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A measure of the equivalent shear stress amplitude from a prismatic hull in the principal coordinate system
International Journal of Fatigue 32 (2010) 1977–1984
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A measure of the equivalent shear stress amplitude from a prismatic hull in the principal coordinate system Khalij Leila a,*, Pagnacco Emmanuel a, Lambert Sylvain b a b
INSA de Rouen, LMR, Technopole du Madrillet, Avenue de l’université, BP 08, 76801 Saint Etienne du Rouvray Cedex, France Universidade Federal de Uberlândia, LMEST, Av. João Naves de Ávila 2121, Campus Santa Mônica, Bloco 1M Uberlândia MG, Brazil
a r t i c l e
i n f o
Article history: Received 16 December 2009 Received in revised form 3 July 2010 Accepted 10 July 2010 Available online 15 July 2010 Keywords: Stress amplitude Non-proportional loading Multiaxial fatigue High-cycle fatigue
a b s t r a c t A measure is proposed in this paper in order to assess the equivalent shear stress amplitude for highcycle multiaxial tests. It is obtained from a prismatic hull oriented along the principal load axes while it encloses the stress path. Harmonic and non-harmonic load examples are presented for illustrating this definition. Moreover, a comparison with various definitions is presented from two same amplitude stress loads. On these examples, the definitions based on the elliptic and prismatic hulls are too safe and this could have a significant impact on the cost of design. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction For high-cycle multiaxial fatigue, authors like Crossland and pffiffiffiffiffiffiffi Sines [1,2] used the equivalent shear stress amplitude J 2;a (amplitude of the square root of the second invariant of the stress deviator) and an equivalent endurance limit (function of the hydrostatic stress and material parameters) to define stress based multiaxial criteria. So, for a suitable criterion, it is necessary to define properly the both quantities. In this work, we focus only on the measure of pffiffiffiffiffiffiffi J 2;a . This quantity can be fully described for in-phase loadings with the minimum hypersphere method proposed by Papadopoulos [3,4]. When the loadings are non-proportional [5,6], this assessment is not a trivial problem because the time-evolution of the principal directions must be taken into account. Various methods based on elliptic [7–13] or prismatic [14,15] hulls have been developed recently in order to give suitable measure of the equivalent shear stress amplitude. This problem is particularly important for evaluating the fatigue behaviour of mechanical structures, in particular when they are submitted to complex random loads [16,17]. The method proposed in this work fits into this logic. The originality concerns the investigation of the principal directions of the stress path in a five-dimensional space E5 in order to obtain a correct value of the amplitude. In this method, these directions constitute the support of the axes defining a prismatic hull which
* Corresponding author. E-mail addresses: [email protected] (L. Khalij), emmanuel.pagnacco@ insa-rouen.fr (E. Pagnacco). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.07.007
circumscribes the stress path history. The measure of the equivalent shear stress amplitude is then defined as the norm of these axes in E5. The idea of using the principal axes is not new [8,16,18]. Indeed, in the reference [18], scalar measures are developed for characterizing the non-proportionality from principal axes of the tensor path. Pitoiset et al. [16] use the principal axes in order to retain pffiffiffiffiffiffiffi the first eigenvalue for assessing J 2;a , similarly to the minimum hypersphere method. At last, Cristofori et al. [8] determine the maximum variance frame of reference for an estimation of the fatigue damage in the presence of complex multiaxial fatigue loadings. The proposed definition can be apply to any load case such as sinusoidal and complex stress loadings in-phase, out-of-phase, synchronous and asynchronous. From our experience, it seems suitable to non empty load paths in the topological sense. To simplify the discussion, we present only a comparison between results of combined tension and torsion tests. The measure obtained in our proposal is confronted to definitions formalised in E5. The first two sections confront these various definitions by (respectively) a state of the art and by a critical analysis from two non-sinusoidal but periodic stresses. The algorithm of the prismatic hull in the principal coordinate system is presented and illustrated in the third section. The measures for the equivalent shear stress amplitude from the prismatic hulls are then compared and we conclude on the choice of the definition in function of the study to be conducted. Note that this work, we are mainly interested with the semi-ductile materials, where damage can be caused by a combination of shear and normal stresses.
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Nomenclature phase shift between the two signals. U closed curve corresponding to the deviator stress path. W closed curve corresponding to the stress path. r(t) stress vector. r,a amplitudes over time of the stress tensor component. r,m means over time of the stress tensor component. rxx(t); ryy(t); rzz(t); rxy(t); rxz(t); ryz stress tensor components. d
2. Measures of equivalent shear stress amplitude in E5 – state of the art The stress tensor can be expressed as the sum of two others stress tensors: (1) a mean hydrostatic stress tensor which tends to change the volume of the stressed body; (2) a stress deviator tensor which tends to distort it. The stress deviator tensor has a set of three invariants, from which J1 = 0 and J21:
J2 ¼
1h 6
i
ðrxx ryy Þ2 þ ðryy rzz Þ2 þ ðrzz rxx Þ2 þ r2xy þ r2yz þ r2zx
Having separated the mean stress from the stress tensor to obtain the deviatoric part, the set of independent stresses could be reduced from six to five-components. A suitable change of variable leads to a five-component vector DðtÞ ¼ ½D1 ðtÞ D2 ðtÞ D3 ðtÞ D4 ðtÞ D5 ðtÞT having its norm equal to J2. This change of variable is a combination of the deviator part of the stress tensor [3,4]. For a periodic load, the tip of D(t) describes in E5 a closed curve U corresponding to the deviator stress path. In high-cycle fatigue, many authors, as for example Sines or pffiffiffiffiffiffiffi Crossland, consider the equivalent shear stress amplitude J 2;a as a major parameter to control the crack initiation. The octahedral plane is then assumed to be the plane of crack initiation (or maximum shear plane) for proportional loading paths. For these proportional loadings, it is easy to demonstrate that the equivalent shear stress amplitude is given by: ffi qffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 J 2;a ¼ ½ðrxx;a ryy;a Þ2 þ ðryy;a rzz;a Þ2 þ ðrzz;a rxx;a Þ2 þ ðr2xy þ r2yz þ r2zx Þ 6
ð1Þ Unfortunately, defining such a measure for general paths is not so easy since stresses do not vary with a constant proportion along the time. The first interesting definition for the equivalent shear stress amplitude is obtained from the radius of the minimum-volume hypersphere [3] circumscribed to the five-dimensional stress deviator path (or a minimum circle in E2 corresponding to a bidimensional load) U as:
qffiffiffiffiffiffiffi J 2;a ¼ max kDðtÞ Dk t2T
ð2Þ
In this expression, the vector D corresponds to the centre of the hypersphere and contains the mean of D(t) (i.e. the mean of each component). This constitutes a difficulty in the procedure since it is obtained as the solution of an optimisation problem D : min max kDðtÞ D0 k . Moreover, it is easy to see that the 0 D
t2T
introduction of a phase shift between two sinusoidal stresses leads in a reduction for the circle diameter. This definition is therefore 1
Since J1 = 0, the stress deviator tensor is in a state of pure shear.
pffiffiffiffiffiffiffi J 2;a Ei I V
amplitude of the second invariant of the stress deviator (or equivalent shear stress amplitude). i-dimensional Euclidean space (i = 1, . . ., 5). identity matrix. matrix of ‘‘mean squares” of . the mean of .
