PSI - Issue 19

Volume 19 • 201 9

ISSN 2452-3216

ELSEVIER

Fatigue Design 2019, International Conference on Fatigue Design, 8th Edition

Guest Editors: Fa b ien L efe b vre P ascal Sou q uet C etim

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Fatigue Design 2019 Preface Fabien Lefebvre* Cetim, 52 avenue Félix Louat, 60304, Senlis, France Fatigue Design 2019 Preface Fabien Lefebvre* Cetim, 52 avenue Félix Louat, 60304, Senlis, France

© 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V.

* Corresponding author. Tel.: +33 340 673 419 E-mail address: Fabien.lefebvre@cetim.fr * Corresponding author. Tel.: +33 340 673 419 E-mail address: Fabien.lefebvre@cetim.fr

2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers.

2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 10.1016/j.prostr.2019.12.001

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Fabien Lefebvre / Procedia Structural Integrity 19 (2019) 1–3

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Peer-review under responsibility of the Fatigue Design 2019 Organizers.

Fatigue Design 2017 Group photo

On behalf of the Fatigue Design 2019 International Scientific Committee and Organizing Committee, we would like to welcome you all to the 8th edition of Fatigue Design 2019 taking place at Cetim, Senlis, France on November 20 & 21 2019. Since 2005, and every two years, the conference takes place at Senlis, the City of culture and history, located at 45km from Paris with more than 2000 years of history. The Notre-Dame Cathedral of 12th century, Saint-Pierre church, Medieval ramparts of 13th - 16th century, the château royal and 4 museums are among the most attractive places to be visited. Organized by Cetim and his partners, the 8th Fatigue Design 2019 International Conference aims to present the most innovative approaches, scientific and technological progress in design methodologies, testing methods and tools to evaluate and extend the fatigue life time of the industrial equipment. The papers are mostly focusing on the industrial applications. Japan has been proposed as th e “partner country” for Fatigue design 201 9 in respect to the japan advance research works in the area of fatigue and fracture mechanics in the last decade. More than 20% of conference speakers are coming from Japan including 2 plenary key speakers.

This edition of the conference will cover the following scientific and technical topics: • additive manufacturing • experimental and numerical design and validation methods, • damage tolerance and fatigue life, • reliability -based approaches and probabilistic methods, • fatigue under severe conditions environment (corrosion, low temperature...), • nonlinear behavior and cumulative damage, • fatigue of assemblies (mechanical, welded, adhesive -bonding, multimaterial...), • composite and elastomers,

Fabien Lefebvre / Procedia Structural Integrity 19 (2019) 1–3 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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• contact fatig ue, • vibration fatigue behavior, • complex loadings, • thermal and thermo -mechanical fatigue, • taking into account manufacturing process in fatigue analysis (effect of microstructure, welding, stress relief techniques...). The presentations will focus on the latest development and most recent experimental, numerical simulation techniques and the associated engineering tools applied to the large domain of the industrial applications. The 8th edition of the Fatigue Design International conference is organized in close collaboration with Elsevier editor for the proceedings ’ publication through Structural Integrity Procedia. The papers are published online on ScienceDirect, which makes available worldwide for a better dissemination and maximum exposure. In this respect the selection and peer review of the papers have been done in collaboration with the International Scientific and Organizing Committees. “Fatigue Design” became the reference conference to address the concerns of industrials on fatigue design of structures and components. It is also considered as the trade crossroads between industry and academia: 84 oral presentations are performed with 50% by industry. We hope that the speakers and the delegates would have a fruitful exchange and discussions on technical and scientific developments and issues during the conferences. The poster sessions and exhibition stands are the complementary opportunities to facilitate the exchanges between the Scientists, industry participants, PhD students and the solution providers. We would like to thank members of the International Scientific and Organizing Committees for their valuable scientific support for the selection and paper reviews, the authors, delegates for their contributions, exhibitors and sponsors, and SF2M and the colleagues of Cetim for the organization and to make this conference successful one.

We wish you all a most pleasant stay in Senlis and hope that you will enjoy the conference.

Fabien Lefebvre Chairman Organizing Committee Fatigue Design 2019 Pascal Souquet Co-Chairman Organizing Committee Fatigue Design 2019

