An equivalent stress process for fatigue life estimation under multiaxial loadings based on a new non linear damage model

Materials Science and Engineering A 538 (2012) 20–27

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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

An equivalent stress process for fatigue life estimation under multiaxial loadings based on a new non linear damage model A. Aid a,b,∗ , M. Bendouba a , L. Aminallah a , A. Amrouche c , N. Benseddiq b , M. Benguediab d a

Laboratoire LPQ3M, BP305, Université de Mascara, Algeria Laboratoire de Mécanique de Lille, Université Lille1 Nord de France, Cité scientifique, 59655 Villeneuve d’Ascq, France Laboratoire de Génie Civil et géo-Environnement LGCgE, EA 4515, Faculté des Sciences Appliquées FSA Béthune, Université d’Artois, France d Laboratoire LMPM, Département de Mécanique, Université de Sidi Bel Abbes, BP89, Cité Ben M’hidi, Sidi Bel Abbes 22000, Algeria b c

a r t i c l e

i n f o

Article history: Received 4 October 2011 Received in revised form 23 December 2011 Accepted 24 December 2011 Available online 20 January 2012 Keywords: Fatigue life Multiaxial loading Damaged Stress Model Random loading

a b s t r a c t Fatigue life estimation or fatigue damage evaluation of mechanical components under a multiaxial state of stress time-history has an important role in virtual design phases, but this evaluation is a real problem to resolve, because this goal is not reached by classical fatigue criteria. In this context, the equivalent uniaxial stress has been chosen as the approach useful for this purpose by combining it with a strength curve for the material (i.e. S–N curve) and damage evaluation (Damaged Stress Model (DMS) and Miner’s rule). So the authors have taken equivalent stresses based on a multiaxial criterion that can be used as a uniaxial stress time history for the application of damage evaluation methods to predict the fatigue life, under multiaxial random loading. The cycles were counted with the Rain Flow algorithm, using equivalent stress as a variable counting. In the case of “asymmetric” histories, the algorithm of transformation of the stress history (because of its global expected value different from zero) is required. In the case of 10HNAP steel, stress amplitude transformation according to the Goodman relationship gives good results. The proposed algorithm was verified during fatigue tests of cruciform specimens made of 10HNAP steel, subjected to biaxial non-proportional random tension-compression. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Most engineering components and structures in service are subjected to a two or three-dimensional stress state with complex loading. The origin of multiaxiality comes from various factors such as multiaxial external loading, complex geometry of the component and residual stresses. Such multiaxial stresses are often non-proportional, that is, the corresponding principal directions and/or principal stress ratios vary with time. The multiaxial fatigue criteria proposed in the literature may be categorized in three groups: stress based methods, strainbased methods, and energy-based approaches [1–3]. For high-cycle fatigue, most of the fatigue criteria are stress-based methods. Although there are numerous multiaxial fatigue criteria in the literature, design engineers are often faced with difficulties in applying these criteria to modern engineering design.

∗ Corresponding author at: Laboratoire LPQ3M, BP305, Université de Mascara, Algeria. Fax: +213 45804169. E-mail address: aid [email protected] (A. Aid). 0921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.12.105

One difficulty is that most of the existing multiaxial fatigue criteria involve in their implementation for general complex multiaxial fatigue loadings. Among current multiaxial fatigue criteria, the Sines [4] and the Crossland [5] criteria are very popular and easy to use for engineering design. They can provide good predictions for proportional loads with mean stress effects [6]. The real loadings are often more complex. Estimation of fatigue life of a material subjected to random loading needs suitable calculation algorithms. In such algorithms, cycle counting [7,8] according to a given method (for example, the Rain Flow method) and damage accumulation according to the assumed hypothesis (for example Palmgren–Miner, Haigh’s diagram) are the main operations. In this paper, our main aim is to extend our Damaged Stress Model (DSM) developed initially for the uniaxial loading, at the biaxial random loading on cruciform specimens. The equivalent stress of von Mises, Sines and Crossland is applied and tested as variables counting in Rain Flow approach. Several biaxial random amplitude fatigue tests that were carried out at the Technical University of Opole allow the validation of the proposed approach with both damage laws. The predictions by the approach adopted in this paper for the prediction of life in multiaxial fatigue and the following models for multiaxial loading (see Smith–Watson–Topper damage parameter used by Bannantine–Socie [9], Fatemi–Socie

