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Probabilistic formulation of the multiaxial fatigue damage of Liu
International Journal of Fatigue 33 (2011) 460–465
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Probabilistic formulation of the multiaxial fatigue damage of Liu S. Calvo, M. Canales, C. Gómez, J.R. Valdés, J.L. Núñez ⇑ Area of Research and Development, Instituto Tecnológico de Aragón, María de Luna, 7-50018 Zaragoza, Spain
a r t i c l e
i n f o
Article history: Received 22 April 2010 Received in revised form 23 September 2010 Accepted 5 October 2010 Available online 11 October 2010 Keywords: Multiaxial fatigue Reliability Probabilistic analysis Cumulative damage
a b s t r a c t The Virtual Strain Energy (VSE) model proposed by Liu to study the multiaxial fatigue damage phenomenon is ordinarily used in its deterministic standard form. This model tries to compensate for the lack of adherence to rigorous continuum mechanics fundamentals of energy-based fatigue models like those of Garud or Ellyin, including not only an energetic approach but also a critical plane approach to the multiaxial fatigue phenomenon. In this paper, a new probabilistic formulation for the multiaxial fatigue damage model proposed by Liu [Int. J. Fatigue 23 (2001) S129] is established. With this random formulation, based on the perturbation method, the two main statistical moments (mean and variance) of the random variable fatigue lifetime (Nf) are obtained. These probabilistic results can be subsequently used to undertake reliability studies of the fatigue behavior of metal components. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In the most active industrial sectors of our economy (for example, the automotive, aeronautics, transport or railway sectors), appropriate uncertainty processing distinguishes a successful and worthwhile mechanical design from a merely commonplace design. It is not surprising that the ability to estimate the uncertainties associated with complex engineering structures subjected to random loads, material properties and geometric parameters is becoming a fundamental aspect in the design and analysis of such structures. In this context, due to the fact that the use of numerical simulation tools is widely used nowadays for predicting the behavior and response of complex components with the final aim of reducing the design and development phases of a product, it is necessary to develop applications and tools able to take into account the stochastic nature of the main parameters of the structural problem. Many engineering components and structures are subjected under normal working conditions to complex multiaxial stress and strain states, produced by cyclic loading histories varying over time, which can lead to failure. These complex stress states in which the principal stresses can be non-proportional, normally appear in geometric discontinuities present in components such as notches or joints. Fatigue under these conditions, known as multiaxial fatigue, is important from the point of view of the initial design and from the necessity of evaluating and assessing the durability of the component once it is in operation. ⇑ Corresponding author. Tel.: +34 976 01 00 00; fax: +34 976 01 18 18. E-mail address: [email protected] (J.L. Núñez). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.10.003
Over recent decades, several studies have demonstrated that the local stress and strain state considerably affects the component fatigue strength and that predictions considering uniaxial theories can be non-conservative. Various authors have developed several multiaxial models based on equivalent stress or strain parameters, plastic work or energy or critical plane approaches [1,2]. However, an accepted universal method does not exist. One of the reasons for this is the complexity of the cyclic stress–strain response under multiaxial loadings depending on the load path [3]. Most models treat fatigue from a deterministic point of view. Fatigue, however, does not respond to deterministic behavior and is sensitive to many different parameters which are rarely determined with sufficient accuracy to be considered as deterministic. Physical magnitudes directly affecting the service life of systems such as loadings, component geometry or material properties have a decisive influence on the fatigue process and, therefore, on the life of the structural component. This suggests that it is advisable to consider the probabilistic character of the different parameters from the initial design stage in order to have a more accurate prediction of the failure probability of structural components. Several authors have studied random fatigue from a probabilistic point of view for the crack propagation stage [4] and for the nucleation phase assuming uniaxiality conditions [5]. A serious problem in studying the reliability of structures under fatigue behavior is the high computational cost of probabilistic analysis. However, techniques such as the Monte Carlo simulation [6,7] or first or second order reliability methods (FORM/SORM) [8] are different alternatives for the problem. FORM and SORM methods can be used directly for problems with an analytical relationship between the state and the indepen-
S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465
dent random variables. This is the case of problems with simple geometries, loads and boundary conditions. For complex problems, however, these relationships are not possible and approximated methods which take into account the random features of the problem have to be used. This paper focuses on the probabilistic formulation of the multiaxial fatigue damage model proposed by Liu [1,9,10], describing the fatigue behavior of metallic components under multiaxial cyclic loadings in the nucleation phase. The values of the first and second statistical moments (mean and variance) of the random variable fatigue lifetime obtained by the use of this formulation enable the life cycles of components to be calculated in terms of failure probability. Section 2 provides a review of the state of the art of the different multiaxial models and a justification of the model selected, including the Liu fatigue damage model. In Section 3, the Liu fatigue damage model is studied from a probabilistic point of view. Input random variables feeding the probabilistic life model correspond to elastoplastic values that are evaluated using an incremental relationship, which relates the elastic and elastic–plastic strain energy densities at the notch tip and the material stress–strain behavior and its denominated equivalence of increments of the total distortional strain energy density. This transformation from elastic to elastic–plastic values will be described in a later paper. In Section 4, several results for an application example are shown and compared with other results, in particular with those of a Monte Carlo simulation. The paper finishes with the main conclusions of the work.