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Fatigue Design 2019 Adapted Locati method used for accelerated fatigue test under random vibrations Yuzhu WANG a* , Roger SERRA a et Pierre ARGOUL b a INSA Centre Val de Loire, Laboratoire de Mécanique G. Lamé, Campus de Blois, Equipe DivS, 3 rue de la chocalaterie, 41000 Blois, France b IFSTTAR, Laboratoire Mast – EMGCU, 14-20 Boulevard Newton, Cité Descartes, 77447 Marne-la-Vallée Cedex 2, France Abstract Fatigue experiments are very important for the study of structural fatigue damage. As we all know, fatigue experiments usually consume a lot of resources and time, slowing down the research progress. The Locati method, which is applied to the frequency domain analysis method of random vibration fatigue, replaces the traditional up-down method and accelerates the progress of the fatigue experiment. This method can effectively improve efficiency and reduce test time and test piece consumption under the premise of ensuring accuracy. It is based on the material S-N curve, after one step-load fatigue test, combined with the FEM, the cumulative damage of the structure under random vibration load is quickly determined, and the life analysis is completed. The specimen was made by AISI304 and designed to make the fatigue damage of the second-order bending mode more significant. The random vibration acceleration defined by the Power Spectral Density method is used as a load and applied on the specimens through the shaker. In this process, the material properties of the structure will be optimally fitted according to the experimental results, so that the finite element model is as close as possible to the real structure for fatigue calculation. Finally, an experiment will be used to verify the accuracy of this method. The method has now been validated under Gaussian distributed signals. Fatigue Design 2019 Adapted Locati method used for accelerated fatigue test under random vibrations Yuzhu WANG a* , Roger SERRA a et Pierre ARGOUL b a IN A Centre Val de Loire, Laboratoire de Mécanique G. Lamé, Campus de Blois, Equipe DivS, 3 rue de la chocalaterie, 41000 Blois, France b IFSTTAR, Laboratoire Mast – EMGCU, 14-20 Boulevard Newton, Cité Descartes, 77447 Marne-la-Vallée Cedex 2, France Abstract Fatigue experiments are very important for the st dy of structural f tigue damage. As we all k ow, fatigue exp riments usually cons m a lot of resources and time, slowing down the r sear h progress. The Locati ethod, which is applied to the frequency do ain analy is method of random vibration fatigue, replaces the traditional up-down method and a celerat s the progress of t fatig e experi ent. This method can effectively improve efficiency and reduce test time and t st piece consumption under t pr mis of ensuring accuracy. It is based on the terial S-N curve, after ne step-load fatigue test, co bined with the FEM, t cumulative damage of the structur under random vibration load is quickly det rmined, and the life analysis is co pleted. The sp cimen was made by AISI304 nd designed to make the fatigu damage of the second-order bending mode more significant. The random vibration acceleration defined by the Power Spectral Density method is us d s a load and applied on th specimens through the shaker. In this process, the material prop rties of the structure will b optimally fitted according to the experimental results, so that the finite element model is as close as possible to the real structure for fatigue calculation. Finally, an experiment will be used to verify the accuracy of this method. The method has now been validated under Gaussian distributed signals.

© 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility f the Fatigue Design 2019 Org nizers. Keywords: Vibration fatigue, Random vibration, Damage cumulative calculation;

* Corresponding author. E-mail address: yuzhu.wang@insa-cvl.fr Keywords: Vibration fatigue, Random vibration, Damage cumulative calculation;

* Corresponding author. E-mail address: yuzhu.wang@insa-cvl.fr

2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers.

2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 10.1016/j.prostr.2019.12.073

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1. Introduction In random vibration fatigue experiments, the results are usually different from theoretical calculations. The fatigue property of the material plays an important role in the results. In addition, the thermal effects of the process and the involvement of stress can affect the fatigue properties of the material. In the process of fatigue damage calculation, re evaluating the fatigue performance of the specimen is also one of the keys to ensuring the accuracy of life prediction. (PU.Nwachukwu & L.Oluwole, 2017) Fatigue limit can be determined using the Up-and-Down method (also known as Staircase) (Lin, Lee, & Lu, 2001). It usually requires testing a large number of specimens to obtain reliable fatigue data (R.Serra & L.Khalij, 2016). Considering that a single experiment requires between 2 to 10 million cycles, the cost of the overall experiment cannot be ignored. L.Locati proposed a method for quickly determining the fatigue limit when the fatigue properties of the material are known. This method uses several empirically to assumed S-N curves and step-load fatigue tests (L.Locati, 1955). It is based on experience to improve the efficiency of the experiment. Compared to the original method, the fixed frequency cyclic load is replaced by a random load around the second bending frequency range of the specimen using a vibration shaker, in order to make it suitable for random vibration fatigue experiments. This method aims to quickly determine the fatigue limit of the specimen. By experimental data, a finite element model is updated to be close to the real structure and used into fatigue calculation and future finite element simulations. According to the fatigue properties of the material, the actual relationship between stress and period is calculated by using the assumed S-N curve of the specimens and the results of the random vibration experiments which the determination of the initial load directly affects the efficiency and progress of the experiment. Then the damage calculation and life prediction are obtained and a new S-N curve is proposed. The acceleration-time (A-T) curve of the specimens is then established and the appropriate initial load can be directly selected for further experiments. In this way, the cost of the experiment (the number of test pieces and the test time) can be greatly reduced.