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27

21

 orientating each material plane P→n at the point M on the surface of the specimen. Fig. 1. Coordinates system used to define the unit normal vector n

[9,10], Socie for HCF [11], Wang–Brown [12,13] and Lagoda–Macha [14–19]) are analyzed and predictions are compared with experiments. The main advantage of our model is the ease of use which request only the Wöhler curve (S–N). This model describes the nonlinear damage and takes into account the loading time history. 2. Linear and non-linear damage laws It should be noted that here two damage and accumulation rules have been used. The first one is the linear Miner’s rule [20], while the second is the non-linear Damaged Stress Model (DSM), which has been developed in our laboratory. For more details, see Refs. [21–25].

The equivalent stress of damage at the level (i + 1) is calculated with the relation Di =

(i)d − i u − i

=

(i+1)d − i+1 u − i+1

(4)

where  (i+1)d , equivalent stress of damage at the level (i + 1) and  i+1 applied stress at the level (i + 1). So  ed is equal to  i to the first cycle, it means D = 0 and  ed is equal at  u at the last cycle D = 1. For the application of DSM model in multiaxial fatigue, we need an equivalent stress that takes into account the state of multiaxial loading. We propose the following criteria: von Mises, Sines and Crossland. The equivalent stress is used as a “counting variable” to extract the random cycles, the calculation of the elementary damage and the accumulated damage.

2.1. Linear Miner’s rule 3. Critical plane approach and definition The damage Di induced by a cycle extracted from the multiaxial sequence is obtained from its fatigue life Ni as: Di =

1 Ni

(1)

The damage accumulation is defined by the sum of the damage Di of all extracted cycles. The fatigue life N of the whole sequence corresponds to the number of repetitions of this stress history up to crack initiation and is obtained by: D=



Di

(2)

The rupture occurs when D = 1. 2.2. Non linear Damaged Stress Model The Damaged Stress Model (DMS) [21] which is developed further is regarded as having many advantages such as taking into account the occurrences order of loading, considering cycles below and over the so-called fatigue limit in different manners, and presenting a non-linear damage accumulation. This damage indicator is easily connected cycle by cycle to the Wöhler’s curve. The expression of the DSM indicator gives the increase of damage Di due to  i applied stress cycles defined by their amplitude  a and their mean value  m , as follows: Di =

(i)d − i u − i

(3)

where  (i)d , stress of damage;  i , applied stress; and  u , ultimate stress.

As shown in Fig. 1, consider a material plane, denoted as Pn , passing through the point under consideration. The plane Pn is located  . This unit vector in turn is described by by its unit normal vector n its spherical angles (, ). All the fatigue life calculation methods using a cycle counting algorithm can be summarized with the same methodology. First a cycle counting variable is chosen in order to extract the cycles (with their amplitude and mean value) from the variable amplitude or random multiaxial loading signal (stresses and/or strains). For the simulations presented hereafter the ASTM Rain Flow algorithm was used [7,8] (with 64 classes). All the random loadings are considered as stationary for life calculation. Secondly, a damage parameter Dp is chosen; it depends on stress–strain quantities. This damage parameter is computed on each material plane  (, ) (Fig. 1), in order to Pn , orientated by the unit normal vector n determine the critical plane. Finally, to quantify the damage generated by each cycle identified with the counting algorithm it is necessary to use an equation relating the damage parameter and the number of cycles to failure Nf . According to the S–N curve of the tested material, it corresponds to stress amplitude smaller to 0.25 af in fully reversed tension [18]. 3.1. Fatemi–Socie model [FS] The Fatemi–Socie model [FS] [10] is widely applied for shear damage model, which predicts the critical plane is the plane orientation  with the maximum F–S damage parameter:



