N d ordre 07 ISAL 0091 Année 2007 THESE. présentée devant DEVANT L INSTITUT NATIONAL DES SCIENCES APPLIQUEES DE LYON. pour obtenir

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1 N d ordre 7 ISAL 9 Année 7 THESE présentée devant EVANT L INSTITUT NATIONAL ES SCIENCES APPLIQUEES E LYON pour obtenr LE GRAE E OCTEUR Spécalté : Mécanque Géne Mécanque Géne Cvl ECOLE OCTORALE ES SCIENCES E L INGENIEUR E LYON : Mécanque, Energétque, Géne cvl, Acoustque (MEGA) par Tong WU EXPERIMENT AN NUMERICAL SIMULATION OF WELING INUCE AMAGE STAINLESS STEEL 5-5PH Soutenue le novembre 7, devant la Commsson d Examen Jury : MM. Y. BRECHET ENSEEG, Grenoble Rapporteur H. MAITOURNAM Ecole Polytechnque, Pars Rapporteur MM. P. PORET AREVA-NP, Chalon Examnateur H. LU SJTU, Shangha Examnateur A. COMBESCURE INSA, Lyon recteur de thèse M. CORET INSA, Lyon Co-drecteur de thèse J.-M. BERGHEAU ENISE, Sant-Etenne Examnateur LaMCoS, INSA-Lyon, CNRS UMR559, F696, France

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3 Acknowledgments I would lke to express my heartfelt thanks to my advser, Professor A. Combescure, for hs contnung encouragement and ntellectual dscussons. Hs scentfc expertse n the numercal smulaton of mechancal problems especally damage s the most mpressve. I apprecate the help of my co-advser: r. M. Coret shared hs knowledge n the thermal, metallurgcal and mechancal smulaton and taught me how to desgn damage experments and then treat the data statstcally. Professor J.-F. Jullen s acknowledged for hs frutful cooperaton wth the unverstes and enterprses n Chna, whch I partcpated n. Hs manageral expertse and the suggestons on weldng experments are apprecated. Mr. Brechet, Mr. Matournam, Mr. Bergheau, Mr. Lu and Mr. Poret are acknowledged for servng as members of on my dssertaton commttee, and makng suggestons to mprove ths dssertaton. I would lke to express my specal thanks to the persons who help me n weldng feld, and they are Professor H. Lu (SJTU, Shangha) and Mr. P. Glles (AREVA-NP). I would lke to thank r. W. El Ahmar (INSA) and r. Y.G. uan (ENISE) for ther nteractve dscussons and r. J. Yang for hs expermental work n mcroscopc observaton, and also lke to thank all of the members of LaMCoS: Prof.. Nelas, r. S. Valance, r. T. Elguedj, r. A. Gravoul, r. P. Clerc, V. Boucly and r. L. Gallego for ther help. The work s part of project INZAT4 sponsored by the followng enterprses or organzatons: EF- SEPTEN, AREVA-NP, ESI-GROUP, EAS-CCR, Rhône-Alps. We would lke to express our grattude for ther fnancal and techncal supports Fnally, I address these last few words to thanks my famly for ther supports.

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5 Abstract The objectve of ths study s the predcton of damage and resdual stresses nduced by hot processng whch leads to phase transformaton n martenstc stanless steel. Ths study frstly concerns the modellng of the damage of materal nduced by a complex hstory of thermoelastoplastc multphase n heat-affected-zone (HAZ) of weldng. In ths work, a two-scale mode of elastoplastc damage multphase was developed n the framework of thermodynamatcs of rreversble process. The consttutve equatons are couplng wth ductle damage, elasoplastcty, phase transformaton, and transformaton plastcty. Besdes, a damage equaton was proposed based on the Lematre s damage model n the framework of contnuum damage mechancs. The experments of 5-5PH were mplemented for the dentfcaton of phase transformaton, transformaton plastcty and damage models. Tensle tests of round specmens were used to dentfy the parameters of damage model as well as mechancal behavours at varous temperatures. Tests of flat notched specmen were desgned to provde the valdaton of damage model and stran localzaton usng three dmensonal mage correlaton technologes. In addton, mcroscopc analyss was performed to provde mcrostructure characterzaton of 5-5PH and to dscover the damage mechansm. Fnally the numercal smulaton was performed n the code CAST3M of CEA. On the one hand, numercal verfcaton of the flat notched plates was mplemented and compared wth expermental results. On the other hand, we used the two-scale model ncludng phase transformaton, transformaton plastcty and damage to smulate the level of resdual stresses of a dsk made of 5-5PH metal heated by laser. The nternal varables, such as stran, stress, damage, were successfully traced n the smulaton of two-scale model. The smulaton results showed the transformaton plastcty changes the level of resdual stresses and should not be neglgble; damage decreases about 8 percent peak value of resdual stresses on upper surface of dsk. KEYWORS: amage, Weldng, Phase Transformaton, Numercal Smulaton, Transformaton Plastcty, Experment

6 Résumé L objectf de cette étude est la prédcton du dommage et des contrantes résduelles ndutes par des procédés haute température condusant à une transformaton de phase martenstque. On s ntéressera plus partculèrement à l acer noxydable 5-5PH. On s est ntéressé premèrement à la modélsaton du dommage ndut par une hstore thermomécanque complexe, comme peuvent en produre les Zones Affectées Thermquement de soudage. Nous proposons dans ce traval un modèle à deux échelles développé dans le cadre de la thermodynamque des processus rréversbles. Les équatons de ce modèle couplent plastcté, endommagement, transformaton de phase et plastcté de transformaton. Le modèle d endommagement résultant étend les modèles de type Lemaître à un matérau multphasé. Nous avons réalsé de nombreux essas sur le 5-5PH en vue de l dentfcaton des transformatons de phase et des los de comportement thermomécanques. es essas de tracton sur éprouvettes rondes nous ont serv à dentfer les paramètres des modèles élasto-plastques endommageables à dfférentes températures. Les essas sur les éprouvettes entallées ont été conçus pour valder les modèles d endommagement ans que la localsaton des déformatons en utlsant la stéréo corrélaton d mages. Nous avons de plus réalsé des analyses mcroscopques de la mcrostructure du 5-5PH et observé les mécansmes d endommagement. Les smulatons numérques ont été effectuées avec le code CAST3M du CEA dans lequel nous avons mplanté le méso modèle. Un premer calcul a été réalsé sur les éprouvettes entallées et a été comparé aux mesures expérmentales. Ensute, nous avons calculé l état de contrantes résduelles dans un dsque de 5-5PH ndutes par un chauffage laser. En sus des contrantes, on peut suvre au cours du calcul les varables nternes telles que l endommagement ou les déformatons anélastques. Les smulatons montrent que la plastcté de transformaton modfe le nveau des contrantes résduelles et peut ne pas être néglgeable. Quand à l endommagement, celu-c fat décroître les valeurs maxmales de contrante résduelles jusqu à hut pourcent dans les zones les plus sollctées. MOTS-CLÉS: Endommagement, Soudage, Transformaton de Phase, Smulaton Numérque, Transformaton Plastcté, Expérmentaton

7 Contents Abstract... Résumé... Contents... Nomenclature...v Chapter... Introducton... Chapter... 7 Martenstc Stanless Steel and Sold-state Phase Transformaton Introducton...7. Martenstc stanless steel and 5-5PH Phase transformaton of martenstc steel Phase transformaton of austente durng heatng Phase transformaton of martenste durng coolng Man factors of nfluence on transformaton Phase transformaton models Austentc gran sze and ts nfluences Gran sze calculaton model Effect on mechancal propertes....5 Measurement of phase proportons Phase transformaton nduced plastcty (TRIP) TRIP mechansms TRIP models Multphase mechancs and models Formulaton of the problem Mechancal models of multphase Summary...36 Chapter amage Mechancs and Weldng amage Introducton Phenomenologcal aspects amage varable Effectve stress Stran equvalence prncple amage measurement Thermodynamcs of sotropc damage State potental sspaton potental Traxalty and damage equvalent stress Threshold and crtcal damage uctle damage models Introducton Models based on porous sold plastcty Models based on contnuum damage mechancs Ansotropc damage...6

8 3.4.5 Concluson Weldng damage Introducton Heat affected zone (HAZ) of weld amage and crackng nduced by weldng Summary...64 Chapter Evoluton amage Multphase Modellng Introducton efnton of damage n multphase A proposed damage equaton Consttutve equatons of mesoscopc model n multphase State potental sspaton potental Consttutve equatons of two-scale multphase model Introducton Stran localzaton Mechancs n martenste Mechancs n austente Memory effect durng phase change Stress and damage homogenzaton An example n one dmenson Concluson...8 Chapter Expermental Study and Identfcaton of amage and Phase Transformaton Models Introducton esgn of specmens Expermental devces Measurement Temperature Force and dsplacement Stress Stran amage gtal mage correlaton (IC) Expermental programs Round bar tests Tensle tests of flat notched specmen Expermental results and dentfcaton of parameters Thermal and metallurgcal data Mechencal data Identfcaton of parameters of transformaton plastcty model Identfcaton of parameters of damage mdoel Comparatve analyss of flat notched specmen Mcrostructure characterzaton of 5-5PH Concluson...4 Chapter Numercal Smulaton and Implementaton of Consttutve Equatons Introducton Metallurgcal calculaton Phase trasnformaton calculaton Gran sze calculaton...9 v

9 6..3 Calculaton of transformaton plastcty Numercal mplementaton of the two-scale model Introducton Algorthm Numercal verfcaton of the models Phase transformaton and transformaton plastcty verfcaton Numercal verfcaton of flat notched bars Smulaton of a dsk heated by laser Introducton Fnte elememt smulaton Thermal and metallurgcal results Mechancal results Concluson...39 Chapter Concluson and Perspectves... 4 Bblography Appendx A Expermental evces A.Stran and stress measurements...57 A. Mcroscopc equpments...59 Appendx B... 6 Expermental Results of 5-5PH n etals... 6 B. Test results of round bar...6 B.. Materal propertes...6 B.. Force vs. dsplacement...6 B.. Stress vs. stran...65 B. amage results and fttng...69 B.3 Test results of flat notched specmen...7 Appendx C Example of Multphase Program n Castem...75 Lst of fgures... 8 Lst of tables v

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11 Nomenclature T ε ref γ α dfference of compactness of phaseα compared to phase γ at T ref z ε e ε volume proporton of phase total mcroscopc stran of phase elastc mcroscopc stran of phase p ε plastc mcroscopc stran of phase thm ε thermo metallurgcal mcroscopc stran of phase E c E e E p E Young s modulus classcal macroscopc stran elastc macroscopc stran plastc macroscopc stran tot E total macroscopc stran pt E phase transformaton plastc stran R X F p α r sotropc stran hardenng knematc stran hardenng yeld functon of phase equvalent plastc stran nternal varable assocated wth knematc hardenng nternal varable assocated wth sotropc hardenng ϕ damage dsspaton potental T temperature T ref reference temperature Σ y γ damage varable of phase macroscopc stress mcroscopc stress of phase yeld strength of phase eq equvalent stress S H devator of stress of phase α Hooke s operator α (T ) dlataton coeffcent of phase (depend on temperature T) v

12 α M s dlataton coeffcent of phase (mean value) martenste start temperature Ac Ac 3 β austentc transformaton start temperature austentc transformaton fnsh temperature coeffcent depend on materal n phase transformaton model pt ε eq equvalent plastc stran nduced by phase transformaton pt ε transformaton nduced plastcty p p R c κ I plastc stran to threshold damage plastc stran at falure threshold damage damage at falure damage exponent depend on materal unt tensor v

13 Chapter : Introducton Chapter Introducton The research on weldng, whch ncludes thermal, metallurgcal and mechancal phenomena, s a both old and new topc because many studes on the subject were started or developed many years ago and some are stll gong on. The weldng processes generally leaves resdual stresses n the weld and ts vcnty vcnty, snce the soldfcaton and contracton of molten materal durng coolng are constraned by the surroundng metal. The exstence of such resdual stresses can sgnfcantly affect subsequent lfetme by augmentng or mpedng fatgue or falure events. The performance n servce of parts that have been subjected to weldng or heat treatment depends on the resdual stress state of the structure. Consequently, for an accurate assessment of engneerng lfetmes, there s a need to determne resdual stress profles. From an expermental pont of vew, a varety of technques such as neutron, X-ray and synchrotron X-ray dffracton, hole drllng, curvature measurements and nstrumented ndentaton are avalable to measure resdual stresses [43][97][77]. Although expermental technques provde practcal solutons to measure resdual stresses, numercal method s necessary and should be emphaszed n ndustral engneerng. Ths s because that desgn and calculaton are no longer separated procedures and the calculaton nowadays s already nvolved wth the desgn process. Nowadays the ndustral needs n dverse felds such as aeronautcs, nuclear power and automoble mpose ncreasngly hgh constrants n order to reduce the manufacturng costs and to ncrease the relablty of the parts. Control of the behavor and the rupture of mechancal Tong WU