Nomenclature N

Number of cycles

N c

Turning point of the S-N curve

S

Stress range

C,m

The material constants are related to the stress ratio and the stress concentration factor

PSD Power Spectral Density in g 2 /Hz PDF Probability Distribution Function G(f) Input acceleration by frequency domain W(f) Response acceleration by frequency domain H(f) Frequency Response Function L e Failure time S f Fatigue limit stress

2. S-N curve The S-N curve can be approximated as linear in double logarithmic coordinates. It is generally composed of three inclined parts and horizontal parts. The N c is the turning point of the S-N curve (NE.Dowlings, 1993).: (1) When the stress ratio is -1, the material is subjected to alternating load, and the fatigue limit is represented by σ �� . For Equation 1, take the logarithm of the two sides and get the linear equation. m S N C  

676 Yuzhu Wang et al. / Procedia Structural Integrity 19 (2019) 674–681 Y.WANG, R.SERRA & P.ARGOUL/ Structural Integrity Procedia 00 (2019) 000–000 (2) Currently measuring the S-N curve is generally a set of specimens (take 6~10 piece), using the lifting method, at different levels, fatigue test, measuring the life of the fracture N i , a series of test points and then connected by a smooth curve. The S-N curve of the material measured by this method is only 50% reliable. Moreover, this curve has a great walk in fatigue performance, and sometimes there are differences for the same batch of materials. After processing, the actual fatigue performance and curve performance of the structure is too different. If we use this curve to design the initial experimental load, it may occur, the initial load is too small , the time is wasted, and the initial load is too large to waste the specimen. 3. Random processes and spectrum analysis Generally, a zero-mean, standard deviation of 1, Gaussian random distributed load is used to perform a random vibration fatigue experiment, usually expressed as G(f) . The process is characterized in the frequency domain by one side with its power spectral density W G (f) , the spectrum parameter could be obtained. The moments of each step are calculated as follows (DE.Newland, 1993).       2 G W f = G f H f (3)   0 0,1, 2, j j G M f W f df j n      (4) Then the according to equation (4) ， the time domain property of W G (f) could be obtained � = �� , � = �� and � = �� (5) After the expected number of upward crossing zero E 0 and the peak rate E p for W G (f) could be obtained , as follows 2 0 0 E = M M 4 p 2 E = M M (6) Further, the spectral and irregular factor � bandwidth parameter � could be obtained. The bandwidth parameter when j =2 is usually used. 2 2 0 4 0 P E M M E M    (7)   2 2 2 ε 1 0,1      ， (8) When ε � is greater than 0.3, the vibration is a broadband random vibration, and when ε � is less than 0.3, the vibration is a narrowband random vibration. When ε � is close to 1, the vibration process approaches white noise. 4. Theory of Fatigue Analysis What is now commonly used for fatigue calculations is the fatigue damage linear accumulation principle from Miner in 1945 (MA.Miner, 1945). 3 o l gN = logC - m logS 

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1 (9) Where � is the number of cycles in the stress amplitude � , which was obtained by rain flow counting. � is the cycles of fatigue failure limit corresponding S-N curve at the same stress amplitude. When D=1 , it means structural failure. When calculating the random vibration fatigue of the probability density function (PDF) as p(S) , the number of cycles of duration T in ( � , � + ∆ ) , can be denoted as follows:   i p n E T p S S       (10)   0 p m E T D S p S dS C      (11) In past studies, scholars used many different statistical models to approximate the distribution of the cycles of specifying stress range during random vibration. There are 8 models that are discussed in the study and verify the actual effect, which is necessary because its effect directly affects the accuracy of life prediction. Bendat proposes a narrow-band signal with as the bandwidth decreases, the probability density function of the peak tends to Rayleigh distribution (EL.Crow, JS.Bendat, & AG.Piersol, 1966). This will be able to easily find the fatigue life, ��� is the root mean square of stress in specifying the frequency range Narrowband approximation of a broadband process can be approximated by narrowband for a wideband process. Many scholars propose a correction factor to correct the fatigue life of a narrowband to obtain a broadband distribution of fatigue life (G.Chaudhury, 1986). The bimodal response spectrum approach considers bimodal spectral density as a superposition of two narrow-band random processes. The specific methods are Wirsching-light method, Ortiz and Chen method, Tovo-Benasciutti method and so on (H. Wirsching, 1980) (T.Dirlik, 1985) (D.Benasciutti, 2019) (D.Benasciutt, A.Cristofori, & R.Tovo, 2013). m i    i i i n D n S N C 

Fig. 1. The probability distribution function of response stress cycles

As shown in the figure 1, the curve of Dirlik using the gamma equation is considered to be the most suitable.

