Prismatic hull: A new measure of shear stress amplitude in multiaxial high cycle fatigue

International Journal of Fatigue 31 (2009) 1144–1153

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Prismatic hull: A new measure of shear stress amplitude in multiaxial high cycle fatigue E.N. Mamiya *, J.A. Araújo, F.C. Castro Department of Mechanical Engineering, Universidade de Brasilia, 70910-900 Brası´lia, DF, Brazil

a r t i c l e

i n f o

Article history: Received 20 October 2008 Received in revised form 9 December 2008 Accepted 19 December 2008 Available online 27 December 2008 Keywords: High cycle fatigue Non-proportional loading Fatigue limit Multiaxial fatigue

a b s t r a c t The largest rectangular prismatic hull enclosing the deviatoric stress history is proposed as a measure of shear stress amplitude in multiaxial high cycle fatigue. The axes of the prismatic hull represent the amplitudes of the stress history along its directions, allowing a proper characterization of non-proportional loadings. The resulting multiaxial fatigue endurance criterion, simple to implement and very cost effective from the computational point of view, provides predictions which are in sound agreement with a wide variety of available fatigue experiments involving combined normal and shear stresses, with harmonic and non-harmonic, synchronous and asynchronous waveforms. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Fatigue is a failure mode experienced by a material point under time-varying solicitation, characterized by the initiation of a visible crack. It is widely believed that this progressive material degradation is mainly driven by accumulated plastic deformations which may occur at different scales. When there exists variations of macroscopic plastic deformation at a material point whose net deformation vanishes for arbitrarily large time [1], then we say that the material undergoes low cycle fatigue (or alternating plasticity). On the other hand, if the accumulated plastic deformation takes place only at mesoscopic level and the macroscopic behavior is elastic, then we have high cycle fatigue [2,3]. Finally, if elastic behavior is observed at both mesoscopic and macroscopic level, or if plastic deformation at mesoscopic level evolve to a state of elastic shakedown, then fatigue failure is not expected to occur or will occur at very long lives, typically higher than 107 cycles. The main difference among the many multiaxial fatigue endurance (or fatigue limit) criteria proposed in the literature concerns the definition of the stress measures which quantify the fatigue damage associated with a given stress history. Critical plane approaches are based on the search of a material plane where a combination of stress quantities (functions of normal and shear stresses) attains its maximum. Representative critical plane models were proposed by Findley [4] and McDiarmid [5], amongst others. Another approach considers measures associated with the history of the stress tensor itself, instead of its manifestation upon * Corresponding author. Tel.: +55 61 3307 3089. E-mail address: [email protected] (E.N. Mamiya). 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.12.010

each material plane. The earlier approaches of this type, proposed by Crossland [6] and Sines [7], are not able to distinct affine stress paths from non-proportional ones (which provide distinct fatigue damage). Deperrois [8] presented an approach which is successful in taking into account the non-proportional nature of the stress path. However, the major criticism upon this model was related to the non-uniqueness of the measure of the shear stress amplitude for some classes of stress paths. Li and co-authors [9] considered the minimum ellipsoid enclosing the deviatoric stress path as a measure of the shear stress amplitude. Nevertheless, as remarked in [10], their procedure to compute the minimum ellipse may fail for some paths (e.g. rectangular paths). Mamiya and co-authors [11,12] propose a definition of the minimum ellipsoid based upon its Frobenius norm. For elliptical paths, it was shown that this norm can be easily computed from the axes of any arbitrarily oriented rectangular hull. For general five-dimensional deviatoric stress paths, the formulation and the computation of the corresponding minimum Frobenius norm ellipsoid was presented by Zouain et al. [10]. Two important criteria that have to be referenced are the ones introduced by Dang Van [2] — written in terms of the instantaneous values of the hydrostatic and the Tresca stresses — and the one proposed by Papadopoulos et al. [14] — which considers the volume-averaged shear and normal stress quantities computed upon all material planes. In this paper, we propose the concept of the largest rectangular prismatic hull enclosing the five-dimensional deviatoric stress history as a measure of shear stress amplitude in multiaxial high cycle fatigue. The axes of the rectangular prismatic hull represent the amplitudes of the stress history along its directions, allowing a proper characterization of non-proportional loadings and resulting

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in a criterion well suited for any periodic stress paths, as shown when confronted with experimental data. From the computational point of view, the implementation of the corresponding algorithm is simple, of very low cost, and can be easily integrated with a finite element code. The paper is organized as follows: the new measure of shear stress amplitude for multiaxial stress histories is defined in Section 2, while in Section 3 we address some numerical aspects of its calculation. In Section 4, two numerical examples illustrate the application of the algorithm described in the previous section: the first one considers a non-harmonic three-dimensional deviatoric stress path enclosed by a cubic convex hull, while the second one addresses a multiaxial stress path resulting from a rolling contact study reported by Bernasconi [15]. Section 5 states the resulting fatigue endurance criterion. The assessment of the resulting model is presented in Section 6, considering eight sets of experimental data describing situations of fatigue limit under harmonic synchronous, harmonic asynchronous and non-harmonic multiaxial loading conditions. Finally, some concluding remarks are presented in Section 7. 2. The prismatic hull as a measure of the shear stress amplitude Let r be the stress path at a given material point along a time interval ½0; T. At each time instant t, the stress tensor rðtÞ can be decomposed into the hydrostatic and the deviatoric stress tensors, which can be represented, respectively, by

rh ðtÞ ¼ rh ðtÞI and SðtÞ ¼ rðtÞ  rh ðtÞ;

