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Influence of material’s ductility and local deformation mode on multiaxial fatigue response
International Journal of Fatigue 33 (2011) 930–947
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Influence of material’s ductility and local deformation mode on multiaxial fatigue response C.M. Sonsino Fraunhofer Institute for Structural Durability and System Reliability LBF, Bartningstr. 47, D-64289 Darmstadt, Germany
a r t i c l e
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Article history: Received 23 August 2010 Received in revised form 13 January 2011 Accepted 16 January 2011 Available online 1 March 2011 Keywords: Multiaxial fatigue Ductility Deformation mode Proportional loading Non-proportional loading
a b s t r a c t The multiaxial fatigue behaviour of components seems to depend mainly on the ductility of the material used determined by applied manufacturing parameters. The ductility steers the damage mechanisms. While, in the case of low-ductility (brittle) materials, the normal stress (strain) is the decisive parameter, in the case of ductile materials, it is the shear stress (strain), and, for semi-ductile materials, a combination of normal and shear stresses (strains). Critical plane oriented hypotheses can consider these different parameters, but the difficulty lies in the definition of ductility and, based on this, the selection of the appropriate hypothesis. Therefore, especially for the evaluation of safety parts, experimental verifications are still necessary, because of the lack of a general multiaxial fatigue hypothesis. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Durability design against complex multiaxial loading is a very old challenge in the history of mankind. A prominent example is the chassis and body adjustment of war carriages, Fig. 1, in ancient times. The major challenges of high speed and manoeuvrability were coupled with fulfilling of the durability requirement of the wheels, hubs and axles, submitted to vertical, lateral, acceleration and braking loadings, pushing the development of lightweight components. The selection and production of the appropriate woods and later metals, their special treatment and, last but not least, the design of the particular components occupied engineers over centuries. This still persists as a task with high responsibility in vehicle design, especially with regard to automotive body and chassis components, Fig. 2. As well as the development of lightweight structures, already being practised in ancient times, there were also machines requiring high functional precision, like printing presses during the Renaissance, Fig. 3. The multiaxial loaded part here was the printing axle submitted to a torque and a compressive axial load causing fatigue failures in the very early presses. The problems were solved by design, thicker wooden axles, and later also by the introduction of castings. Also nowadays, in cases where lightweight is not primarily the focus, but rather stability aspects, e.g. tilting axles of steel converters, Fig. 4, certainly thick components are designed. However,
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besides the geometry, material selection and the assessment of the material behaviour under multiaxial loading also remain important. The design of multiaxial loaded components in almost all industrial sectors; automotive, wind power plants, steel manufacturing, chemical processing, storage and handling, aeronautical and many others still require large efforts in the numerical assessment, i.e. valid hypotheses. Some selected examples of multiaxial loaded components from these application fields are displayed in Figs. 2 and 4–6. However, to date all proposed fatigue strength hypotheses reveal limitations and possess no general validity. Thus, there is a continuing motivation for and process of development of so-called new hypotheses, always with observed limitations. These limitations seem mainly to arise from features steering the local deformations and from particular material characteristics, which are not properly taken into account; it is not sufficient to know only the fatigue behaviour under uniaxial loading, i.e. SN-curves, and to have data on crack propagation. Therefore, the present paper presents and contrasts examples to advise design engineers and scientists with regard to the influence of local deformations and of material ductility on the multiaxial fatigue behaviour.
2. Experimental prerequisites In developing multiaxial fatigue hypotheses, the experimental background is very decisive, because this delivers the verification basis. In this context, the conception of multiaxial tests requires
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Nomenclature Stresses and deformations r axial, bending or normal stress s shear stress e axial, bending or normal strain c shear strain Indexes a a, b, t el, pl tot max x, y, z eq 1,2
amplitude axial, bending, torsion elastic, plastic total maximum coordinates equivalent principal directions
E G Kt N Ps R Rm Rp0.2 T e f k
u d
Young’s modulus shear modulus stress concentration cycle probability of survival ratio between minimum and maximum ultimate tensile strength 0.2% yield strength scatter elongation frequency slope of the SN- curve direction of an interference plane, direction of principle stress phase angle
Other symbols A area reduction
Fig. 1. Antique war carriage of pharaoh Tutanchamun (1250 B.C.) with multiaxial loaded wheels, hubs and axle (source: British Museum, London).
the simulation of real local deformation conditions. In the past, many multiaxial tests were carried out with unnotched specimens in obtaining the first material related data, without considering component related geometrical details like notches, which are of prime importance. Here, it must be mentioned that multiaxial load control is much easier to perform than multiaxial deformation control. However, it was noticed that load controlled multiaxial tests with unnotched and notched specimens from the same material did not result in the same tendencies with regard to the influence of constant or changing principal stress directions, simulated by in- and out-of-phase loading (proportional and non-proportional loading), on fatigue life [1]. For a long time, axial load controlled
multiaxial tests with unnotched specimens were also carried out in other research works [e.g. 2–5], without being aware of the consequences of the selected loading mode. Namely, while axial load controlled multiaxial tests with unnotched specimens [1–5] under non-proportional (out-of-phase) loading deliver a prolongation of fatigue life, Figs. 7 and 8, tests with notched specimens [1,6] render a significant decrease of fatigue life, Figs. 9 and 10. The reason for this contradictory behaviour was clarified by load and deformation controlled tests with unnotched specimens [7], Figs. 11 and 12. The deformation control of the unnotched specimens result clearly in the reduction of fatigue life under out-of-phase (non-proportional) loading, Fig. 12, as observed with notched specimens, Figs. 9 and 10, because, in notches, the deformation is also controlled by the stress gradients, as long as the structural yield point is not exceeded [8] or a ratcheting or a cyclic creep is not activated. The decrease of fatigue life under non-proportional loading arises from a multiaxial cyclic hardening, in comparison to proportional loading [7,8]. This multiaxial cyclic hardening is a consequence of the forcing of energy into the material under deformation control and causes an earlier damage than under in-phase loading [7,8]. The multiaxial cyclic hardening under deformation control is not related to the cyclic softening or hardening characteristic of the material under uniaxial pure axial straining or pure torsional shear straining [7,8]. In Figs. 8 and 13, left, a tempered cyclic softening high strength, quenched and tempered ductile steel and in Figs. 8 and 13, right, two similar austenitic cyclic hardening ductile steels are presented. Despite the differences in the uniaxial cyclic behaviour, Fig. 14, the response to multiaxial loading, either under multiaxial load control of unnotched specimens [2,3], Fig. 8, or under component/notch imposed multiaxial deformation control of unnotched specimens [7,8], Fig. 13, reveals the same tendencies, i.e. an increase of fatigue life under non-proportional load control, Figs. 7, 8 and 11, or the component related decrease of fatigue life under non-proportional deformation control, Figs. 12 and 13. This behaviour was attributed to the observed multiaxial cyclic hardening, Fig. 15, and the imposed higher energy under non-proportional loading [8]. In this context it should be mentioned that there are also materials like Ti–6.5Al–3.4Mo which reveal a fatigue life decrease under deformation controlled out-of-phase multiaxial loading without a
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Fig. 2. Chassis of a seven series BMW.
Fig. 3. Printing press from the Renaissance (Original press of G. and N.I. Sonsino (1480)).
multiaxial cyclic hardening compared to in-phase loading [9]. This means that under non-proportional loading a fatigue life decrease can be also caused by other damaging mechanisms than multiaxial cyclic hardening and related energy increase. Until this point, the multiaxial fatigue response of mainly cyclic softening quenched and tempered and of cyclic hardening austenitic ductile steels, determined with unnotched specimens, has been discussed. The non-component related multiaxial behaviour of load controlled unnotched specimens is also related to plasticity effects depending on the applied load and deformation level, which are not controlled by stress/strain gradients, as they exist in the
notches of components. As soon as the multiaxial load/deformation level is above the branching point of the cyclic and monotonic stress–strain curves, Fig. 16, on top, a multiaxial load control will cause an uncontrolled increase of the deformations (axial strain and shear strain) under in-phase (proportional) loading, which is higher than the out-of-phase (non-proportional) loading [7,8]. This causes the observed higher fatigue life for out-of-phase loading of the displayed ductile steels. However, if the deformations are below the branching point, they remain globally elastic and, in this case, load control of the unnotched specimen corresponds to the deformation control. The consequence is that, at such load levels,
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Fig. 4. Converter with bearing axles.
