Recent updates in fatigue analysis with LS-DYNA

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1 Livermore Sotware Technology Corp. New Feature February 2018 Recent update in atigue analyi with LS-DYNA Yun Huang, Zhe Cui Livermore Sotware Technology Corporation Abtract Thi paper review the recent development in atigue olver in LS-DYNA. Fatigue analyi i critical to the deign and aety evaluation o metal tructure and component. Both requency domain and time domain atigue olver have been implemented to LS-DYNA, to meet the requirement o uer rom dierent indutry area. Example are provided in thi paper, to illutrate how to ue thee atigue olver or real problem and do pot-proceing on the reult. The plan on uture development o the atigue olver in LS-DYNA, i alo dicued. Keyword: LS-DYNA, atigue analyi 1 Introduction Fatigue o metal tructure and part i an important topic in indutrie. Fatigue caued 80% o the catatrophic ailure in the hitory. Even today, atigue i till a igniicant and challenging problem or engineer, epecially or thoe who deign tructure uch a aircrat, railroad vehicle, automotive, teel bridge, preure veel, and crane. Since verion 971 R6, a erie o requency domain atigue olver have been provided in LS-DYNA, including random vibration atigue (*FREQUENCY_DOMAIN_RANDOM_VIBRATION_FATIGUE) and SSD (Steady State Dynamic) atigue (*FREQUENCY_DOMAIN_SSD_FATIGUE). They have been ucceully ued or ome indutry problem [1, 2]. Sometime uer need to run atigue analyi in time domain. For example, they may need to tudy the lie cycle o gun pipe due to cyclic thermal loading. Another example i the atigue o the aircrat turbine blade, which i made o alloy and ubject to combined load o high temperature and high air preure. In thee cae, the loading level i high and the platic deormation can be oberved in the material. So it i appropriate to ue the low cycle atigue criterion and olve the problem in time domain. To provide olution or thee problem, time domain atigue olver are alo developed in LS-DYNA. A new keyword *FATIGUE i introduced to run the time domain atigue. Thi keyword provide uer with the opportunity to perorm atigue and durability analyi ollowing tranient analyi in LS-DYNA. The tranient analyi, which provide tre or train cycle or atigue analyi, can be linear or nonlinear, can ue a wide variety o material model (elatic, platic,...), and can work with a variety o loading condition (thermal, mechanical,...). The reult o atigue analyi, in both requency domain and time domain, are provided in the binary plot databae D3FTG, which can be eaily acceed by LS-PrePot. Some example are provided in the paper to demontrate the atigue analyi capabilitie o LS-DYNA, and how how to ue thee requency domain or time domain atigue olver toward dierent problem. Some o the example mentioned in thi paper have been reported in previou publication or preentation [3]. 2 Databae and pot-proceing or atigue analyi The atigue analyi reult are aved in a binary plot databae D3FTG. A keyword *DATABASE_FREQUENCY_BINARY_D3FTG i ued to activate the output o D3FTG databae. Currently 1

2 thi keyword i alo ued even i it i a time domain atigue analyi. D3FTG can be eaily acceed by LS-PrePot. A typical GUI or D3FTG i hown in Figure 1. For random vibration atigue analyi, the ollowing reult are provided in D3FTG: 1. Cumulative damage ratio 2. Expected atigue lie 3. Zero-croing requency with poitive lope 4. Peak-croing requency 5. Irregularity actor 6. Expected atigue cycle Figure 1. GUI in LS-PrePot or atigue analyi reult For atigue analyi baed on *FREQUENCY_DOMAIN_SSD, and atigue analyi in time domain, only the cumulative damage ratio i provided in D3FTG. The other ringe component are zero. More detail about the pot-proceing o the atigue analyi reult can be ound in [3]. 3 Frequency domain atigue analyi with LS-DYNA A erie o requency domain analyi eature have been implemented to LS-DYNA, ince verion 971 R5. A comprehenive review o the requency domain eature, and their application in auto NVH and acoutic problem can be ound in [4, 5]. Solution or both determinitic and non-determinitic vibration are provided in requency domain. For non-determinitic vibration, one can deine the load in term o PSD (Power Spectral Denity) and run random vibration analyi (*FREQUENCY_DOMAIN_RANDOM_VIBRATION); For determinitic vibration, one can run teady tate dynamic (*FREQUENCY_DOMAIN_SSD) and get the repone or each excitation requency. 2

With the knowledge o tre cycle rom the vibration analyi, one can proceed to atigue analyi, upon receiving material atigue propertie (S-N atigue curve).