 2



1+k

nmax y



(5) max

22

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27

3.3. Socie’s proposal for HCF regime [So] In the case of high cycle fatigue and ductile materials most of the fatigue life is consumed by crack nucleation on the planes where the shear stress is maximum. In this case Socie [11] proposes the following stress based approach by using the linear combination of the shear stress amplitude  a and the maximum normal stress  nmax acting on the critical plane, both during the load cycle (Fig. 2) as damage parameter: DP-So =  a + k2  nmax . The cycle counting algorithm has to be applied on two variables: the shear stresses nx (t) and ny (t). The equation linking the damage parameter and the number of cycles to failure is given by the following relation: DP-So = f (2Nf )b

Fig. 2. Definition of the mean and maximum normal stress during an extracted cycle (from t1 to t2 ) with the Rain Flow algorithm.

where /2 is the maximum shear strain amplitude on a plane ,  nmax is the maximum normal stress on that plane and  y is the material yield strength. The critical plane is, for each of these two counting variables, the plane experiencing the highest shear strain range. For the materials where the fatigue crack initiation is dominated by plastic shear strains, [FS] recommend to use the following damage parameter: DP-FS =  a (1 + k nmax / y ) related to the Manson–Coffin curve in torsion by equation: DP-FS =

f G

(2Nf )b + f (2Nf )c

(6)

For each loading cycle extracted by the Rain Flow method,  a is the shear strain amplitude,  nmax is the maximum of the normal stress on the critical plane (Fig. 2). In this damage parameter, k is a material constant, which can be found by fitting fatigue data from simple uniaxial tests to fatigue data from simple torsion tests, k = 1.0 (for 42CrMo4 steel) and k = 0.4 (for CK45 steel) [26,27]. Eq. (6) is the description of the strain-life Manson–Coffin curve in torsion. When the strain-life Manson–Coffin torsion curve is not known, [FS] propose to approximate this curve from the tensile strain-life curve. The algorithm used to apply this fatigue life calculation method is detailed in [18]. 3.2. Bannantine and Socie model [Ba] The Smith–Watson–Topper (SWT) parameter proposed by Smith et al. (1970) is a direct development of the strain-life equation which seeks to address the effect of mean stress on fatigue life. The SWT parameter is based on the combination of the Coffin–Manson low cycle fatigue equation, Basquin’s high cycle fatigue equation and consideration of the peak stress to account for the mean stress effect, as follows: DP-SWT,BS = εn,a nmax =

  2f E

(2Nf )2b + f εf (2Nf )b+c

(7)

where f is the fatigue strength coefficient, E is Young’s modulus, Nf is the number of cycles to crack initiation, b is the fatigue strength exponent, εf is the fatigue ductility coefficient and c is the fatigue ductility exponent. Bannantine and Socie [9] recommend using the (SWT) damage parameter DP-SWT = εn,a  nmax . In Eq. (7) and for each load cycles extracted by the Rain Flow method, εn,a is the amplitude of the normal strain and  nmax is the maximum normal stress during the current cycle of the counting variable εn (t) (Fig. 2).

(8)

where f (2Nf )b is the elastic part of the strain-life curve. k2 is a material parameter identified by fitting tension and torsion fatigue data. The algorithm used to apply this method is illustrated [19].

3.4. Wang and Brown’s model [W–B] Wang and Brown put up their criterion first in 1993 in [12,13] restricted to low (LCF) and medium (MCF) cycle fatigue according to the assumption that fatigue crack growth is controlled by the maximum shear strain. Brown and Wang introduced their specific way of load history decomposition in [28]. The equivalent strain is a combination of the shear strain amplitude and efficient normal strain range: a + Sεn = (1 + e + S(1 − e ))

f − 2nmean E

+S(1 − p ))εf (2Nf )c

(2Nf )b + (1 + p (9)

where e and p are respectively the elastic and plastic Poisson’s ratio of the material.  nmean is the mean normal stress on the critical plane during each extracted cycle (Fig. 2) of counting variable. The efficient normal strain range εn is the range between upper and lower values of normal strain in one shear strain half-cycle. The critical plane is the plane experiencing the highest shear strain range. The Rain Flow cycle counting method has to be applied on two counting variables: the shear strains nx (t) and ny (t). The algorithm used to compute the fatigue life according to the Wang–Brown model is detailed in [19]. Assuming that, during one cycle the normal strain εn plays an important additional role, Wang and Brown propose the following expression as damage parameter: DP-WB =  a + Sεn where  a is the shear strain amplitude and S is a material parameter identified by fitting tension against torsion fatigue data.