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5. Locati method The Locati method was proposed by L. Locati in 1955 to accelerate fatigue experiments and obtain fatigue property curves for materials fast (L.Locati, 1955). This method uses a step load to replace a very constant load, which greatly reduces the test time and also reduces the number of test pieces required. With Locati method, it is possible to approximately estimate the fatigue limit when some fatigue parameters are known in advance, e.g. the slope of the S-N curve’s left branch. The Locati method is based on Miner law. This method uses the gradual loading of the tested specimen. The initial stress value should not exceed the lowest expected value of fatigue limit. The fixed increment of the gradual load amplitude is assumed in the method. Similarly, the assumed number of cycles n i = ΔN to be executed at individual levels is also fixed. Three curves are assumed based on the slope of the material S-N curve. At the bottom is the minimum estimate, the top is the maximum estimate, and the middle is the estimated mean. These three curves will have three intersections with the previous experimental plots. Record the intersection point S and N , and calculate the damage caused by each load S i (i=1,2,3) when the duration is N , and record it as D i (i=1,2,3) . The D-S curve is then plotted based on the S i and D i and S-N curve shapes. When D =1, through the curve, the fatigue limit of the structure S f can be obtained.

a

b

Fig. 2. (a) Step-stress applied in Locati method, (b)Damage-Stress curve based on step stress load

6. Experimental random vibration fatigue test According to the adapted Locati method, before the simulation, one complete step load fatigue test should be performed, the life L e was recorded and used into fatigue analysis. The fatigue experiment of random vibration is usually performed by a vibration shaker and o close loop control system. A specific input acceleration signal usually expressed by PSD within the specified frequency range. In order to speed up the experiment and ensure the damage to appear in the observable position, the specimen is designed with two notches and the stress is concentrated at the notch position by changing the cross-sectional area, which means that the crack initiation and expansion area has been knowing. Through the CAE design, the length of the specimen is adjusted so that the displacement of the second bending mode is at the maximum position in the middle of the first notch.

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Accelemeter

Fig. 3.Specimen geometry The specimen is made of AISI304 and Young's modulus (E=1.69e11 Pa), density (ρ=8000kg/m3) and another property of material can be found from the material card. According to the measurement of the specimens, the actual quality can be obtained, the average of 5 specimens is 113.6g. After the actual material density can be calculated as 7942.22kg/m3. It can be seen that there are subtle differences in the material properties of the universal material card. Specific analysis of each test piece is the key to ensuring an accurate test.

C losed loop

Fig. 4.Schematic diagram of the test Through the modal analysis, the bending mode of the specimen under the clamping-free state can be obtained.

Table 1. Bending mode of the specimen by experiment

Mode

1

2

3

Natural frequency (Hz)

17.1

84.4

229.9

For Gaussian random vibration, PSD used to describe the level of acceleration. This time, the acceleration for a constant RMS value within the range of the second bending mode evaluation is selected. According to the modal analysis results, load frequency range choose 40-130 Hz (84.4±45 Hz), so the bandwidth is 90Hz. For the initial load, choose S 0 =0.25 g 2 /Hz, choose a step size of λ =0.05 g 2 /Hz until its failure. Choose the load shape as follows according to the Fig.5. Through PSD calculation, it could obtain the equivalent acceleration with unit g for all the level of the load as Table 2, it can be found that these two parameters are linearly related in the sample PSD (DE.Newland, 1993).

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Table 2. PSD-gRMS table

PSD : g 2 /Hz 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 gRMS : g 4.75 5.20 5.62 6.01 6.37 6.72 7.04 7.36 7.66

After one complete test, the specimen failed after a total of averaged 8.5 hours and the test was finished in load level of 0.65g 2 /Hz.

a

b

Fig. 5. PSD shape of the initial test (a) input acceleration, (b) response acceleration

7. Simulations In order to perform virtual test for response analysis, a finite element model was built based on the material card and geometry (ALTAIR Hypermesh). After, the modal analysis was performed using Abaqus on this model. Through the comparison with the experimental results, the relevant parameters are updated to obtain a finite element model close to the experimental specimen responses. After validation and verification, the refined FE model is more closed to the real specimen. For the harmonic analysis, the stress frequency response function (FRF) for the unit acceleration load (1g) is obtained in the specified frequency range. In the vibration fatigue analysis, one of the nodes with the highest stress in the second-order bending mode in the notch is selected for the evaluation index of the PSD response using ABS MAX principle stress. Then, according to each level of the experiment loading, each amplitude stress under the load is defined. Further, the spectral coefficient of the PSD corresponding to the load can be calculated, and the damage value determined according to the previously mentioned distribution model. Comparing the eight previous methods by difference from the average value, the best method is selected. According with the initial S-N curve of the material card, the limit value of the low-cycle fatigue portion is 1729 MPa, and the m value is 17.92. Obviously, these parameters are a bit high relative to the actual material. In fact, after 8.5 hours (as obtained experimentally), the damage calculation result is 0.035502 < 1. When the slope of the original S-N curve is kept constant, different fatigue limit values are considered. When the ultimate strength value equal to 1421 MPa, after 8.5 hours, the damage value is 0.99 which obtained by interpolation calculation. Then the rest parameter of S-N curve calculated based on fatigue limit. Finally, the corrected S-N curve for the specimen is as shown in Fig.6.

