
(f) OA, in vacuum
Fig. 9. SEM fracture surfaces after fatigue loading of UA-OA alloy, in air and vacuum.
374
M.FONTEETAL
DISCUSSION Studies of the mechanisms governing fatigue behaviour in aluminium alloys rationalised accelerated crack growth rates in moist media (as compared to those in vacuum or inert environments) in terms of conventional corrosion fatigue processes such anodic dissolution and/or hydrogen embrittlement [26,50,]. Apart from environmental effects, certain intrinsic metallurgical phenomena, in particular those related to slip characteristics, are also considered to cause pronounced differences in near-threshold crack growth behaviour between different alloys. In addition to environment and microstructurally influenced growth mechanisms, crack closure processes can significantly affect fatigue behaviour in the near-threshold regime [8,18,32]. Microstructural features directly influence material properties. The toughness, for example, may be reduced by large fractions of GB precipitates produced by inefficient quenching and by aging [14,18,19,22]. The concept of strain locahsation in planar slip bands appears to be significant in both monotonic and fatigue testing. Environment and microstructure also strongly influence the fatigue crack growth resistance of high strength aluminium alloys [57] with crack deflection and branching leading to important consequences for the mechanical behaviour [10]. Local microstructure and the applied AAT primarily control the slip mode being responsible for crack propagation. In addition, crack advance can be significantly altered by the presence of the environment [51]. As a result, both microstructural and environmental factors have a strong effect on the near threshold fatigue crack growth behaviour. Aiming to contribute for the understanding of these phenomena, the discussion will lay in these two areas: (a) environment and (b) microstructure via slip characteristics. First of all one needs to compare the fatigue results in ambient air to the results in vacuum, in order to distinguish the role of microstructure and environment. In vacuum, the planar slip alloy exhibits a significant fatigue resistance in comparison with to the wavy slip OA alloy microstructure shown by the increased threshold in both AK th and Ar*max. Moreover, due to slip reversibility in the UA alloy, both AAr*th and K max can have independently different contributions to the crack growth process: crack branching e.g. can occur in planar slip materials and the crack path can be tortuous, in zigzag, with crystallographic facets. Figures 5 (a) shows the AATth versus /?-ratio relationship of the OA and UA alloy with decreasing of/^-ratios. The resulting curve is similar to the systematic curve in Fig. 2 (a) [49] for the OA alloy. In compression (R=-l), the UA alloy looses its fatigue resistance in contrast to the OA alloy. This anomalous behaviour of the UA alloy could be due to compressive parts of loading, causing shear loads that induce tensile stresses, which result in secondary cracks parallel to the compression axis. The AATth versus ATmax plot in Fig.5 (b) shows the expected Lshaped curves [52,53] according to Fig. 1 (a) and Fig. 2 (b). It may be seen again that the UA alloy looses the expected L-shape under compression loading probably due to shear loads which can induce tensile stresses. Figures 6 (a) and (b), AAT versus R, show that d^/d/V^ is mainly controlled by ATmax at Rvalues up to -0.5 and by AAT above R=0.5, see also Fig. 2 (a). However, this behaviour is not as pronounced for the UA alloy, with a fatigue resistance at negative 7?-ratios, which results in almost constant AA' values and slightly increasing ATmax values at negative /^-ratios. These results are in principle similar for both environments, although the magnitudes differ. Figures 7 (a) and (b) show a similar dependence of AAT on R of the OA alloy for specified constant crack growth rates as in the threshold regime for tests in vacuum. Therefore, again Lshaped curves result, which points to a class Ilia behaviour according to Vasudevan and
The Environment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at...
Sadananda [40,48]. The UA alloy shows an L-shape at positive /^-ratios too, whereas a loss of fatigue crack propagation resistance may be recognised at negative /^-ratios again. AAT reaches a plateau for RO and /Cmax therefore decreases with decreasing R to comply with constant AAT required for crack growth. This means that the controlling mechanism switches from a ATmax controlled behaviour for positive 7?-ratios to a A/T controlled for negative R. This implies that reversed cyclic plasticity may become the governing factor for fatigue crack growth. Observing the experimental results, the role of environment and microstructure on the threshold AATth-A'max curve for the same 7049 alloy and summarising the contributions of several investigators [3,19,20,22,25,46,47], it is clear that the introduction of moist air environment reduces strongly the AAr*th values of the UA and OA alloys. The wavy slip mode in the OA alloy probably is the reason for the reduced AAr*th in comparison to the UA alloy in moist air and in vacuum. The two microstructures likewise show different fatigue crack growth behaviour in moist air, with higher thresholds of the UA structure than those of the OAalloy. Figures 8 (a) shows the better fatigue crack growth properties of both alloys in vacuum than in humid air which has been found in a similar extend by Kirby and Beevers [25,36]. The data of Kirby and Beevers, resulting in a AATth-ATmax curve with a slope of 1, point to the prevailing influence of fatigue loading and microstructure. The different resuhs of the present study probably are caused by two facts. First, the studied alloy (Al 7049 alloy instead of 7075) is more susceptible to corrosive influences, and second, the vacuum was not a high vacuum (-2.6x10'^ Pa). Figures 9 (a) to (f) show the fracture surfaces typical for fatigue loading of the UA-OA alloy in air and vacuum at R=-l. The UA alloy shows planar slip in air in contrast to the rather ductile fracture surface in vacuum (Figs 9 (c) and (d)). In ambient air both the UA and OA alloy (Figs 9 (a) and (c)) shows a brittle crystallographic fracture mode. As an additional example for the influence of the environment. Fig. 9 (d) shows that fatigue loading of the UA alloy in vacuum leads to a rather ductile fracture surface, whereas humid air causes some embritteling, as visible in Fig. 9 (c) and (e): The main influence seems to come from the load ratio, showing extensive crystallographic brittle fracture features at R = -1 in air. The UA alloy shows in addition crack branching, and the crack advance profile is zig-zag like. The main differences in FCGR behaviour of the OA and UA microstructure indeed arise from different slip deformation behaviour: homogeneous and wavy slip in the OA alloy (more brittle in ambient air than in vacuum, probably induced by hydrogen) and localised planar slip in the UA microstructure. The present results for 7049 aluminium alloy tested in ambient air show a distinct trend of lower threshold A/Tth values and higher near threshold growth rates with increasing aging treatment. These features can be rationalised in terms of several competing mechanistic processes: intrinsic and microstructural effects and microstructure environment interactions. In the absence of any environment effect, in vacuum, the crack propagation mechanism is governed only by microstructural factors whose action in turn is governed by the loading conditions [54,55]. Crack propagation is intergranular, controlled by slip in one or many active planes. In the crack growth range where the Paris law is valid, i.e., in stage II, the crack tip loading conditions permit at least two slip systems to be active which in turn leads to a plane crack growth path affected only by the presence of large inter-metallic precipitates [14]. The chemisorption phenomenon describes the formation of hydrogen by the dissociation of the absorbed water molecules. In such a case a hydrogen embrittlement mechanism can be brought into action [26,50,53]. Accordingly the thresholds AATth for both aging conditions are higher in vacuum than in humid air at all load ratios. Moreover, the differences between the
375
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threshold A^th values in vacuum and humid air decrease with increased extent of aging, over the entire range of load ratios. The thickness of oxidation products (which has not been determined in this paper) on the near-threshold fracture surfaces in the overaged structure may be considered as an indication of conventional corrosion fatigue processes, i.e. active path corrosion and hydrogen embrittlement, which will tend to accelerate crack growth. The extend of such embrittlement is found to depend on both the rate of transport of water vapour to the crack tip and on the surface reactions kinetics [50,54,56]. It has also been pointed out that the cathodic hydrogen produced concomitantly with the crack tip oxidation process may be a significant source of embrittlement in 7075-OA structure [57]. For both aluminium alloy conditions (UA, OA), the crack growth curves. Figs 3 and 4, are shifted towards lower AK values with increasing /?-values. It is interesting to note that both microstructures show the /^-effect on AK, even at higher crack growth rates of 10'^ m/cycle. Extending these results to much higher growth rates probably leads to observe the /^-effects to be independent of AK which is commonly observed in many alloys. This experimental observation is consistent with early investigations on the same type of alloy [21-24]. The threshold stress intensity range AA^th value of the 7049-UA and 7049-OA material, which was fatigue tested in ambient air, decreases with increasing load ratio, as mentioned already. At all ratios, the magnitude of AA'th decreases with increased aging. Comparison of the near-threshold fatigue crack growth behaviour obtained in ambient air with the data for vacuum, however, shows that the presence of humidity leads to a larger reduction of AA^th for the UA microstructure than for the OA condition, at all load ratios. The apparent differences in the resistance to near-threshold fatigue crack growth of the two aging conditions are attributed to a complex interplay among several concurrent mechanisms involving moistureinduced embrittlement, slip characteristics, crack deflection processes and crack closure due to environment and microstructures factors [18]. This favourable property of the UA alloy seems to arise from its capacity to produce a highly nonlinear crack profile. The microstructural differences (UA-OA) manifested in terms of its deformation slip mode of planar versus wavy, indicate that the resistance to crack growth in planar slip alloy is significantly better than that of the overaged alloy due to the contributions from crack branching and environment in the tension-tension load ratio region. In the compressiontension region, the underaged alloy shows a loss in the fatigue resistance due to a change in the slip andfracturemodes. The apparent differences in fatigue crack growth resistance of the two aging conditions are ascribed to a complex interaction of several mechanisms: the embrittling effect of humid air resulting in conventional corrosion fatigue processes, the role of microstructure and slip mode in inducing crack deflection, and - in an unknown extent crack closure arising from a combination of environment and microstructural contributions. Crack tip branching, deflection and secondary cracking observed in 7049-UA affect crack tip driving force because Mode II and Mode III components are superimposed on Mode I [35]. The mechanisms are important for materials with significant planarity of slip and these mechanisms can be accentuated by certain environments or microstructures. Thus one can infer that the role of environment is strongly more significant in the near-threshold regime than any mechanical contributions such as plasticity, roughness, oxide, closure, etc. From this, one may conclude that AKth decreasing with /? is an intrinsic fatigue property of the material for that environment [53]. Results in Table 5 show that the C and m parameters in the Paris law which are traditionally considered as a specific property of each material, are significantly different for each microstructure and environment, either in ambient air or in vacuum.
The Environment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at...
CONCLUSIONS The fatigue crack growth threshold behaviour of an Al 7049 (UA-OA) alloy was studied by comparing differently aged materials with identical chemical composition and yield strength, but different microstructure. These two alloys exhibit the same crystallographic texture and grain morphology, but differ in precipitate microstructure. Experiments at different /?-ratios and different environment (ambient air and vacuum) showed that thresholds depend on the different microstructure and the associated deformation mechanisms. These are homogeneous slip in the overaged (OA) condition and localised slip with crystallographic cracking and a tendency to crack branching in the underaged (UA) material. The microstructural differences manifested in terms of planar vs. wavy slip, indicate that the resistance to crack growth in the planar slip alloy is significantly higher than that of the overaged alloy (OA) due to the contributions from crack branching and environment in the tension-tension load region. In the compression-tension region, the underaged alloy shows a loss in the fatigue resistance, which is probably caused by a change in the slip and fracture modes. The second most important influence comes from the environment. It shows that the threshold cyclic stress intensity factor is reduced by approximately 50% probably mainly by hydrogen embrittlement. The overall behaviour is due to the complex relationship between the effect of environment with microstructure and loading. The key to the understanding such complex mechanisms lies in quantifying the role of crack tip chemistry and decoupling the role of time dependent environmental effects on crack growth from the cycle dependent fatigue loading. Such understanding is only possible through careful systematic measurements of fatigue data under high vacuum and in the selected environments. The parameters C and m of the Paris law, traditionally considered as a specific property of the material, significantly differ for each microstructure and environment.
REFERENCES 1. 2.
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Davidson, D.L. and Lankford, J. (1985) The Effects of Aluminum Alloy Microstructure on Fatigue Crack Growth. Matls. Sci. Engng. 74, 189-199. Lafarie-Frenot, M.C. and Gasc, C. (1983) The influence of age-hardening of fatigue crack propagation behaviour in 7075 aluminium alloy in vacuum. Fat. Fract. Engg. Matls. Structures 6, 329. Stanzl, S. and Tschegg, E.(1981) Influence of environment on fatigue crack growth in the threshold region. Acta Metall 29, 21-32. Hombogen, E. and Starke, J.R.(1993) Theory assisted design of high strength low alloy aluminum (overview 102). Acta Metall. Mater. 41, 1-16. Beevers, C.J. (1981) Some aspects of the influence of microstructure and environment on AK thresholds. Fatigue Thresholds Proceedings, Stokholm. Backlund, Blom & Beevers Eds, EMAS, Wasrley, UK, 257-276. Petit, J. Zeghloul A. (1986) On the Effect of Environment on short crack Growth Behaviour and Threshold. In: The Behaviour of Short Fatigue Cracks, EOF Pub. 1 (Edited by Miller and Rios, Mech. Engng Publications, London, 163-177. Lankford, J. (1983) The effects of environment on crack growth of small fatigue cracks. Fatigue Engng. Mater. Struct. 6, 15-32. Suresh, S., Vasudevan, A. K. and Bretz, P.E.(1984), Mechanisms of Slow Fatigue Crack Growth in High Strength Aluminium Alloys: Role of Microstructure and Environment. Metall. Trans. A., 15 A, 369-379.
