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1 AD LOCAL STRESS-STRAIN APPROACH TO CUMULATIVE FATIGUE DAMAGE ANALYSIS JDean Mrrw, et al Illinis University Prepared f r: Naval Air Develpment Center January 1974 DISTRIBUTED BY: KJOl Natinal Twhnlcal InfrmtiM Stnrica U. S. DEPARTMENT OF COMMERCE 5285 Prt Ryal Rad. Springfield Va > *.,t

2 ii FOREWORD This reprt summarizes mre than five years f research n the prblem f estimating the fatigue life f spectrum laded members. Three cperative research cntracts between the Aer Structures Department f the Naval Air Develpment Center and the H. F. Mre Fracture Research Labratry f the Department f Theretical and Applied Mechanics f the University f Illinis, at Urbana, made this investigatin pssible. The mst recent f these was Cntract N. N C and this reprt cnstitutes the final reprt fr that cntract. Backgrund research was perfrmed under Cntract Ns. N and N C Messrs. M. S. Rsenfeld, R. E. Vining and F. F. Brriell were instrumental in inaugurating this line f research and prvided technical liaisn fr the Navy. A number f staff members and graduate students have made valuable cntributins t the prcedure fr cumulative fatigue damage analysis which is utlined in this reprt. An attempt is made in the text t acknwledge the individuals and their cntributins by citing their published wrks. Thse papers that were prduced as a part f this prgram are indicated by an asterisk in the list f references: Refs. (1-18). ACCBMB»» ITU IK «NMKSilMEl JUSTiriUTIM... Mltl SNtM MlKtlH * IT OlSTRIBUTIOMMIUIIlin CMO OUt. XMIL md/ir SPECIAL * V ;

3 - I JL T UNCLASSIFIED A.O 777 DOCUMENT CONTROL DATA -R&D untv i litssifirättn f titlt', htidy %tt mbsttavl asni indexing.tnntittitm must lit enfercti tv/jfn th vvrull rvpnrt is ctussificdt St-iuritv Cltissifuatin C " it.tn A rtng ACTIVITY ( Crprate authr) Department f Theretical and Applied Mechanics University f Illinis Urbana, Illinis H T P O R T TITLE 2a. REPORT SECURITY CLASSIFICATION Unclassified Zh. GROUP Lcal Stress-Strain Apprach t Cumulative Fatigue Damage Analysis 4 DETSCRIPTIVE NOTES (Type f reprt and,inc/u.siv» 1 dates) Final Reprt (26 September 1969 t 21 June 1971) 5 AU THORtS» ff irsf name, middle initial, last name) JDean Mrrw, J. F. Martin and N. E. Dwling 6 REPORT DATE April 1973 (Printed January 1974) 8a. CONTRACT OR GRANT NO. NOO C PROJEC T NO 7a. TOTAL NO OF P A G 1. S 78 9a. ORIGINATOR'S REPORT NUMBER(SI T & A M Reprt N NO. OF REFS 48 9h. OTHER REPORT NO(S) (Any ther numbers that may be assigned this reprt) Nne 10 DISTRIBUTION STATEMENT Apprved fr public release--distributin unlimited. Rpprnduced by NATIONAL TECHNICAL INFORMATION SERVICE U S Department f CmmeTce Sprmfjficld VA II. SUPPLEMENTARY NOTES 12 SPONSORING MILI TARY ACTIVITY Aer Structures Department Naval Air Develpment Center Jhnsville, Warminster, Pa ABSTRACT A cumulative fatigue damage prcedure fr estimating the fatigue crack initiatin life f nfched structural members subjected t knwn lad histries is utlined. This prcedure assumes that a knwledge f the lcal cyclic stress-strain respnse f the metal at the mst severely strained regin in a member is sufficient t predict when a crack will frm there. Sme f the steps in this prcedure that are f current interest and which are especially applicable t a lcal stress-strain apprach are discussed. Alternative, apprximate and/r abbreviated steps in the cumulative fatigue damage prcedure are given wherever pssible. Limitatins f the methd and areas where research is needed are pinted ut. Cumulative fatigue test results fr smth specimens, ntched plates and built-up bx beams are cmpared t life calculatins made using the lcal stress-strain apprach. Cyclic defrmatin and fracture prperties, used in the analysis, were btained frm tests n a limited number f axially laded unntched specimens. These examples indicate that a cumulative fatigue damage analysis based n the lcal stress-strain apprach emplying a minimum amunt f materials test data can be used t make reasnable life estimates fr members similar t many practical structural members/ While there appears t be n serius limitatins n the cmplexity f lad histries that can t e analyzed, the fatigue analysis f parts with cmplex gemetric frm is presently impractical withut direct measurements f critical strains in the part. DD '» 1473 S/N I (PAGE 1) UNCLASSIFIED Security Classificatin A-3I408

4 r ""T UNCLASSIFIED Security Classificatin KEY WORDS Metal fatigue Cyclic stress-strain Stress cncentratin Strain cycling Cumulative fatigue damage Cmputer based fatigue analysis V DD FORM S/N 0101-e07-6«2t I NOV 1473 (BACK, 7 O- X UNCLASSIFIED Security Classificatin > ' «l «I

5 T 111 SUMMARY \ A cumulative fatigue damage prcedure fr estimating the fatigue crack initiatin life f ntched structural members subjected t knwn lad histries is utlined. This prcedure assumes that a knwledge f the lcal cyclic stress-strain respnse f the metal at the mst severely strained regin in a member is sufficient t predict when a crack will frm there. Sme f the steps in this prcedure that are f current interest and which are especially applicable t a lcal stress-strain apprach are discussed. Alternative, apprximate and/r abbreviated steps in the cumulative fatigue damage prcedure are given wherever pssible. Limitatins f the methd and areas where research is needed are pinted ut. Cumulative fatigue test results fr smth specimens, ntched plates and built-up bx beams are cmpared t life calculatins made using the lcal stress-strain apprach. Cyclic defrmatin and fracture prperties, used in the analysis, were btained frm tests n a limited number f axially laded unntched specimens. These examples indicate that a cumulative fatigue damage analysis based n the lcal stress-strain apprach emplying a minimum amunt f materials test data can be used t make reasnable life estimates fr members similar t many practical structural members. While there appears t be n serius limitatins n the cmplexity f lad histries that can be analyzed, the fatigue analysis f parts with cmplex gemetric frm is presently impractical withut direct measurements f critical strains in the part. Ja.V

6 c- iv TABLE OF CONTENTS FOREWORD ii SUMMARY iii LIST OF FIGURES v LIST OF SYMBOLS vii INTRODUCTION 1 LOCAL STRESS-STRAIN APPROACH 2 RELATING LOCAL AND NOMINAL STRESS-STRAIN BEHAVIOR 4 SMOOTH SPECIMEN STRESS-STRAIN BEHAVIOR 7 SMOOTH SPECIMEN FATIGUE BEHAVIOR 10 FATIGUE DAMAGE ANALYSIS 14 CONCLUSIONS, LIMITATIONS AND FUTURE DIRECTIONS 17 APPENDIX - COMPUTER BASED FATIGUE LIFE PREDICTIONS FOR NAVY BOX BEAMS 19 REFERENCES 28 FIGURES Page V -. «-j.. /

7 " um 7 TZr LIST OF HGURES N Title Smth Specimen Representatin f Material at the Critical Zne Cyclic Sftening and Relaxatin f Mean Stress under Neuber Cntrl (Ti-8Al-lM-lV, K f = 1.75), (12) Cumulative Fatigue Damage Prcedure Recrded Data frm a Smth Specimen Simulatin f the Lcal Stress- Strain Behavir f a Ntch (1) Stress-Strain Respnse f a Smth Specimen Ntch Rt Simulatin (7075-T6 Aluminum, K f = 1.95), (12) Stress-Strain Respnse f a Smth Specimen Ntch Rt Simulatin (7075-T6 Aluminum, K f = 1.95), (12) Cmparisn f Ntched Fatigue Data with Life Curves Predicted frm Smth Specimen Data (6) Distributin f Failure Predictins (12) Stress-Strain Respnse fr 2024-T4 Aluminum (11) Hysteresis Lps at Varying Strain Amplitudes fr Man-Ten Steel (27) Typical Stress-Inelastic Strain Hysteresis Lps during a Test with Overs; rains every 10 5 Cycles fr SAE 4340 Steel (14) Cyclic Relaxatin f Mean Stress fr 7075-T6 Aluminum (11) Actual and Simulated Stress-Strain Respnse f 2024-T4 Aluminum Representatin f Elastic, Plastic and Ttal Strain Amplitude-Fatigue Life Relatins Inelastic Strain versus Life fr SAE 4340 Steel (14) Ttal Strain versus Life fr SAE 4340 Steel (14) Equivalent Strain Amplitude-Life Cmparisn fr SAE 4340 Steel (10) Stress-Strain Recrding fr Initial Overstraining Stress versus Life fr SAE 4340 Steel (14) Mean Stress Data fr SAE 4340 Steel Cmpared t Eqs. 11 and 12 kjh *..f

8 " r T- VI N Title Distributin f Failure Predictins fr 2024-T4 Aluminum (15) Distributins f Life Calculatins fr SAE 4340 Steel (14) Lad-Life Curve fr Plates Subjected t Lad Spectra Shwn (11) Cnstant Amplitude Bx Beam Predictins and Data: lg t P Lading Cnstant Amplitude Bx Beam Predictins and Data: lg ± P Lading Cnstant Amplitude Bx Beam Predictins and Data: Lad Spectrum B Bx Beam Predictins and Data Bx Beam Specimen Design Highly Stressed Area f Bx Beam Cver Plate Strain Spectrum fr One Specimen Simulatin Mntnie Stress-Strain Curve f 7075-T6 Aluminum lg 6 t P Lading max Set f Stable Hysteresis Lps fr Determining the Cyclic Stress-Strain Curve Plastic Strain versus Stress Amplitude fr 7075-T6 Aluminum Cyclic Relaxatin Data fr a Strain Range frm 0.0 t fr 7075-T6 Aluminum Strain-Life Data fr 7075-T6 Aluminum Rhelgical Mdel fr Cmputer Simulatin Mdel Apprximatin f Stress-Strain Curve Example f Cyclic Hardening Used in the Mdel Flw Chart f Cmputer Prgram iiw»».,- vt..«...««.«., / - -' J i Jh w * *>

9 T er Vll LIST OF SYMBOLS a b b r c d E E ä E E i E i 00 CT E.* e,ae e, Ae Material cnstant indicating sensitivity t mean stress Fatigue strength expnent Minimum width f a ntched specimen Fatigue ductility expnent Material cnstant Mdulus f elasticity Secant mdulus fr the metal at a ntch rt Secant mdulus fr the metal away frm a ntch rt Stiffness f i spring, superscript a signifies stress dependence Cnstant relating t i th Effective mdulus f i spring Nminal strain, nminal strain range True strain, true strain range segment f stress-strain curve True fracture ductility Fatigue ductility cefficient P P Ae_ Plastic strain, plastic strain range Equivalent cmpletely reversed strain range Maximum strain attained ver i segment f stress-strain curve 6e, i Change in strain ver i th Change in strain ver i element segment K', K* Cyclic strength cefficient, transient strength cefficient Theretical elastic stress cncentratin factr Stress cncentratin factr Strain cncentratin factr Fatigue ntch factr i t Jli j/ *. * M

10 T Jfc. VI n M N, 2N, N, 2N. n. i n*, n* P r S,AS S u a, Aa Cnstant in relaxatin functin Number f cycles t failure Reversals r half cycles t failure fr a cnstant amplitude test Life fr a given strain amplitude and mean stress Transitin fatigue life Number f cycles at a given strain amplitude and mean stress Transient strain hardening expnent; n' is the stable value Lad Ntch radius Nminal stress, nminal stress range Ultimate tensile strength True stress, true stress range Fatigue strength cefficient True fracture strength a, a a' max Stress amplitude, mean stress Maximum stress Equivalent cmpletely reversed stress amplitude Cnstant stress exerted by i frictinal slider Residual stress exerted by i spring Final stress attained fr a particular reversal Maximum stress attained ver i segment f stress-strain curve 6* t Change in stress fr i Specimen thickness segment fr stress-strain curve V n Highly stressed vlume fr ntched specimen Highly stressed vlume fr unntched specimen