questionable especially when considering the experiments, since such a phase shift is known to be generally with no effects for semi-ductile materials in combined tension–torsion tests [3]. Kueppers and Sonsino [20] show that the effect of non-proportional loads is dependant on stress path and material. Again in the high-cycle fatigue context, some authors [3,20] define metal classes from a ratio between the fatigue limits in fully reversed torsion and in fully reversed bending. Each of them is differently influenced by the phase shift. In order to better take into account the pffiffiffiffiffiffiffi stress path for the assessment of J 2;a , the following definition is suggested by Deperrois [21]:
qffiffiffiffiffiffiffi 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 2 l J 2;a ¼ i¼1 i 2
ð3Þ
The components li are evaluated in E5 by determining first the longest chord of U denoted l5. This procedure is repeated for next li by searching again the longest chord in a reduced and orthogonal subspace to the subspace generated by the previous chord. However, the definition (3) suffers from non-uniqueness of the longest chord in some cases [3] (the triangular load path for example [19]) which leads to a non-uniqueness of the mean equivalent shear stress. To avoid the lack of uniqueness of the Deperrois’ definition, Li and De Freitas [7] formalise the definition of the minimum ellipse in E2. The ellipse offers a better measure than the circle because it takes into account the effects of phase shift. The equivalent shear stress amplitude is obtained through semi-axes as:
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2;a ¼ R21 þ R22
ð4Þ
R1 and R2 are the major and minor semi-axes of the ellipsoid (determined from the minimum hypersphere procedure). But this choice is criticised by the Araújo’s research team because it leads to the same results for two different triangular equivalent shear stress paths [10]. They suggest defining the equivalent shear stress amplitude from another elliptic convex hull, those corresponding to the minimum Frobenius norm of the transformation which charpffiffiffiffiffiffiffi acterizes this ellipsoid [9,10,13]. The measure of J 2;a is then computed from:
qffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 a2 J 2;a ¼ i¼1 i
ð5Þ
where ai is the semi-axes of the ellipsoid which circumscribes U. However, for this last method and when a synchronous and outof-phase sinusoidal load is considered, it is noted that ai components are more easily evaluated from component amplitudes of the rectangular prism which circumscribes the ellipsoid [9–11]:
ki ¼
1 maxðDi ðtÞÞ minðDi ðtÞÞ i ¼ 1; . . . ; 5 t2T 2 t2T
This observation has led the Araújo’s research team to propose pffiffiffiffiffiffiffi also another variant of the J 2;a definition: the concept of a largest rectangular prismatic hull [14,15] (or rectangle in E2). In this case, rotations of the stress deviator path are performed around each of the five axes in E5 in order to find the maximum-volume of
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pffiffiffiffiffiffiffi prismatic hull. The measure of J 2;a is then directly defined from semi-axes of this prismatic hull. The difficulty of this task is due to the algorithm which necessitates iterations for finding the optimal rotation. In another way, a modified procedure of the minimum ellipsoid is recently developed by Li et al. [12]. For 2D loads, the authors propose to find R2 in the MCE method as the maximum distance between stress points along the normal to the major semi-axes R1. This procedure overcomes the inconvenient of the original definition raised by Gonçalves et al. [10]. One has to note that a common procedure difficulty which is valid for all the definitions based on elliptic hull is the necessity to find the centre of the loading path. 3. Critical analysis of various definitions with an illustrative example pffiffiffiffiffiffiffi In this section, we compared the various definitions of J 2;a on two non-sinusoidal periodic out-of-phase tension–torsion stress tests. Forms and amplitudes are defined in Fig. 1a and c. In a first time, we define a change of variable over the stress in a five-dimensional space E5 thanks to a transition matrix P [17] as:
2 pffiffi
3 2
6 60 6 SðtÞ ¼ P rðtÞ with P ¼ 6 60 6 40 0
pffiffi 3 6 1 2
pffiffi 3 6 1 2
0
0
0
0
0
3
0
0
0
0
7 07 7 1 0 07 7 7 0 1 05
0
0
0
ð6Þ
0 1
The tip of S(t) describes a curve corresponding to the stress path and the projections of this vector in the bi-dimensional space are described in the right of Fig. 1 for the two examples. These ones have been proposed by Li et al. in [7] and correspond to triangular and rectangular stress paths. They should lead to the same result pffiffiffiffiffiffiffi for J 2;a since they act in tension–torsion with same amplitude. In these conditions, Eq. (1) is applicable and the analytical exprespffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sion leads to J 2;a ¼ 13 r2xx;a þ r2xy;a , as it is also obtained in following Section 4.2 for tension–torsion stress case. Knowing these, we can assess the following proposals for these load examples: (1) The measurements of equivalent shear stress amplitude should be the same for the triangular and the rectangular paths (2) The values of the pffiffiffi equivalent shear stress amplitude should be equal to 2a 2. Fig. 2 shows the constructions and measurements associated with the major definitions exposed in Section 2. From the resulting pffiffiffiffiffiffiffi value for J 2;a (Table 1), we note that the measurements are very different for the two stress paths. Thus, none of the definitions satisfy both previous proposals. From another point of view, we observe on these examples that pffiffiffiffiffiffiffi the Li et al. definition produces the maximal values for J 2;a . Note 1. Li et al. do not arrive at the same conclusion because their results do not satisfy the first previous proposal [7]. To bring the study to a successful conclusion, they considered the maxipffiffiffiffiffiffiffi mum value obtained for J 2;a . 2a
σ xx (t )
2a 3
a
σ xy (t )
S 2 (t )
2a
0
-a
− 2a − 2a 3
-2a
-2a
-a
0
a
2a
S1 (t )
(a)
(b) 2a
σ xx (t )
2a 3
2a
σ xy (t ) − 2a
S 2 (t )
a
0
-a
− 2a 3
-2a -2a
-a
0
a
2a
S1 (t )
(c)
(d)
Fig. 1. Tension–torsion stress tests of (a)(b) triangular; (c)(d) rectangular paths; and (b)(d) their associated projection in the bi-dimensional space of S(t) (see expression 6).
2.5a 2.38a
2.12a
2a
S′2 (t )
2a
S′1 (t )
2.59a
2a
2a
2a
a
2.5a
2.5
2.12a
L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
S′2 (t )
1980
2a
2a
S′1 (t )
(a)
(b)
(c)
(d)
Fig. 2. Definitions of the (a) minimum circle [3], (b) minimum ellipse [12], (c) elliptic hull with the minimum Frobenius norm [11], and (d) largest rectangular hull [14,15] which circumscribes the triangular (up) and rectangular (down) paths.
Table 1 pffiffiffiffiffiffiffi Measurements of J 2;a for various definitions.
Triangular path Rectangular path
Minimum circle
Minimum ellipse
Ellipse with the minimum Frobenius norm
Largest rectangular hull
2.5a 2a
3.535a pffiffiffi 2a 2
3.517a pffiffiffi 2a 2
3a pffiffiffi 2a 2
Stress path
4. A new measure based on the prismatic hull
Pri the smati pri c hu nci pal ll alon dir g ect ion
s Prismatic hull in the transformed stress space
In the next section, we propose a new definition for assessing pffiffiffiffiffiffiffi the measure of J 2;a . The methodology is direct (no iteration to perform) and does not need to seek for the hull centre, unlike the hypersphere or the ellipsoids. We propose to retain only the amplitudes evaluated along the principal axes of the stress path in E5. We call this definition ‘‘the prismatic hull in the principal coordinate system”.
S 2 (t )
4.1. The methodology According to the change of the Eq. (6), we propose to make an analysis of the eigenvectors of the ‘‘mean squares” matrix VS of the vector S(t) in E5 as:
1 VS ¼ lim T!1 T
Z
T=2
S1′ (t )
S1 (t ) Fig. 3. Representation of two prismatic hulls enclosing the stress path; the blue one is oriented along the E5 directions while the red one is oriented along the principal directions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ðSðtÞ SÞ T dt ðSðtÞ SÞ
T=2
ðSðtÞ SÞ T ¼ E½ðSðtÞ SÞ
ð7Þ
stands for the mean of S(t), i.e. S ¼ E½S with E[] the mean where S operator. A new evaluation S0 ðtÞ ¼ ½S01 ðtÞ S02 ðtÞ S03 ðtÞ S04 ðtÞ S05 ðtÞT is obtained from an orthogonal transformation of S(t) as:
S0 ðtÞ ¼ TT SðtÞ
ð8Þ 2
T is a matrix containing the five-eigenvectors of VS . Hence, a coordinate rotation is performed around the mean stresses in order to align the transformed axes with the directions of maximum mean squares. These five vectors correspond to the principal components 2
S′2 (t )
Since TTT = I, det (T) = ±1. When the determinant of T is equal to 1 (by a suitable choice of these vectors signs), T can also be interpreted as a rotation matrix.
of the stress in E5, from which we take the amplitude to define the prismatic hull (Fig. 3). Thus, this new measure for estimating the amplitude of the equivalent shear stress is based on a prismatic hull which is defined along the principal coordinate system. Note 2. For particular case such as the circular load path, there is an infinity choice for the directions of the eigenvectors. However, they have the same eigenvalues and then lead to an invariance of pffiffiffiffiffiffiffi the measure of J 2;a . Denoting R ¼ ½R1 R2 R3 R4 R5 T the amplitudes reached by the stresses along the principal directions, we obtain a unique measure pffiffiffiffiffiffiffi of J 2;a :
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2;a ¼ R21 þ R22 þ R23 þ R24 þ R25 ¼ kRk
ð9Þ
1981
250
200
200
150
150
100
100
50
50
0
0
-50
-50
-100
-100
-150
-150
-200
-200
-250 -250 -200 -150 -100 -50
0
50 100 150 200 250
Rectangular hull
250
S′3 (t )
S 3 (t )
L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
-250 -250 -200 -150 -100 -50
0
50 100 150 200 250
S1 (t )
S′1 (t )
(a)
(b)
Fig. 4. Stress paths in the bi-dimensional space of (a) S(t) for dxy ¼ p3 and (b) the associated orthogonal transformation S0 (t) for non-proportional load.