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Ritchie, R.O. and Zaiken, E. (1985) On the Development of Crack Closure and the Threshold Condition for Short and Long Fatigue Cracks in 7150 Aluminum Alloy, Met. Trans A, 16A, 1467-1477. McEvily, AJ, Ritchie R.O.(1998) Crack closure and the fatigue-crack propagation threshold as funtion of load ratio. Fat Fract Engg Matls Struct 21, 847-855. Suresh, S. (1983) Micromechanisms of fatigue crack growth retardation following overloads. Engng. Fracture. Mechanics, 18, 577-593. Pippan, R., Kolednik, O., Lang, M.(1994) A mechanism for plasticity-induced crack closure under plane strain conditions. Fat. Fract. Engng Mater. Struct. 17, 6, 721-726. Vasudevan, A.K., Sadananda, K and Louat, N.(1993) Two critical stress intensities for threshold fatigue crack propagation. Scripta Metall et Materialia, 28, 65-70. Vasudevan, A.K., Sadananda, K. and Louat, N.(1994) A review of crack closure, fatigue crack threshold and related phenomena. Invited Review. Mat. Sci. Engng. A188, 1-22. Vasudevan, A.K., Sadananda, K. and Glinka, G (2001) Critical parameters for fatigue damage. Int. J. Fatigue, 23, S39-S53. Beevers, C.J.(1974) Metall Trans 5A, 391-398. Cooke, R.J., Irving, P.E, Booth, G.S, Beevers, C.J.(1975) Engg Fract Mech, 7, 69-77. Paris, P. (1998). In: Fatigue Damage of Structural Materials II: Service load fatigue damage-A historical perspective. Cape Cod, Massachusetts. Sadananda, K., Vasudevan, A., Hohz, R., Lee, E. (1999) Analysis of overload effects and related phenomena. Int. J. Fatigue 21, S233-S246. Vasudevan, A.K., Sadananda, K.(1995) Classification of Fatigue Crack Growth Behavior. Metall. Matls. Trans. A, 26 A, 1221-1234. Sadananda, K. and Vasudevan, A.K. (1997) Short crack growth and internal stresses. Int. J. Fatigue, 19, 1, S99-S108. Doker, H. (1997) Fatigue crack growth threshold: implications, determination and data evaluation. Int. J. Fatigue, 19, 1, S145-S149. Vasudevan, A.K., Sadananda, K. (2001) Analysis of fatigue crack growth under compression-compression loading. Int. J. Fatigue, 23, S365-S374. Ling, M. and Schijve, J.(1992) Fact Fract Eng Mat Struct 15, 421-430. Schmidt, R.A and Paris, P.C.(1973) Threshold for fatigue crack propagation and the effects of load ratio and frequency. In: Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, ASTM, Philadelphia, PA, 79-94. Doker, H. and Marci, G. (1983), Int. J. Fatigue 5, 187. Doker, H.and Peters, M. (1985) In: 2nd Int. Conf. on Fatigue thresholds, Birmingham, UK: EMAS Publns, 1, 275. Sadananda, K., Vasudevan, A.K., Holtz, R.L.(2001) Extension of the Unified Approach to fatigue crack growth to environmental interactions. Int. J. Fatigue 23, S277-S286. Sadananda, K. and Vasudevan, A.K. (1997) Analysis of high temperature fatigue crack growth behavior. Int. J. Fatigue, 19, 1, S183-S189. Harris, T.M. and Latanision, R.M. (1991) Grain Boundary Diffusion of Hydrogen in Nickel. Metall. Trans. A, 22 A, 351. Wei, R.P. and Simmons, G.W. (1981) Recent progress in understanding environmentassisted fatigue crack growth. Int. J. of Fracture, 17 (2), 235-247. Vasudevan, A.K., Sadananda, K.(1999) Application of unified fatigue damage approach to compression-tension region. Int. J. Fatigue, 21, S263-S274. Vasudevan AK, Sadananda K, Rajan K.(1997) Int. J. Fatigue 19, 1, S151-S160. Petit, J. and Maillard, J.L.(1980) Environment and load effects on fatigue crack propagation near threshold conditions. Script Metall 14, 163-166.
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Petit J.(1984) Some aspects of near threshold crack growth: microstructural and environmental effects. In: Fatigue Crack Growth Threshold Concepts, Davidson and Suresh (Eds), TMS AIME pub, Philadelphia, 3-25. Ricker, R.E. and Duquette, D.J.(1988) The role of Hydrogen in Corrosion Fatigue of High Purity Al'Zn-Mg Exposed to Water Vapor. Metall Trans 19 A, 1775-1783. Vasudevan, A.K. and Suresh, S. (1982) Influence of corrosion deposits on nearthreshold fatigue crack growth behavior in 2xxx and 7xxx series aluminum alloys. Metall Trans A, 13 A, 2271-2280.
Appendix: NOMENCLATURE a = crack length da/dN = fatigue crack growth rate FCGR = fatigue crack growth rate GB = grain boundary GP = grain precipitate K = stress intensity factor AK = stress intensity range AKth = values of A^ at threshold K max and AT th = two threshold parameters Kmax, Kmm = valucs ofK Corresponding to Pmax, Pmin N = number of load cycles ^max, ^min = maximum and minimum loads applied in each cycle ^
" -» min''' max
SIF = Stress intensity factor
5. LOW CYCLE MULTIAXIAL FATIGUE
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Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
383
A MULTIAXIAL FATIGUE LIFE CRITERION FOR NON-SYMMETRICAL AND NON-PROPORTIONAL ELASTO-PLASTIC DEFORMATION
Mauro FILIPPINl' , Stefano FOLETTl\ loannis V. PAPADOPOULOS^ and Cetin Morris SONSINO^ Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy 2 European Commission, JRC, IPSC, Ispra, Italy Fraunhofer-Institute for Structural Durability LBF, Darmstadt, Germany
ABSTRACT A new low-cycle multiaxial fatigue life prediction methodology based on the concept of an effective shear strain is proposed. This effective shear strain is derived by averaging the total shear strains acting on all planes passing through a material point. The proposed model, which is formulated as a generalised equivalent strain, takes into account the effect of nonsymmetrical loading cycles. The main advantage of the model relies on the small number of material parameters to be identified. The axial cyclic stress-strain curve, the basic strain-life curve (Manson-Coffin) and an additional life curve obtained under zero to tension strain controlled axial fatigue tests are sufficient to allow application of the proposed criterion in all loading conditions. The experimentally observed fatigue lives of proportional and nonproportional multiaxial strain controlled low-cycle fatigue tests from un-notched tubular specimens, have been compared with the predicted lives of the proposed approach showing in all cases a good agreement. KEYWORDS Multiaxial fatigue criteria, strain-controlled fatigue, mean strain, Inconel 718 alloy, steel.
INTRODUCTION Since many mechanical components are subject to cyclic multiaxial loading, fatigue evaluation is becoming one of the major issues in the lightweight design of structures. Many methods have been proposed to reduce the complex multiaxial stress/strain state to an equivalent uniaxial condition, namely empirical formulas, stress or strain invariants, strain energy, critical plane approaches and space average of stress or strain. Historically, the first multiaxial lowcycle fatigue criteria have been based on the extension of static criteria, e.g. maximum principal strain, maximum shear strain or maximum octahedral shear strain criteria: the main
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disadvantage of these criteria is that their appUcation is limited to the case of fixed principal stress or strain directions during the loading cycle. Modified versions of such criteria, so that application to out-of-phase loadings is made possible, have been also proposed [1]. In the so-called critical plane approaches, quantities related to the mechanism of formation of fatigue cracks under multiaxial loading are inserted explicitly in the formulation of the criteria: a combination of normal and shear stresses or strains acting on particularly oriented planes, on which fatigue cracks are likely to nucleate, is chosen as the critical parameter for assessing the fatigue life of components submitted to multiaxial cyclic loading. Among critical plane approaches, a distinction between criteria formulated in terms of strain or in terms of both stress and strain is also possible. Following the proposal of Brown and Miller [2] (Fplane) and successive contributions [3,4], the shear and the normal strain acting on the plane of maximum alternating shear strain are used. Though these criteria employ exclusively strainrelated quantities, they should be classified in the category of critical plane approaches, rather than in the strain based criteria (see Socie and Marquis [5]). The proposals of Socie [6], where combinations of stress and strain acting on critical planes are used to predict fatigue life, have been applied for predicting fatigue behaviour in the intermediate life region. The critical plane approach is given a physical justification based on the observations of nucleation and early growth of fatigue cracks but, in most cases, its adoption is limited by the need of developing complex multiaxial material models. The observation of hysteresis loops in low-cycle fatigue testing have suggested many authors the formulation of criteria based on the relationship between the total or the plastic energy in a loading cycle and the fatigue life. These criteria are usually grouped under the name of energy criteria: among many others, the proposals [7,8,9] may be considered. However, the major obstacle to the application of criteria based on strain energy is either the necessity of the complete loading histories of all the components of stress and strain tensors or the availability of a material model able to reproduce the stress-strain loading paths experienced by the material. More detailed review of multiaxial fatigue criteria can be found in references [5,10,11,12]. In this paper a new approach based on a space average of the tensor of total strains reducing the complex loading history to an effective equivalent strain is presented. The proposed approach, based on an extension of the Sonsino-Grubisic methodology [13], takes into account the effect of shear strains on crack initiation, expanding the investigation of the interaction of shear strains on all different interference planes. This new approach makes possible to link the advantages of a strain based criterion with the possibility of taking into account the different material behaviour due to out-of-phase loads and the modifying effect of superimposed mean strains. In general, the advantage of criteria based on total strain is that they may be easily applied without making use of an elasto-plastic multiaxial model, at least in the case of simple components or specimens. In the case of complex geometry structures, the strains at the critical points have still to be calculated by means of finite element method in combination with a suitable material model. Alternatively, measured strains by means of strain gauges may be employed in combination with the criterion presented in this paper for predicting the fatigue life of a component. Moreover, the possibility of taking into account the effect of a mean strain allows extending the use of the new criterion to the range of intermediate fatigue life (about 10^ cycles). The effect of mean strains on the fatigue life may be neglected in the low-cycle fatigue range; nevertheless it may seriously affect fatigue life in the intermediate life range up to the highcycle fatigue regime. This effect is more evident in the case of superalloys and hard metals, where the mean strains are closely related to the mean stresses, even at shorter lives.
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ...
385
A SELECTIVE REVIEW OF STRAIN-BASED CRITERIA Many mechanical components and structures are often subject to complex elasto-plastic strain states, particularly at stress concentration zones such as notches. For a uniaxial stress state, in the low and intermediate range of life, a fatigue life prediction may be obtained by the Manson-Coffm equation: £.
•.^[2N,)\e',{2N,)
(1)
where a'j^ and h are the fatigue strength coefficient and exponent respectively, s^ and c are the fatigue ductility coefficient and exponent respectively, and E is the Young modulus. Clearly, the above relation is not able to take into account the effect of multiaxial loading. In the last years different multiaxial fatigue life prediction methods have been proposed [10] for assessing the fatigue life under complex loads. Strain-based criteria are obtained by casting a multiaxial strain state into an equivalent uniaxial strain. Some of the strain-based fatigue life prediction methodologies are briefly reviewed in the following. von Mises criterion One of the most common equivalent strain-based criteria is the maximum octahedral shear strain amplitude criterion. For a multiaxial strain state, this hypothesis defines an equivalent strain amplitude through the relationship:
^^^'^
(l + v)V2
(^.,a - £ y j
+[£y,a " ^ . . a f + ( ^ . , a " ^.,a ) ' + T ( ? i , a + 1^,0 +
2'
f^,a)
(2)
where ^^^ and y.^^ denote respectively normal and shear strain amplitudes and v is the Poisson's ratio. In the following, this criterion will be named after von Mises, even if the original proposal by von Mises, currently employed in plasticity for determining the onset of yielding, is based on the strain energy density of distortion. According to this approach, one obtains a fatigue life prediction replacing into the Manson-Coffin relationship the axial strain amplitude £^ with the equivalent strain amplitude €^^ ^, given by Eq. (2). Let us consider two load states both having the same axial and shear strain amplitudes; in the first state the strains are in-phase whereas in the second they are out-of-phase. The major drawback resulting from the hypothesis of von Mises is that it produces the same equivalent strain for both the in-phase and out-of-phase load states above. Consequently, both states would result to the same fatigue life according to von Mises approach. Several experimental results contradict this prediction, showing that, for strain controlled fatigue tests, the fatigue life under out-of-phase loading is lower than the fatigue life under in-phase loading at the same applied strain amplitudes. ASMECode The ASME Boiler and Pressure Vessel Code Procedure [1] is based on the von Mises hypothesis. An equivalent strain range is defined through the relationship:
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M FILIPPINIET AL
(3)
+6 (Af^^ + Af"^ + A£*^) r > maximized with respect to time The terms Af., A^^ have to be calculated as strain differences between two generic instants t\ and ti, e.g. A^"^ = e^ (/,) - £*^ (^2), A^"^ = ^"^^ (/,) - 6"^^ (^2) etc. The equivalent strain range A^^^, Eq. (3), is calculated by varying t\ and ti such as to obtain its maximum value. This criterion produces a lower equivalent strain for the out-of-phase than for the in-phase loading, predicting an increase of the fatigue life, in contradiction with the experimental results. The application of this criterion may lead to unconservative predictions, as shown by Tipton and Nelson [14]. Criterion ofSonsino and Grubisic The criterion of Sonsino and Grubisic [13] assumes that the fatigue damage is caused by the interaction of shear strains acting on different elementary material planes, called interference planes. An interference plane is completely defined by the spherical coordinates, t^ and (p, of its unit normal vector n (Fig. 1).
Fig. 1. Definition of interference plane: dA represents the free material surface; n is the unit normal vector of the generic interference plane According to Sonsino and Grubisic [13], in order to simplify the calculation procedure the shear strain is calculated only on the interference planes defined by a constant value of (p = 90°, corresponding to the planes normal to the surface. The shear strain on these planes can be obtained at each time in the following way: y{i},t) = [e^(t)-eXt)\
s\n{2t}) + r,^(t)cos{2i})
The shear amplitudes Yai^,) ^^ calculated for each plane:
(4)
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ...
7'^(z^) = — max;^(2^,/)-min7(?^,/)
387
(5)
Then, the arithmetic mean value is determined by taking into account the interaction of shear strains as following:
ra.^..=]-lyMW
(6)
The equivalent axial strain amplitude is calculated according to: 5 4(1 + 1/)
Ya..
(7)
With this equivalent strain amplitude the fatigue life can be derived from the Manson-Coffm curve obtained under uniaxial strain. THE NEW CRITERION Effective shear strain The new criterion adopts a complete calculation procedure, extending the investigation of the interaction of shear strains to all different interference planes at a material point. For each interference plane, completely defined by the spherical coordinates 2? and cp of the unit normal vector n (Fig. 2), a shear strain vector denoted as l/2y„ may be obtained: 1
y,(^,z^,/) = 8(r)-n-[n-8(r)-n]i
(8)
where z(t) is the total strain tensor at instant t. In the appendix, the squared intensity of l/2y^, i.e. l/4(^^) =(l/2y„)-(l/2yj is calculated for the most general case where all the six components of ^(t) are present. Clearly, [y^) is a scalar quantity.