11 TT INTRODUCTION Mre than 100 years f research n the subject f metal fatigue has been cncerned primarily with the effect f the many factrs influencing fatigue damage. A number f specific practical prblems have been slved and much has been learned abut the basic mechanisms f the fatigue prcess. Hwever, nly limited prgress has been made tward develping prcedures that can be used t estimate fatigue lives f cmpnents still in the design r develpment stage. In mst cases, it is necessary t subject a trial design f a cmpnent t simulated service testing r t rely heavily upn actual service experience fr similar parts. Cmpnent testing and the cllecting f service data are expensive and time cnsuming. It is desirable, therefre, that these activities be augmented by analytical prcedures fr calculating fatigue lives. It has been clearly established fr sme time that lcal repeated plastic strain is respnsible fr fatigue damage. Yet, the engineering apprach t fatigue prblems has usually been t cnsider the nminal stresses caused by the maximum frces, mments and/r trques acting n a part. "Stress" calculatins are usually based n the assumptin that the material respnds in a linear elastic fashin. The research discussed here is based n the premise that an understanding f the cyclic stress-strain behavir and fatigue resistance f the material at the mst highly strained regin in a member shuld be sufficient t predict when that regin will fail by fatigue. Fatigue damage generally nucleates in the rt f the mst severe ntch in a regin f high cyclic nminal stress. An extensin f this simple apprach t elements ahead f the crack, nce it is frmed, culd be applied t imprve understanding f fatigue crack prpagatin. This cupled with the fracture mechanics cncept f an unstable fast crack culd prvide an analytic basis fr the entire fatigue sequence f crack initiatin, prpagatin and final fracture. Here we have nt separated the initiatin, prpagatin and fracture stages f fatigue. Instead ur effrts have been cncentrated n arriving at the materials and mechanics cncepts needed t predict when ne small element f metal will fail by fatigue. Fr the sake f simplicity, many factrs that may have a large effect n the fatigue life f a member in service have been ignred. Amngst these are the effects f temperature, crrsin, machining and fabricatin stresses and multiaxial states f stress which may be present at the lcal pint where cracks initiate. * Purpse and Scpe The aim f this final reprt is t utline a prcedure fr cumulative fatigue damage analysis which has evlved at Illinis and elsewhere ver the past several years. During this perid rapid prgress has been made in fatigue analysis, i articularly in the areas f better characterizing the cyclic defrmatin and fatigue resistance f metals and in applying high speed cmputers t d the extensive bkkeeping necessary in a fatigue damage analysis. We will draw mainly frm ur wn wrk and experience and that f clse assciates. Examples are given at the end f the paper t illustrate the applicatin f current fatigue analysis techniques. Finally, the weak pints in the prcedure which need additinal systematic research are enumerated and briefly discussed Jl I)

12 r CT- LOCAL STRESS-STRAIN APPROACH The cyclic stress-strain respnse and the fatigue resistance f a metal may be characterized by perfrming a series f labratry tests n smth uniaxial specimens. A member made f this metal may be analyzed fr fatigue resistance by cnsidering the behavir f the material at the site f crack initiatin t be analgus t the respnse f a smth specimen subjected t the same cyclic strains. The basic idea f this apprach is depicted in Fig. 1 fr a simple ntched plate. It is assumed that the critical zne is in an uniaxial stress state and that the smth specimen is representative cf the metal where fatigue cracks initiate. T estimate the fatigue crack initiatin life f structural cmpnents, the lcal cyclic stresses and strains affecting the critical zne must be related by the methds f mechanics t the knwn lad r nminal stress histry n the part. In practice, this requirement is perhaps the mst difficult t fulfill because f the gemetric cmplexity f mst structural parts and because the critical lcatin usually experiences cyclic plastic defrmatin. Mechanics methds fr elast-plastic analysis f structural cmpnents subjected t cyclic lading have nt as yet develped t the level required fr cycle t cycle determinatins f lcal stresses and strains in cmplex parts. As an alternative, experimental stress analysis techniques can be used in sme cases. Hwever, even if the cmpnent has been fabricated and the type f lading experienced in service simulated, strain gages may nly be useful fr measuring the nminal strains since the lcatin f the critical regin may be unknwn r may be s small r inaccessible that meaningful strain measurements are impractical. Simple Ntched Members The viability f the lcal stress-strain apprach t fatigue damage analysis has been demnstrated fr simple ntched members such as shwn in Fig. 1. Fr such members the lcal and nminal stresses and strains can be related by an elastplastic "strength f materials" type f analysis based n wrk by Neiiber (19). Details f this analysis are given in the next sectin. The analysis prvides a rule that specifies the cyclic stress-strain cntrl cnditins necessary t simulate n a smth specimen the cnstraint f the material surrunding the metal at a ntch rt in a member subjected t a knwn lad histry. An example f this type f cntrl is shwn in Fig. 2. This is an actual X-Y plt f the stress-strain respnse f a smth specimen cntrlled accrding t Neuber's rule t simulate the material respnse at the rt f a ntch in a plate subjected t a zer t tensin lad histry. Althugh the nminal stress n the plate is always tensile (see the inset in Fig. 2), the lcal stress at the ntch rt is seen t cyclically shift tward cmpressin tending tward cmpletely reversed lcal stress. The lcal strain is seen t cyclically increase in the tensile directin. The cycle-dependent relaxatin f stress, cyclic hardening, sftening, effects f verlad n a ntched member and ther factrs which will affect its fatigue crack initiatin life can be simulated n smth specimens using this technique (12). In additin, the specimen is subjected t cumulative fatigue damage at the same rate j/

13 T as at the ntch rt in the simulated member. Thus, the fatigue life f the smth specimen shuld be nearly equal t the crack initiatin life f the simulated ntched member. In Ref. (12) it is shwn that the simulated smth specimen lives agree with actual ntched specimen lives within abut a factr f tw in mst cases. Cmputer Based Simulatin A majr bjective f the fatigue damage prcedure reviewed in this reprt is t eliminate the need fr extensive testing and t reduce the time and expense necessary fr evaluating the fatigue life f structures subjected t large numbers f cycles and cmplicated lad spectra. The type f smth specimen simulatin test just described is time cnsuming and requires sphisticated testing apparatus and persnnel. In fact, nly simple lading sequences can be simulated at the present time. A mre practical and efficient apprach can be emplyed by "simulating" the smth specimen simulatin using a high speed digital cmputer. Martin et al. (ii) have demnstrated the feasibility f this apprach. The necessary materials prperties fr the metal f interest are estimated frm ther prperties and taken frm the literature and/r varius "data banks" that are nw becming available. If vital materials infrmatin is nt available, minimal smth specimen testing may be necessary t fully characterize the cyclic defrmatin and fracture resistance f the metal. Cyclic as well as mntnic prperties f the metal f which the member is made are needed since the stress-strain respnse f mst metals changes significantly during cyclic straining int the plastic range. These prperties alng with a digital cmputer mdel permit the cycle t cycle respnse f the material t be rapidly evaluated fr virtually any histry f input. Once the cyclic stress-strain prperties f the metal are embdied in the cmputer mdel, a methd f estimating the lcal stresses and strains frm the lad histry n a member can be emplyed in the cmputer. Of the available methds, Neuber's rule seems t be the mst practical fr a ntched member. At present, hwever, the effects f gemetry and ntch size are treated in the Neuber analysis by means f a fatigue ntch factr, K*. Fr small ntches, cmplicated shapes r welds, Kf is difficult t estimate withut fatigue testing f actual parts. Frm the cmputed lcal stresses and strains, fatigue damage r expected life can be calculated if strain amplitude and mean stress versus fatigue life infrmatin are available fr the metal f interest. Such infrmatin requires apprpriate fatigue testing r can be estimated frm ther knwn prperties. Lcal fatigue damage may be assayed fr each reversal r half cycle f strain via a cycle cunting methd, r by prperly summing the damage due t small increments f the stress-strain histry in cnjunctin with the cmputer mdel f the cyclic respnse. Of the well knwn cycle cunting methds, nly the range pair and rain flw methds give ratinal cunting results if there are minr cycles superimpsed n changes in the mean level. The cumulative fatigue damage analysis prcedure which is briefly described abve is utlined in die schematic shwn in Fig. 3. Steps in the prcedure will be discussed in mre detail in the next fur sectins and an example f the applicatin f this methd t estimate the fatigue life f Navy bx beams is presented in the Appendix.! ml k y

14 RELATING LOCAL AND NOMINAL STRESS-STRAIN BEHAVIOR If the mst highly stressed lcatin in a cmpnent subjected t variable lading is knwn, it may be pssible in sme situatins t munt a strain gage at that lcatin. In this way a recrd is btained f the strain histry fr a typical duty cycle r perid f usage f the vehicle, machine, r aircraft f which the cmpnent is a part. The number f repetitins f this strain histry required fr crack initiatin can then be estimated, prvided the state f stress at the critical lcatin can be treated as uniaxial and its cyclic value can be estimated s that the effect f mean stress n fatigue damage can be included in the analysis. If a suitable test facility is available, the knwn strain histry can be enfrced upn an axially laded unntched specimen until it fails. Thus it is pssible t btain a direct estimate f the fatigue life which bypasses all f the analysis. Alternatively, the stress histry frm the smth specimen test culd be measured during a representative blck f the strain histry and the results used t calculate the fatigue life taking int accunt the effects f mean stress. In this way the specimen need nt be tested t failure. If a cmputer based mdel is available that is capable f simulating the cyclic stress-strain respnse f the metal, the stress histry crrespnding t the knwn strain histry can be determined s that a life estimate can be made. Such mdels have been develped by Martin et al. (11,20) and may include the effects f cyclic hardening r sftening and cyclic relaxatin f mean stress. Sme Pssible Simplificatins in Fatigue Analysis Fr a typical sectin f strain histry measured n a cmpnent, it is reasnable t assume, prvided there has been a perid f shakedwn, that the material behavir is nearly stable r that the cyclic hardening r sftening is n lnger significant. The fatigue damage prcedure may therefre be simplified. Cyclic relaxatin f mean stress and cyclic creep can ften be mitted frm the mdel since these will nt affect fatigue life calculatins significantly unless there are lng perids f nearly cnstant cyclic strain amplitude present in the histry. If the strain histry is simple r shrt and if cyclic hardening/sftening, relaxatin, and creep can be neglected the stable cyclic stress-strain curve f the metal may be used t graphically estimate the stress respnse. It is necessary t knw the residual stress at the beginning f a perid f typical strain histry nly if the typical histry des nt include at least ne strain cycle large enugh t set up a system f residual stresses that "verride" the initial residual stresses (18). If mst f the fatigue damage is due t sufficiently large strain cycles, the mean stress present during these cycles will be insignificant due t cyclic plasticity and may be neglected in the fatigue damage analysis. It may be determined if mean stress can be neglected by making trial damage calculatins using life data fr zer mean stress. The results f this calculatin can then be cmpared t the cyclic stress-strain curve t determine what fractin f the calculated damage is due t cycles during which n significant mean stress wuld be expected t be present. If this fractin is large, the life estimate will nt be significantly affected by the mean stresses during any smaller cycles which might be present and mean stress can be neglected. j/ *>,f

15 - -' r c Neuber Analysis Usually the cyclic strains cannt be measured at the fatigue crack initiatin site in a member. Rather, nminal strains are measured near t the critical lcatin r the applied external frces are measured and nminal stresses are calculated. In either case, a mechanics analysis f the member must be perfrmed t relate the nminal values t the lcal stresses and strains. Several types f ntch analyses (6,19,21,22) have been prpsed fr relating nminal and lcal cyclic stresses and strains. Mst f the applicatins have been fr cnstant amplitude r blck type lading. Als, the cmpnents which have been tested were usually labratry samples. Few fatigue analyses have been perfrmed n cmpnents subjected t realistic lad spectra (23). Neuber's rule (19) prvides a relatin between lcal strain and stress cncentratin factrs and the theretical stress cncentratin factr. K t = (K e Kp i (1) where K, K, and K are the lcal strain, lcal stress and theretical elastic stress cncentrltin factrs, respectively. This relatin was mdified fr fatigue applicatins (6) int the fllwing frm: K: f (AS Ae E)2 = (ACT Ae E)* (2) where E is the elastic mdulus, AS and Ae are the nminal stress and strain ranges. ACT and Ae are the lcal stress and strain ranges, and Kf is the fatigue ntch factr. With the exceptin f the change f K t t Kf, Eq. 2 is derived frm the definitins f the terms in Eq. 1. The substitutin f K* fr K t is necessary t crrect fr size effects in fatigue, especially when the ntch is small (16,17). If the lad;? are small enugh that the behavir f the ntched member is nminally elastic, i.e., AS/Ae = E, Neuber's rule reduces t > als (K f ASr = ACT Ae K f AS = (ACT Ae E) i (3) (3a) Equatins 2 and 3 imply that the material surrunding the critical zne impses bth a strain and stress restrictin n the material there. This wuld seem reasnable in that the surrunding material neither acts as a rigid wall nr des it simply lad the highly stressed regin. Anther expressin relating lcal and nminal behavir that is based n Stwell's wrk (22) has been prpsed fr use in cyclic analyses (24) *. J