14 Thess: Experment and Numercal Smulaton of Weldng Induced amage structures becomes paramount wth the development of smulaton technology. Consequently, numercal smulaton leads to varous benefts for ndustry: Smulaton should help avod long test perods. Vrtual technques lead to maxmzaton of performances and optmzaton of processes. Computatonal approaches provde addtonal nsghts nto the expermental data and offer desgn capabltes. Smulaton could predct and then prevent cracks or further damage nduced by weldng process. For these reasons the numercal smulaton plays an ncreasngly domnant role n certan felds of ndustry. It was well known that the fnte element method (FEM) s a very effectve approach to calculate resdual stresses and damages, and ncreasng applcatons emerge n varous research and ndustral felds wth the development of computatonal or electronc technology. In partcular, n the fled of metal hot process, vrtual process becomes more popular n many nsttutes and enterprses to predct the results and tune the parameters before mplement of the actual experments or manufacture processes. The hstory of fnte element smulatons of the thermal and mechancal behavour durng weldng could be traced back to the 97s [9][94][75][66][9]. Computatonal Weldng Mechancs (CWM) s concerned wth the thermo-mechancal response as well as the changes n materal propertes due to the thermal cycles n weldng. The three most mportant conferences n CWM are Numercal Analyss of Weldablty [], Trends n Weldng Research [] and Modellng of Castng, Weldng, and Advanced Soldfcaton [3]. Lndgren [3] ntroduced dfferent modellng aspects of the applcaton of the fnte element method to predct the thermal, materal and mechancal effects of weldng. Besdes, he ntroduced complexty of weldng smulaton, materal modellng of weldng and computatonal strateges of weldng smulaton n [7][8][9]. In general, the predcton of weldng resdual stresses can be classfed as followng: statstcal and emprcal curves or formula, analytc method, nherent stran method [9][39][3] and thermo vscoelastoplastc fnte element analyss. The frst two methods are based on many smplfcatons and hypotheses, whch lead to lmted applcatons n ndustry. The concepts of nherent stran and stress were developed n Japan decades ago. Ths method meets the requrements of predctng weldng resdual stresses of carbon steels. The thermo elasto-plastc fnte element analyss seems to be the most promsng approach, and t was well developed by many researchers [45][6][84][46]. In our study, we use ths method to mplement the models. Therefore, our study ams to develop numercal weldng model under phase transformaton ncludng damage predcton. A successful model needs to consder thermal, mechancal and metallurgcal phenomena (Fgure. ) that are nvolved durng the weldng process. Ths coupled problem s usually solved by usng fnte element (FE) models where the thermal, mechancal and metallurgcal analyses are ether performed smultaneously or staggered. In pror lteratures [98][89][7][49],

15 Chapter : Introducton couplng were studed between thermcs and mechansm, between thermcs and metallurgy as well as the nfluence of metallurgy on mechancs. 4 Heat nduced by mechancal work Mechancs 5 Transformaton stran, stress 3 Thermal stress 6 Stress nduced transformaton Thermal Latent heat Phase transformaton Metallurgy Fgure. Couplng mechansms [98] When a model has been created, how could t be evaluated? For a successful model wth applcaton to realstc materals, one should not neglect the followng two aspects: frst, calbratons of parameters; second, model valdatons. Valdaton s the process durng whch the accuracy of the model s evaluated by comparson wth expermental results whereas calbraton to match the parameters wth some benchmark (see Fgure. ). Thus the same measurements should not be used for both calbraton and valdaton purposes. An adequate model thus should have suffcent accuracy for such purpose. Here, verfcaton s performed by assumng that the fnte element model s correct wth respect to the conceptual model and qualfcaton s the smlar process but between conceptual model and realty. Physcal Problem Qualfcaton Valdaton Mathematcal Model FE Soluton Verfcaton Fgure. Valdaton and verfcaton n fnte element modelng [44] Our work s a part of Project INZAT, whch s dedcated to study on smulaton and experment of weldng of metallc materals. The project s carred out n our laboratory (LaMCoS, INSA de Lyon) and lasted more than years, and s supported by varous ndustral enterprses of France: AREVA, EF, EAS and ESI. Some recent work of INZAT are shown n the references [53][6][63][93][97][99][7][8][9]. Our study extends M. Coret s work that s dedcated to expermental study and numercal smulaton of the weldng resdual stress of 6MN5 and SA533 [45][46][47][48]. Another prevous study could be traced back to the thess work of M. Martnez on weld jont of 6MN5-INCONEL [35]. In our study, the numercal model ncorporates more factors such as damage and s appled for 5-5PH nstead of 6MN5 stanless steel. The metal 5Cr-5N (a martenstc precptaton hardenng Tong WU 3

16 Thess: Experment and Numercal Smulaton of Weldng Induced amage stanless steel) whch s examned n our research undergoes the phase transformaton of martenste to austente durng heatng or vce verse durng coolng of weldng process. In the model, we wll take the three most mportant phenomena nto account, namely damage, phase transformaton and transformaton plastcty. In prevous studes, much work was devoted to the study of phase transformaton and TRansformaton Induced Plastcty (TRIP). Several authors have developed consttutve models for the mechancal behavor of dual-phase TRIP steels [][7][46][59]. Leblond et al. [6][7] consders that n a mxture of two phases wth dfferent specfc volumes, volume ncompatbltes generate a mcroscopc plastcty n the weaker phase, namely the one that has the lower yeld stress. He then proposed a proper TRIP model. On the other sde, the mechansm of damage has been extensvely studed n the lterature from the metallurgcal to the modelng pont of vew, such as Lematre-Chaboche damage model for elastoplastc materal [4][5], Rousseler ductle fracture model [63] and Gurson's ductle fracture model [87]. The lterature concernng ths topc s too broad to make complete quotatons [8][68][34][4][58][9][86]. However, there are few evdences for the exstence of models couplng both damage and phase transformaton (even TRIP) based on the same framework of thermodynamcs. Ths mples to cope wth a dversty of damage models and ncrease complexty to standard consttutve equatons. In our study, a new model wll be developed. Ths model contans three man ngredents, contnuum damage mechancs, transformaton plastcty and multphase behavor. Ths model wll be appled to calculate the weldng resdual stresses of 5-5PH under phase transformaton and damage condtons. As mentoned above, calbraton and valdaton are necessary for practcal applcatons. Thus, detaled processes of calbraton and valdaton for the damage and phase transformaton models, whch are two man dffcultes n numercal smulaton, wll be gven after the descrptons of two models n the thess. Next, the numercal smulatons of several cases wll be provded to llustrate the verfcatons and valdatons of consttutve models. In summary, the presented study wll be performed n the framework of the thermodynamcs of damage. Its gudelne prncples are to combne a detaled expermental analyss of the consdered materals and of ther specfc damage mechansms, a realstc modelng of these mechansms and the mplementaton of these models nto a numercal smulaton of the response of the structural components under nvestgaton. Chapters after the ntroducton secton are summarzed as followng: Chapter presents the specfc materal studed and ts metallurgcal aspect. Phenomena of phase transformaton, varous models of phase transformaton, TRIP models, several mechancal models of multphase are ntroduced n detals. Chapter 3 gves an ntroducton to Contnuum amage Mechancs (CM). efnton of damage varables and consttutve equatons of damage evoluton n the framework of thermodynamcs are ntroduced. 4

17 Chapter : Introducton The above two chapters are the ntroductons of some exsted theores and models related to our study, whch are selected from numerous lteratures. Chapter 4 s dctated to develop the couplng between damage and phase transformaton, and then to provde the coupled consttutve equatons n mesoscopc scale and two-scale n the framework of thermodynamcs. In addton, a damage equaton was proposed based on the Lematre s damage model. Chapter 5 focuses on expermental study and parameter dentfcaton of the stanless steel 5-5PH. These parameters nclude classcal materal propertes, parameters of phase transformaton, transformaton plastcty and damage models. The means of experments nclude the crcularly tensle tests of round bars, the tensle tests of flat notched specmens and the mcro observatons. The three dmensonal dgtal mage correlaton (IC) s used to study stran localzaton on surface of notched specmen. In Chapter 6, the numercal smulaton s mplemented n CEA CAST3M fnte element code [7]. The numercal verfcaton of the flat notched bars s performed to compare wth expermental results. Another example s the smulaton of dsk heated by laser, n whch the phase transformaton, transformaton plastcty and damage are ncluded n two-scale model. The damage and resdual stress wll be predcted n macro, meso and two-scale models. At the end, some concludng remarks are found n Chapter 7. Tong WU 5

18 Thess: Experment and Numercal Smulaton of Weldng Induced amage 6

19 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton Chapter Martenstc Stanless Steel and Sold-state Phase Transformaton. Introducton The phase transformatons often play a domnatng role n the modellng of certan thermomechancal problems. General concept of thermodynamc and phase transformaton s presented n [48] wthn the framework of sold-state physcs. Ths chapter s dedcated not only to the mechancal behavour and consttutve modellng but also to physcal and metallurgcal aspects. Introductons to concepts and models of transformaton-nduced volumetrc stran and TRansformaton-Induced Plastcty (TRIP) wll be found n ths chapter.. Martenstc stanless steel and 5-5PH There are three types of martenstc stanless steels: tradtonal (Cr3 s typcal one), low-carbon and super (or soft) martenstc stanless steels. The tradtonal type contans 4-7 wt(cr)% and - wt(c)%. espte of ts good toughness and hardness, the tradtonal type of martenstc stanless steels has lmted applcaton and t s thus replaced by new materals due to ts low ductlty and poor weldng ablty. The low-carbon martenstc stanless steels are melted through addng small quantty of alloyng element (e.g. Mo, T) and reducng the carbon content. Ths mproves the workablty and wt( )% means weght percent. Tong WU 7

20 Thess: Experment and Numercal Smulaton of Weldng Induced amage gans good corroson resstance. Therefore, ths type of martenstc stanless steel s wdely appled n hydro-turbnes and water pumps. The supermartenstc stanless steel wth super low-carbon and low ntrogen was developed durng recent years. Its nckel content s restrcted between 4% and 7%, and some alloyng elements (e.g. Mo, T, S and Cu) are added. Ths type of martenstc stanless steel s well appled n ppe and ol ndustres. Some typcal low-carbon and super martenstc stanless steels are gven n Table.. Another mportant type of martenstc steel s a martenstc precptaton hardenng stanless steel, such as 7-4PH, 5-5PH etc. Table. lsts chemcal ntegratons of some typcal knds of martenstc precptaton hardenng stanless steels. The alloy nvestgated n ths research s 5-5PH stanless steel (AMS5659), a knd of martenstc precptaton hardenng stanless steels steel. It offers hgh strength and ts common applcatons are components of aerospace ncludng valves, shafts, fasteners, fttngs and gears. The poston of 5-5PH n metal famly s shown n Fgure.. The specfc 5-5PH whch we used n experment s H5 (Condton H5) whch means that t s precptaton of age hardened and heat soluton-treated materal at specfed temperature (5 F) +/-5 F for 4 hours and ar coolng. The element compostons of 5-5PH H5 are shown n Table. 3 (as provded by the factory). In term of mcroscopc scale, ths steel s martenste at low temperature whereas t s n austentc state at hgh temperature. The mcroscopc observaton of 5-5PH n ambent temperature s gven n Fgure.. Type Standard C Mn S Cr N Mo Others ZGCr3N4Mo (Chna) JB/T N,S:.3 ZGCr3N5Mo (Chna) JB/T N,S:.3 CA-6NM (USA) Z4 CN 3-4-M (France) Z4 CN 6-4-M (France) ASTM A734 /ASTM-998 AFNOR NF A AFNOR NF A N,S: N,S: N,S:.3 Cr-4.5N-.5Mo (France) CLI Company Cr-6.5N-.5Mo (France) CLI Company Note: Sngle value means maxmum. Table. Some typcal knds of low-carbon and super martenstc stanless steels [36] N:. S:. N:. S:. C Mn S Cr N Cu T Nb+Ta 7-4PH PH Mo Al Mo.-.5 AM AM Table. Chemcal compostons of some typcal martenstc precptaton hardenng stanless steels [36] 8

21 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton 5-5PH Fgure. Classfcaton of 5-5PH n metal famly 5Cr-5N alloy s readly machned under both soluton treated and varous age-hardened condtons. In the soluton-treated condton, t performs smlarly to stanless Types 3 and 34. The machnablty wll mprove as the hardenng temperature s ncreased. 5Cr-5N can be satsfactorly welded by the shelded fuson and resstance weldng processes. Normally, weldng n the solutontreated condton has been satsfactory; however, where hgh weldng stresses are antcpated, t may be advantageous to weld n the overaged (H 5) condton. Usually, preheatng s not requred to prevent crackng. If welded n the soluton-treated condton, the alloy can be drectly aged to the desred strength level after weldng. However, the optmum combnaton of strength, ductlty and corroson resstance s obtaned by soluton treatng the welded part before agng. If welded n the overaged condton, the part must be soluton treated and then aged [96]. C SI Mn P S Cr Mo N Cu Al N NB Fe Balance Table. 3 Chemcal compostons of 5-5PH stanless steel (wt%) um Fgure. Mcroscopc observaton ( X) of 5-5PH steel Tong WU 9

22 Thess: Experment and Numercal Smulaton of Weldng Induced amage.3 Phase transformaton of martenstc steel At hgh temperatures (over typcal temperature of 75 C), steel atoms are arranged accordng to a crystal lattce form known as austente (FCC structure, Fgure. 3 ). As steel cools, austente changes nto pearlte, ferrte, bante or martenste, whch have a dfferent lattce structure (BCC structure, Fgure. 3). A schematc dagram of phase transformaton of martenstc steel under heatng and coolng condtons s gven n Fgure. 4. A transton lke ths, from one type of lattce to another, s referred to as a phase transformaton. Sold-state phase transformaton causes the macroscopc geometrc change because these dfferent types of crystallne structures have dfferent denstes, whch s the so-called transformaton-nduced volumetrc stran. urng a process of weldng, the transformatons of phase n a sold state, whch are the consequences of the mposed thermomechancal loadng, result from the combnaton of the change of crystal lattce of the ron and the dsplacement of the atoms of aqueous soluton. The crystallne states are functons of the temperature, the level and the stress type. The transformatons of phase of steels are often studed by usng the system of ron-carbon. Fgure. 5 shows the equlbrum dagram for combnatons of carbon n a sold soluton of ron. Face Centered Cubc (FCC) Body Centered Cubc (BCC) Fgure. 3 Crystal lattce structures of FCC and BCC Heatng Coolng Stran T ε ref α γ M f M s Ac Ac 3 Temperature Fgure. 4 Schematc dagram of phase transformaton under heatng and coolng condtons In steels, austente can transform to ferrte, pearlte and martenste ether by a reconstructve or by a dsplacve mechansm.