Fig. 5. Main stresses on a crankcase wall.
the multiaxial fatigue behaviour corresponds to the behaviour of the notched specimens, i.e. the decrease of fatigue life under non-proportional loading of unnotched ductile specimens. In the case of low-ductility (or non-ductile (brittle)) materials, the relation between the branching point and the load levels from 104 cycles upwards is much more favourable with regard to global elastic behaviour, Fig. 16, bottom. Therefore, for such materials, like cast steels or cast iron [10,11], the multiaxial fatigue behaviour may be also investigated using unnotched specimens under load control and result in component like tendencies. Beside the metallic materials discussed, some preliminary information on the multiaxial fatigue behaviour of fibre reinforced materials will also be presented in the following. The material response under multiaxial loading for metallic materials with unnot-
ched and notched specimens under load control is also observed for the short-fibre reinforced polyamide PA 66-GF35 with 35% content of glass fibres (e = 3%) [12]. Load controlled tests with unnotched thin tubular specimens at room temperature reveal an increase of fatigue life under out-of-phase loading in contrast to in-phase loading [13], Fig. 17. However, with notched specimens under multiaxial load control a remarkable decrease of fatigue life under out-of-phase loading is obtained [14], Fig. 18. This is due to the different local deformations in the critical unnotched and notched cross sections. Obviously, due to the different microstructure of reinforced plastics in contrast to metallic materials different damage mechanisms [15] are present which have not yet been investigated in details. Independently of acting damage mechanisms, the design relevant results are those obtained with notched
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Fig. 6. Stirrers of a fertilizer plant.
Fig. 7. SN-curves for uniaxial and combined load controlled loading.
specimens (local deformation control due to stress gradients) as in the case of metallic materials, because unnotched components do not exist. If a recommendation has to be made here with regard to specimen geometry and loading mode, then it can be stated that multiaxial load controlled tests with unnotched specimens should be avoided. Instead, multiaxial deformation controlled tests with unnotched specimens should be carried out, in order to determine a more component related material response. However, industrial reality will be approached in a more realistic way if the tests are carried out with components or component like specimens containing the fatigue critical areas, i.e. notches, in which the local multiaxial deformation is deformation controlled, as long as the structural yield point is not exceeded and a cyclic creep or ratcheting does not occur.
3. Multiaxial loading and crack behaviour Most of the results presented in the previous section were determined for the failure criterion of initiation of a technically detectable crack with a defined depth (a = 0.25–1.50 mm) or surface length (l = 0.50–3.00 mm). This criterion is important for components for which a crack propagation is not allowed, e.g. automotive safety components like stub axles, hubs, brakes, etc. However, there are also applications where the crack formation and propagation have to be considered, from stand points of material development as well as of safety aspects (damage tolerance) or of determining inspection intervals. With regard to crack initiation and propagation, the microstructure plays an important role. A banded microstructure of a structural fine grained ductile ferritic steel, as displayed in Fig. 19,
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Fig. 8. Results of load controlled tests under pure and combined axial and torsional loading.
Fig. 9. Fatigue testing results obtained with notched specimens under combined bending and torsion (u = 0° and 90°).
influences the direction of early crack propagation especially significantly. The early cracks do not lie in the expected direction when the plane of maximum axial or shear strains are calculated [7]. Certainly, the formation direction of early cracks is determined not only by the interaction between the grain orientation and the microstructure itself (austenitic, ferritic, tempered, grain structure, i.e. body- or face-centred cubic, hexagonal, etc.), Fig. 20, and the ductility but also by the load level [16]. Also, the interaction between the chemical composition and the surface treatment can influence the crack formation significantly. Fig. 21 shows the different crack appearances of a case hardened SAE 1050 carbon steel (188–226 Brinell hardness in the core, 543–606 BHN in the case of about 2.5 mm depth) under shear strain controlled torsion. The hardness values and the related particular ductilities were determined by the lower and higher sulphur contents between 0.009% and 0.0039% [17].
Investigations with unnotched specimens certainly make sense for understanding the material behaviour, for drawing conclusions on the consequences for possible microstructural improvements and for recognizing tendencies with regard to the influence of multiaxial stress/strain states on the material behaviour. However, they will not give an indication of the component related influence of the material on the crack propagation behaviour, because of the missing stress/strain gradients when unnotched specimens are used. For this, propagation investigations with multiaxial loaded components or component like specimens are necessary. The results in Fig. 9 with the ductile quenched and tempered steel SAE 1045 were presented for the failure criterion of a first technical crack, but the tests were continued until the final rupture of the specimens. So, the crack propagation phases for in- and out of-phase loadings were also determined. The fatigue lives for crack initiation, propagation and total failure are displayed in Fig. 22, and also the ratios between the lives for crack propagation and total
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Fig. 10. Fatigue testing results obtained with notched specimens under combined axial loading and torsion (u = 0° and 90°).
Fig. 11. SN-curves for uniaxial and combined load controlled loading of unnotched hollow specimens.
Fig. 12. Strain controlled determined SN-curves with unnotched hollow specimens.
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Fig. 13. SN-curves for combined axial and shear strain (failure criterion: initial crack a 61 mm).
Fig. 14. Stress–strain curves of a cyclic softening and of a cyclic hardening steel.
Fig. 15. Stabilized hystereses at Ncr/2 for combined axial and shear strain.
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Fig. 16. Influence of multiaxial load controlled tests on the fatigue and deformation behaviour of unnotched specimens in relation to the cyclic stress–strain curve.
Fig. 17. Multiaxial fatigue behaviour of unnotched short-fibre reinforced polyamide tubes under load control (u = 0 and 90°).
failure. It can be seen clearly that not only the fatigue life to crack initiation but also the crack propagation life is reduced significantly by the non-proportional (out-of-phase) loading with the changing principal stress directions. At the lower load level, the crack propagation phase decreases compared to the higher load level. The results in Fig. 10 [6] also confirm that the crack propagation phase under out-of-phase loading is reduced compared to in-phase loading. These tests were carried out with specimens having a higher stress concentration than the specimens displayed in Fig. 9. Therefore, it can be assumed that the major parts of the determined fatigue lives consist of crack propagation. These results possess a relevance because they were obtained on component like notched solid specimens, Figs. 9 and 10. Unfor-
tunately, crack propagation results obtained with comparable loading conditions and geometries were not available for other materials, especially for less ductile materials, for which neutral behaviour or an increase of fatigue life under non-proportional loading is observed (see next chapter). It will remain an open question if both the crack initiation and the crack propagation life will be influenced with the same tendency. In relation to the crack propagation behaviour, multiaxial load controlled tests with fillet notched solid specimens, Fig. 9, and unnotched hollow specimens [18] do not show the same tendency. Tests carried out with thin cylindrical hollow specimens of the ductile fine grained steel StE 460 (S460N) show a prolongation of the crack propagation life for non-proportional loading in comparison to proportional axial loading and torsion [18], Fig. 23. This
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Fig. 18. Multiaxial fatigue behaviour of notched short-fibre reinforced polyamide tubes under load control (u = 0 and 90°).
Fig. 19. Position of a crack on an unnotched cylindrical hollow specimen under deformation controlled in-phase loading.
example underlines again that specimen geometry related stress gradients in radial (core) direction and different friction effects on crack propagation surfaces especially under torsion may lead to opposing results. With regard to the material influence on the crack propagation under equi-biaxial proportional loading, results are available from aeronautical applications [19] and are displayed in Fig. 24 for the aluminium–lithium alloy AlLi2.5Cu1.5Mg1 T81 (EN AW-8090 T81) (e = 7%) and the conventional aluminium alloy AlCu4Mg1 T3 (EN AW-2024 T3) (e = 14%). The ratio between the applied stresses was rx/ry = 0 for uniaxial loading and rx/ry = 1 for biaxial loading and the load ratio for the stress components was R = 0.1. pffiffiffiffiffiffiffiffiffi At lower stress intensities, DK < 1200 MPa mm, the biaxial loading does not influence the crack propagation behaviour significantly compared to uniaxial loading. However, at higher stress intensities, a marked decrease of the crack propagation rate is observed for both alloys. This is due to the crack closing influence of the second stress component rx. The crack growth resistance of 2024 T3 is lower than that of alloy 8090 T81 under uniaxial as well pffiffiffiffiffiffiffiffiffi as biaxial loading at stress intensities DK < 1600 MPa mm. How-
ever, due to the higher fracture toughness, i.e. higher ductility, of the conventional 2024, it reveals lower crack propagation rates than the 8090 alloy at higher stress intensities. 4. Influence of material’s ductility on multiaxial fatigue response and damage causes 4.1. Non-welded material states The initial examples of the influence of material ductility on crack formation, Figs. 20 and 21, have already been presented. In this section, this issue will be addressed more from the view point of component behaviour, but first for non-welded material states, i.e. wrought, sintered and cast materials. Figs. 9 and 10 have already demonstrated that non-proportional loading significantly decreases the fatigue life of ductile steels, here SAE 1045 (e = 22%) [1] and SAE 1040 (e = 32%) [6], up to 106 cycles; at higher fatigue lives, the SN-curves intersect with each other [20]. In contrary to this, materials with low-ductility, like the cast aluminium piston alloy (e < 2%), Fig. 25 [20], or sintered steels even at elevated
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Fig. 20. Early crack formation under pure torsion of thin hollow specimens.