2. Multiaxial fatigue review Multiaxial fatigue theories developed over the years have been constantly evolving. The first methodologies to predict multiaxial fatigue life were based on equivalent stress or strain parameters related to static yield criteria [11]. The three most common criteria are the maximum normal stress theory, the maximum shear stress theory and the octahedral shear stress theory. These equivalent parameters consider the failure as a volumetric phenomenon. They are defined by a global parameter value and do not consider the stress state effects, the mean stress or the stress–strain path. Several researchers have demonstrated the influence of the stress and strain state on fatigue strength and that fatigue predictions according to uniaxial models may be non-conservative. The physical parameters used in damage models have been adapted to the increasing knowledge of the problem. The initial extrapolation from the static yield theories was performed choosing the stress states as characteristic parameters. Later, the importance of elastoplastic behavior on the notch tip and the strain controlled situation on the material were noted. This led to the use of the strain as the basic parameter in the correlations. In recent decades, the importance of the elastoplastic multiaxial behavior under non-proportional loading histories has been researched, showing the advisability of using both parameters, stresses and strains, in the correlation of the phenomenon. A natural way of achieving this integration is by means of the cyclic strain energy. At the same time, the parallel progress of fracture mechanics has made possible the understanding of the physical nature of fatigue as a phenomenon principally of crack propagation. There is no consensus about the classification of the different multiaxial damage models developed in recent years. One proposed classification distinguishes between high cycle fatigue (HCF) and low cycle fatigue (LCF) models. In Ref. [12–15] different classifications for HCF theories are described. In [2] four categories are distinguished: stress based criteria, strain based criteria, energy theories and critical plane approximations. Socie and Marquis [1]
461
refer to stress-based approximations, energy models and critical plane approximations. In the current work, multiaxial fatigue models have been classified in three main groups: – Stress-based models (HCF) – Strain-based models (LCF) – Energy-based models (both H&LCF) Five different types of stress-based models can be distinguished in the literature for metals. First, there are equivalent parameter theories defined as an extension of the static yield criteria. Second, until the end of the 1950s several empirical formulae were proposed to synthesize many fatigue data (Haigh, Gerber, Marin, Gough and Pollard [1,2]). The third group includes the criteria based on the concept of critical plane (McDiarmid, Findley [1,2,13,16]). From the observation that after nucleation a microcrack propagates first along a shear plane, many authors assumed that crack initiation is governed by the shear stress. The fourth category encompasses criteria based on the second invariant of the deviatoric stress tensor (Sines, Crossland [2,11]). Finally, after the criterion proposed by Dang Van [17], other micro–macro approaches appeared (Papadopoulos [18], Deperrois, Morel [19,20]). They consider that elastic shakedown is the condition needed to avoid fatigue crack initiation in unfavourably oriented grains. Fatigue crack initiation in polycrystalline metal is determined from the critical plane containing the easiest slip directions of the grain and experiencing the largest shear strain amplitude. Strain-based approximations include equivalent parameter criteria and critical plane models. The first group was introduced in the 1940s in an attempt to correlate low cycle multiaxial fatigue tests. The second group, in a similar manner as the stress-based models, tries to reflect the physical nature of damage fatigue in the formulation. The critical plane concept is based on the fracture mode or the crack initiation mechanism. Typical models in this category were developed by Brown and Miller [9,16,17], Fatemi and Socie [16,21] or Smith, Watson and Topper [1,9,21]. There is currently a general consensus about the inability of the classic theories, related to an equivalent parameter, to explain properly all multiaxial fatigue cases, especially when nonproportional loadings and mean stresses are present. For this reason, the critical plane concept based on the fracture mode has gained great acceptance. Critical plane stress-based approximations are used in the prediction of high cycle fatigue, assuming that for long lives the loads applied on the components are low and linearity between stresses and strains exists. These criteria propose the maximum principal stresses, maximum shear stresses or the maximum octahedral stresses as the parameters driving the multiaxial fatigue process. In contrast, critical plane strain-based approximations are typically associated with low cycle fatigue where significant plasticity may occur. Energy-based criteria use the product of strains and stresses to quantify fatigue damage [22]. Several proposals for defining the energy parameter controlling multiaxial fatigue have been analysed. Three main groups may be noted in function of the strain energy density per cycle assumed as the damage parameter under multiaxial fatigue. These are criteria based on the elastic strain energy for high cycle fatigue (HCF), on the plastic strain energy for low cycle fatigue (LCF) and finally on the sum of elastic and plastic strain energies valid for HCF and LCF. Morrow’s researches into cyclic plasticity serves as a basis for many energy models such as those proposed by Garud [16,22] or Ellyin [1]. However, energy-based models have been criticized for their lack of adherence to rigorous continuum mechanics fundamentals since the use of a scalar variable (energy) leaves crack orientation analysis out of the model.
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S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465
To compensate for this lack, Liu [1,9,10], Chu et al. [1,9] and Glinka [1,9,16] used the energy criterion in conjunction with the critical plane approach. Liu calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes and Mohr’s circles for stress and strain. Chu et al. formulated normal and shear energy components based on the Smith– Watson–Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. Glinka et al. proposed a multiaxial life parameter based on the summation of the products of normal and shear strains and stresses on the critical shear plane. They consider only the shear work and add a mean stress correction similar to the Morrow mean stress model to account for crack surface sliding and opening. In the current work the Liu model serves as the base for the probabilistic fatigue estimations. Original expressions proposed by Liu have been transformed to obtain the principal statistical moments of the life random variable. The selection of this model was motivated by the fact that the combined type of models (critical plane and energy) arises naturally from considering the elastic energy required to propagate a fatigue crack. This model overcomes the drawback of the Garud model for considering plastic work exclusively being very small at long lives and unable to be computed with sufficient accuracy and is valid for multiaxial non-proportional loading. In the Liu model the damage summation is considered in a specific plane and failure is expected to occur on the plane in the material having the maximum VSE quantity. For all these qualities this model has a wide field of application, this being an important motivation of the investigation. The VSE model proposed by Liu includes both elastic and plastic parts and can be considered as a critical plane model since the working terms are defined in specific planes inside the material. For multiaxial loading, this model takes into account two possible failure modes: a tensile failure and a shear failure, with the latter divided into two types of cracks, Types A and B. The failure appears in the material plane reaching the maximum value of the VSE parameter, defined through expressions (1) and (2), in which e0f is the fatigue ductility coefficient, c is the fatigue ductility exponent, r0f is the fatigue strength coefficient, b is the fatigue strength exponent, Drn and Den are the normal stress and strain range on a specific plane and Ds and Dc are the shear stress and strain range on a specific plane, respectively. The subscript max indicates the maximum value over all planes of the quantity in the parenthesis.