Fig. 2. Definition of the generic interference plane by means of spherical coordinates
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An instantaneous effective shear strain may be calculated at every time t by averaging the (scalar) squared shear strain intensity [Y^{(p,i},t)) in the following way:
rerr„„s.(0 = ^ i £ : X o ( ^ » ( « ' ' ^ ' ' ) ^ ' " ^ ^ ^ ^ ^
(9)
Taking into account a full loading cycle, an effective shear strain amplitude is defined as follows:
Since the procedure adopted to derive the effective shear strain does not depend on the particular shape of the loading cycle, the criterion may be also applied to non-sinusoidal, proportional and/or non-proportional multiaxial loading histories. Fully reversed loading Let us consider a multiaxial loading the mean strains of which are zero. We seek to establish an axial equivalent strain amplitude, based on the effective shear strain amplitude introduced before, such as to be able to use directly the Manson-Coffm relationship to make fatigue life predictions. Formally, we write:
where ^ is a material function, which depends on the relative amount of plastic and elastic strains present in a load cycle. It is determined by imposing the e^ to reduce to the e^^^ in the case of axial strain loading, i.e.:
/eff,fl
If a sinusoidal axial strain e^=£^^sm{ax), which remains within the elastic range of the behaviour of the material, i.e. s^ = e, = -v^i£^, is considered, the effective shear strain amplitude calculated according to Eqs (9) and (10) is:
Therefore, the material function K reduces to a constant for the elastic case, denoted as K^J :
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ...
389
For an elastic-plastic axial strain, e^ = e^' + ef, one has:
where v^i and v^i are respectively the elastic and plastic Poisson's ratio. This would imply that a material model should be used to separate the elastic and plastic strains once the total strain is given. Instead, a simplified approach will be adopted here. In order to determine the material function fc in the general case, it is found useful to introduce an effective Poisson ratio V, defined in the elastic-plastic range as following:
where E is the Young's modulus and E^ ^a^^je^^ one always can write e -£^-
is the secant modulus. With this definition
-Vf, • The equivalent stress amplitude, a^^, may be determined
by setting the equivalent strain amplitude, Eq. (2), in the uniaxial cyclic stress-strain curve, that is described in mathematical form by the Ramberg-Osgood equation: 1/1-
= —^ —^} I F + {1 fr'
equivalently ^ or ' ^'^^»^'"^«^""/
€^aa~— *^eq,u j^
'^
eq,a
K'
(15)
For numerically determining the value of the material function K for each value of the effective shear strain /^^^, Eq. (10), a convergent iterative procedure has to be employed. First, an initial guess value of K = v^i is set, by which the transverse strains may be evaluated as £^ =£^= -V£^, for a given value of longitudinal strain. Then, the equivalent strain is calculated and, by employing the cyclic stress-strain curve, Eq. (15), the equivalent stress and the secant modulus E^ are determined. Finally, the new estimated value of the effective Poisson ratio is calculated, Eq. (14). The procedure is repeated, by employing the calculated value as initial guess, until the difference between the guess and the obtained value is sufficiently small. Values of K calculated for a range of effective shear strain amplitudes are shown as hollow circles in Fig. 3-a, b and c, for the Inconel 718 alloy, the Mild Steel and the SAE 1045 steel, respectively. Until the material response is elastic, a constant value of /r, i.e. /r^,, Eq. (13), is achieved while in the plastic region /r is a decreasing function of the strain. In order to speed up the evaluation of the parameter K when predicting the fatigue life, the following interpolating expression has been adopted for defining the material function K:
K=\
g
l [ 4 (reff,. + C j ' - D ,
(16) f o r reff,,, > f^„,a
M FILIPPINIET AL
390
For the three materials considered in this paper, the values of the constants used to define the material function K are shown in Table 1. The interpolating curves are shown in Fig. 3. K PARAMETER vs. EFFECTIVE SHEAR STRAW
" 0
0.005
0.01
0.015
0.02
0.025
0.03
K PARAMETER vs. EFFECTIVE SHEAR STRAIN
EFFECTIVE SHEAR STRAIN
0.'
0.006
Effective stiear strain [mm/mm]
0.01
0.015
0.02
0.005
0.025
(a)
0.01
0.015
0.02
0.02
Effective stiear strain [mm/mm]
Effective st)ear strain [mm/mm]
(b)
(c)
Fig. 3. Values of the material function x" for Inconel718 alloy (a), Mild Steel (b) and SAE 1045 steel (c) Table 1. Constants used in the definition of the material function /r Material
Kei
A,
B,
c.
D^
INCONEL718 MILD STEEL SAE 1045
L655 1.655 1.667
L0610' 3.3610-^ 1.05-10-^
-2.35 -9.49-10' -8.2910'
1.26-10-^ 9.7510-^ 8.8510'
-1.46 -1.44 -1.44
With the above development the procedure for the fatigue life prediction under fully reversed loading is summarised in three steps; first calculate y^^^ through Eqs (9) and (10), second evaluate an equivalent axial strain €^^^ - Ky^^^ with K given by Eq. (16) and third, introduce £gq ^ in the Manson-Coffin relationship in the place of e^ and solve for the number of loading cycles. It is noticed that the experimental data upon which the above procedure is based are limited in: 1) the cyclic stress-strain curve, which serves to establish the material function K and, 2) in the Manson-Coffin curve, which allows estimating the number of load cycles to crack initiation. Mean strain effects For predicting the fatigue life in the intermediate life range up to the high-cycle fatigue regime, the new criterion has to take into account the effect of a superimposed mean strain. If for a given multiaxial loading the mean strains are not zero, the strain cycles may be partitioned into their mean and alternating components, so that two effective shear strains, y^^^^, y^^,^ are calculated. In order to take into account the effect of mean strain components, a general formula, obtained as a second order power expansion in terms of the effective shear strains, may be tentatively proposed:
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elas to-Plastic ...
(17)
= yl{r.ff,af +(«^reff,Jreff,.+>^(nff,.)'+>ireff,. +/^reff,. where a,fi,/i,ju are four material parameters that must be identified for each material by fitting experimental results. In most cases this approach, though seeming sound from a mathematical point of view, is impractical and adds little to the description of the real material behaviour. Actually, the identification of the four material constants requires additional test campaigns, which not only increase the costs but also introduce severe mathematical complexities in building a model for fatigue life prediction. Instead, a few considerations regarding the form of Eq. (17) will reduce this formula to what is believed to be the essential needs of fatigue analyses. In Eq. (17) the effect of the alternating strain is considered twice, by the quadratic term [/^ff a) ^^^ ^V ^^^ linear term A7eff,a- ^^^^ terms being positive it is believed that retaining only one of them would be sufficient to capture the effect of y^^^ ^. The quadratic term is retained here. If experiments under strain control with and without mean strain are compared (e.g. axial low-cycle fatigue experiments on cylindrical smooth specimens with strain ratios Re=0 and Re^'-l, respectively), it may be observed that for higher strain ranges the mean strain has less influence on the fatigue life than for smaller strain ranges. This behaviour may be partially explained by the fact that for higher strain ranges mean stress relaxation is usually observed, so that mean stresses disappear after the first few cycles, while for smaller applied strain these stresses are kept constant during the life of a component and they do affect its durability. In the light of this, it seems a reasonable choice to neglect the quadratic term y^(;^eff m)
^^ ^^- (^^)- Further inspecting the effect of
y^ff,„, it is noticed that two possibilities are now left. First, one may hold both the interaction term {cxy^ffa)7cff,m ^ Fig. 4 Axial-torsional loading paths All biaxial tests have been conducted on a servo-controlled closed-loop system with computer control and data acquisition using tubular specimens. The failure was defined as a 10% axial load drop from the previous logarithmic interval of data acquisition for any axialtorsional test with a cyclic axial loading. For torsion-only histories or torsional histories with static axial stress or strain, a torque drop was applied. Comparison with experimental results The life predictions given by the new criterion are presented in Fig. 5 for Inconel 718 alloy, in Fig. 6 for MILD STEEL and in Fig. 7 for SAE 1045 steel (bottom part of the figures). Predictions obtained by the Sonsino-Grubisic approach are also shown in the same figures (upper part).
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic . 393
SONSINO'S CRITERION — INCONEL718
SONSINO'S CRITERION — INCONEL718
10'^
I
'
'
;^ ^v
N.
\.
/ /\ / / * // / ^^ ^ i
1 + Axial
Coffin-Manson 1 + Axial n * Torsional 1 O Ax-tors in-ph X Ax-tors out-of-ph 0 Axial (mean strain) M V Torsional (mean strain) •6- Ax-tors in-ph (mean strain) D Ax-tors out-of-ph (mean strain)
* Torsional O Ax-tors in-ph X Ax-tors out-of-ph 0 Axial (mean strain) V Torsional (mean strain) •tf Ax-tors in-ph (mean strain) 1 • Ax-tors out-of-ph (mean strain)
/
]
•510"
UNSAFE
•55
/•
^
-
/ ^
^OS5 ^
/ // ^ /'/ '
/
/^^/
^""^^--^ ^"^"^^^
/
SAFE
fU^/
/ / ^ ^ ^
/
10
,
10'
10
I
xN^^^d?
.510"
.
.
10="
.
•
10°
Experimental fatigue life N^ [cycles]
Fatigue life [cycles]
NEW CRITERION — INCONEL718
NEW CRITERION — INCONEL718 — + * O X 0 V •AD
10'
Coffin-Manson Axial Torsional Ax-tors in-ph. Ax-tors out-of-ph. Axial (mean strain) Torsional (mean strain) Ax-tors in-ph (mean strain) Ax-tors out-of-ph (mean strain)
+ * O X 0 V * n
I] : [
"oT
1"1
^
'V^^ / / 10
10^
10
10
NEW CRITERION — SAE 1045
NEW CRITERION — SAE 1045
+ * 0 X
Coffin-Manson Axial Torsional Ax-tors in-ph Ax-tors out-of-ph [
1 + Axial
*
0 10^
1 X
'"" / / / 7'
Torsional Ax-tors in-ph. Ax-tors out-of-ph
/ //
// *
Vi q>
y//
^
'^.^5
'
/^d^
^ ^° • ^
UNSAFE /
3
jr
.2> /
^ ^ N * *
0 •6
^"^^^*^
SJ
t^iu
3
^*^ '
' 10
10
Fatigue life [cycles]
/
/ .n2
/
/ ^Tc p ' *X
y
/^>0f A4 /
^£*59^
45 10 13
I
10
Experimental fatigue life N^ [cycles]
Fatigue life [cycles]
E^
1^ %
^'>^ji» ^2^^'
^' -^vod^
A7^ /
AM^"
SAFE
/ y
/ / /
#**y^
ym ''
/ 10
10
10
10
Experimental fatigue life NJcycles]
Fig. 7: Comparison of experimental and calculated fatigue lives for SAE 1045 steel: SonsinoGrubisic criterion (top), new criterion (bottom)
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ...
A good agreement with the predictions of the new proposal is observed: 93% of 62 data points of the SAE 1045 steel, 84% of the 26 data points of the MILD STEEL and 84% of the 55 data points of the Inconel 718 alloy fall within a range of factor 2 in life. The new criterion has also been checked for other experimental data sets, not shown in the present paper, obtaining in all cases a good correlation [15]. In Figs 8, 9 and 10 the logarithmic error index frequency histograms for the new criterion and the Sonsino-Grubisic criterion are shown: E...%'''
\ogNf-\ogN'p
(19)
•100
In Eq. (19) A^^ represents the experimental fatigue life and A^^ the calculated fatigue life. The logarithmic error index gives a quantitative evaluation of the agreement between the predictions of each criterion and the experimentally determined fatigue lives. NEW CRITERION — INCONEL718
SONSINO'S CRITERION — INCONEL718
-20
-10
0
10
Logarithmic error index [%]
20
-20
-10
0
10
20
Logarithmic error index [%]
Fig. 8: Logarithmic error index for Inconel 718 alloy: Sonsino-Grubisic criterion (left), new criterion (right) SONSINO'S CRITERION — MILD STEEL
NEW CRITERION — MILD STEEL
Logarithmic error index [%]
Logarithmic error index [%]
-10
0
10
Fig. 9: Logarithmic error index for MILD STEEL: Sonsino-Grubisic criterion (left), new criterion (right)
395
396
M. FILIPPINIET
AL
NEW CRITERION — SAE 1045
SONSINO'S CRITERION — SAE 1045
-20
Logarithmic error index [%]
-10
0
10
20
30
Logarithmic error index [%]
Fig. 10: Logarithmic error index for SAE 1045 steel: Sonsino-Grubisic criterion (left), new criterion (right) For the examined materials, the new proposal gives errors between predicted and experimental fatigue lives that pile up in the classes around 0% with a rather small scatter. It is observed that, by employing the Sonsino-Grubisic method, the majority of the logarithmic error values concentrate on the positive side, showing a more conservative prediction. This could be attributed to the fact that the beneficial effect of the stress gradients is not taken into account, thus predicting a fatigue life in torsion lower than the experimental one. It is noticed that the basic material curve used in the new proposal is the axial strain/life curve. An improved version of the Sonsino criterion, including the effect of stress gradients [20], may be used instead, where the fatigue life- of notched components could be also assessed. Another source of the slight discrepancy observed in torsion tests might be the anisotropy in the material, particularly in the case of the SAE 1045 steel [18]. COMPARISON BETWEEN CRITERIA As mentioned previously, the von Mises approach, the Sonsino-Grubisic method and the new proposal are, in principle, all applicable under in-phase and out-of-phase loading conditions. However, it is reminded that for two loads with the same strain amplitudes, the one being inphase the other being out-of-phase, the von Mises approach produces the same equivalent strain, leading thus to the same predicted life. This contradicts the experimental findings. Actually, several experimental results show a fatigue life reduction for the out-of-phase case due to the variation of the principal strain directions and the interaction of the deformations acting along different directions. Both the Sonsino-Grubisic criterion and the new proposal are able to correctly model this behaviour, predicting a higher equivalent strain value and thus a fatigue life reduction for the out-of-phase case in comparison to the in-phase condition. For the new criterion the study of the interaction of the deformation in different direction is extended to all possible material interference planes, obtaining thus an equivalent strain lower than the value predicted by the Sonsino-Grubisic criterion. In Fig. 11 a combined axial and shear strain loading is examined. On the left, the equivalent strains calculated according to the three approaches above are shown for the case where the axial and shear strains are in-phase. It is noticed that the new proposal leads to the same value of equivalent strain as the von Mises method. On the right, the axial and shear strains are out-
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ...
