1. Fatigue life assessment of structural component with a non-uniform stress distribution based on a probabilistic approach Aleksander KAROLCZUK. Thierry Palin-Luc Department of Mechanics and Machine Design, Opole University of Technology, Poland 17th European Conference on Fracture (ECF17) Brno. 2 – 5 September 2009 Arts et Métiers ParisTech, Université Bordeaux 1, France

2. 2 Plan of the presentation Introduction A short review of the fatigue life calculation approaches applied for structural elements with non-uniform stress distributions - summary A Weibull based mathematical model Implementation of the model for fatigue life calculations Experimental tests and results Numerical simulations and results Conclusions

3. 3 Introduction Element with a hole Elements under bending Elements with a non-uniform stress distribution Spherical defect Alternating heterogeneous stress distribution …. complex mechanisms of fatigue failure …

4. 4 A short review of approaches applied for elements with non-uniform stress distributions (i)Empirical approaches Murakami i Endo (defects): (ii)Approaches based on fatigue notch factor Mitchel (defects): Deterministic concepts (iii) Non-local approaches The so-called non-local approaches consider the interaction between locally damaged points located in the vicinity of the hot spot by normal or weighted integration process of local damage parameters. Line:  = L Area:  = A Volume:  = V

5. 5 Probabilistic concepts The fatigue failure could start at any elementary domain in element but the probability of such event depends on the local stress history. Thus. the fatigue failure of the whole element is a function of the cyclic stress volumetric distribution and dimensions of the considered structural element. This mechanism is usually taken into account by probabilistic methods based on the weakest link concept. (i) Bomas, Linkewitz, Mayr (for fatigue limit regime) (ii) Thomas Delahay, Thierry Palin-Luc A short review of approaches applied for elements with non-uniform stress distributions PfPf

6. 6 Deterministic concepts do not consider the statistical nature of fatigue Most of the probabilistic concepts based on weakest link concept are limited to fatigue strength regime There is necessity (and possibility) to build a probabilistic model for fatigue life calculation in the fatigue crack initiation regime Summary of review PfPf P f = 0.632 N, cycles

7. 7 A Weibull based mathematical model Aim: Fatigue life calculation N cal using probability approach under arbitrary fatigue regime (LCF-HCF) for a given (or requested) failure probability level P f. Main hypothesis: The weakest link concept might be applied in any arbitrary fatigue regime if the appropriate crack length defining the failure as a crack initiation is specified.

8. 8 A Weibull based mathematical model Weibull model The usual (Weibull - 1939) form of failure probability P f =1-P s is as follows Where :  0 reference domain (surface or volume).  0,  u, m – parameters of stress shift, stress scale and shape New concept The general form of probability distribution is analogous to the Weibull expression. However. the former stress function of ”risk of rupture” g(  ) becomes also function of lifetime N and the failure probability takes the general form as follows General form: The failure probability P f increases with increasing the stress level  a but P f (for a given  a ) also increases with the number of cycles N. The longer structural element is in service the failure probability is higher.

9. 9 Some researchers lean towards a view that the Weibull distribution describes the scatter of fatigue life (under a given stress amplitude  a ) in logarithmic scale fairly well. It could be expresses by the following equation where  is the lifetime scale parameter and m is the shape parameter. A Weibull based mathematical model Fatigue characteristic  = log( N f ) For another stress level  a the mean value of fatigue lives N(  a ) would be moved. Thus. the scale parameter  would also change its value. Relation between expected fatigue life N f and stress level is captured by fatigue characteristic, e.g. Wöhler curve. As a result the scale parameter  becomes function of expected fatigue life N f taken from fatigue reference curve:

10. 10 +++ aa NfNf where p is a constant parameter. The coefficient p can be seen as a quality factor of both element manufacturing process and material (internal defect for instance). Finally, the failure probability distribution of the component, before N under the stress amplitude  a, takes the form where  =V (volume) or  =A (surface). Some experimental results shows that the magnitude of fatigue life scatter depends on the stress level  a. Thus, the scale parameter m should be function of the stress level  a or expected fatigue life N f taken from reference fatigue curve. This relation is modeled by a proposed simple function (1) A Weibull based mathematical model

11. 11 Fig. 1. (a) Simulated two-dimensional distribution of the failure probability P f for the element made of 18G2A steel (p=580); (b) Fatigue reference curve  a -N f with experimental points and fatigue scatter bands for P f = 0.05 and P f = 0.95 (p=580). In the case of a uniform stress distribution, the previous equation reduces to Figure 1a shows an example of two-dimensional failure probability distribution according to Eq. (2). using the fatigue reference curve (  a -N f ) of 18G2A steel with p = 580. Fig. 1b illustrates experimental fatigue test data used to identify the reference Wöhler curve  a -N f along with scatter band obtained for P f = 0.05 and P f = 0.95. (2) A Weibull based mathematical model

12. 12 Implementation of the model for fatigue life calculations The free surface of the considered element is divided into sub-domains A (i) of sizes which allows appropriate integration process In each sub-domain A (i) multiaxial stress state is reduced to an equivalent stress amplitude state by using a multiaxial fatigue crack initiation criterion The equivalent stress amplitude and the fatigue reference curve  a -N f are used for calculations of a number of cycles to failure for each sub-domain A (i) Then. the survival probability distribution is determined as follows

13. 13 Implementation of the model for fatigue life calculations For each fatigue life N. exponents of e natural logarithm are summed along all the sub-domains A (i) and the survival probability distribution P s (N) for the whole structural element is obtained The fatigue life calculation N cal is performed for P f (N cal ) = 0.63. Fatigue life for any other probability level, i.e. the scatter of results, can be calculated in a similar way.