Fig. 21. Crack formation of case hardened steel bars under shear strain controlled torsion.
densities (e = 13%), Fig. 26 [20,21], display a significant increase of fatigue life under non-proportional (out-of-phase) loading. These investigations were carried out with the same component like geometries as in Fig. 9 under load control. The fatigue life increasing influence of low-ductility materials is also displayed in Figs. 27 and 28. The ferritic cast nodular iron ENGJS-400 (e 6 11%) shows a significant increase of fatigue life under out-of-phase fully reversed as well as under pulsating multiaxial loading [10]. This is observed under both constant and variable amplitude loadings. Nevertheless, for low-ductility materials like the mentioned cast aluminium piston alloy, the sintered steels or cast nodular iron, it can be assumed that both the crack initiation and the crack propagation lives are larger for out-of-phase loading compared to in-phase loading. A few deformation controlled tests with unnotched hollow cast magnesium AZ 91 (G–MgAl9Zn1) specimens with low-ductility
(e 6 2.5%) have also shown an increase of fatigue life under nonproportional loading [22], Fig. 29. Last but not least, the behaviour of semi-ductile materials is also addressed by the example of the cast steel G-X 5 CrNi 13 4 (1.4313) with a martensitic–ferritic microstructure (e 6 20%) [11], Fig. 30. This material behaves in the same way under both proportional and non-proportional loadings; the changing of principal stress directions by the out-of-phase loading does not affect the behaviour obtained under in-phase loading. 4.2. Welded material states The multiaxial fatigue behaviour of welded joints under proportional and non-proportional loading is also determined by material ductility. Flange-tube connections (t = 10 mm) of seam welded ferritic fine grained steel FeE 460 (e = 25%) reveal a significant decrease
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Fig. 22. Influence of phase difference on crack propagation of a ductile steel.
Fig. 23. Crack propagation lives under in- and out-of-phase combined axial loading and torsion.
of fatigue life under both constant and variable amplitude out-ofphase loading in contrast to in-phase loading [23,24], Fig. 31. Also,
after N = 106 cycles, the fatigue life curves do not intersect, as is observed for the non-welded quenched and tempered steels SAE 1045 and 1040, Figs. 9 and 10. Also the laser beam welded overlapped thin tubes (t = 1.0 mm) of the ferritic steel S235 G2T (St 35) (e = 26%) behave in the same way under non-proportional loading [25,26], Fig. 32, as the seam welded thicker flange-tube connections [23], Fig. 31, without any intersection of the SN-curves after N = 106 cycles. In contrast to the seam welded steel flange-tube connections, similar thickness joints (t = 10 mm) of the age hardened aluminium alloy AlSi1MgMn T6 (AW-6082 T6) (e = 14%) render a neutral behaviour under both out-of-phase constant and variable amplitude loading in comparison to in-phase loading [24,27], Fig. 33. This means that this alloy possesses a semi-ductile material behaviour with regard to multiaxial fatigue. However, new results show that this observed behaviour cannot be generalized for the alloy AlSi1MgMn T6. Laser beam welded thin (t = 1.5 mm) overlapped tubes from nominally the same alloy but in a more ductile condition (e = 17%) display a decrease of fatigue life under non-proportional out-of-phase loading [28], Fig. 34. This behaviour is not due to the difference between seam welding of a thick joint and laser beam welding of a thin joint, as displayed
Fig. 24. Crack propagation rates of different aluminium alloys under uni- and biaxial loading.
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Fig. 25. SN-curves for pure and combined bending and torsion of a cast aluminium.
Fig. 26. SN-curves for pure and combined bending and torsion of a sintered steel.
Fig. 27. Influence of multiaxial alternating loading on the fatigue behaviour of the nodular cast iron GGG-40, fully reversed loading (R = 1).
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Fig. 28. Influence of multiaxial pulsating loading on the fatigue behaviour of the nodular cast iron GGG-40, pulsating loading (R = 0).
Fig. 29. SN-curves of pure and combined axial and torsional loading of cast magnesium alloy AZ 91 obtained under deformation control.
for the ductile steel joints in Figs. 31 and 32, but due to the differences in material ductility. Obviously, the manufacturing process and the heat treatment of thick and thin tubes (10 mm versus 3 mm thickness before machining) lead to differences in ductility despite the use of the same nominal material. Also the laser beam welded aluminium alloy AlMg 3.5 Mn (AW-5042) (e = 20%) results in a decrease of fatigue life under non-proportional loading compared to proportional loading, Fig. 35 [28]. However, the reduction of fatigue life is restricted for both aluminium alloys in the fatigue life region N < 106 cycles. At higher fatigue lives, an intersection or a crossing of the SN-lines occurs. The non-proportional loading may even become more advantageous in this region. This may be a result of contact of the overlapped tubes at low load levels. However, more tests in this region are necessary to clarify the course of the SN-lines. 4.3. Damage causes Obviously, the ductile metallic materials react in a different way to non-proportional loading with changing principal stress or
strain directions than low-ductility (or brittle) and semi-ductile metallic materials. What is generally accepted is that, for low-ductility (or brittle) materials, damage is activated by normal stresses or strains and, for ductile materials, damage is activated by shear stresses or strains. It is logical to assume that, in the case of semi-ductile materials, an interaction between shear and normal stresses or strains will be responsible for the damage. These three possibilities are displayed in Fig. 36, in terms of stresses on a surface element; this can also be expressed in terms of strains [7]. As experiences with fibre-reinforced plastic materials in different ductility states are not yet available the possible damage mechanisms under complex multiaxial loading will not be discussed here. But in this context it has to be mentioned that anisotropy effects will influence the damage mechanisms considerably [15]. What is also known is that, in the case of changing principal stress or strain directions in ductile materials, dislocations are activated in different directions blocking each other. This leads to the so-called multiaxial stress hardening in contrast to in-phase loading with constant principal stress or strain directions even if the material reveals a cyclic softening under uniaxial strain controlled
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Fig. 30. SN-curves of a cast steel under uni- and multiaxial loading.
Fig. 31. Fatigue testing results obtained with as-welded thick (t = 10 mm) steel specimens under combined bending and torsion (u = 0° and 90°).
cyclic loading [7]. The multiaxial stress hardening under out-ofphase deformation control causes the fatigue life decrease compared with in-phase multiaxial deformation controlled loading. However, it has to be mentioned again, that there are also materials which reveal a fatigue life decrease without a multiaxial cyclic hardening compared to in-phase loading [9]. Several critical plane and integral hypotheses have been developed for considering the described damage causes in past years e.g. [1–4,8–11,23–37], but none of them were introduced in advance without having a knowledge of the multiaxial behaviour and the estimation of the ductility related damaging parameters. The selection of the appropriate hypothesis requires an assessment of the ductility of the material to be evaluated, but the example with the seam welded and laser beam welded aluminium alloy, Figs. 33 and 34, shows the difficulty in deciding which level of ductility
may cause a change of the multiaxial fatigue response for nominally the same material. In addition to the classical ductility values of elongation or area reduction, the ratio between the local fatigue strength amplitudes sa/ra to be determined by pure axial loading or bending and by pure torsion is also often used as an indicator for multiaxial fatigue behaviour [1,4,8–11,33–37]: pffiffiffi sa/ra = 1/2 = 0.5, according to Tresca, or 1/ 3 = 0.58, according to von Mises, indicate that the shear stress (strain) will be the dominating parameter, and sa/ra = 1.0 (principal stress/strain hypothesis) points to the normal stress as being the damage steering parameter. Intermediate values suggest an interaction of shear and normal stresses/strains. However, this ratio also depends on several parameters, which restrict its applicability as an indicator for the selection of an appropriate hypothesis:
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Fig. 32. Laserbeam welded overlapped thin (t = 1.0 mm) steel tube-tube specimens under constant amplitude multiaxial loading.
Fig. 33. Fatigue testing results obtained with as-welded thick (t = 10 mm) aluminium specimens under combined bending and torsion (u = 0° and 90°).
Fig. 34. Laserbeam welded overlapped thin (t = 1.5 mm) aluminium tube–tube specimens under constant amplitude multiaxial loading.
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Fig. 35. Laserbeam welded overlapped thin (t = 1.5 mm) aluminium tube–tube specimens under constant amplitude multiaxial loading.
Fig. 36. Interference plane stresses and damage models.
Firstly, its determination requires samples with the same highly stressed material volume for the fatigue tests under pure axial (bending) loading and torsion in order to avoid or eliminate the influence of stress (strain) gradients on the fatigue strength; for this, thin hollow specimens are suitable. Secondly, anisotropy effects must be excluded, because the critical crack formation planes are different for axial loading (bending) and torsion. Therefore, the resulting fatigue strength must not be influenced by an anisotropy. Thirdly, the failure criterion for pure axial loading (bending) and torsion must be the same. For this, crack initiation for a defined crack depth, e.g. 0.25–0.50 mm, is recommended. The criterion of total rupture is not suitable because the crack propagation phases for axial loading (bending) and torsion are completely different. Fourthly, ratios obtained even when respecting the afore-mentioned three conditions for fully reversed loading (R = 1) cannot be applied for pulsating loading (R = 0), because the mean-stress sensitivity under torque is much less than under axial loading (bending). Neither the elongation nor the ratio sa/ra are values which may generally help as criteria for selecting the suitable hypotheses. Their relation with regard to the mentioned damage scenarios (normal, shear or their combination) can only be determined by multiaxial fatigue tests contrasting proportional and non-proportional loading.
