BiaxiaVMultiaxial Fatigue and Fracture
Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.)
0 Elsevier Science Ltd. and ESIS. All rights reserved. 83
MULTIAXIAL FATIGUE LIFE ESTIMATIONS FOR 6082-T6 CYLINDRICAL SPECIMENS UNDER IN-PHASE AND OUT-OF-PHASE BIAXIAL LOADINGS
Luca SUSMEL' and Nicola PETRONE2 'Department of Engineering, University of Ferrara,
Via Saragat 1, 44100 Ferrara, Italy
'Department of Mechanical Engineering, University of Padova, Via Venezia I , 35131 Padova, Italy
ABSTRACT
Fully reversed bendinghorsion fatigue tests were conducted on 6082-T6 solid cylindrical specimens under force control. Specimens were subjected to pure bending, pure torsion, in-phase and out-of-phase bendinghorsion loadings and the investigated fatigue lives ranged between lo4 and 2-106 cycles to failure. The actual strains were measured by means of strain gauges positioned in correspondence of critical points. Experimental strain measurements highlighted that all the tests were conduced in pure elastic stress conditions. The material fatigue behaviour was studied by analysing the cracks pattern due to the considered biaxial loadings. All the tests showed that crack initiation was always MODE I1 dominated (that is, it occurred on the plain of maximum shear stress amplitude), whereas the crack propagation was MODE I governed. Just in the presence of pure torsional loadings cracks grew under MODE I1 loadings. A good correlation with measured fatigue lives was obtained by applying the Susmel and Lazzarin's criterion valid for homogeneous and isotropic materials, despite the slight degree of anisotropy showed by the material.
KEYWORDS
Biaxial fatigue loadings, multiaxial fatigue life prediction, cracking behaviour.
INTRODUCTION
In the most genera1 situations mechanical components are subjected to complex fatigue loadings that generate multiaxial stress states in correspondence of critical points. During the last 60 years the problem of multiaxial fatigue assessment has been extensively investigated by researchers in order to provide engineers with safe methods for the fatigue life prediction in the presence of complex stress states. The state of the art shows that the problem can be faced by using two different approaches [l, 21: in the low cycle fatigue ambit (that is, when the damage contribution
84 L. SUSMEL AND N. PETRONE
due to the plasticity cannot be disregarded), strain based methodologies are always suggested, whereas in the high cycle fatigue field, stress based techniques have to be employed to predict the multiaxial fatigue limit.
Criteria valid for the fatigue liietime calculation can be classified in three different categories:
strain based methods, straidstress based methods and energy based approaches.
Brown and Miller [3] observed that the fatigue life prediction could be performed by considering the strain components normal and tangential to the crack initiation plane. Moreover, the multiaxial fatigue damage depends on the crack growth direction: different criteria are required if the crack grows on the component surface or inside the material. In the first case they proposed a relationship based on a combined use of a critical plane approach and a modified Manson-Coffin equation, where the critical plane is the one of maximum shear strain amplitude. Subsequently, Wang and Brown [4] introduced in the criterion formulation the mean stress normal to the critical plane in order to account for the mean stress influence.
Socie [5,6] observed that the Brown and Miller’s idea could be successfully employed even by using the maximum stress normal to the critical plane, because the growth rate mainly depends on the stress component normal to the fatigue crack. Starting from this assumption, he proposed two different formulations according to the crack growth mechanism: when the crack propagation is mainly MODE I dominated, then the critical plane is the one that experiences the maximum normal stress amplitude and the fatigue lifetime can be calculated by means of the uniaxial Manson-Coffin curve [ 5 ] ; on the other hand, when the growth is mainly MODE I1 governed, the critical plane is that of maximum shear stress amplitude and the fatigue life can be estimated by using the torsional Manson-Coffin curve [6].