3 With the knowledge o tre cycle rom the vibration analyi, one can proceed to atigue analyi, upon receiving material atigue propertie (S-N atigue curve). A Miner rule o cumulative damage ratio, i ued: ( D) = i E (1) i n N i Where, E(D) i the expected cumulative damage ratio, ni i the number o cycle at tre level Si, and Ni i the number o cycle or ailure at tre level Si, given by a material S-N curve. 3.1 Fatigue analyi in random vibration environment In thi ection, we introduce the atigue analyi method in random vibration environment. With random vibration analyi, PSD o tre component are computated (and aved in D3PSD). Variou method (Dirlik method, Narrow band method, Wirching method, etc.) have been developed to calculate the PDF (Probability Denity Function) o tree. The number o tre cycle can be calculated or a given time o expoure [3]. The material S-N curve can be deined by *MAT_ADD_FATIGUE. More detail about deining the material S-N curve can be ound in [3]. The example in thi ection conider a metal bracket hown in Figure 2. It i contrained to a haker table through the two mall hole (marked a red in Figure 2) or a random vibration tet. Figure 2. A bracket model contrained to haker table The S-N curve i repreented by the equation N S (2) = where, N i the number o cycle or ailure (or endurance) under the tre level S. The model i ubject to bae acceleration. The acceleration PSD curve i hown in Figure 3. 3

Acl. PSD (g^2/hz) 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 10 100 1000 10000 Frequency (Hz) Figure 3. The bae acceleration PSD curve The duration o excitation i 1 hour.

4 Acl. PSD (g^2/hz) 1.E-02 1.E-03 1.E-04 1.E-05 1.E Frequency (Hz) Figure 3. The bae acceleration PSD curve The duration o excitation i 1 hour. For the irt loading cae, we conider the acceleration in z-direction only. Figure 4. Cumulative damage ratio or the cae o z-acceleration Figure 5. Sae / Failed zone or the cae o z- acceleration Dirlik method i employed to perorm atigue analyi in thi example. Von-mie tre i ued a the tre index. Figure 4 how the cumulative damage ratio plot under the z-acceleration excitation. The peak value o the cumulative damage ratio i around Thi indicate that the model will ail due to atigue in the tet. The peak value appear at the edge o one contrained hole. Thi ugget that the initial crack will take place at that location. Uing the keyword *INITIAL_FATIGUE_DAMAGE_RATIO, one can accumulate the damage on a tructure rom multiple loading cae. For example, the ame tructure hown in Figure 2 can be ubject to bae acceleration in x-, y- and z-direction equentially. When we run atigue analyi or the lat loading cae (e.g. acceleration in z-direction), we can deine the keyword *INITIAL_FATIGUE_DAMAGE_RATIO and et the path and name or the binary databae or the previou atigue reult (due to acceleration in x- and y-direction). The combined cumulative damage ratio plot, due to the three loading cae, can be ound in Figure 6. 4

5 Figure 6. Cumulative damage ratio or x-, y- and z- acceleration loading cae Figure 7. Sae / Failed zone or x-, y- and z- acceleration loading cae Figure 7 how the ae / ailed zone a the inal reult o the three loading cae. A we can ee, the hole edge area ha a higher chance or ailure than other area in thi tet. Sometime it i important to account or the eect o pre-tre or nonzero mean tre on atigue analyi. For random vibration analyi, one can ue a predeined initial tre a the mean tre. The initial tre i introduced by keyword *INITIAL_STRESS_SHELL, *INITIAL_STRESS_SOLID, etc. To deine the mean tre correction method, ue the keyword *FATIGUE_MEAN_STRESS_CORRECTION. The ollowing mean tre correction method are available in LS-DYNA. Goodman method Soderberg method Gerber method S S S 1- / = a (3) m m u 1- / = a (4) y = (5) a 1- m u ( / ) 2 Goodman tenion-only method, and Gerber tenion-only method are alo available. For the two teniononly method, LS-DYNA perorm mean tre correction or tenile mean tre only, uing the equation (3) and (5). a i the original tre In the equation (3)-(5), S i the corrected tre magnitude or alternating tre; magnitude or alternating tre; m i the mean tre; u i the ultimate tenile trength and yield trength o the material. For the model hown in Figure 2, an initial tre (mean tre) tate i aumed (ee Figure 8). y i the Figure 8. Initial Von mie tre (mean tre) 5