Also, other damaging causes with regard to multiaxial fatigue are suggested, e.g. the hydrostatic stress state [38,39]. However, this suggestion also has its limitation and is not of general use [29]. Furthermore, the introduction of particular factors into the Gough–Pollard equation [40] considering the influence of ductility on multiaxial fatigue, e.g. in design codes [41–43,38], is only an experience based consideration. 5. Summary and conclusions For the multiaxial fatigue response of materials, one important parameter is the mode of local deformation under multiaxial loading. Load control of unnotched specimens, if macroscopic plasticity occurs, results in a quite different fatigue life response than strain control of unnotched specimens. However, load control of notched specimens or of components results in a similar behaviour to unnotched specimens under strain control, because in the critical areas, i.e. notches, the local deformations are strain controlled as long as ratcheting or cyclic creep does not occur. Only under locally multiaxial elastic deformations does the loading mode play no role: the deformations are also locally elastic under load control. In this case, the right component related material behaviour is also determined with unnotched specimens. While ductile wrought steels, non-welded as well as welded, under changing principal stress directions, e.g. caused by an
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out-of-phase non-proportional loading between normal and shear stresses, show a significant fatigue life reduction compared to an in-phase multiaxial loading with constant principal stress directions, less ductile (semi-ductile) cast steels reveal an insensitive (neutral) behaviour and the even less ductile (low-ductility) cast nodular iron even show an increase of fatigue life under out-ofphase non-proportional loading. In this context, cyclic softening (quenched and tempered bainitic) steels or cyclic hardening (austenitic) steels reveal no influence with regard to the fatigue life response under non-proportional loading. Also, for porous sintered steels, a behaviour like cast nodular iron is determined. Similar tendencies are also observed for aluminium and magnesium alloys. While cast aluminium alloys with a very low elongation result in an increase of fatigue life under non-proportional loading, as does cast magnesium, welded semi-ductile aluminium extrusions show a neutral behaviour and more ductile welded extrusions even show a decrease of fatigue life. A further difficulty with regard to fatigue life calculations is that the slopes of the SN-curves for proportional and non-proportional loading are not always parallel. For wrought non-welded ductile steels, they are significantly apart in the finite fatigue life region (N < 106), but they intersect in the high-cycle region (N > 106). This means that a single hypothesis would not be able to describe the multiaxial fatigue behaviour along the whole SN-curve. A similar behaviour is also observed for laser beam welded thin aluminium joints, but not for seam or laser beam welded steel connections. It is obvious that ductility decides the failure mechanisms. However, a hypothesis suggesting ductility, e.g. expressed by area reduction or elongation, as a steering parameter has not yet been proposed. Despite this fact, there are suggestions for considering the influence of ductility on failure mechanisms, e.g. critical plane approaches or modifications with regard to the evaluation of shear and normal stresses or strains. However, as a general multiaxial fatigue hypothesis does not yet exist, the selection of the appropriate hypothesis still requires much testing effort. Therefore, in the case of safety components, e.g. axle systems or complete body and chassis, experimental verifications are still indispensible. References [1] Simbürger A. Festigkeitsverhalten zäher Werkstoffe bei einer mehrachsigen phasenverschobenen Schwingbeanspruchung mit körperfesten und veränderlichen Hauptspannungsrichtungen Fraunhofer-Institut für Betriebsfestigkeit (LBF), Darmstadt, LBF-Bericht Nr. FB-121; 1975. [2] Sanetra C. Untersuchungen zum Festigkeitsverhalten bei mehrachsiger Randombeanspruchung unter Biegung und Torsion. Dissertation TU Clausthal; 1991. [3] Hug J. Einfluss mehrachsiger Beanspruchung auf die Lebensdauer im Zeitfestigkeitsgebiet. Dissertation TU Clausthal; 1994. [4] Esderts A, Zenner H. Multiaxial fatigue under radom loading: experiments and lifetime prediction: In: Fatigue ‘96: proceedings of the 6th international fatigue congress, Berlin, 6.-10.5.1996; 1996. p. 1019–24. [5] Löwisch G, Bomas H. Ermüdung des Stahles 42CrMo 4 unter mehrachsiger phasenverschobener BeanspruchungAmtliche Materialprüfungsanstalt (MPA), Bremen. Abschlussbericht zum AiF-Forschungsvorhaben Nr. 10058 1997; un published. [6] Atzori B, Berto F, Lazzarin P, Quaresimin M. Multiaxial fatigue behaviour of a severely notched carbon steel. Int J Fatigue 2006;28:485–93. [7] Sonsino CM. Influence of load and deformation controlled multiaxial tests on fatigue life to crack initiation. Int J Fatigue 2001;23(2):116–25. [8] Sonsino CM, Grubisic V. Fatigue behaviour of cyclically softening and hardening steels under multiaxial elastic–plastic deformation. In: ASTM STP 853; 1985. p. 586–605. [9] Fatemi A, Shamsaei N. Multiaxial fatigue modelling and some simple approximations. Int J Fatigue 2011 [present Special Issue]. [10] Müller A. Zum Festigkeitsverhalten von mehrachsigen stochastisch beanspruchten Gusseisen mit Kugelgraphit und Temperguss. LBF-Bericht Nr. FB-203; 1994. [11] Müller A. Schwingfestigkeitsverhalten von Stahlguss G-X5CrNi 13 4 unter mehrachsigen Spannungszuständen. Fraunhofer-Institut für Betriebsfestigkeit (LBF), DarmstadtLBF-report no. 6389/E 1988; un published. [12] Sonsino CM, Moosbrugger E. Fatigue design of highly loaded short-glass–fibre reinforced polyamide parts in engine compartments. Int J Fatigue 2008;30:1279–88.
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[13] Jaschek K, Büter A. Short fibre reinforced thermoplastics under multiaxial cyclic stress – a method for the fatigue life estimation. In: Proceedings of fatigue design. Paper no. 6.1 Senlis, ISBN No.: 2-85400-802-2; 2007. p. 01–6. [14] Fleckenstein J, Sonsino CM, Büter A. Internal investigations on multiaxial fatigue of notched short-glassfibre reinforced polyamidesfraunhofer. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, LBF-report no. 570257 2011; un published. [15] Tsai SW. Strength and life of composites. ISBN 978-0-9819143-0-5. Stanford/ USA: Stanford University; 2008. [16] Bannantine JA, Socie DF. Observation of cracking behaviour in tension and torsion low cycle fatigue low cycle fatigue. ASTM STP 942; 1988. p. 899–921. [17] Fatemi A. Personal communication; December 2009. [18] Zerres P, Brüning J, Vormwald M. Fatigue crack growth behaviour of fine-grained steel S460N under proportional and non-proportional loading. Engineering Fracture Mechanics 2010;77:4822–34. [19] Ohrloff N. Crack growth behaviour of the Al–Li alloy 8090 T81 and of the conventional alloy 2024 T3 BRITE EURAM project. Partial final report. Deutsche Airbus report no. TN-Tk 536-4/92 1992; un published. [20] Sonsino CM, Grubisic V. Mechanik von Schwingbrüchen an gegossenen und gesinterten Konstruktionswerkstoffen unter mehrachsiger. Beanspruchung. Konstruktion 1985;37(72):61–269. [21] Sonsino CM, Grubisic V. Multiaxial fatigue behaviour of sintered steels under Combined in- and out-of-phase bending and torsion. Zeitschrift für Werkstofftechnik 1987;18(5):148–57. [22] Renner F. Einflussgrößen auf die Schwingfestigkeit von Magnesium-Guss- und -Knet-Legierungen und Lebensdauerrechnungen. Dissertation TU Clausthal, Clausthal –Zellerfeld; 2004. [23] Sonsino CM. Multiaxial fatigue of welded joints under in-phase and out-ofphase local strains and stresses. Int J Fatigue 1995;17(1):55–70. [24] Sonsino CM. Einfluss der Werkstoffzähigkeit auf das Festigkeitsverhalten unter mehrachsiger Beanspruchung am Beispiel von Schweißverbindungen aus Stahl und Aluminium. Materialwissenschaft und Werkstofftechnik 2003;24(2):189–97. [25] Sonsino CM, Küppers M, Eibl M, Zhang G. Fatigue strength of laser beam welded thin steel structures under multiaxial loading. Int J Fatigue 2006;28:657–62. [26] Störzel K, Wiebesiek J, Bruder T, Hanselka H. Betriebsfeste Bemessung von mechanisch belasteteten Laserstrahlschweißverbindungen aus Stahlfeinblechen des Karosseriebaus. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt/Germany. Report no. FB235; 2008. [27] Küppers M, Sonsino CM. Critical plane approach for the assessment of the fatigue behaviour of welded aluminium under multiaxial loading. Fatigue Fract Eng Mater Struct 2003;26(6):507–14. [28] Wiebesiek J. Research project of the German Research Community (DFG, Bonn), project no. So 180, 2007–2011, un published. [29] Nieslony A, Sonsino CM. Comparison of some selected multiaxial fatigue assessment criteria. Report no. FB-234. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt/Germany; 2008. [30] Garud YS. Multiaxial fatigue. A survey of the state of the art. J Test Eval 1981;9(3):165–78. [31] Brown MW, Miller KJ. Two decades of progress in the assessment of multiaxial low-cycle. Fatigue life. In: ASTM STP 770; 1982. p. 482–99. [32] Socie D. Marquis G. Multiaxial Fatigue SAE. Warrendale, PA/USA; 2000. [33] Susmel L. Multiaxial notch. Cambridge: Fatigue Woodhead Publishing in Materials; 2009. [34] Zenner H, Simbürger A, Liu J. On the fatigue limit of ductile metals under complex multiaxial loading. Int J Fatigue 2000;22(2):137–45. [35] Grubisic V, Simbürger A. Fatigue under combined out of phase multiaxial stresses. In: Fatigue testing and design, vol. 2, proceedings of SEE international conference, vol. 2. 5–9th April 1976, City University, London. p. 27.1–28. [36] Grubisic V, Neugebauer J. Festigkeitsverhalten von Gusseisen mit Kugelgraphit unter mehrachsiger Schwingbeanspruchung. Giessereiforschung 1979;31(4): 123–8. [37] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Int J Fatigue 2009;31(1):188–96. [38] Dang Van K, Griveau B, Message O. On a new multiaxial fatigue limit criterion: theory and application biaxial and multiaxial fatigue, EGF 3. In: Braun MW, Miller KJ, editors. London: Mechanical Engineering Publications; 1989. p. 479–96. [39] Weick M, Aktaa J. Microcrack propagation and fatigue lifetime under nonproportional cyclic loading. Int J Fatigue 2003;25:1117–24. [40] Gough HJ, Pollard HV. The strength of metals under combined alternating stresses. Proc Inst Mech Eng 1935;131:1–101. [41] Sonsino CM, Wiebesiek J. Assessment of multiaxial spectrum loading of welded steel and aluminium joints by modified equivalent stress and Gough– Pollard algorithms. IIW-Doc. No. XIII-2158r1-07/XV-1250r1-07, Cavtat/ Croatia. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt/Germany; 2007. [42] Wiebesiek J, Sonsino CM. IIW-Doc. No. XIII-2314-10/XV-1349-10, Istanbul. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany; 2010. [43] Hobbacher A, editor, Recommendations for fatigue design of welded joints and components. IIW Doc. No. XIII-1823-07 (2007), update July 2008.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Influence of material’s ductility and local deformation mode on multiaxial fatigue response C.M. Sonsino Fraunhofer Institute for Structural Durability and System Reliability LBF, Bartningstr. 47, D-64289 Darmstadt, Germany