Criteria based on energetic parameters calculation are founded on the assumption that the energy density is the unique parameter really significant of the fatigue damage. The employment of this quantity has a crucial advantage over the methodologies discussed above: theoretically, the amount of energy required for the fatigue failure is independent from the complexity of the stress state present at the critical point, therefore just a uniaxial fatigue curve is enough to predict fatigue lives even in the presence of complex loadings.
Garud [7] suggested that the multiaxial fatigue assessment could be performed by using only the energy due to the plastic deformation. Subsequently, Ellyin [8, 91 stated that not only the plastic energy but also the positive elastic energy is crucial to estimate properly the fatigue damage. In particular, fatigue depends on the elastic energy due to the tensile stress components, because experimental investigations clearly showed that fatigue failures can occur in the high cycle fatigue field, where the plastic contribution is negligible. Moreover, it is well known that a positive mean stress component reduces the fatigue life more than a negative one. For these two reasons an energy based approach can give satisfactory results only when it can account for both the plastic and the positive elastic contribution.
Recently, Lazzarin and Zambardi [lo] employed successfully a linear-elastic energy method together with a critical distance approach for the static and fatigue assessment of sharply notched components under mixed mode loadings.
In the ambit of high cycle fatigue most of the established theories are based on the use of stress components. The oldest high-cycle fatigue assessment method has been proposed by Gough [ll]
and it is based on an extensive experimental investigation conduced on smooth and notched specimens of different materials subjected to in-phase bending and torsion loadings. By reanalysing these results Gough proposed two empirical formulations valid for brittle and ductile materials.
Multiaxial Fatigue Life Estimations f o r 6082-T6 Cylindrical Specimens Under ... 85 Several criteria were subsequently proposed by other researchers adopting the critical plane approach. Among the most employed in the case of out-of-phase loadings, it is worth mentioning the criteria proposed by Findley [12], by Matake [13] and by McDiarmid [14]. The first method considers as critical plane the one where a linear combination of the shear and the normal stresses reaches its maximum value. The other two are based on the assumption that the critical plane is the one experiencing the maximum shear stress amplitude and the fatigue damage depends even on the stress normal to this plane. These criteria allow the evaluation in a component of “non-failure”
conditions, but they are not suitable for the fatigue lifetime calculation.
By starting from a mesoscopic approach, Dang Van [15] proposed an original criterion completely different if compared to the methodologies above discussed. In particular, the simplest version of this method is formalised as a linear combination of the shear and the hydrostatic stresses. This criterion postulates that the fatigue failure is avoided if, in each instant of the load history, the critical stress parameter is lower than a reference value depending on two fatigue limits experimentally determined in simple loading conditions (typically, the use of the uniaxial and the torsional fatigue limits is suggested).
Papadopoulos [16] as well formulated a criterion founded on the mesoscopic approach fundamentals. By using rigorous mathematical procedures applied to the elastic shakedown state concept, he introduced two different formulations valid for brittle and ductile materials, which gave satisfactory results in the fatigue limit estimation of plane components subjected to in-phase and out-of-phase loadings [ 171. Recently, Papadopoulos proposed a new development of this method for the fatigue life calculation in the medium-high cycle fatigue field [18].
Carpinten and Spagnoli [19, 201 formulated a new method founded on an original reinterpretation of the classical critical plane approach. Their new criterion correlates the critical plane orientation with the weighted mean principal stress directions and the multiaxial fatigue assessment i s performed by using a non-linear combination of the maximum normal stress and the shear stress amplitude acting on the critical plane.
By using the theory of cyclic deformation in single crystal, Susmel and Lazzarin [21] presented a medium-high cycle multiaxial fatigue life prediction method developed under the hypothesis of homogeneous and isotropic material and based on the combination of a modified Wohler curve and a critical plane approach. This criterion correlated very well with the experimental results referred both to smooth and blunt notched specimens subjected to either in-phase or out-of-phase loads [21,22].
All the approaches discussed above can be employed for the multiaxial fatigue assessment of mechanical components of homogeneous and isotropic materials, but they do not give satisfactory results in the presence of anisotropy, as it happens with composites. For this reason different theories have been formulated to predict the fatigue endurance of this kind of materials.