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a

b

Fig. 6. (a) Corrected S-N curve of the specimen, (b) A-T curve of vibration test According to the corrected S-N curve, the A-T curve of the specimen in the experimental frequency range could be determined. It is used as a reference to guide the relevant fatigue experiments of similar models and to specify the initial load and reference range of the experiment. 8. Conclusion By using adapted Locati method, virtual test by the finite element simulations can be performed after one completed test, and the S-N curve of the specimen can be corrected. Obtaining an A-T curve for a specified PSD spectral shape based on the corrected S-N curve of the specimen could improve the accuracy and efficiency for similar experiments. Acknowledgments This contribution has been elaborated under the China Scholarship Council. References EL.Crow, JS.Bendat, & AG.Piersol. (1966). Measurement and analysis of random data. G.Chaudhury. (1986). Spectral fatigue of broad-band stress spectrum with one or more peaks. H. Wirsching& C. Light, MarkP. (1980). Fatigue under wideband random stress. Journal of the Structural Division. L.Locati. (1955). Le prove di fatica come ausilio alla progettazione ed alla produzione. Matallurgia.It. LinS., LeeY., & LuM. (2001). Evaluation of the staircase and the accelerated test methods for fatigue limit distributions. International Journal of Fatigue, 75-83. MA.Miner. (1945). Cumulative damage in fatigue Jour. Journal of Applied Mechanics. NE.Dowlings. (1993). Mechanical behavior of materials. Prentice Hall: Englewood Cliffs(NJ). PU.Nwachukwu, & L.Oluwole. (2017). Effects of rolling process parameters on the mechanical properties of hot-rolled St60Mn steel. Case Studies in Construction Materials, 134-146. R.Serra, & L.Khalij. (2016). Effets de l’environnement vibratoire sur la durée de vie d’une éprouvette en aluminum. SF2M, 35èmes Journées de Printemps. T.Dirlik. (1985). Application of computers in fatigue analysis. D.Benasciutt, A.Cristofori, & R.Tovo. (2013). Analogies between spectral methods and multiaxial criteria in fatigue damage evaluation. Probabilistic Engineering Mechanics. Probabilistic Engineering Mechanics. D.Benasciutti. (2019). Fatigue analysis of random loadings. A frequency-domain approach. DE.Newland. (1993). An Introduction to Random Vibration. Spectral Analysis & Wavelet Analysis.

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Fatigue Design 2019 An alternative approach to calibrate multiaxial fatigue models of steels with small defects L.C. Araujo a *, P.V.S. Machado a , M.V.S Pereira b , J.A. Araújo a a University of Brasília, Departament of Mechanical Engineerinng, Brasília, DF 70910-900, Brazil b Catholic University of Rio de Janeiro, Department of Chemical and Materials Engineering, Rio de Janeiro, RJ 22453-901, Brazil Fatigue Design 2019 An alternative approach to calibrate multiaxial fatigue models of steels with small defects L.C. Araujo a *, P.V.S. Machado a , M.V.S Pereira b , J.A. Araújo a a University of Brasília, e rtament of Mechanical Engineerinng, Brasília, DF 70910-900, Brazil b Catholic University of Rio de Janeiro, Department of Chemical and Materials Engineering, Rio de Janeiro, RJ 22453-901, Brazil In this work the authors propose a new methodology to estimate threshold loading conditions for naturally defective steels under complex stress states. In this setting, multiaxial fatigue models based on the critical plane approach and in the stress invariant approach are adapted to assess the effect of non-metallic inclusions in the material. In order to do so, it will be used a method that relates the √ parameter of a small defect and the Vickers hardness of the material ( ) with the nominal fatigue limit for this material either under uniaxial ( ) or under torsional ( ) loading. Therefore, uniaxial and torsional fatigue limits usually required to calibrate the multiaxial criteria will be calculated by the √ parameter. With this methodology the multiaxial fatigue criteria utilized are not changed. However, calibrating them is considerably cheaper and faster compared to traditional methods. To assess the new methodology, combined axial-torsional multiaxial fatigue data in DIN 42CrMo6 steel generated by the authors will be used. The proposed methodology is compared to the experiments and agreement within 5% error bands were obtained. In this work the authors propose a new methodology to estimate threshold loading conditions for naturally defective steels under complex stress states. In this setting, multiaxial fatigue models based on the critical plane approach and in the stress invariant approach are adapted to assess the effect of non-metallic inclusions in the material. In order to do so, it will be used a method that relates the √ parameter of a small defect and the Vickers hardness of the material ( ) with the nominal fatigue limit for this material either under uniaxial ( ) or under torsional ( ) loading. Therefore, uniaxial and torsional fatigue limits usually required to calibrate the multiaxial criteria will be calculated by the √ parameter. With this methodology the multiaxial fatigue criteria utilized are not changed. However, calibrating the is considerably cheaper and faster compared to traditional methods. To assess the new methodology, combined axial-torsional multiaxial fatigue data in DIN 42CrMo6 steel generated by the authors will be used. The proposed methodology is compared to the experiments and agreement within 5% error bands were obtained. Abstract Abstract

© 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. Keywords: non-metallic inclusions; defective materials; multiaxial fatigue; small defects. Keywords: non-metallic inclusions; defective materials; multiaxial fatigue; small defects.