For the loading cae with z-acceleration, the updated cumulative damage ratio reult, with mean tre correction by Goodman method (equation 3), and by Soderberg method (equation 4), are hown in Figure 9

6 For the loading cae with z-acceleration, the updated cumulative damage ratio reult, with mean tre correction by Goodman method (equation 3), and by Soderberg method (equation 4), are hown in Figure 9 and 10. Figure 9. Cumulative damage ratio or z- acceleration PSD (uing Goodman equation or mean tre correction) Figure 10. Cumulative damage ratio or z- acceleration PSD (uing Soderberg equation or mean tre correction) Table 1 how the maximum cumulative damage ratio with and without mean tre correction. Mean tre correction method Maximum damage ratio No mean tre correction Goodman equation Soderberg equation Gerber equation Table 1: Maximum damage ratio with and without mean tre correction One can ee, with the mean tre correction, the damage ratio o the tructure can be larger, which mean the atigue lie o the tructure become horter due to the exitence o the mean tre. 3.2 Fatigue analyi in teady tate vibration environment Sometime we have teady tate vibration environment, or example, the vibration in a ine weep tet. With the keyword *FREQUENCY_DOMAIN_SSD_FATIGUE one can run atigue analyi baed on teady tate vibration. Thi approach i alo baed on Miner rule o cumulative damage and we um up the atigue damage rom dierent tre level. The example hown here i about the atigue analyi o a olid tructure (a impliied auto ront bumper model), in a teady tate vibration tet. Figure 11. A ront bumper model in teady tate vibration 6

The bumper i ubjected to continuou vibration rom ground excitation during driving. It i aumed that the bumper i contrained to the auto rame by the edge o the two hole (ee red node hown in Figure 11).

7 The bumper i ubjected to continuou vibration rom ground excitation during driving. It i aumed that the bumper i contrained to the auto rame by the edge o the two hole (ee red node hown in Figure 11). The material S-N curve i deined *MAT_ADD_FATIGUE uing a curve hown in Figure E+02 S: tre range (MPa) 1.E+00 1.E-02 1.E-04 1.E-06 1.E+03 1.E+05 1.E+07 1.E+09 N: number o cycle Figure 12. S-N curve or the bumper model The bae acceleration pectrum i given in Table 2 (ee below). Frequency (Hz) Amplitude (g) Duration (ec.) Table 2. Bae acceleration pectrum The cumulative damage ratio ringe plot can be ound in Figure 13. Figure 13. Cumulative damage ratio plot or the bumper 7

To locate the ailed zone quickly, one can ue the Sae / Failed zone check box and get the ailed zone indicated by red element a below (Figure 14). Figure 14.

8 To locate the ailed zone quickly, one can ue the Sae / Failed zone check box and get the ailed zone indicated by red element a below (Figure 14). Figure 14. Sae / Failed zone under the teady tate vibration One can ee that the ailed element are almot the ame element which are contrained to haker table directly. For thoe element we expect higher tre concentration due to the contraint and that i the reaon or the higher chance or atigue ailure at the ame location. 4 Time domain atigue analyi with LS-DYNA For many application, the loading condition are deined in time domain and the repone o the tructure or part are alo computed in time domain. Thi i true or door lam, and or tructure with thermal loading, and or many other cae where the nonlinear eect i igniicant and the problem can not be linearized. To provide olution or thoe atigue analyi problem, time domain atigue olver are being implemented in LS-DYNA. The olver can be put into 2 categorie: one i baed on tre and the other i baed on train. A new keyword or general time domain atigue analyi ha been implemented *FATIGUE $# pid ptype 1 20 $# dt 0.01 $# trn index retrt To do mean tre correction, the keyword *FATIGUE_MEAN_STRESS_CORRECTION i needed. A imple way to run atigue analyi in time domain i to ue the tre/train data aved in ELOUT (ee *DATABASE_ELOUT and *DATABASE_HISTORY_SHELL, *DATABASE_HISTORY_SOLID, etc.). To ue ELOUT, one jut need to add option _ELOUT to *FATIGUE and kip the irt two card in thi keyword. To ave train reult to ELOUT, one need to et STRFLG = 1 in *DATABASE_EXTENT_BINARY. The Material atigue propertie can be deined by keyword *MAT_ADD_FATIGUE (or S-N curve) or *MAT_ADD_FATIGUE_EN (or E-N curve). More detail o thee keyword can be ound in LS-DYNA Uer Keyword manual [6]. 4.1 Stre-baed atigue analyi For tre-baed time domain atigue olver, a rain low counting method i adopted to get the number o tre cycle or each tre level. One can alo make mean tre correction baed on the Goodman 8