14. 14 Parameters identification Implementation of the model for fatigue life calculations Taking advantage of the empirical analytic equation of the reference Wöhler curve  a -N f. the two-dimensional distribution parameters has only two parameters to be identified. i.e. A 0 and p. The reference surface area A 0 is the free surface area of the specimen applied for determining the reference Wöhler curve. The parameter p responsible for the distribution of the fatigue life scatter can be identified from the tests of specimens having the same distribution of defects (kind and morphology) as the considered element. However, manufacturing qualities of elements and specimens are usually different. In such a case, distribution parameters should be fitted based on one series of tests of a real element subjected to simple fatigue loadings. In the present paper, the authors applied different values of the parameter p to find the best correlation between experimental fatigue lives N exp and the calculated fatigue lives N cal.

15. 15 (i) In the first set of experiments. cruciform specimens made of 18G2A steel with a central hole as stress concentrator were subjected to biaxial fatigue loading. Experimental tests and results Experimental results obtained from testing two steels with different specimen geometries were used for analyzing and verification of the proposed probabilistic method.

16. 16 Test conditions and results Speci men d, mm h, mm F xa kN F ya kN N i cycles A i, mm P023.01.4013.3013.10397000.22 P033.01.5413.5013.30311000.37 P043.01.8613.5513.30600480.07 P052.51.5010.219.902466950.25 P082.41.209.309.101670500.10 P073.01.7511.2010.801407000.20 N i is the numbers of cycles to crack initiation which corresponds to the crack length a i Experimental tests and results Time, s Force, kN

17. 17 The second set of experimental results comes from the work of Fatemi et al [1]. Circumferentially notched round bar made of AISI 1141 steel, in both the as-forged (AF) and quenched and tempered (QT) conditions were subjected to tension-compression loading. [1] Fatemi A., Zeng Z., Plaseied A. (2004), Fatigue behavior and life predictions of notched specimens made of QT and forged microalloyed steels, Int. J. Fatigue 26, pp. 663–672. Geometry with two notch radius were tested R=0.529 mm and R=1.588 mm. Experimental tests and results

18. 18 Numerical simulations and results The strain and stress distribution in the specimens were calculated using the 3D finite element analysis on COMSOL software. In all the computations a cyclic constitutive model with linear kinematic hardening was applied. The Lagrange elements (tetrahedrons) of order 2 with higher mesh density in the vicinity of the notch were used in the computations. Because of the symmetry of loading and geometry of the specimens, only 1/8 part of the cruciform specimens was modeled (Fig) and 1/32 part of round notched specimen was modeled. Careful analysis of mesh size influence on stress and fatigue life calculation has been performed.

19. 19 where n i is the unit normal vector to the plane experiencing the maximum normal stress or strain 1/8 of geometry of the cruciform specimen with distribution of the equivalent stress amplitude  eqa on the notch surface (specimen P05, F xa =10.21 kN, F ya =9.90 kN) 1/32 of geometry of the notched round bar with the distribution of the equivalent strain amplitude  eqa on the notch surface (state AF, R=0.529 mm, under nominal stress amplitude in net section  anom = 225 MPa) The criterion of maximum normal stresses (cruciform specimens) or strains (round specimen) on the critical plane was assumed as the criterion of multiaxial fatigue crack initiation. The equivalent stresses or strains are calculated according to the following equations Numerical simulations and results

20. 20 Numerical simulations and results Comparison of the experimental fatigue lives N exp with the calculated lives N cal for (a) p = 400 and (b) p = 580 for the cruciform specimens in 18G2A steel under biaxial tension (a) (b) Fatigue lives to crack initiation N cal were calculated for three failure probability levels: P f = {0.05; 0.63; 0.95} and for different values of the p parameter Additional two scatter bands (  2 and  3) around the solid line N cal =N exp of perfect results consistency are also shown in figures

21. 21 Numerical simulations and results Comparison of the experimental lives N exp with the calculated lives N cal for (a) p = 200 and (b) p = 580 for the notched round specimens in AISI 1141 steel under fully reversed tension (a)(b)

22. 22 Conclusions A procedure for determining the two-dimensional failure probability distribution P f -N-  /  of structural elements has been proposed. It allows the calculation of fatigue life of structural elements under any stress amplitude and probability levels. The presented approach allows calculating global fatigue life at any requested probability levels for elements with heterogeneous stress fields. The calculated fatigue lives N cal for failure probability equal to 63% are well correlated with the experimental fatigue lives N exp for notched round specimens made of AISI 1141 steel (with p =340) independently on the notch radius and material state (QT or AF); and for the notched cruciform specimens made of 18G2A steel (with p =580). The proposed probability distribution function to the fatigue macro crack initiation needs to identify only one additional parameter ( p ). This parameter can be seen as a quality factor of both element manufacturing process and material (internal defects for instance).