FATIGUE LIFETIME EVALUATION OF NOTCHED COMPONENTS: IMPLEMENTATION OF THE CONTROL VOLUME CONCEPT IN A STRAIN-BASED LCF CRITERION

The goal of the present paper is to discuss the reliability of a strain-based multiaxial Low-Cycle Fatigue (LCF) criterion, recently proposed by some of the present authors, in estimating the fatigue lifetime of metallic structural components weakened by sharp notches. Such a criterion, based on the critical plane approach, is formulated according to the control volume concept related to the Strain Energy Density (SED) criterion: a material point located at a certain distance from the notch tip is assumed to be the verification point where to perform the fatigue assessment. The above distance is assumed to be a function of both the biaxiality ratio (applied shear stress amplitude over normal stress amplitude) an the with the curve.


INTRODUCTION
The state of the art clearly shows that the fatigue problem of mechanical components characterised by geometrical irregularities (such as notches, fillets and key-sets) has widely been examined in order to propose reliable methodologies for estimating the fatigue strength/lifetime under both uniaxial and multiaxial loadings [1][2][3][4][5][6][7][8].
Dealing with the fatigue failures of metallic structural components weakened by notches, the average Strain Energy Density (SED) criterion [9][10][11][12], originally proposed by Lazzarin and Zambardi [13], can be considered one of the most powerful engineering tool suitable for accurately performing the fatigue assessment of the above components.
Taking as starting point both the Neuber concept of an 'elementary material volume' [14,15] and the Sih criterion [16,17], Lazzarin and Zambardi adopted the average value of SED, evaluated over a control volume surrounding the notch tip, as a damage parameter for notched structural components. In particular, the radius of the control volume, over which the energy is averaged, was assumed to be a function of the notch geometry, the material fatigue limit evaluated on unnotched specimens, the threshold stress intensity factor range and the Poisson ratio. Moreover, according to such a criterion, fracture of brittle materials is expected to occur when the average value of SED is equal to a critical value, which is a material property [18].
The original version of the SED criterion [13] has been proposed for fatigue strength assessment of components weakened by sharp V-notches subjected to tensile loading (Mode I), and blunt U and Vnotches under Mode I loading have been examined in 2005 [19].
Subsequently, the SED criterion has been extended to notched components under mixed Mode loading [20][21][22][23]. Moreover, the aforementioned criterion has successfully been used also to perform the fatigue strength evaluation of both welded joints [24] and notched specimens under high-temperature conditions [25].
Then, an extension of the SED criterion to mechanical and structural components experiencing non-localised creep deformations has been proposed in 2016 [26]. Further, the criterion has recently been applied together with the Equivalent Material Concept (EMC) in order to estimate the failure loads of both U and V-notched aluminum plates characterised by large plastic deformations near the notch tip [27,28].
Note that the overall effectiveness of the above criterion when used to evaluate fatigue failures in notched metallic components has been analysed in Ref. [12].
In that paper, about one thousand experimental data taken from the literature for different materials and notches geometries have been examined, and a satisfactory agreement between experimental and theoretical results has been noticed.
In light of the efficiency and wide applicability of the SED criterion, some of the present authors have recently implemented the concept of the control volume in a strain-based criterion in order to estimate the fatigue life of severely notched specimens under Low-Cycle Fatigue (LCF) [29].
In particular, such a criterion, based on the critical plane approach, is a reformulation of its counterpart for smooth metallic structural components [30,31]  The aim of the present paper is to discuss the accuracy and reliability of the strain-based criterion together with the control volume concept (proposed in [29]) in estimating multiaxial LCF lifetime of structural components weakened by notches. Firstly, the analytical main points of such a criterion are outlined in Section 2. Then the fatigue experimental campaign reported in Refs [32,33] for V-notched round bars made of titanium grade 5 alloy (Ti-6Al-4V) is briefly summarised in Section 3. In Section 4, the criterion reported in Ref [29] is applied to the experimental fatigue data and, finally, some conclusions are provided in Section 5.

STRAIN-BASED CRITERION FORMULATION FOR SHARP V-NOTCHES
Now the analytical main points of the criterion proposed in Ref. [29] are briefly outlined.
The fatigue life assessment is carried out at point P (verification point), which is distant r from the V-notch tip ( Fig.   1).
More precisely, such a distance r , measured along the notch bisector line, is assumed to depend on both the biaxiality ratio   [20,21,32] through the following expressions: for Mode III), such strengths being all referred to the same reference number a N of loading cycles to failure (for example, ).