ð1Þ

where rh ðtÞ :¼ is the hydrostatic stress, I is the identity tensor and tr denotes the trace operator. We consider the following orthonormal basis for the deviatoric space: 1 trrh ðtÞ 3

0

2 B N1 ¼ p1ffiffi6 @ 0 0 0 0 B N3 ¼ p1ffiffi2 @ 1 0

0 0 B C 1 0 A; N2 ¼ p1ffiffi2 @ 0 0 1 0 1 0 1 0 0 B C 0 0 A; N4 ¼ p1ffiffi @ 0 0

1

0

2

0

0

0

0

0 1

0

0

1

C 1 0 A; 0 1 1 0 0 C 0 1 A;

ð2Þ

jjsjj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s21 þ s22 þ s23 þ s24 þ s25 ;

ð7Þ

produce the same result. Fig. 1 illustrates a stress path, projected into the deviatoric space, and its corresponding convex hull C (i.e. the smallest convex set containing all the points of the stress history). The basic idea in this work is to associate the stress states belonging to the boundary @ C of the convex hull to a measure of the amplitude of the stress path. Such stress states can be identified by considering rectangular prismatic hulls enclosing the stress path. As illustrated in the same figure, the stress path is tangent to the boundary of the rectangular prismatic hull with orientation H at points ðiÞ

smax ðHÞ ¼ sðtðiÞ max ðHÞ; HÞ; ðiÞ

ðiÞ

smin ðHÞ ¼ sðtmin ðHÞ; HÞ;

  t ðiÞ si ðt; HÞ ; max ðHÞ :¼ arg max t   ðiÞ t min ðHÞ :¼ arg min si ðt; HÞ ;

ð8Þ

t

for each ith component of s, i ¼ 1; . . . ; 5. When all the orientations of the rectangular prism enclosing the stress path are considered, the points of the stress path belonging to the boundary of the convex hull C are recovered (Fig. 1). Within the setting of fatigue endurance, for a given orientation H of the prismatic hull, we define the ith component of the amplitude of the stress path as

1 ðmax si ðtÞ  min si ðtÞÞ; t 2 t

ai ðHÞ :¼

ð9Þ

and the corresponding H-amplitude as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 5 uX sa ðHÞ :¼ t a2i ðHÞ:

ð10Þ

The terms ai ; i ¼ 1; . . . ; 5 in expression (9) can be understood as a shear stress amplitudes defined in five mutually orthogonal directions, with resultant sa given by (10). We define the shear stress amplitude associated with multiaxial stress paths as

ð11Þ

H

and hence the deviatoric stress SðtÞ can be represented in component form as 5 X

and

sa :¼ max sa ðHÞ:

B C N5 ¼ p1ffiffi2 @ 0 0 0 A; 1 0 0

SðtÞ ¼

ð6Þ

i¼1

0 1 0

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjSjj ¼ S2x þ S2y þ S2z þ 2S2xy þ 2S2xz þ 2S2yz

si ðtÞNi ;

For the particular case of elliptic stress paths, Mamiya and co-workers [11,12] showed that the H-oriented shear stress amplitude, as defined in Eq. (10), is invariant with respect to the orientation H. This means that there exists an infinite number of prismatic hulls

ð3Þ

prismatic hull

i¼1

s(1)max( )

where

qffiffi

  ¼ p1ffiffi6 2rx ðtÞ  ry ðtÞ  rz ðtÞ ;     s2 ðtÞ ¼ p1ffiffi2 Sy ðtÞ  Sz ðtÞ ¼ p1ffiffi2 ry ðtÞ  rz ðtÞ ; pffiffiffi pffiffiffi s3 ðtÞ ¼ 2Sxy ðtÞ ¼ 2rxy ðtÞ; pffiffiffi pffiffiffi s4 ðtÞ ¼ 2Sxz ðtÞ ¼ 2rxz ðtÞ; pffiffiffi pffiffiffi s5 ðtÞ ¼ 2Syz ðtÞ ¼ 2ryz ðtÞ: s1 ðtÞ ¼

3 S ðtÞ 2 x

ð4Þ

s

In expressions (4), Sx ; Sy ; . . . are the components of the deviatoric tensor with respect to a Cartesian framework, while rx ; ry ; . . . are the components of the Cauchy stress tensor. In what follows, the deviatoric stress tensor S will be represented by the 5-vector

s ¼ ½ s1

s2

s3

s4

T

s5  :

Expressions for the norm of the deviatoric stress,

s(2)max( )

(1) min

convex hull ( ) s(2)min( )

s2( ) s1( )

ð5Þ Fig. 1. Stress path s, its convex hull, a H-oriented prismatic hull and the tangent points of its boundary to the path.

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enclosing the elliptic stress path, leading to the very same measure, i.e. the solution is unique. In the case of more general, non-elliptic stress paths, the orientation invariance is no longer observed but the multiplicity of enclosing hulls resulting in the same measure is not necessarily eliminated. However, it is not the orientation of the hull (eventually not unique), but simply the associated quantity (11) (with a unique global maximum) which has to be considered as a measure of shear solicitation to fatigue. 3. Computation of the shear stress amplitude Whenever the stress path is defined by up to two non-zero components, the prismatic hull is reduced to a rectangle in a two-dimensional space, its orientation can be characterized by a scalar angle h and hence the search for the maximum in Eq. (11) poses no additional difficulties. On the other hand, when more general cases are considered, then the description of the orientation H of the prismatic hull in the five-dimensional deviatoric space becomes a more complex issue. As a first approach, we could consider H as a five-dimensional quantity describing the angles around the axes of a given basis of the deviatoric space. However, in the general case, the matrices necessary to perform rotations — and hence to describe the orientation of the prismatic hull — are quite elaborate. In what follows, we consider an alternative approach, based on the so called Givens rotations (also known as Jacobi rotations), usually considered in algorithms of matrix diagonalizations (see Golub & Van Loan [16], for instance). Instead of rotating four-dimensional subspaces around each of the five axes in the deviatoric space, we perform rotations within each of the ten two-dimensional subspaces 1–2, 1–3, 1–4, 1–5, 2–3, 2–4, 2–5, 3–4, 3–5 and 4–5. Although the number of rotations increases (from 5 to 10), the expression for each Jacobi rotation is very simple. A rotation in a plane p–q is given by