Initially, it is important to highlight that the fatigue damage of composites under multiaxial loadings is mainly influenced by the biaxiality ratios [23]; in particular it depends on the shear stress components and it holds true both for plane and notched components. Another important role is played by the off-axis angle (lay-up), whereas the influence on the fatigue strength of the out-of-phase angle between load components seems to be negligible, at least for the glass/polyester laminates [23].
By reanalysing about 700 experimental data take from the literature Susmel and Quaresimin [24] showed that the best accuracy in the multiaxial fatigue prediction of composite materials can be obtained by using the criteria proposed by Tsai-Hill [25], by Fawaz & Ellyin [26] and by Smith
& Pascoe [27]. In particular, the first one is founded on the application of the classical polynomial failure criterion due to Tsai-Hill to the multiaxial fatigue assessment. The second one is based on
86 L. SUSMEL AND N. PETRONE
an extension of the Mandell semi-log linear equation and, finally, the criterion due to Smith &
Pascoe is capable of modelling three different fatigue damage mechanisms: the rectilinear cracking, the shear deformation along the fibre plane and the combined rectilinear cracking and matrix shear deformation.
These approaches are clearly developed for composites, but when the degree of anisotropy is not so high, different methods are again suggested. The criterion proposed by Lin et al. [28, 291 can be mentioned as a valid criterion to apply in this kind of situations. This method is based on an original employment of the strain vector.
Aim of the present paper is the validation of the Susmel and Lazzarin's criterion on a widely used industrial aluminium alloy showing a slight degree of anisotropy to confirm if the employment of specific models for anisotropic materials can be avoided when subjected to in- phase and out-of-phase multiaxial loadings.
FUNDAMENTALS OF THE MODIFIED WOHLER CURVES METHOD
The theoretical frame of the Susmel and Lazzarin's criterion is based on a combined use of a modified Wohler diagram and the initiation plane concept [21].
The theory of deformation in single crystal has been employed to give a physical interpretation of the fatigue damage: in a polycrystal this depends on the maximum shear stress amplitude, tar (determined by the minimum circumscribed circle concept [30]) and on the maximum stress u",,,,~~
normal to the plane of maximum shear stress amplitude (initiation plane) [21].
A
ZA,Ref(P1)
' ZA,Ref (pi)
increasing p J 1
!
I
Fig. 1. Frames of reference and definition of the polar co-ordinates I# and 9.
Fig. 2. Modified Wohler diagram.
Consider a cylindrical specimen subjected to a multiaxial cyclic load (Figure 1). With reference to a maximum shear stress amplitude plane, located by $* and 8* angles, it is possible to define for this plane the stress ratio p as follows:
Multiaxial Fatigue Life Estimations f o r 6082-T6 Cylindrical Specimens Under ._.
where in Eq. 1 the maximum value of normal stress is able to include the influence of mean stress on the fatigue strength, according to the Socie’s fatigue damage model [6].
Consider now a log-log Wohler diagram where in the abscissa there are the number of cycles to failure Nf and in the ordinate the shear stress amplitude ta(+*, e*) calculated on the initiation plane (Figure 2).
It can be demonstrated [21] that different fatigue curves are generated in the modified Wohler diagram by changing the p values. Each single curve is identified by the inverse slope k,(p) and by the reference shear stress amplitude %&Ref (p) corresponding to NRef cycles (usually 2.106 cycles in several design codes). Moreover, experimental results showed [21, 221 that as the p ratio increases the fatigue curve moves downwards in the modified Wohler diagram (Figure 2).