method, the Soderberg method, the Gerber method and tenion-only verion o the Goodman method, and the Gerber method (ee keyword *FATIGUE_MEAN_STRESS_CORRECTION).

9 method, the Soderberg method, the Gerber method and tenion-only verion o the Goodman method, and the Gerber method (ee keyword *FATIGUE_MEAN_STRESS_CORRECTION). The example or the tre baed time domain atigue analyi i a metal pipe model (Figure 15), which i ubject to temperature ocillation (ee Figure 16). For real lie application, thi pipe can be a cannon or a gun pipe and the temperature ocillation can be due to continuou gun ire. The thermal tre cycle due to the temperature ocillation induce atigue damage to the metal. Ater uicient number o gun ire, the cumulative damage ratio can reach 1 and indicate the end o the ervice lie o the gun pipe. The pipe i modeled by *MAT_ELASTIC_PLASTIC_THERMAL. The thermal load i deined by *LOAD_THERMAL_LOAD_CURVE. Figure 15. The metal pipe model Figure 16. The temperature load curve The material atigue property i deined by *MAT_ADD_FATIGUE. The S-N curve look like Figure 17. S-N atigue curve o the material The cumulative damage ratio plot i given in Figure 18. One can ee that the peak value o the cumulative damage ration appear in the middle part o the pipe, which agree with the location or peak value o the von mie tre (Figure 19). 9

Figure 18. cumulative damage ratio Figure 19. ditribution o von mie at time Figure 20 how the xy- hear tre cycle at an element in the middle o the pipe.

10 Figure 18. cumulative damage ratio Figure 19. ditribution o von mie at time Figure 20 how the xy- hear tre cycle at an element in the middle o the pipe. One can ee a lot o tre cycle have taken place in the 1000 econd period. Figure 20. the xy-tre hitory at one element 4.2 Strain-baed atigue analyi Although many tructure and engineering part are deigned o that tree and train are elatic under normal ervice loading, yielding can occur due to a local tre concentration, and reult in crack initiation. The train-baed atigue analyi require a decription o the material repone to cyclic elatic-platic train, and a relationhip between thee train and atigue lie to crack initiation. For train-baed atigue analyi, we ue the local train lie equation De = 2 E b ( 2N ) + e ( 2N ) c With Smith-Waton-Topper mean tre correction, equation (6) i changed to De 2 max = 2 ( ) ( ) 2b ( ) b+ c E 2N + e 2N With Morrow mean tre correction, equation (6) i changed to De = 2 - E m b ( 2N ) + e ( 2N ) c (6) (7) (8) 10

Here i the deinition o the variable ued in the equation: D e N b train amplitude endurance, or number o cycle or ailure the atigue trength exponent (Baquin exponent) the atigue trength coeicient max

11 Here i the deinition o the variable ued in the equation: D e N b train amplitude endurance, or number o cycle or ailure the atigue trength exponent (Baquin exponent) the atigue trength coeicient max maximum tre c m mean tre the atigue ductility exponent (the Coin-Manon exponent) e the atigue ductility coeicient, which i the platic train amplitude at 2 N = 1 E the elatic modulu I pure elatic material are ued in the tre-train cycle computation (e.g. uing an elatic inite element analyi), Neuber rule i ued to convert the elatic tre/train to local true tre/train e = e e e (9) Where, and e are the local tre and train, e and e e are the elatic tre and train. For illutration purpoe, we conider a bracket model hown in Figure 21. The bracket i modeled by *MAT_PIECEWISE_LINEAR_PLASTICITY. Figure 21 a bracket model with inuoidal nodal orce excitation The bracket model i contrained at the two ring (the red point in Figure 21). A inuoidal nodal orce F(t) i applied on the other end to produce high requency vibration and tre-train cycle. The nodal orce i deined a ( t) 5 in( t) = N (10) F 1000 The total loading time i 1 econd. In the example, the material atigue contant are deined a E = 209,000 MPa ; K = 1230 MPa ; n = 0.161; = 1006 MPa b = ; e = ; c =

The cumulative damage ratio due to the inuoidal nodal orce i computed and plotted in Figure 24.