1. Introduction 1. Introduction

The effect of small defects and nonmetallic inclusions on fatigue strength of materials has been the object of study The effect of small defects and nonmetallic inclusions on fatigue strength of materials has been the object of study

* Corresponding author. Tel.: +55 61 31071148; fax: +55 61 31075707. E-mail address: lucasc.araujo@aluno.unb.br (L.C. Araujo), pedromachado@aluno.unb.br (P.V.S. Machado), marcospe@puc-rio.br (M.V.S Pereira), alex07@unb.br (J.A. Araújo). * Correspon ing author. Tel.: +55 61 31071148; fax: +55 61 31075707. E-mail address: lucasc.araujo@aluno.unb.br (L.C. Araujo), pedromachado@aluno.unb.br (P.V.S. Machado), marcospe@puc-rio.br (M.V.S Pereira), alex07@unb.br (J.A. Araújo).

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of several researchers over the last decades (Murakami (2002), Endo and Ishimoto (2006)). The importance of this type of study is due to the detrimental nature of these defects and inclusions, that can be observed in the recent failures in crankshafts of Brazilian thermal electric power-plants. The steel of the crankshafts contains non-metallic inclusions that can be observed in the region of initiation of cracks, as can be seen in the Fig. 1.

Fig. 1. Non-metallic inclusion in the region of initiation of the crack, (a) fracture surface and (b) inclusion detail. A parameter that has been widespread and that relates the fatigue strength of internally defective metallic materials or with small defects and inclusions is the √ , proposed by Murakami (2002). This method has been used to estimate the uniaxial fatigue limit, , successfully in many cases without the need of fatigue tests. And it was expanded by Yanase and Endo (2014) in order to also estimate the torsional fatigue limit, . Recently, researchers have been working on developing new models to predict the effect of these small defects on fatigue resistance of materials subjected to complex multiaxial loading conditions (Endo and Ishimoto (2006)). However, in this work, the effect of small defects in the fatigue strength of materials will be modeled without the use of any new criteria. The feasibility of combining classic models of multiaxial fatigue with the fatigue limit estimated from the √ parameter will be tested. For this, the models of Crossland (1956), Findley (1959) and Susmel and Lazzarin (2002) are used. 2. Multiaxial fatigue models The Crossland (1956) model which is a multiaxial fatigue strength model based on stress invariants evaluates the fatigue resistance from the history of the deviatoric stress tensor. This is expressed as follows: √2 + = (1) where is the amplitude of the history of the deviatoric stress tensor, is the maximum hydrostatic stress, and are parameters of the material and can be obtained from Eq. 1, with the fatigue limit obtained from simple uniaxial tests ( 0 ) and simple torsional tests ( 0 ), both with loading ratio R of -1, and are obtained as follows: = 0 0 − √3 (2) and = 0 (3) The critical plane models of Findley and Susmel and Lazzarin consider that for metallic materials the driving force

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for nucleation of a crack is the shear stress and that the propagation is strongly influenced by the normal stress. Findley (1959) was the first researcher to use the concept of critical plane with his model, which makes a linear relationship between the shear stress amplitude, , and the maximum normal stress, , . It considers the critical plane as the one with maximum combination of these two parameters. The Findley model for fatigue strength is expressed by Eq. 4. ( , ) ( + , ) = (4) where the critical plan ( , ) is defined by the maximum value of the term in brackets found from all plans ( , ) that pass through a material point of a component submitted to a stress history. The and are parameters of the material and can be calculated from Eq. 4, with the fatigue limit obtained from simple uniaxial tests ( 0 ) and simple torsional tests ( 0 ), both with loading ratio R of -1, as shown below: = 1 √ − − 0.5 1 (5) and = 0 2√ − 1 (6) where = 0 / 0 . Susmel and Lazzarin (2002) proposed the Modified Wöhler Curves Method (MWCM). In this model the critical plane is one of the planes that experience the maximum amplitude of the shear stress and among those the one with the highest normal stress. In the condition of the fatigue strength the MWCM is represented as follows: ( , ) + ( , ) = (7) as in the Findley ’s model and are parameters of the material and can be obtained from Eq. 7, with the fatigue limit obtained from simple uniaxial tests ( 0 ) and simple torsional tests ( 0 ), both with loading ratio R of -1 and are defined as in Eq. 8 and 9. = 0 − 0 2 (8) = 0 (9) Susmel, Tovo and Lazzarin (2005) also defined the variable of Eq. 7, which represents the influence of normal stress on fatigue strength, limiting the use of the model, which applies up to a limit value, , above which it has no meaning and therefore the MWCM cannot be applied. The values of and are obtained as follows: = , (10) = 0 2 0 − 0 (11) There are several known ways of finding the values of , for the Crossland model, and of finding the values of , which is applied in both the Findley model and the MWCM. Therefore in this work, the Maximum Rectangular Hull (MRH) method proposed by Araújo et al. (2011) were used, which in addition to being easy to implement considers the effect of non-proportionality of stress histories. This method considers as the amplitude value the diagonal of the largest rectangle capable of enveloping the shear stress history.