On the basis of this observation and by evaluating the functions t&Ref(p) and k, (p) by a best fit procedure performed using experimental data, it is possible to predict the fatigue life for a multiaxial cyclic stress state by applying the following expression:
By reanalysing systematically experimental data taken from the literature [21, 221, it has been observed that a good correlation with experimental results can be obtained just by expressing
q R e f (p) and k,(p) as linear functions and by using uniaxial and torsional fatigue data for their calibration. In particular, under this assumption ThRef(p) and kr (p) can be expressed as follows:
The presented method can also be used to estimate the fatigue life of notched components by applying the correction based on the fatigue notch factor Kf to the fatigue curves used in the model calibration, and by performing the assessment in terms of nominal stresses [21,22].
Given that the multiaxial fatigue behaviour can be described by a single modified Wohler curve as the p value changes, it can be highlighted that a multiaxial notch factor can be always defined as function of the p ratio. By applying this idea, it has been demonstrated that the multiaxial &
factor is always a linear function of the stress ratio p [22].
Finally, it is interesting to mention the fact that this method can be reinterpreted even in terms of critical distance approach [31]. In particular, by using as critical distance a length depending on the El’Haddad short crack constant [32], it has been shown that this method is capable of predictions within an error of about 15%, when employed to estimate the fatigue limit of V-notched specimens of low carbon steel subjected to in-phase MODE I and MODE I1 loads [31].
88 L. SUSMEL AND N. PETRONE
EXPERIMENTAL DETAILS
Fatigue tests were carried out on solid cylindrical specimens subjected to both in-phase and out-of-phase bendingkorsion loadings. The material used in this investigation was aluminium 6082 T6, supplied in 30-mm-diameter bars. The material chemical composition is reported in Table 1. The bars were produced by means of an extrusion process, which introduced a certain degree of anisotropy by orientating grains mainly along the bar axis.
The specimen (Figure 3) was machined in a CNC lathe and its 50mm long gauge length surface was successively polished down to a 6-pm diamond compound to obtain a mirror-like finish. The average values of the experimentally determined mechanical properties of the used material, along the longitudinal extrusion axis, were: tensile strength 343 MPa, yield stress 301 MPa and Young’s modulus 69400 MPa. In Figure 4 it has been plotted the lowest axial stress-strain curve recorded from a test conducted to evaluate the material static properties. Strains reported on this diagram were directly measured by means of an axial/torsional extensometer.
The fully reversed bending and torsion moments were generated by using two hydraulic actuators, which loaded the specimen by means of a friction clamped loading arm (Figure 5). The contemporary use of a LabVIEW software and two MTS 407 digital controllers allowed to generate and control the applied forces during each test; axial and shear strains were monitored by means of strain gages (Figures 3 and 5) applied at known positions with respect to the nominal maximum bending stress point. Signals were gathered by a National Instruments SCXI-1000DC data acquisition system. Length and orientation of the macro-cracks were measured by means of a Leika stereoscope. Finally, the fatigue failure was defined as 2% bending or torsion stiffness drop and the frequency of each test was equal to 4 Hz.
c , 106
270
Fig. 3. Specimen geometry (a) and picture of a polished specimen with rosette (b).
Table 1. Chemical composition of the tested 6082-T6 aluminium alloy [ %].
Si ME Mn Fe Cr Zn c u Ti
~ ~~
0.9;l.l 0.8~1.0 OSi-0.9 0.5 0.25 0.20 0.1 0.10
Multiaxial Fatigue Life Estimations for 6082-T6 Cylindrical Specimens Under .._ 89
0 0.01 0.02 0.03 0.04 E 0.05
Fig. 4. Lowest recorded static stress-strain curve (strain measured by exstensometer).
Fig. 5. Biaxial test machine.
90 L. SUSMEL AND N PETRONE
EXPERIMENTAL RESULTS
The multiaxial fatigue behaviour of the considered aluminium alloy was studied by testing 44 different cylindrical specimens. Experimental tests were subdivided into four different groups:
pure bending tests, pure torsional tests, biaxial tests characterised by h = ~ ~ ~ , $ o , , , less than 1 and biaxial tests with h greater than 1. Moreover, each single biaxial experimental investigation was performed by applying in-phase loadings and out-of-phase loading having a nominal phase angle equal to 90" for h<l and 126" for b l .