3.5. The strain energy density parameter [Eng] A change of strain energy density, applied in theory of plasticity, has been proposed as a parameter to describe multiaxial fatigue [14–17]. The strain energy density versus time is given by [16,17]: W (t) =

1 (t) · ε(t) 2

(10)

If the cyclic stresses and strains reach their maximum values  a and εa , the maximum value of the energy parameter and its amplitude are the same: Wa = Wmax . Wa =

1 a εa 2

(11)

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27

Assuming W(t) as the fatigue damage parameter according to Eq. (10), we can rescale the standard characteristics of cyclic fatigue ( a − Nf ) and (εa − Nf ) and obtain a new one, (Wa − Nf ) by: Wa =

(f )2 2E

(2Nf )

2b

1 + εf f (2Nf )b+c 2

(12)

The fatigue limit is determined from Eq. (11). We obtain: Waf =

2 af

(13)

2E

Fatigue characteristic Eq. (12) for high cycle fatigue takes the form Wa =

(f )2 2E

(2Nf )

2b

(14)

When the energy density history at the given plane had been determined, the energy cycles were counted with the Rain Flow method; next damage was accumulated according to Palmgren–Miner hypothesis [20] taking into account energy cycle amplitudes below the fatigue limit [14,15,17]. The life time, TRF is calculated from cycles and half-cycles in the time observation T0 of stress history (t), TRF,W =

T0 = S(T0 )

S(T0 ) = 0

T0

J

n /(N0 (Waf /Wai ) i=1 i

for Wai < a · Waf ,

m

)

for Wai ≥ a · Waf

a = 0.25

where S(T0 ) is material damage up to time T0 ; j is number of class intervals of the histogram of the amplitudes of the strain energy density; Waf is the fatigue limit expressed by strain energy density; a is a coefficient allowing to include amplitudes below Waf in the damage accumulation process; m is the slope of fatigue curve expressed by energy: log(Nf ) = A − m log(Wa )

(16)

N0 is a number of cycles corresponding to the fatigue limit Waf ; ni is a number of cycles with amplitude Wai (two the same halfcycles from one cycle). The algorithm used to apply this method is illustrated in [15]. 3.6. Sines, Crossland and von Mises criterion

Hmean

oct,amax + ˛Cr ·

˛Si =

√  −1 6

Crossland recommends use of its maximum value.

(17)

≤ ˇCr

(18)

√ − 2 ,

0



ˇSi =

And for Crossland: ˛Cr =

√  −1 6

−1

−

√ 2 ,

2 −1 3

ˇCr =

(19)

2 −1 3

(20)

 a is the square root of J2,a ,  that is the alternate component of the deviatoric second invariant, Hmean is the mean value of the hydrostatic stress and Hmax is the maximum value of the hydrostatic stress. If the two S–N curves (the stress amplitude corresponding to failure at N cycles), i.e. the uniaxial repeated bending fatigue strength  0 (N) at N cycles and the reversed torsion fatigue strength  −1 (N) at N cycles are available, then the Sines formulation Eq. (17) becomes:



Hmean

≤ ˇSi (N)

where ˛Si =

√  (N) −1 6

0 (N)

−



√ 2 ,

ˇSi =

(21)

2 −1 (N) 3

(22)

Eq. (21) is an extension of the Sines fatigue limit formulation to high-cycle fatigue. If the two S–N curves, the uniaxial reversed bending fatigue strength  −1 (N) at N cycles and the reversed torsion fatigue strength  −1 (N) at N cycles are available, then the Crossland formulation, Eq. (18) becomes: oct,amax + ˛Cr (N) ·



Hmax

≤ ˇCr (N)

where

√  (N) −1 6

−1 (N)

√ − 2 ,

ˇCr =

(23)

2 −1 (N) 3

(24)

Eq. (24) is an extension of the Crossland fatigue limit formulation to high-cycle fatigue. For Eqs. (21) and (23), the key step is the evaluation of the equivalent shear stress amplitude oct,amax and hydrostatic stresses  Hmax , throughout a general multiaxial loading. 3.6.3. Compute the hydrostatic and deviatoric stresses Split the stress tensor (t) into its deviatoric and spherical parts (t) =   (t) +