12 Figure 22. Eective train at lower layer integration point Figure 23. Eective train at upper layer integration point Figure 22 and 23 how the eective train at lower and upper integration point layer near the end o loading. The cumulative damage ratio due to the inuoidal nodal orce i computed and plotted in Figure 24. The location or the peak value o the cumulative damage ratio matche well with the location or highet train value. Figure 25 how the train time hitory or the element which ha the highet eective train value near the end o loading. The reult how that there i a larger chance o atigue ailure or the area next to the lower ring (which i ixed) where high train value are oberved. Figure 24. Cumulative damage ratio Figure 25. Eective train at upper layer integration point or element Concluion and uture work Some baic requency domain and time domain atigue analyi method have been implemented to LS- DYNA. For requency domain atigue analyi, the index variable i tre and the method are baed on S-N curve; or time domain analyi, both tre and train can be ued a the index variable o that both the S-N curve and E-N curve can be ued. Combining with the requency domain and time domain vibration olver in LS-DYNA, thee atigue analyi method provide uer the opportunitie to run eamle vibration-atigue analyi, or a variety o phyical problem and loading condition. Real lie atigue problem are complicated. To provide more robut and powerul olution to atigue problem, ome urther development and improvement are needed. Firt o all, more benchmark tet and validation are needed to check the reliability and accuracy o the current atigue olver. Second, a more comprehenive approach to conider multi-axial tre/train tate i needed, or a more realitic imulation o atigue ailure proce. The current approach i baed on impliication o the tre/train tenor to a principal, calar value and can give very optimitic lie etimate, epecially or component which experience biaxial / multi-axial tree, oten a a reult o combined bending and torion. Third, in time domain atigue analyi, to how the evolution o the atigue cumulative damage, a progreive ringe plot databae i preerred. Currently D3FTG how only the inal cumulative damage ratio plot at 12

13 the end o loading and uer may loe ome important time intant when the crack initiate. With a progreive D3FTG, the dynamic procedure o material atigue ailure can be preerved and uer can get more inormation about the intermittent crack initiation and atigue ailure. 6 Reerence [1] Arnaud Ringeval, Yun Huang, Random Vibration Fatigue Analyi with LS-DYNA. Proceeding o the 12th International LS-DYNA Uer Conerence. [2] Al-Bahkali Eam, Elkenani Hiham, Souli Mhamed, NVH and Random Vibration Fatigue Analyi o a Landing Gear' Leg or an Un-Manned Aerial Vehicle Uing LS-DYNA. Proceeding o the 9th European LS-DYNA Conerence 2013 [3] Yun Huang, Zhe Cui, Philip Ho, Chengju Zhang, Application o LS-DYNA in Structural Fatigue Analyi, Proceeding o the 14th International LS-DYNA Uer Meeting & Conerence 2016, June 12-14, 2016, Detroit. [4] Yun Huang, Zhe Cui, Recent update in LS-DYNA requency domain olver, proceeding o the 10th European LS-DYNA Conerence 2015, Wurzburg, Germany. [5] Yun Huang, Zhe Cui, Updated requency domain analyi in LS-DYNA, proceeding o the 13th LS-DYNA German Forum 2014, Bamberg, Germany. [6] Yun Huang, Zhe Cui, Application o LS-DYNA or Auto NVH Problem. Proceeding o the 13th International LS-DYNA Uer Conerence 2014, Detroit, USA. [7] Praanna S. Kondapalli, Tyler Jankowiak, Yun Huang, Application o LS-DYNA to NVH Solution in the Automotive Indutry. Proceeding o the 14th International LS-DYNA Uer Conerence 2016, June 12-14, 2016, Detroit. [8] LS-DYNA Keyword Uer' Manual, Livermore Sotware Technology Corporation,

CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS

CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3