The amplitude and the mean value of stress components as well as the phase angle were calculated by using the average value of all the peaks of forces recorded during each single test.
The experimental results have been summarised in Tables 2-5, where the listed values refer to:
bending stress amplitude, sx,,; mean bending stress, sx,m; shear stress amplitude, z,~,~; mean shear stress, z ~ ~ , ~ ; biaxial stress ratio, h = ~ ~ ~ , $ o ~ , ~ ; phase angle between the applied stress components, 6;
number of cycles to failure, Nf,2%, defined using the 2% bending or torsion stiffness decrease criterion.
By observing Table 3 it can be seen that a small bending stress component was always present even under pure nominal torsional loadings. The value of measured ox,, ranged from 13 MPa up to 24 MPa and it was practically independent of the applied torsional stress amplitude. In any case, the influence of the bending has been always disregarded because its value was in general negligible if compared to the applied torque. The presence of this unwanted stress component under pure torsional loadings was due to intrinsic plays within the ball-socket joints used to connect the loading arm to the actuators.
The presence of these plays generated further complications under out-of phase loadings. In particular, when h was greater than 1 the measured out-of-phase angle ranged between 125" and 129", that is, it was greater than the imposed nominal angle of 90".
As an example, in Figure 6 the T~~ vs. ox diagrams for two different biaxial tests are reported.
By observing these diagrams some signal perturbations generated by the joints plays can be noticed.
- 1 5 0 1 , , , 1 , , , I - l 5 0 $ , , , 1 , , , I
-200 -200
-200 -150 -100 -50 0 50 100 150 200 -200 -150 -100 -50 0 50 100 150 200
0, [MPal ox [ m a l
Fig. 6. Crossplots of tests named P23BT4 (oX,,=147 MPa, zxY,,=90 MPa, 6=-So) and P32BT8 (ox.,=149 MPa, ~ , ~ , , = 6 8 MPa, 6=93")
Multiaxial Fatigue Life Estimations for 6082-T6 Qlindrical Specimens Under ... 91
Table 2. Experimental results: fully-reversed bending tests.
Specimen ,a ,,a txllr z~,,,, 6 A Nf2%
Code [MPa] [MPa] [MPa] [MPa] ["I [Cycles]
P5B5 224 -1 4 0 0 0 52990
P l B l 190 0 5 7 0 0 159000
P7B1 188 -1 4 0 0 0 197275
P2B2 180 -4 4 -1 0 0 244403
P8B3 162 0 3 1 0 0 421560
P3B3 165 -2 4 1 0 0 437636
P4B4 145 -1 4 1 0 0 1060730
P6B4 145 -1 4 0 0 0 1235690
Table 3. Experimental results: fully-reversed torsion tests.
Specimen U, Ux,m 'Cxyr Zxy,m 6 A Nf$%
Code [MPa] [MPa] [MPa] [MPa] ["I [Cycles]
P14T2 14 1 138 0 0 9.9 14695
P10T2 18 3 139 0 0 7.7 23052
P l l T 3 15 1 111 0 0 7.4 67690
P12T3 16 1 111 0 0 6.9 113455
P13T1 13 3 99 0 0 7.6 196555
P9T1 24 0 98 0 0 4.1 449997
P16T4 15 2 86 1 0 5.7 497990
P15T4 15 1 87 0 0 5.8 11OoooO
CRACKING BEHAVIOUR
The complex cracking behaviour showed by the material under complex fatigue loadings was a consequence of the morphological structure of the studied 6082-T6 generated by the extrusion manufacturing process. This orientated grains mainly along the extrusion axis and it favoured the initiation of small cracks evenly distributed on the specimen surface and oriented along the specimen axis.
Under bending loading, shear crack nucleation was followed by growth on the plane of maximum principal stress (MODE I growth), independently of the bending stress amplitude levels.
As expected, the crack initiation occurred in correspondence of the maximum bending stress points, so two different cracks were present on the surface of each specimen.