1 tr((t))I 3

(25)

where tr((t)) is the first stress invariant given by tr((t)) = xx (t) + yy (t) + zz (t) The stress deviator



⎛ 2xx − yy − zz ⎜ ⎝

 = ⎜ ≤ ˇSi

Hmax

The coefficients ˛ and ˇ in both equations can be set through evaluation of the formulas at fatigue limits in torsion and tension. The Sines formula does not allow alternating fatigue limit in tension to be used due to a singular solution. Thus another load condition – a repeated tension – has to be used. The appropriate value for Sines is:

˛Cr =

These methods depend on the invariants of the stress tensor. They have a less onerous computational effort than the critical plane methods (they do not need of extensive projections on physical planes), but its efficiency is not good as for critical plane methods. Indeed stress-invariants criteria work better only if principal axes variation is large. The numerical approach for evaluating the effective shear stress amplitude throughout a loading cycle makes it possible to extend the Sines and the Crossland multiaxial fatigue criteria for finite fatigue life prediction under general multiaxial loadings. Both Crossland [29] and Sines [30,31] published their works throughout the fifties of the last century. Their criteria are very much alike, using the amplitude of second invariant of stress tensor deviator (which corresponds to the von Mises stress) as the basis. Another term is added to the equation in order to cope with the mean stress effect – while Sines prefers the mean value of first invariant of stress tensor (i.e. hydrostatic stress): 3.6.1. Sines criterion  oct,amax + ˛Si ·

criterion 3.6.2. Crossland

oct,amax + ˛Si (N) · (15)

23

3 yz zx

(26)

is expressed as: xy 2yy − xx − zz 3 zy

xz 2zz

yz − xx − yy 3

⎞ ⎟ ⎟ ⎠ (27)

24

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27

Fig. 3. Cruciform specimen descriptions.

The hydrostatic stress

 H

(t) =



H (t)

is calculated as:

1 tr((t)) 3

(28)

For a cycle loading, the mean value of stress is computed as: []mean



1 = T

T

[(t)] dt

(29)

0

the tensor of amplitude value of stress as: [a (t)] = [(t)] − []mean

3.6.4. von Mises criterion The method presented here is based on a generalization of the theory of plasticity generally applied in the case of static loads. According to this conjecture, damage under multiaxial loading can be estimated by determining a uniaxial equivalent stress or strain [32]. According to the literature, we note that the amplitude of the equivalent von Mises stress is usually used as a first approximation. When dealing the study of a vehicle component, Heyes et al. [33] have used this approach for the components subjected to complex multiaxial loadings. It should be noted that the von Mises equivalent stress is calculated by the formula:

(30)

and the shear octahedral stress as: oct,a =

1 3



(a11 (t) − a22 (t))2 + (a22 (t) − a33 (t))2 + (a33 (t) − a11 (t))2 + 6(a23 (t)2 + a13 (t)2 + a12 (t)2 )



(31)

For a loading cycle, the mean value of hydrostatic stress, the maximum values of hydrostatic and alternating octahedral shear stress are calculated by:

1 eq = √ 2



4. Validation with biaxial variable amplitude sequences

Hmean

 Hmax

=

1 T



= max t

t1 +T

 H

t1

 H

(32)

(t) dt

(t)

oct,amax = maxoct,a (t) t

The Sines and Crossland equivalent stresses are calculated by the equations: tr((t)) = xx (t) + yy (t) + zz (t)



eq

Cr

= Cr oct,amax + ˛Cr ·



 Hmax

(33) (34)

Si and Cr are the correction coefficients of the equivalent stress, which can use the S–N curve with alternating tension.