16 > i Jii. IM)I lj E a = S(l+(K t -l)) (4) where E is the secant mdulus at the stress raiser and E is the secant mdulus fr the material far remved frm the stress raiser. In Ref. (24) this relatin is successfully emplyed t predict the lcal cyclic stresses near the ntch rt f edge ntched plates. Fr these predictins it was necessary t determine :he stress-strain behavir f the material frm smth specimens. Hwever, fatigue predictins were nt made using Eq. 4 and smth specimen fatigue data. Mansn and Hirschberg (25) discuss bth the Neuber and Stwell equatins, but nly Neuber's relatin was used t make life predictins. Few cmparisns between experimentally and analytically determined stresses and strains at the ntch rt have been made. Wetzel (1) used Neuber's relatin t estimate the experimental data f Ref. (24). Agreement was als gd. Mst applicatins f Neuber's rule have cncerned predicting fatigue lives. These life predictins are made either by subjecting a smth specimen t cntrl cnditins which bey Eq. 3 fr the nminal stresses n the simulated ntened specimen r by using Eq. 3 fr each reversal alng with smth specimen stress-strain and fatigue data. Examples f the Applicatins f Neuber's Rule Wetzel (1) manually cntrlled the stress and strain limits n smth specimens s as t bey Neuber's rule fr zer t tensin lading f ntched plates. Figure 4 shws a typical result frm these tests. Attempts t enfrce Neuber cntrl n smth specimens by means f an analg cmputer have failed. Hwever, digital cmputer cntrl f a materials test system prduced the results shwn in Fig. 2 (12). Figures 5 and 6 illustrate the use f Neuber cntrl t simulate the residual stresses prduced by verlading ntched > members. Fr any type f ntch simulatin n a smth specimen, the real time f expended in determining failure can be quite large. I Tpper, et al. (6) shwed that cmpletely reversed cnstant amplitude fatigue data fr bth smth and ntched specimens can be represented by a single band n the same life curve by emplying Eq. 2. Fr ntched members the left side f the equatin is used as a life parameter. Fr smth specimens the right side f the equatin is used. Figure 7 is an example f the crrelatin that is pssible using this apprach. The line is based n smth specimen data (pints nt shwn) and the pints are ntched fatigue results. Whether Stwell r Neuber equatins are used t relate nminal t lcal stresses and strains, the stress-strain respnse f the metal f interest must als be emplyed in the analyse. This is dne mst directly by simulatin testing f a smth specimen. Hwever, the sphisticatin needed in equipment and test technique is quite large. Stadnick and Mrrw (12) listed fatigue predictins resulting frm three different means f cntrlling the Neuber parameter. An apprximate type f Neuber cntrl that is suitable fr an analg cmputer was used fr sme f the tests. It is termed "sliding line cntrl." It apprximates the hyperblic frm f Neuber's rule (see Fig. 4) by a straight line. The accuracy f the predictins reprted in Ref. (12) is summarized in Fig

17 ZT SMOOTH SPECIMEN STRESS-STRAIN BEHAVIOR Since the relatin between lcal and nminal behavir invlves ne equatin and tw unknwns, mre infrmatin is needed relating lcal stress, a, and strain, e. Such a relatin is derived frm experimental data, mst f which is frm cmpletely reversed strain r stress cntrl. It is necessary t restrict the treatment t the uniaxial state f stress. Biaxial and triaxial cyclic defrmatin and fatigue behavir f metals have nt been sufficiently investigated t allw their cnsideratin. Frtunately, many practical situatins invlve an apprximately uniaxial state f stress since the lcatin where fatigue damage initiates is usually at a free surface. Axially laded unntched specimens with cylindrical r hurglass shaped test sectins are apprpriate fr studies f materials behavir. Fr such specimens it can be assumed that the states f stress and strain are nearly unifrm, and als these quantities can be cnveniently measured. Mre detailed infrmatin n testing techniques fr such specimens is given in Ref. (7). Static and Stea jy State Cyclic Stress-Strain Behavir The mst ppular test fr assessing mechanical resistance f a material is the mntnic (static) tensin test. At least the engineering tensile prperties that can be btained frm this test are listed fr mst structural metals in material handbks. Fr this reasn attempts are made t assciate cyclic and mntnic prperties (7). Such parameters as the true fracture strength, a«, true fracture ductility, ef, strain at necking, and the ultimate strength, S u, can all be f use fr making fatigue life estimatins. End and Mrrw (3) list the mntnic prperties f the fur aircraft metals f majr interest in this investigatin. Fr mst metals, the stable cyclic stress-strain curve is quite different frm the mntnic. References (4) and (26) demnstrate this difference. Reference (4) als lists the cyclic prperties f the fur aircraft meials used here. There are at least fur ways f determining the cyclic stress-strain curve: 1) a curve passing thrugh the tips f stable lps frm cmpanin specimens tested at different strain ranges, 2) the curve passing thrugh the tips f lps frm an incremental step strain test (4), 3) mntnic tensin after a stable state has been btained, and 4) analysis f the curves frm stable individual hysteresis lp. The cyclic stress-strain curve in mst cases will be quite different frm the stress-strain curve btained frm a tensin test. A mathematical expressin fr the cyclic stressstrain curve which is satisfactry fr mst engineering metals may be btained if the cyclic strain is separated int its elastic and inelastic cmpnents. i«2 Ae. ACT 2E (5) where the prime (') dentes cyclic, therefre, K' and n* are the cyclic strength cefficient and cyclic strain hardening expnent, respectively. The cnstants fr Eq. 5 are usually determined as the slpe and intercept f a lg-lg plt f cyclic stress amplitude and cyclic plastic strain amplitude.,v J. II *B*M

18 JLJ* rftafc Tf By whatever means the cyclic stress-strain curve is determined, it is nly useful if it can reprduce the actual cyclic stress-strain behavir f the material fr which the cnstants were determined. Figure 9 is a plt f a set f stable hysteresis lps f 2024-T4 aluminum with the lp tips at a cmmn rigin (11). Equatin 5 fits the data quite well. Figure 10 is the same type f illustratin fr a structural steel (27). Fr this steel the curve described by Eq. 5 des nt fit the actual lp shape very accurately. Cyclic Hardening and Sftening In a cnstant strain cntrlled test, cyclic hardening is bserved as an increase in the stress range and sftening as a decrease. Sme f the hardening and sftening characteristics f the metals f interest are discussed in Refs. (3) and (26). Mrrw (28) cites sme extreme examples f hardening and sftening f cpper. DwUng (14) bserved cyclic sftening f quenched and tempered SAE 4340 steel. An initially hanx metal, such as 4340 steel, can prgress frm an elastic cnditin t a state resulting in significant inelastic strain while being cycled at a stress r strain range that initially prduced nly elastic behavir. Many structural steels have been bserved t either harden r sften, depending n the strain range (29). Steels that are verstrained and then cycled at a lw strain level may cyclically harden due t dynamic strain aging as seen in Fig. 11 fr the SAE 4340 steel tested by Dwling (14). Attempts have been made t predict whether a metal will harden r sften by the value f the mntnic strain hardening expnent, n (26,28,30). Metals with high initial n values tend t cyclically harden, while thse with lw values f n tend t sften. If lp shape is t be described by an equatin f the frm f Eq. 5, fr each lp during cyclic hardening r sftening, the strength cefficient and/r the strain hardening expnent must be altered with each cycle. Martin, et al. (11) accmplished this fr 2024-T4 aluminum by changing the strength cefficient and hlding the strain hardening expnent cnstant. This same type f apprach wuld prbably wrk fr any material which nly sftens r hardens. Fr a metal, such as mild steel, which hardens and sftens, mdeling wuld be mre difficult. The need t mdel the transients f hardening r sftening wuld depend n the accuracy f the rest f the stress-strain mdel, i.e., if stame lp shape cannt be mdeled with gd accuracy, there is n need t include the transients. Unlike cyclic relaxatin f mean stress, cyclic hardening r sftening is nly related t fatigue damage in an indirect manner. V Cyclic Relaxatin f Mean Stress An example f the cycle-dependent relaxatin f mean stress is shwn in Fig. 12. Being able t predict this type f relaxatin fr estimating fatigue lives f actual structures is particularly imprtant when large residual stresses are induced by ccasinal verlads. At lwer lads there may be enugh plastic defrmatin at the critical lcatin t cause cyclic relaxatin. This situatin culd ccur even if the majrity f the structure is behaving elastically.

19 r Nt unlike cyclic hardening and sftening, cyclic relaxatin is difficult t mathematically characterize. As a testing prblem, it is difficult t separate cyclic relaxatin frm hardening r sftening. The presence r absence f a large mean stress at the critical lcatin in a structure can ptentially alter the life by a factr f a hundred r mre. Thus, it is imprtant t accurately mdel cyclic relaxatin. Effrts t estimate cyclic relaxatin rates have nt been cmpletely successful (31). Several methds have been used t simulate cyclic relaxatin f mean stress (11,29, 20), all f which are utilized in cnjunctin with a cmputer based mdel. Mdels f Cyclic Stress-Strain Behavir The ideal mdel f cyclic stress-strain respnse shuld be capable f generating individual stress-strain lps s as t simulate cyclic hardening r sftening and cyclic relaxatin f mean stress. Anther characteristic f metals which must be mdeled is the histry dependence f the material. This is an essential characteristic t be simulated if randm r blck lading is present. Several mdels have been develped which accunt fr memry (histry dependence), cyclic hardening r sftening, and cyclic relaxatin f mean stress (11,20,32). Martin, et al. (11) and Wetzel (20) simulate the stress-strain curve by small linear segments. Jhansale (32) utilizes smth curves and prgrams the bserved behavir, such as memry. If fatigue damage is t be viewed as a cntinuus prcess, the amunt f "bkkeeping" is very large. Mdels f stress-strain behavir which can be prgrammed fr the digital cmputer are desirable. The mathematics f a mdel similar t that prpsed by Martin, et al. (11) is described in detail in the Appendix. An example f this mdel's ability t simulate the initial cyclic stress-strain behavir f 2024-T4 aluminum is shwn in Fig. 13. A general stress-strain mdel that includes transients des nt exist. Each mdel relies n empirical express'ns fr cyclic hardening r sftening and relaxatin. These expressins vary fi each material. Cnceivably enugh simplifying assumptins culd be made t derive a general mdel. If hardening r sftening wew» cnsidered t be a tw-step prcess by which the material jumped frm the mniqnic t the stable cyclic stress-strain curve, a general mdel might be cnceived. Jhe mdel prpsed by Martin, et al. (11) emplyed a relaxatin expressin which cntained nly ne cnstant. This expressin wuld certainly wrk fr any material if the taite at which mean stress relaxed was nt imprtant. The mdel wuld need nly six experimentally determined cnstants t simulate the stress-strain respnse f a metal. These are: elastic mdulus, mntnic and stable cyclic strength cefficients and strain hardening expnents, and the mean stress relaxatin cnstant. v The purpse f predicting stress-strain respnse is t determine the parameters necessary fr making fatigue predictins. Such infrmatin as stress amplitude, mean stress, ttal strain, elastic strain, plastic strain, etc., are needed fr each reversal r increment in the lad histry. ±A. *.,» «i