a r t i c l e
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Article history: Received 23 August 2010 Received in revised form 13 January 2011 Accepted 16 January 2011 Available online 1 March 2011 Keywords: Multiaxial fatigue Ductility Deformation mode Proportional loading Non-proportional loading
a b s t r a c t The multiaxial fatigue behaviour of components seems to depend mainly on the ductility of the material used determined by applied manufacturing parameters. The ductility steers the damage mechanisms. While, in the case of low-ductility (brittle) materials, the normal stress (strain) is the decisive parameter, in the case of ductile materials, it is the shear stress (strain), and, for semi-ductile materials, a combination of normal and shear stresses (strains). Critical plane oriented hypotheses can consider these different parameters, but the difficulty lies in the definition of ductility and, based on this, the selection of the appropriate hypothesis. Therefore, especially for the evaluation of safety parts, experimental verifications are still necessary, because of the lack of a general multiaxial fatigue hypothesis. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Durability design against complex multiaxial loading is a very old challenge in the history of mankind. A prominent example is the chassis and body adjustment of war carriages, Fig. 1, in ancient times. The major challenges of high speed and manoeuvrability were coupled with fulfilling of the durability requirement of the wheels, hubs and axles, submitted to vertical, lateral, acceleration and braking loadings, pushing the development of lightweight components. The selection and production of the appropriate woods and later metals, their special treatment and, last but not least, the design of the particular components occupied engineers over centuries. This still persists as a task with high responsibility in vehicle design, especially with regard to automotive body and chassis components, Fig. 2. As well as the development of lightweight structures, already being practised in ancient times, there were also machines requiring high functional precision, like printing presses during the Renaissance, Fig. 3. The multiaxial loaded part here was the printing axle submitted to a torque and a compressive axial load causing fatigue failures in the very early presses. The problems were solved by design, thicker wooden axles, and later also by the introduction of castings. Also nowadays, in cases where lightweight is not primarily the focus, but rather stability aspects, e.g. tilting axles of steel converters, Fig. 4, certainly thick components are designed. However,
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besides the geometry, material selection and the assessment of the material behaviour under multiaxial loading also remain important. The design of multiaxial loaded components in almost all industrial sectors; automotive, wind power plants, steel manufacturing, chemical processing, storage and handling, aeronautical and many others still require large efforts in the numerical assessment, i.e. valid hypotheses. Some selected examples of multiaxial loaded components from these application fields are displayed in Figs. 2 and 4–6. However, to date all proposed fatigue strength hypotheses reveal limitations and possess no general validity. Thus, there is a continuing motivation for and process of development of so-called new hypotheses, always with observed limitations. These limitations seem mainly to arise from features steering the local deformations and from particular material characteristics, which are not properly taken into account; it is not sufficient to know only the fatigue behaviour under uniaxial loading, i.e. SN-curves, and to have data on crack propagation. Therefore, the present paper presents and contrasts examples to advise design engineers and scientists with regard to the influence of local deformations and of material ductility on the multiaxial fatigue behaviour.
2. Experimental prerequisites In developing multiaxial fatigue hypotheses, the experimental background is very decisive, because this delivers the verification basis. In this context, the conception of multiaxial tests requires
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Nomenclature Stresses and deformations r axial, bending or normal stress s shear stress e axial, bending or normal strain c shear strain Indexes a a, b, t el, pl tot max x, y, z eq 1,2
amplitude axial, bending, torsion elastic, plastic total maximum coordinates equivalent principal directions
E G Kt N Ps R Rm Rp0.2 T e f k
u d
Young’s modulus shear modulus stress concentration cycle probability of survival ratio between minimum and maximum ultimate tensile strength 0.2% yield strength scatter elongation frequency slope of the SN- curve direction of an interference plane, direction of principle stress phase angle
Other symbols A area reduction
Fig. 1. Antique war carriage of pharaoh Tutanchamun (1250 B.C.) with multiaxial loaded wheels, hubs and axle (source: British Museum, London).
the simulation of real local deformation conditions. In the past, many multiaxial tests were carried out with unnotched specimens in obtaining the first material related data, without considering component related geometrical details like notches, which are of prime importance. Here, it must be mentioned that multiaxial load control is much easier to perform than multiaxial deformation control. However, it was noticed that load controlled multiaxial tests with unnotched and notched specimens from the same material did not result in the same tendencies with regard to the influence of constant or changing principal stress directions, simulated by in- and out-of-phase loading (proportional and non-proportional loading), on fatigue life [1]. For a long time, axial load controlled
multiaxial tests with unnotched specimens were also carried out in other research works [e.g. 2–5], without being aware of the consequences of the selected loading mode. Namely, while axial load controlled multiaxial tests with unnotched specimens [1–5] under non-proportional (out-of-phase) loading deliver a prolongation of fatigue life, Figs. 7 and 8, tests with notched specimens [1,6] render a significant decrease of fatigue life, Figs. 9 and 10. The reason for this contradictory behaviour was clarified by load and deformation controlled tests with unnotched specimens [7], Figs. 11 and 12. The deformation control of the unnotched specimens result clearly in the reduction of fatigue life under out-of-phase (non-proportional) loading, Fig. 12, as observed with notched specimens, Figs. 9 and 10, because, in notches, the deformation is also controlled by the stress gradients, as long as the structural yield point is not exceeded [8] or a ratcheting or a cyclic creep is not activated. The decrease of fatigue life under non-proportional loading arises from a multiaxial cyclic hardening, in comparison to proportional loading [7,8]. This multiaxial cyclic hardening is a consequence of the forcing of energy into the material under deformation control and causes an earlier damage than under in-phase loading [7,8]. The multiaxial cyclic hardening under deformation control is not related to the cyclic softening or hardening characteristic of the material under uniaxial pure axial straining or pure torsional shear straining [7,8]. In Figs. 8 and 13, left, a tempered cyclic softening high strength, quenched and tempered ductile steel and in Figs. 8 and 13, right, two similar austenitic cyclic hardening ductile steels are presented. Despite the differences in the uniaxial cyclic behaviour, Fig. 14, the response to multiaxial loading, either under multiaxial load control of unnotched specimens [2,3], Fig. 8, or under component/notch imposed multiaxial deformation control of unnotched specimens [7,8], Fig. 13, reveals the same tendencies, i.e. an increase of fatigue life under non-proportional load control, Figs. 7, 8 and 11, or the component related decrease of fatigue life under non-proportional deformation control, Figs. 12 and 13. This behaviour was attributed to the observed multiaxial cyclic hardening, Fig. 15, and the imposed higher energy under non-proportional loading [8]. In this context it should be mentioned that there are also materials like Ti–6.5Al–3.4Mo which reveal a fatigue life decrease under deformation controlled out-of-phase multiaxial loading without a
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Fig. 2. Chassis of a seven series BMW.