23 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton In the case of reconstructve nucleaton, t s formed n the mother phase startng from local nhomogeneousness, an embryo wth the composton and the crystallne structure of the new phase. The partton of the aqueous solutons to the nterface between the mother phase and the embryo s supposed near to where thermodynamc balance s. In order that the transformaton s carred out, t s necessary that the embryo reaches a crtcal sze, and there s a mechansm of growth of the new phase to complete the transformaton, n partcular of the transport of atoms by dffuson. In the case of dsplacve nucleaton, t appears n the austente of local confguratons of dslocaton whch, by slp n a correlated way, causes the dsplacement of the atoms from ther stes on the retculum of the mother phase to new nearby stes whch have the confguraton of the crystal lattce of the new phase. The growth of the germ s carred out n the form of slats or of plates, wthout transport of matter at long dstances compared to the nteratomc dstances. In steels, ths type of nucleaton ntervenes only when, followng a forced coolng, the transformaton of austente s carred out at a temperature qute lower than the Ac3 temperature. Martenstc transformaton s dsplacve and can occur at temperatures where dffuson s nconcevable wthn the tme scale of the process. Transformaton starts only after coolng to a partcular temperature called martenste-start temperature or M s. The fracton transformed ncreases wth the under coolng below M s. A martenste-fnsh temperature or M f s usually defned as the temperature where 95% of the austente has decomposed. Unlke M s, M f has no fundamental sgnfcance. Fgure. 5 Fe-Fe 3 C phase dagram [36].3. Phase transformaton of austente durng heatng When the austentzaton s processed at a suffcently slow rate of heatng under condtons of near stably thermodynamc balance, and the ntal and end temperatures of transformaton are respectvely Tong WU

24 Thess: Experment and Numercal Smulaton of Weldng Induced amage noted Ac and Ac3 (Fgure. 4). When the heatng rate s faster, and the ntal and end temperatures of transformaton are moved towards hgher values noted Ac' and Ac3' and the beach of transformaton s extended. Thus, the heatng rate of temperature nfluences the knetcs of transformaton and the ponts of begnnng and end of transformaton. urng the austentzaton, two other parameters play an essental role: the maxmum temperature reached and duraton perod at ths temperature. These two parameters determne the parameter of austentzaton whch leads to a balance between tme and the temperature. The hgher value of ths parameter means the larger sze of the austentc gran. The austentc sze of gran has an nfluence on the transformatons durng coolng. All the transformatons carry out nucleaton and growth. Nucleaton s done preferably on dscontnutes of the retculum and for ths reason the gran boundares represent a prvleged place. To modfy the sze of the austentc gran can modfy the spatal dstrbuton of the gran boundares and thus nfluence the knetcs of transformaton durng coolng..3. Phase transformaton of martenste durng coolng The martenstc transformaton occurs very far from thermodynamc equlbrum. When austente s cooled very quckly, carbon does not have tme to dffuse and ferrte cannot appear. Transformaton happens at temperatures well below eutectod temperature and occurs extremely fast,.e. at the speed of sound. Just lke ferrte and the pearlte, martenste s formed by a mechansm of nucleaton and growth. The martenstc transformaton s wth characterstcs of nstantaneousness nucleaton, quck growth and lmted sze of martenstc plate, and ts process depends more on appearance of new martenstc plates than on growth of old martenstc plates. New martenstc plates often appear boundares between new phase (martenste) and old phase (austente), and t means that locatons of nucleaton are nsde of grans, not lke phase transformaton of dffuse type, whch nucleaton occurs on the gran boundares (see Fgure. 6). urng martenstc transformaton, there s no dffuson, and the transformaton progresses through local atomc rearrangement. It stops transformng the rest austente when the suppled energy becomes nsuffcent. In ferrtc steels, t s n partcular of the martenstc transformaton and to a less extent about the bantc transformaton. Under ansothermes condtons, for fast rates of coolng, the ferrtc dffuson transformaton s stopped (or cannot occur). Contrary to the bantc transformaton, the martenstc transformaton occurs n hgher coolng rate. However, n some specfc martenstc steels, when austente s cooled, there s only martenstc transformaton occurs n low temperature, whereas no phase transformaton happens n hgh temperature or mesothermal state [][57]. In steels, martenste exsts n many forms, accordng to dfferent morphologcal characterstcs and sub-structure, namely: lath martenste, flake martenste, butterfly shaped martenste, thn slced martenste and thn plate martenste, etc. The lath martenste and flake martenste are often the cases. Usually, the former appears wth substructure of hgh-densty dslocatons, and the substructure of the latter s twn crystal.

25 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton Martenste (α') s hndered by austente gran boundares whereas allotromorphc ferrte (α) s not Fgure. 6 Schematcs for the formaton of martenste plates.3.3 Man factors of nfluence on transformaton Thermal effect In terms of thermal effect on phase transformaton, there are four factors that should be taken nto account: heatng rate, hghest temperature reached, duraton of hgh temperature stay and coolng rate. Fgure. 7 CCT dagram of martenstc steel Fv5(B) [57] The phase transformaton strongly depends on the coolng rate and the composton of alloy elements. Two types of dagrams are used by the researchers of the heat treatments to represent these transformatons smply: Tme-temperature transformaton (TTT) dagrams are obtaned by fast coolng of austente then mantenance at constant temperature; contnuous coolng transformaton (CCT) dagrams represents the transformatons durng coolng at constant speed. It s known that the decomposton of austente happens n ansothermal condton and more or less coarse mxture of ferrte-cementte, bante and martenste accordng to the speed of coolng. The physcal mechansm durng the nucleaton of the new phases depends on the temperature of start Fv5(B)- w(%):.5c,.4s,.8mn, 4.5Cr, 5.5N,.8Cu,.7Mo,.35Nb,.7P,.6S Tong WU 3

26 Thess: Experment and Numercal Smulaton of Weldng Induced amage transformaton, whch s a functon the coolng rate. Hgher speed of coolng, the ntal temperature of transformaton s lower. In martenstc steel, there are only martenste and austente durng heatng and coolng, so these three factors have no nfluence on the types of phase transformaton (Fgure. 7). However, the thermal parameters affect to the phase stablty and propertes. General speakng, the ncreasng of heatng rate can loss the stablty of austente, and make more dffcult to absorb carbondes nto austente. Improvng the top temperature ncreases the gran sze of austente and the stablty of austente. The heatng rate nfluence on the austente transformaton can be explaned n Fgure. 8. For a very slow heatng rate, there s enough tme to make the austente fracton to reach equlbrum state for each temperature. For an ncreasng heatng rate, t becomes more and more far away equlbrum state, and the ncreasng of austente fracton s slower and wth hysteress effect. Ths s well n accordance wth physcal realty. z T T T T 3 A A 3 T < < T < T T3 T Fgure. 8 Schematc dagram of heatng rate nfluence on austente phase transformaton [4] Gran sze of anstente The austentc transformaton as well as others phase-changes processes through nucleaton and growth. Usually the nucleaton of austentc transformaton occurs preferably on dscontnutes of texture and for ths reason the gran boundares present a preferred ste. The sze of gran of austente plays an mportant role n the transformatons durng coolng. Several works [44][35], resultng from the lterature, evaluated the nfluence of the sze of austentc gran on the transformatons durng coolng. These studes show the gran szes modfy not the thermodynamc equlbrum but the knetcs. In ferrte steels, t can not modfy thermodynamcs equlbrum but change both the knetcs of phase transformaton and the morphologes of phases formed. However, n supermartenstc stanless steels, the gran sze of austente does not affect the types of phase transformaton because martenste s only phase of output of phase transformaton (retaned austente possble) durng coolng, but t changes the rate of nucleaton and growth, and temperature of transformaton to some extent. 4

27 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton Percentage of carbon and other elements of addton Supermartenstc stanless steels and martenstc precptaton hardenng stanless steels are essentally alloys based on ron but contanng chromum and nckel. They owe ther name to ther room temperature martenstc mcrostructure. To understand the metallurgy of ths famly of steels, attenton s pad to the effects that the key alloyng elements have on phase stablty and propertes. - Carbon and ntrogen The transformatons of phase are related to the possblty of dffusng carbon, and the carbon concentraton modfes the transformatons of phase. Carbon and ntrogen are strong austente stablsers n Fe-Cr alloys. More percentage of carbon leads to more ferrte to be formed n the ferrte steel. However, n supermartenstc stanless steels, the carbon and ntrogen have lmted nfluence, because ther contents have to be kept as low as possble, about. wt%. Ths s because martenste hardness ncreases sharply wth carbon concentraton and therefore rases the probablty of sulfde stress corroson-crackng and hydrogen-nduced cold-crackng. - Chromum Chromum s a ferrte stablser. Low carbon Fe-Cr stanless steels have a ferrtc or martenstc, possbly sem-ferrtc, mcrostructure dependng on composton. When the chromum content s below approxmately wt%, t s possble to obtan a martenstc mcrostructure snce the steel can be made fully austentc at elevated temperatures. Such steels soldfy as δ-ferrte and are completely transformed to austente (γ) at hgh temperature, followed by relatvely rapd coolng to transformaton nto non-equlbrum martenste. A chromum content greater than approxmately 4 wt% gves a completely ferrtc stanless steel over the whole temperature range correspondng to the sold state and hence cannot be hardened on quenchng. Between the austente phase feld and the fully ferrtc doman, there s a narrow range of compostons whch defnes the sem-ferrtc alloys, wth a mcrostructure consstng partly of δ-ferrte whch remans unchanged followng soldfcaton, the remander beng martenste (Fgure. 9). - Nckel Austente can be stablsed usng substtutonal solutes. Nckel has the strongest effect n ths respect (Fgure. ) and also a tendency to mprove toughness. Nckel also has nfluence on Ms (martenste-start temperature) as Fgure.. Cr, N and Mo concentraton has nfluence on boundares of the austente, ferrte and martenste phases (Fgure. ). In addton, other elements have ther own functons n metallurgy, for example, manganese and copper are austente stablsng elements whereas slcon and ttanum are ferrte stablsng elements. For the sake of completeness, the synthess of Maruyama [36] about the role of elements s gven n Table. 4 where the role of the newly used alloy elements as W or Re, are also reported. Tong WU 5

28 Thess: Experment and Numercal Smulaton of Weldng Induced amage Fgure. 9 Range of lqud, austente and ferrte (α and δ) phase n the ron-chromum consttuton dagram wth a carbon content below. wt%. Fgure. Influence of nckel on the range of the austente phase feld n the ron-chromum system [74]. Ms temperature ( C) N % Fgure. Martenste start temperature (Ms) plotted aganst nckel content for 8Cr wt% -.4C wt% steel [] Fgure. Expermental dagram showng the boundares of the austente, ferrte and martenste phases as a functon of Cr, N and Mo concentraton for. wt% C after austentzaton at 5 C and ar coolng [9]. 6

29 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton Element Mert emert B C Improve creep strength and quench hardenablty. Stablze M3C6 partcles and delay ther coarsenng. Necessary to make M3C6 and NbC. Reduce mpact toughness. Co Suppress δ-ferrte. ecrease. Cr Improve oxdaton resstance. Lower Ms. Rase A. Man element of M3C6. Increase. Cu Suppress δ-ferrte. Promote precptaton of Fe M. Mn Increase and reduce creep strength. Lower A. Promote M 3 C 6. Mo Lower Ms. Rase A. Sold soluton hardenng. Accelerate growth of M 3 C 6. N Necessary to make VN. Nb Form MX and contrbute to strengthenng. Promote precptaton of z phase. N Increase and reduce creep strength. Lower A. Re Prevent the loss of creep rupture strength. Lower Ms. Lower A. S Improve oxdaton resstance. Increase and reduce creep strength. V Form MX and contrbute to strengthenng. Lower Ms. Rase A. elay coarsenng of M3C6 partcles. Sold W soluton hardenng. Note: desgnates the austente gran sze. Table. 4 Role of elements [36] Carbon and some elements of addton have nfluence on Ms. An emprcal formula as followng can explan the detals [9]. Ms( C) = C 6.3Mn 63.3N.8Cr 46.6Mo( wt%) (- ) It can be deduced from the above equaton that the Ms of materal 5-5PH s 88.5 C. The temperature predcated by the formula s found n good agreement wth our expermental result. Stress Appled stress modfes the energy stored n materal and the structure of the retculum. The nfluence of stress on the transformatons of phases s obvous. These couplngs were studed n the Insttut Polytechnque de Lorrane of France. Many work shows that the mechansms of nucleaton are most affected [5][5][78]. The appled stress s ether hydrostatc or unaxal. On the one hand, n term of the hydrostatc pressure, only the extremely hgh value about the hundred MPa can make ther nfluence notable on the knetcs of transformaton. On the other hand, the unaxal pressure, even weak, can affect the transformaton. Thus Gauter [78] shows that the tmes of begnnng and end of transformaton depend on the appled pressure n the case of an sothermal transformaton n eutectod Tong WU 7