397
of-phase. It is seen that the von Mises criterion produces the same value of equivalent strain as for the in-phase case. COMPARISON BETWEEN CRITERIA
t" ~ - " - '
1 — Axial strain 1 ^•\ — Shear strain j
5 = 90° 0
7^"^^^"
•"^-^_ 1
2
3
4
5
6
"
L i
—
2p.002
vonMises vonMises (timed) - • - Sonsino-Grubisic - - New Criterion (timed) o New Criterion (equivalent strain)
(At [rad]
O O 0~"OQ,0 O O O O O O JD-O'b O O O O O O O Q O O O O O O
OJXfO
O O O
— gp.002 UJ
vonMises [ vonMises (timed) Sonsino-Grubisic 1 - - New Criterion (timed) [ o New Criterion (equivalent strain) |
(Of [rad]
Fig. 11 Comparison betv^een criteria The new proposal leads to an equivalent strain higher than that of the von Mises value but lower than the Sonsino-Grubisic approach. This means that for out-of-phase loading the new proposal predicts a shorter life than for the in-phase case. However, longer lives are predicted by the new approach than by the Sonsino-Grubisic methodology. In view of the analysis of the experimental data presented in the previous section, which showed that fatigue life predictions of the Sonsino-Grubisic method were conservative, one can conclude that the new proposal represents an improvement with respect to the Sonsino-Grubisic criterion.
CONCLUSIONS The most commonly used multiaxial LCF strain-based criterion (i.e. von Mises) leads to the same equivalent strain for in-phase and out-of-phase loading of the same imposed total strain amplitudes, resulting thus in unsafe life predictions. The original proposal of Sonsino-Grubisic represents a certain improvement of the preceding criteria because it summarises all the advantages of criteria based on equivalent strain, correctly predicting the fatigue life reduction observed under out-of-phase loading conditions. However, in view of the analysis of the experimental results examined before, it seems that the Sonsino-Grubisic approach leads to some extent to conservative life predictions. This could be possibly attributed to the lack of taking into account the beneficial influence of stress gradients. The stress gradient effect is not modelled even in the approach proposed here. Nevertheless, the new methodology developed in this work has proved to give life predictions that are in good agreement with the experimental results. The advantage of the new criterion relies on the fact that no elasto-plastic material model is required and, since mean strain effects are explicitly included in the proposed methodology, its applicability may be also extended to the intermediate fatigue region.
REFERENCES 1.
ASME (1988). Cases ofASME Boiler and Pressure Vessel Code. Sec. Ill, Div. 1, Code Case n°47-23, "Class 1 Components in Elevated Temperature Service", Appendix T, NY.
398
2. 3. 4. 5. 6.
7.
8.
9.
10. 11. 12. 13.
14.
15.
16.
17.
18.
19.
20.
M. FILIPPINIET AL
Brown, M.W. and Miller, K.J. (1973). A Theory for Fatigue Under Multiaxial StressStrain Conditions. Proc. of the IMechE, 187, 745-756. Lohr, R.D. and Ellison, E.G. (1980). A Simple Theory for Low Cycle Multiaxial Fatigue. Fatigue ofEng. Mater. Struct., 3, 1-17. Wang, C.H. and Brown, M.W. (1993). Fatigue Fract. Engng Mater. Struct., 16, 12851298. Socie, D.F. and Marquis, G.B. (2000). Multiaxial Fatigue. SAE, Warrendale, PA. Socie, D.F. (1993). Critical Planes Approaches for Multiaxial Fatigue Damage Assessment. In: Advances in Multiaxial Fatigue, ASTM STP 1191, pp. 7-36, McDowell, D.L. and Ellis, R. (Eds). ASTM, Philadelphia, PA. Garud, Y.S. (1981). A New Approach to the Evaluation of Fatigue Under Multiaxial Loadings. J. of Engineering Materials and Technology, Trans, of the ASME, 103, 118125. Ellyin, F. and Xia, Z. (1993). A General Fatigue Theory and Its Application to Out-ofPhase Cyclic Loading. J. of Engineering Materials and Technology, Trans, of the ASME, 115,411-416. Park, J. and Nelson, D. (2000). Evaluation of an energy-based approach and a critical plane approach for predicting constant amplitude multiaxial fatigue life. Int. J. Fatigue, 22, 23-39. Garud, Y.S. (1981). Multiaxial Fatigue: A Survey of the State of the Art. Journal of Testing and Evaluation, JTEVA, 9, 165-178. McDowell, D.L. (1996). In: Fatigue and Fracture, ASM Handbook Vol. 19, pp. 263-273, ASM Int., Materials Park, OH. Ellyin, F. (1997). Fatigue Damage, Crack Growth and Life Prediction. Chapman & Hall, London. Sonsino, C M . and Grubisic, V. (1989). Fatigue Behavior of Cyclically Softening and Hardening Steels Under Multiaxial Elastic-Plastic Deformation. In: Multiaxial Fatigue, ASTM STP 853, pp. 586-605, Miller, K.J. and Brown, M.W. (Eds). ASTM, Philadelphia, PA. Tipton, S.M. and Nelson, D.V. (1985). Fatigue Life Predictions for a Notched Shaft in Combined Bending and Torsion. In: Multiaxial Fatigue, ASTM STP 853, pp. 514-550, Miller, K.J. and Brown, M.W. (Eds). ASTM, Philadelphia, PA. Foletti, S. and Passerini, M. (2000). Modelli elasto-plastici e fatica a basso numero di cicli in stato di sollecitazione multiassiale. Eng. Degree Thesis, (in Italian), Politecnico di Milano, Milano. Fatemi, A., Kurath, P. (1988). Multiaxial Fatigue Life Prediction Under the Influence of Mean-Stresses. J. of Engineering Materials and Technology, Trans, of the ASME, 110, 380-388. Socie, D., Kurath, P. and Koch, J. (1989). A Multiaxial Fatigue Damage Parameter. In: Biaxial and Multiaxial Fatigue, EGF 3, pp. 535-550, Brown, M.W. and Miller, K.J. (Eds). MEP, London. Kurath, P., Downing, S.D. and Galliart, D.R. (1989). Summary of Non-Hardened Notched Shaft Round Robin Program. In: Multiaxial fatigue: analysis and experiments, pp. 13-31, Leese, G.E. and Socie, D.F. (Eds). SAE, Warrendale, PA. Doquet, V. and Pineau, A. (1991). Multiaxial Low-Cycle Fatigue Behavior of a Mild Steel. In: Fatigue under Biaxial and Multiaxial Loading, ESIS 10, 81-101, Kussmaul, K., McDiarmid, D. and Socie, D.F. (Eds). MEP, London. Sonsino, C M . (2001). Influence of load and deformation-controlled multiaxial tests on fatigue life to crack initiation. Int. J. Fatigue, 23, 159-167.
A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ...
2 1 . Simburger, A. (1975). Festigkeitsverhalten zaher Werkstoffe bei einer mehr-achsigen, phasenverschobenen Schwingbeanspruchung mit korper-festen und verdnderlichen Hauptspannungsrichtungen. LBF F B - 1 2 1 , Fraunhofer-Institut fiir Betriebsfestigkeit (LBF), Darmstadt. NOMENCLATURE ^K, ^K, CK, D^
Constants used to define the material function K
b
Uniaxial fatigue strength exponent
c
Uniaxial fatigue ductility exponent
E, Es
Axial elastic and secant modulus
£'iog%
Logarithmic error index
K'
Cyclic hardening coefficient
n
Cyclic hardening exponent
Nf, Nc
Experimental and calculated fatigue life
R
Cycle ratio
t
Time
a, p , X, ju £eq fgq^ £f y^ 12 £^,€y,£^^£^,£y^^€^ ^x,a' ^y,a' ^z,a
Material constants Total equivalent strain Equivalent strain amplitude Uniaxial fatigue ductility coefficient Shear strain vector in each interference plane of normal n (^, ^?) Strain tensor components Normal straiu amplitudes
y{i^,^)
Shear strain in an interference plane z? at time t
7^{A)
Shear strain amplitude in an interference plane 7}
?^,arith
Arithmetic mean value of y^ (z^)
?^eff,inst ( 0
Instantaneous effective shear strain at time /
7^ii,a' ^eff,/n
Effcctivc shcar strain alternating and mean component
Xeff 7xy,a' Yyz^a' 7zx,a
Total effective shear strain Shcar Strain amplitudes
K
Material function
ic^i
Elastic material constant
399
400
M. FILIPPINIET AL V, Vgj, Vpi
Effective, elastic and plastic Poisson's ratio
^, (p
Spherical coordinates
;) sinV cos^!^ sinz^+(4£-^^^) sinV cos^ cos^ z^+ -^[A£y£^y) sinV sin^z^ cos2?+(4£'^£*^) sinV cos^ sin^ z^+ + (4£-,£^) cosV sin^ sin2^+(4£,^^) cosV sin^ cosz^+ + [^£x£y,-^^£.^£^y) sinV cos^ cos^ z^ sinz?+ + (4^^£-,^+8£*^^,£:^J sinV cos^ sin^ ^? cosz^+ •^{^£,£xy-^^£yz£zx) sinV cos^ ^ sinz^ cosz^l w^here for the off-diagonal components of the strain tensor the standard equality £.j = y^jjl between the mathematical and engineering notation holds.
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
401
CYCLIC BEHAVIOUR OF A DUPLEX STAINLESS STEEL UNDER MULTIAXIAL LOADING: EXPERIMENTS AND MODELLING
Veronique AUBIN, Philippe QUAEGEBEUR and Suzanne DEGALLAIX Laboratoire de Mecanique de Lille, Ecole Centrale de Lille, Cite scientifique, BP 48, 59651 Villeneuve d'Ascq Cedex, France
ABSTRACT The low^-cycle fatigue behaviour of a duplex stainless steel, 60 % ferrite - 40 % austenite, is studied under tension-compression/torsion loading at room temperature. It is shown that the duplex stainless steel has an isotropic behaviour under cyclic proportional loading. The nonproportional loading paths induce an extra-hardening, but lower on duplex stainless steel than on austenitic stainless steels. Three models able to account for the extra-hardening are identified and tested on the experimental data base. Two of them give accurate predictions.
KEYWORDS Duplex stainless steel, cyclic plasticity, biaxial loading, extra-hardening, experimental study, low-cycle fatigue, constitutive modelling INTRODUCTION The use of austeno-ferritic stainless steels (duplex stainless steels) in branch of industry with severe conditions in terms of corrosion and mechanical resistance widely developed for about thirty years. Duplex stainless steels are notably used for applications in the power, offshore, petrochemical and paper industries. The combination of their austenitic and ferritic phases bring them an excellent resistance to corrosion, particularly to intergranular and chloride corrosion, and very high mechanical properties, in terms of Yield Stress and Ultimate Tension Strength as well as in terms of ductility. These properties result from their "composite" nature and from their very small grain size (10 |am). A nitrogen addition, essentially concentrated in the austenitic phase, is nowadays a usual practice. It enables to improve the corrosion resistance and to increase the Yield Stress [1-3]. The properties of duplex stainless steels are closely linked to the two-phase nature of these materials, in terms of crystallographic structure (FCC for austenite and BCC for ferrite), volume fraction and morphology of each phase, interactions between phases. These various parameters influence the cyclic mechanical behaviour of that composite material and modify its stress response to variable loading, with variations of amplitude and/or direction in time and in space. Especially, the austenitic phase (FCC) of duplex stainless steels has a low stacking fault energy which favours the planar slip of dislocations. This phase is consequently very sensitive to non-proportional cyclic loadings, the obtained extra-hardening is clearly observed in austenitic stainless steels such as AISI 304L or 316L [4-12]. Moreover, this phase is sensitive
402
V AUBIN, P. QUAEGEBEUR AND S. DEGALLAIX
to loading history, in terms of loading amplitude and loading path [13-14]. On the other hand, the individual ferritic phase (BCC) shows a low sensitivity to non-proportional loadings and to loading history [15-16]. The purpose of the present work is firstly to establish the mechanical behaviour of a forged duplex stainless steel under uniaxial and biaxial cyclic loadings and secondly to test the ability of a class of constitutive models to account for the behaviour observed. A large number of phenomenological models have been developed during the last two decades to describe the extra-hardening under non-proportional cyclic loadings [17-27, 12]. These models consist more often in modifying the isotropic and/or the kinematic rule through the introduction of a non-proportionality parameter defined as a relationship between stress, plastic strain or backstress, or through the introduction of a structural tensor. The base model studied in the present work is a cyclic plasticity model with non-linear isotropic and kinematic rules of the type initially proposed by Armstrong and Frederick [28]. Three modifications of this base model are tested to improve the description of the extra-hardening under non-proportional cyclic loading. The first model was proposed by Benallal and Marquis [20] and modified by Calloch [10, 12], the second model is a modification of the first one proposed by Abdul-Lafif et al. [25]. The third model was developed by Tanaka [26].
MATERIAL The material studied is a X2 Cr Ni Mo N 25-07 duplex stainless steel. This steel contains approximately 60 % ferrite and 40 % austenite. The composifion is given in Table 1. It was suppUed in rolled bars of 70 mm diameter, solution treated for an hour at 1060°C and then water-quenched before machining the specimens. The resulting microstructure consists of long austenitic rods ( 0 10 |im x 1 mm) in a ferritic matrix (Fig. 1). The microstructure seems to be transverse isotropic.