2

2

2

2 2 2 (xx − yy ) + (yy − zz ) + (xx − zz ) + 6(xy + yz + xz )

(35)

The validation of the method has been obtained by the use of biaxial tension-compression variable amplitude tests results. Seven biaxial random stress histories are considered. They are composed of 177.000 up to 190.413 events. These tests were carried out in the laboratory of Professor E. Macha (Bedkowski [34]) in Opole (Poland) kindly provided by Professor T. Palin-Luc (Laboratoire LAMEFIP, Bordeaux, France). The specimens have a cruciform shape as shown in Fig. 3. All test specimens were made of low carbon steel 10HNAP. Both the full chemical composition and the mechanical properties of the material used in this investigation are given in Tables 1 and 2 [15,19,25] respectively. The required material fatigue data are three S–N curves  −1 (N),  −1 (N) and  0 (N). These give the material fatigue strengths versus the number N of cycles for a reversed tensile test (R = −1), a reversed torsion test (R = −1) and a zero to maximum tensile test (R = 0) respectively. Bedkowski and Macha obtained the following S–N

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27

25

Table 1 Chemical composition of the steel tested in (%). Steel

C

Mn

Si

P

S

Cr

Cu

Ni

Fe

10HNAP

0.115

0.71

0.41

0.082

0.028

0.81

0.30

0.50

Rest

Table 2 Mechanical properties of 10HNAP steel. Steel

Young modulus E (GPa]

Poisson ratio ( )

Yield stress Re (MPa)

Tensile strength Rm (MPa)

Z (%)

A10 (%)

10HNAP

215

0.29

414

566

60

32

curves that are shown in Fig. 4. The mechanical property used by the Damaged Stress Model for this material is  u = Rm = 566 MPa. According to [3,25], the applied loads are stationary and ergodic with normal centred distributions and zero mean value. These loads are generated by a random signal generator. Seven tests have been provided with each temporal multiple records (3–7 records) local deformations (εxx , εyy , εxy ), measurements are taken with a frequency of 480 Hz. The total number of measurement points is 59,420 points per sequence, each sequence with duration of 123.8 s. Hook’s equations is used for calculated the stress ( xx ,  yy ,  xy ) [25]. Note that  xx and  yy are out of phase (ϕ ≈ /2). The shear stress  xy are virtually low values (between −29.3 and 8.4 MPa) compared with others values (between −225 and 244.6 MPa for  xx and between −287.9 and 277.3 MPa for  yy ). Applied loading are determined by the equivalent stresses formulas (Eqs. (33)–(35)) of the chosen criteria for this study. The characteristics of the equivalent stress are given in Table 3. We can see in Table 3 that the von Mises value and the amplitude are more important than Crossland and Sines ones. Consequently, the von Mises equivalent stress is more damaging than Crossland and Sines equivalent stresses. Table 4 summarizes experimental and numerical results. Experimental lives of the biaxial sequences are compared with theoretical ones (assessed using Miner and DSM model). They are expressed as the number of predictions up to crack initiation. The results of Table 4 are plotted in Figs. 5–7. These figures show the comparison of the calculated fatigue life by ([Ba], [FS], [W–B], [So] and [Eng] criterion), experimental data and predicted life by (VM-Mi, VM-DSM, Cr-M, Cr-Dsm, Si-Mi and Si-Dsm) by biaxial random loading for 10HNAP steel.

Fig. 4. Reversed tension-compression (S–N) curve  −1 (N), reversed torsion (S–N) curve  −1 (N) and zero maximum tensile curve  0 (N) [35].

Fig. 5. Assessed lives by ([Ba], [FS], [W–B], [So] and [Eng]) criterion against (VM[Mi] , VM[DSM] ) ones for biaxial random loading.

For all Figs. 5–7, it is noticed that for lives ranging between 104 and 106 , Socie’s method [So] is very conservative. Fatemi–Socie [FS] and Bannantine [Ba] methods are closest to the experiment. Wang–Brown’s [W–B], method leads to non-conservative predictions on the whole of these simulations. Application of the

Fig. 6. Assessed lives by ([Ba], [FS], [W–B], [So] and [Eng]) criterion against (Cr[Mi] , Cr[DSM]) ones for biaxial random loading.

26

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27

Table 3 Equivalent stress characteristics.