Table 2 Measurements of equivalent shear stress amplitudes for illustrative examples. d (rad)
R
Tension–torsion
8dxy
Biaxial
dyy ¼ p3
R1 ¼ 118:70 R2 ¼ 209:64 R1 ¼ 39:3 R2 ¼ 136:5
pffiffiffiffiffiffiffi J 2;a PHPS (MPa)
kRk ¼ 240:91
pffiffiffiffiffiffiffi J2;a PAP (MPa) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 3 rxx;a þ rxy;a ¼ 240:91 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 3 ðrxx;a þ ryy;a rxx;a ryy;a cos dyy Þ ¼ 142:1
kRk ¼ 142:1
PHPS: prismatic hull in the principal coordinate system; PAP: Papadopoulos’expression [22].
with:
and:
1 maxðS0i ðtÞÞ minðS0i ðtÞÞ Ri ¼ i ¼ 1; . . . ; 5 t2T 2 t2T
rxx ðtÞ ¼ rxx;a sinðxtÞ; ryy ðtÞ ¼ ryy;a sinðxt dyy Þ ð10Þ
The complete methodology is described as follows: Step 1. Transformation in E5 of the stress history r(t) (Eq. (6)) Step 2. Evaluation of the matrix VS (Eq. (7)). In practice, we only know S(t) at N discrete time ti that we denote Si (Si is a 5 1 vector). Hence, S is a 5 N matrix. Then, firstly the mean ¼ 1 PN Si of S(t) is computed in order to remove it for each S i¼1 N ~i ¼ Si S (for i = 1, . . ., N). Next, we estimate recorded time S the matrix VS by using the Eq. (7) as:
VS ¼
1 ~ ~T SS N1
Step 3. Evaluation of S0 (t) (Eq. (8)) the vector associated to S(t) from an orthogonal transformation such that TTT = I. Step 4. Component assessment of R from the maxima and minima of S0 (t) (Eq. (10)). Step 5. Calculation of the norm of the vector R to produce the pffiffiffiffiffiffiffi measure of J 2;a (Eq. (9)). 4.2. Illustrative examples
rxx ðtÞ ¼ rxx;a sinðxtÞ; rxy ðtÞ ¼ rxy;a sinðxt dxy Þ
Let rxx,a = 315 MPa, rxy,a = 158 MPa, and ryy,a = 100 MPa. Fig. 4 is an illustration of the vectors S(t) and S0 (t) in the bi-dimensional space for the first illustrative example (tension–torsion state of stress). The measurements of tension–torsion (Eq. (11)) and biaxial (Eq. (12)) cases are reported on Table 2 with the Papadopoulos, expression.3 The results obtained for tension–torsion are the same for any values of phase shift. Figs. 5 and 6 represent the semi-axes and the associated norms respectively for the tension–torsion and biaxial examples. The measurements are recovered from the stress path along the principal directions for various phase shifts. For the tension–torsion test, we can write:
VS ¼ and
1 2
"
r2xx;a 1 pffiffi rxx;a rxy;a cos dxy 3
ð11Þ
1 3
p1ffiffi 3
rxx;a rxy;a cos dxy r2xy;a
#
2n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio1=2 3 7 7 R ¼ pffiffiffi 4 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio1=2 5 6 2 r2xx;a þ 3r2xy;a þ 12r2xx;a r2xy;a cos2 dxy þ ðr2xx;a 3r2xy;a Þ
r2xx;a þ 3r2xy;a 12r2xx;a r2xy;a cos2 dxy þ ðr2xx;a 3r2xy;a Þ2 1 6 6
which leads to
To illustrate simply the method, we consider two experimental bi-dimensional standard examples of the literature [3]: a tension– torsion (Eq. (11)) and a biaxial (Eq. (12)) state of stress. They are such that:
ð12Þ
pffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J2;a ¼ 13 r2xx;a þ r2xy;a . We can notice that the phase
shift is not involved in this result (see Fig. 5). This behaviour can be confirmed experimentally for materials such as 34Cr4 for example (for the tests of lines 21 and 23 of the ref [3], the results are the same). 3
See the note 3.
L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
200
200
150
R2 = 201.1972
250
100
S′2 (t )
50 0
100 50
S′2 (t )
150
-50
0
R1 = 51.5287
-50
-100
-100
-150
-150
-200 -250
R2 = 194.4868
1982
-1
-0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-200 -60
-40
-20
S1′ (t )
J 2,a = 201.2MPa 200
150
150
R2 = 151.8431
200
50 0
50
R1 = 132
-50
0
-100
-150
-150
-200 -150
-100
-50
0
50
100
150
-200 -80
-60
-40
S1′ (t )
(c)
60
R1 = 76.0943
-50
-100
40
J 2,a = 201.2MPa
100
S′2 (t )
100
S′2(t )
(b)
20
R2 = 186.2525
(a)
0
S1′ (t )
-20
0
20
40
60
80
S1′ (t )
J 2,a = 201.2MPa
(d)
J 2,a = 201.2MPa
Fig. 5. The stress path S0 (t) and measurements of the semi-axes for (a) dxy = 0°, (b) dxy = 30°, (c) dxy = 90°, and (d) dxy = 135°.
For the biaxial test, we have: "1
2 2 1 3 ð4rxx;a 4rxx;a ryy;a cos dyy þ ryy;a Þ VS ¼ p1ffiffi ð2rxx;a ryy;a cos dyy r2 Þ 8 yy;a 3
p1ffiffi ð2 3
rxx;a ryy;a cos dyy r2yy;a Þ r2yy;a
#
" 2 # 1=2 2 1 frxx;a þ ryy;a rxx;a ryy;a cos dyy þ Pg R ¼ pffiffiffi 6 fr2xx;a þ r2yy;a rxx;a ryy;a cos dyy Pg1=2
4.3. Comparison of definitions based on the prismatic hulls
with P¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4 cos2 dyy 1Þr2xx;a r2yy;a þ r4xx;a þ r4yy;a 2 cos dyy ðr3xx;a ryy;a þ rxx;a r3yy;a Þ
and finally
pffiffiffiffiffiffiffi q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2;a ¼ 3 ðr2xx;a þ r2yy;a rxx;a ryy;a cos dyy Þ. In this case,
the expression is function of the phase shift (see Fig. 6) with a maximum amplitude for dyy = 180°. Note 3. The proposed method enables to recover the expression proposed by Papadopoulos for 3D problems [22]:
qffiffiffiffiffiffiffi 1 h r2 þ r2yy;a þ r2zz;a rxx;a ryy;a cos ðdyy Þ J 2;a ¼ 3 xx;a
rxx;a rzz;a cos ðdzz Þryy;a rzz;a cos ðdyy dzz Þ þ r2xy;a þ r2xz;a þ r2yz;a
This interpretation is correct only for sinusoidal and synchronous loads. Indeed, if we look for the principal axes amplitudes of complex load paths and asynchronous, it is more interesting to use the prismatic hull.
1=2 ð13Þ
In this section, we compare the prismatic hull along the principal directions with the others prismatic hulls, from which the maximum one. This comparison fits again in the aim to verify the proposals of the Section 3 and coherence of the definition which we propose. The cases of triangular and rectangular paths are still used in this section. In order to make these comparisons, we performed rotation around the mean stresses of the prismatic hulls within a range ±180°. The reference positions (0°) correspond to measurements pffiffiffiffiffiffiffi of J 2;a along the E5 directions. The equivalent shear stress amplitude is plotted versus these angles in Fig. 7 for the triangular (left) and the rectangular (right) paths. Two sets of the equivalent shear stress amplitude are illustrated for each case. They correspond to the prismatic hulls leading to minimum and maximum values of pffiffiffiffiffiffiffi J 2;a : – The minimum values are obtained for rotation angles of 0° [mod p ] and 45° [mod p ] of prismatic hulls for the triangular and the 2 2 rectangular paths (respectively).