2 ½ðDrn Den Þmax þ ðDsDcÞ rn;min 1 rn;max ¼ 4r0f e0f ð2Nf Þbþc þ
4r02 f E
½ðDrn Den Þ þ ðDsDcÞmax ¼ 4s0f c0f ð2Nf Þbc þcc þ
ð2N f Þ2b
Tensile failure
r0f 0 rf rn;mean
4s02 f G
!
ð2N f Þ2bc
ð1Þ
!
Shear failure
ð2Þ
3. Random formulation of the LIU damage model The objective of the random formulation of a general fatigue damage model is to obtain at least the two main statistical moments (mean and variance) of the random variable fatigue lifetime, Nf. Next, it is shown how to compute the random properties of the variable Nf to be considered in the construction of the reliability B-model [23]. In this study the tensile failure mode is considered for the estimation of the fatigue life. However, the same procedure can be applied for the shear failure mode. The most correct use of each mode will be determined in function of the material characteristics and the type of loading acting on the components. Replacing expressions (3)–(6) the multiaxial fatigue damage model of Liu indicated in Eq. (1) is transformed to expression (7).
Dr ¼ rn;max rn;min
ð3Þ
De ¼ en;max en;min
ð4Þ
Ds ¼ smax smin
ð5Þ
Dc ¼ cmax cmin
ð6Þ
½ðrn;max rn;min Þðen;max en;min Þ þ ðsmax smin Þðcmax cmin Þ ! 4r02 2 f bþc 0 0 þ ð2Nf Þ2b rn;min ¼ 4rf ef ð2N f Þ E 1 rn;max
ð7Þ
As it is not possible to obtain a direct expression of the variable Nf from Eqs. (1) and (7) and in order to obtain the random properties of this variable, the perturbation approach is used. This method approximates a random function by its Taylor expansion
Fig. 1. Thin walled tubular sample used in the validation of the proposed methodology.
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S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465 Table 1 Principal moments of the random variables considered in the analyses. Variable
Mean
Typical deviation
E |p| |t|
2.10 E5 N/mm2 28.50 E3 N 10.00 E3 N mm
1.05 E4 N/mm2 1.425 E3 N 0.50 E3 N mm
up to a given order around the means of the different random variables on which it depends. From this, the approximate mean and variance of the fatigue life can be computed in terms of the moments of the different random variables. Isolating the maximum normal strain variable, en,max, from Eq. (7) and establishing the Taylor expansion of this variable up to order one around the mean of the different random variables, ai, the following expression is obtained.
en;max ¼ en;max j0 þ
X @ en;max ðai li Þ þ @ ai i
where Eqs. (10)–(22) define the coefficients included in Eq. (9). 0
a ¼ en;max j f en;min j g rn;max j h rn;min j j smax j
0
k smin j0 l cmax j0 m cmin j0 n Nf j0 p r0f j0 q e0f j0 r bj0 s cj0 f ¼1
g¼
ð11Þ
2r2n;max ðrn;max rn;min Þ
4r02 1 f ð2N f Þ 4r0f e0f ð2Nf Þbþc þ 2 E
1
rn;max rn;min
! 2b
1
rn;min rn;max
4r02 ð2Nf Þ
j¼
ð12Þ
2b
4r0f e0f ð2Nf Þbþc þ f E 1 þ 2rn;max rn;max rn;min rn;max rn;min 2 ! 2b 4r02 1 rn;min f ð2N f Þ 4r0f e0f ð2Nf Þbþc þ 1 2 E rn;max 0 ðsmax smin Þðcmax cmin Þ 0 cmax þ cmin rn;max rn;min
2
ðsmax smin Þðcmax cmin ÞÞj0 h¼
ð13Þ
ð14Þ
c cmin 0 k ¼ max rn;max rn;min 0 smax þ smin l¼ rn;max rn;min
ð15Þ
ð16Þ
m¼
smax smin 0 rn;max rn;min
bþc rn;min 0 2r0f 2Nf 1 rn;max q¼ rn;max rn;min
ð20Þ
1 r¼ 2 rn;max rn;min
s¼
2b 8r02 f ð2N f Þ
E
1
rn;min 0 0 4rf ef ð2N f Þbþc ðlog 2 þ log Nf Þ rn;max
!!0 ðlog 2 þ log Nf Þ
1 2ðrn;max rn;min Þ
1
ð21Þ
0 rn;min ð4r0f e0f ð2N f Þbþc ðlog 2 þ log N f ÞÞ rn;max
The approximate mean and variance of Nf are obtained by isolating this variable from Eq. (4) and applying the expectation and variance operators.
E½Nf ¼
Xn
aX i¼1 i i
var½Nf ¼ ð10Þ
4r02 ð2N f Þ2b 4r0f e0f ð2Nf Þbþc þ f E rn;min
ð19Þ
ð22Þ
ð9Þ
0
0 8r0 ð2N f Þ2b rn;min 1 rn;max 4e0f ð2Nf Þbþc þ f E p¼ 2ðrn;max rn;min Þ
þ
en;max ¼ a þ f en;min þ g rn;max þ h rn;min þ j smax þ k smin þ l cmax þ m cmin þ n Nf þ p r0f þ q e0f þ r b þ s c
0
ð18Þ
ð8Þ
being ai = en,min, rn,max, rn,min, smax, smin, cmax, cmin, Nf, r0f , e0f , b, c. The term ai j0 is a random variable evaluated in the mean value of ai and li the mean value corresponding to each random variable. Eq. (9) shows the extended expression of Eq. (8).