8 cosðhpq Þ > > > > > sinðh > pq Þ < Q ðhpq Þij ¼  sinðhpq Þ > > > > cosðhpq Þ > > : dij

if i ¼ p;

j ¼ p;

if i ¼ p;

j ¼ q;

if i ¼ q;

j ¼ p;

if i ¼ q;

j ¼ q;

ð12Þ

otherwise:

with 0 6 hpq < 90 , p ¼ 1 : 4; q ¼ ðp þ 1Þ : 5 and dij accounting for the Kronecker delta. For instance, a rotation h25 in plane 2–5 is given by

0

1 B B0 B Q ðh25 Þ ¼ B B0 B @0

0 c25 0 0

0 s25

0

0

0

1

C s25 C C 0 C C; C 0 1 0 A 0 0 c25

0 0 1 0

ð13Þ

where c25 :¼ cosðh25 Þ and s25 :¼ sinðh25 Þ. Thus, a rotation of a vector within a plane p–q can be computed as

v pq ¼ Q ðhpq Þv :

ð14Þ

In practice, the matrix-vector multiplication in expression (14) is provided by the following scalar operations:

v pq p ¼ cosðhpq Þv p þ sinðhpq Þv q ; v pq q ¼  sinðhpq Þv p þ cosðhpq Þv q ; v pq for i–p and i–q: i ¼ vi

ð15Þ

which are very cost effective from the computational point of view. Thus, a general rotation of a 5-vector v can be defined as

v ðHÞ ¼ Q ðHÞv ;

ð16Þ

where the orthogonal operator Q ðHÞ is a cyclic composition of rotations within each of the two-dimensional planes

Q ðHÞ ¼ Q ðh45 ÞQ ðh35 Þ . . . Q ðh12 ÞQ ðh45 ÞQ ðh35 Þ . . .

ð17Þ

It should be emphasized that, although H denotes the orientation of the prismatic hull, in practice its calculation needs not to be formally performed. In fact, it is not the orientation of the prismatic that has to be quantified, but the maximum value of the quantity (10). With the aforementioned strategy to perform rotations of the basis which describes the stress path (or, equivalently, the prismatic hull), we are in condition to present the algorithm which provides the solution for Eq. (11): Algorithm. Given the discretized stress history frðt k Þ; k

1 : Ng:

1 : Ng onto the deviatoric stress space. 1. Project frðtk Þ; k 0 2. sa 3. conv FALSE 4. while conv ==FALSE conv TRUE for each plane p–q, p 1:4, q (p+1):5 0 : Dhpq : 90 for hpq (a) str Q ðhpq ÞsðHÞ 1 tr ðmaxk str (b) atr i i  mink si Þ; i ¼ 1 : 5 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q P 5 2 tr (c) str a i¼1 ðai Þ (d) if str a > sa ð1 þ eÞ then sa stra , s str conv FALSE end end s sðHÞ end end

Remarks  The projection onto the deviatoric space in step (1.) can be performed by considering any orthonormal basis (e.g. the one described by expressions (2)).  Steps (4.a)–(4.c) compute trial values, respectively, for s, ai ; i ¼ 1; . . . ; 5 and sa . In (3.d), the value of the stress amplitude is updated if its computed trial value is greater than the current estimate sa (multiplied by a perturbation factor 1 þ e, where e is a small value which can be set, for instance, equal to 105 ). The new description sðHÞ of the stress path is updated only after the scanning process for the maximum is concluded within plane p– q. Meanwhile, candidates for newly oriented stress paths are denoted s in (4.d).  As stopping criterion, the boolean variable conv is set as TRUE at the beginning of each scan over all ten two-dimensional planes. If sa is updated (step 4.d), then conv is set to FALSE and the scan over the planes is repeated. On the other hand, if sa is not updated, then conv remains TRUE and the condition checked at the command while is no longer verified, meaning that the maximum value for sa has been attained.

4. Numerical examples For situations where the stress paths are contained in a twodimensional plane, the calculation of the prismatic hull which leads to the maximum value in expression (11) is a simple task, since rotation around only one axis has to be performed. Neverthe-

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less, there are situations, as for instance those involving mechanical contact, where the stress path may be embedded in a space with dimension greater than two. For these more general situations, rotations in more than one plane are necessary so that we have to rely on the algorithm described in the previous section. In this setting, we explore the capabilities of the algorithm by studying two situations, namely: (i) a hypothetical loading path defining a cubic convex hull and (ii) a loading path obtained from a repeated rolling contact study, as reported by Bernasconi [15]. In what follows, processing times associated with computation of (11) are compared with those corresponding to calculation of the shear stress amplitude, as proposed in [13,14] by Papadopoulos,

!1=2 Z Z p Z 2p qffiffiffiffiffiffiffiffiffi 1 5 2p 2 2 hT a i :¼ ðT ðu; h; vÞ dv sin h dh du ; p 8 u¼0 h¼0 v¼0 a