20 i i Jh 7 " -r 10 SMOOTH SPECIMEN FATIGUE BEHAVIOR Axially laded unntched specimens with cylindrical r hurglass test sectins can be subjected t cmpletely reversed cycling between cnstant strain limits t determine the strain-life curve. Sme metals underg cycle-dependent hardening while thers cyclically sften. The transient hardening r sftening ccurs rapidly at first then gradually slws until, at abut half f the fatigue life, further change in the size and shape f the hysteresis lp is difficult t detect. The behavir can then be cnsidered stable and a stable stress-strain hysteresis lp can be said t exist. TTie width f the lp at zer stress is the plastic strain range, A e, and the height f the lp is the stress range. ACT. Each time the directin f straining is reversed the initial unlading is elastic. The elastic mdulus, E, des nt change significantly during cyclic lading fr mst engineering metals. Fatigue Prperties A lg-lg plt f strain range versus reversals t failure fr mst engineering metals will be similar t that shwn in Fig. 14. At shrt lives the plastic strain range is large cmpared t the elastic strain range. At lng lives the elastic strain is larger. There is a transitin regin where the elastic and plastic strain ranges are nearly the same. The fatigue life when they are equal is called the transitin fatigue life, 2 N t. A mathematical relatinship fr the strain-life curve may be btained if the elastic and plastic cmpnents f the strain range are pltted separately as indicated in Fig. 14. Ae Ae CT.' e _ e. p _ f (2Nf ) D + e (2N f ) 1 (6) The cnstants b and CTf'/E are the slpe and intercept f the elastic strain versus life line, and e' and c are similar cnstants fr the plastic strain line. Fr a cnstant amplitude test the number f reversals t failure, 2 Nf, is equal t twice the number f cycles. If sufficient fatigue data are nt available t establish the fatigue prperties (cnstants in Eq. 6) they can be estimated frm tensin prperties. The fatigue strength cefficient CT is nearly equal t the true fracture strength, 0*, and the fatigue ductility cefficient, e', is apprximately equal t the true fracture ductility, ej. The fatigue ductility expnent, c, is usually a cnstant value f apprximately -0.6 fr mst engineering metals. It is mre difficult t estimate the value f the fatigue strength expnent, b. Fr wrught metals it can usually be shwn that the f atigue strength at 10 reversals is apprximately ne half f the ultimate tensile strength, S. This estimate, cupled with the apprximatin that CT' ~ CTf f gives an apprximate value f b f -1/6 lg(2ct f /S u ). The abve apprximatins shuld be used nly fr preliminary analyses and with great cautin. Actual fatigue test data such as shwn fr SAE 4340 steel in Figs. 15 thrugh 20 shuld be used t determine the fatigue prperties. The cst and time needed t generate such data can be justified by the fact that nce the cyclic prperties fr a metal in a particular cnditin have been determined they can be used as a basis fr fatigue damage analysis f a variety f cmpnents made f the metal subjected t virtually any histry f lading.

21 r T- 11 Mre details n the cyclic behavir f metals are given elsewhere (3,4,7,26, 28,30). Prcedures fr apprximating the cyclic stress-strain prperties frm varius tests n a single specimen are described in the Appendix. Effect f Cyclic Overstrain If fatigue specimens are first verstrained a few cycles at a strain range between ne and tw percent (Fig. 18), the life will generally be shrter at intermediate and lng lives than if n verstraining had been applied (14,15). The magnitude f the initial cyclic verstraining des nt seem t affect the resulting strain-life curve as lng as it is large enugh t cause cyclic plastic strain. Fr life calculatins where verstrain may ccur during cmpnent fabricatin r at any time during service, the fatigue data shuld be determined frm initially verstrained test specimens. Fr sme steels and pssibly ther metals the cyclic inelastic strains bserved at intermediate and lng lives are affected by ccasinal cycles f larger magnitude. Greater inelastic strains result fr mre frequently applied r larger verstrains (14). This effect eliminates the endurance limit fr steel as shwn in Fig. 19. In such cases where the stress-strain relatinship is nt unique, the use f a strain-life curve determined as described abve is questinable. It must be decided whether ttal strain, plastic strain, r stress are mre significant fr calculating fatigue lives. The use f the plastic strain-life equatin. (2N f r (7) is supprted by mechanistic arguments and by experimental evidence (14). It is recmmended that the plastic strain versus life relatinship at intermediate and lng lives be determined frm tests with peridic verstrains. If the inelastic strains that ccur at lng lives are t small t measure, a straight line extraplatin n the lg-lg plt f the shrter life inelastic strain versus life data shuld be made (see Pig. 15). Als, the cyclic stress-strain curve determined frm an incremental step test may differ frm that determined frm cnstant amplitude tests (14). The lwer curve shuld be used in fatigue analysis. The implicatins f the detrimental influence f verstrain in the fatigue analysis f structures in service are bvius. Many cmpnents experience plastic strain during fabricatin, r are verladed in service, which is similar t an verstrain n a smth specimen. Thus, it is apprpriate t peridically verstrain all test samples used t establish base line fatigue prperties fr purpses f cumulative fatigue damage analysis. V Effect f Mean Stress If a tensile mean r residual stress is present during fatigue, the life will be shrter than if it is nt present. A cmpressive mean r residual stress will have the ppsite effect. Fatigue life calculatins at intermediate and lng lives must include the effect f mean and residual stresses. A.p * **

22 12 At shrt lives the plastic strains are sufficiently large t cause an accumulatin f defrmatin if there is a mean stress. If a test is cnducted in stress cntrl, the mean strain will increase as the specimen is cycled. This behavir is called cycle-dependent creep. Fr strain cntrlled cnditins the mean r residual stress will relax (5,31) in a cycle-dependent manner if the inelastic strain is sufficiently large. Cycle-dependent relaxatin f mean stress is illustrated in Fig. 12. If the inelastic strain range is f the same rder r larger than the elastic strain range, n significant mean stress will exist after a few cycles (5). A mean stress can therefre be remved by strain cycling ne r tw cycles at a large strain range and then gradually reducing the strain range t zer. This technique is used after cyclic verstraining discussed abve t insure that the specimen is returned t zer stress and strain befre cntinuing the test at a small level f cyclic stress r strain. Fig. 18. Several extensive studies f mean stress have been carried ut fr 2024-T4 aluminum and SAE 4340 steel (10,13) as a part f this investigatin. Wilsn (13) generated data n 2024-T4 aluminum fr cnstant mean stress and varying mean stress lading. Using the cnstant stress data, life estimatins were made fr the varying mean stress tests. Mst f the life estimatins ver-estimated the life f the specimens. Tpper and Sandr (10) later studied the effect f mean stress with cnstant amplitude data based n prestrained specimens. Tensile and cmpressive mean stress tests were perfrmed n smth specimens that were cyclically prestrained. They fund the fllwing equatin t fit cmpletely reversed and mean stress data reasnably well. a r _ Ac 2 2 E (8) where Ae /2 is the "equivalent cmpletely reversed strain amplitude. " Equatin 8 defines a Fully reversed strain amplitude that wuld give the same life as a strain amplitude f Ae/2, cexisting with a mean stress, a. The expnent, a, was reprted as 0.73 fr 2024-T4 aluminum and 0.89 fr SAE 4340 steel. The crrelatin fr the cnstant amplitude data, including that with mean stress, was gd. Fig. 17. Equatin 8 als was used t estimate the fatigue lives f the varying mean stress data f Ref. (13). These life estimatins were sufficiently accurate when prestrained data was used. Damage summatins were frm 0.55 t 1.3. Thus, sme f the effects bserved by Wilsn (13) are believed t be due t the prestrain caused by impsing a mean stress. V Dwling (14,15) studied the effect f mean stress n SAE 4340 steel and 2024-T4 aluminum. Varius mean stress parameters were investigated. All specimens were initially verstrained and mst f the steel specimens with lives lnger than 10 cycles were peridically verstrained. Several mean stress parameters were cnsidered fr calculating an equivalent cmpletely reversed amplitude f stress. Fur f these parameters are listed belw: CT r = a 1 - ajs 7S ' u Smith (33) (9) Jt_

23 13 a 1 - a /a, ' f Mrrw (30) (10) a = ff a + -7? r a a Stulen (34) (ID a = [(a +a ) ( )] 2 r LX v a a /J Smith, et al., (35) (12) where S is the ultimate strength, CT, is the true fracture strength and d is a cnstant fr a particular material. Dwlinp reprted that Eq. 11 gave the best agreement fr bth metals. Figure 20 shws the calculated values frm Eqs. 11 and 12 pltted as a /a versus a /a. a' r r In anther investigatin f this nature (36) the effect f prestrain and mean stress was studied fr SAE 1015 steel. It was fund that bth Eqs. 10 and 12 agree reasnably well with the data. The accurate fit f the mean stress data f Eqs. 8 and 11 is due t the cnstants which are determined frm actual mean stress data. T utilize either f these equatins, mean stress data must be generated. Equatins 9 and 10 have mntnic tensin prperties as input. Equatin 12 des nt rely n any materials prperties but relates directly t cmpletely reversed fatigue data. The final chice f the mean stress parameter will depend n the data available. In mst practical cases the mean stress present at the critical lcatin in a part can nt be very large because f the presence f cyclic plas icity. Fr this reasn nearly any mean stress parameter is satisfactry since they d nt deviate significantly frm each ther fr small mean stresses. < f

24 A M 14 FATIGUE DAMAGE ANALYSIS Assuming that an accurate relatin exists between nnänal and lcal stressstrain behavir, that the stress-strain respnse f the metal can be simulated, and that apprpriate smth specimen fatigue data are available, a methd f cmbining these elements int an efficient cumulative fatigue damage prcedure is necessary. This prcedure must be able t calculate and keep accunt f the many variables which ccur in realistic lading situatins. One such prcedure which has been develped (11) is based n a mdel which is suitable fr digital prgramming. Fatigue damage analysis f simple histries is pssible withut the aid f a cmputer but might be very time cnsuming. Tucker (18) used such a methd wherein the mntnic and cyclic stress-strain curves were used t estimate mean stress and plastic strain fr cmplicated, but shrt, lading spectra. Fatigue Damage Criteria By knwing the stress-strain respnse f the material at the critical lcatin, fatigue estimates can be made by cmputing the damage frm the fatigue prperties f smth specimens. Since at any time during a lad spectrum the stress and strain at the critical lcatin is knwn, a large amunt f freedm is pssible in selecting the fatigue damage criteria t be used. Nearly every parameter imaginable has been used as a basis fr summing fatigue damage. During the curse f this investigatin, cyclic strain mdified t accunt fr mean stress effects, was fund t be the mst cnvenient parameter fr the type f prblems analyzed. Strain is a particularly attractive basis fr damage summatins, since a lg-lg linear relatin usually exists between plastic strain and life (37,38) and elastic strain and life (39). Tlius, a mathematical expressin can be used fr damage which is desirable if a cmputer based mdel is used. Hwever, it is pssible t stre fatigue data in tabular frm and interplate fr damage estimates. Fatigue damage is summed n the abve basis in the linear manner prpsed by Palmgren (40) and Miner (41). Many f the errrs which were nce attributed t the methd f summing damage using nminal stress-life relatins have been since traced t ther surces. Serius errrs have been due t nt prperly accunting fr mean stress and the effect f verstrain. Fr certain types f lading patterns, the way in which cycles are cunted can als be imprtant. The chice f a cycle cunting methd is particularly critical if the histry invlves ccasinal shifts in the mean strain that are large cmpared t the cyclic strain at each mean level. The grund-air-grund cycle in aircraft is an example. >» Cycle Cunting Prcedures fr interpreting cmplex lad, stress, r strain versus time histries are called cycle cunting methds. A number f different cycle cunting methds have been reviewed by Dwling (15). Only the range pair and the rain flw methds cunt strain cycles that crrespnd t clsed stress-strain hysteresis lps. These tw cycle cunting methds are described and cmpared by Dwling (15). /