0
0 16r02 bð2N f Þ2b1 rn;min 1 rn;max 8r0f e0f ðb þ cÞð2Nf Þbþc1 þ f E n¼ 2ðrn;max rn;min Þ
ð17Þ
Xn
¼ N0f
a2 var½X i i¼1 i
ð23Þ þ
Xn Xn i¼1
j¼1 i–j
ai aj cov½X i ; X j
2 2 1 f var½en;max þ var½en;min ¼ n n 2 g 2 h var½rn;max þ var½rn;min þ n n 2 2 2 j k l þ var½smax þ var½smin þ var½cmax n n n m2 p2 q 2 þ var½cmin þ var½r0f þ var½e0f n n n r 2 s 2 þ var½b þ var½c n n
ð24Þ
where the terms including cov½X i ; X j vanish due to the independence of the random variables. For non-independent variables, expression (24) becomes a little more complex than the one used in the following approach since the correlation function of each pair of variables has to be known. Finally, it should be noted that in Eqs. (23) and (24) the values related with fatigue material properties are known, but the means and variances of the structural strain and stress behavior of the component must be evaluated by means of a probabilistic calculation method such as the PFEM, Monte Carlo Simulation or similar. 4. Example of numerical application of the probabilistic formulation In this section, a complete reliability analysis in terms of fatigue life is applied to the cylindrical specimen shown in Fig. 1. An axial force of 28,500 N and a torque of 10,000 N mm act on the sample reproducing a sinusoidal cycling loading that produces a multiaxial stress–strain state on the component. The module of these loads (|p| and |t|) together with the Young modulus (E) characterizing the mechanical properties of the material are considered as
464
S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465
Table 2 Elastic stress tensor in terms of expectation and typical deviation for the element numbered 19668. Element 19668
Mean value Typical deviation
r1
r2
r3
r12
r23
r31
394.7 19.07
23.2 1.19
18.4 0.9
363.1 18.21
4.6 0.22
11.5 0.57
Fig. 2. Location of nodes in the sample used to compare results.
Table 3 Mean and typical deviation in terms of fatigue life. Life
Monte Carlo simulation
PFEM + Liu prob.
Difference (%)
Mean Typical deviation
9105 cycles 2860 cycles
8205 cycles 2915 cycles
10 2
random variables in the basic probabilistic analysis carried out on the specimens. The mean values and the typical deviations of those variables are shown in Table 1. The problem is solved according to a PFEM approach developed by the authors (whose description is not the objective of this paper) and the two first statistical moments of the random stress and strain variables are obtained. Table 2 shows the elastic stress tensor in terms of expectation (mean value) and typical deviation for the element numbered 19,668 in the model (located in the central zone of the specimen and corresponding to the maximum stress area, as can be seen in Fig. 2) in which the multiaxial fatigue damage model proposed by Liu will be applied in tensile mode but formulated in a probabilistic way to determine the variable fatigue lifetime (Nf) of the sample. With the aim of verifying the statistical moments computed via PFEM, Monte Carlo simulations have been carried out considering a total of 1500 samples and varying the input variables. The results showed differences between the methods of less than 4%. The elastic stress and strain tensors are corrected to their related elastic–plastic values according to the equivalence of increments of the total distortional strain energy density method, which relates the elastic and elastic–plastic strain energy densities at the notch tip and the material stress–strain behavior [17]. In this
way, mechanical and fatigue properties characterizing the material are known. Table 3 compares the means and typical deviations achieved in the estimation of fatigue life using both methods, Monte Carlo simulations and the combination of the PFEM and the probabilistic Liu fatigue model proposed in this research. From the results shown in Table 3, it can be observed that the percentage difference between both sets of results is low, being 10% for the mean values and about 2% for the typical deviation results, this new procedure thus being more conservative. 5. Conclusions This work presents a probabilistic model to describe the phenomenon of crack initiation in metal multiaxial fatigue under stochastic conditions, based on Liu’s fatigue damage model. Besides the external load, additional random variables such as the cyclic fatigue material parameters, r0f , e0f , b and c, and elastic material parameters have been considered. The accuracy of the fatigue lifetime values obtained, when compared to an equivalent analysis performed with a Monte Carlo simulation approach, proves the good performance of the model. It is remarkable that results of the probabilistic model developed in this paper are contrasted with numerical data to demonstrate the accuracy of the first order expressions. Experimental data are not use in the comparison because is out of its reach to verify the suitability of the Liu model to solve this kind of multiaxial fatigue problems. Acknowledgement The authors acknowledge funding of this work by the Spanish Ministry of Science and Education through Project DPI200766903-C02-02.
S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465
References [1] Socie DF, Marquis GB. Multiaxial fatigue. Society of Automotive Engineers, Inc. Warrendale (PA);2000. [2] Stephens R, Fatemi A, Stephens R, Fuchs H. Metal fatigue in engineering, 2nd ed. Wiley-Interscience Publication; 2000. [3] Buczynski A, Glinka G. Multiaxial stress–strain notch analysis multiaxial fatigue and deformation: testing and prediction. In: Kalluri S, Bonacuse PJ, editors. ASTM STP 1387: American Society for Testing and Materials; 2000. p. 82–98. [4] Bea JA. Simulación del crecimiento de grietas por fatiga aleatoria mediante elementos finitos probabilistas. PhD thesis, Universidad de Zaragoza; 1997. [5] Núñez JL. Análisis del fenómeno de fatiga en metales en etapa de nucleación mediante la utilización de modelos estadísticos de daño acumulado y elementos finitos probabilistas. PhD thesis, Universidad de Zaragoza; 2003. [6] Rubinstein RY. Simulation and the Monte Carlo method. New York: Willey; 1981. [7] ANSYS theory reference: probabilistic design [chap. 21]. [8] Riha DS. Capabilities and applications of probabilistic methods in finite element analysis. In: Fifth ISSAT international conference on reliability and quality in design, Las Vegas; 1999. [9] Han C, Chen X, Kim KS. Evaluation of multiaxial fatigue criteria under irregular loading. Int J Fatigue 2002;24:913–22. [10] Liu KC, Wang JA. An energy method for predicting fatigue life, crack orientation, and crack growth under multiaxial loading conditions. Int J Fatigue 2001;23:S129–34. [11] Garud YS. Multiaxial fatigue: a survey of the state of the art. J Test Eval JTEVA 1981;9(3):165–78.