22 L.C. Araujo et al. / Procedia Structural Integrity 19 (2019) 19–26 L.C. Araújo et al. / Structural Integrity Procedia 00 (2019) 000 – 000 3. Fatigue limt from √ parameter The estimated value of the uniaxial fatigue limit, , with loading ratio R of -1 can be found from the relation of Murakami (2002), which is expressed by Eq. 12. = 1.41( + 120) (√ ) 1 6 (12) Yanase and Endo (2014) proposed that the torsional fatigue limit, , also with R of -1, is obtained from Eq.13. = 1.19( + 120) (√ ) 1 6 (13) where in Eq. 12 and 13, is the Vickers hardness of the material. The √ parameter corresponds to the square root of the projected area of a micro defect in the cross-section perpendicular to the maximum principal stress for the uniaxial and torsion cases. Considering the non-metallic inclusions of naturally defective materials, the value of √ can be obtained through a sample analysis and with a method of statistics of extreme value that are described in detail on Murakami (1994). With this statistical method it is possible to predict which is the largest √ that one expects to find from an internal inclusion. The √ is a function of the volume of material, which may be of the test specimen or of an entire mechanical component. The maximum estimated value of √ is renamed √ and this value that will be applied in Eq. 12 and 13. 4. Adapting the models of multiaxial fatigue to consider the small defects The purpose of this work is to combine the classical models of multiaxial fatigue with the fatigue limits estimated from the √ parameter, obtained for internally defective materials, considering the non-metallic inclusions of the material. For this, instead of calibrating the multiaxial models with data obtained from fatigue tests, it is suggested the use of the estimated limits, and , in the calibration of the models. So, in determining the material constants of each model the following relationships must be assumed: 0 = (14) 0 = (15) The great advantage in the use of the constants obtained from the estimated limits is that these limits are obtained relatively easily, quickly and at a reduced cost when compared to the experimental methods of lifting the fatigue limits. Another, is that with the use of the theory of the statistics of extreme value it is possible to consider the effect of size on the design of mechanical components, which is not feasible experimentally since tests would have to be made with components in real scale. 5. Material and experimental procedures The material used in the manufacture of the specimens is the DIN 42CrMo6 (AISI 4140) steel oil quenched and tempered at 600 °C, taken from crankshafts of generator sets, whose the mechanical properties are shown in Table 1. The specimens are smooth and round with diameter of 10 mm in the test section and have been designed according to international standard ASTM E466-15 (2015). 4

Table 1. Mechanical properties of analysed DIN 42CrMo6 steel. Yield Strength (MPa) Tensile Strength (MPa)

Elongation (%)

Vickers hardness

710

900

20

320

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The tests were conducted in the laboratories of the Department of Mechanical Engineering of the University of Brasília. The equipment used was a servo-hydraulic axial-torsional fatigue testing machine (Fig. 2 (a)) equipped with 100 kN and 1100 Nm load cell operating between 5 and 15 Hz and capable of applying combined loading in phase and out of phase. During the tests the ambient temperature was controlled between 20 and 23 °C. All the tests were conducted in load control with fully reverse sinusoidal load, that is, with loading ratio R of -1. The values of the shear stress and normal stress ratio used were, / , of 0, 0.5, 1, 2 and ∞ . The tests with combined loading were conducted in-phase, that is, with phase angle, , of 0. The complete rupture of the specimen was considered as failure, as seen in Fig. 2 (b). The stop criterion was when a specimen did not fail when reaching 2 × 10 6 cycles.

Figure 2. Axial-torsional equipment used, (a) axial-torsional test system, (b) test mount detail with a broken specimen.