30 Thess: Experment and Numercal Smulaton of Weldng Induced amage steel. In the same way, Patel [5] studes the nfluence of stresses on the martenste start temperature. Hydrostatc pressure delays the transformaton whereas compresson or tensle stresses make the transformaton to start earler. However, n the case of the transformatons wth contnuous coolng, f the coolng rate s relatvely hgh (> C/s), t seems that low unaxal pressures have few nfluence on the knetcs of transformaton..3.4 Phase transformaton models Knetc models The frst model of transformaton of phase was proposed by Johnson and Mehl [], then Avram [][], n order to predct evoluton of the proportons of pearlte. π NG t z = e (- ) wth z : volumnal proporton of pearlte N : rate of nucleaton G : rate of growth t : tme These models supposed that the pearlte appears through nucleaton then growth, dependng on austente. If the mechansms are dfferent, these models are used for the transformaton of ferrte and bante [][5]. The equaton of Johnson-Mehl-Avram s wrtten for sothermal transformaton, whereas the requrements n calculaton come manly from the heat treatments, whch are ansothermal. That s why some authors propose modfcatons of the precedng model to take nto account the ansothermal effects. Thus we wll note the models of Inoue [98], and then these models were further developed n INPL (Insttut Polytechnque de Lorane) [65][88][7]. These models consdered not only the effects of ansothermy but also the effects due to the appled stresses and the percentage of carbon. The martenstc transformatons are treated separately, because consdered as ndependent of tme. The emprcal law of Kostnen and Marburger [8] gves the volumnal fracton of martenste accordng to the temperature. The theoretcal justfcaton of ths equaton was gven by Magee [3]: β < Ms T > zα = zγ ( e ) (- 3) where z α and z γ are volumnal proporton of martenste and austente respectvely; M s s martenste start temperature; β s coeffcent depend on materal; T represents temperature. 8

31 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton Phenomenologcal models The generalzaton of the models based on laws of the Johnson-Mehl-Avram was done on phenomenologcal consderatons and dd not brng a really satsfactory applcaton, because these models on the physcal bases are dffcult to use n practce. Ther prncpal dsadvantage comes from ther specfcty to descrbe sngle type of transformaton. However n the case of the heat treatments or of weldng, a same structure s prone to varous transformatons dctated manly by the feld of temperature. Ths s why one seeks for ndustral applcatons and smple equatons of evoluton, whch are applcable to vared solutons and easy to establsh n a computer code. Other types of purely phenomenologcal models were developed. Thus Leblond [4][5] proposed a model based on a law of smplfed evoluton utlzng a proporton of transformed phase n equlbrum and a constant of tme: zeq( T) z z = (- 4) τ ( T) wth z : volumnal proporton of new phase z eq (T) : volumnal proporton of phase n equlbrum τ (T) : constant of tme Ths model, dentfed from CCT dagram, gves good results for the ferrtc and perltc transformatons but t s not ready to reproduce the faster transformatons correctly. Leblond ntroduces a dependence on dt/dt, whch enables to make ts model usable for the bantc and martenstc transformatons. However, the prncpal dffculty of use of the Leblond model les n ts dentfcaton. Therefore, Waeckel [][] proposed another possblty, an easly dentfable model starng from CCT dagram and able to reproduce thermal hstores of weldng. The purpose to lmt the dentfcaton to a CCT dagram led Waeckel to lmt the ntervenng varables n the transformatons of phase. A law of evoluton of ths type s proposed: z = f ( T, T, z, d) (- 5) wth z : proporton of consdered phase d : gran sze of austente Contrary to the model of Leblond, n the model of Waeckel, T s an nternal varable, whch seems to rather close to the realty n ths type of model owng to the fact that the knetcs s ansothermal. Also t s noted that the temperature Ms s not a metallurgcal state, but as t can vary durng transformaton. It was blended n the metallurgcal varables. Moreover the volumnal proporton of martenste s treated separately and follows the equaton of Kostnen and Marburger [8]. The gran sze of Tong WU 9

32 Thess: Experment and Numercal Smulaton of Weldng Induced amage austente s a parameter but not a varable. The functon f s known n a certan number of states, and one can deduce the volumnal factons of the phases by lnear nterpolaton. Ths model descrbes well the transformatons of phases durng coolng at a constant rate. We wll fnsh ths revew of the varous models of phase transformaton by work of Martnez, whch was carred out at the CEA de Grenoble and from whch we serve for calculatons of thermalmetallurgcal-mechancs. Martnez extends the model of Waeckel and adds the dependence on the gran sze of austente d and of the percentage of carbon c. Many tests were carred out to trace the CCT of a nuance of steel accordng to condtons of austentzaton and carbon contents durng coolng. Martnez proposes a law of evoluton of ths type: where c : carbon concentraton d : gran sze of austente z γ = f ( T, T, zγ, d, c) (- 6) z γ In ths model, the varables c and d are calculated from the laws dependng only on materal and thermal. Ths model was establshed [] n the calculaton code of the CEA Cast3M [94]. Thermalmetallurgcal procedure was suggested to calculate the quanttes T, T, z γ, d and c, whch are an teratve procedure n fve stages:. T and T are calculated by a nonlnearly thermal calculaton dependng on the boundary condtons (conducton, convecton, radaton), nput sources and characterstcs of materal.. The carbon concentraton s calculated by an equaton of dffuson dependng on T and the coeffcent of actvty of carbon. 3. The gran sze d s determned by the equaton of Grey Hggns dependng on the temperature T. 4. The proportons of phases are calculated ether by lnear nterpolatons, or by an explct equaton n the case of the martenstc transformaton. The result depends on T, T, d and c. 5. Fnally one calculates the latent heats of transformatons. In concluson of ths part we propose a small assessment of these varous models n Table. 5. Knetc model Phenomenologcal model Advantages Smply use. Exhaustve descrpton of the varous transformatons.. ependence on the carbon concentraton and the sze of gran. sadvantages Adapted to only one type of transformaton. Few physcal concerns.. The valdty for of nonconstant T s hypothetcal. 3. Calculaton of nterpolaton s long. Table. 5 Advantages and dsadvantages of knetc and phenomenologcal models

33 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton.4 Austentc gran sze and ts nfluences In welds, gran sze s greatest at the fuson zone and then gradually gets smaller n the heat-affected zone (HAZ) untl the base metal s reached. Therefore, gran sze s mportant for study of mechancal behavours (resdual stress, damage ). Calculaton of the gran sze can be useful for three reasons as followng: Ths parameter s mportant n tself (t s nvolved n hot falure phenomena). Gran sze can nfluence the transformatons as mentoned above. Gran sze has nfluences on mechancal behavour of metal..4. Gran sze calculaton model Changes n gran sze only concern the austentc phase and depend on temperature changes and the proporton of ths phase. The evoluton of gran sze s calculated by usng an extenson of the conventonal equaton proposed by Alberry & Jones [6] and Ikawa et al []: Q = C exp( ) RT (- 7) G a where G : gran sze T : absolute temperature R : constant for gases 4 a, C, Q are postve constants. (Generally, a = 4, C =.4948 mm 4 / sec, Q R = 639 gves satsfactory results.) In addton, Ashby and Easterlng have proposed a more general equaton: Q G = f ( G) exp( ) RT (- 8) Ths conventonal equaton s lmted to cases n whch the proporton of austente s constant or decreasng. In fact, f the proporton of austente ncreases, two phenomena can be observed physcally: a) the sze of the grans already exstng ncreases; b) new grans are formed wth an ntal gran sze of zero (or, at least, much lower than the exstng sze). Generalzaton of the conventonal equaton was gven by Leblond [4] as followng expressons: d dt ( G a ) = C exp( Q RT C exp( z ) G z Q ) RT a f f z > z (- 9) Tong WU

34 Thess: Experment and Numercal Smulaton of Weldng Induced amage.4. Effect on mechancal propertes Gran sze has mportant nfluence on mechancal behavours. Generally, a small gran sze n HAZ s beneft to strength and toughness. The dependence of the yeld strength on gran sze s gven by the Hall-Petch relatonshp [5]: k y y = + (- ) d where d s the gran dameter, y s the yeld stress, s the frcton stress opposng the movement of dslocatons n the grans and k y s the fttng parameter (a materal constant). In face, the Hall-Petch relatonshp or Hall-Petch Law s a relaton n materals scence that deals wth the connecton between the gran sze, or crystallte sze, and the yeld pont of a materal. Ths relaton says that the larger the gran sze of a crystallne materal, the weaker t s. The dervaton of the Hall-Petch equaton reles on the formaton of a dslocaton ple-up at a gran boundary, one whch s large enough to trgger dslocaton actvty n an adjacent gran. Yeld n a polycrystallne materal s n ths context defned as the transfer of slp across grans. A larger gran s able to accommodate more dslocatons n a ple-up, enablng a larger stress concentraton at the boundary, thereby makng t easer to promote slp n the nearby gran [7]. For martenstc structures, however, dslocaton sources are found at gran boundares, whch are dfferent from Hall-Petch approach that consders dslocaton sources wthn ndvdual grans. The ncrease n strength due to martenste lath sze s gven by: = 5( L) (- ) G where L s the mean lnear ntercept taken on random sectons and at random angles to the length of any lath secton. In the case of phase transformaton, n low carbon mcroalloyed steels, ferrte gran szes and precptaton states are mportant factors, whch affect the strength toughness relatonshp. The ferrte gran sze s a functon of the austente gran sze after austente recrystallzaton, the amount of retaned stran n the austente before the start of transformaton, and the coolng rate through the transformaton regme [56]. There s no general mechansm, by whch gran refnement mproves toughness. The argument for steels s that carbde partcles are fner when the gran sze s small. Fne partcles are more dffcult to crack and any resultng small cracks are dffcult to propagate, thus leadng to an mprovement n toughness.

35 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton.5 Measurement of phase proportons Ths part ntroduces the measurement of the proportons of phase durng a thermal cycle. There exst three possbltes to determne the proportons of phase. Resstvty as well as the magnetc permeablty are very dfferent for the γ and α phases, and the consttuton of the two measurable quanttes leads to the proportons of phases [9][9]. Here, we use the dlatometry to measure the proportons of phase. A test of dlatometry conssts of heatng and coolng of a sample, and measurement of ts extenson. Fgure. 3 gves an example of dlatometry. Proportons of phase can be deduced from ths curve because of the dfference between the dlaton coeffcents of γ phase and that of α phase. Fgure. 3 Test of dlatometry on A58 steel (heatng: 3 C/s; coolng: - C/s) [6] Leblond shows, by consderng two elastoplastc phases whose elastc characterstcs are the same, whch the partton of the total deformaton s equal to: E + tot e thm p = E + E E (- ) =< ε ε ε (- 3) thm e thm p E > v + < > v + < > v where, <> v represents the average volume. E thm thm th th T =< ε > = ( z ) ε + z ( ε + ε ref ) (- 4) v γ α γ γ α γ The above equaton can be wrtten: E thm = [( z ) + ]( ) + (- 5) γ T α ref α zγα γ T Tref zγ ε α, γ z ( T ) γ E α ( T T thm α ref = (- 6) ( α α )( T T γ α ref ) ) + ε T ref α, γ We can calculate volume proporton of austente from the above equatons. Tong WU 3

36 Thess: Experment and Numercal Smulaton of Weldng Induced amage.6 Phase transformaton nduced plastcty (TRIP) Transformaton nduced plastcty (TRIP), pt ε n Fgure. 4, can be defned as the anomalous plastc stran observed when metallurgcal transformaton occurs under an external stress much lower than the yeld lmt. From a technologcal pont of vew, t plays an essental role n many problems, n partcular for the understandng of resdual stresses resultng from weldng operatons and for ther predcton n practcally sgnfcant cases. From an expermental pont of vew, t s usually characterzed n a TRIP test where a constant stress s appled durng transformaton under prescrbed coolng condtons. a) Free dlataton b) lataton under pressure Fgure. 4 lataton of unaxal tests of phase transformaton wthout/wth appled stress [45] y <.6. TRIP mechansms From a phenomenologcal pont of vew, the transformaton plastcty results from the couplng between an external stress and the evoluton of proportons of phase nvolved. A transformaton, whch s accompaned by a varaton of volume, causes a plastc stran of the softest phase. In general, several faces of transformaton move n dfferent drectons, so that the plastc strans of transformaton are not added from one to others and t appears a null macroscopc resdual stress feld. Ths s why, there s not reman of resdual deformaton at the end of a cycle of free dlaton. On the other hand, varous work for a long tme showed that the appled stress, even low, durng the transformaton, could lead to an addtonal deformaton wth the deformaton of transformaton, n the drecton of the loadng of pressure. It was called phase transformaton nduced plastcty (TRIP). Two mechansms can explan ths deformaton of transformaton. Greenwood & Johnso s mechansm The frst model s the mechansm of Greenwood & Johnson. It expresses the fact that the transformaton plastcty s nduced by the appled pressure. Ths knd transformaton plastcty arses 4