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[12] Banvillet A, Palin-Luc T, Lasserre S. A volumetric energy based high cycle multiaxial fatigue criterion. Int J Fatigue 2003;25:755–69. [13] 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. [14] Lagoda T, Macha E. Energy approach to fatigue life estimation under combined tension with torsion. VII Summer School of Fracture Mechanic. Pokrywna, Poland; 2001. [15] Palin-Luc T, Lasserre S. An energy based criterion for high cycle multiaxial fatigue. Eur. J. Mech A/Solids 1998;17(2):237–51. [16] Pan W, Hung C, Chen L. Fatigue life estimation under multiaxial loadings. Int J Fatigue 1999;21:3–10. [17] MSC/Fatigue theory book:multiaxial fatigue [chap. 6]. [18] Papadopoulos IV. Long life fatigue under multiaxial loading. Int J Fatigue 2001:23 839–849. [19] Morel F. A critical plane for life prediction of high cycle fatigue under multiaxial variable amplitude loading. Int J Fatigue 2000;22:S101–19. [20] Morel F, Palin-Luc T, Froustey C. Comparative study and link between mesoscopic and energetic approaches in high cycle multiaxial fatigue. Int J Fatigue 2001;23:317–27. [21] Das J, Sivakumar SM. An evaluation of multiaxial fatigue life assessment methods for engineering components. Int J Pressure Vessels Pip 1999;76:741–6. [22] Varvani-Farahani A. A new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-ofphase multiaxial fatigue loading conditions. Int J Fatigue 2000;22:295–305. [23] Bogdanoff JL, Kozin F. Probabilistic models of cumulative damage. New York: Wiley; 1985.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Probabilistic formulation of the multiaxial fatigue damage of Liu S. Calvo, M. Canales, C. Gómez, J.R. Valdés, J.L. Núñez ⇑ Area of Research and Development, Instituto Tecnológico de Aragón, María de Luna, 7-50018 Zaragoza, Spain
a r t i c l e
i n f o
Article history: Received 22 April 2010 Received in revised form 23 September 2010 Accepted 5 October 2010 Available online 11 October 2010 Keywords: Multiaxial fatigue Reliability Probabilistic analysis Cumulative damage
a b s t r a c t The Virtual Strain Energy (VSE) model proposed by Liu to study the multiaxial fatigue damage phenomenon is ordinarily used in its deterministic standard form. This model tries to compensate for the lack of adherence to rigorous continuum mechanics fundamentals of energy-based fatigue models like those of Garud or Ellyin, including not only an energetic approach but also a critical plane approach to the multiaxial fatigue phenomenon. In this paper, a new probabilistic formulation for the multiaxial fatigue damage model proposed by Liu [Int. J. Fatigue 23 (2001) S129] is established. With this random formulation, based on the perturbation method, the two main statistical moments (mean and variance) of the random variable fatigue lifetime (Nf) are obtained. These probabilistic results can be subsequently used to undertake reliability studies of the fatigue behavior of metal components. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In the most active industrial sectors of our economy (for example, the automotive, aeronautics, transport or railway sectors), appropriate uncertainty processing distinguishes a successful and worthwhile mechanical design from a merely commonplace design. It is not surprising that the ability to estimate the uncertainties associated with complex engineering structures subjected to random loads, material properties and geometric parameters is becoming a fundamental aspect in the design and analysis of such structures. In this context, due to the fact that the use of numerical simulation tools is widely used nowadays for predicting the behavior and response of complex components with the final aim of reducing the design and development phases of a product, it is necessary to develop applications and tools able to take into account the stochastic nature of the main parameters of the structural problem. Many engineering components and structures are subjected under normal working conditions to complex multiaxial stress and strain states, produced by cyclic loading histories varying over time, which can lead to failure. These complex stress states in which the principal stresses can be non-proportional, normally appear in geometric discontinuities present in components such as notches or joints. Fatigue under these conditions, known as multiaxial fatigue, is important from the point of view of the initial design and from the necessity of evaluating and assessing the durability of the component once it is in operation. ⇑ Corresponding author. Tel.: +34 976 01 00 00; fax: +34 976 01 18 18. E-mail address: [email protected] (J.L. Núñez). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.10.003
Over recent decades, several studies have demonstrated that the local stress and strain state considerably affects the component fatigue strength and that predictions considering uniaxial theories can be non-conservative. Various authors have developed several multiaxial models based on equivalent stress or strain parameters, plastic work or energy or critical plane approaches [1,2]. However, an accepted universal method does not exist. One of the reasons for this is the complexity of the cyclic stress–strain response under multiaxial loadings depending on the load path [3]. Most models treat fatigue from a deterministic point of view. Fatigue, however, does not respond to deterministic behavior and is sensitive to many different parameters which are rarely determined with sufficient accuracy to be considered as deterministic. Physical magnitudes directly affecting the service life of systems such as loadings, component geometry or material properties have a decisive influence on the fatigue process and, therefore, on the life of the structural component. This suggests that it is advisable to consider the probabilistic character of the different parameters from the initial design stage in order to have a more accurate prediction of the failure probability of structural components. Several authors have studied random fatigue from a probabilistic point of view for the crack propagation stage [4] and for the nucleation phase assuming uniaxiality conditions [5]. A serious problem in studying the reliability of structures under fatigue behavior is the high computational cost of probabilistic analysis. However, techniques such as the Monte Carlo simulation [6,7] or first or second order reliability methods (FORM/SORM) [8] are different alternatives for the problem. FORM and SORM methods can be used directly for problems with an analytical relationship between the state and the indepen-
S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465
dent random variables. This is the case of problems with simple geometries, loads and boundary conditions. For complex problems, however, these relationships are not possible and approximated methods which take into account the random features of the problem have to be used. This paper focuses on the probabilistic formulation of the multiaxial fatigue damage model proposed by Liu [1,9,10], describing the fatigue behavior of metallic components under multiaxial cyclic loadings in the nucleation phase. The values of the first and second statistical moments (mean and variance) of the random variable fatigue lifetime obtained by the use of this formulation enable the life cycles of components to be calculated in terms of failure probability. Section 2 provides a review of the state of the art of the different multiaxial models and a justification of the model selected, including the Liu fatigue damage model. In Section 3, the Liu fatigue damage model is studied from a probabilistic point of view. Input random variables feeding the probabilistic life model correspond to elastoplastic values that are evaluated using an incremental relationship, which relates the elastic and elastic–plastic strain energy densities at the notch tip and the material stress–strain behavior and its denominated equivalence of increments of the total distortional strain energy density. This transformation from elastic to elastic–plastic values will be described in a later paper. In Section 4, several results for an application example are shown and compared with other results, in particular with those of a Monte Carlo simulation. The paper finishes with the main conclusions of the work.