Table 1 Vertices of the load path enclosed within a cube.

s1 s2 s3 s4 s5

1

2

3

4

5

6

7

8

1 1 1 0 0

1 1 1 0 0

1 1 1 0 0

1 1 1 0 0

1 1 1 0 0

1 1 1 0 0

1 1 1 0 0

1 1 1 0 0

Table 2 Shear stress amplitudes and processing times, as a function of discretization in orientation increments, as proposed by Papadopoulos et al. [14] and in this paper, for the cubic loading path. Papadopoulos

ð18Þ which is the volumetric average of the resolved shear stress amplitude T a over all material planes and gliding directions (parameterized by ðu; hÞ and v, respectively). The procedures for the computation of both measures of shear stress amplitude were implemented in the C++ language and compiled with the Visual C++ 2008 Express Edition [17]. The results were obtained with an IntelÒCoreTM 2 Quad processor running at 2.4 GHz. 4.1. Cubic stress path The first numerical example considers a hypothetical loading path which defines a cubic convex hull in the deviatoric space, as illustrated in Fig. 2. The vertices of this path are listed in Table 1. Table 2 lists shear stress amplitudes and processing times for the shear measures following the integral approach, proposed by Papadopoulos, and the prismatic hull measure, proposed in this paper. In both cases, angle increments of 1 , 10 , 20 and 30 were considered. It is worth observing that, in both approaches, even very coarse angle discretizations produce very good results. Indeed, for angle increments of 30 , the computed shear stress amplitudes differ in only 1.6% (for the prismatic hull measure) and in only 2.5% (for the mesoscopic approach), when compared to those obtained under finer angle increments of 1 . In favor of the coarsest discretizations comes also the processing times: in the prismatic hull approach, consideration of Dh ¼ 30 leads to a processing time about 30 times as fast as the elapsed time for Dh ¼ 1 . For the integral approach, the time corre-

Prismatic hull

Dh; Du; Dv ð Þ

Elapsed time (s)

pffiffiffiqffiffiffiffiffiffiffiffiffi 2 hT 2a i (MPa)

Dh ð Þ

Elapsed time (s)

(MPa)

1, 1, 1 10, 10, 10 20, 20, 20 30, 30, 30

9.0 1:6  102 3:0  103 1:2  103

2.611 2.606 2.595 2.545

1 10 20 30

4:2  104 4:3  105 2:4  105 1:4  105

2.887 2.881 2.881 2.841

sa

sponding to ðDu; Dh; DvÞ ¼ ð30 ; 30 ; 30 Þ results in a processing time 7500 times faster than the one associated with the finest discretization. When the coarsest discretization is considered in both cases, the prismatic hull approach is about 85 times faster than the integral one. This can be an advantage when considering fatigue analyses within the setting of finite elements, when several thousands of nodes are involved. 4.2. Rolling contact In this example, we consider a stress path obtained from a repeated rolling contact study, as reported by Bernasconi in [15]. Fig. 3 describes the components of the stress path, while Fig. 4 represents its projections on each of the ten two-dimensional planes 1–2, 1–3, . . . , 4–5 of the deviatoric space. (Notice that Fig. 4 and the one reported in [15] present differences due to the consideration of distinct bases for the deviatoric space). The loading history was represented in this example by stress states described along 60 time instants. Table 3 lists the results obtained by considering, in the case of the prismatic hull model, four discretizations of the rotation angle in each of the two-dimensional planes: Dhpq ¼ 1 , 10 , 20 and 30 . The same table lists the results corresponding to four discretiza-

100

Stress components (MPa)

50 1

s3

0

0

-50 -100 -150

xx yy

-200

zz xy

-1 -1

-1

-250

xz yz

s2

0

0 1

s1

1

Fig. 2. Loading path, described in the deviatoric space, enclosed in a cube.

-300 0

10

20

30 40 Loading steps

50

Fig. 3. Stress history for the example of rolling contact [15].

60

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E.N. Mamiya et al. / International Journal of Fatigue 31 (2009) 1144–1153

s1 x s2

200

s1 x s3

200

0

0

-200 200

s2 x s4

200

0

-200 200

0

-200 -200 200

0

s3 x s5

200

0

-200 -200 200

0

s3 x s4

200

0

0

0

-200 -200 200

0

s4 x s5

0

-200 -200 200

0

-200

0

200

0

-200 -200 200

s2 x s3

200

0

-200 -200 200

s2 x s5

200

s1 x s5

200

0

-200 -200 200

0

s1 x s4

200

-200 -200 200

0

-200

0

Fig. 4. Stress paths within each of the two-dimensional planes 1–2, 1–3, . . . ,4–5 for the example of rolling contact [15].

From the phenomenological point of view, it is important to remark that the above formulation, when applied to uniaxial solicitation, correctly captures the well known experimental observations: the fatigue limit in torsion is independent of a mean shear stress

600

400

200 xy

tions of the integration angles u, h and v when the integral model is considered. Here again little influence of the discretization on the quality of the result is observed: when considering the prismatic hull model, comparison of results with angle discretizations Dh ¼ 30 and Dh ¼ 1 produced a difference of only 0:18%, while the coarsest and the finest angle discretizations in the integral approach resulted in a difference of the calculated values of the shear stress of only 2%. The prismatic hull approach computed the shear stress amplitude from expression (11) 29 times faster for Dh ¼ 30 when compared with the finest angle discretization Dh ¼ 1 , while computation of the shear stress (18) from the integral approach with ðDu; Dh; DvÞ ¼ ð30 ; 30 ; 30 Þ was 3970 times faster than for ðDu; Dh; DvÞ ¼ ð1 ; 1 ; 1 Þ. In this example, when the coarsest angle discretizations were considered, the prismatic hull approach was 104 times faster than the integral one.