37 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton from mcro plastcty n the weaker austentc phase nduced by the dfference of specfc volume between the phases. Ths plastc flow s orented by the external load. Many expermental results show that for low levels of stresses, the rate of deformaton of transformaton plastcty s proportonal to the appled pressure, the varaton of volume assocated wth the phase shft, the development of the transformaton and nversely proportonal to the yeld stress of the softest phase. For levels of stress hgher than half of the elastc lmt of the austentc phase, the relaton of lnearty between the rate of deformaton and the external pressure ceases. The deformaton becomes more mportant. On the study of the relaton between the appled pressure and the deformaton of transformaton plastcty, one can quote, n the chronologcal order, the work completed by [55][85][43] [77][7][95], and lately the expermental work done by [8] and [53]. All ths work confrms the lnearty between the appled stress and the deformaton of transformaton plastcty as long as the pressure does not exceed a certan fracton of the elastc lmt of the austentc phase. Ths fracton depends on study of ferrous alloy. Some expermental work [77][95][8][53][53] was dedcated to study the relaton between the rate of deformaton of transformaton plastcty and the development of ths transformaton. To study ths relaton means to study the knetcs of the transformaton plastcty: n other words t s the study of the varaton of the deformaton accordng to the proporton of daughter phase nduced by coolng. Expermental work shows a relaton of non-lnearty between these two quanttes, the varaton of the transformaton plastcty beng ncreasngly hgher at the begnnng of the transformaton. Ths relaton changes accordng to the metallurgcal type of transformaton durng coolng. It was verfed that the mechansm ntally proposed by Greenwood & Johnson s manly appled to descrbe the deformaton of transformaton plastcty n the case of the ferrtc transformaton and, to a more extent, n the case of the bantc transformaton. Indeed n the case of the phase shfts wthout dffuson,.e. the martenstc transformaton, only ths mechansm cannot be responsble for the phenomenon of transformaton plastcty. Therefore the second mechansm proposed by Magee [33] s presented. It explans the orentaton of the transformed zones by external stresses. Magee s mechansm Mechansm of Magee corresponds to the formaton of selected martenstc varants resultng from the appled stress. Mcrographc analyss shows that martenste has a crystallographc structure derved from that of austente. Its fnal structure s obtaned by a deformaton of the ntal phase, n partcular a shearng. It has morphology of slats or plates and presents the relatons of prvleged orentaton and the parent phase. When pressure s not appled, the auto-accommodatng arrangement of the plates s lke that the energy stored n materal s mnmzed and the deformaton measured at the end of the transformaton s practcally null. When the transformaton occurs under pressure, the martenste plates are orentated by the external pressure. It s the phenomenon of transformaton plastcty of Magee. Tong WU 5

38 Thess: Experment and Numercal Smulaton of Weldng Induced amage In the case of a monocrystal of parent phase transformed durng coolng under constant and unform external pressure, the deformaton assocated wth the transformaton has a lmted value and does not depend on the appled stress [95]. Ths deformaton s reversble when the transformaton s reversed. However t s subject to the nfluence of the crystallne orentaton. For the polycrystal, the contrbuton of ths mechansm depends on the dstrbuton of orentaton of the martenste plates. In general, the measured deformaton n the drecton of the stress s weaker than that obtaned n the monocrystal. It s a functon of the orentaton of the products. There s no more lnearty between ths deformaton and the development of the transformaton, and t depends on the condtons of loadng [77][95][68][69][5]. The relatve mportance of these two mechansms depends on the materal and the transformaton under consderaton. Strctly speakng, both mechansms are generally present n dffusonal and dffusonless transformatons. However, general speakng, the preponderance of the mechansm of Magee s more than that of the mechansm of Greenwood & Johnson when the transformaton s martenstc, dependng on the dfference n specfc volume between the parent phases and daughter phases. The dlaton of martenstc transformaton s weak, and should not generate any transformaton plastcty of the type of Greenwood & Johnson. It can be observed n thermoelastc alloys. For lowalloy ferrtc steels, the dlaton of transformaton s drven and the mechansm of Greenwood & Johnson s domnatng n stead of that of Magee..6. TRIP models Models based on Greenwood & Johnson s mechansm Practcally all the models of transformaton plastcty proposed and based on the mechansm of G&J are put n the form of a product of three functons: wth V V ( V y pt ; ) f ( z; z) f 3 ( S χ ) = (- 7) V pt ε f γ ; : volumnal varaton assocated wth the transformaton y γ : yeld stress of the mother phase z : proporton of daughter phase transformed S : devatorc tensor of the appled stress pt χ : tensor of the nternal stresses assocated wth plastcty wth transformaton The functon f ntroduces the dependence of the rate of transformaton plastcty wth respect to the yeld stress of the mother phase and the change of volume nduced by transformaton. The second functon f expresses the dependence accordng to the rate of development of transformaton. Fnally the last functon f 3 ntroduces a relaton between the appled pressure, the nternal stress and the rate of transformaton plastcty. 6

39 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton In the absence of nternal stress and f the load appled durng the transformaton s constant and weaker than the elastc lmt of the mother phase, the precedng relaton s ntegrable and can then be put n the followng form: ε pt ( V γ y ) g ( z) 3( ) = g ; g (- 8) V After a short recall of the hstorcally undmensonal models, n partcular developed by G&J, Abrassart and esalos, we wll present the prncpal multaxal models. - Greenwood & Johnson [85] Its expresson s gven as followng: 5 V g = ; g γ = ; g 3 = (- 9) 6 V y Ths unaxal model can only envsage the fnal value of the deformaton of transformaton plastcty for a constant load appled (weak) durng phase changes. - Abrassart [4] 3 V 3 g = ; g γ = 3z z ; g = 4 V 3 (- ) y The ntroducton of a functon g, dependng on the proporton of daughter phase, authorzes the smulaton of the deformaton of transformaton plastcty durng phase transformaton. - esalos [54] g = k ; g = z ( z) ; g 3 = (- ) Ths relaton uses a constant coeffcent k. The functon g s dfferent from that of the precedng relaton. The expressons above developed to descrbe the evoluton of the deformaton of transformaton plastcty present mportant gaps. These models cannot deal wth the multaxal problems and ther applcaton s not possble for low levels of stresses. Thus, we ntroduce the followng multaxal models. - Leblond Leblond establshed a multaxal and ncremental formulaton of the deformaton of transformaton plastcty. The modelng retaned for ths phenomenon s founded on theoretcal and numercal studes of the mechansm of Greenwood and Johnson, where the mechansm of Magee s neglected. The mathematcal model smulates the phenomenon of transformaton plastcty n the Tong WU 7

40 Thess: Experment and Numercal Smulaton of Weldng Induced amage case of deally plastc phases [6][8]. The extenson of ths model to the case of materals to sotropc or knematc work hardenng s presented n reference [9]. Leblond supposes that the daughter phase s a sphercal ncluson whch grows nsde an austentc sphere. It consders the behavour of the mother phase as plastc perfect whereas the daughter phase remans elastc. It obtans fnally the expresson below. The followng formalsm takes nto account the sotropc work hardenng of the phases. Rate of deformaton of transformaton plastcty s: E pt Tref eq 3 ε γα Σ = S h ln y eff y γ ( Eγ ) Σ ( zγ ) z γ γ [.3 ] z (- ) wth Σ eq 3 = S j Sj macroscopc equvalent stress (von Mses) eq p p E = E j E j rate of macroscopc equvalent plastc stran 3 y eff y eff [ f ( z) ] ( E ) + f ( z) ( E ) y Σ = γ γ α α homogenzed macroscopc ultmate stress f (z) : correctonal functon resultng from dgtal smulatons (see Table. 6 ) y : elastc lmt of the phase γ γ y : elastc lmt of the phasesα α E : elastcty modulus, consdered common to all the phases S : homogenzed macroscopc devatorc tensor eff E γ : macroscopc effectve nternal varable of sotropc hardenng of the mother phase eff E α : macroscopc effectve nternal varable of sotropc hardenng of the daughter phase T ε ref γα : dfference of compactness of phasesα compared to phase γ at Tref Σ h Σ eq y : term whch translates the non-lnearty of the transformaton plastcty accordng to the equvalent macroscopc stress appled. Leblond dstngushed the condton of the low stresses from that of the hgh stresses. The rate of deformaton of transformaton plastcty presented above does not ntervene f the homogenzed macroscopc equvalent stress s strctly lower than the ultmate stress of the mxture of multphase contanng austente. Ths ultmate stress s a functon of the elastc lmts of each metallurgcal component whch depends on the effectvely nternal varables of hardenng. A functon h s also ntroduced n the formulaton of the rate of deformaton of transformaton plastcty. 8

41 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton eq Σ eq h = Σ + Σ 3.5 f y y Σ Σ Σ Σ Σ eq y eq y.5 >.5 It ntroduced the nonlnearty of the rate of deformaton wth the level of the appled pressure. Ths functon results from numercally mcromechancal tests of transformaton plastcty [8][55]. Note: To crcumvent an awkward sngularty when z, Leblond consders that the rate of deformaton of transformaton plastcty s null as long as the fracton of daughter phase s less than 3%. Models based on Magee s mechansm Ths very complex mechansm s often neglected even for the martenstc transformaton, except where t s always domnatng. Fscher starts the prncple whch the transformaton plastcty results from the two mechansms. It uses a mcromechancally analytcal approach for ts modellng. It ntroduces a spatal arrangement of the martenste varants whch t descrbes by the Euler angles. A functon of dstrbuton G takes nto account the effect of orentaton of the martenste varants. By a rather complex mathematcal process, based on a random dstrbuton of the varants, t establshes the followng formulaton [69]: pt 5 V 3 ε = + γ S (- 3) * y V 4 wth * y γ α ( ) α y y y α ln y y = γ : Average elastc lmt of γ α phase mxture γ : Constant of shearng nduced by the deformaton of transformaton One can also quote another model of ths phenomenon, the mcromechancal models developed by Cherkaou & Berveller [34]..7 Multphase mechancs and models The smulaton of the problems nvolved n the heat treatments or weldng, lke presented n the ntroducton, should take nto account thermal, metallurgcal and mechancal phenomena, more or less coupled. Thus ths part lsts varous equatons to be solved as well as the type of model used. At last, we wll present two classes of multphase mechancal models n detal. Tong WU 9

42 Thess: Experment and Numercal Smulaton of Weldng Induced amage.7. Formulaton of the problem Thermal problem The thermal problem (see Fgure. 5) whch one seeks to solve follows the Furrer s law. In multphase materals, latent heats of transformaton are ntroduced nto the equaton of heat. In addton, structures can exchange heat by convecton or radaton. The problem thus conssts of determnng temperature feld T and the heat flow q, such as: - Boundary condtons T = Td on Ω (- 4) q n = qd on Ω (- 5) Convecton q = h T T ) (- 6) c ( ext 4 4 Radaton q = h ( T ) (- 7) r T ext - Heat equaton - Consttutve relaton ρ ct = dvq ρlz + ρr (- 8) q = k grad(t ) (- 9) Where q s the heat flux per unt; r s the heat flux per unt volume generated wthn body; ρ s the mass densty of materal; c presents the specfc heat capacty; k s the coeffcent of thermal conductvty. L represents the latent heat of phase transformaton. Fgure. 5 Thermal boundary condtons Fgure. 6 Mechancal boundary condtons 3

43 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton Metallurgcal problem The metallurgcal problem s strongly coupled wth the thermal problem, whch determnes the evoluton of the volumnal proportons of phases. The formulaton takes the models of the type Waeckael or Martnez, whch was ntroduced n Chapter.3.4. z = f ( T, T, z, d, c) and z = (- 3) n Mechancal problem The mechancal problem (see Fgure. 6) s more complex than thermal problem and our efforts wll focus on ths problem of multphase. - Boundary condtons U = Ud on 3 Ω (- 3) F = n on 4 Ω (- 3) - Equlbrum dv ( M ) = M Ω (- 33) - Consttutve relaton It s to be determned..7. Mechancal models of multphase There are many models whch we wll not present here. The followng theses and books [56][69][53][35] propose a broadly bblographcal revew of these models. We wll ntroduce some partcular models here. Phlosophy adopted to deal wth the mechancal problem s n order to obtan an explct relaton of the mxture behavour, based on the behavour of each phase. Thus, two methods are exploted to obtan ts macroscopc behavour of materal. The frst method uses the thermodynamcs of the rreversble processes [8]. The second method ams to determne macroscopc laws through mcro-macro method. Models based on mxture of energy - Inoue & Wang [98][99] The model of Inoue s one of the smplest mechancal models of multphase. Ths model s elaborated based on lnear laws of mxture of nternal energes and the potentals of dsspaton [4]. Inoue does not separate stran of transformaton plastcty from the total tran. The model s developed n plastcty or n vscoplastcty and the mechancal propertes depend on the proportons of phases. Tong WU 3

44 Thess: Experment and Numercal Smulaton of Weldng Induced amage The partton of the strans s as follows: E e = E E (- 34) The recoverable and rreversble strans are wrtten: wth E e n + υ n n n υ t z z Tr II z dt z z + II t + o E = E ( ) α β ( ) = = = = (- 35) K( T,, z ) E κ = ( S X ) (- 36) µ J ( S X ) β : rate of varaton of volume due to transformaton µ : vscosty of homogeneous materal K : varable of hardenng κ, X : sotropc and knematc hardenng - INPL The INPL (Insttut Polytechnque de Lorane) proposed many models. These models cover a broad feld snce they supposed plastc [5][88][7] or vscoplastc [7][8][43][78][6] behavours, whch can be coupled wth the metallurgy [5] and the concentraton of carbon [5][78] or gran sze of austente [6]. The behavour of materal s descrbed by aggregate varables and the parameters of the behavour law are calculated by lnear mxtures of the characterstcs of the phases [7][43][5]. Another type of model was developed where one consders a law of evoluton of the plastc flow partcular for each phase. It s supposed, n these models, that the plastc stran s the same for all the phases [6][78][5]. We present here the prncpal equatons of the vscoplastc model [7]. The partton of the strans can be wrtten as: E + where metallurgcal dlaton s gven by: e vp th met pt = E + E + E + E E (- 37) n met tr E = ε z II (- 38) o = the transformaton plastcty by: E pt = f ( z ) z S (- 39) the vscoplastc stran by: γ γ 3