Fig. 3. Printing press from the Renaissance (Original press of G. and N.I. Sonsino (1480)).
multiaxial cyclic hardening compared to in-phase loading [9]. This means that under non-proportional loading a fatigue life decrease can be also caused by other damaging mechanisms than multiaxial cyclic hardening and related energy increase. Until this point, the multiaxial fatigue response of mainly cyclic softening quenched and tempered and of cyclic hardening austenitic ductile steels, determined with unnotched specimens, has been discussed. The non-component related multiaxial behaviour of load controlled unnotched specimens is also related to plasticity effects depending on the applied load and deformation level, which are not controlled by stress/strain gradients, as they exist in the
notches of components. As soon as the multiaxial load/deformation level is above the branching point of the cyclic and monotonic stress–strain curves, Fig. 16, on top, a multiaxial load control will cause an uncontrolled increase of the deformations (axial strain and shear strain) under in-phase (proportional) loading, which is higher than the out-of-phase (non-proportional) loading [7,8]. This causes the observed higher fatigue life for out-of-phase loading of the displayed ductile steels. However, if the deformations are below the branching point, they remain globally elastic and, in this case, load control of the unnotched specimen corresponds to the deformation control. The consequence is that, at such load levels,
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Fig. 4. Converter with bearing axles.
Fig. 5. Main stresses on a crankcase wall.
the multiaxial fatigue behaviour corresponds to the behaviour of the notched specimens, i.e. the decrease of fatigue life under non-proportional loading of unnotched ductile specimens. In the case of low-ductility (or non-ductile (brittle)) materials, the relation between the branching point and the load levels from 104 cycles upwards is much more favourable with regard to global elastic behaviour, Fig. 16, bottom. Therefore, for such materials, like cast steels or cast iron [10,11], the multiaxial fatigue behaviour may be also investigated using unnotched specimens under load control and result in component like tendencies. Beside the metallic materials discussed, some preliminary information on the multiaxial fatigue behaviour of fibre reinforced materials will also be presented in the following. The material response under multiaxial loading for metallic materials with unnot-
ched and notched specimens under load control is also observed for the short-fibre reinforced polyamide PA 66-GF35 with 35% content of glass fibres (e = 3%) [12]. Load controlled tests with unnotched thin tubular specimens at room temperature reveal an increase of fatigue life under out-of-phase loading in contrast to in-phase loading [13], Fig. 17. However, with notched specimens under multiaxial load control a remarkable decrease of fatigue life under out-of-phase loading is obtained [14], Fig. 18. This is due to the different local deformations in the critical unnotched and notched cross sections. Obviously, due to the different microstructure of reinforced plastics in contrast to metallic materials different damage mechanisms [15] are present which have not yet been investigated in details. Independently of acting damage mechanisms, the design relevant results are those obtained with notched
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Fig. 6. Stirrers of a fertilizer plant.
Fig. 7. SN-curves for uniaxial and combined load controlled loading.
specimens (local deformation control due to stress gradients) as in the case of metallic materials, because unnotched components do not exist. If a recommendation has to be made here with regard to specimen geometry and loading mode, then it can be stated that multiaxial load controlled tests with unnotched specimens should be avoided. Instead, multiaxial deformation controlled tests with unnotched specimens should be carried out, in order to determine a more component related material response. However, industrial reality will be approached in a more realistic way if the tests are carried out with components or component like specimens containing the fatigue critical areas, i.e. notches, in which the local multiaxial deformation is deformation controlled, as long as the structural yield point is not exceeded and a cyclic creep or ratcheting does not occur.
3. Multiaxial loading and crack behaviour Most of the results presented in the previous section were determined for the failure criterion of initiation of a technically detectable crack with a defined depth (a = 0.25–1.50 mm) or surface length (l = 0.50–3.00 mm). This criterion is important for components for which a crack propagation is not allowed, e.g. automotive safety components like stub axles, hubs, brakes, etc. However, there are also applications where the crack formation and propagation have to be considered, from stand points of material development as well as of safety aspects (damage tolerance) or of determining inspection intervals. With regard to crack initiation and propagation, the microstructure plays an important role. A banded microstructure of a structural fine grained ductile ferritic steel, as displayed in Fig. 19,
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Fig. 8. Results of load controlled tests under pure and combined axial and torsional loading.
Fig. 9. Fatigue testing results obtained with notched specimens under combined bending and torsion (u = 0° and 90°).
influences the direction of early crack propagation especially significantly. The early cracks do not lie in the expected direction when the plane of maximum axial or shear strains are calculated [7]. Certainly, the formation direction of early cracks is determined not only by the interaction between the grain orientation and the microstructure itself (austenitic, ferritic, tempered, grain structure, i.e. body- or face-centred cubic, hexagonal, etc.), Fig. 20, and the ductility but also by the load level [16]. Also, the interaction between the chemical composition and the surface treatment can influence the crack formation significantly. Fig. 21 shows the different crack appearances of a case hardened SAE 1050 carbon steel (188–226 Brinell hardness in the core, 543–606 BHN in the case of about 2.5 mm depth) under shear strain controlled torsion. The hardness values and the related particular ductilities were determined by the lower and higher sulphur contents between 0.009% and 0.0039% [17].
Investigations with unnotched specimens certainly make sense for understanding the material behaviour, for drawing conclusions on the consequences for possible microstructural improvements and for recognizing tendencies with regard to the influence of multiaxial stress/strain states on the material behaviour. However, they will not give an indication of the component related influence of the material on the crack propagation behaviour, because of the missing stress/strain gradients when unnotched specimens are used. For this, propagation investigations with multiaxial loaded components or component like specimens are necessary. The results in Fig. 9 with the ductile quenched and tempered steel SAE 1045 were presented for the failure criterion of a first technical crack, but the tests were continued until the final rupture of the specimens. So, the crack propagation phases for in- and out of-phase loadings were also determined. The fatigue lives for crack initiation, propagation and total failure are displayed in Fig. 22, and also the ratios between the lives for crack propagation and total
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Fig. 10. Fatigue testing results obtained with notched specimens under combined axial loading and torsion (u = 0° and 90°).
Fig. 11. SN-curves for uniaxial and combined load controlled loading of unnotched hollow specimens.
Fig. 12. Strain controlled determined SN-curves with unnotched hollow specimens.
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Fig. 13. SN-curves for combined axial and shear strain (failure criterion: initial crack a 61 mm).
Fig. 14. Stress–strain curves of a cyclic softening and of a cyclic hardening steel.
Fig. 15. Stabilized hystereses at Ncr/2 for combined axial and shear strain.
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Fig. 16. Influence of multiaxial load controlled tests on the fatigue and deformation behaviour of unnotched specimens in relation to the cyclic stress–strain curve.
Fig. 17. Multiaxial fatigue behaviour of unnotched short-fibre reinforced polyamide tubes under load control (u = 0 and 90°).
failure. It can be seen clearly that not only the fatigue life to crack initiation but also the crack propagation life is reduced significantly by the non-proportional (out-of-phase) loading with the changing principal stress directions. At the lower load level, the crack propagation phase decreases compared to the higher load level. The results in Fig. 10 [6] also confirm that the crack propagation phase under out-of-phase loading is reduced compared to in-phase loading. These tests were carried out with specimens having a higher stress concentration than the specimens displayed in Fig. 9. Therefore, it can be assumed that the major parts of the determined fatigue lives consist of crack propagation. These results possess a relevance because they were obtained on component like notched solid specimens, Figs. 9 and 10. Unfor-
tunately, crack propagation results obtained with comparable loading conditions and geometries were not available for other materials, especially for less ductile materials, for which neutral behaviour or an increase of fatigue life under non-proportional loading is observed (see next chapter). It will remain an open question if both the crack initiation and the crack propagation life will be influenced with the same tendency. In relation to the crack propagation behaviour, multiaxial load controlled tests with fillet notched solid specimens, Fig. 9, and unnotched hollow specimens [18] do not show the same tendency. Tests carried out with thin cylindrical hollow specimens of the ductile fine grained steel StE 460 (S460N) show a prolongation of the crack propagation life for non-proportional loading in comparison to proportional axial loading and torsion [18], Fig. 23. This
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Fig. 18. Multiaxial fatigue behaviour of notched short-fibre reinforced polyamide tubes under load control (u = 0 and 90°).