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L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
R2 = 132.7516
0
50 0
-50
-50
-100
-100
-150 -1 -0.8 -0.6 -0.4 -0.2 0
0.2 0.4 0.6 0.8 1
-150 -30
-20
-10
0
10
20
S′1 (t )
S′1 (t )
(a) J 2,a = 132.7 MPa
(b) J 2,a = 137.1MPa
30
200
R2 = 187.5002
200
100 50
150 100 50
S′2 (t )
150
S′2 (t )
R1 = 28.2968
0 -50
0
-100
-150
-150 0.2 0.4 0.6 0.8 1
R1 = 7. 0496
-50
-100
-200 -1 -0.8 -0.6 -0.4 -0.2 0
R2 = 187.0119
S′2 (t )
50
100
S′2 (t )
100
R2 = 134.1522
150
150
-200 -8
-6
-4
(c)
0
-2
S′1 (t )
2
4
6
8
S′1 (t )
(d) J 2,a = 187.1MPa
J 2,a = 187.5 MPa
3a
2.9a
2a 2
2.8a −180
−90
0
45
90
180
Shear stress amplitude J 2,a ((MPa))
S′1 (t )
S′1 (t )
2a 2
2.4 a
2a −180
−90
0
45
90
180
Angle (°) S′2 (t )
Angle (°)
3a
S′2 (t )
Shear stress amplitude J 2,a ((MPa))
S′2 (t )
S′2 (t (t )
Fig. 6. The stress path S0 (t) and measurements of the semi-axes for (a) dyy = 0°, (b) dyy = 30°, (c) dyy = 180°, and (d) dyy = 190°.
S′1 (t )
S′1 (t )
(a)
(b)
Fig. 7. Measurements of equivalent shear stress amplitude vs. rotation angle (in degrees) for (a) triangular and (b) rectangular stress path.
– The maximum values are obtained for rotation angles of 45° [mod p2 ] and 0° [mod p2 ] of prismatic hulls for the triangular and the rectangular paths (respectively).
pffiffiffiffiffiffiffi Table 3 puts together the J 2;a values for each rotation angle. As the principal directions coincide with the reference position, we can observe from this table that the application of the proposed
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L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
Table 3 pffiffiffiffiffiffiffi Values of J2;a and the associated hull sizes. Stress paths
Triangular
Angles pffiffiffiffiffiffiffi J 2;a Hull size
0° [mod p2 ]
pffiffiffi 2a 2 Minimal
Rectangular 45° [mod p2 ] 3a Maximal
0° [mod p2 ] pffiffiffi 2a 2 Maximal
45° [mod p2 ] 2a Minimal
pffiffiffi strategy leads to a 2a 2 amplitude value for both stress path cases. Thus, measurements along the principal directions satisfy both proposals of Section 3. Moreover, if we are interested in the prismatic hull size (Fig. 7 and Table 3), we note that in the principal coordinate system, the rectangular prism have a minimal size for the triangular path while it have a maximal size for the rectangular path. Consequently, the prismatic hull in the principal coordinate system is not systematically the maximal hull and then does not pffiffiffiffiffiffiffi always lead to a maximum value of J 2;a . More generally, for arbitrary stress paths, the principal directions can be oriented along any rotation angle to leads to a prismatic hull which is not necessarily the largest or the smallest. To conclude this study, we can note that looking for the largest prismatic hull is not the best way if the aim of the study is to respect the previous proposal (Section 3). On the contrary, the largest prismatic hull is interesting if a better prediction of fatigue pffiffiffiffiffiffiffi strength is investigated since the values of J 2;a are expected to be maximal for this situation. 5. Conclusion In this work, we have proposed a simple definition of the equivalent shear stress amplitude. The stress path is enclosed by a prismatic hull in the principal coordinate system. As a first step, a variable change of stress is performed in a five-dimensional space. The measure of the equivalent shear stress amplitude is obtained from the axes of this prismatic hull. The methodology is illustrated by simple examples of tensile shear and biaxial tension. It leads to the well validated results known for 2D or 3D sinusoidal loads. Furthermore, the proposed methodology is compared with various definitions for two non-sinusoidal out-of-phase signals with the same amplitudes. These loads have a triangular and a rectangular stress path in the stress space. From the example chosen, it is found that: The definition proposed by Li et al. [12] can be employed for a multiaxial high-cycle fatigue study when a good prediction of pffiffiffiffiffiffiffi the fatigue damage is sought, since the measurements of J 2;a are maximal. The prismatic hull in the principal coordinate system is interesting to use for a study based on a cost-efficient design because these two loads of same amplitudes induce the same values of pffiffiffiffiffiffiffi J 2;a .
Finally, note that to study the high-cycle multiaxial fatigue, the pffiffiffiffiffiffiffi equivalent shear stress amplitude J 2;a is not sufficient and an equivalent endurance limit is necessary to define a suitable multiaxial criterion. This would be the subject of a further study. References [1] Crossland B. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proceeding of the international conference on fatigue of metals, London, New York, 1956, p. 138–49. [2] Sines G. Behaviour of metals under complex static and alternating stress. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw-Hill; 1959. p. 145–69. [3] Papadopoulos IV, Davoli P, Gorla C, Filippini M, Bernasconi A. A comparative study of multiaxial high-cycle fatigue criteria for metals. Int J Fatigue 1997;19:219–35. [4] Bernasconi A. Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications. Int J Fatigue 2002;24:649–57. [5] Chamat A, Abbadi M, Gilgert J, Cocheteux F, Azari Z. A new non-local criterion in high-cycle multiaxial fatigue for non-proportional loadings. Int J Fatigue 2007;29(8):1465–74. [6] Chamat A, Azari Z, Jodin Ph, Gilgert J, Dominiak S. Influence of the transformation of a non-proportional multiaxial loading to a proportional one on the high cycle fatigue life. Int J Fatigue 2008;30(7):1189–99. [7] Li B, Santos LT, De Freitas M. A computerized procedure for long-life fatigue assessment under complex multiaxial loading. Fatigue Fract Engng Mater Struct 2001;24:165–77. [8] Cristofori A, Susmel L, Tovo R. A stress invariant based criterion to estimate fatigue damage under multiaxial loading. Int J Fatigue 2008;30:1646–58. [9] Gonçalves CA, Araújo JA, Mamiya EN. A simple multiaxial fatigue criterion for metals. CR Mécanique 2004;332:963–8. [10] Gonçalves CA, Araújo JA, Mamiya EN. Multiaxial fatigue: a stress based criterion for hard metals. Int J Fatigue 2005;27:177–87. [11] Mamiya EN, Araújo JA. Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress. Mech Res Commun 2002;29:141–51. [12] Li B, Reis L, De Freitas MA. Comparative study of multiaxial fatigue damage models for ductile structural steels and brittle materials. Int J Fatigue 2009;31:1895–906. [13] Zouain N, Mamiya EN, Comes F. Using enclosing ellipsoids in multiaxial fatigue strength criteria. Euro J Mech A/Solids 2006;25:51–71. [14] Mamiya EN, Araújo JA, Castro FC. Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue. Int J Fatigue 2009;31:1144–53. [15] Castro FC, Araújo JA, Mamiya EN, Zouain N. Remarks on multiaxial fatigue limit criteria based on prismatic hulls and ellipsoids. Int J Fatigue 2009;31:1875–81. [16] Pitoiset X, Rychlik I, Preumont A. Spectral methods to estimate local multiaxial fatigue failure for structures undergoing random vibrations. Fatigue Fract Engng Mater Struct 2001;24(11):715–27. [17] Lambert S, Pagnacco E, Khalij L. A probabilistic model for the fatigue reliability of structures under random loadings with phase shift effects. Int J Fatigue 2010;32(2):463–74. [18] Bishop JE. Characterizing the non-proportional and out-of-phase extent of tensor paths. Fatigue Fract Engng Mater Struct 2000;23:1019–32. [19] Bernasconi A, Papadopoulos IV. Efficiency of algorithms for shear stress amplitude calculation in critical plane class fatigue criteria. Comput Mater Sci 2005;34:355–68. [20] Kueppers M, Sonsino CM. Assessment of the fatigue behaviour of welded aluminium joints under multiaxial spectrum loading by a critical plane approach. Int J Fatigue 2006;28:540–6. [21] Ballard P, Dang Van K, Deperrois A, Papadopoulos IV. High cycle fatigue and a finite element analysis. Fatigue Fract Engng Mater Struct 1995;18(3):397–411. [22] Papadopoulos IV. A high-cycle fatigue criterion applied in biaxial and triaxial out-of-phase stress conditions. Fatigue Fract Engng Mater Struct 1995;18(1):79–91.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