45 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton n J ( S X ) S X E vp κ = (- 4) K J ( S X ) - Hamata [89][9] As what we presented prevously, Hamata proposes a bphasc model for cast ron GS. Here also, the vscoplastc flow of each phase s dctated by ts own threshold functon. Moreover, Hamata dentfes a vscotransformaton plastcty: E pt 3 eq = λ n S eq z (- 4) - Vdeaul [95][96] In ts study, Vdeau s nterested n the nteracton between classcal plastcty and transformaton plastcty. Its model consders a classcal law of vscoplastc flow and a law of vscoplastc flow related to the transformaton plastcty. Lastly, the varous parameters knematcs are bounded by a matrx of nteracton. The macroscopc behavour s generally obtaned through a lnear mxture on the characterstcs of the phases. Then, ther phenomenologcal approach s constraned to an expermental dentfcaton for all the proportons of phases and all the temperatures. Models based on mcro-macro method - Leblond [6][7][8][9] Leblond undertakes a theoretcal study of the problem, from whch t brngs new results, whether the behavour of the phases s ether perfect elastoplastc or knematc elastoplastc. The object of ts study s A58 steel. By supposng that all the phases have the same characterstcs thermoelastc, Leblond shows that the macroscopc rates of deformatons can be put n the followng form: The rates of elastc stran E E E + E e thm p = + (- 4) e E and thermal thm E follow usually lnear mxture law. On the other hand, the rate of plastc stran s the sum of three terms proportonal to S, T and z. The transformaton plastcty appears here naturally. In Leblond model, the classcal plastc stran rate two terms: the frst one, cp E constant temperature; the second one, plastcty at constant appled stress. cp E s a sum of, proportonal to the stress rate, represents classcal plastcty at cp E T, proportonal to the temperature, represents classcal If eq < y E + p cp cp tp = E + ET E (- 43) Tong WU 33

46 Thess: Experment and Numercal Smulaton of Weldng Induced amage E cp 3( zγ ) g( zγ ) = S z E y γ γ eq (- 44) E 3( α cp γ αα ) T = z ln( z ) ST y γ γ (- 45) γ and the formula of plastcty of phase transformaton E Equaton -7). tp was gven n Chapter.6. (see If eq = y where Λ E p = ΛS (- 46) s unspecfed. However, certan results are dffcult to obtan analytcally wthout many assumptons. Ths s why, calculaton by fnte elements (mplemented n Sysweld) s necessary to know the yeld stress of the mxture. Leblond proposes then the functon f to computer the law of mxture (see Table. 6). z f(z) g(z) Table. 6 Values of functons f(z) and g(z) n Leblond model [7] Leblond contnues ts study by ntroducng hardenng on the level of the behavour of each phase. The case of knematc hardenng s treated as that of sotropc hardenng. Hardenng has an nfluence on the transformaton plastcty because t modfes the yeld stress of the phases (Equaton -7). Ths model can also gve an account of an effect of memory of hardenng between mother phases and daughter phases. Leblond model ncludng sotropc hardenng: If eq < y E cp 3( z = ( E y γ γ eff γ ) g( zγ ) S ) z E γ eq (- 47) E 3( α cp γ αα ) T = z ln( z ) ST y eff γ γ (- 48) ( E ) γ γ Tref eq g z E ε eff α γ ( γ ) eq z h z ( αγ αα ) ln( ) ln( z ) T γ = γ y γ + + γ z γ E zγ (- 49) 34

47 Chapter : Martenstc Stanless Steel and Sold-state Phase Transformaton z z eff eff eff E γ γ α = Eα + θ Eγ (- 5) z z γ γ If eq = y E p = 3 E eq eq S (- 5) E = eff eq γ E (- 5) z zγ eff = θ Eγ (- 53) z eff eq eff E γ α Eα + zγ E γ - Coret & Combescure [45][46] The model of Coret & Combescure s much more numercally orented, and gnores an a pror consttutve law for each phase. Ths model s developed on the bass of four assumptons: The rate of total macroscopc stran conssts of the rate of mcroscopc stran of the phases and the transformaton plastcty. They uncouple classcal plastcty from the transformaton plastcty. That means neglectng the nteractons between these two parts. The local and macroscopc mechancal magntudes are connected va the assumpton of Taylor. In other words the rate of homogenzed macroscopc stran wthdrawn from the rate of stran of transformaton plastcty s equal to the mcroscopc rate of stran of each phase. The homogenzed macroscopc stress s obtaned by a lnear law of mxture balanced by the volume fracton of the phases. That amounts neglectng nonlnearty (accordng to the proporton of mother and daughter phase) of the mechancal behavour of a mxture of phases. The stran rate equatons are E E + E tot cp pt = (- 54) cp = ε (- 55) E The stress equaton s ε = ε + ε + ε (- 56) e = n z = thm vp (- 57) - Others Tong WU 35

48 Thess: Experment and Numercal Smulaton of Weldng Induced amage. Taleb and Sdoroff [8][53] The authors extend the study of Leblond and modfy some assumptons. They do not neglect the elastc stran of austente and do not consder t only plastc, whatever the proporton of formed phase. They arrve fnally at an expresson very close to that of Leblond.. Fscher [69][7] Fscher adopts a mcromechancal analyss ncludng not only the mechansm of Greenwood and Johnson but also the mechansm of Magee. Its very thorough work leads to the followng expresson: 5 V 3 = S * + γ 4 V 4 (- 58) γ E pt * where s calculated from the yeld stresses of the elastc phase and the ferrtc phases. γ γ represents the devatorc part of the transformaton. 3. an [56][58] Ths study s to treat the transformaton plastcty n the case of fast loadngs. an undertakes a theoretcal study based on work of Strngfellow [75] concernng memory-shape alloys. Its man contrbuton relates to the modellng of the couplng between appled pressure and evoluton of the martenstc transformaton..8 Summary We ntroduce the materal studed, 5-5PH, a martenstc precptaton hardenng stanless steel, whch undergoes phase transformaton durng heatng and coolng. Our numercal model to be developed s based on the study of ths martenstc steel. Hence, there are detaled ntroductons of the materal structure and the behavours of phase transformaton n ths chapter. It s shown that the comprehenson and the mechancal modellng of the problems wth phase change are partcularly complex. The phase transformaton must be mperatvely taken nto account for the correct smulaton of the mechancal problem consdered. Therefore, we present the concepts and mechansms of phase transformaton and transformaton nduced plastcty. And then the varous models of phase transformaton and transformaton nduced plastcty are lsted and explaned to wden and further the understandng of the achevement and development of ths feld. Several partcular models wll be used and studed n our succeedng work of numercal smulaton. In addton, we also presented varous mechancal models of multphase bult ether on an energy law of mxture, or on a mcrophone-macro approach. The common pont of these two approaches s to explctly seek the law of behavour of multphase materal. 36

49 Chapter 3: amage Mechancs and Weldng amage 3 Chapter 3 amage Mechancs and Weldng amage 3. Introducton amage and crackng deterorate the coheson because of volumnal or superfcal dscontnutes nduced by weldng n materals. These damage phenomena are assocated wth the decrease n the materal propertes due to the nucleaton and growth of mcrovods and mcrocracks. From the physcal pont of vew, the damage or deteroraton of materals s an rreversble and progressve physcal process. The entropy s ncreasng durng such damagng process. ependng on the nature of the materal, the type of loadng, and the temperature, the damage appears n varous ways ncludng brttle damage, ductle damage, creep damage, low cycle fatgue damage and hgh cycle fatgue. The cause of these materal mperfectons could be due to several processes: The creep damage due to the vscoplastc stran, whch leads to ntergranular decohesons. The hgh cycle fatgue damage where the local mcroplastcty occurs. The fragle cleavage damage. The ductle damage for whch large plastc strans nduce dslocaton stackng and mcro cavtes. In terms of dfferent levels of sze studed, damages mentoned above are classfed to three categores: mcroscale, mesoscale, and macroscale. At the mcroscale, damage s caused by the accumulated deteroraton due to mcrostresses n the neghborhood of defects, nterfaces and by breakng of bonds. When the sze of these Tong WU 37

50 Thess: Experment and Numercal Smulaton of Weldng Induced amage cracks reaches dmensons of the order of the mcron, they could be regarded as mcroscopc cracks. At the mesoscale,.e. the level of the so-called representatve volume element (RVE), damage s assocated wth the growth of mcrocracks or mcrovods that ntate the fnal macroscopc crack. The sze of the RVE for metals s. mm 3. At the macroscale, t s the growth of ths macroscopc crack. In terms of mcroscale and mesoscale, we can study them by means of damage varables and mechancs of contnuous meda defned at the mesoscale level. The thrd stage s usually studed usng fracture mechancs wth varables defned at the macroscale level. The mcro or meso mechancal approach s much more physcally orentated, and t nvestgates mcroscopc behavours and elementary mechansms (gran, system of slp and dslocaton) of metallc materals. The smaller the scale s, the more ansotropc unt studed s. Based on the analyss of nternal varables at mcroscale, the macromechancal behavour can be deduced by usng homogenzaton method. Although mcro approach seems to be more realstc, t s less lkely to be modelled successfully because mcroscopc materal parameters are dffcult to determne and the consderable number of equatons lead to extremely long computng tmes. Another modellng approach s phenomenologc. It s founded on the ntroducton of state varables assocated wth the varous phenomena, whch are revealed by the experments. These phenomena are descrbed wthn the framework of thermodynamcs of rreversble processes [8]. Some models have been developed by studyng cavtes or porous sold plastcty [6][87][63]. Ths knd of approach s based on a growth rate of the cavtes nsde a matrx wth the elastoplastc behavor [6]. Ths theory assumes the sotropy of damage and uses a scalar varable to descrbe the volumnal fracton of the cavtes. Others are based contnuum damage mechancs (CM) [5][58][9][4]. The theory assumes that damage s one of the nternal varables of consttutve equatons that govern the rreversble processes n the materal. In our study, the ductle damage s focused on metallc materals. The lterature, current stuaton and varous models of ductle damage wll be presented n ths chapter. amage behavour n sngle phase wll be emphaszed whle the behavour of ductle damage for multphase materals wll be nvolved n next chapter. 3. Phenomenologcal aspects The dffculty of defnng a mechancal damage varable les n the fact that there s lttle dfference between a damaged volume element and a vrgn one n macroscopc scale. Therefore, t s necessary to use nternal varables whch are representatve of the deterorated state of the matter. Followng are some types of damage measurement envsaged [4]: 38

51 Chapter 3: amage Mechancs and Weldng amage Measurements at the scale of mcrostructure (densty of mcrocracks or cavtes) that lead to mcroscopc models. These can be ntegrated nto macroscopc behavour va homogenzaton technque. Global physcal measurements (densty, resstvty etc.) that can be converted nto mechancal propertes of materal by usng proper model. Global mechancal measurements (modfcaton of elastc or plastc propertes) that are easer to nterpret n terms of damage varable va usng the concept of effectve stress ntroduced by Rabotnov [58]. 3.. amage varable Representatve volume element Contnuum mechancs deals wth quanttes defned for the mathematcal pont. From a physcal pont of vew ths means some knd of homogenzaton n a certan volume. Ths Representatve Volume Element (RVE) must be small enough to avod smoothng of hgh gradents; meanwhle they have to be large enough to represent an average of the mcroprocesses. Certanly the sze of the RVE depends on the chosen materal, e.g.. mm 3 for metals and ceramcs, mm 3 for polymers and compostes, and mm 3 for wood [4]. efnton of damage varable The damage may be nterpreted at the mcroscale as the creaton of mcrosurfaces of dscontnutes: breakng of atomc bonds and plastc enlargement of mcrocavtes. At the mesoscale, the number of broken bonds or the pattern of mcrocavtes may be approxmated n any plane by the area of the ntersectons of all the flaws wth that plane. In order to manpulate a dmensonless quantty, ths area s scaled by the representatve volume element. Ths sze s very mportant n the defnton of a varable contnuous n terms of contnuum mechancs. The damage varable, as ntroduced by Kachanov [5], was defned on the bass of the rreversble processes leadng to nucleaton and growth of mcrovods and mcrocracks n the entre volume of the sample. All types of vods and cracks (nter- and transgranular) that deterorate the ntegrty of the materal are accounted for. Consder a damaged part wth a RVE n one pont orentated by a normal n. δs s the area of ntersecton of the consdered plane wth the representatve volume element (RVE). δs s the effectve area of ntersectons of all mcrocracks and mcrocavtes n δs ( see Fgure 3. ). The damage value (n) attached to ths pont s defned as: δ S ( n) = (3- ) δ S Tong WU 39