Fig. 19. Position of a crack on an unnotched cylindrical hollow specimen under deformation controlled in-phase loading.
example underlines again that specimen geometry related stress gradients in radial (core) direction and different friction effects on crack propagation surfaces especially under torsion may lead to opposing results. With regard to the material influence on the crack propagation under equi-biaxial proportional loading, results are available from aeronautical applications [19] and are displayed in Fig. 24 for the aluminium–lithium alloy AlLi2.5Cu1.5Mg1 T81 (EN AW-8090 T81) (e = 7%) and the conventional aluminium alloy AlCu4Mg1 T3 (EN AW-2024 T3) (e = 14%). The ratio between the applied stresses was rx/ry = 0 for uniaxial loading and rx/ry = 1 for biaxial loading and the load ratio for the stress components was R = 0.1. pffiffiffiffiffiffiffiffiffi At lower stress intensities, DK < 1200 MPa mm, the biaxial loading does not influence the crack propagation behaviour significantly compared to uniaxial loading. However, at higher stress intensities, a marked decrease of the crack propagation rate is observed for both alloys. This is due to the crack closing influence of the second stress component rx. The crack growth resistance of 2024 T3 is lower than that of alloy 8090 T81 under uniaxial as well pffiffiffiffiffiffiffiffiffi as biaxial loading at stress intensities DK < 1600 MPa mm. How-
ever, due to the higher fracture toughness, i.e. higher ductility, of the conventional 2024, it reveals lower crack propagation rates than the 8090 alloy at higher stress intensities. 4. Influence of material’s ductility on multiaxial fatigue response and damage causes 4.1. Non-welded material states The initial examples of the influence of material ductility on crack formation, Figs. 20 and 21, have already been presented. In this section, this issue will be addressed more from the view point of component behaviour, but first for non-welded material states, i.e. wrought, sintered and cast materials. Figs. 9 and 10 have already demonstrated that non-proportional loading significantly decreases the fatigue life of ductile steels, here SAE 1045 (e = 22%) [1] and SAE 1040 (e = 32%) [6], up to 106 cycles; at higher fatigue lives, the SN-curves intersect with each other [20]. In contrary to this, materials with low-ductility, like the cast aluminium piston alloy (e < 2%), Fig. 25 [20], or sintered steels even at elevated
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Fig. 20. Early crack formation under pure torsion of thin hollow specimens.
Fig. 21. Crack formation of case hardened steel bars under shear strain controlled torsion.
densities (e = 13%), Fig. 26 [20,21], display a significant increase of fatigue life under non-proportional (out-of-phase) loading. These investigations were carried out with the same component like geometries as in Fig. 9 under load control. The fatigue life increasing influence of low-ductility materials is also displayed in Figs. 27 and 28. The ferritic cast nodular iron ENGJS-400 (e 6 11%) shows a significant increase of fatigue life under out-of-phase fully reversed as well as under pulsating multiaxial loading [10]. This is observed under both constant and variable amplitude loadings. Nevertheless, for low-ductility materials like the mentioned cast aluminium piston alloy, the sintered steels or cast nodular iron, it can be assumed that both the crack initiation and the crack propagation lives are larger for out-of-phase loading compared to in-phase loading. A few deformation controlled tests with unnotched hollow cast magnesium AZ 91 (G–MgAl9Zn1) specimens with low-ductility
(e 6 2.5%) have also shown an increase of fatigue life under nonproportional loading [22], Fig. 29. Last but not least, the behaviour of semi-ductile materials is also addressed by the example of the cast steel G-X 5 CrNi 13 4 (1.4313) with a martensitic–ferritic microstructure (e 6 20%) [11], Fig. 30. This material behaves in the same way under both proportional and non-proportional loadings; the changing of principal stress directions by the out-of-phase loading does not affect the behaviour obtained under in-phase loading. 4.2. Welded material states The multiaxial fatigue behaviour of welded joints under proportional and non-proportional loading is also determined by material ductility. Flange-tube connections (t = 10 mm) of seam welded ferritic fine grained steel FeE 460 (e = 25%) reveal a significant decrease
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Fig. 22. Influence of phase difference on crack propagation of a ductile steel.
Fig. 23. Crack propagation lives under in- and out-of-phase combined axial loading and torsion.
of fatigue life under both constant and variable amplitude out-ofphase loading in contrast to in-phase loading [23,24], Fig. 31. Also,
after N = 106 cycles, the fatigue life curves do not intersect, as is observed for the non-welded quenched and tempered steels SAE 1045 and 1040, Figs. 9 and 10. Also the laser beam welded overlapped thin tubes (t = 1.0 mm) of the ferritic steel S235 G2T (St 35) (e = 26%) behave in the same way under non-proportional loading [25,26], Fig. 32, as the seam welded thicker flange-tube connections [23], Fig. 31, without any intersection of the SN-curves after N = 106 cycles. In contrast to the seam welded steel flange-tube connections, similar thickness joints (t = 10 mm) of the age hardened aluminium alloy AlSi1MgMn T6 (AW-6082 T6) (e = 14%) render a neutral behaviour under both out-of-phase constant and variable amplitude loading in comparison to in-phase loading [24,27], Fig. 33. This means that this alloy possesses a semi-ductile material behaviour with regard to multiaxial fatigue. However, new results show that this observed behaviour cannot be generalized for the alloy AlSi1MgMn T6. Laser beam welded thin (t = 1.5 mm) overlapped tubes from nominally the same alloy but in a more ductile condition (e = 17%) display a decrease of fatigue life under non-proportional out-of-phase loading [28], Fig. 34. This behaviour is not due to the difference between seam welding of a thick joint and laser beam welding of a thin joint, as displayed
Fig. 24. Crack propagation rates of different aluminium alloys under uni- and biaxial loading.
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Fig. 25. SN-curves for pure and combined bending and torsion of a cast aluminium.
Fig. 26. SN-curves for pure and combined bending and torsion of a sintered steel.
Fig. 27. Influence of multiaxial alternating loading on the fatigue behaviour of the nodular cast iron GGG-40, fully reversed loading (R = 1).
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Fig. 28. Influence of multiaxial pulsating loading on the fatigue behaviour of the nodular cast iron GGG-40, pulsating loading (R = 0).
Fig. 29. SN-curves of pure and combined axial and torsional loading of cast magnesium alloy AZ 91 obtained under deformation control.
for the ductile steel joints in Figs. 31 and 32, but due to the differences in material ductility. Obviously, the manufacturing process and the heat treatment of thick and thin tubes (10 mm versus 3 mm thickness before machining) lead to differences in ductility despite the use of the same nominal material. Also the laser beam welded aluminium alloy AlMg 3.5 Mn (AW-5042) (e = 20%) results in a decrease of fatigue life under non-proportional loading compared to proportional loading, Fig. 35 [28]. However, the reduction of fatigue life is restricted for both aluminium alloys in the fatigue life region N < 106 cycles. At higher fatigue lives, an intersection or a crossing of the SN-lines occurs. The non-proportional loading may even become more advantageous in this region. This may be a result of contact of the overlapped tubes at low load levels. However, more tests in this region are necessary to clarify the course of the SN-lines. 4.3. Damage causes Obviously, the ductile metallic materials react in a different way to non-proportional loading with changing principal stress or
strain directions than low-ductility (or brittle) and semi-ductile metallic materials. What is generally accepted is that, for low-ductility (or brittle) materials, damage is activated by normal stresses or strains and, for ductile materials, damage is activated by shear stresses or strains. It is logical to assume that, in the case of semi-ductile materials, an interaction between shear and normal stresses or strains will be responsible for the damage. These three possibilities are displayed in Fig. 36, in terms of stresses on a surface element; this can also be expressed in terms of strains [7]. As experiences with fibre-reinforced plastic materials in different ductility states are not yet available the possible damage mechanisms under complex multiaxial loading will not be discussed here. But in this context it has to be mentioned that anisotropy effects will influence the damage mechanisms considerably [15]. What is also known is that, in the case of changing principal stress or strain directions in ductile materials, dislocations are activated in different directions blocking each other. This leads to the so-called multiaxial stress hardening in contrast to in-phase loading with constant principal stress or strain directions even if the material reveals a cyclic softening under uniaxial strain controlled
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Fig. 30. SN-curves of a cast steel under uni- and multiaxial loading.
Fig. 31. Fatigue testing results obtained with as-welded thick (t = 10 mm) steel specimens under combined bending and torsion (u = 0° and 90°).
cyclic loading [7]. The multiaxial stress hardening under out-ofphase deformation control causes the fatigue life decrease compared with in-phase multiaxial deformation controlled loading. However, it has to be mentioned again, that there are also materials which reveal a fatigue life decrease without a multiaxial cyclic hardening compared to in-phase loading [9]. Several critical plane and integral hypotheses have been developed for considering the described damage causes in past years e.g. [1–4,8–11,23–37], but none of them were introduced in advance without having a knowledge of the multiaxial behaviour and the estimation of the ductility related damaging parameters. The selection of the appropriate hypothesis requires an assessment of the ductility of the material to be evaluated, but the example with the seam welded and laser beam welded aluminium alloy, Figs. 33 and 34, shows the difficulty in deciding which level of ductility
may cause a change of the multiaxial fatigue response for nominally the same material. In addition to the classical ductility values of elongation or area reduction, the ratio between the local fatigue strength amplitudes sa/ra to be determined by pure axial loading or bending and by pure torsion is also often used as an indicator for multiaxial fatigue behaviour [1,4,8–11,33–37]: pffiffiffi sa/ra = 1/2 = 0.5, according to Tresca, or 1/ 3 = 0.58, according to von Mises, indicate that the shear stress (strain) will be the dominating parameter, and sa/ra = 1.0 (principal stress/strain hypothesis) points to the normal stress as being the damage steering parameter. Intermediate values suggest an interaction of shear and normal stresses/strains. However, this ratio also depends on several parameters, which restrict its applicability as an indicator for the selection of an appropriate hypothesis:
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Fig. 32. Laserbeam welded overlapped thin (t = 1.0 mm) steel tube-tube specimens under constant amplitude multiaxial loading.