2. Multiaxial fatigue review Multiaxial fatigue theories developed over the years have been constantly evolving. The first methodologies to predict multiaxial fatigue life were based on equivalent stress or strain parameters related to static yield criteria [11]. The three most common criteria are the maximum normal stress theory, the maximum shear stress theory and the octahedral shear stress theory. These equivalent parameters consider the failure as a volumetric phenomenon. They are defined by a global parameter value and do not consider the stress state effects, the mean stress or the stress–strain path. Several researchers have demonstrated the influence of the stress and strain state on fatigue strength and that fatigue predictions according to uniaxial models may be non-conservative. The physical parameters used in damage models have been adapted to the increasing knowledge of the problem. The initial extrapolation from the static yield theories was performed choosing the stress states as characteristic parameters. Later, the importance of elastoplastic behavior on the notch tip and the strain controlled situation on the material were noted. This led to the use of the strain as the basic parameter in the correlations. In recent decades, the importance of the elastoplastic multiaxial behavior under non-proportional loading histories has been researched, showing the advisability of using both parameters, stresses and strains, in the correlation of the phenomenon. A natural way of achieving this integration is by means of the cyclic strain energy. At the same time, the parallel progress of fracture mechanics has made possible the understanding of the physical nature of fatigue as a phenomenon principally of crack propagation. There is no consensus about the classification of the different multiaxial damage models developed in recent years. One proposed classification distinguishes between high cycle fatigue (HCF) and low cycle fatigue (LCF) models. In Ref. [12–15] different classifications for HCF theories are described. In [2] four categories are distinguished: stress based criteria, strain based criteria, energy theories and critical plane approximations. Socie and Marquis [1]
461
refer to stress-based approximations, energy models and critical plane approximations. In the current work, multiaxial fatigue models have been classified in three main groups: – Stress-based models (HCF) – Strain-based models (LCF) – Energy-based models (both H&LCF) Five different types of stress-based models can be distinguished in the literature for metals. First, there are equivalent parameter theories defined as an extension of the static yield criteria. Second, until the end of the 1950s several empirical formulae were proposed to synthesize many fatigue data (Haigh, Gerber, Marin, Gough and Pollard [1,2]). The third group includes the criteria based on the concept of critical plane (McDiarmid, Findley [1,2,13,16]). From the observation that after nucleation a microcrack propagates first along a shear plane, many authors assumed that crack initiation is governed by the shear stress. The fourth category encompasses criteria based on the second invariant of the deviatoric stress tensor (Sines, Crossland [2,11]). Finally, after the criterion proposed by Dang Van [17], other micro–macro approaches appeared (Papadopoulos [18], Deperrois, Morel [19,20]). They consider that elastic shakedown is the condition needed to avoid fatigue crack initiation in unfavourably oriented grains. Fatigue crack initiation in polycrystalline metal is determined from the critical plane containing the easiest slip directions of the grain and experiencing the largest shear strain amplitude. Strain-based approximations include equivalent parameter criteria and critical plane models. The first group was introduced in the 1940s in an attempt to correlate low cycle multiaxial fatigue tests. The second group, in a similar manner as the stress-based models, tries to reflect the physical nature of damage fatigue in the formulation. The critical plane concept is based on the fracture mode or the crack initiation mechanism. Typical models in this category were developed by Brown and Miller [9,16,17], Fatemi and Socie [16,21] or Smith, Watson and Topper [1,9,21]. There is currently a general consensus about the inability of the classic theories, related to an equivalent parameter, to explain properly all multiaxial fatigue cases, especially when nonproportional loadings and mean stresses are present. For this reason, the critical plane concept based on the fracture mode has gained great acceptance. Critical plane stress-based approximations are used in the prediction of high cycle fatigue, assuming that for long lives the loads applied on the components are low and linearity between stresses and strains exists. These criteria propose the maximum principal stresses, maximum shear stresses or the maximum octahedral stresses as the parameters driving the multiaxial fatigue process. In contrast, critical plane strain-based approximations are typically associated with low cycle fatigue where significant plasticity may occur. Energy-based criteria use the product of strains and stresses to quantify fatigue damage [22]. Several proposals for defining the energy parameter controlling multiaxial fatigue have been analysed. Three main groups may be noted in function of the strain energy density per cycle assumed as the damage parameter under multiaxial fatigue. These are criteria based on the elastic strain energy for high cycle fatigue (HCF), on the plastic strain energy for low cycle fatigue (LCF) and finally on the sum of elastic and plastic strain energies valid for HCF and LCF. Morrow’s researches into cyclic plasticity serves as a basis for many energy models such as those proposed by Garud [16,22] or Ellyin [1]. However, energy-based models have been criticized for their lack of adherence to rigorous continuum mechanics fundamentals since the use of a scalar variable (energy) leaves crack orientation analysis out of the model.