Table 1. Chemical composifion of the duplex stainless steel studied (in wt %).
c
Cr
Ni
Mo
Mn
Si
N
Cu
P
S
Fe
0,024
24,68
6,54
2,84
0,79
0,62
0,17
0,07
0,021
/3T
407
(MPa)
400
O.Ol
Fig. 6. Responses in different planes {a: (a,V5T), b: (S^,Y7V3), C: (T,Y ), d: (a,8 )} during the first 50 cycles of a test conducted with a hourglass path, a strain amplitude of 0.5 % and a zero mean strain.
800
(7aeq(MPa)
700
600
500 50
100
150
Number of cycles Fig. 7. Cyclic hardening-softening curves with a strain amplitude of 0.5 %, for seven different loading paths.
V AUBIN, P. QUAEGEBEUR AND S. DEGALLAIX
408
^aeq (MPa)
O
800 700 X
—'
^R-
600 -
+ • D O D • X
500 400 1 300 200 -
0.002
0.004
0.006
Monotonic tension Tension-compression Torsion Proportional 45° Circle Square Hourglass Clover
0.008 ^.aeq
Fig. 8. Cyclic stress-strain responses at the stabilized or quasi-stabilized cycles for seven different loading paths.
The fatigue lives of the specimens loaded under tension-compression and circle path are drawn as a function of the equivalent plastic strain amplitude in Fig. 9. This figure confirms the classical result, extensively studied elsewhere in the present ESIS Special Technical Publication: non-proportional loadings highly reduce the fatigue life in low-cycle fatigue of materials. A ratio of up to 7 exists here between the fatigue life obtained for a circle path and that in tension-compression with a same plastic strain amplitude of 0.25 %.
aeq
Number of cycles 0.1 100
1000
10000
0.01 0.001
0.0001 D Tension-compression 0.00001 J
O Circle
Fig. 9. Equivalent plastic strain-life diagram.
100000
1000000
Cyclic Behaviour of a Duplex Stainless Steel Under Multiaxial Loading: Experiments and Modelling 409
CONSTITUTIVE MODELING OF CYCLIC BEHAVIOR Different constitutive laws were used with the aim at checking their abilities to describe the experimental data presented above. The base model for the simulations is a cyclic plasticity model with one non-linear isotropic hardening rule and two non-linear kinematic hardening rules, initially proposed by Armstrong and Frederick [28] (model NLK). It has been shown that the non-linear Armstrong-Frederick rule does not consider the extra-hardening induced by nonproportional loadings in tension-torsion tests conducted on austenitic stainless steels [17-28]. Therefore, the three other models tested, called non-proportional models, are derived from this base model and propose modifications either of the isotropic or of the kinematic rules to improve the description of non-proportional hardening. For all these models, the elastic behaviour takes the following form for an orthotropic material under tension-torsion loading:
(4)
E^ = A:a
where ~±
'
600500 400 'xnn -
JUU
i
() 0.01
0.02
Model
'
100
0.03
1
200
1
1
300 400 Number of cycles
Experiment
Fig. 12. Comparison between experiments and simulation for model NLK. (a) Cyclic sircssstressstrain response in tension-compression tests (8a = 0.35; 0.5; 0.8; 1.0 %); (b) cyclic hardening/softening curves in tension-compression tests (Sa = 0.35; 0.5; 0.8; 1.0 %); (c) monotonic stress-strain response; (d) cyclic hardening/softening curve for a circle path (8a = 0.5 %).
V AUBIN, P. QUAEGEBEUR AND S. DEGALLAIX
416
Table 7. Material parameters of model NPl. E 190 GPa b, 3.2
G 65 GPa
V
0.3
k 390 MPa
Qi
b2
Qoo
-84.5 MPa
30
150 MPa
^lAi
Yi
180 MPa Qo OMPa
1900 n 18
800 n^(MPa)
C2/Y2 21209 MPa Qi
946 MPa
72 0.786 d 150
CO
Cpoo
12.79 f 2.8
0.728 g 0.157
800 .or (MPa)
^ T (MPa)
^ T (MPa)
O
0
100
200
300 400 Number of cycles
100
200
Model Experiment
300 400 500 Number of cycles
Fig. 13. Comparison between experiments and simulation for model NPl. (a) Stabilized stress response for a circle path (Sa = 0.5%); (b) stabilized stress response for a square path (8a = 0.5 %); (c) stabilized stress response for a hourglass path (8a = 0.5%); (d) stabiHzed stress response for a clover path (8a = 0.5%); (e) cyclic hardening/softening curve for a circle path (8a = 0.5%); (f) cyclic hardening/softening curve for a torsion path following a circle path (8a = 0.5%).
Cyclic Behaviour of a Duplex Stainless Steel Under Multiaxial Loading: Experiments and Modelling 417
Table 8. Material parameters of model NP2. E 190 GPa
G 65 GPa
bi
Qi
3.2
-84.5 MPa
0.3
k 390 MPa
Y3 200
50 MPa
V
CxAs
c,/y,
Yi
180 MPa
1900
C00A3 570 MPa
Co OMPa
800 -iO^(MPa)
CO CPoo Y2 21209 MPa 0.786 12.79 0.728 n COx ^. Tlx
C2A2
3.2
100
0.2
10
800 . a (MPa)
A^T (MPa)
800 ^^aeq ^^P^) 750 M£:;I:^^ 700 7
400
650 I 600 550 500 450-1 400 0
- Model O Experiment
(f) -800 440 n^aeq ^^^^^ 420 380 360 340 320 300
100
200
300 400 Number of cycles
0
100
200
300 400 500 Number of cycles
Fig. 14. Comparison between experiments and simulation for model NP2. (a) Stabilized stress response for a circle path (Sa = 0.5%); (b) stabilized stress response for a square path (Sa = 0.5 %); (c) stabilized stress response for a hourglass path (8a = 0.5%); (d) stabilized stress response for a clover path (Sa = 0.5%); (e) cyclic hardening/softening curve for a circle path (Sa = 0.5%); (f) cyclic hardening/softening curve for a torsion path following a circle path (8a = 0.5%).
418
V AUBIN. P. QUAEGEBEUR AND S. 'DEGALLAIX
Table 9. Material parameters of model TANA. E 190 GPa
G 65 GPa
0.3
k 390 MPa
V
CiAi
Yi
180 MPa
1900
^ihi
Y2
21209 MPa
0.786
CO
Cpoo
12.79 0.728
bi
Qi
b2
Cc
ry
ap
bp
Cp
aN
bN
CN
3.2
-84.5 MPa
3.2
1
80
0
0
1
62000
40
5000
800 n or (MPa)
800 -,<J(MPa)
>/3T
0
100
200
(MPa)
300 400 Number of cycles
rV^T (MPa)
300 400 500 Number of cycles
Fig. 15. Comparison between experiments and simulation for model TANA, (a) Stabilized stress response for a circle path (Sa = 0.5); (b) stabilized stress response for a square path (8a = 0.5); (c) stabilized stress response for a hourglass path (Sa= 0.5%); (d) stabilized stress response for a clover path (Sa=0.5%); (e) cyclic hardening/softening curve for a circle path (8a = 0.5%); (f) cyclic hardening/softening curve for a torsion path following a circle path (8a = 0.5 %).
Cyclic Behaviour of a Duplex Stainless Steel Under Multiaxial Loading: Experiments and Modelling 419
1000 900 800 700 600 500 400 300 200 100 0
Cyacq(MPa)
O » ^ ^ « •
W D
g aa•D
• D A O X
B
Experiment NLK NPl NP2 TANA
Q
a o
o CO
1
C
d,, 2 " di - d„ , Nff = Nfif = oo
(for Case 1, 2) (12) (for Case 3, 4)
the total damage for cruciform straining can be rewritten as
D-d,=d„=-^=-^
N,
(for Case 1 - 4 )
(13)
NL
Box and circular loading. In tests with box (Case 10-12) and circular (Case 14) straining, di and dii are given by Eq.( 10) as in the cruciform straining tests. In addition, the relationship between di and dn is expressed in Case 10, 12 and 14 by Eq.(12), so that the total damage is given by Eq.(13). In these cases, however, the value of the nonproportional intensity factor, fyp, is approximately doubled in comparison with those of the cruciform straining tests as listed in Table 2, resulting in the larger total damage.
Step loading. In the step straining tests (Case 6-9), the strain path is basically composed by one main strain range with large amplitude on the I-plane and several strain ranges with small amplitudes on the Il-plane. As for example shown in Fig.6, the strain path of Case 7 consists of the maximum strain amplitude with one cycle on the I-plane, and the medium strain amplitude with three cycles, and further small strain amplitudes with two cycles on the Il-plane. In this case, the damages on I and Il-planes can be expressed by,
A Damage Model for Estimating Low Cycle Fatigue Lives Under Nonproportional Multiaxial Loading 435
dii =
3Nii ^ 2N^ Nlif N'II f
(14)
As shown by the equation, by increasing the number of steps in the step loadings, also the number of terms in the second equation increase, but the damage for each term is decreased, which results in a smaller value of dn. Therefore, in the step-straining tests with very small length of steps such as Case 6, dn takes values small enough to be ignored, so that the total damage can be approximated by that of proportional straining tests. The above showed the equations of nonproportional LCF damage for typical cases of strain paths in this model. In the next section, by comparing nonproportional LCF lives obtained from experiments and calculations in which the total damage in Eq.(7) takes unity, i.e., D=l, the applicability of the presented model for nonproportional LCF life evaluation will be discussed.
CORRELATION OF NONPROPORTIONAL LCF DATA The applicability of the proposed damage model for the evaluation of nonproportional LCF lives is examined with data of 304SS and 606 lAl tested together with different materials such as stainless steels, copper, aluminum alloy, chromium-molybdenum and carbon steels that were obtained from various research institutes. Table 4 lists the materials employed together with the
Table 4. List of materials used in data correlation. Material
Temperature
a
Author
Ref.
SUS 304
R.T.
0.8
Itoh
4
SUS 304
R.T.
0.8
Socie
10
SUS 304
923K
0.4
Hamada
5
OFHC Cu
R.T.
0.5
Socie
10
6061 Al
R.T.
0.4
Itoh
8
llOOAl
R.T.
0.0
Socie
10
42CrMo
R.T.
0.5
Chen
7
S45C
R.T.
0.2
Kim
19
R.T.: Room temperature in air. 923K: 923Kinair.
436
T. ITOHAND T. MIYAZAKI
test temperatures, the values of a and reference numbers. Figure 9 (a) contains the comparison of the experimental and the predicted nonproportional LCF lives. All data are correlated within a factor of two scatter band, so it demonstrates that the present damage model has a possibility to
10'^
10^
10^
10
10^
10'
Estimated fatigue life Nf, cycles (a) Damage model based on nonproportional LCF strain range.
CO
o o S.
10' | i iiiiiii—1 1 iiiiiii—1 1IImil 1 1 lllllll 1 1 P O i u S 304 (Itoh) ^ t • BUS 304 (Socie) 10^ I d SUS 304 (Hamadal f A OFHC Cu ^ / / E O 6061 Al / /x/ h 4 1100AI A / / A / 10' t-® 42CrMo p A S45C /
CD
/ /
o
/ i A'
CO
iiiyi]—TTTi rrn /
y^
^
^ '4
Y
*•—
/
•£ 10^1 03
E
1 ^' ^!mW
Y/^/ 1
X
LU
/ ^ /
Y^/
^
n iiiuil
10^
Factor of 2 Factor of 10
1 1 ~3
J
ASME strain range\ ^
/
pTi/nml
J
"3
1 1 iiiiiii 1 1 lllllll
10^
10^
1 iiiiiJil
10^
11 iiiiil
10^
10'
Estimateted fatigue life Nf, cycles (b) Damage model based on ASME strain range. Fig.9. Comparison of LCF data between experimental and estimated results.
A Damage Model for Estimating Low Cycle Fatigue Lives Under Nonproportional Multiaxial Loading 437
become the appropriate model for LCF life prediction under complex nonproportional straining. In Fig.9 (b) where the life prediction is made by ASME strain instead of the nonproportional strain parameter, some of the data under nonproportional straining are obviously underestimated by more than a factor of two. The larger scattering of the data can be seen for the larger/A^/> tests, such as Case 10, 12 and 14. The maximum scattering of the data almost reached a factor of 20. In the data correlation for S45C, on the other hand, several data are correlated too conservatively and the scattering of the data tends to be larger with increasing fatigue life. The longer fatigue lives in the experiments can bee seen in the results of pure cyclic torsion tests. Conservative life estimation in pure torsion tests under a constant Mises' equivalent strain condition were also reported for 304SS cruciform specimen [20] and tube specimen subjected to tension/torsion [21].
CONCLUSION This paper developed a simple damage model for the evaluation of LCF lives under complex biaxial/multiaxial loadings. This model was developed by combining the equivalent strain, based on the maximum principal strain, as the nonproportional LCF strain parameter with Miner's law. The model was able to correlate most of all the fatigue data for different materials within a factor of two scatter band and was demonstrated to be effective for the nonproportional LCF data correlations of various materials.
REFERENCES 1. ASME Code Case N-47 (1978), Case of ASME Boiler and Pressure Vessel Code, Case N-47, Class 1, Section 3, Division 1. 2. Nitta A., Ogata T. and Kuwabara K. (1987), The Effect of Axial-Torsional Strain Phase on Elevated-Temperature Biaxial Low-Cycle Fatigue Life in SUS304 Stainless Steel, J. Society of Materials Science, Japan, 36, 376. (Japanese). 3. Socie D. F. (1987), Multiaxial Fatigue Damage Models, ASME J. of Engng. Mater. Tech., 109, 293. 4. Itoh T, Sakane M, Ohnami M. and Socie D. F. (1995), Nonproportional Low Cycle Fatigue Criterion for Type 304 Stainless Steel, ASME J. of Engng. Mater. Tech., 117, 285. 5. Hamada N. and Sakane M. (1997), High Temperature Nonproportional Low Cycle Fatigue Using Fifteen Loading Paths, Proc. of 5th Int. Conf on Biaxial/ Multiaxial Fatigue and Fracture, I, 251. 6. Wang C. H. and Brown M. W. (1993), A Path-Independent Parameter for Fatigue Under Proportional and Non-proportional Loading, Fatigue and Fract. Engng. Mater. Struct., 16, 1285. 7. Chen X, Xu S. and Huang D. (1999), A Critical Plane-Strain Energy Density Criterion for Multiaxial Low-Cycle Fatigue Life under Nonproportional Loading, Fatigue Fract. Engng. Mater. Struct., 22, 619.