52 Thess: Experment and Numercal Smulaton of Weldng Induced amage In the general case of ansotropc damage consstng of cracks and cavtes wth preferred orentatons, the scalar varable depends on the orentaton of the normal. If mcrocracks and cavtes are unformly dstrbuted, the damage varable wll not depend on the orentaton of n. For homogeneous damage, damage varable could be smplfed, and onedmensonal varable s defned as: S = (3- ) S Ths s the case of sotropc damage. We wll restrct our study to the case of sotropc damage for almost cases. From the defnton of damage varable, the value of the scalar varable followng: s lmted as (3-3) When =, t means that the materal n the RVE s undamaged, and we may call such materal as vrgn materal. = means that the RVE materal s broken nto two parts. Fgure 3. efnton of the damage varable 3.. Effectve stress The ntroduced damage varable leads drectly to the concept of effectve stress. In the case of unaxal tenson, f all the defects are open, the cross secton of sotropc damaged materal s no longer S but S S. Thus the effectve stress becomes: ~ F = (3-4) S S It wll be therefore wrtten n terms of the effectve stress as: ~ F = = (3-5) S S( ) S 4

53 Chapter 3: amage Mechancs and Weldng amage Evdently ~, ~ = for a vrgn materal and ~ at the moment of facture Stran equvalence prncple We ntend to avod the mcromechancal analyss for each type of defect and damage mechansm. Therefore, a themodynamcal descrpton s used n a mesoscale postulatng the followng prncple. Lematre proposed the prncple of stran equvalence [4]: Any deformaton behavour, whether unaxal or multaxal, of a damaged materal s represented by the consttutve laws of the vrgn materal n whch the usual stress s replaced by the effectve stress. The unaxal lnear elastc law of a damaged materal s wrtten as: ~ ε e = = (3-6) E ( ) E where E s Young s modulus amage measurement amage s not easy to measure drectly. Its quanttatve evaluaton, lke any physcal value, s lnked to the defnton of the varable chosen to represent the phenomenon. We can measure damage by measurng the modfcaton of the mechancal propertes of elastcty, plastcty and vscoplastcty. The method of varaton of the modulus of elastcty, whch s a non-drect measurement, s preferred because of ts smplcty (see Fgure 3. ). Recall effectve stress n the case of unaxal tenson, whch we mentoned n the prevous secton as: ~ = = Eε e (3-7) ~ E( ) = E could be nterpreted as the elastc modulus of the damaged materal. amage varable could be wrtten as: = E ~ / E (3-8) where E s Young s modulus of undamaged materal. E ~ s Young s modulus of damaged materal. Tong WU 4

54 Thess: Experment and Numercal Smulaton of Weldng Induced amage Stress u E = E > E > E 3 y 3 4 E E E E 3 ε ε R Stran Note: Zone : elastc stage; Zone : elastoplastc stage; Zone 3: elastoplastc stage n whch damage occurs; Zone 4: ntaton and propagaton of cracks leadng to fnal rupture. Fgure 3. Schematc dagram of stress vs. stran and varablaton of Yound s modulus Although the prncple s very smple, ths measurement s rather trcky to perform for the followng reasons: The measurement of modulus of elastcty requres precse measurement of very small stran. amage s usually localzed, thus requres a very small base n the order of.5-5mm. The best straght lne n the stran-stress graph representng elastc loadng or unloadng s dffcult to defne. At hgh temperature, metal becomes weak and stran-stress curve ndcates strongly non-lnear. Besdes the very useful method of varaton of the Young s modulus, thanks to the development of technology, there are some other approaches for the choce to measure damage [5]: Ultrasonc waves propagaton: a technque based on the varaton of the elastcty modulus conssts n measurng the speed of ultrasonc waves. Varaton of the mcrohardness: based on the nfluence of damage on the plastcty-yeld crteron through the knetc couplng. Varaton of densty: based on the measurement of decrease of densty wth apparatuses of Archmedean prncple. Varaton of electrcal resstance: the effectve ntensty of the electrcal current can be defned n the same way as the defnton of effectve stress. Acoustc emsson: a good method to detect the locaton of the damage zone, but the results reman qualtatve as far as the values of varable are concerned. 4

55 Chapter 3: amage Mechancs and Weldng amage 3.3 Thermodynamcs of sotropc damage In the framework of thermdynamcs, two potentals are ntroduced n the rreversble processes: the state potental and the potental of dsspaton: The state potental defnes the state laws, and t can be descrbed as the functon of the state varable. The potental of dsspaton s descrbed by the assocated varable whch accounts for the knetc laws. In the damage consttutve equaton, the damage rate s a functon of ts assocated varable. In ths secton, only damageable materals exhbtng elastc, elastoplastc or elastovscoplastc behavours are consdered. The damage varable s the scalar whch wll be consdered as a state varable amenable to the thermodynamc representaton State potental The state potental s ntroduced through the method of local state n the thermodynamcs of rreversble process. In our study, thermal factor s taken nto account, whereas t s restrcted n sothermal process n Lematre s consttutve equatons. The Helmholtz free energy s adopted and defned by the followng equaton, whch s a convex functon of the state varables. p Ψ = Ψ( T, ε e, ε, V, ) (3-9) k where Ψ Ak = ρ (3- ) V k Ψ V k A k ρ : Helmholtz free energy : state varable : assocated varable : densty In Table 3., state varables and ther dual varables (assocated varables) are lsted for the behavour of hardenng and damage. The state varables can be dvded nto observable and nternal varables. The temperature and stran tensor are observable, whle others can not be measured drectly and called nternal varables. Tong WU 43

56 Thess: Experment and Numercal Smulaton of Weldng Induced amage State Varables Observable Internal Assocated Varables Thermal Temperature T s Specfc entropy Elastcty eformaton ε Cauchy stress tensor Plastcty Plastc stran ε p - Cauchy stress tensor Isotropc hardenng Accumulated plastcty r, p R Isotropc stran hardenng Knematc hardenng Back stran tensor α X Back stress tensor egeneraton amage varable -Y amage energy release rate Table 3. Varables of thermodynamcs To nterpret the Equ. 3-, t can be wrtten wth the ntroducton of each state varable: Ψ s = (3- ) T Ψ = ρ (3- ) e ε Ψ R = ρ (3-3) r Ψ X = ρ (3-4) α Y Ψ = ρ (3-5) When the states are coupled between mechansms I and mechansms J, and state varable V s couplng wth another varable V J. Snce A J Ψ = ρ, the coupled states are: V J I A V J I = Ψ ρ (3-6) V V I J When there s no couplng among dfferent states, t becomes: A V J I = Ψ ρ = (3-7) V V I J For example, there s no nteracton between elastc behavor and plastc behavor: ε e Ψ r = (3-8) Ψ = (3-9) ε e α 44

57 Chapter 3: amage Mechancs and Weldng amage Whle the damage varable s coupled wth elastc varable or varable of phase transformaton, t means: ε e Ψ (3- ) Table 3. shows the couplngs and uncouplng between dfferent varables. ε p α r T ε p - α - r - T - - Note: means uncouplng; presents couplng Table 3. Couplng between state varables The free energy for each phase s dvded nto two parts when the elastc behavor s uncoupld wth the plastc behavor: where e ρψ = ρψ ( T, ε, ) + ρψ ( T, r, α, ) (3- ) e p Ψ e : thermoelastc free energy Ψ p : blocked energy by the mechansm of sotropc and knematc hardenng When the sotropc hardenng and knematc hardenng are decoupled: Ψ = Ψ ( r,, T ) + Ψ ( α,, T ) (3- ) p p pk Therefore, the free energy s derved as: e ρψ = ρψ ( T, ε, ) + ρψ ( r,, T ) + ρψ ( α,, T ) (3-3) e p pk The thermoelastc free energy s: e e e e ρ Ψ e ( T, ε, ) = ( ) ε : A( T ) : ε + B( T ) : ε + C ( T )( T T ) (3-4) where A (T ) : elastc tensor depended on Young s Module E (T ) and Posson coeffcent v(t ) Tong WU 45

58 Thess: Experment and Numercal Smulaton of Weldng Induced amage B (T ) : tensor related to the thermal expanson consdered as sotropc α (T ) : thermal dlataton coeffcent C (T ) : scalar dependent on the purely thermal behavor T : reference temperature The blocked energy by sotropc hardenng can be expressed by: ρ Ψ p = c( T )( ) p + exp( γp) (3-5) γ The blocked energy by knematc hardenng s: ρ Ψ pk = g( T )( ) α : α (3-6) 3 where γ s hardenng constant. c (T ) and g(t ) are hardenng functons depended on materal and temperature. Therefore, the equatons lnk between state varables and assocated varables are: Ψ e = ρ = ( ) A( T ) : ε + B( T ) e ε (3-7) Ψ R = ρ = c( T )( ) [ exp( γp) ] p (3-8) X Ψ = ρ = g( T )( ) α (3-9) α 3 Ψ Y = ρ = Y e + Y p (3-3) Y e e = A( T ) ε (3-3) Y p = g( T ) α : α + c( T ) p (3-3) sspaton potental The second prncple of thermodynamcs or the Clausus-uhem nequalty s wrtten as: where q grad T : ε ρ ( Ψ + st ) q (3-33) T s the heat flux vector assocated wth the temperature gradent. 46

59 Chapter 3: amage Mechancs and Weldng amage The rate of the Helmholtz free energy s obtaned by the dfferentaton wth respect to the state varables: Ψ e Ψ Ψ Ψ Ψ = ε + r + α + e ε r α (3-34) If the dsspaton of thermodynamcs s uncoupled wth the dsspaton of mechancs, we get: Φ = Φ m + Φth (3-35) Φ m p = : ε A V + Y (3-36) k k Φ th = grad q / T (3-37) ue to the plastc flow n the absence of damage (vce versa), the two nequaltes are wrtten as: p p : ε A V = : ε X : α Rr (3-38) k k Y (3-39) Here, -Y s a postve quadratc functon, so that the damage rate must be a nonnegatve functon. The pseudo-potental of dsspaton ϕ s then ntroduced: then: e ϕ = ϕ(, A, gradt, Y; ε, V, T, ) (3-4) k k V ϕ k = and A k q T ϕ = gradt (3-4) The pseudo-potental of dsspaton s the sum of the thermal dsspaton and vscoplastc dsspaton, whch s coupled wth damage: ϕ = ϕ (, X, R,, T ) + ϕ ( gradt, T ) (3-4) vp d th gradt gradt ϕ th( gradt, T ) = k( T ) (3-43) T T If the materal propertes of dffuson are sotropc, the Fourer law drectly wrtes: q = k( T ) gradt (3-44) The damage dsspaton ϕ (, Y ) could be chosen to be an exponental functon [4]: d Tong WU 47

60 Thess: Experment and Numercal Smulaton of Weldng Induced amage Y S ( s + )( ) Y S s+ ϕ d (, ) = ( ) (3-45) where s and S are materal coeffcents characterzng the ductle damage evoluton: they are functons of temperature T. In order to nterpret the vscoplastc-damage dsspaton, a yeld functon (plastcty convex) f and a plastc potental (non assocatve theory) F can be ntroduced [68], for nstance, as followng: f J ( X ) R = y( T ) (3-46) a( T ) b( T ) S( T ) Y s( T ) + F = f + J ( X ) + R + [ ] (3-47) ( ) ( ) [ s( T ) + ]( ) S( T ) where a (T ) and b(t ) are materal coeffcents dependent on the temperature to descrbe the nonlnear sotropc and knematc hardenng respectvely. The scalar J s defned as followng: J 3 : ( ) = (3-48) J ( 3 X ) = ( X ) : ( X ) (3-49) = Tr( ) I (3-5) 3 The accumulated plastc stran p s defned by ts rate: p : 3 p p = ε ε (3-5) Therefore, the vscoplastc-damage dsspaton s wrtten as: µ ( T) n( T ) + ϕ vp d = ϕvp d (, X, R, Y,, T) = < F > (3-5) + n( T) where µ (T ) and n (T ) are coeffcents of vscosty. < > presents the Heavsde step functon, also called unt step functon, whch s a dscontnuous functon whose value s zero for negatve argument and one for postve argument. For specfc materals (the generalzed normalty), we can ntroduce the Lagrange multpler λ leads to the flux of state varables: whch p f 3 λ X ε = λ = (3-53) J ( X ) 48

61 Chapter 3: amage Mechancs and Weldng amage λα λ λ α a X J X X F = = ) ( 3 (3-54) ) ( br R F r = = λ λ (3-55) s S Y Y F ) ( = = λ λ (3-56) For vscoplastcty, the Lagrange multpler s [68]: n vp k f = = λ λ (3-57) where and n characterze the materal vscosty. k Traxalty and damage equvalent stress To formulate an sotropc crteron, t s postulated that the mechansm of damage s governed by the total elastc stran energy, whch conssts of dstorton energy and volumetrc energy [4]: e H e e ω ω ω + = (3-58) + = = e H e e H H e e e d d d ε ε ε ε ε ε ω : 3 : : (3-59) Hooke s law coupled wth damage contans the devatorc and the hydrostatc parts: E e + = ν ε, E H H = ν ε (3-6) Therefore, Equ changes to: + + = E E H e 3 : ν ν ω (3-6) the von Mses equvalent stress s defned as: : 3 = eq (3-6) Smlar to the equvalent stress n plastcty, the damage equvalent stress s defned by that ths energy n a multaxal state s equal to that n an equvalent unaxal state [4]. + + = = ) 3( ) ( 3 ) ( eq H eq e E Y ν ν ω (3-63) * ) 3( ) ( = eq H eq ν ν (3-64) Tong WU 49