A measure of the equivalent shear stress amplitude from a prismatic hull in the principal coordinate system Khalij Leila a,*, Pagnacco Emmanuel a, Lambert Sylvain b a b
INSA de Rouen, LMR, Technopole du Madrillet, Avenue de l’université, BP 08, 76801 Saint Etienne du Rouvray Cedex, France Universidade Federal de Uberlândia, LMEST, Av. João Naves de Ávila 2121, Campus Santa Mônica, Bloco 1M Uberlândia MG, Brazil
a r t i c l e
i n f o
Article history: Received 16 December 2009 Received in revised form 3 July 2010 Accepted 10 July 2010 Available online 15 July 2010 Keywords: Stress amplitude Non-proportional loading Multiaxial fatigue High-cycle fatigue
a b s t r a c t A measure is proposed in this paper in order to assess the equivalent shear stress amplitude for highcycle multiaxial tests. It is obtained from a prismatic hull oriented along the principal load axes while it encloses the stress path. Harmonic and non-harmonic load examples are presented for illustrating this definition. Moreover, a comparison with various definitions is presented from two same amplitude stress loads. On these examples, the definitions based on the elliptic and prismatic hulls are too safe and this could have a significant impact on the cost of design. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction For high-cycle multiaxial fatigue, authors like Crossland and pffiffiffiffiffiffiffi Sines [1,2] used the equivalent shear stress amplitude J 2;a (amplitude of the square root of the second invariant of the stress deviator) and an equivalent endurance limit (function of the hydrostatic stress and material parameters) to define stress based multiaxial criteria. So, for a suitable criterion, it is necessary to define properly the both quantities. In this work, we focus only on the measure of pffiffiffiffiffiffiffi J 2;a . This quantity can be fully described for in-phase loadings with the minimum hypersphere method proposed by Papadopoulos [3,4]. When the loadings are non-proportional [5,6], this assessment is not a trivial problem because the time-evolution of the principal directions must be taken into account. Various methods based on elliptic [7–13] or prismatic [14,15] hulls have been developed recently in order to give suitable measure of the equivalent shear stress amplitude. This problem is particularly important for evaluating the fatigue behaviour of mechanical structures, in particular when they are submitted to complex random loads [16,17]. The method proposed in this work fits into this logic. The originality concerns the investigation of the principal directions of the stress path in a five-dimensional space E5 in order to obtain a correct value of the amplitude. In this method, these directions constitute the support of the axes defining a prismatic hull which
* Corresponding author. E-mail addresses: [email protected] (L. Khalij), emmanuel.pagnacco@ insa-rouen.fr (E. Pagnacco). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.07.007
circumscribes the stress path history. The measure of the equivalent shear stress amplitude is then defined as the norm of these axes in E5. The idea of using the principal axes is not new [8,16,18]. Indeed, in the reference [18], scalar measures are developed for characterizing the non-proportionality from principal axes of the tensor path. Pitoiset et al. [16] use the principal axes in order to retain pffiffiffiffiffiffiffi the first eigenvalue for assessing J 2;a , similarly to the minimum hypersphere method. At last, Cristofori et al. [8] determine the maximum variance frame of reference for an estimation of the fatigue damage in the presence of complex multiaxial fatigue loadings. The proposed definition can be apply to any load case such as sinusoidal and complex stress loadings in-phase, out-of-phase, synchronous and asynchronous. From our experience, it seems suitable to non empty load paths in the topological sense. To simplify the discussion, we present only a comparison between results of combined tension and torsion tests. The measure obtained in our proposal is confronted to definitions formalised in E5. The first two sections confront these various definitions by (respectively) a state of the art and by a critical analysis from two non-sinusoidal but periodic stresses. The algorithm of the prismatic hull in the principal coordinate system is presented and illustrated in the third section. The measures for the equivalent shear stress amplitude from the prismatic hulls are then compared and we conclude on the choice of the definition in function of the study to be conducted. Note that this work, we are mainly interested with the semi-ductile materials, where damage can be caused by a combination of shear and normal stresses.
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Nomenclature phase shift between the two signals. U closed curve corresponding to the deviator stress path. W closed curve corresponding to the stress path. r(t) stress vector. r,a amplitudes over time of the stress tensor component. r,m means over time of the stress tensor component. rxx(t); ryy(t); rzz(t); rxy(t); rxz(t); ryz stress tensor components. d
2. Measures of equivalent shear stress amplitude in E5 – state of the art The stress tensor can be expressed as the sum of two others stress tensors: (1) a mean hydrostatic stress tensor which tends to change the volume of the stressed body; (2) a stress deviator tensor which tends to distort it. The stress deviator tensor has a set of three invariants, from which J1 = 0 and J21:
J2 ¼
1h 6
i
ðrxx ryy Þ2 þ ðryy rzz Þ2 þ ðrzz rxx Þ2 þ r2xy þ r2yz þ r2zx
Having separated the mean stress from the stress tensor to obtain the deviatoric part, the set of independent stresses could be reduced from six to five-components. A suitable change of variable leads to a five-component vector DðtÞ ¼ ½D1 ðtÞ D2 ðtÞ D3 ðtÞ D4 ðtÞ D5 ðtÞT having its norm equal to J2. This change of variable is a combination of the deviator part of the stress tensor [3,4]. For a periodic load, the tip of D(t) describes in E5 a closed curve U corresponding to the deviator stress path. In high-cycle fatigue, many authors, as for example Sines or pffiffiffiffiffiffiffi Crossland, consider the equivalent shear stress amplitude J 2;a as a major parameter to control the crack initiation. The octahedral plane is then assumed to be the plane of crack initiation (or maximum shear plane) for proportional loading paths. For these proportional loadings, it is easy to demonstrate that the equivalent shear stress amplitude is given by: ffi qffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 J 2;a ¼ ½ðrxx;a ryy;a Þ2 þ ðryy;a rzz;a Þ2 þ ðrzz;a rxx;a Þ2 þ ðr2xy þ r2yz þ r2zx Þ 6
ð1Þ Unfortunately, defining such a measure for general paths is not so easy since stresses do not vary with a constant proportion along the time. The first interesting definition for the equivalent shear stress amplitude is obtained from the radius of the minimum-volume hypersphere [3] circumscribed to the five-dimensional stress deviator path (or a minimum circle in E2 corresponding to a bidimensional load) U as:
qffiffiffiffiffiffiffi J 2;a ¼ max kDðtÞ Dk t2T
ð2Þ
In this expression, the vector D corresponds to the centre of the hypersphere and contains the mean of D(t) (i.e. the mean of each component). This constitutes a difficulty in the procedure since it is obtained as the solution of an optimisation problem D : min max kDðtÞ D0 k . Moreover, it is easy to see that the 0 D
t2T
introduction of a phase shift between two sinusoidal stresses leads in a reduction for the circle diameter. This definition is therefore 1
Since J1 = 0, the stress deviator tensor is in a state of pure shear.
pffiffiffiffiffiffiffi J 2;a Ei I V
amplitude of the second invariant of the stress deviator (or equivalent shear stress amplitude). i-dimensional Euclidean space (i = 1, . . ., 5). identity matrix. matrix of ‘‘mean squares” of . the mean of .