438
T. ITOH AND T. MIYAZAKI
8. Itoh T., Sakane M., Ohnami M. and Socie D. F. (1999), Nonproportional Low Cycle Fatigue of 6061 Aluminum Alloy under 14 Strain Paths, In: Multiaxial Fatigue & Fracture, pp.41-54, E. Macha, W. B^dkowsky and T. Lagda (Eds). ESIS-25. 9. McDowell D. L. (1983), On the Path Dependence of Transient Hardening and Softening to Stable States under Complex Biaxial Cyclic Loading, Proc. Int. Conf. on Constitutive Laws for Engng. Mater., Desai and Gallagher, (Eds), 125. 10. Doong S. H., Socie D. F. and Robertson, I. M. (1990), Dislocation Substructures and Nonproportional Hardening, A^Mf" 7. of Engng. Mater. Tech., 112, 456. 11. Kida S., Itoh T., Sakane M., Ohnami M. and Socie D. F. (1997), Dislocation Structure and Nonproportional Hardening of Type 304 Stainless Steel, Fatigue Fract. Engng Mater. Struct.,!^, 1375. 12. Smith R. N., Watson P. and Topper T. H. (1970), A Stress-Strain Function for the Fatigue of Materials, J. Materials JMLSA, 5, 767. 13. Fatemi A. and Socie D. F. (1988), A Critical Plane Approach to Multiaxial Fatigue Damage Including Out-of-Phase Loading, Fatigue and Fracture of Engng. Mater, and Struct., 11, 149. 14. Krempl E. and Lu H. (1983), Comparison of the Stress Responses of an Aluminum Alloy Tube to Proportional and Alternate Axial and Shear Strain Paths at Room Temperature, Mechanics of Materials, 2, 183. 15. Benallal A. and Marquis D. (1987), Constitutive Equations for Nonproportional Cyclic Elasto-Viscoplasticity, A5M£ 7. of Engng. Mater. Tech., 109, 326. 16. Doong S. H. and Socie D. F. (1991), Constitutive Modeling of Metals under Nonproportional Loading, A^Mf" 7. of Engng. Mater. Tech., 113, 23. 17. Wang C. H. and Brown M. W. (1996), Life Prediction Techniques for Variable Amplitude Multiaxial Fatigue-Part 1: Theories, A5'M£'7. of Engng. Mater. Tech., 118, 367. 18. Bannantine J. A. and Socie, D. F. (1991), A Variable Amplitude Multiaxial Fatigue Life Prediction Method, In: Fatigue under Biaxial and Multiaxial Loading, pp.35-51, K.F. Kussmaul, D. L. McDiarmid and D. F Socie (Eds). ESIS-10. 19. Kim K. S. and Park J. C. (1999), Shear Strain Based Multiaxial Fatigue Parameters Applied to Variable Amplitude Loading, International 7. of Fatigue, 21, 475. 20. Itoh T, Sakane M. and Ohnami M. (1994), High Temperature Multiaxial Low Cycle Fatigue of Cruciform Specimen, A^SME 7. of Engng. Mater. Tech., 116, 90. 21. Sakane M. Ohnami M. and Sawada M. (1987), Fracture Modes and Low Cycle Biaxial Fatigue Life at Elevated Temperature, ASME J. of Engng. Mater. Tech., 109, 236.
Appendix : N O M E N C L A T U R E
a
Parameter for degree of additional hardening due to nonproportional straining
fNP
Nonproportional intensity factor for nonproportional straining
£i(t)
Maximum amplitude of principal strain at time t
A Damage Model for Estimating Low Cycle Fatigue Lives Under Nonproportional Multiaxial Loading 439
£i max
M a x i m u m value of ei(t) during one cycle
^(t)
Direction change of principal strain axis at time t
A8i
Maximum principal strain range under nonproportional straining
A8NP
Equivalent strain range for nonproportional low cycle fatigue life prediction
e, Y
Axial and shear strains, respectively
£n, 7n
Normal and shear strains on eimax-plane
di, dii
L C F damage for ei and en, respectively
D
Total L C F Damage
N'l, N'li
Number of cycles for / orj-th strain range on I and H-planes, respectively
N'lf, N\if
Number of cycles to failure for / or y-th strain range on I and Il-planes, respectively
This Page Intentionally Left Blank
Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
441
MICROCRACK PROPAGATION UNDER NON-PROPORTIONAL MULTIAXIAL ALTERNATING LOADING
Matthias WEICK and Jarir AKTAA Institutfur Materialforschung II, Forschungszentrum Karlsruhe Postfach 3640, 76021 Karlsruhe, Germany
ABSTRACT Non-proportional multiaxial fatigue tests of tubular specimens were performed under purely alternating strain-controlled loading. Both the load paths and the phase shifts were varied. With increasing phase shift at the same equivalent load, the lifetime was found to increase. Based on the Manson-Coffin law, a model for lifetime prediction was developed, which takes the hydrostatic loading part and, thus, the phase shift into account. Consequently it was possible to compare the results of non-proportional multiaxial fatigue tests at different phase shifts with the results of uniaxial tests. To obtain more information about the microcrack behaviour under multiaxial non-proportional loading, sonic-emission studies and fractographic analyses were performed. The results of the sonic emission studies and fractographic analyses suggest a discontinuous microcrack growth with the propagation rate correlating with the event rate. Further more, good agreement has been found between these observations and some microcrack propagation models known from the literature. On the base of both a simplified model for micro and short crack propagation was proposed in terms of J-Integral range AJ and microstructural measures. When applying the crack propagation model to our multiaxial experiments for lifetime prediction fairly well results were obtained. KEYWORDS Multiaxial, non-proportional fatigue, microcrack, short crack, fracture, crack growth, sonic emission,AISI 316 L (N), AJ INTRODUCTION For lifetime prediction in the case of multiaxial loading, no satisfying models exist for the universal case in spite of a multiplicity of proposed failure hypotheses. Most of the known concepts are only applicable to special loading conditions. Hence they are not able to cover a wide spectrum of multiaxial loading. In particular, it is not possible to sufficiently explain the influence of the phase shift. To attain a better understanding of the influence of the phase shift, experiments with different out-of-phase loadings were performed. In this research, biaxial test facility was used instead of the tension-torsion test facility, usually used by other experimenters. In case of the tension-torsion loading, the directions of principal stresses are a function of the ratio between the axial stress and the shear stress, which is not constant for non-proportional loading during a cycle. Due to the rotating principal
442
M WEICK AND J. AKTAA
stresses system, slip bands are activated in all directions. Hence, preferred directions of the orientation of cracks cannot be expected. Consequently, it is impossible in this case to determine an explicit correlation between the phase shift and fatigue. With our biaxial machine the multiaxial loading was generated as follows: While the load in axial direction is raised by a conventional tension feature, the circumferential strain is generated by a pressure difference of the surrounding media. The advantage of this method are the fixed directions of principal stresses and strains during the non-proportional loading cycles. Thus, the orientation of a crack can be assigned to the directions of principal stresses. To obtain information about the microcrack nucleation and propagation, we use a sonic emission system which records the current sonic emissions during the experiment. From these signals, conclusions are drawn with regard to the propagation behaviour of microcracks under multiaxial loading. With the developed microcrack model we calculate the lifetime on the basis of the experimental obtained AJ's. To verify this model we compare the results to the measured lifetime of the specimens.
EXPERIMENTAL DETAILS Testfacility As mentioned above, the existing test facility allows to perform non-proportional cyclic fatigue tests with fixed directions of principal stresses and strains [1, 2]. The mechanical components of the biaxial test facility are illustrated schematically in Fig. 1.
servo-hydraulic compressor
pressure vessel
specimen
servo-hydraulic test-machine
-•r Fig. 1. Mechanical components of the biaxial test facility.
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
443
Axial loading is built up by the plunger of the servo-hydraulic test machine. The circumferential strain is generated by a pressure difference of the surrounding media. While the outer pressure is generated by a nitrogen compressor, the inner pressure is built up by a servo-hydraulic compressor that uses fully demineralised water as pressure medium. During the test, only the inner pressure is cycled, while the outer pressure is held constant (pa=20 MPa). As a result of this special layout we are able to perform non-proportional multiaxial fatigue tests under pure alternating loading. This is different from other experimenters, e.g. Dietmann et. al. [3] who investigated non-proportional fatigue, too. In cause of their test facility, which uses no outer pressure, all their experiments were afflicted with positive mean stress and positive minimum to maximum stress ratio (R-ratio). All experiments are fully strain-controlled.
The specimen Ascertaining an optimised geometry of the specimen for this special case of loading was not trivial. It was necessary to find a geometry having a sufficient resistance to instability, but not too high a stiffness, so that we could achieve sufficient strains. Furthermore, we had to ensure that the distribution of forces in the measurement area was homogeneous. So, we decided in favour of a waisted form of the specimen. The real measurement area was a small part in the middle of the whole specimen only. In this area, smallest wall thickness was encountered. The first version of our specimens had a linear connection from the measurement zone to the rest of the specimen. As a result of this geometry chosen we had a preferential crack initiation at the transition point. FE analyses revealed an obvious stress increasing at this transition point. With the aid of FE optimisation a new transition geometry was created which had no stress increasing in the transition point. This was achieved by using a large transition radius. Unfortunately, this new geometry was susceptible to instability, so that we could not realise sufficient strain amplitudes. In a third step, we therefore reduced the measurement area and used a somewhat smaller transition radius to achieve a sufficient stiffness of the specimen, which was high enough to resist instability. The resulting minimum stress increase at the transition point as compared to version two, has not yet caused any preferential crack initiation. The final geometry of specimens used is illustrated in Fig. 2.
Fig. 2. Shape and dimension of used specimens (mm).
444
M. WEICK AND J. AKTAA
The observation area is a 25 mm wide cylindrical zone in the middle of the specimen, where the stress distribution is nearly homogeneous. As an advantage compared to a fully cylindrical specimen, this specimen is less susceptible to buckling and the most probable location of crack nucleation, the middle of the specimen, is restricted considerably. Thus, measurement of the radial extensometers near the location of crack nucleation is ensured. In addition, there are not any sharp changes of the outline, which could provoke notch effects, due to the geometry chosen. The chosen material for this tests was the high-alloy stainless steel X2CrNiMoN1712 ( German standard 1.4909, American standard AISI 316 L (N)). This material is in discussion for the first wall of a fusion reactor. The specimens were made out of seamless and textureless tubes. After the turning of the specimens the surface of the measurement area were fine ground. The chemical composition and the basic material properties are shown in table 1 and table 2.
Table 1. Chemical composition of the used material
1
Chemical Element Cr Ni Mo Mn N Cu Si Co C Nb+Ta+Ti P S
Concentration [%] 17.34 12.35 2.41 1.69 0.053 0.048 0.037 0.029 0.026 0.025 0.01 0.003
B
0.0015
Table 2. Ascertained basic material properties of the used material
1
Young's Modulus Yield Strength Tensile Strength Average diameter of grains
ISOGPa 343 MPa 630 MPa 30-40 |Lim
1
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
445
Measurement instrumentation The measurement of the axial strain was performed by a conventional clip extensometer (s. Fig. 3). To determine the circumferential strain, six radial extensometers were distributed at the circumference of the specimen. From the average value of the radial displacement, the circumferential strain was calculated. With this, a more precise registration of the circumferential strain compared to a diametrical layout was possible. To obtain information about the sonic emissions transmitted during the experiment, two sonic emission sensors were mounted outside the pressure vessel. These two sensors have a registration spectrum from 600 kHz up to 1200 kHz. In this frequency range, most signals due to crack nucleation and crack propagation are recorded. Signals induced by plastic deformations have frequencies lying below this spectrum.
1. 2. 3. 4.
Specimen Axial strain extensometer Radial displacement extensometers -^circumferential strain Sonic emission sensors Fig. 3. Measurement of the experimental parameters.
446
M WEICK AND J. AKTAA
Loading procedure The tests performed were strain-controlled with the strain in axial and circumferential direction following sin functions as illustrated in Fig. 4 for one cycle with a time period of 10 seconds.
axial circumferential radial
Fig. 4. Example for a phase shift between axial and circumferential loading.
For all tests the amplitude of the axial strain was selected equal to the amplitude of the circumferential strain. With the phase shift between axial and circumferential strain, nonproportionality of biaxial loading could be specified. Thus, loading during a test was determined by the parameters of strain amplitude and phase shift. For a given phase shift, the strain amplitude was selected in a way that the equivalent plastic strain range to be expected in the test - introduced below in the next section - was comparable to the plastic strain ranges of the uniaxial fatigue tests considered as reference. Therefore, the viscoplastic model by Chaboche [4] fitted to the behaviour of AISI 316 L(N) at room temperature was used to predict the equivalent plastic strain range for a given phase shift and strain amplitude. Thus, the strain amplitude to be selected for a test could be calculated iteratively for a given phase shift and a desired equivalent plastic strain range.
RESULTS AND DISCUSSION Lifetime behaviour and its modelling Definition of the equivalent plastic strain range. To compare results of experiments at different phase shifts, an equivalent plastic strain range Ae^ was introduced, which is defined as:
-K
Ae'p?=J-^l maxmax||ep|(t)-ep,(to)|
(1)
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
447
Here £p,(t) is the plastic strain tensor at a time t and Bp,(to) is the plastic strain tensor at a time IQ. TO obtain the equivalent plastic strain range, the magnitude (Euclidean norm: ||x|| = Vx:x = ^Trace(x-x) ) of the difference tensor 8p,(t) - ep,(to) must be calculated during a cycle for every reading point couple. The maximum magnitude of the difference tensor is then multiplied by the factor ^ for reduction to the uniaxial case delivering the equivalent plastic strain range. This procedure is illustrated in Fig. 5.
Fig. 5. Graphical determination of Ae^
The curve shown in Fig. 5. results from the connection of all reading points ascertained during a cycle. This curve is located in the plane of volume constancy during plastic deformation (8p, +8p, +8p" =0). The dashed line represents an arbitrary absolute value of the differential tensor, while the solid line represents the maximum of the absolute value of the differential tensor, just as the diameter of the circumscribing circle. Fatigue lifetime and its description. Following the test procedure presented above, multiaxial fatigue tests have been performed with the phase shifts of 45, 90, and 135°. The observed fatigue lifetime data are listed in Table 3 and plotted in Figure 6 in comparison with the reference data from uniaxial tests as well as tests with proportional multiaxial tests performed by Windelband on the same facility with similar specimens [5].