25 15 The mean stress fr each clsed stress-strain hysteresis lp shuld be included in the damage analysis. The mean stress is simply the average f the maximum and minimum stresses ccurring during the cycle. If strain-life curves fr varius values f mean stress are available, the life fr any cmbinatin f strain range and mean stress can be fund by interplating between the curves. One f the mean stress parameters described in the last sectin can als be used t accunt fr mean stress. Each clsed stress-strain hysteresis lp can be assumed t be equivalent t ne cycle during a cnstant amplitude test. If linear accumulatin f fatigue damage is assumed, then the fatigue damage fr each clsed hysteresis lp is equal t the reciprcal f the fatigue life fr a cnstant amplitude test at the same strain range and mean stress. If a similar damage calculatin is made fr each clsed hysteresis lp, the results can be added t determine the ttal fatigue damage fr ne repetitin f the histry analyzed. Failure is expected when the strain histry is repeated a sufficient number f times fr the accumulated damage t reach unity. Sme Applicatins f the Fatigue Damage Analysis Utilizing the rain flw cycle cunting methd, prestrained and mean stress data, Dwling predicted the fatigue lives f smth 2024-T4 aluminum (15) and SAE 4340 steel (14) specimens subjected t a variety f cmplicated histries. The results f these predictins are shwn in Figs. 21 and 22. Fr the 2024-T4 aluminum the stress-strain respnse was calculated by hand, and the effect f mean stress n damage was taken directly frm plts fr 2024-T4 aluminum. Fr the SAE 4340 steel, a cmputer prgram f the type used in Ref. (11) was emplyed t determine the stress-strain respnse. Mean stress effects were estimated by Eq. 11. These results shw that with gd stress-strain simulatin, a prper methd f accunting fr mean stress and prestrain, and with a reasnable cycle cunting technique, accurate fatigue predictins can be made fr smth specimens. Nte in Fig. 22 the imprtance f using peridic verstrained life data rather than cnstant amplitude results. Martin (11) used the cmputer based mdel f stress-strain behavir in cnjunctin with Neuber's rule t predict the fatigue lives f plates with a center hle. Sme f the results are shwn in Fig. 23. > V Wetzel (20) als utilized a cmputer based mdel t estimate fatigue lives f ntched plates. In his analysis, damage is summed ver small segments f the cmputer based cyclic mdel. This prcedure eliminates the need t cunt cycles but prduces similar results as the rain flw methd f cunting. Althugh this methd was easier t prgram than the rain flw cunting methd, it is cnvenient nly if a cmputer based mdel is emplyed. Mst f the applicatins f the prpsed fatigue damage analysis have been fr small labratry specimens. As a mre realistic example, this analysis has been used t estimate the fatigue lives f sme built-up bx beams which have been tested by the Navy. M K f M*

26 J 16 Applicatin t Navy Bx Beams As an illustrative prblem, a cmputer based fatigue damage analysis was used t estimate the fatigue lives f sme f the built-up bx beams that have been fatigue tested by the Navy ver the past several years (42-46). Fatigue lives were cmputed fr the cnstant amplitude lad and test spectrum B fatigue tests reprted in Ref. (42). Additinal life estimatins were made fr ther cnstant lad amplitude fatigue tests reprted in Ref. (43). The bx beams are built-up structures. General design, fabricatin techniques and materials are representative f cmpnents used in aircraft structures. Fur metals were used in the beams. An aluminum ally, 7075-T6, made up the tp and bttm plates where failure ccurred. The beams were tested in three pint bending as described in Ref. (42). A brief summary f the prcedure used fr the bx beam life estimatins is presented here. A mre detailed explanatin is given in the Appendix. Cyclic stress-strain data cnsisted f the mntnic and cyclic steady state stress-strain curves. Cyclic hardening and cyclic relaxatin f mean stress were simulated by the cmputer mdel in much the same manner as stated in Ref. (11). Damage was based n either elastic r plastic strain. Mean stress was accunted fr by Eq. 10. Each reversal was individually cunted. A mre elabrate methd f cunting was nt emplyed at the time, since a prgram was nt available which culd stre the large amunt f data used fr the blck lading spectra. Figures 24 thrugh 26 shw the fatigue life estimatins made by the cmputer based fatigue damage analysis fr the cnstant amplitude tests. Agreement is gd fr mst f the data. After the estimatins had been made, it was fund that the tests were at times shutdwn. This releases the lad t zer. Upn relading, the mean stress wuld be altered frm its previus value. In Ref. (47) an attempt is made t accunt fr this by putting in ccasinal zer minimum lads which imprved the life estimatins. The variatin f Kj with life, which is reprted in Ref. (43), wuld als cntribute t the discrepancies between actual and estimated lives. The fatigue ntch factr is usually determined frm lng life fatigue data. Fr this prblem, K was estimated by first determining K t, frm published phtelastic results. This value was cnverted t Kf, by the highly stressed vlume apprach discussed in Ref. (16). Cnsidering that n previus testing was necessary fr the analysis, the results are quite encuraging. Figure 27 shws fatigue life data and estimatins fr the beams subjected t test spectrum B f M1L-A Fr the zer margin f safety, a definite blck size effect is evident in the tests. The estimated lives did nt shw the blck size effect exhibited by the actual test data. This blck size effect culd, in part, be attributed t the cyclic relaxatin f mean stress, which wuld be mre prbable fr the lnger blck sizes.

27 k p-» I Jl IJ - 17 CONCLUSIONS, LIMITATIONS AND FUTURE DIRECTIONS A prcedure fr perfrming cumulative fatigue damage analyses has been utlined. Several examples are presented f the successful applicatin f this analysis t estimate the fatigue life f varius specimens and parts made frm representative aircraft metals. Inputs t the analysis cnsist f a knwn lad, nminal strain r lcal strain histry, the gemetry f the part including its ntches and stress raiser, and mechanical prperties f the material frm which the part is made. Cyclic Histry Virtually any cmplex cyclic histry can nw be analyzed due t imprved methds f cycle cunting, better understanding f the effects f verstrain n the cyclic a-e and fatigue resistance f metals, and the use f cmputers t perfrm cycle t cycle analysis f lcal stress and strain and t sum damage. Difficulties wuld be encuntered in the fatigue analysis f parts that are laded frm mre than ne surce, particularly if the lads are nt in phase. This wuld cause cyclic changes in the directin f the maximum stress at the critical lcatin and precludes the use f nly uniaxial materials data in the analysis. Material Prperties The material input cnsists f a set f mntnic and cyclic stress-strain prperties and fatigue prperties including the effects f verstrain and mean stress. Metals with a yield pint may nt have unique cyclic prperties and s are smewhat mre difficult t accurately analyze fr fatigue purpses. Cast metals r parts with large initial flaws (such as weldments) wuld als be difficult t analyze, since their cyclic defrmatin and fracture resistance cannt be related directly t their bulk behavir in a "smth" specimen. Part Gemetry * The gemetry f the member t be analyzed is the mst limiting f the inputs t the analysis. Presently, nly simply ntched members that can be assumed t bey Neuber's rule can be treated withut resrting t methds f experimental stress analysis. Als, the critical lcatin must be subjected t essentially uniaxial cyclic,, stress since the present analysis des nt treat multiaxial stress states. Research is needed n the fatigue behavir f metals under cyclic multiaxial and nnprprtinal. stressing. \ Fatigue Life The utput f the fatigue damage analysis is the life fr die initiatin f a crack f the same rder f size as the smth specimens used t determine the base line fatigue data. This is a serius limitatin n the analysis since the size f this crack is arbitrary and many structures and parts satisfactrily perfrm in fatigue lng after cracks have initiated. The analysis shuld be extended t treat fatigue crack prpagatin as a sequence f crack initiatin events ahead f the crack.

28 ' ' ' wm TZ7 " 18 Despite the limitatin discussed abve as well as ther simplificatin such as ignring time and temperature effects, crrsin, and s n, it is felt that the prcedure utlined here can be helpful in estimating the useful fatigue life f simple metal parts. As methds are develped t treat mre f the cmplexities f real parts under actual service envirnments, the analysis can be altered and expanded t prvide mre accurate and meaningful life predictins. T extend the analysis t treat mre cmplex gemetries, a mre generally applicable methd fr analytically relating nminal and lcal stresses and strains needs t be develped. Perhaps an extensin f elast-plastic finite element analysis t cyclic prblems will be pssible in the future.

29 y 19 APPENDIX COMPUTER BASED FATIGUE LIFE PREDICTIONS FOR NAVY BOX BEAMS by J. F. Martin INTRODUCTION In the bdy f this reprt varius alternative methds fr predicting the lives f ntched members are presented. This appendix uses sme f these methds t predict the fatigue lives f built-up bx beam structures. References 42 thrugh 46 describe the bx beams and the type f lading emplyed. Only cnstant amplitude lading and Flight Spectrum B were analyzed. The bx beam was chsen because f its resemblance t aircraft structures designed by the Navy and because f the large amunt f data available. Available smth specimen materials data and existing cmputer prgrams are used fr making the fatigue life estimatins. Minimal specimen testing was cnsidered imprtant. These minimal data were used fr determining bth stress-strain behavir characteristics and fatigue resistance. Als, an effrt was made t minimize the amunt f cmputer time needed t make the life predictins. GENERAL APPROACH The apprach taken t the bx beam analysis can be brken int fur steps: (1) Calculate the nminal stresses near the critical lcatin. (2) Use an apprpriate mechanics analysis t relate nminal stresses t the lcal stresses and strains at the critical lcatin. (3) Determine, frm smth specimens, the uniaxial stress-strain behavir and the fatigue resistance f the metal where the crack initiates. V (4) Incrprate the abve int a prcedure t perfrm the cmputatins necessary fr the large number f cycles invlved. The analysis was made using techniques that were apprpriate fr the available smth specimen data r that culd quickly be generated. All full scale fatigue tests n the bx beams were perfrmed by the Navy and NBS (42-46). The beam was subjected t three pint bending. Simple beam thery was used t calculate the nminal stress near the critical lcatin fr a particular lad. Materials prperties and fatigue data were either taken frm reprts frm this cntract r generated specifically fr this prblem. Neuber's rule was used as ne relatin between the ntch rt stresses and strains and the nminal stresses. A cmputer prgram, similar t that described in Ref. 11, incrprated all the necessary data int a fatigue damage prcedure fr the bx beam prblem. j* i Jl *i

30 r TT 20 CALCULATION OF NOMINAL STRESSES The bx beam cnfiguratin and details are shwn in Fig. 28. Failure was assumed t ccur in the tensin side at the center set f rivets. Nminal stresses, S, fr the varius lad histries were reprted in Refs. 42 and 43. These stresses were btained frm strain gage readings that were cnverted t stresses frm the stress-strain curve f the 7075-T6 aluminum. The simple flexure frmula, Eq. 13, prduced stresses apprximately ten percent higher than experimentally determined. S = Mment x Half Depth f Bx Beam Mment f Inertia (13) Fr the fatigue analysis, nly the experimentally determined nminal stresses were used. MECHANICS ANALYSIS FOR RELATING NOMINAL STRESSES TO LOCAL STRESSES AND STRAINS Equatin 3 relates the prduct f the change in lcal stresses and strains t the change in nminal stresses. This equatin is nly valid when the nminal regin respnds elastically. The nly cnstants invlved are the elastic mdulus and the fatigue ntch factr, K f. Determinatin f K, The fatigue ntch factr can be determined by cmparing the stress-life data fr the beam and smth specimens. Ibis cmparisn is usually dne at lng live and necessitates cnstructin and testing f the structure. Fr this prblem, K is determined by phtelastic and analytical means. By this technique, it wuld nt have been necessary t fabricate and test the beam t determine K.. The value f the theretical stress cncentratin factr, K t, was taken frm Ref. 48 where values f K t are reprted fr flat plates with varius arrangements f circular hles. The bx beam did nt have cntinuus array f hles as des the phtelastic mdel. Figure 29 shws the prtin f the bx beam under cnsideratin. The b/a rati fr the phtelastic mdel was 0.63 as cmpared t 0.66 fr the bx beam. Fr a d/b rati f 0.19, Ref. 48 reprted a 1 f 2.9. The fatigue ntch factr, Kf, was determined by a highly stressed vlume apprach discussed in Ref. 8. Fr mst metals, Eq. 14 was fund t be accurate. U (14) where V and V are the highly stressed vlumes f the ntched and unntched specimens, respectively. Fr axially laded plates with shallw ntches,».1 Jl it