questionable especially when considering the experiments, since such a phase shift is known to be generally with no effects for semi-ductile materials in combined tension–torsion tests [3]. Kueppers and Sonsino [20] show that the effect of non-proportional loads is dependant on stress path and material. Again in the high-cycle fatigue context, some authors [3,20] define metal classes from a ratio between the fatigue limits in fully reversed torsion and in fully reversed bending. Each of them is differently influenced by the phase shift. In order to better take into account the pffiffiffiffiffiffiffi stress path for the assessment of J 2;a , the following definition is suggested by Deperrois [21]:
qffiffiffiffiffiffiffi 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 2 l J 2;a ¼ i¼1 i 2
ð3Þ
The components li are evaluated in E5 by determining first the longest chord of U denoted l5. This procedure is repeated for next li by searching again the longest chord in a reduced and orthogonal subspace to the subspace generated by the previous chord. However, the definition (3) suffers from non-uniqueness of the longest chord in some cases [3] (the triangular load path for example [19]) which leads to a non-uniqueness of the mean equivalent shear stress. To avoid the lack of uniqueness of the Deperrois’ definition, Li and De Freitas [7] formalise the definition of the minimum ellipse in E2. The ellipse offers a better measure than the circle because it takes into account the effects of phase shift. The equivalent shear stress amplitude is obtained through semi-axes as:
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2;a ¼ R21 þ R22
ð4Þ
R1 and R2 are the major and minor semi-axes of the ellipsoid (determined from the minimum hypersphere procedure). But this choice is criticised by the Araújo’s research team because it leads to the same results for two different triangular equivalent shear stress paths [10]. They suggest defining the equivalent shear stress amplitude from another elliptic convex hull, those corresponding to the minimum Frobenius norm of the transformation which charpffiffiffiffiffiffiffi acterizes this ellipsoid [9,10,13]. The measure of J 2;a is then computed from:
qffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 a2 J 2;a ¼ i¼1 i
ð5Þ
where ai is the semi-axes of the ellipsoid which circumscribes U. However, for this last method and when a synchronous and outof-phase sinusoidal load is considered, it is noted that ai components are more easily evaluated from component amplitudes of the rectangular prism which circumscribes the ellipsoid [9–11]:
ki ¼
1 maxðDi ðtÞÞ minðDi ðtÞÞ i ¼ 1; . . . ; 5 t2T 2 t2T
This observation has led the Araújo’s research team to propose pffiffiffiffiffiffiffi also another variant of the J 2;a definition: the concept of a largest rectangular prismatic hull [14,15] (or rectangle in E2). In this case, rotations of the stress deviator path are performed around each of the five axes in E5 in order to find the maximum-volume of
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L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
pffiffiffiffiffiffiffi prismatic hull. The measure of J 2;a is then directly defined from semi-axes of this prismatic hull. The difficulty of this task is due to the algorithm which necessitates iterations for finding the optimal rotation. In another way, a modified procedure of the minimum ellipsoid is recently developed by Li et al. [12]. For 2D loads, the authors propose to find R2 in the MCE method as the maximum distance between stress points along the normal to the major semi-axes R1. This procedure overcomes the inconvenient of the original definition raised by Gonçalves et al. [10]. One has to note that a common procedure difficulty which is valid for all the definitions based on elliptic hull is the necessity to find the centre of the loading path. 3. Critical analysis of various definitions with an illustrative example pffiffiffiffiffiffiffi In this section, we compared the various definitions of J 2;a on two non-sinusoidal periodic out-of-phase tension–torsion stress tests. Forms and amplitudes are defined in Fig. 1a and c. In a first time, we define a change of variable over the stress in a five-dimensional space E5 thanks to a transition matrix P [17] as:
2 pffiffi
3 2
6 60 6 SðtÞ ¼ P rðtÞ with P ¼ 6 60 6 40 0
pffiffi 3 6 1 2
pffiffi 3 6 1 2
0
0
0
0
0
3
0
0
0
0
7 07 7 1 0 07 7 7 0 1 05
0
0
0
ð6Þ
0 1
The tip of S(t) describes a curve corresponding to the stress path and the projections of this vector in the bi-dimensional space are described in the right of Fig. 1 for the two examples. These ones have been proposed by Li et al. in [7] and correspond to triangular and rectangular stress paths. They should lead to the same result pffiffiffiffiffiffiffi for J 2;a since they act in tension–torsion with same amplitude. In these conditions, Eq. (1) is applicable and the analytical exprespffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sion leads to J 2;a ¼ 13 r2xx;a þ r2xy;a , as it is also obtained in following Section 4.2 for tension–torsion stress case. Knowing these, we can assess the following proposals for these load examples: (1) The measurements of equivalent shear stress amplitude should be the same for the triangular and the rectangular paths (2) The values of the pffiffiffi equivalent shear stress amplitude should be equal to 2a 2. Fig. 2 shows the constructions and measurements associated with the major definitions exposed in Section 2. From the resulting pffiffiffiffiffiffiffi value for J 2;a (Table 1), we note that the measurements are very different for the two stress paths. Thus, none of the definitions satisfy both previous proposals. From another point of view, we observe on these examples that pffiffiffiffiffiffiffi the Li et al. definition produces the maximal values for J 2;a . Note 1. Li et al. do not arrive at the same conclusion because their results do not satisfy the first previous proposal [7]. To bring the study to a successful conclusion, they considered the maxipffiffiffiffiffiffiffi mum value obtained for J 2;a . 2a
σ xx (t )
2a 3
a
σ xy (t )
S 2 (t )
2a
0
-a
− 2a − 2a 3
-2a
-2a
-a
0
a
2a
S1 (t )
(a)
(b) 2a
σ xx (t )
2a 3
2a
σ xy (t ) − 2a
S 2 (t )
a
0
-a
− 2a 3
-2a -2a
-a
0
a
2a
S1 (t )
(c)
(d)
Fig. 1. Tension–torsion stress tests of (a)(b) triangular; (c)(d) rectangular paths; and (b)(d) their associated projection in the bi-dimensional space of S(t) (see expression 6).
2.5a 2.38a
2.12a
2a
S′2 (t )
2a
S′1 (t )
2.59a
2a
2a
2a
a
2.5a
2.5
2.12a
L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
S′2 (t )
1980
2a
2a
S′1 (t )
(a)
(b)
(c)
(d)
Fig. 2. Definitions of the (a) minimum circle [3], (b) minimum ellipse [12], (c) elliptic hull with the minimum Frobenius norm [11], and (d) largest rectangular hull [14,15] which circumscribes the triangular (up) and rectangular (down) paths.
Table 1 pffiffiffiffiffiffiffi Measurements of J 2;a for various definitions.
Triangular path Rectangular path
Minimum circle
Minimum ellipse
Ellipse with the minimum Frobenius norm
Largest rectangular hull
2.5a 2a
3.535a pffiffiffi 2a 2
3.517a pffiffiffi 2a 2
3a pffiffiffi 2a 2
Stress path
4. A new measure based on the prismatic hull
Pri the smati pri c hu nci pal ll alon dir g ect ion
s Prismatic hull in the transformed stress space
In the next section, we propose a new definition for assessing pffiffiffiffiffiffiffi the measure of J 2;a . The methodology is direct (no iteration to perform) and does not need to seek for the hull centre, unlike the hypersphere or the ellipsoids. We propose to retain only the amplitudes evaluated along the principal axes of the stress path in E5. We call this definition ‘‘the prismatic hull in the principal coordinate system”.
S 2 (t )
4.1. The methodology According to the change of the Eq. (6), we propose to make an analysis of the eigenvectors of the ‘‘mean squares” matrix VS of the vector S(t) in E5 as:
1 VS ¼ lim T!1 T
Z
T=2
S1′ (t )
S1 (t ) Fig. 3. Representation of two prismatic hulls enclosing the stress path; the blue one is oriented along the E5 directions while the red one is oriented along the principal directions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ðSðtÞ SÞ T dt ðSðtÞ SÞ
T=2
ðSðtÞ SÞ T ¼ E½ðSðtÞ SÞ
ð7Þ
stands for the mean of S(t), i.e. S ¼ E½S with E[] the mean where S operator. A new evaluation S0 ðtÞ ¼ ½S01 ðtÞ S02 ðtÞ S03 ðtÞ S04 ðtÞ S05 ðtÞT is obtained from an orthogonal transformation of S(t) as:
S0 ðtÞ ¼ TT SðtÞ
ð8Þ 2
T is a matrix containing the five-eigenvectors of VS . Hence, a coordinate rotation is performed around the mean stresses in order to align the transformed axes with the directions of maximum mean squares. These five vectors correspond to the principal components 2
S′2 (t )
Since TTT = I, det (T) = ±1. When the determinant of T is equal to 1 (by a suitable choice of these vectors signs), T can also be interpreted as a rotation matrix.
of the stress in E5, from which we take the amplitude to define the prismatic hull (Fig. 3). Thus, this new measure for estimating the amplitude of the equivalent shear stress is based on a prismatic hull which is defined along the principal coordinate system. Note 2. For particular case such as the circular load path, there is an infinity choice for the directions of the eigenvectors. However, they have the same eigenvalues and then lead to an invariance of pffiffiffiffiffiffiffi the measure of J 2;a . Denoting R ¼ ½R1 R2 R3 R4 R5 T the amplitudes reached by the stresses along the principal directions, we obtain a unique measure pffiffiffiffiffiffiffi of J 2;a :
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2;a ¼ R21 þ R22 þ R23 þ R24 þ R25 ¼ kRk
ð9Þ
1981
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Rectangular hull
250
S′3 (t )
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L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
-250 -250 -200 -150 -100 -50
0
50 100 150 200 250
S1 (t )
S′1 (t )
(a)
(b)
Fig. 4. Stress paths in the bi-dimensional space of (a) S(t) for dxy ¼ p3 and (b) the associated orthogonal transformation S0 (t) for non-proportional load.