0

5. Multiaxial fatigue criterion -200

The shear stress amplitude sa is now combined with the maximum hydrostatic stress rh max in order to define a multiaxial fatigue endurance criterion as follows

1 pffiffiffi sa þ jrh max 6 k; 2

-200

0

200

Fig. 5. Stress path for data produced by Froustey and Lasserre [20]: 30CND16 steel, f1 ¼ 660 MPa; t1 ¼ 410 MPa; rxa ¼ 470 MPa; rxm ¼ 300 MPa; rxya ¼ 270 MPa; rxym ¼ 0 MPa; b ¼ 60 .

Table 4 Synchronous harmonic bending-torsion. Material: f1 ¼ 313:9 MPa; t1 ¼ 196:2 MPa, Nishihara and Kawamoto [18]. Table 3 Shear stress amplitudes and processing times, as a function of discretization in orientation increments, as proposed by Papadopoulos et al. [14] and in this paper, for the example of rolling contact. Papadopoulos

Prismatic Hull

Dh; Du; Dv ð Þ

Elapsed time (s)

pffiffiffiqffiffiffiffiffiffiffiffiffi 2 hT 2a i (MPa)

Dh ð Þ

Elapsed time (s)

sa

1,1,1 10,10,10 20,20,20 30,30,30

2:7  101 7:9  102 1:6  102 6:8  103

168.6 168.2 167.2 165.2

1 10 20 30

1:9  103 1:9  104 1:1  104 6:5  105

170.7 170.4 170.6 170.4

(MPa)

800

x

ð19Þ

where j and k are material parameters which have to be identified with two independent fatigue limit data. This inequality defines a safe region in the sa  rh max space: infinite life (i.e. life > 107 cycles) is expected whenever the loading history satisfies such inequality.

600

400

hard

steel,

rxa

rxm

rxya

rxym

(MPa)

(MPa)

(MPa)

(MPa)

b ð Þ

IDV (%)

IP (%)

IPH Dh¼1 (%)

IPH Dh¼30 (%)

138.1 140.4 145.7 150.2 245.3 249.7 252.4 258.0 299.1 304.5

0 0 0 0 0 0 0 0 0 0

167.1 169.9 176.3 181.7 122.6 124.8 126.2 129.0 62.8 63.9

0 0 0 0 0 0 0 0 0 0

0 30 60 90 0 30 60 90 0 90

1.0 0.3 2.4 6.8 4.0 2.4 6.8 17.8 1.7 3.0

2.3 0.6 3.1 6.3 1.4 3.3 4.4 6.7 0.9 2.7

2.3 0.6 3.1 6.3 1.4 3.3 4.4 6.7 0.9 2.7

2.3 0.6 3.1 6.3 1.4 3.3 4.4 6.7 0.9 2.7

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E.N. Mamiya et al. / International Journal of Fatigue 31 (2009) 1144–1153 Table 5 Synchronous harmonic tension–torsion. f1 ¼ 398 MPa; t 1 ¼ 265 MPa, Lempp [19].

Material:

42CrMo4

steel,

Table 7 Synchronous harmonic tension–torsion. f1 ¼ 410 MPa; t1 ¼ 256 MPa, Zenner et al. [21].

Material:

34Cr4

steel,

rxa

rxm

rxya

rxym

IPH Dh¼1 (%)

IPH Dh¼30 (%)

rxym

(MPa)

IP (%)

rxya

(MPa)

IDV (%)

rxm

(MPa)

b ð Þ

rxa

(MPa)

(MPa)

(MPa)

(MPa)

(MPa)

b ð Þ

IDV (%)

IP (%)

IPH Dh¼1 (%)

IPH Dh¼30 (%)