Fig. 33. Fatigue testing results obtained with as-welded thick (t = 10 mm) aluminium specimens under combined bending and torsion (u = 0° and 90°).
Fig. 34. Laserbeam welded overlapped thin (t = 1.5 mm) aluminium tube–tube specimens under constant amplitude multiaxial loading.
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Fig. 35. Laserbeam welded overlapped thin (t = 1.5 mm) aluminium tube–tube specimens under constant amplitude multiaxial loading.
Fig. 36. Interference plane stresses and damage models.
Firstly, its determination requires samples with the same highly stressed material volume for the fatigue tests under pure axial (bending) loading and torsion in order to avoid or eliminate the influence of stress (strain) gradients on the fatigue strength; for this, thin hollow specimens are suitable. Secondly, anisotropy effects must be excluded, because the critical crack formation planes are different for axial loading (bending) and torsion. Therefore, the resulting fatigue strength must not be influenced by an anisotropy. Thirdly, the failure criterion for pure axial loading (bending) and torsion must be the same. For this, crack initiation for a defined crack depth, e.g. 0.25–0.50 mm, is recommended. The criterion of total rupture is not suitable because the crack propagation phases for axial loading (bending) and torsion are completely different. Fourthly, ratios obtained even when respecting the afore-mentioned three conditions for fully reversed loading (R = 1) cannot be applied for pulsating loading (R = 0), because the mean-stress sensitivity under torque is much less than under axial loading (bending). Neither the elongation nor the ratio sa/ra are values which may generally help as criteria for selecting the suitable hypotheses. Their relation with regard to the mentioned damage scenarios (normal, shear or their combination) can only be determined by multiaxial fatigue tests contrasting proportional and non-proportional loading.
Also, other damaging causes with regard to multiaxial fatigue are suggested, e.g. the hydrostatic stress state [38,39]. However, this suggestion also has its limitation and is not of general use [29]. Furthermore, the introduction of particular factors into the Gough–Pollard equation [40] considering the influence of ductility on multiaxial fatigue, e.g. in design codes [41–43,38], is only an experience based consideration. 5. Summary and conclusions For the multiaxial fatigue response of materials, one important parameter is the mode of local deformation under multiaxial loading. Load control of unnotched specimens, if macroscopic plasticity occurs, results in a quite different fatigue life response than strain control of unnotched specimens. However, load control of notched specimens or of components results in a similar behaviour to unnotched specimens under strain control, because in the critical areas, i.e. notches, the local deformations are strain controlled as long as ratcheting or cyclic creep does not occur. Only under locally multiaxial elastic deformations does the loading mode play no role: the deformations are also locally elastic under load control. In this case, the right component related material behaviour is also determined with unnotched specimens. While ductile wrought steels, non-welded as well as welded, under changing principal stress directions, e.g. caused by an
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out-of-phase non-proportional loading between normal and shear stresses, show a significant fatigue life reduction compared to an in-phase multiaxial loading with constant principal stress directions, less ductile (semi-ductile) cast steels reveal an insensitive (neutral) behaviour and the even less ductile (low-ductility) cast nodular iron even show an increase of fatigue life under out-ofphase non-proportional loading. In this context, cyclic softening (quenched and tempered bainitic) steels or cyclic hardening (austenitic) steels reveal no influence with regard to the fatigue life response under non-proportional loading. Also, for porous sintered steels, a behaviour like cast nodular iron is determined. Similar tendencies are also observed for aluminium and magnesium alloys. While cast aluminium alloys with a very low elongation result in an increase of fatigue life under non-proportional loading, as does cast magnesium, welded semi-ductile aluminium extrusions show a neutral behaviour and more ductile welded extrusions even show a decrease of fatigue life. A further difficulty with regard to fatigue life calculations is that the slopes of the SN-curves for proportional and non-proportional loading are not always parallel. For wrought non-welded ductile steels, they are significantly apart in the finite fatigue life region (N < 106), but they intersect in the high-cycle region (N > 106). This means that a single hypothesis would not be able to describe the multiaxial fatigue behaviour along the whole SN-curve. A similar behaviour is also observed for laser beam welded thin aluminium joints, but not for seam or laser beam welded steel connections. It is obvious that ductility decides the failure mechanisms. However, a hypothesis suggesting ductility, e.g. expressed by area reduction or elongation, as a steering parameter has not yet been proposed. Despite this fact, there are suggestions for considering the influence of ductility on failure mechanisms, e.g. critical plane approaches or modifications with regard to the evaluation of shear and normal stresses or strains. However, as a general multiaxial fatigue hypothesis does not yet exist, the selection of the appropriate hypothesis still requires much testing effort. Therefore, in the case of safety components, e.g. axle systems or complete body and chassis, experimental verifications are still indispensible. References [1] Simbürger A. Festigkeitsverhalten zäher Werkstoffe bei einer mehrachsigen phasenverschobenen Schwingbeanspruchung mit körperfesten und veränderlichen Hauptspannungsrichtungen Fraunhofer-Institut für Betriebsfestigkeit (LBF), Darmstadt, LBF-Bericht Nr. FB-121; 1975. [2] Sanetra C. Untersuchungen zum Festigkeitsverhalten bei mehrachsiger Randombeanspruchung unter Biegung und Torsion. Dissertation TU Clausthal; 1991. [3] Hug J. Einfluss mehrachsiger Beanspruchung auf die Lebensdauer im Zeitfestigkeitsgebiet. Dissertation TU Clausthal; 1994. [4] Esderts A, Zenner H. Multiaxial fatigue under radom loading: experiments and lifetime prediction: In: Fatigue ‘96: proceedings of the 6th international fatigue congress, Berlin, 6.-10.5.1996; 1996. p. 1019–24. [5] Löwisch G, Bomas H. Ermüdung des Stahles 42CrMo 4 unter mehrachsiger phasenverschobener BeanspruchungAmtliche Materialprüfungsanstalt (MPA), Bremen. Abschlussbericht zum AiF-Forschungsvorhaben Nr. 10058 1997; un published. [6] Atzori B, Berto F, Lazzarin P, Quaresimin M. Multiaxial fatigue behaviour of a severely notched carbon steel. Int J Fatigue 2006;28:485–93. [7] Sonsino CM. Influence of load and deformation controlled multiaxial tests on fatigue life to crack initiation. Int J Fatigue 2001;23(2):116–25. [8] Sonsino CM, Grubisic V. Fatigue behaviour of cyclically softening and hardening steels under multiaxial elastic–plastic deformation. In: ASTM STP 853; 1985. p. 586–605. [9] Fatemi A, Shamsaei N. Multiaxial fatigue modelling and some simple approximations. Int J Fatigue 2011 [present Special Issue]. [10] Müller A. Zum Festigkeitsverhalten von mehrachsigen stochastisch beanspruchten Gusseisen mit Kugelgraphit und Temperguss. LBF-Bericht Nr. FB-203; 1994. [11] Müller A. Schwingfestigkeitsverhalten von Stahlguss G-X5CrNi 13 4 unter mehrachsigen Spannungszuständen. Fraunhofer-Institut für Betriebsfestigkeit (LBF), DarmstadtLBF-report no. 6389/E 1988; un published. [12] Sonsino CM, Moosbrugger E. Fatigue design of highly loaded short-glass–fibre reinforced polyamide parts in engine compartments. Int J Fatigue 2008;30:1279–88.
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[36] Grubisic V, Neugebauer J. Festigkeitsverhalten von Gusseisen mit Kugelgraphit unter mehrachsiger Schwingbeanspruchung. Giessereiforschung 1979;31(4): 123–8. [37] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Int J Fatigue 2009;31(1):188–96. [38] Dang Van K, Griveau B, Message O. On a new multiaxial fatigue limit criterion: theory and application biaxial and multiaxial fatigue, EGF 3. In: Braun MW, Miller KJ, editors. London: Mechanical Engineering Publications; 1989. p. 479–96. [39] Weick M, Aktaa J. Microcrack propagation and fatigue lifetime under nonproportional cyclic loading. Int J Fatigue 2003;25:1117–24. [40] Gough HJ, Pollard HV. The strength of metals under combined alternating stresses. Proc Inst Mech Eng 1935;131:1–101. [41] Sonsino CM, Wiebesiek J. Assessment of multiaxial spectrum loading of welded steel and aluminium joints by modified equivalent stress and Gough– Pollard algorithms. IIW-Doc. No. XIII-2158r1-07/XV-1250r1-07, Cavtat/ Croatia. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt/Germany; 2007. [42] Wiebesiek J, Sonsino CM. IIW-Doc. No. XIII-2314-10/XV-1349-10, Istanbul. Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany; 2010. [43] Hobbacher A, editor, Recommendations for fatigue design of welded joints and components. IIW Doc. No. XIII-1823-07 (2007), update July 2008.