31 < i- i ji i u 21 V «0.018 trrb - (15) n n Here, t is the thickness, r the ntch radius, and b the ntched minimum width. The ntched minimum width fr the bx beam was taßen t be the width f the shaded area shwn in Fig. 29. The unntched specimen vlume was fr a cylindrical \" diameter specimen with a straight sectin f 0.66". Frm this infrmatin the apprximate value f K* was calculated t be 2.4. This value is an intermediate value f the varius estimatins reprted in Ref. 43. Lcal Stress-Strain Equatins 3 and 13 establish a relatin between an applied lad range and the prduct f the lcal stress and strain range. The slutin t Eq. 3 is nt unique fr a given nminal stress range, AS. Withut the materials stress-strain respnse, an infinite number f slutins exist fr the prduct Aa Ac. One means f btaining a cmplete slutin wuld be t defrm a uniaxial specimen in a manner s as t satisfy Eq. 3. The smth specimen wuld supply the needed relatin between stress and strain range. Hwever, since a testing machine capable f cntrlling under these cnditins was nt available, an analytical prcedure was emplyed. This prcedure required knwledge f the stress-strain behavir and fatigue resistance f the T6 aluminum at the critical lcatin. UNIAXIAL STRESS-STRAIN AND FATIGUE DATA In Ref. 3 the mntnic and cyclic stress-strain prperties and fatigue data fr 7075-T6 aluminum are reprted. Hwever, the majrity f the data were fr large t intermediate cyclic strains. It was estimated that mst f the strain cycles experienced at the rivet hle wuld be belw this range. Therefre, additinal data were btained at lwer strain amplitudes. The stress-strain prperties were determined frm a single uniaxial specimen, subjected t the strain histry shwn in Fig. 30. Frm this test the mntnic and cyclic prperties are characterized. Figure 31 shws the mntnic stress-strain curve up t a ttal strain f The cyclic stress-strain curve was determined frm the set f lps shwn in «Fig. 32. A lg-lg plt f these daw i s shwn in Fig. 33. f Mean Stress Relaxatin Cyclic mean stress relaxatin data were als prduced frm the single specimen test. Cyclic relaxatin was bserved during the strain limits with cmpress!ve mean strains. Sme f the hysteresis lps were shwn in Fig. 12. The assumptin is made that mean stress relaxatin can be islated frm cyclic hardening r sftening. If a sample is cycled fr a sufficient number f cycles, a stable cnditin is usually established in-s-far as hardening is cncerned. Hwever, by bserving the initial set f hysteresis lps. Fig. 12, it can be seen that the lwer lp tips d nt change at the same rate as the psitive peaks. Thus, hardening and relaxatin f mean stress are ccurring simultaneusly. Figure 34 shws the rate f mean stress relaxatin. These relaxatin data were used fr establishing the relaxatin parameter in the mdel f cyclic stress-strain behavir. y

32 - ' X ' " ' 22 Fatigue Resistance The last prtin f the single specimen test was under cnstant strain until final fracture. This was dne t btain ne data pint fr a strain-life plt fr cumulative damage analysis. Strain-life curves culd have been apprximated b\ the data frm this single specimen, had a means been available fr determining the slpes f the lines invlved. This was nt necessary since sufficient strainlife data were available. Figure 35 shws strain-fatigue life data, which were taken frm Refs. 3 and 17 and frm tests cnducted fr this investigatin. The strain-life data at small strain ranges withut verstrain were quite scattered. Hwever, the verstrain data had less scatter and enabled a better linear lg-lg fit fr the verstrain and large strain range data. All the verstrain data were generated specifically fr this prblem. It is speculated that the effect f verstrain n smth specimens is similar t the effect f fabricatin n structural cmpnents. Fr this reasn, nly the verstrain data were used at lng lives fr this analysis. COMPUTER FATIGUE DAMAGE ANALYSIS The infrmatin thus far gathered is cmbined and utilized t frm a fatigue damage analysis which is suitable fr the digital cmputer. The cmputer prgram fr this analysis is similar t that utlined in Ref. 11. The majr differences are in the damage summatin and the type f cyclic hardening emplyed. < The cmputer based damage analysis cnsists f three sectins: (1) the simulatin f uniaxial cyclic stress-strain respnse, which includes cyclic hardening and cyclic relaxatin f mean stress, (2) the cntrlling limits fr the stress-strain mdel, Neuber cntrl, and (3) damage summatin. These three subdivisins frm the cmplete fatigue damage analysis prgram. Cyclic Stress-Strain Respnse The basic frm f the stress-straii- simulatin prtin f the analysis fllws the rules fr a "series" rhelgical mdel. This mdel is shwn in Fig. 36. Memry capabilities are an inherent feature f the mdel. Memry is an essential characteristic fr simulating irregular lad histries. The stress-strain curve is apprximated by a series f linear segments. The slpe f any ne segment is determined by the stiffnesses f the springs, E. a, f the elements which are active. Whether r nt an element is active depends n the frictinal slider yield stress, a., and the residual stress,, lcked in the element due t previus lading. Elements are always activated in ascending rder. The details f adjusting the cnstants fr the mdel t simulate a particular stress-strain curve, and then utilizing the mdel t apprximate that curve are discussed belw. Determinatin f a and E: The yield stresses, Oj, and the spring stiffnesses, E, för N elemenrs are chsen In such a manner as t apprximate the smth stressstrain curve described by Eq. 5. This is illustrated in Fig. 37. The number f elements, N, is chsen large enugh t cnstruct a hysteresis lp fr any lading that might be encuntered during the analysis... E..

33 w T? 23 The cnstants fr simulating the stress-strain behavir f 7075-T6 aluminum were established by first setting the frictinal slider stresses and then the spring stiffnesses. The superscript a n E signifies that each spring stiffness is stress dependent. These stress dependent spring stiffnesses are the means by which mean stress is relaxed. The smth curve, which the mdel apprximates, represents a stress-strain relatin fr a particular degree f hardness. Thus, hardening r sftening is accmplished by determining a new stress-strain curve and adjusting the cnstants f the mdel t apprximate it. The first yield stress, which ends the first linear segment as shwn in Fig. 37, is chsen as * 2 = K»(e/ (16) where n* is either the mntnic r cyclic strain hardening expnent. The remaining yield stresses are incremented by a predetermined amunt. Fr this analysis, tp was assigned a value f and the yield stresses were incremented by 1 ksi. Since the equatin f the smth curve is knwn and the frictinal stresses have been determined, the slpes f each segment, E:*, can be calculated. E r = 5a.* i ST 7 ' (i = 2, 3,..., N) (17) Fr the first segment, the slpe E,* is the elastic mdulus. Already knwing the frictinal stresses 6 V = a i + i (18) The crrespnding changes in strain are 6CT* Vn* <j l/n* R * _ i. / i + U,1 v (19) f Fr this case K* and n* are either the mntnic r cyclic strength cefficient and strain hardening expnent, respectively. The reciprcal f the slpe f the i linear segment is the sum f the reciprcals f the stiffnesses f the elements that have been activated (20) fr i = 2,3 N..:/ Ji.

34 9 r 1 T? 24 T save cmputer time, the sum f the reciprcals f the slpes f elements are stred as a single number, Sl/E. Equatin 20 is rewritten as E E ff 1 (21) The value f E. a is nt a cnstant, but is dependent n the stress level at which the element is beirfg used. Equatin 22 is the relatin fr E chsen fr this simulatin E. CT = E i f( c ) (22) where Ej is a cnstant fr a particular stress-strain curve and relaxing functin, f( cr ). The value f la I is calculated at the midpint f each linear line segment. If f(, ) is f a frm that will reduce E with increasing cr, the mdel will cyclically relax mean stress. Such a functin is f(m) = i - i Sf (23) where M is the relaxing cnstant which is determined frm e:npirical data and des nt change during the simulatin. Equatin 23 is rewritten as 1 1 (S1/E.) E7E + -TnFlT (24) Once the values f E* are determined frm Eqs. 17 thrugh 19, the cnstants fr each spring, S(l/Ej), are calculated frm Eqs. 23 and 24. At this pint the cyclic stress-strain behavir f the metal whse stress-strain curve is described by Eq. 5, culd be simulated if a value fr the relaxatin cnstant, M, is knwn. Cyclic relaxatin is inherently assciated with the spring stiffnesses. The relaxatin cnstant is determined by trial and errr t fit the available relaxatin data. Cyclic hardening is accmplished by re-evaluating the cnstants f the parameters in the rhelgical mdel t fit a different stress-strain curve as described by Eq. 5. In determining the cnstants Ej and cf,, all calculatins are made ignring any residual stresses frm previus reversals. In this case, hardening was accmplished in tw steps. The mntnic stress-strain curve was used fr the initial stress-strain path. Once a predetermined plastic strain, = , was exceeded fr any ne cycle, the stable cyclic stress-strain curve was used as a basis fr cnstructing the frictinal slider stresses, Q., and the spring stiffnesses, E, as utlined befre. This is represented in Fig. 38. It shuld be nted that the residual stresses f all the elements which were left frm previus reversals are still lcked in. Althugh the spring stiffnesses and frictinal slider stresses are changed, the residual stresses remain until the element is activated again..-,...-;.' ---'-«"«Jl «I * *«

35 i - i a r 25 Simulating Stress-Strain Behavir with the Mdel; Once the parameters ä. and E i have been determined, the mathematical frm f the rhelgical mdel may be used t apprximate the cyclic stress-strain behavir f the metal fr which the cnstants were derived. The mdel cnstructs each segment f a hysteresis lp by determining the stress, a, *, at which the next element will be activated and calculating the slpe f the curve frm the spring stiffnessts f the elements that are activated. Elements are activated in ascending rder. At the beginning f each reversal, the last pint f stress and strain frm the previus reversal is set as a,* and e,*. The signs f the frictinal stresses are set. If stress and strain are increasing, the values f V- are negative. Fr a reversal where the stress and strain values are decreasing, sfj are psitive. TTie values f the residual stresses set by all previus reversals are knwn. Starting with subscript 2, the stresses fr each increment are calculated frm The change in stress fr that increment is Fr each segment, the reciprcal f slpe is a.* = -?r, - a. J. 1 (25) i 1+1 i + 1 s ' 6a.* = a.* - a*, (26) i i i - 1 v ' lx 1 E = F +(S E) rufiy (27) The change in strain fr each segment is The ttal strain is be.* = E.* (6a.*) (28) i i x i K e i* = e i* c i* (29) Thus, the mdel recnstructs the stress-strain curve, including the effect f memry, V by linear apprximatins. :. Fr each segment, a check is made t determine if the limits have been exceeded. If the limits were exceeded, a linear interplatin is perfrmed. Fr a final stress f Of*, all residual stresses activated fr the reversal are reset at I CT i ""f* ' ü i (30)

36 26 T7" n The same prcedure is fllwed fr the next reversal, unless a plastic strain f was just exceeded fr the first time. Exceeding this plastic strain limit signals the prgram t again calculate the cnstants fr the mdel using the cyclic stress-strain curve. This same prcedure is fllwed fr each reversal. At this pint the mdel was prgrammed fr the digital cmputer with strain as the cntrlling limits. The inputs t the prgram were the elastic mdulus, mntnic and cyclic strength cefficients and strain hardening expnents, and the relaxing cnstant. Initially, a high value f M = 10 4 ksi, was used. The mdel was checked t insure it culd reprduce the mntnic and steady state cyclic stressstrain behavir. T determine the relaxing cnstant, the prgram was run fr the strain limits shwn in Fig. 30 with varius values f M. A sample f the actual data and simulatin results are shwn in Fig. 34. Only the data during which mean stress relaxatin ccurred were f interest. A relaxing - ~ cnstant ~ - - f ~* 3» x 10 tn ksi i was chsen. Neuber Cntrl Limits: Fr each linear segment f the stress-strain curve prduced by the mdel, a check is perfrmed t determine if the desired limits have been exceeded. Fr the bx beam analysis the cntrlling limits were impsed by Eq. 3 fr a particular AS. In Ref. 42 AS frm Eq. 13 is calculated and the values are tabulated. These values were input int the prgram. Each time the prduct f lcal stress and strain range fr a particular nminal stress range was exceeded, a linear interplatin was perfrmed t satisfy Eq. 3 and the stress-strain relatin. Cumulative Fatigue Damage Summatin Damage summatin was accmplished by linearly summing damage fr each reversal. The rain flw cunting methd was nt emplyed, since the data fr this reprt were generated befre a suitable prgram was available. The damage summed fr any ne reversal was determined either by the elastic strain r the plastic strainlife curve frm Fig. 35. The plastic strain-life curve was used if the plastic strain exceeded Fr smaller plastic strains, the elastic strain-life curve was used. The equatins shwn in Fig. 35 were prgrammed int the analysis. Mean stress was accunted fr by an equivalent cmpletely reversed stress amplitude expressed by Eq. 10. Fr reversals with large plastic strains, mean stress was nt accunted fr. A cmputer prgram was written as shwn in the schematic in Fig. 39. This prgram was written t accmmdate the large amunt f input data fr the varius flight spectra which were simulated. With the materials stress-strain and fatigue data as input alng with the lads, fatigue life estimatins were made fr the cnstant amplitude and Right Spectrum B tests. The results f these estimatins are shwn in Figs and are discussed in the main text. SUMMARY AND DISCUSSION As an illustrative prblem, specific prcedures, which were discussed in the main bdy f this reprt, were applied t the bx beam prblem. The prcedures selected were thse which culd mst readily be utilized fr this prblem. The prcedure used fr this example can be discussed under the headings f the general utline presented earlier. M JL

37 27 1) Determinatin f the nminal stresses near the critical lcatin Experimentally determined nminal stresses as a functin f applied lad were taken frm Refs. 42 and 43. 2) Mechanics analysis fr relating nminal and lcal stress-strain behavir Neuber's rule was used t relate nminal and lcal stresses and strains. Phtelasticity data was used t determine a K t fr a structure similar t the tp plate f the bx beam. The highly stressed vlume apprach was used t estimate Kf frm Kj.. 3) Uniaxial stress-strain and fatigue behavir Since the Neuber relatin des nt independently determine lcal stress and strain but nly a relatin invlving the prduct f lcal stress and strain range, characterizatin f the stress-strain behavir f the metal at the critical lcatin is necessary. Data were generated t establish the mntnic and stable cyclic stress-strain behavir and the cyclic relaxatin characteristics. T incrprate the metal's stress-strain prperties int a cumulative damage prcedure, the strain-life data shwn in Fig. 35 were used. Only prestrain data were included in the analysis and the effect f mean stress was accunted fr by Eq ) Prcedure fr estimating the fatigue lives f the bx beams Neuber's rule and the cmputer based mdel f the metal's stress-strain behavir were cmbined t determine the stress-strain respnse at the critical lcatin. The stress-strain mdel primarily fllws the rule gverning the rhelgical mdel shwn in Fig. 36. The mdel's cnstants are initially set t simulate the mntnic stress-strain curve. Hardening is accmplished by changing the mdel's cnstants t fit the cyclic stress-strain curve. Cyclic relaxatin f mean stress is accmplished by making the spring stiffnesses stress dependent. c Fr each reversal, damage is summed based n the simulatin results. y The results f the applicatin f this analysis t the bx beam are encuraging. Future applicatins wuld help determine the extent t which this type f prcedure will prduce satisfactry estimates f fatigue life.