6. Comparison of predictions with the experimental results The values of the fatigue limit obtained from the √ parameter with Eq. 12 and 13 are shown in Table 2. They were estimated considering two volumes of material to exemplify the possibility of considering the size of the component for mechanical designs. The first volume refers to the useful section of the specimen used during the tests and the second represents the crankshaft from which the material for producing the specimens was removed. Table 2. Results of statistics of extreme value theory applied to inclusion’s √ . Prediction Volume Volume ( 3 ) (MPa) (MPa) √ ( ) at 90° √ ( ) at 45° Cumulative distribution (%) 1 Specimen 2.4 × 10 3 360 301 18.5 27.6 99.893 2 Crankshaft 7.9 × 10 9 311 260 63.5 65.4 99.9999996 Fig. 3 shows the fatigue limit prediction obtained from the Crossland ’s model and the experimental results of uniaxial, torsional and combined in-phase loads. The √ parameter values used in the calibration of the models for the comparisons made below correspond to the volume of the specimen shown in Table 2. Error bands of 5% where drawn on the graphs for comparison. At least two tests were performed at each stress levels, because of this there is overlap in the experimental points, which is evidenced where a failure and a run-out at the same stress level was obtained.

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Figure 3. A comparison of predictions with Crossland ’s model and experimental results of uniaxial, torsional and combined in-phase loads. It can be observed a good agreement of the experimental results with the prediction of the model. The same comparison is made in Fig. 4 and 5 with the Findley ’s model and the MWCM, respectively. The values of and , used to plot the experimental results of the MWCM model, Fig.5, were obtained as demonstrated by Araújo et al. (2011). They consider a range of 99% of the maximum value of and within of this range searches in all planes the value of the maximum normal stress , , and thus the definition of the critical plane is made. In the Findley ’s model for pure torsional loads, / = ∞ , the forecast is somewhat non-conservative, the opposite happens with the MWCM in uniaxial cases, / = 0 , where the forecast is more conservative and this can be attributed to the fact that the value of in this loading condition is approaching the value of . However, it can be observed that overall, results sit within 5% error bands.

Figure 4. A comparison of predictions with Findley ’s model and experimental results of uniaxial, torsional and combined in-phase loads.

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Figure 5. A comparison of predictions with MWCM and experimental results of uniaxial, torsional and combined in-phase loads. In Fig. 6, the difference in the fatigue limit prediction with volume variation is exemplified. For this Crossland ’s model and the data referring to the √ parameter for the specimen volume and the crankshaft volume, available in Table 2, were used.

Figure 6. Difference on the fatigue strength, predicted with the √ parameter with the variation of material volume. This decrease in predicted fatigue strength applies to all multiaxial models since with increasing volume a larger defect is expected to exist in the material and thus the estimated limits, e , decrease.

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7. Conclusions As it was observed, it is possible to calibrate known multiaxial fatigue models using the √ parameter obtained for internally defective materials, specifically DIN 42CrMo6 steel, considering the area of the non-metallic inclusions of the material. It was shown that modeling the effect of natural defects in this steel is possible with classical models. The use of this method spends much less time and resources in the calibration of multiaxial models, when compared to traditional methods. The experimental data of combined in-phase axial-torsional loads appears to follows the model’s predictions trend as shown by the fact that most data points are within 5% error bands. The proposed model is useful to consider the size of a component in the determination of the fatigue limit with the √ parameter, which for mechanical design purposes can be considered as an advantage, once with a more conservative approach the risk of failure reduces. Acknowledgements The authors of this work would like to acknowledge the financial support provided by FAP-DF by means of the project entitled “ Fadiga em Virabrequins de Grupos Geradores ”. Araújo, J. A. et al. (2011) ‘On the characterization of the criti cal plane with a simple and fast alternative measure of the shear stress amplitude in multiaxial fatigue’, International Journal of Fatigue , 33(8), pp. 1092 – 1100. Crossland, B. (1956) ‘Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel’, International conference on fatigue of metals , 6(3), p. 12. E466- 15 (2015) ‘Practice for conducting force controlled constant amplitude axial fatigue tests of metallic materials’, in ASTM Book of Standards . Endo, M. and Ishimoto, I. (2006) ‘The fatigue strength of steels containing small holes under out -of- phase combined loading’, International Journal of Fatigue , 28, pp. 592 – 597. Findley, W. N. (1959) ‘A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending.’, Journal of Engineering for Industry , 81, pp. 301 – 306. Murakami, Y. (1994) ‘Inclusion rating by statistics of extreme values and its application to fatigue strength prediction and quality control of materials’, Journal of Research of the National Institute of Standards and Technology , 99(4), p. 345. Murakami, Y. (2002) Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions . Ekevier Science Ltd. Susmel, L. and Lazzarin, P. (2002) ‘A bi -parametric Wöhler curve for hig h cycle multiaxial fatigue assessment’, Fatigue and Fracture of Engineering Materials and Structures , 25(1), pp. 63 – 78. Susmel, L., Tovo, R. and Lazzarin, P. (2005) ‘The mean stress effect on the high -cycle fatigue strength from a multiaxial fatigue point of view’, International Journal of Fatigue , 27(8), pp. 928 – 943. Yanase, K. and Endo, M. (2014) ‘Multiaxial high cycle fatigue threshold with small defects and cracks’, Engineering Fracture Mechanics . Elsevier Ltd, 123, pp. 182 – 196. References