328 286 233 213 266 283 333 280 271

0 0 0 0 0 0 0 280 271

157 137 224 205 128 136 160 134 130

0 0 0 0 128 136 160 0 0

0 90 0 90 0 90 180 0 0

6.2 28.1 9.9 21.1 13.7 28.9 8.0 8.2 4.8

3.8 9.5 6.4 2.7 15.7 10.3 5.5 2.1 5.1

3.8 9.5 6.4 2.7 15.7 10.3 5.5 2.1 5.1

3.8 9.5 6.4 2.7 15.7 10.3 5.5 2.1 5.1

314 315 316 315 224 380 316 314 315 279 284 355 212 129

0 0 0 0 0 0 0 0 0 279 284 0 212 0

157 158 158 158 224 95 158 157 158 140 142 89 212 258

0 0 0 0 0 0 158 157 158 0 0 178 0 0

0 60 90 120 90 90 0 60 90 0 90 0 90 90

2.0 10.8 22.9 10.8 11.6 7.3 2.7 11.2 23.2 4.3 16.9 5.2 6.0 1.0

0.6 0.1 0.1 0.1 5.2 0.4 0.1 0.6 0.1 6.4 4.8 6.2 3.4 7.3

0.6 0.1 0.1 0.1 5.2 0.4 0.1 0.6 0.1 6.4 4.8 6.2 3.4 7.3

0.6 0.1 0.1 0.1 5.2 0.4 0.1 0.6 0.1 6.4 4.8 6.2 3.4 7.3

and the endurance limit is influenced by the presence of a static normal stress. 6. Comparison with experimental data A wide variety of experimental data available in the literature [18–24] describing multiaxial high cycle fatigue endurance were considered for the assessment of the fatigue criterion based on the proposed measure of the shear stress amplitude. The analysis was performed separately for synchronous harmonic (Section 6.1) and for more general — asynchronous harmonic and non-harmonic — (Section 6.2) load patterns. The first set of data produces affine or elliptical stress paths, while the second one produces other patterns like parabolic, rectangular, eight shaped, cross shaped and so on. The collectedpdata correspond to experiments on hard metals, ffiffiffi 1:3 6 f1 =t1 6 3, as defined by Papadopoulos et al. [14], where f1 and t 1 account for fatigue limits under fully reversed normal and shear stresses, respectively. The quality of the results provided by the model based on the prismatic hull is assessed in terms of an error index I, defined as:

I

PH

  1 1 pffiffiffi sa þ jrh max  k  100ð%Þ: :¼ k 2

ð20Þ

This index provides a measure of how close the estimation of the criterion is with respect to the experimental data. A negative I yields a non-conservative fatigue strength prediction — since it indicates that the stress solicitation has not attained a critical value — while the experimental data represent limiting situations. On the other hand, a positive I provides a conservative estimate while I = 0 means an exact prediction for the observed fatigue strength. Parameters j and k were identified by calibration of the criterion (19) so as to coincide with the fatigue limits obtained from fully reversed tension-compression and torsion experiments (with fatigue limits f1 and t 1 , respectively). This leads to expressions

Table 6 Synchronous harmonic bending–torsion. Material: f1 ¼ 660 MPa; t1 ¼ 410 MPa, Froustey and Lasserre [20]. DV

P

30CND16

steel,

IPH Dh¼1

IPH Dh¼30

rxa

rxm

rxya

rxym

(MPa)

(MPa)

(MPa)

(MPa)

b ð Þ

I (%)

I (%)

(%)

(%)

485 480 480 480 470 473 590 565 540 211

0 0 300 300 300 300 300 300 300 300

280 277 277 277 270 273 148 141 135 365

0 0 0 0 0 0 0 0 0 0

0 90 0 45 60 90 0 45 90 0

4.7 25.8 12.5 4.6 3.5 18.0 6.8 0.9 9.3 7.8

1.8 0.7 3.9 3.9 1.6 2.4 0.1 4.1 8.1 0.7

1.8 0.7 3.9 3.9 1.6 2.4 0.1 4.1 8.1 0.7

1.8 0.7 3.9 3.9 1.6 2.4 0.1 4.1 8.1 0.7

j¼3

t 1 pffiffiffi  3; f1

k ¼ t 1 :

ð21Þ

For the sake of comparison, Dang Van [2,3] and Papadopoulos et al. [14] criteria were also assessed against the experimental data. In this context, the quality of the prediction of fatigue endurance from Dang Van criterion was performed by considering the error index

IDV :¼

1 max ½rT ðSðtÞ  AÞ þ jrh ðtÞ  k  100%; k t

ð22Þ

where rT is the Tresca stress corresponding to the stress state SðtÞ  A, SðtÞ is the deviatoric stress at time instant t, A is the center of the minimum hypersphere enclosing the stress path in the deviatoric space, while j and k are material parameters which once again can be identified from fully reversed tension-compression and torsion experiments, leading to



j¼3

 t1 1  ; f1 2

k ¼ t 1 :

ð23Þ

Errors in estimation of endurance condition provided by the Papadopoulos criterion were quantified by the index

IP :¼

1 k

qffiffiffiffiffiffiffiffiffi  hT 2a i þ jrh max  k  100%;

ð24Þ

qffiffiffiffiffiffiffiffiffi where hT 2a i is the volumetric average of the resolved shear stress amplitude T a over all material planes and gliding directions (parameterized by ðu; hÞ and v, respectively), as defined by Papadopoulos

Table 8 Asynchronous harmonic tension–torsion. Material: f1 ¼ 415 MPa; t1 ¼ 259 MPa, Heidenreich et al. [22].

34Cr4

steel,

rxa

rxm

rxya

rxym

(MPa)

(MPa)

(MPa)

b ð Þ

g

(MPa)

IDV (%)

IP (%)

IPH Dh¼1 (%)

IPH Dh¼30 (%)

263

0

132

0

0

4

18.8

0.6

10.2

6.6

Table 9 Asynchronous harmonic tension–torsion. f1 ¼ 340 MPa; t1 ¼ 228 MPa, Kaniut [23].

Material:

25CrMo4

steel,

rxa

rxm

rxya

rxym

(MPa)

(MPa)

(MPa)

b ð Þ

g

(MPa)

IDV (%)

IP (%)

IPH Dh¼1 (%)

IPH Dh¼30 (%)

210 220 242 196

0 0 0 0

105 110 121 98

0 0 0 0

0 0 90 0

0.25 2 2 8

21.5 26.3 18.1 25.4

4.2 6.3 4.9 8.9

4.5 0.03 2.0 0.3

1.3 2.5 3.6 3.4

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E.N. Mamiya et al. / International Journal of Fatigue 31 (2009) 1144–1153

200

test 1:

= 0.25,

200

= 0°

xy

0

= 0°

0 -100

-100 -200 -300

-200

-100

0

100

200

-200 -300

300

-200

-100

0

x 200

test 3:

100

200

300

x

= 2,

200

= 90°

test 4:

= 8,

= 0°

100 xy

100 xy

= 2,

100

100 xy

test 2:

0 -100

0 -100

-200 -300

-200

-100

0

100

200

300

-200 -300

-200

-100

0

100

200

300

x

x

Fig. 6. Stress paths for data produced by Kaniut [23]: 25CrMo4, f1 ¼ 340MPa; t1 ¼ 228 MPa; g ¼ 0:25; 2 and 8; b ¼ 90 in case 3.

et al. [14] and expressed by (18). For this criterion, parameters and k can be described in terms of limits f1 and t 1 as

j¼3

t 1 pffiffiffi  3; f1

k ¼ t 1 :

j

ð25Þ

which happens to coincide with expressions (23). Further details of this criterion can be found in [14]. 6.1. Synchronous harmonic solicitation The first set of data considers harmonic loading histories with the same frequency of solicitation imposed on both stress components. Stress histories associated with these data can be described in terms of expressions:

rx ðtÞ ¼ rxm þ rxa sinðxtÞ; rxy ðtÞ ¼ rxym þ rxya sinðxt  bÞ;

where rx and rxy are normal and shear stresses, subscripts a and m stand, respectively, for the amplitude and the mean value, while b is the phase angle. Fig. 5 depicts a typical non-proportional stress history with normal mean stress. Tables 4–7 describe data and corresponding error indexes for tests under in-phase (proportional) and out-of-phase (non-proportional) sinusoidal loadings, with and without mean (shear or normal) stress, reported, respectively, by Nishihara & Kawamoto [18], Lempp [19], Froustey and Lasserre [20] and Zenner et al. [21]. Two levels of discretization of the rotation increments Dhpq in each two-dimensional plane were considered: Dhpq ¼ 1 and Dhpq ¼ 30 . Examination of results show that the error index associated with the prismatic hull criterion varied between 15:7% and 7:3%, being exactly the same for both discretizations. As a matter of fact, for all cases in Tables 4–7, no strategy to search

ð26Þ 300 x

34Cr4

rxm

rxya

rxym

(MPa)

(MPa)

(MPa)

(MPa)

b ð Þ

IDV (%)

IP (%)

IPH Dh¼1 (%)

IPH Dh¼30 (%)

240

0

120

0

90

22.4

5.1

4.2

0.8

100 0

x

,

xy

rxa

xy

200

steel,

(MPa)

Table 10 Trapezoidal out-of-phase tension–torsion. Material: f1 ¼ 415 MPa; t 1 ¼ 259 MPa, Heidenreich et al. [22].

-100 Table 11 Piecewise proportional tension–torsion. f1 ¼ 340 MPa;t1 ¼ 228 MPa, Mielke [24].

Material:

25CrMo4

steel,

-200 DV

P

IPH Dh¼1

IPH Dh¼30

rxa

rxm

rxya

rxym

(MPa)

(MPa)

(MPa)

(MPa)

b ð Þ

I (%)

I (%)

(%)

(%)

250 250 296/206 179

0 0 0 0

145 145 102 165/116

0 0 0 0

0 0 0 0

26.5 26.5 13.9 22.6

8.9 8.9 2.4 8.9

0.04 0.04 4.6 1.9

0.04 0.04 3.8 2.6

-300 0

1

2

3

4

5

6

t Fig. 7. Out-of-phase trapezoidal axial-torsional loading corresponding to tests produced by Heidenreich et al. [22] (Table 10).

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E.N. Mamiya et al. / International Journal of Fatigue 31 (2009) 1144–1153

the maximum in expression (11) is actually necessary since, for elliptic paths, the quantity defined by expression (10) is invariant with respect to the orientation of the prismatic hull, as was shown by Mamiya and Araújo [11]. Notice that, for synchronous harmonic loading conditions, the results provided by the Papadopoulos criterion are exactly the same as the one obtained by considering the prismatic hull. Dang Van criterion produced estimation errors varying between 28:9% and 12:5% for the same set of data.

200 100 0

xy

(MPa)

Next, we consider more intricate load patterns, namely asynchronous harmonic and non-harmonic time-varying solicitations. Tables 8 and 9 describe data and error indexes for asynchronous harmonic loadings, carried out, respectively, by Heidenreich et al. [22] on 34Cr4 steel and by Kaniut [23] on 25CrMo4 steel. Fig. 6 illustrates the stress paths considered in experiments provided by Kaniut. Expressions for stress histories are now given by

rx ðtÞ ¼ rxm þ rxa sinðxtÞ; rxy ðtÞ ¼ rxym þ rxya sinðgxt  bÞ;

300

-100 -200 -300 -300

6.2. Asynchronous harmonic and non-harmonic solicitation

-200

-100

0 x

100

200

300

(MPa)

Fig. 8. Rectangular stress path corresponding to the trapezoidal loading path in Fig. 7.

where g accounts for the frequency ratio between normal and shear components, which varied between 0:25 and 8. Since these stress paths are not elliptical, the invariance of the quantity sa ðHÞ defined in (10) is no longer valid and hence the maximum of sa ðHÞ among all orientations H has to be searched, as stated in (11). As long as these stress paths define only two non-zero stress components s1 and s3 in the deviatoric space, rotations hpq in the algorithm above had to be performed only within the 1–3 plane. Results from the prismatic hull criterion produced errors between 2% and 10:2% when considering Dhpq ¼ 1 and between 3:6% and 6:6% when setting Dhpq ¼ 30 . Dang Van criterion produced estimation errors between 26:3% and 18:1%, while for the Papadopoulos criterion the errors were bounded between 8:9% and 0:6%. To further investigate the applicability of the proposed model against complex load conditions, data considering non-harmonic

300

x

xy

200

200

100

100 xy

xy

300

0

-100

-200

-200

-300

-300 0

100

200

xy

x

0

-100

-300 -200 -100

300

-300 -200 -100

x

300

ð27Þ

0

100

200

300

100

200

300

x

x

300

xy

x

200

200

100

100 xy

xy

xy

0

0

-100

-100

-200

-200

-300

-300 -300 -200 -100

0 x

100

200

300

-300 -200 -100

0 x

Fig. 9. Piecewise proportional stress paths for data (Table 11) produced by Mielke [24]: 25CrMo4, f1 ¼ 340 MPa; t 1 ¼ 228 MPa.