38 n 28 REFERENCES (Asterisk dentes papers prduced as a part f this prgram.) 1. * R. M. Wetzel, "Smth Specimen Simulatin f Fatigue Behavir f Ntches, " Jurnal f Materials, Vl. 3, N. 3, September 1968, pp See als: R. M. Wetzel, "Smth Specimen Simulatin f Fatigue Behavir f Ntches, " T. &A.M. Reprt N. 295, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, May * R. M. Wetzel, JDean Mrrw, and T. H. Tpper, "Fatigue f Ntched Parts with Emphasis n Lcal Stresses and Strains, " Naval Air Develpment Center Reprt N. NADC-ST-6818, September * T. End and JDean Mrrw, "Cyclic Stress-Strain and Fatigue Behavir f Representative Aircraft Metals, " Jurnal f Materials, Vl. 4, March 1969, pp See als: T. End and JDean Mrrw, "Mntnie and Cmpletely Reversed Cyclic Stress-Strain and Fatigue Behavir f Representative Aircraft Metals," Aernautical Structures Labratry Reprt N. NAEC-ASL- 1105, June * R. W. Landgraf, JDean Mrrw, and T. End, "Determinatin f the Cyclic Stress-Strain Curve, " Jurnal f Materials, Vl. 4, N. 1, March 1969, pp * T. H. Tpper, B. 1. Sandr, and JDean Mrrw, "Cumulative Fatigue Damage under Cyclic Strain Cntrl, " Jurnal f Materials, Vl. 4, N. 1, March 1969, pp See als: T. H. Tpper, B. I. Sandr and JDean Mrrw, "Cumulative Fatigue Damage under Cyclic Strain Cntrl, " Aernautical Structures Labratry Reprt N. NAEC-ASL-1115, June * T. H. Tpper, R. M. Wetzel, and JDean Mrrw, "Neuber's Rule Applied t Fatigue f Ntched Specimens," Jurnal f Materials, Vl. 4, N. 1, March 1969, pp See als: T. H. Tpper, R. M. Wetzel, J. Mrrw, "Neuber's Rule Applied t Fatigue f Ntched Specimens," Aernautical Structures Labratry Reprt N. NAEC-ASL-1114, June * D. T. Raske and JDean Mrrw, "Mechanics f Materials in Lw Cycle Fatigue Testing," Manual n Lw Cycle Fatigue Testing, ASTM STP 465, American Sciety fr Testing and Materials, 1969, pp See als: D. T. Raske and JDean Mrrw, "Appendix A - Lw Cycle Fatigue Testing frm the Mechanics f Materials Viewpint," Appendix t Naval Air Develpment Center Reprt N. NADC-ST-6818, September 1968, pp r 8. * T. H. Tpper and JDean Mrrw, Editrs, "Simulatin f the Fatigue Behavir at the Ntch Rt in Spectrum Laded Ntched Members (U)," T. & A. M. Reprt N. 333, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, January 1970 (Final Reprt fr Aer Structures Department, Naval Air Develpment Center). I '.Ai..

39 * JDean Mrrw, R. M. Wetzel, and T. H. Tpper, "Labratry Simulatin f Structural Fatigue Behavir, " Effect f Envirnment and Cmplex Lad Histry n Fatigue Life, ASTM STP 462, American Sciety fr Testing and Materials, 1970, pp See als Ref. (2). 10. * T. H. Tpper and B. I. Sandr, "Effects f Mean Stress and Prestrain n Fatigue Damage Summatin, " Effects f Envirnment and Cmplex Lad Histry n Fatigue Life, ASTM STP 462, American Sciety fr Testing and Materials, 1970, pp See als: T. H. Tpper and B. I. Sandr, "Effects f Mean Stress n Fatigue Damage Summatin, " T. &A. M. Reprt N. 318, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, August See als Chapter 111 f Ref. (8). 11.* J. F. Martin, T. H. Tpper, and G. M. Sinclair, "Cmputer Based Simulatin f Cyclic Stress-Strain Behavir with Applicatins t Fatigue," Materials Research and Standards, Vl. 11, N. 2, February 1971, p. 23. See als: J. F. Martin, T. H. Tpper, and G. M. Sinclair, "Cmputer Based Simulatin f Cyclic Stress-Strain Behavir, " T. &A. M. Reprt N. 326, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, July See als Chapter V f Ref. (8). 12. * S. J. Stadnick and JDean Mrrw, "Techniques fr Smth Specimen Simulatin f the Fatigue Behavir f Ntched Members, " Testing fr Predictin f Material Perfrmance in Structures and Cmpnents, ASTM STP 515, American Sciety fr Testing and Materials, 1972, pp See als: S. J. Stadnick, "Simulatin f Overlad Effects in Fatigue Based n Neuber's Analysis," T. &A.M. Reprt N. 325, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, See als Chapter IV f Ref. (8). - t», 13. * R. B. Wilsn, "Influence f Variable Mean Stress n Fatigue Damage," T. &A.M. Reprt N. 672, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, June See als: R. B. Wilsn, "Influence f Variable Mean Stress n Fatigue Damage, " M.S. Thesis, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, * N. E. Dwllng, "Fatigue Life and Inelastic Strain Respnse under Cmplex Histries fr an Ally Steel," T. & A. M. Reprt N. 354, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, April See als: Jurnal f Testing and Evaluatin, Vl. 1, N. 4, July 1973, pp I. 15. * N. E. Dwllng, "Fatigue Failure Predictins fr Cmplicated Stress-Strain Histries, " Jurnal f Materials, Vl. 7, N. 1, March 1972, pp See als: N. E. Dwllng, "Fatigue Failure Predictins fr Cmplicated Stress- Strain Histries, " T. &A.M. Reprt N. 337, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, January * D. T. Raske, "The Variatin f the Fatigue Ntch Factr with Life," M.S. Thesis, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, See als Chapter 11 f Ref. (8). 4 J/ K.f - -

40 " ' Jl 'U * D. T. Raske, "Sectin and Ntch Size Effects in Fatigue, " T. & A. M. Reprt N. 360, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, August * L. E. Tucker, "A Prcedure fr Designing against Fatigue Failure f Ntched Parts, " Preprint f Sciety f Autmtive Engineers, frm Autmtive Engineering Cngress, Detrit, Michigan, January 10-14, See als: L. E. Tucker, "A Prcedure fr Designing against Fatigue Failure f Ntched Parts, " M.S. Thesis, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, H. Neuber, "Thery f Stress Cncentratin fr Shear-Strained Prismatical Bdies with Arbitrary Nnlinear Stress-Strain Law, " Transactins, American Sciety f Mechanical Engineers, Jurnal f Applied Mechanics, Vl. 8, December 1961, pp R. M. Wetzel, "A Methd f Fatigue Damage Analysis, " Technical Reprt N. SR , Scientific Research Staff, Frd Mtr Cmpany, August See als: R. M. Wetzel, "A Methd f Fatigue Damage Analysis, " Ph. D. Thesis, Department f Civil Engineering, University f Waterl, Waterl, Ontari, September, C. R. Smith, "Small Specimen Data fr Predicting Life f Full-Scale Structures, " Fatigue Tests f Aircraft Structures: Lw-Cycle, Full-Scale, and Helicpters, ASTM STP 338, American Sciety fr Testing and Materials, 1962, pp E. Z. Stwell, "Stress and Strain Cncentratin at a Circular Hle in an Infinite Plate," NACA TN 2073, J. Schijve, "The Accumulatin f Fatigue Damage in Aircraft Materials and Structures, " AGARD-AG-157, Advisry Grup fr Aerspace Research & Develpment, Nrth Atlantic Treaty Organizatin, January J. H. Crews, Jr. and H. F. Hardrath, "A Study f Cyclic Plastic Stresses! *f at a Ntch Rt," Experimental Mechanics, Vl. 6, N. 6, June 1966, pp IF i 25. S. S. Mansn and M. H. Hirschberg, "Crack Initiatin and Prpagatin in Ntched Fatigue Specimens, " Prceedings f the First Internatinal Cnference n Fracture, The Japanese Sciety fr Strength and Fracture f Materials, Vl. 1, Sendai, Japan, September 1965, pp R. W. Landgraf, "The Resistance f Metals t Cyclic Defrmatin," Achievement f High Fatigue Resistance in Metals and Allys, ASTM STP 467, American Sciety fr Testing and Materials, 1970, pp ' 27. P. C. Rsenberger, "Fatigue Behavir f Smth and Ntched Specimens f Man- Ten Steel, M.S. Thesis, Department f Theretical and Applied Mechanics, University f Illinis, Urbana, 1968., S>. w*«. -

41 A - m «p JDean Mrrw, "Cyclic Plastic Strain Energy and Fatigue f Metals, " Internal Frictin, Damping, and Cyclic Plasticity, ASTM STP 378, American Sciety fr Testing and Materials, 195, pp H. R. Jhansale and T. H. Tpper, "Cyclic Defrmatin and Fatigue Behavir f Axial and Flexural Members - A Methd f Simulatin and Crrelatin," Prceedings f the First Internatinal Cnference n Structural Mechanics in Reactr Technlgy, Berlin, Germany, September 1972, Vl. 6, Part L, pp JDean Mrrw, "Fatigue Prperties f Metals," Sectin 3.2 f Fatigue Design Handbk, Sciety f Autmtive Engineers, JDean Mrrw and G. M. Sinclair, "Cycle-Dependent Stress Relaxatin, " Sympsium n Basic Mechanisms f Fatigue, ASTM STP 237, American Sciety fr Testing and Materials, 1958, pp H. R. Jhansale, "Inelastic Defrmatin and Fatigue Respnse f Spectrum Laded Strain Cntrlled Axial and Flexural Members, " Ph. D. Thesis, Department f Civil Engineering, University f Waterl, Waterl, Ontari, Canada, March J. O. Smith, "The Effect f Range f Stress n the Fatigue Strength f Metals, " Bulletin N. 334, University f Illinis, Engineering Experiment Statin, Urbana, February F. L. Stulen, "Fatigue Life Data Displayed by a Single Quantity Relating Alternating and Mean Stress," AFML TR , Air Frce Materials Lab., Wright-Pattersn AFB, Ohi, July K. N. Smith, P. Watsn, and T. H. Tpper, "A Stress-Strain Functin fr the Fatigue f Metals, " Jurnal f Materials, Vl. 5, N. 4, December 1970, pp P. Watsn and T. H. Tpper, "Fatigue Damage Evaluatin fr Mild Steel Incrprating Mean Stress and Overlad Effects," Experimental Mechanics, Vl. 12, N. 1, January 1972.? 37. L. F. Cffin, Jr., "A Study f the Effects f Cyclic Thermal Stresses n a Ductile Metal," Transactins, American Sciety f Mechanical Engineers, Vl. 76, August 1954, pp S. S. Mansn, "Behavir f Materials under Cnditins f Thermal Stress, " Heat Transfer Sympsium, University f Michigan Engineering Research Inst., 1953, pp See als: NACA TN 2933, \. 39. O. H. Basquin, "The Expnential Law f Endurance Tests," Prceedings, American Sciety f Testing and Materials, Vl. 10, Part II, 1910, pp" A. Palmgren, "Die Lebensdauer vn Kugellagern," Zeitschrift des Vereines Deutscher Ingenieure, Vl. 68, N. 14, April 1924, pp ~~"~