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To compensate for this lack, Liu [1,9,10], Chu et al. [1,9] and Glinka [1,9,16] used the energy criterion in conjunction with the critical plane approach. Liu calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes and Mohr’s circles for stress and strain. Chu et al. formulated normal and shear energy components based on the Smith– Watson–Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. Glinka et al. proposed a multiaxial life parameter based on the summation of the products of normal and shear strains and stresses on the critical shear plane. They consider only the shear work and add a mean stress correction similar to the Morrow mean stress model to account for crack surface sliding and opening. In the current work the Liu model serves as the base for the probabilistic fatigue estimations. Original expressions proposed by Liu have been transformed to obtain the principal statistical moments of the life random variable. The selection of this model was motivated by the fact that the combined type of models (critical plane and energy) arises naturally from considering the elastic energy required to propagate a fatigue crack. This model overcomes the drawback of the Garud model for considering plastic work exclusively being very small at long lives and unable to be computed with sufficient accuracy and is valid for multiaxial non-proportional loading. In the Liu model the damage summation is considered in a specific plane and failure is expected to occur on the plane in the material having the maximum VSE quantity. For all these qualities this model has a wide field of application, this being an important motivation of the investigation. The VSE model proposed by Liu includes both elastic and plastic parts and can be considered as a critical plane model since the working terms are defined in specific planes inside the material. For multiaxial loading, this model takes into account two possible failure modes: a tensile failure and a shear failure, with the latter divided into two types of cracks, Types A and B. The failure appears in the material plane reaching the maximum value of the VSE parameter, defined through expressions (1) and (2), in which e0f is the fatigue ductility coefficient, c is the fatigue ductility exponent, r0f is the fatigue strength coefficient, b is the fatigue strength exponent, Drn and Den are the normal stress and strain range on a specific plane and Ds and Dc are the shear stress and strain range on a specific plane, respectively. The subscript max indicates the maximum value over all planes of the quantity in the parenthesis.
2 ½ðDrn Den Þmax þ ðDsDcÞ rn;min 1 rn;max ¼ 4r0f e0f ð2Nf Þbþc þ
4r02 f E
½ðDrn Den Þ þ ðDsDcÞmax ¼ 4s0f c0f ð2Nf Þbc þcc þ
ð2N f Þ2b
Tensile failure
r0f 0 rf rn;mean
4s02 f G
!
ð2N f Þ2bc
ð1Þ
!
Shear failure
ð2Þ
3. Random formulation of the LIU damage model The objective of the random formulation of a general fatigue damage model is to obtain at least the two main statistical moments (mean and variance) of the random variable fatigue lifetime, Nf. Next, it is shown how to compute the random properties of the variable Nf to be considered in the construction of the reliability B-model [23]. In this study the tensile failure mode is considered for the estimation of the fatigue life. However, the same procedure can be applied for the shear failure mode. The most correct use of each mode will be determined in function of the material characteristics and the type of loading acting on the components. Replacing expressions (3)–(6) the multiaxial fatigue damage model of Liu indicated in Eq. (1) is transformed to expression (7).
Dr ¼ rn;max rn;min
ð3Þ
De ¼ en;max en;min
ð4Þ
Ds ¼ smax smin
ð5Þ
Dc ¼ cmax cmin
ð6Þ
½ðrn;max rn;min Þðen;max en;min Þ þ ðsmax smin Þðcmax cmin Þ ! 4r02 2 f bþc 0 0 þ ð2Nf Þ2b rn;min ¼ 4rf ef ð2N f Þ E 1 rn;max
ð7Þ
As it is not possible to obtain a direct expression of the variable Nf from Eqs. (1) and (7) and in order to obtain the random properties of this variable, the perturbation approach is used. This method approximates a random function by its Taylor expansion
Fig. 1. Thin walled tubular sample used in the validation of the proposed methodology.
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S. Calvo et al. / International Journal of Fatigue 33 (2011) 460–465 Table 1 Principal moments of the random variables considered in the analyses. Variable
Mean
Typical deviation
E |p| |t|
2.10 E5 N/mm2 28.50 E3 N 10.00 E3 N mm
1.05 E4 N/mm2 1.425 E3 N 0.50 E3 N mm
up to a given order around the means of the different random variables on which it depends. From this, the approximate mean and variance of the fatigue life can be computed in terms of the moments of the different random variables. Isolating the maximum normal strain variable, en,max, from Eq. (7) and establishing the Taylor expansion of this variable up to order one around the mean of the different random variables, ai, the following expression is obtained.
en;max ¼ en;max j0 þ
X @ en;max ðai li Þ þ @ ai i
where Eqs. (10)–(22) define the coefficients included in Eq. (9). 0
a ¼ en;max j f en;min j g rn;max j h rn;min j j smax j
0
k smin j0 l cmax j0 m cmin j0 n Nf j0 p r0f j0 q e0f j0 r bj0 s cj0 f ¼1
g¼
ð11Þ
2r2n;max ðrn;max rn;min Þ
4r02 1 f ð2N f Þ 4r0f e0f ð2Nf Þbþc þ 2 E
1
rn;max rn;min
! 2b
1
rn;min rn;max
4r02 ð2Nf Þ
j¼
ð12Þ
2b
4r0f e0f ð2Nf Þbþc þ f E 1 þ 2rn;max rn;max rn;min rn;max rn;min 2 ! 2b 4r02 1 rn;min f ð2N f Þ 4r0f e0f ð2Nf Þbþc þ 1 2 E rn;max 0 ðsmax smin Þðcmax cmin Þ 0 cmax þ cmin rn;max rn;min
2
ðsmax smin Þðcmax cmin ÞÞj0 h¼
ð13Þ
ð14Þ
c cmin 0 k ¼ max rn;max rn;min 0 smax þ smin l¼ rn;max rn;min
ð15Þ
ð16Þ
m¼
smax smin 0 rn;max rn;min
bþc rn;min 0 2r0f 2Nf 1 rn;max q¼ rn;max rn;min
ð20Þ
1 r¼ 2 rn;max rn;min
s¼
2b 8r02 f ð2N f Þ
E
1
rn;min 0 0 4rf ef ð2N f Þbþc ðlog 2 þ log Nf Þ rn;max
!!0 ðlog 2 þ log Nf Þ
1 2ðrn;max rn;min Þ
1
ð21Þ
0 rn;min ð4r0f e0f ð2N f Þbþc ðlog 2 þ log N f ÞÞ rn;max
The approximate mean and variance of Nf are obtained by isolating this variable from Eq. (4) and applying the expectation and variance operators.