Figure 1.
Once the position of the verification point P is determined according to Eq. (1), the strain tensor at the above point is = load phase angles. The subscripts a and m refer to amplitude and mean value, respectively.
The strain tensor ) t ( ε (at point P ) with respect to the fixed reference system Prtz ( Fig. 2(a)) is hence given by: Then, the critical plane orientation, which is linked to the averaged principal strain directions, has to be determined by using ) attains its peak value over the loading cycle.
The orientation of the critical plane, which is the verification plane for fatigue life evaluation, is defined by taking an off-angle  (in the principal averaged plane 1 3 ) formed by the normal w to the critical plane and the averaged direction 1 . The off-angle  is assumed to be expressed as follows: where f N is the number of loading cycles to failure, and  [29][30][31]).
After defining the normal w to the critical plane, a local reference system uvw P is taken into account, where the unit vector u is on the intersection line between the critical plane and the plane defined by the normal vector w and the z -axis; further, v is normal to u so that uvw P forms a right-handed reference system ( Fig. 2(b)).
The directions cosines of the normal w can be computed with respect to the Prtz as a function of two angles,  and  , in a spherical Fig. 2(c)) [35]: Furthermore, the direction cosines of the u -and v -axis are given by the following expressions [35]: Then, by taking into account the strain tensor   t ε at point P (Eq. (4)) with respect to the reference system uvw P , the displacement vector w η related to the critical plane ( Fig. 2(b)) can be expressed as follows:   (Fig. 2(b)), is obtained from Eq. (10): By recalling Eqs (9) and (13), the tangential displacement vector C η lying on the critical plane ( Fig. 2(b)) can be computed according to the following expression:   [36,37]). In the present paper, a min-max procedure to define m , C  and a , C  is applied [38] by examining the components of C η along both u -and v -axis: Recalling Eqs (3) and (16), the above expressions become: where the mean values, The mean value

FATIGUE EXPERIMENTAL CAMPAIGN
The strain-based multiaxial LCF criterion formulated in conjunction with the control volume concept is applied to a set of data recently published in the literature [32,33]. Uniaxial and multiaxial fatigue tests on circumferentially V-notched round bars made of Ti-6Al-4V titanium alloy are briefly described in the present Section.
Each specimen presents a V-notch with depth equal to 6 mm, opening angle equal to  90 and notch root radius equal to about 1 0.

mm.
All specimens have been polished in order to remove surface scratches before performing the tests.
The experimental uniaxial and multiaxial fatigue tests have been carried out by means of a MTS 809 servo-hydraulic biaxial machine.
All tests have been conducted under load control at a frequency value from 5 to 10 Hz, depending on the applied load. Details of the loading conditions being examined are reported in Tables 1-3, where exp , f N is the experimental fatigue life.

CRITERION VALIDATION AND DISCUSSION
The present strain-based multiaxial LCF criterion is here applied to the experimental data described in the previous Section. (1) two series of tests under pure tension (specimens No. 1-6 in Table 1) and pure torsion (specimens No. 7-11 in Table 1) fatigue loading; (2) two series of tests under combined in-(    0 ) and out-ofphase (    90 ) tension and torsion loading, with constant biaxiality ratio  equal to 6 0. (Table 2); (3) two series of tests under combined in-(    0 ) and out-ofphase (    90 ) tension and torsion loading, with constant biaxiality ratio  equal to 0 . 2 (Table 3). .
Note that Eq. (1) has been obtained from a best-fit procedure by considering some values of  related to the experimental data reported in Ref. [32,33]. In particular, the following error index, I , has been optimised in order to determine the data-points to be interpolated: where a  and a  are defined by means of Eqs (6a) and (6b), respectively.
As is previously discussed, in order to estimate the fatigue lifetime of the V-notched specimens, the strain tensor at the verification point P has been estimated from a finite element analysis by means of the commercial software Straus7® [39]. More precisely, the strain state at the verification point, for every investigated fatigue test, has been determined through a tridimensional model by using both 6-and 8-node finite elements.
Taking advantage of the geometric symmetry, only one-half of the specimen has been modelled, as is shown in Figure 4. Furthermore, the mesh size has been gradually refined nearing the region containing the notch tip ( Fig. 4(b)).
The results in terms of strains have been determined by running a series of linear transient dynamic analyses in order to simulate the experimental tests performed by Berto et al. [32,33].    The accuracy of the present fatigue lifetime estimation can also be evaluated by means of the root mean square error method [40]. In more detail, the value of the root mean square logarithmic error is computed as follows: where j is the total number of data, and the mean square error   Therefore, the present criterion seems to be a promising tool to assess the fatigue lifetime of notched structural components, although both more complex loading configurations (characterised by nominal load ratio different from 1  ) and different notch geometries (i.e. blunt notches) need to be processed in order to devise a robust procedure suitable for practical applications.
Submitted to Special Issue "Energy Density"   (c) angles  and  in the spherical coordinate system.           Ti-6Al-4V Titanium Alloy