Von Mises (Eq. (35)) Crossland (Eq. (34)) Sines (Eq. (33))

Maximum value (MPa)

Minimum value (MPa)

Amplitude (MPa)

Mean value (MPa)

535.85 439.47 459.2

15.32 9.67 −3.26

520.53 429.8 462.46

222.69 186.62 173.84

Table 4 Experimental and predicted lives ([Ba]: Bannantine, [F–S]: Fatemi–Socie [W–B]: Wang and Brown’s, [So]: Socie, [Eng]: strain energy density parameter, [Exp]: experimental). No. of specimen B10H03 B10H04 B10H05 B10H06 B10H07 B10H08 B10H09

T[Ba] 61,820 70,560 163,972 217,375 669,208 354,745 317,088

T[F–S] 54,863 74,397 119,073 135,921 246,094 226,009 178,257

T[W–B] 104,800 133,012 245,475 307,989 515,362 529,573 372,954

T[So]

T[Eng]

13,344 20,598 41,209 51,694 120,274 102,807 73,036

32,185 14,948 28,617 34,001 105,520 43,574 53,498

T[exp]

VM[Mi]

70,500 58,800 102,200 188,000 141,800 309,600 298,800

1630 3080 12,348 19,059 82,717 35,857 69,174

VM[DSM]

Cr[Mi]

Cr[DSM]

Si[Mi]

Si[DSM]

1128 2600 10,950 17,053 76,568 31,966 62,793

39,753 53,749 228,830 392,813 1,065,149 487,364 1,311,443

36,034 49,818 214,336 371,030 1,014,596 458,272 1,242,472

25,238 36,479 168,342 329,850 728,253 409,648 1,078,017

22,541 33,450 157,534 293,846 692,829 383,984 1,032,698

suitability of the proposed methods. The ratio between experimental lives and expected ones fall in a scatter band of a factor of 5. The methods of Fatemi–Socie [FS] and Socie [So] and must be used with caution. They are often very conservative. They can lead to over sizing. Von Mises equivalent stress is very conservative and its predictions are far from experimental results. Crossland and Sines equivalent stress are in good agreement with experimental results. We note that the equivalent stress calculated according to the Sines criterion gives good results with the Damaged Stress Model (DSM), for calculation of life and the accumulation of damage. References

Fig. 7. Assessed lives by ([Ba], [FS], [W–B], [So] and [Eng]) criterion against (Si[Mi] , Si[DSM]) ones for biaxial random loading.

energy parameter [Eng] shows that the calculation results for 10HNAP steel fall in a scatter band of a factor of 3 [15]. The majority of predictions using the equivalent stress of von Mises combined with linear or nonlinear damage is very conservative and far from experimental reality. Using the Crossland and Sines equivalent stress (assessed by using linear and non-linear damage models) gives predictions closest to the experimental reality. For all simulations, the predictions   are generally all contained in an interval 0, 2Texp , 5Texp . 5. Conclusions A fatigue life prediction method has been proposed for multiaxial random loading stress histories. It is based on a counting variable representative of the stress states and of the evolution versus in order to identify and extract multiaxial cycles. The counting variable used here is an equivalent stress defined by the multiaxial criteria selected for this study to solve the problem of damage accumulation for the nonlinear Damaged Stress Model (DSM). Both models DSM and Miner’s model are usable for the fatigue assessment for biaxial random stress histories issued from test carried out on cruciform specimens allow validation the