42 Jl I A - ~ > -is M. A. Miner, "Cumulative Damage in Fatigue, " Transactins, American Sciety f Mechanical Engineers, Vl. 67, 1945, p. A W. Breyan, "Effects f Spectrum Blck Size and Stress Level n Fatigue Characteristics f Aluminum Ally Bx Beams under Fixed-Sequence Unidirectinal Lading, " Reprt N. NADC-ST-6811, Naval Air Develpment Center, Jhnsville, Pennsylvania, September W. Breyan, "Cnstant-Lad-Amplitude Fatigue Characteristics f Aluminum Ally Bx Beams under Partially Reversed Lading," Reprt N. NADC-ST-7011, Naval Air Develpment Center, Warminster, Pennsylvania, Octber W. Breyan and E. P. Reser, "Effects f Spectrum Blck Size and Stress Level n Fatigue Characteristics f Aluminum Ally Bx Beams under Randm- Sequence Unidirectinal Lading," Reprt N. NADC-ST-7013, Naval Air Develpment Center, Warminster, Pennsylvania, December W. Breyan, "Summary f Test Results fr Aluminum Ally Bx Beam Fatigue Prgram, Test Phases 1-IV, " Reprt N. NADC VT, Naval Air Develpment Center, Warminster, Pennsylvania, June W. Breyan, "Effects f Blck Size, Stress Level, and Lading Sequence n Fatigue Characteristics f Aluminum-Ally Bx Beams," Effects f Envirnment and Cmplex Lad Histry n Fatigue Life, ASTM STP 462, American Sciety fr Testing and Materials, pp L. F. Impellizzeri, "Cumulative Damage Analysis in Structural Fatigue," Effects f Envirnment and Cmplex Lad Histry n Fatigue Life, ASTM STP 462, American Sciety fr Testing and Materials, 1970, pp J. W. Dally and A. J. Durelli, "Stresses in Perfrated Panels, " Prduct Engineering, Vl. 27, N. 3, March 1956, pp !,

43 - " - '-r r TZr 33 Critical Zne Ntch Smth Specimen Fig. I Smth Specimen Representatin f Material at the Critical Zne

44 turn T w-r tr 2>'l <D E T3 C t t c 0) If) II 'S c X a: c I < 00 I > _.E c t CO Ü 3 0) il f * i Jh *am

45 " y Si 0) k. 3 D 0) O O O» O E Q 0> g» (U > 3 E 3 Ü r APPLIED LOADS L J/

46 " y 31- Lcal Strain 2034.T4 2 N ( = 3/400, Reversals Fig. 4 Recrded Data frm Smth Specimen Simulatin f the Lcal Stress-Strain Behavir f a Ntch. (I) r O.J;.HV».?- v --'' Jl.V it

47 37 \ a».- " V E O v II r M 4 - J c > > 5 «0 'S CVJ z «#» c 0) If) E en WM Ü 0> II a H- V) _ ; * E 3 C E en 3 < H- CO H a> i W ID c N O a (A 0) **-' a: c g c ' «'5 L. D 55 E»MM i CO (A </) 0) cn a:» V in d» b. J/. A It

48 c OJ E if) c ii 0) <*- E «E 0) 3 C 2-6 D O O < E CO i in M- N N 0) c c. Ui 0) a: E 3 c <ö 5 _ (0 i a) (A JC a> u w. 5) z CD d» L. I s,'/ A. *. c

49 3? T3 (A a> > a> «4- (0 12 c S.i ü Ü c 0 w E a I S fc«!s).pövsv)** Li.

50 n -i O -i c O "5 3 I 15 C =.E* 2 1 ii n en Q l(/)l lt (n Q 5 1 c Q 5 1 Q 1 5 ta 1 Q si (/> 5 O in 0» s p c > g "a 3 c E <u (/) E c <u (U Q. E cn <u Q-l 0) cn E if) (A C 0) w OL a> 2 c g CM Ö 5 J L (M X 00 (0 «(M GO Ü I.,,,....u *---* - J/ - ' Jl «I

51 r *? TZT' Cyclic cr-c Curve Dubled y * Strain Measured frm Lp Tip Fig. 9 Stress-Strain Respnse fr 2024-T4 Aluminum (II) Ji j/

52 T - r O c I C 3 a E < 5 1 c V) a 8 _J (A (A X,...4*iS ;»v'-'* #;J ' Ä; " *.;/

53 I ll y i5 J2 CM O O s O m i AGMIOksi O J r > Fig. II Typical Stress-Inelastic Strain Hysteresis Lps during a Test with Overstrains every I0 5 Cycles fr SAE 4340 Steel (14) 4-,v

54 -.A T1 Cycles Fig. 12 Cyclic Relaxatin f Mean Stress fr 7075-T6 Aluminum m fvc- ä.fu» i.***»*«***' ' :

55 mm T T" vi c if) 0) en c a (/> a: c k. 00 I (A * I S i II CO V) c i CVJ < r * i

56 . jl MMI T? t$ W 0) Ü 0) 0) - ". E < c w CO Reversals t Failure (lg scale) Fig. 14 Representatin f Elastic, Plastic and Ttal Strain Amplitude-Fatigue Life Relatins V

57 i7 A f O trl k 4i - /. / «/ < / /» cn c c / «u 8 S * < M 0) c '5 J2 - y* w / * "55 * C / a> /!2» 3 c y<4 ' «mmrn '6 ««jl O Lü w " <» w 5:»- A& 0> V c «A Ü. < M > 5 r w O / _ «0 CO > c / O /. Ns v s «in \- > w /» IHM a) rf 0} 0) H- 9mm <n / J. -J O a < < / O 0) 1 Q» 3 «*- tf) Z / 0) > / / J. z i cvi n a U) <I T Hb > i a6uy ujdjts us 9U,d 37 _ - K 'fe t 0) a) 5) O c 1 astic S CM > Fig. 15 In. > Jl «I

58 rt "b 0> O a> 2 en O r Lü < (A 0) 0) 0.a C 1 S 5.E </> c 2 * O 0) g 0 V> S 2 1 l2 UJ c Q. '25 H- 0> -J (/) D (/) > c O 5) 2 l-«_i - I grj,i i i ' - ' fb * d6uy U1DJ4S '37 JL. *.

59 :L ~TZ7- f Ik "5.1 1 V A S CVJ 2 0 Mean Stress- 21 ksi Q Mean Stress = 50 ksi Nte-. All specimens prestrained 10 cycles at Ac/2'10.02 prir t testing -*- U) < I0 4 I0 9 10* Nf Cycles t Failure Fig. 17 Equivalent Strain Amplitude-Life Cmparisn fr SAE 4340 Steel (10)

60 50 r Fig. 18 Stress-Strain Recrding fr Initial Overstraining - *..P taal. * I Jh atm

61 m T t) u c? 8 I S c M - 2?., ö s ü > s c _ 6.2 (0 c O a c - s % S a> i! 1 a> > (/) O r < CO -J (0 a> v \ (A a> CO i 11 js) '»Buy ss9jis '07 8 2? , I w

62 - ia er O 5X 0 3 <N f <5X 0 4 5XI0 4 <N f <5X 0 5 A 5XI0 5 <N f <5xi0 6 Of"OQ + Tk 4.00 Fig. 20 Mean Stress Data fr SAE 4340 Steel Cmpared t Eqs. II and 12 V :/ 4. J

63 r- 9 r T? 3 I CSJ CM c Ü 5 0) w CL!2 it a < V CM S4S91 P JaqujnN i».1 Jl. -'

64 T3 I5r- 10 Using Cnstant Amplitude Data RVV! I J L aj n E 3 ZIO Accunting fr Histry Dependence 3i J I LJL SSSSSH I Rati f Actual Life t Calculated Life Fig. 22 Distributins f Life Calculatins fr SAE 4340 Steel (14) Ji

65 ...1 xr- fs 801- Tim«-80-I- 0n«Blck f Lad Spectra u a (0 I.Or I c. u «cc c 0.2- «w a> a. Test Data Ntched Plates 2024-T4, K f * ID* 10* 10* 10* N fl Cycles t Failure 10 I0 T 0.1 DU. V * Fig. 23 Lad-Life Curve fr Plates Subjected t Lad Spectra Shwn (II)

66 1 T TT Bx Beam E X 52 Simulatin I I0 2 I0 3 I0 4 I I0 7 r. Reversals t Failure, 2N f Fig. 24 Cnstant Amplitude Bx Beam Predictins and Data: lg t P max Lading i ' I i t. Ha_ tej it M

67 - - -» a* ftm mm TT ?5 -J P/2 Bx Beam P/2 X I Simulatin -Test Data g Time 10* 10 I0 1 I0 : \0 { Reversals t Failure, 2N f 10 * Fig. 25 Cnstant Amplitude Bx Beam Predictins and Data: lg±p Lading

68 7 -r- % Bx Btm r 25 P/2 S4 Test Data S Tim«5 2 Simulatin 10' 10' icy i Reversals t Failure, 2N f I0 ] r Fig. 26 Cnstant Amplitude Bx Beam Predictins and Data: -lg t P m(j% Lading ' ;.,.>,:'--T.'A*W.,Vr*H ; --'' Ji ii

69 7 7 5/ 24 r 20 Cmputer Simulatin Actual Data k y k. D O I O IOK 0% Margin f Safety 0) 11% Margin f Safety V Blck Size, Hurs 100 Fig. 27 Lad Spectrum B Bx Beam Predictins and Data «i

70 T T? a M a) r 0,1H.i-l :< < H 1 * CM O 0 I < i-l < H i H m «O U rh i H i i-i u-i (U ro a) O 4J rh HI <u IU M ih 1 vo Q Q <: gi H n 4f 4J ih IU c a 4= < m rh 1 rh ITl rh % 3 SB v m ON H t 1 ih N M >» in U-l u a. c <u a c a> 6 0) CL cn E a> GQ x CD GO CVJ d> iz v - ' i A Ml mmm

71 'r T- i>i a=l d = 0J9" Ö 0=1.0".J: Highly Stressed Regin Fig. 29 Highly Stressed Area f Bx Beam Cver Plate V

72 n leä 0) 6 " 6 (?) c a> E ü. CO a> c O w. O «- E UIDJ4S 4.,/i

73 *? wm U3 2C 0) (/) Strain Fig. 31 Mntnie Stress-Strain Curve f 7075-T6 Aluminum t«- - 3* it M

74 Jl Mi d 20ksi Strain Fig. 32 Set f Stable Hysteresis Lps fr Determining the Cyclic Stress-Strain Curve CJ

75 TT CM ID.0 ' 'S 00 b < i i a> "O 3. E < (0 0) 0) k. 5) 5 e M c 2. I < S CD V 8 r r d» il - * _A_ft «I

76 lh -r ~*zr 100 r 50 Test Results " v 1 * 0 -Peak Stresses A-Mean Stresses 0 Simulatin M = M = I x IQ 3 ksi ix IQ 4 ksi <5 a a- "tr i I 10 --tr O 5 10 I0 a Reversals, 2N ± j I 10' 10 Fig. 34 Cyclic Relaxatin Data fr a Strain Range frm 0.0 t fr 7075-T6 Aluminum i i Jh *.

77 1*7 E 3 C E < ID I in * 8pn;i dujv UIDJIS Q a> l c i- - V) lo r d» i

78 Jx. hi ww T Ef \ element E? / Ef E IYA/S TV Y A > a Fig. 36 Rhelgical Mdel fr Cmputer Simulatin ith segment Fig. 37 Mdel Apprximatin f Stress-Strain Curve

79 c 6 c 0) 0) 3 T3 0) tf) D SS9J4S 9 nj, i *E a k. X ü >% C 5 0) 3 a E K ÜJ F ssajs 9 njl l GO fo d» iz L A

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