E½Nf ¼
Xn
aX i¼1 i i
var½Nf ¼ ð10Þ
4r02 ð2N f Þ2b 4r0f e0f ð2Nf Þbþc þ f E rn;min
ð19Þ
ð22Þ
ð9Þ
0
0 8r0 ð2N f Þ2b rn;min 1 rn;max 4e0f ð2Nf Þbþc þ f E p¼ 2ðrn;max rn;min Þ
þ
en;max ¼ a þ f en;min þ g rn;max þ h rn;min þ j smax þ k smin þ l cmax þ m cmin þ n Nf þ p r0f þ q e0f þ r b þ s c
0
ð18Þ
ð8Þ
being ai = en,min, rn,max, rn,min, smax, smin, cmax, cmin, Nf, r0f , e0f , b, c. The term ai j0 is a random variable evaluated in the mean value of ai and li the mean value corresponding to each random variable. Eq. (9) shows the extended expression of Eq. (8).
0
0 16r02 bð2N f Þ2b1 rn;min 1 rn;max 8r0f e0f ðb þ cÞð2Nf Þbþc1 þ f E n¼ 2ðrn;max rn;min Þ
ð17Þ
Xn
¼ N0f
a2 var½X i i¼1 i
ð23Þ þ
Xn Xn i¼1
j¼1 i–j
ai aj cov½X i ; X j
2 2 1 f var½en;max þ var½en;min ¼ n n 2 g 2 h var½rn;max þ var½rn;min þ n n 2 2 2 j k l þ var½smax þ var½smin þ var½cmax n n n m2 p2 q 2 þ var½cmin þ var½r0f þ var½e0f n n n r 2 s 2 þ var½b þ var½c n n
ð24Þ
where the terms including cov½X i ; X j vanish due to the independence of the random variables. For non-independent variables, expression (24) becomes a little more complex than the one used in the following approach since the correlation function of each pair of variables has to be known. Finally, it should be noted that in Eqs. (23) and (24) the values related with fatigue material properties are known, but the means and variances of the structural strain and stress behavior of the component must be evaluated by means of a probabilistic calculation method such as the PFEM, Monte Carlo Simulation or similar. 4. Example of numerical application of the probabilistic formulation In this section, a complete reliability analysis in terms of fatigue life is applied to the cylindrical specimen shown in Fig. 1. An axial force of 28,500 N and a torque of 10,000 N mm act on the sample reproducing a sinusoidal cycling loading that produces a multiaxial stress–strain state on the component. The module of these loads (|p| and |t|) together with the Young modulus (E) characterizing the mechanical properties of the material are considered as
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Table 2 Elastic stress tensor in terms of expectation and typical deviation for the element numbered 19668. Element 19668
Mean value Typical deviation
r1
r2
r3
r12
r23
r31
394.7 19.07
23.2 1.19
18.4 0.9
363.1 18.21
4.6 0.22
11.5 0.57
Fig. 2. Location of nodes in the sample used to compare results.
Table 3 Mean and typical deviation in terms of fatigue life. Life
Monte Carlo simulation
PFEM + Liu prob.
Difference (%)
Mean Typical deviation
9105 cycles 2860 cycles
8205 cycles 2915 cycles
10 2
random variables in the basic probabilistic analysis carried out on the specimens. The mean values and the typical deviations of those variables are shown in Table 1. The problem is solved according to a PFEM approach developed by the authors (whose description is not the objective of this paper) and the two first statistical moments of the random stress and strain variables are obtained. Table 2 shows the elastic stress tensor in terms of expectation (mean value) and typical deviation for the element numbered 19,668 in the model (located in the central zone of the specimen and corresponding to the maximum stress area, as can be seen in Fig. 2) in which the multiaxial fatigue damage model proposed by Liu will be applied in tensile mode but formulated in a probabilistic way to determine the variable fatigue lifetime (Nf) of the sample. With the aim of verifying the statistical moments computed via PFEM, Monte Carlo simulations have been carried out considering a total of 1500 samples and varying the input variables. The results showed differences between the methods of less than 4%. The elastic stress and strain tensors are corrected to their related elastic–plastic values according to the equivalence of increments of the total distortional strain energy density method, which relates the elastic and elastic–plastic strain energy densities at the notch tip and the material stress–strain behavior [17]. In this
way, mechanical and fatigue properties characterizing the material are known. Table 3 compares the means and typical deviations achieved in the estimation of fatigue life using both methods, Monte Carlo simulations and the combination of the PFEM and the probabilistic Liu fatigue model proposed in this research. From the results shown in Table 3, it can be observed that the percentage difference between both sets of results is low, being 10% for the mean values and about 2% for the typical deviation results, this new procedure thus being more conservative. 5. Conclusions This work presents a probabilistic model to describe the phenomenon of crack initiation in metal multiaxial fatigue under stochastic conditions, based on Liu’s fatigue damage model. Besides the external load, additional random variables such as the cyclic fatigue material parameters, r0f , e0f , b and c, and elastic material parameters have been considered. The accuracy of the fatigue lifetime values obtained, when compared to an equivalent analysis performed with a Monte Carlo simulation approach, proves the good performance of the model. It is remarkable that results of the probabilistic model developed in this paper are contrasted with numerical data to demonstrate the accuracy of the first order expressions. Experimental data are not use in the comparison because is out of its reach to verify the suitability of the Liu model to solve this kind of multiaxial fatigue problems. Acknowledgement The authors acknowledge funding of this work by the Spanish Ministry of Science and Education through Project DPI200766903-C02-02.
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