448
M. WEICK AND J. AKTAA
Table 3. Experimental results of the non-proportional fatigue tests performed Phase shift [°]
1
Nf
A8^ [%] 0.45 0.4 0.25 0.46 0.35 0.33 0.32 0.40 0.40 0.36 0.33 0.31 0.28
45 45 45 90 90 90 90 135 135 135 135 135
135
1
1063 40700 86274 8074 13702 72000 52914 11238 19970 33319 49023 29065
15051
J
uniaxial
;>( •
-
-
.
.
.
.
.
•
•
•
*
\^
,
•
0°
X
45°
#
•
90° 135°
# " .,
^ ••••»•-..
o\o
•-..
•
180°
CO
< AISI316L(N)
0,1 1E+03
1E+04
1E+05
1E+06
Nf Fig. 6. Correlation of non-proportional life with the equivalent plastic strain range for various phase shifts. The results for 0° and 180° phase shift are taken from Windelband [5]. The behaviour can be described by the Manson-Coffin relation. In our case, it is: (2)
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
449
Nf is the number of cycles to failure, while C and z are material parameters. N^ is defined by the point of time when the crack has penetrated the wall. In our facility we are monitoring the gradient of the inner pressure and the outer pressure. If we have a wall penetrating cracking the gradient of the inner pressure various very quickly and the outer pressure increases. So we can determine the point of time of the wall-penetrating of the crack and out of this N^. It can be seen that for different phase shifts the material parameters are also different. Furthermore, it can be observed that an increasing phase shift causes a higher life at the same equivalent load. This indicates that the failure process is not controlled by shear stress exclusively. Hydrostatic stress plays an important role. Similar results were observed by Mouguerou [6] and Ogata [7]. For a phase shift of 0 degree the hydrostatic stress reaches its maximum, while for a phase shift of 180 degrees the hydrostatic stress is minimum. If the loading amplitudes, at a phase shift of 180 degrees have the same value, the hydrostatic stress even vanishes. It may therefore be supposed that a higher hydrostatic stress at the same equivalent stress reduces the lifetime. To obtain a correlation between the multiaxial experiments with different phase shifts on the one hand and the uniaxial reference experiments on the other hand, a multiaxiality factor f^ was introduced. Therewith, the Manson-Coffin relation was modified as follows: Ae^=f,-C*-Nf
(3)
C* and z* are material parameters which are extracted from uniaxial fatigue experiments (C* =: 12.63,z* =-0.3395).f^ is a function of the multiaxial hydrostatic stress range Aa^'"^'"'^' referring to the uniaxial hydrostatic stress range Aa^"'*'"^', arising for the same equivalent plastic strain range A8pJ:
/
Ao:
(4)
^^un,ax,
with Aaf'"^"^' = j[max Trace(a) - min Trace(d)]
and A^r'^''' = ^ Aa""'^'^' = f 2 • u|
(5)
(6)
V 2 y
From the uniaxial deformation data determined, it results u = 425.7 and p = 0.2. The multiaxiality factors out of the different multiaxial experiments were investigated with Eq. (3). When plotting these multiaxiality factors over the ratio Aaf^""^'"' / Aaf'^•'' it can be seen that a linear function could describe this relation quite well: ^ Arr"*"'''^*^' ^
f. = h - k
A —uniaxial
(7)
with h=1.396 and k=0.396. Thus it results a rather good description of the lifetime by the Ae"^ / modified Manson-Coffin relation (Eq. 3). In Fig. 7 the ratio yr is plotted over the number of cycles to failure on a double logarithmic scale. It can be seen that the results lie within an
450
M WEICK AND J. AKTAA
acceptable scatter band. Thus, it is possible to predict the lifetime of multiaxially loaded components using data obtained from uniaxial fatigue tests.
uniaxial
•
0°
:^
45°
#•
90°
135°
•
o \ ^ x
CO
. ) - ^ + t,(:^)7cn Vn
(12)
and X = —
(13)
Aa^ denotes the von Mises equivalent stress range. Its calculation is analogous to the determination of Ae^. The dimensionless function J takes into account the increased plastic deformation in front of the crack tip in comparison to the global plastic deformation. The function s{X) takes into account the mode mixity of the crack loading and decreases as X increases. For pure Mode I loading ( X = 0) s =3.85. For the pure Mode II (k -> °°) s=l,45 [9]. For a mixed mode loading we propose an interpolation in the following form:
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
455
(14)
s = 2.4e"'' + 1.45
The dependence of t^ on X is determined by the energy release rate in terms of the equivalent stress and strain in the linear elastic case [9]: \ + X'
t„ =
(15)
((1 + 3^^)1 l + - ( l + v)'*>.^ 3 Thus and since the regime of short cracks starts at a crack length a=kd, AJ can be written as AJ = AJ
(16)
kd
and Eq. (8) as
dN
'^
(17)
Mkd
with AJ* denoting the value of AJfor a crack of the length a=kd. Now, McDowell and Bennet had ideas how Eq. (8) can be extended to describe crack propagation also in the regime of micro cracks [8]. We adopted some of these ideas and modified Eq. (17) for the description of crack propagation in both, the regime of micro cracks (a < kd) and the regime of short cracks (a > kd), as follows:
^dN = C^(Ajtf'^ M k d
(18)
with m„ =\ AJ. AJ
for
a>kd
for
a0) nearly independent on the crack length a (m^ - ^ 0 ) like it is discussed in the previous section. Assuming that the lifetime fractions for micro crack nucleation and for long crack propagation can be neglected in comparison to fractions for micro and short crack propagation Eq. (18) can be used for lifetime prediction. For our material, the sonic emission observations indicate that the lifetime fraction up to micro crack nucleation is rather small in comparison to whole lifetime. Due to the thin wall thickness of the used specimens the cracks leading to failure and wall penetration, respectively, can be considered to be small so that the lifetime fraction for long crack propagation is assumed to be negligible in comparison to the whole lifetime. So we
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M WEICK AND J. AKTAA
applied Eq. (18) to calculate the lifetime for our multiaxial tests. The results of this application are presented in the next section.
Lifetime calculation using crack propagation model. For a cyclic test the lifetime fraction for micro and short crack propagation and therewith the lifetime is calculated by integrating Eq. (18) as follows: N,
led/
x-'Mj'"^
jC^(AJTdN = J A
a^Y
(kd)^-'"^'"^-a;,-'
=>N,:
c-^i^yp
da
(19)
a^'-Ckd)'" (l-mp(kd)-'"^
(20)
d a . j kd U
(l-m^m^)(kd)-"'^'"^
For ao = d = 0.02 mm, k = 5, and a^ =0.5 mm (half wall thickness) the material parameters Cj, m-, c^, and b^ are determined by fitting the numbers of cycles to failure calculated using Eq. 20 to the numbers of cycles to failure observed experimentally. They are independent from the loading and should be valid for other cases. However they are not verified for other cases as yet. The resulting values are listed in Table 4. The quality of the fit is illustrated in Table 5 as well as in Fig. 13 where the calculated lifetimes are compared with the observed ones.
Table 4. Microcrack propagation constants
ao kd a. C^
1
0.1 mm
1
0.5 m m
3.573 10"'' mm(N/m)"'"' 2.715 4.643
m.
S
1
0.02 m m
^
2.736
Table 5. Experimental ascertained AJQ 's (AJ for a=ao) and herefrom calculated lifetime.
1
AJo [N/m]
1 ^ calculated
1 ^ measured
359.8 345.2 320.7 299.2 240.5 234.9
11812 12092 8185 19231 39569 37104 22848
11238 19970 8074 33319 29065 49023 13702
219,6
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
217.3 198.1 197.6
1
159.3
21773 30863 31331 50584
457
40700 72000 52914
86274
1
In Fig. 13 all points lie in a scatter band of factor two on lifetime. A reason for the variance downward of the calculated values, for high lifetimes, may be the load independent critical crack length a^. For low load-levels the critical crack length a^ is probably higher than that for high load-levels. This may be taken into account in further reasonable modifications of the crack propagation model.
10"
Factor of two on lifetime
10^
10 measured N f
Fig. 13. Comparison of experimentally observed and calculated life.
By using the above determined parameters for the microcrack propagation model we are able to calculate the quantitative trend of the crack growth rate above the crack length. For the 45° and 90° out-of-phase tests the crack growth rate in the regime of microcracks is, onto this model, nearly constant. For the 135° out-of-phase tests we have in contrast also in this regime a dependence of the crack growth rate on the crack length (cf. Fig. 14).
M WEICK AND J. AKTAA
458 1.0E-03
- - 90°,45° : 135°
/ /
;
1 .OE-04
^
10E-05
/
CD •o
..^''m
" f
4
*
1 .OE-06
ao
kd
ac
1 .OE-07 1.0E-02
1.0E-01
1.0E+00
crack length a
Fig. 14. Typical trends of the crack growth rates for the different phase shifts.
CONCLUSIONS The present investigations show that the lifetime under multiaxial non-proportional loading increases, at the same equivalent load, with an increasing phase shift. The reason for this behaviour is that the hydrostatic part of stresses decreases with an increasing phase shift. By using a modified Manson-Coffin law , which takes the hydrostatic loading part into account, the lifetime can be predict quite well. The sonic emission observations confirm the meanwhile well known behaviour of microcracks and the influence of microstructure on it. With the AJ based model for micro and short crack growth, we proposed a second approach to predict the lifetime under multiaxial conditions. This model also yields satisfying results where the increasing lifetime with increasing phase shift is predicted taking into account changes of the crack nucleation plane and therewith of the crack loading. This results in changes on the behaviour of the crack propagation in the regime of microcracks. The crack propagation model seems to be a promising tool for lifetime prediction also for cyclic loading with varying amplitudes, phase shifts ...etc. what should be verified in further applications. ACKNOWLEDGEMENTS The present work was financially supported by the Deutsche Forschungsgemeinschaft within the framework of the "Schwerpunktprogramm mechanismenorientierte Lebensdauervorhersage fiir zyklisch beanspruchte metallische Werkstoffe". Stefan Knaak, employee of the Institut fur Materialforschung II in the Forschungszentrum Karlsruhe, is acknowledged for his assistance in the performance of the experiments.
Microcrack Propagation Under Non-Proportional Multiaxial Alternating Loading
459
REFERENCES 1.
M. Weick, J. Aktaa and D. Munz, Micro Crack Nucleation and Propagation under Nonproportional Low Cycle Fatigue of AISI 316 L(N), Proceeding of the Sixth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Lisboa, 2001, Vol I, pp. 495502.
2.
M. Weick, J. Aktaa, D. Munz, Inbetriebnahme und Optimierung einer biaxialen Priifmaschine zur Durchfuhrung von nichtproportionalen, mehrachsigen Ermiidungsversuchen an Rohrproben, Berichtsband, KoUoquium des DVM, Bremen, 2000, pp. 179-186.
3.
H. Dietmann, T. Bhonghibhat and A. Schmid, Multiaxial Fatigue Behaviour of Steels under In-phase and Out-of-phase Loading Including Different Wave Forms and Frequencies, Proceeding of the 3'^^ International Conference on Biaxial/Multiaxial Fatigue, 3-6 April, 1989, Stuttgart, FRG, pp. 61.1-61.7.
4.
J. L. Chaboche, (1977) Viscoplastic constitutive equations for the description of cyclic and anisotropic behaviour of metals. Bull, de I'Acad. Polonaise des Sciences, Sc. et Tech. 25(1), 33-42.
5.
B. Windelband, Mehrachsige Ermiidungsversuche an Rohrproben aus dem austenitischen Stahl 1.4909. Dissertation, Universitat Karlsruhe, 1996.
6.
A. Moguerou, R. Vassal, G. Vessiere, and J. Bahuaud, (1982), Low-Cycle Fatigue under Biaxial Strain, Low-Cycle Fatigue and Life Prediction, ASTM STP 770, C. Amzallag, B. N. Leis, P. Rabbe, Eds., ASTM, pp. 519-546.
7.
T. Ogata, A. Nitta, K. Kuwabara, (1991) Biaxial Low-Cycle Fatigue Failure of Type 304 Stainless Steel under In-Phase and Out-of-Phase Straining Conditions, Fatigue under Biaxial and Multiaxial Loading, K. F. Kussmaul, D. L. McDiarmid D.F. Socie, Eds., ESIS Publication 10, Mechanical Engineering Publications Ltd., London, pp. 377-392.
8.
D.L. McDowell and V.P. Bennet, (1996) A microcrack growth law for multiaxial fatigue. Fatigue Fract. Engng Mater. Struct. 19 (7), 821-837.
9.
T. Hoshide and D. Socie (1987) Mechanics of mixed mode small fatigue crack growth. Engng. Fract. Mech. 26, 841-850.
10.
K. J. Miller (1993) The two thresholds of fatigue behaviour. Fatigue Fract. Engng Mater. Struct. 16 (9), 931-939.
Appendix : NOMENCLATURE a h k
Crack length Fit parameter Fit parameter
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M WEICK AND I AKTAA
s
Mode factor
t^
Modulation factor
u
Fit parameter
p C
Fit parameter
C*
Material parameter, uniaxially ascertained
Cj
Microcrack propagation coefficient
S
Material parameter
K
Material parameter
da/ /dN
Crack growth rate
E
Young's modulus
fm
Multiaxial coefficient
AJ
Cyclic J-integral range
AJ*
AJ fora=kd
AJe,
Elastic part of the cyclic J-integral
AJp.
Plastic part of the cyclic J-integral
Material parameter
J
Plastic deformation increasing function
mj
Crack length exponent
m^
Crack length extra exponent for a < kd
n
Hardening exponent
Nf
Number of cycles until failure
N/Nf
Normalised number of cycles
z
Material parameter
z*
Material parameter, uniaxially ascertained
^pl'^pl.^pl
Plastic strains in the three main directions
^p.
Ae;?