[1] Y.S. Garud, J. Test. Eval. 9 (1981) 165–178. [2] I.V. Papadopoulos, in: K. Dang Van, I.V. Papadopoulos (Eds.), Advanced Course on High-Cycle Metal Fatigue, ICMS, Udine, Italy, 1997. [3] T. Lagoda, E. Macha, Proceedings of Fracture from Defects, ECF12, Sheffield, 1998, pp. 73–79. [4] G. Sines, in: G. Sines, J.L. Waisman (Eds.), Metal Fatigue, McGraw Hill, New York, 1959, pp. 145–169. [5] B. Crossland, Proceedings of the International Conference on Fatigue of Metals, London, 1956, pp. 138–149. [6] M. De Freitas, B. Li, J.L.T. Santos, in: P.J. Kalluri, Bonacuse (Eds.), Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, American Society for Testing and Materials, West Conshohocken, PA, USA, 2000, pp. 139–156. [7] C. Amzallag, J.P. Gerey, J.L. Robert, J. Bahuaud, Int. J. Fatigue 16 (1994) 287–293. [8] S. Dowing, D. Socie, Int. J. Fatigue 1 (1982) 31–40. [9] J.A. Bannantine, D. Socie, in: K. Kussmaul, D. McDiarmid, D. Socie (Eds.), A Variable Amplitude Multiaxial Fatigue Life Prediction Method, ESIS 10, 1991, pp. 35–51. [10] A. Fatemi, D.F. Socie, Fatigue Fract. Eng. Mater. Struct. 11 (3) (1988) 149–165. [11] D. Socie, in: D.L. McDowell, R. Ellis (Eds.), Critical Plane Approaches for Multiaxial Fatigue damage Assessment, ASTM STP 1191, 1993, pp. 7–36. [12] C.H. Wang, M.W. Brown, Fatigue Fract. Eng. Mater. Struct. 16 (12) (1993) 1285–1298. [13] C.H. Wang, M.W. Brown, J. Eng. Mater. Technol. 118 (1996) 367–370. [14] T. Łagoda, E. Macha, in: S. Kalluri, P.J. Bonacuse (Eds.), Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, 2000, pp. 173–190. [15] T. Łagoda, E. Macha, W. Bedkowski, Int J. Fatigue 21 (5) (1999) 431–443. [16] T. Łagoda, Int. J. Fatigue 23 (6) (2001) 467–480. [17] T. Łagoda, Int. J. Fatigue 23 (6) (2001) 481–489. [18] A. Banvillet, T. Łagoda, E. Macha, A. Niesłony, T. Palin-Luc, J.F. Vittori, Int. J. Fatigue 26 (4) (2004) 349–363. [19] A. Banvillet, Prévision de durée de vie en fatigue multiaxiale sous chargements réels: vers des essais accélérés. PhD thesis, ENSAM CER de Bordeaux, France, 2001, p. 274. [20] M.A. Miner, J. Appl. Mech. 12 (1945) 159–164. [21] G. Mesmacque, S. Garcia, A. Amrouche, C. Rubio-Gonzalez, Int. J. Fatigue 27 (5) (2005) 461–467. [22] S. Garcia, A. Amrouche, G. Mesmacque, X. Decoopman, C. Rubio, Int. J. Fatigue 27 (10–12) (2005) 1347–1353. [23] A. Aid, J. Chalet, A. Amrouche, G. Mesmacque, Fatigue Des. (2005). [24] A. Aid, A. Amrouche, B. Bachir Bouiadjra, M. Benguediab, G. Mesmacque, Mater. Des. 32 (1) (2011) 183–191. [25] A. Aid, Cumul d’endommagement en fatigue multiaxiale sous sollicitations variables. PhD thesis, Universitè de Sidi-Bel Abbes, Algérie, 2006, p. 195.

A. Aid et al. / Materials Science and Engineering A 538 (2012) 20–27 [26] L. Li., B. Reis, M. Leite, M. Freitas, Fatigue Fract. Eng. Mater. Struct. 28 (2005) 445–454. [27] L. Li., B. Reis, M. Leite, M. Freitas, Fatigue Fract. Eng. Mater. Struct. 27 (2004) 775–784. [28] C.H. Wang, M.W. Brown, J. Eng. Mater. Technol.-Trans. ASME 118 (1996) 367–374. [29] B. Crossland, Proc. Int. Conf. Fatigue of Metals, Ins. Mech. Eng., London, 1956, pp. 138–149. [30] G. Sines, Failure of Materials Under Combined Repeated Stresses with Superimposed Static Stresses. NACA-TN-3495, NACA, Washington, 1955.

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[31] G. Sines, in: G. Sines, J.L. Waisman (Eds.), Metal Fatigue, McGraw Hill, 1959, pp. 145–169. [32] X. Pitoiset, A. Preumont, Int. J. Fatigue 22 (7) (2000) 541–550. [33] P.J. Al Heyes, 4th Int. Conf. Biaxial/Multiaxial Fatigue, Saint-Germaint, France, 1994. [34] W. Bedkowski, 4th Int. Conf. Biaxial/Multiaxial Fatigue, ESIS, France, 1994, pp. 435–447. [35] B. weber, G. Vialaton, A. Carmet, J.L. Robert, 5th Int. Conf. Biaxial/Multiaxial Fatigue & Fracture Cracow, ESIS, Poland, 1997, pp. 218–231.