Table 2 Measurements of equivalent shear stress amplitudes for illustrative examples. d (rad)
R
Tension–torsion
8dxy
Biaxial
dyy ¼ p3
R1 ¼ 118:70 R2 ¼ 209:64 R1 ¼ 39:3 R2 ¼ 136:5
pffiffiffiffiffiffiffi J 2;a PHPS (MPa)
kRk ¼ 240:91
pffiffiffiffiffiffiffi J2;a PAP (MPa) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 3 rxx;a þ rxy;a ¼ 240:91 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 3 ðrxx;a þ ryy;a rxx;a ryy;a cos dyy Þ ¼ 142:1
kRk ¼ 142:1
PHPS: prismatic hull in the principal coordinate system; PAP: Papadopoulos’expression [22].
with:
and:
1 maxðS0i ðtÞÞ minðS0i ðtÞÞ Ri ¼ i ¼ 1; . . . ; 5 t2T 2 t2T
rxx ðtÞ ¼ rxx;a sinðxtÞ; ryy ðtÞ ¼ ryy;a sinðxt dyy Þ ð10Þ
The complete methodology is described as follows: Step 1. Transformation in E5 of the stress history r(t) (Eq. (6)) Step 2. Evaluation of the matrix VS (Eq. (7)). In practice, we only know S(t) at N discrete time ti that we denote Si (Si is a 5 1 vector). Hence, S is a 5 N matrix. Then, firstly the mean ¼ 1 PN Si of S(t) is computed in order to remove it for each S i¼1 N ~i ¼ Si S (for i = 1, . . ., N). Next, we estimate recorded time S the matrix VS by using the Eq. (7) as:
VS ¼
1 ~ ~T SS N1
Step 3. Evaluation of S0 (t) (Eq. (8)) the vector associated to S(t) from an orthogonal transformation such that TTT = I. Step 4. Component assessment of R from the maxima and minima of S0 (t) (Eq. (10)). Step 5. Calculation of the norm of the vector R to produce the pffiffiffiffiffiffiffi measure of J 2;a (Eq. (9)). 4.2. Illustrative examples
rxx ðtÞ ¼ rxx;a sinðxtÞ; rxy ðtÞ ¼ rxy;a sinðxt dxy Þ
Let rxx,a = 315 MPa, rxy,a = 158 MPa, and ryy,a = 100 MPa. Fig. 4 is an illustration of the vectors S(t) and S0 (t) in the bi-dimensional space for the first illustrative example (tension–torsion state of stress). The measurements of tension–torsion (Eq. (11)) and biaxial (Eq. (12)) cases are reported on Table 2 with the Papadopoulos, expression.3 The results obtained for tension–torsion are the same for any values of phase shift. Figs. 5 and 6 represent the semi-axes and the associated norms respectively for the tension–torsion and biaxial examples. The measurements are recovered from the stress path along the principal directions for various phase shifts. For the tension–torsion test, we can write:
VS ¼ and
1 2
"
r2xx;a 1 pffiffi rxx;a rxy;a cos dxy 3
ð11Þ
1 3
p1ffiffi 3
rxx;a rxy;a cos dxy r2xy;a
#
2n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio1=2 3 7 7 R ¼ pffiffiffi 4 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio1=2 5 6 2 r2xx;a þ 3r2xy;a þ 12r2xx;a r2xy;a cos2 dxy þ ðr2xx;a 3r2xy;a Þ
r2xx;a þ 3r2xy;a 12r2xx;a r2xy;a cos2 dxy þ ðr2xx;a 3r2xy;a Þ2 1 6 6
which leads to
To illustrate simply the method, we consider two experimental bi-dimensional standard examples of the literature [3]: a tension– torsion (Eq. (11)) and a biaxial (Eq. (12)) state of stress. They are such that:
ð12Þ
pffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J2;a ¼ 13 r2xx;a þ r2xy;a . We can notice that the phase
shift is not involved in this result (see Fig. 5). This behaviour can be confirmed experimentally for materials such as 34Cr4 for example (for the tests of lines 21 and 23 of the ref [3], the results are the same). 3
See the note 3.
L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
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R2 = 201.1972
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S′2 (t )
50 0
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0
R1 = 51.5287
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R2 = 194.4868
1982
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1
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S1′ (t )
J 2,a = 201.2MPa 200
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R2 = 151.8431
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S1′ (t )
(c)
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R1 = 76.0943
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40
J 2,a = 201.2MPa
100
S′2 (t )
100
S′2(t )
(b)
20
R2 = 186.2525
(a)
0
S1′ (t )
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0
20
40
60
80
S1′ (t )
J 2,a = 201.2MPa
(d)
J 2,a = 201.2MPa
Fig. 5. The stress path S0 (t) and measurements of the semi-axes for (a) dxy = 0°, (b) dxy = 30°, (c) dxy = 90°, and (d) dxy = 135°.
For the biaxial test, we have: "1
2 2 1 3 ð4rxx;a 4rxx;a ryy;a cos dyy þ ryy;a Þ VS ¼ p1ffiffi ð2rxx;a ryy;a cos dyy r2 Þ 8 yy;a 3
p1ffiffi ð2 3
rxx;a ryy;a cos dyy r2yy;a Þ r2yy;a
#
" 2 # 1=2 2 1 frxx;a þ ryy;a rxx;a ryy;a cos dyy þ Pg R ¼ pffiffiffi 6 fr2xx;a þ r2yy;a rxx;a ryy;a cos dyy Pg1=2
4.3. Comparison of definitions based on the prismatic hulls
with P¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4 cos2 dyy 1Þr2xx;a r2yy;a þ r4xx;a þ r4yy;a 2 cos dyy ðr3xx;a ryy;a þ rxx;a r3yy;a Þ
and finally
pffiffiffiffiffiffiffi q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2;a ¼ 3 ðr2xx;a þ r2yy;a rxx;a ryy;a cos dyy Þ. In this case,
the expression is function of the phase shift (see Fig. 6) with a maximum amplitude for dyy = 180°. Note 3. The proposed method enables to recover the expression proposed by Papadopoulos for 3D problems [22]:
qffiffiffiffiffiffiffi 1 h r2 þ r2yy;a þ r2zz;a rxx;a ryy;a cos ðdyy Þ J 2;a ¼ 3 xx;a
rxx;a rzz;a cos ðdzz Þryy;a rzz;a cos ðdyy dzz Þ þ r2xy;a þ r2xz;a þ r2yz;a
This interpretation is correct only for sinusoidal and synchronous loads. Indeed, if we look for the principal axes amplitudes of complex load paths and asynchronous, it is more interesting to use the prismatic hull.
1=2 ð13Þ
In this section, we compare the prismatic hull along the principal directions with the others prismatic hulls, from which the maximum one. This comparison fits again in the aim to verify the proposals of the Section 3 and coherence of the definition which we propose. The cases of triangular and rectangular paths are still used in this section. In order to make these comparisons, we performed rotation around the mean stresses of the prismatic hulls within a range ±180°. The reference positions (0°) correspond to measurements pffiffiffiffiffiffiffi of J 2;a along the E5 directions. The equivalent shear stress amplitude is plotted versus these angles in Fig. 7 for the triangular (left) and the rectangular (right) paths. Two sets of the equivalent shear stress amplitude are illustrated for each case. They correspond to the prismatic hulls leading to minimum and maximum values of pffiffiffiffiffiffiffi J 2;a : – The minimum values are obtained for rotation angles of 0° [mod p ] and 45° [mod p ] of prismatic hulls for the triangular and the 2 2 rectangular paths (respectively).