Plastic strain tensor Equivalent plastic strain range
a
Stress tensor
C^n
Normal stress
T
Shear stress
^
Stress ratio ^Xon
V
Poisson's ratio
A -.uniaxial
Aa, A -.multiaxial
Aa, Aa
Uniaxial hydrostatic stress range Multiaxial hydrostatic stress range Stress range
6. APPLICATIONS AND TESTING METHODS
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Biaxial/Multiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) © Elsevier Science Ltd. and ESIS. All rights reserved.
463
FATIGUE ASSESSMENT OF MECHANICAL COMPONENTS UNDER COMPLEX MULTIAXIAL LOADING
Jose L.T. SANTOS, M. de FREITAS, B. LI and T.P. TRIGG Dept. of Mechanical Engineering, Instituto Superior Tecnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal
ABSTRACT This paper addresses an integrated FEM based approach for crack initiation life assessment of components under complex multiaxial loading. Generally, there are many sources of error in the computational fatigue damage assessments, including uncertainties in analysing complex service environments, complex geometries, and lack of usable material information, etc. This paper is focused in the methodology for handling the effect of non-proportional multiaxial loading, and in improvements in computational algorithms for reducing the computation time for fatigue assessments. Since the effective shear stress amplitude is an important parameter for crack initiation life prediction, the recent approaches on evaluating the effective shear stress amplitude under comlex loading paths are studied and compared by examples. The MCE approach developed on the basis of the MCC approach is described in detail, and it is shown that this approach can be easily implemented as a post-processing step within a commercial FEM code such as ANSYS. Fatigue assessments of two application examples are shown, using the computational procedure developed in this research. The predicted fatigue damage contours are compared for proportional and non-proportional loading cases, it is concluded that the fatigue critical zone and fatigue damage indicator vary with the combined conditions of multiaxial fatigue loading. Advanced multiaxial fatigue approaches must be applied for fatigue assessments of components/structures under complex multiaxial loading conditions, to avoid unsafe design obtained from the conventional approaches based on the static criteria. KEYWORDS Multiaxial fatigue, fatigue damage evaluation, computational durability assessment, fatigue life prediction.
INTRODUCTION Due to the increasing pressure of market competition for light weight design and fuel economy, computational durability analysis of engineering components/structures is more and more used in today's industrial design for reducing prototype testing and shortening the product development cycle [1]. Since it is widely recognized that about 80% of mechanical/structural component failures are related to fatigue, structural fatigue life has become the primary concern in design for durability.
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J.L.T. SANTOS ETAL
In real service, engineering components and structures are generally subjected to multiaxial fatigue loading conditions, in which the cyclic loads act in various directions, with different frequencies and/or different phases [2]. In these non-proportional multiaxial loading conditions, the corresponding principal directions and/or principal stress ratios vary during a loading cycle or block. Advanced engineering designs require efficient, accurate and easy-ofuse methods for durability assessment of components/structures under complex multiaxial loading. Current fatigue design approaches treat both proportional and non-proportional loading with the maximum principal or equivalent stress range, and then, they refer to the design S-N curve obtained under uniaxial loading condition [3]. The Eurocode 3 design code recommends that the maximum principal stress range may be used as a fatigue life damage parameter if the loading is proportional. For non-proportional loading, the components of damage for normal and shear stresses are assessed separately using the Palmgren-Miner rule and then combined using an interaction equation. Maximum shear stress range is used as an equivalent stress for non-proportional loading in the ASME code. However, conventional multiaxial fatigue criteria were based on proportional fatigue data, and hence not applicable to non-proportional loading, due to the changes in direction and/or ratio of the principal stresses. This has led to a number of research studies on the multiaxial fatigue problem over the past 20 years. Much progress has been made in understanding the cracking modes under complex loading, and various multiaxial fatigue damage parameters have been proposed. Although many multiaxial fatigue models have been proposed in the literature, there still exist gaps between the theoretical models and engineering applications. Generally, there are many sources of error in the computational fatigue damage assessments, including uncertainties in analysing complex service environments, complex geometries, and lack of usable material information, etc. It is imperative to study the accuracy and improve the computational algorithms for every step of the fatigue evaluation process. The objective of this paper is to study the engineering approaches for crack initiation life assessment of components under complex multiaxial loading. Firstly, current multiaxial fatigue models are briefly reviewed and compared. Then the recent approaches for evaluating the effective shear stress amplitude under complex loading paths are studied and compared with example problems. It is shown that the minimum circumscribed ellipse (MCE) approach, developed on the basis of the minimum circumscribed circle (MCC) approach, is an easy and efficient way to take into account of the non-proportional loading effect for fatigue evaluations. The stress invariants based multiaxial criterion, coupled with the minirnum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude, are shown to be a simple and efficient methodology for handling the complex loading effects. The implementation of the minimum circumscribed ellipse (MCE) approach in the commercial FEM code ANSYS is discussed. Applications of the developed procedure for engineering problems are shown for two examples: an automotive suspension torque arm, and a train car. In the integrated FEM based fatigue assessment procedure, the quasi-static FE analyses are used to obtain the stress-time histories at each nodal point by stress superimposition due to each individually applied load. Then the minimum circumscribed ellipse (MCE) approach is used for multiaxial fatigue life evaluation at each nodal point, requiring only the knowledge of basic material fatigue parameters.
Fatigue Assessment of Mechanical Components Under Complex Multiaxial Loading
CURRENT METHODOLOGffiS FOR FATIGUE DAMAGE EVALUATIONS The fatigue life of a mechanical component or structure depends on the interaction of at least three physical and mechanical phenomena: the material behaviour, the geometry of the component, and the service loading of the component or structure [1]. The fatigue damage assessment methods can be categorized as two groups: global approach and local approach [4]. The global approach uses directly the amplitudes of the nominal stresses or the acting forces/moments, and compares them with the nominal stress S-N curve for fatigue limit evaluation or fatigue life prediction. The local approach evolved from the global approaches, and proceeds from local stress and strain parameters, consists of different types: structural stress approach, notch root approach, and so on. The structural stress approach proceeds from the structural stress amplitudes in the component/structure, and compares them with a structural stress S-N curve. The structural stresses (also called hot spot stresses) are generally the results of finite element analysis of welded or nonwelded structures, without consideration of the actual notches (such as the welding geometry, etc.) in the finite element modelling. Commonly, the structural stresses are elastic and indicate the macro-geometrical influences. The notch root approach proceeds from the elastic-plastic strain amplitudes at the notch root and compares them with the strain S-N curve of the material in the unnotched comparison specimen. The notch root approach is also called the local strain approach, and is based on the hypothesis that the mechanical behaviour of the material at the notch root in respect of local deformation, local damage and crack initiation is similar to the behaviour of a miniaturized, axially loaded, unnotched specimen in respect of global deformation, global damage and complete fracture. Different views exist between experts concerning how detailed the local consideration must be in the fatigue assessment procedure, based on structural stresses only or on notch stresses also. No general answer is possible. The choice of the approach must be made based on the circumstances of the case considered. Generally, the structural stress analysis is always required because the notch stresses/strains are based on structural stresses. If the scatter range of the local notch geometry, caused by the manufacturing process, is small or if the scatter range can be passed over by a worst-case consideration, the step from the structural stress approach to the notch stress approach is justified. However, if the scattering of the notch geometry is very significant such as the case of non-machined welded joints, the notch stress analysis is not well suited because the notch geometry cannot be accurately modelled. Due to the complex geometry of engineering components and structures, the nominal stresses cannot meaningfully be defined. The local approach is widely used in the computational fatigue assessment procedures, which involves isolating each potential critical location and independently determining its fatigue life. By isolating each potential fatigue critical location, the complex component is regarded as a number of individual fatigue specimens. The most fatigue-critical location is then the location with the shortest fatigue crack formation life. The fatigue life of the component is therefore defined by the fatigue life of the most fatigue-critical location. Computer aided fatigue evaluation of engineering components/structures consists of two main steps: dynamic stress computation and fatigue life prediction. Dynamic stress histories can be obtained either from experiments (mounting sensors or transducers on a physical component) or from computer simulation. The simulation-based approach is usually done by performing finite element analysis of the component under the specified set of applied loads.
465
466
J.L.T.
SANTOSETAL.
Then, fatigue life prediction is carried out as a post-processing step of finite element output results. A general flow chart of computational fatigue assessment is shown in Fig. 1.
Multi-body Dynamic Analysis
Component Finite Element Analysis
ii
ji Static Stresses for Unit Loads
Load Time Histories
Superposition Principle
a
Stress Time Histories
d Multiaxial Fatigue Criteria
1 Fatigue Life Prediction Fig. 1. Schematic flowchart of computational fatigue assessment. Uncertainties in computational fatigue assessments are attributable to many sources, such as uncertainties in analysing complex service environments, complex geometries, and lack of usable material information, etc. Among the many sources of errors in the computational fatigue assessment, the effect of non-proportional multiaxial loading is one of the important considerations, since recent researches have shown that the non-proportional loading causes additional fatigue damage and the conventional methodologies of multiaxial fatigue life assessment may lead to unsafe design.
EVOLUTION OF MULTIAXL\L FATIGUE PREDICTION METHODS The multiaxial fatigue criteria proposed in the literature may be categorized in three groups: stress-based, strain-based and energy-based methods. For high-cycle fatigue problems, most of the multiaxial fatigue criteria are stress-based. Early works on multiaxial fatigue include the extension of the von Mises criterion to the S-N curve, which has been widely used for proportional cyclic stresses where ratios of principal stresses and their directions remain fixed during cycling.
Fatigue Assessment of Mechanical Components Under Complex Multiaxial Loading
467
In order to handle non-proportional loading effect on fatigue resistance, many new methodologies have been developed and are based on various concepts such as the critical plane approach [5], integral approach [6], mesoscopic scale approach [7], etc. A common feature of many high-cycle multiaxial fatigue criteria is that they are expressed as a general form and include both shear stress amplitude ta and normal stress CJ during a loading cycle: T^ + k{N)cj=?i{N)
(1)
where k{N) and X{N) are material parameters for a given cyclic life N. Multiaxial fatigue models differ in the interpretation of how shear stress and normal stress terms in Eq. (1) are defined. For non-proportional cases, a stress-based version of the ASME boiler and pressure vessel code, case N-47-23 [8], may be used as an extension of the von Mises criterion, in which an equivalent stress amplitude parameter, SEQA. is defined from stress ranges AGX, Aay, AGZ, Aixy, Axyz, ATZX, in the form
SEQA =—P^(ACT^. - A c r J ' +(A(T,, -ACT.)' + (ACT^ - A C T J ' +6(Ar_^,' + Ar,,' + Ar_^/)
(2)
where Aax=ax(/7)-ax(^2), ^^y=^y{ti)-^y{t2), etc. SEQA is maximized with respect to two time instants, t\ and ti, during a fatigue loading cycle. For constant amplitude bending and torsional stresses such as
^xy-
Ttsm\a)t-Sj^y)
Eq.(2) becomes
where K=2Tt/ab. When Tt/Cb=0.5 and 5xy=0 (proportional loading case), Eq. (4) gives S^Q^ =\323CJ^^.
When
Tt/ab=0.5 and 5xy=90° (out-of-phase loading case), Eq. (4) gives S^Q^ - cr^,, which means that out-of-phase load case is predicted to be less damaging than the proportional load case with the same stress amplitudes. However, experimental results showed that the prediction by Eq. (4) for out-of-phase load case is inconsistent and non-conservative. Hitherto, many approaches have been proposed for treating the non-proportional effects, among them the critical plane approach and the integral approach are two important concepts.
Critical Plane Approaches Critical plane approaches are based upon the physical observation that fatigue cracks initiate and grow on certain material planes. The orientation of the critical plane is commonly defined as the plane with maximum shear stress amplitude. The linear combination of the shear stress
468
J.LT. SANTOS ETAL'
amplitude on the critical plane and the normal stress acting on that plane is defined as the fatigue damage correlation parameter. For complex loading histories, the principal directions may rotate during a loading cycle (e.g. see Ref. [9]). Therefore, Bannantine and Socie [5] suggested that the critical plane should be identified as the plane experiencing the maximum damage, and the fatigue life of the component is estimated from the damage calculations on this plane. The approach proposed by Bannantine and Socie [5] defines the critical plane as the plane of maximum damage rather than the plane of maximum shear stress (strain) amplitude, as defined by previous authors. This approach evaluates the damage parameter on each material plane. The plane with the greatest fatigue damage is the critical plane, by definition. For general random loading conditions, with six independent stress components, the critical plane approaches have to be carried out for plane angles 0 and (j) varying from 0 to n. These procedures demand a great deal of calculations, especially when small angle increments are used. In the last decades, the critical plane approaches have found wide applications and also received some criticism. The critical plane approach assumes that only the stress (strain) acting on a fixed plane is effective to induce damage, and then, no interaction of the damages on the different planes occurs. These assumptions are not always valid, and may considerably underestimate fatigue damage. Zenner et al. [10] also indicated by a typical example that the hypotheses of the critical plane approach are not suitable for describing the effect of the phase difference. The example considers the stress waves of Eq. (3), with phase shift angle 5xy= 90° and stress amplitude Tt= 0.5ab- Under this load case, the shear stress amplitude has the same magnitude in all planes.
Integral Approaches Integral approaches are based on the Novoshilov's integration formulation, as a mean square value of the shear stresses for all planes [10]:
,=12
j ^rj^sinydydcp
(5)
Equivalent-stress amplitude is yielded by an integration of the square of the shear stress amplitude over all planes ycj) for fully reversed stresses. Further developments of the integral approaches led to various hypotheses such as the effective shear stress hypothesis, the shear stress intensity hypothesis (SIH), etc. Generally, the integral approach [10] uses the average measure of the fatigue damage by integrating the damage over all the planes. The integral approach considers all damaged planes of a specific critical volume. The averaged stress amplitude of the shear stress intensity hypothesis (SIH) is formulated as: r 1 S ^ 2;r ' y=0(p=0
Papadopoulos' mesoscopic approach [11] is also formulated as an average measure, by integration of the plastic strains accumulated in all the crystals, within the elementary volume:
Fatigue Assessment of Mechanical Components Under Complex Multiaxial Loading
M=
li^C
U \yMr,V)\-r.rdy.dyd