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E.N. Mamiya et al. / International Journal of Fatigue 31 (2009) 1144–1153

50 40

50 Prismatic hull -

40 30 20

20 15%

10 0 -10

-15%

-20

15%

10 0 -10 -15%

-20

-30

-30

-40

-40 -50

-50 4

5

6

7 8 Table

9

10

4

11

Fig. 10. Estimation errors for the prismatic hull criterion. Results produced with discretization Dhpq ¼ 1 .

50 Prismatic hull -

elliptical stress paths non-elliptical stress paths

= 30°

30

Error index (%)

Error index (%)

Error index (%)

30

40

elliptical stress paths non-elliptical stress paths

Papadopoulos

elliptical stress paths non-elliptical stress paths

= 1°

20

15%

10 0 -10

-15%

5

6

7 8 Table

9

10

11

Fig. 13. Estimation errors for the Papadopoulos criterion.

illustrated in Fig. 7, which leads to the rectangular stress path in Fig. 8. The specimens made of 25CrMo4 steel were submitted to four piecewise proportional loading programs, as illustrated in Fig. 9. Estimation errors from the prismatic hull criterion produced errors between between 1:9% and 4:6% for Dhpq ¼ 1 and between 2:6% and 3:8% for Dhpq ¼ 30 . Dang Van criterion produced errors between 26:5% and 13:9%, while the errors associated with the Papadopoulos criterion were bounded between 8:9% and 2:4%. 6.3. Discussion

-20 -30 -40 -50 4

5

6

7 8 Table

9

10

11

Fig. 11. Estimation errors for the prismatic hull criterion. Results produced with discretization Dhpq ¼ 30 .

50 40

elliptical stress paths non-elliptical stress paths

Dang Van

Error index (%)

30 20 15%

10 0 -10 -15%

-20 -30 -40

Figs. 10–13 show the error indexes, as defined in expressions (20, 22 and 24), for the complete set of 53 experimental data listed in Tables 4–11. It can be noticed that the criterion based on the prismatic hull produces consistently good predictions, within the interval ½15:7; 10:2 ð%Þ, performing equally well for elliptic as well as for non-elliptical stress paths. The Papadopoulos criterion also generates good results, with errors within the same interval. However, for non-elliptical stress paths, it exhibits a trend to produce non-conservative estimations when compared to the prismatic hull criterion, but still within a 15% error band. The Dang Van criterion, on the other hand, leads to differences between estimation and experimental data within the interval ½28:9; 12:5 ð%Þ. The same criterion produces non-conservative predictions with errors above 15% for nine of the considered non-elliptical stress paths. It is worth observing that, in the searching process of the maximum prismatic hull, although the use of a coarser scanning (Dhpq ¼ 30 ) demands about one hundredth of the time required for a finer degree-by-degree scan, the corresponding results do not decrease significantly in quality. Indeed, among all the results, the error index differed in only up to 3:6%, as can be observed in Tables 4–11 and is illustrated in Figs. 10 and 11. 7. Conclusions

-50 4

5

6

7 8 Table

9

10

11

Fig. 12. Estimation errors for the Dang Van criterion.

combination of shear and normal stress histories were selected from works published by Heidenreich et al. [22] on 34Cr4 steel (Table 10) and by Mielke [24] on 25CrMo4 steel (Table 11). The specimens made of 34Cr4 steel were subjected to the loading program

A new measure for the shear stress amplitude was proposed within the setting of multiaxial fatigue endurance. The basic idea was to consider the maximum rectangular prismatic hull enclosing the stress path to quantify the fatigue damage produced by a multiaxial loading history. The issue of computing the prismatic hull in the five-dimensional deviatoric space was addressed. A simple algorithm, based on Givens rotations (classical in matrix computations) performed in up to ten two-dimensional planes was proposed.

E.N. Mamiya et al. / International Journal of Fatigue 31 (2009) 1144–1153

A linear combination of the proposed shear stress amplitude with the maximum hydrostatic stress was considered in order to establish an endurance criterion for multiaxial loading conditions. The resulting criterion performed well when assessed against a wide variety of experimental data, including affine, elliptic (with and without mean stresses) as well as non-elliptical stress paths. Numerical examples showed that a coarse discretization of the rotation leads to accurate results and, at the same time, are very cost effective in terms of computation time. Thus, the fatigue endurance criterion based on the concept of prismatic hull is well suited to be integrated in post-processing procedures of finite element analysis. For the sake of illustration, if we consider a finite element mesh composed of one hundred thousand nodes, a loading history discretized in 60 time steps and rotation increments Dh ¼ 30 , then the computation of the fatigue criterion over all the nodes would take only 6.5 s (see Table 3). Acknowledgements This project was supported by CNPq under Contracts 308916/ 2006-9 and 303279/2007-9. The third author was supported by CAPES under the Contract PRODOC 53001010053PO. These supports are gratefully acknowledged.

[7]

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[15]

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[19]

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