1983
L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
R2 = 132.7516
0
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S′1 (t )
S′1 (t )
(a) J 2,a = 132.7 MPa
(b) J 2,a = 137.1MPa
30
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S′2 (t )
R1 = 28.2968
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R1 = 7. 0496
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S′2 (t )
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3a
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Shear stress amplitude J 2,a ((MPa))
S′1 (t )
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Angle (°) S′2 (t )
Angle (°)
3a
S′2 (t )
Shear stress amplitude J 2,a ((MPa))
S′2 (t )
S′2 (t (t )
Fig. 6. The stress path S0 (t) and measurements of the semi-axes for (a) dyy = 0°, (b) dyy = 30°, (c) dyy = 180°, and (d) dyy = 190°.
S′1 (t )
S′1 (t )
(a)
(b)
Fig. 7. Measurements of equivalent shear stress amplitude vs. rotation angle (in degrees) for (a) triangular and (b) rectangular stress path.
– The maximum values are obtained for rotation angles of 45° [mod p2 ] and 0° [mod p2 ] of prismatic hulls for the triangular and the rectangular paths (respectively).
pffiffiffiffiffiffiffi Table 3 puts together the J 2;a values for each rotation angle. As the principal directions coincide with the reference position, we can observe from this table that the application of the proposed
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L. Khalij et al. / International Journal of Fatigue 32 (2010) 1977–1984
Table 3 pffiffiffiffiffiffiffi Values of J2;a and the associated hull sizes. Stress paths
Triangular
Angles pffiffiffiffiffiffiffi J 2;a Hull size
0° [mod p2 ]
pffiffiffi 2a 2 Minimal
Rectangular 45° [mod p2 ] 3a Maximal
0° [mod p2 ] pffiffiffi 2a 2 Maximal
45° [mod p2 ] 2a Minimal
pffiffiffi strategy leads to a 2a 2 amplitude value for both stress path cases. Thus, measurements along the principal directions satisfy both proposals of Section 3. Moreover, if we are interested in the prismatic hull size (Fig. 7 and Table 3), we note that in the principal coordinate system, the rectangular prism have a minimal size for the triangular path while it have a maximal size for the rectangular path. Consequently, the prismatic hull in the principal coordinate system is not systematically the maximal hull and then does not pffiffiffiffiffiffiffi always lead to a maximum value of J 2;a . More generally, for arbitrary stress paths, the principal directions can be oriented along any rotation angle to leads to a prismatic hull which is not necessarily the largest or the smallest. To conclude this study, we can note that looking for the largest prismatic hull is not the best way if the aim of the study is to respect the previous proposal (Section 3). On the contrary, the largest prismatic hull is interesting if a better prediction of fatigue pffiffiffiffiffiffiffi strength is investigated since the values of J 2;a are expected to be maximal for this situation. 5. Conclusion In this work, we have proposed a simple definition of the equivalent shear stress amplitude. The stress path is enclosed by a prismatic hull in the principal coordinate system. As a first step, a variable change of stress is performed in a five-dimensional space. The measure of the equivalent shear stress amplitude is obtained from the axes of this prismatic hull. The methodology is illustrated by simple examples of tensile shear and biaxial tension. It leads to the well validated results known for 2D or 3D sinusoidal loads. Furthermore, the proposed methodology is compared with various definitions for two non-sinusoidal out-of-phase signals with the same amplitudes. These loads have a triangular and a rectangular stress path in the stress space. From the example chosen, it is found that: The definition proposed by Li et al. [12] can be employed for a multiaxial high-cycle fatigue study when a good prediction of pffiffiffiffiffiffiffi the fatigue damage is sought, since the measurements of J 2;a are maximal. The prismatic hull in the principal coordinate system is interesting to use for a study based on a cost-efficient design because these two loads of same amplitudes induce the same values of pffiffiffiffiffiffiffi J 2;a .
Finally, note that to study the high-cycle multiaxial fatigue, the pffiffiffiffiffiffiffi equivalent shear stress amplitude J 2;a is not sufficient and an equivalent endurance limit is necessary to define a suitable multiaxial criterion. This would be the subject of a further study. References [1] Crossland B. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proceeding of the international conference on fatigue of metals, London, New York, 1956, p. 138–49. [2] Sines G. Behaviour of metals under complex static and alternating stress. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw-Hill; 1959. p. 145–69. [3] Papadopoulos IV, Davoli P, Gorla C, Filippini M, Bernasconi A. A comparative study of multiaxial high-cycle fatigue criteria for metals. Int J Fatigue 1997;19:219–35. [4] Bernasconi A. Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications. Int J Fatigue 2002;24:649–57. [5] Chamat A, Abbadi M, Gilgert J, Cocheteux F, Azari Z. A new non-local criterion in high-cycle multiaxial fatigue for non-proportional loadings. Int J Fatigue 2007;29(8):1465–74. [6] Chamat A, Azari Z, Jodin Ph, Gilgert J, Dominiak S. Influence of the transformation of a non-proportional multiaxial loading to a proportional one on the high cycle fatigue life. Int J Fatigue 2008;30(7):1189–99. [7] Li B, Santos LT, De Freitas M. A computerized procedure for long-life fatigue assessment under complex multiaxial loading. Fatigue Fract Engng Mater Struct 2001;24:165–77. [8] Cristofori A, Susmel L, Tovo R. A stress invariant based criterion to estimate fatigue damage under multiaxial loading. Int J Fatigue 2008;30:1646–58. [9] Gonçalves CA, Araújo JA, Mamiya EN. A simple multiaxial fatigue criterion for metals. CR Mécanique 2004;332:963–8. [10] Gonçalves CA, Araújo JA, Mamiya EN. Multiaxial fatigue: a stress based criterion for hard metals. Int J Fatigue 2005;27:177–87. [11] Mamiya EN, Araújo JA. Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress. Mech Res Commun 2002;29:141–51. [12] Li B, Reis L, De Freitas MA. Comparative study of multiaxial fatigue damage models for ductile structural steels and brittle materials. Int J Fatigue 2009;31:1895–906. [13] Zouain N, Mamiya EN, Comes F. Using enclosing ellipsoids in multiaxial fatigue strength criteria. Euro J Mech A/Solids 2006;25:51–71. [14] Mamiya EN, Araújo JA, Castro FC. Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue. Int J Fatigue 2009;31:1144–53. [15] Castro FC, Araújo JA, Mamiya EN, Zouain N. Remarks on multiaxial fatigue limit criteria based on prismatic hulls and ellipsoids. Int J Fatigue 2009;31:1875–81. [16] Pitoiset X, Rychlik I, Preumont A. Spectral methods to estimate local multiaxial fatigue failure for structures undergoing random vibrations. Fatigue Fract Engng Mater Struct 2001;24(11):715–27. [17] Lambert S, Pagnacco E, Khalij L. A probabilistic model for the fatigue reliability of structures under random loadings with phase shift effects. Int J Fatigue 2010;32(2):463–74. [18] Bishop JE. Characterizing the non-proportional and out-of-phase extent of tensor paths. Fatigue Fract Engng Mater Struct 2000;23:1019–32. [19] Bernasconi A, Papadopoulos IV. Efficiency of algorithms for shear stress amplitude calculation in critical plane class fatigue criteria. Comput Mater Sci 2005;34:355–68. [20] Kueppers M, Sonsino CM. Assessment of the fatigue behaviour of welded aluminium joints under multiaxial spectrum loading by a critical plane approach. Int J Fatigue 2006;28:540–6. [21] Ballard P, Dang Van K, Deperrois A, Papadopoulos IV. High cycle fatigue and a finite element analysis. Fatigue Fract Engng Mater Struct 1995;18(3):397–411. [22] Papadopoulos IV. A high-cycle fatigue criterion applied in biaxial and triaxial out-of-phase stress conditions. Fatigue Fract Engng Mater Struct 1995;18(1):79–91.