62 Thess: Experment and Numercal Smulaton of Weldng Induced amage where H eq : traxalty rato, whch plays an mportant role n damage evoluton. The term n square brackets s called traxalty functon H f ( ) : eq H f ( eq ) ( ) 3( ) H = + ν + ν 3 eq (3-65) When a traxalty rato H eq = 3, t leads to H f ( ) =. eq From the above formulatons, the damage equvalent stress dffers from the von Mses equvalent stress by the traxalty functon. Plastcty s connected to slps, whch are caused by shear stress. There s no dependency on hydrostatc observed. amage s connected wth debondng whch s nfluenced by hydrostatc or traxalty rato. In our followng experment of notched specmen, the radus of notch controls the dstrbutons of traxalty rato, thus dentfes damage parameters by usng varous raxalty ratos Threshold and crtcal damage From a mcroscopc pont of vew ductle plastc damage conssts of the nucleaton, growth and coalescence of cavtes nduced by larger plastc strans. Secton 3..4 presented that the varable could be measured by varatons of elastcty modulus. From the experment of many metallc materals subjected to a unaxal monotoncally ncreasng load, t s notced that no damage takes place when stran s smaller than some specfc value; when the accumulated stran reaches a large enough value, rupture ntates. Such phenomena can be summarzed n a schematc dagram of damage evoluton (see Fgure 3. 3), and three man damage parameters are threshold damage stranε, falure stranε R, and crtcal damage c. Lematre model Bonora model c p p R Plastc stran Fgure 3. 3 Schematc dagram of damage evoluton and damage parameters 5

63 Chapter 3: amage Mechancs and Weldng amage Startng from the dsspaton potental, we can deduce the threshold and crtcal damage of a multaxal model from expermental results of unaxal model n the framework of thermodynamcs. The Equ.3-64 can be wrtten as: K H Y = f ( ) (3-66) E eq wth eq = K = const. Wth the hypotheses of sotropc damage and sotropc hardenng, n Lematre-Chaboche model, damage rato s wrtten as: F Y = λ = ( ) s p (3-67) Y S Insertng Equ.3-66 to the above equaton, we obtan: In ntegral form, t s wrtten as: s K H s = f ( ) p (3-68) ES eq s K H s = f ( ) ES eq p p (3-69) The smplfcaton of fracture condton s: p = and =. c pr Therefore, the Equ.3-69 s wrtten as: p s K H s R p = f ( ) ES eq c (3-7) In unaxal condton, = 3 and t s assumed that no dstncton exsts between total stran ε and plastc stran p and H eq pr p = ε ε. R s K pr = ε R = 3 ES c ε ε R (3-7) In multaxal case, we get: p R H = eq H f ( ) eq s ε R s K = ES H f ( ) eq s c ε ε R (3-7) Tong WU 5

64 Thess: Experment and Numercal Smulaton of Weldng Induced amage 3.4 uctle damage models 3.4. Introducton The damage s called ductle when t occurs smultaneously wth plastc strans. It results from the nucleaton of cavtes due to decohesons between nclusons and the matrx followed by ther growth and ther coalescence through the phenomenon of plastc nstablty. In plastc deformatons nduced by weldng process, the prncpal mechansm of falure s due to the ductle damage. Consderable studes have been carred out n the topc of ductle damage, and many dfferent models have been proposed. All these formulatons n the lterature could be categorzed n three man categores: The frst approach s the abrupt falure crtera. Falure s predcted to occur when one external varable, whch s uncoupled from other nternal varables, reaches ts crtcal value. McClntock [38] frstly recognzed the role of mcrovods n ductle falure process and tred to correlate the mean radus of the nucleated cavtes to the overall plastc stran ncrement. Rce and Tracey [6] analytcally studed the evoluton of sphercal vods n an elastcperfectly plastc matrx. In these poneerng studes, the nteracton between mcrovods, the coalescence process and hardenng effects were neglected and falure was postulated to occur when the cavty radus reachs a crtcal value specfc for the materal. Second, porous sold plastcty. The effect of ductle damage, as proposed by Gurson [87] and Rousseler [63], was taken nto account n the yeld condton by a porosty term that progressvely shrnks the yeld surface. Later, Needleman and Tvergaard [4] as well as Koplk and Needleman [] extended the ntal formulaton proposed by Gurson n order to nclude the acceleraton n the falure process nduced by vod coalescence (GTN model). More recently, a number of fnte element unt cell based mcromechancal studes were performed n order to correlate vods evoluton and nteracton wth the resultng macroscale materal yeld functon. Tvergaard and Nordoson [9] nvestgated the role of smaller sze vods n a ductle damage materal by usng nonlocal plastcty model as proposed by Acharya and Bassan [5]. Schacht et al. [7], used the 3 voded unt cell based approach to nvestgate the role and the effects assocated wth the crystallographc orentaton of the matrx materal, and they found a substantal dependency of the growth and coalescence phase wth the ansotropy of the materal surroundng the vods. The thrd s contnuum damage mechancs (CM). In ths approach, damage s assumed to be one of the nternal varables that accounts for the effects on the materal consttutve response nduced by the rreversble processes that occur n the materal mcrostructure. As the startng pont of contnuum damage mechancs (CM), Kachanov was the frst to ntroduce a contnuous damage varable n 958 [3]. Then, the concept of effectve stress was 5

65 Chapter 3: amage Mechancs and Weldng amage ntroduced by Rabotnov n 968 [58]. The CM framework for ductle damage was later developed by Lematre and Chaboche [4][5]. More recently, startng from the consderaton that the gradent effect s mportant when the characterstc dmenson of the plastc stran or damage s of the same order of the materal ntrnsc length scale, a number of so-called non-local theores have been proposed by Fleck et al. [7][73] and Bammann et al. [3]. Relatve to the ntal framework proposed by Lematre, several damage models, based on the use of specal expressons for the damage dsspaton potental, have been derved by dfferent authors [3][8][8]. In addton, Bonora [8][9][] proposed a damage model formulaton n the framework of CM. Ths formulaton s materal ndependent, and t allows the descrpton of dfferent damage evoluton laws wth plastc stran wthout the need to change the choce of the damage dsspaton potental. Later, Prond and Bonora [54] reformulated ths model, and ponted out how the expermentally determned flow stress curve does not requre the substtuton of the effectve stress n ts expresson. In term of the frst approach, t s so smple that t s far away from the realty n many stuatons, thus leads to the lmted applcaton. In the followng secton, we wll ntroduce the later two branches of approaches, whch are developed based on the framework of rreversble thermodynamcs Models based on porous sold plastcty Gurson [87][88][39] Concerned wth the ductle falure of porous materals, Gurson s model [87] s wdely accepted. Assumng that vods are ntally sphercal and reman sphercal n the growth process, the approach to descrbe the mcromechancal effects of damage n ductle metals was frst proposed by Gurson and then t was extended to GTN model by Tvergaard and Needleman [88]. The model s nterpreted as an extenson of conventonal von Mses plastcty, and t taks nto account the expermental observaton that ductle degradaton processes consst of the nucleaton, the growth and the coalescence of mcrovods. The plastc flow wth cavtes proposed by Gurson rses from a plastc potental of form [87]: wth = Tr( ) 3 H = eq 3 H Φ = + f cosh ( + f ) (3-73) J( ) 3 3 eq = S : S = J ( ) Tong WU 53

66 Thess: Experment and Numercal Smulaton of Weldng Induced amage The above flow potental Φ characterzes the porosty n terms of a sngle scalar nternal varable, where s the vod volume fracton defned by: f f V V = V t ρ t = J = ρ J (3-74) J ρ = ρ Vt = V (3-75) where V : volume of orgnal matrx V : volume of matrx at tme t t ρ : densty of orgnal matrx ρ : densty of matrx at tme t It s notceable that when there s no cavty n matrx, V t = V f and f the matrx s totally ruptured, ρ and f. = * In GTN model, the damage varable f s replaced by and three addtonal parameters ( q, q q ) are ntroduced [88]. The yeld functon s wrtten as: f, 3 eq * 3 q H * Φ = q cosh ( + f + q3 f ) YM (3-76) YM f * = f q + f F f f f c c f f ( f fc ) f > fc c (3-77) where the parameter f characterzes the begnnng of vod nucleaton and f denotes the fnal falure. c The parameters, and q make the predctons of the Gurson model agree wth hs numercal q q 3 studes of materals contanng perodcally dstrbuted crcular cylndrcal and sphercal vods. F The ncreasng rate of the mcrovod volume fracton s gven by: f = f n + f g (3-78) where f and f are, respectvely, the nucleaton and the growth rate of mcrovods. n g Assumng that the matrx materal s plastcally ncompressble, the rate of ncreasng of the mcrovod volume fracton, due to the growth of exstng mcrovods, s gven by: f g p = ( f ) Tr( ε ) (3-79) 54

67 Chapter 3: amage Mechancs and Weldng amage The nucleaton of vods s a very complex physcal process dependng on the mcrostructure of the materal. Chu and Needleman [39] mproved the Gurson model by a statstcal approach: f n = s N p f N ε eq ε N exp π s N ε p eq (3-8) The hardenng of the matrx materal depends on the equvalent plastc stran power law ncludng three materal parameters: n p ε eq and t s gven by a p ε p p eq YM eq ( ε ) = + (3-8) ε The ntal yeld stress and ε and n are hardenng parameters. Usng the equvalence of mcroscopc and macroscopc plastc work, one obtans an evoluton equaton for the nternal hardenng varable: where λ s the plastc multpler. p ε = Φ λ ( f ) : eq (3-8) YM In the represented form, the GTN model contans materal parameters: P =, ε, n, q, q, q, f, f, f, f, ε, s ) (3-83) ( 3 c F N N N Gurson model was studed and then extended by many researchers. After Tvergaard [88] uses emprcal parameters to take nto account cavty growth and the cavtes coalescence, the Rchmond and Smelser [6] crteron ntroduces a parameter that consders shear bands, whch could domnate the plastfcaton process. The Sun and Wang [78] crteron s the only one to be based on an nner approach of the Gurson model under a macroscopc stress on the hollow sphere boundary. It s notced that many processed materals, such as rolled plates, have non-sphercal vods. Even for materals havng ntally sphercal vods, the vods wll change to prolate or oblate shape after deformaton, dependng on the state of the appled stress. Because of the lmted applcaton of Gurson model due to the assumpton of sphercal vods, Gologanu et al. [8][8][83] extended the Gurson model to the spherodal vod ncorporatng vod shape effects, namely the GL (Gologanu-Leblond-evaux) model. The GL model provdes an mportant mprovement to the wdely adopted GT model n descrbng vod growth and the correspondng materal behavor durng the ductle fracture process [49][76]. Brunet [4][5]extended GL model for prolate vods, plane-stress state and non-lnear knematc hardenng ncludng ntal quadratc ansotropy of the matrx or base materal. In order to predct the damage evoluton nduced by porosty ncreasng n bulk formng processes, Staub and Boyer [73] proposed a vod growth model, based upon the Rce and Tracey analyss and suted for thermo-elasto-plastc fnte-element modellng. Tong WU 55

68 Thess: Experment and Numercal Smulaton of Weldng Induced amage Rousseler [63][64][65] Although Rousseler model [63][64] was proposed n the form of a thermodynamcally based contnuum damage mechancs method, we classfy t nto the secton of porous sold damage (secton 3.4.), because ths ductle fracture model s related to vod nucleaton, growth, and coalescence n metals. The Rousseler crteron [65] s based on a macroscopc thermodynamc approach of rreversble phenomena under the local equlbrum assumpton,.e. the state of a system can be defned locally usng the same varables as at the equlbrum state. The parameter values of ths crteron must be determned expermentally. In the Rousseler model, several hypotheses are adopted: Hypothess : The nternal hardenng p and damage varable β are scalar, and the assocated varables wth p and β are P and B. Hypothess : In the free energy the varables are separated. The thermodynamc potental Φ s of the from: e Φ = Φ ( ε ) + Φ ( p) + Φ ( β ) e p β (3-84) Hypothess 3: The plastc potental F s of the form: F = F ~, P) + F ( ~, ) (3-85) ( eq m B wth F = ~ eq + P( ) (von Mses yeld crteron) (3-86) p F = B( β ) g( ~ ) (3-87) m ~ ~ m g ( ) = K exp( ) (3-88) m n whch K s the constant of ntegraton, and s a constant too. Based on the concepts of prncple of smplcty and standard model, fnally, the plastc potental s wrtten as: ~ Φ = ~ m ( ) + ( ) exp eq R p B β K (3-89) where R( p) s the hardenng curve of the materal; p s the hardenng varable (the cumulated plastc stran); β s the damage varable. Both p and β are scalar. The damage varable β s defned by: v β = (3-9) v where v s the average volume of the cavtes. 56

69 Chapter 3: amage Mechancs and Weldng amage The functon B(β) s defned by: f exp( β ) ( β ) = = f f + f exp( β ) (3-9) B where f s a constant ncludng the ntal volume fracton of cavtes, and f s the actual volume fracton of cavtes. In Rousseler model, t dd not ntroduce a crtcal value of damage varable because the stresses decrease abruptly and vansh, and zone of stran and damage localzaton were assmlated to a crack. Therefore, three parameters are ntroduced: f, related to the ntal volume of nclusons., related to the resstance of the metal matrx to the growth and coalescence of cavtes., does not depend on the materal and can be taken to be equal to, at least for the ntal volume fracton of f equal to or smaller to than -3. The Rousseler model can be extended to account for vod nucleaton effects and by ntroducng a vod volume fracton at whch falure occurs [65]. f F In addton, based on the study on vods and cracks of the porous materals, numerous models have been developed, such as Km and Kwon model [6]. More recently, Bgon and Pccolroaz [7] have proposed a macroscopc yeld functon to agree wth a varety of expermental data relatve to sol, concrete, rock, metallc and composte powders, metallc foams, porous materals and polymers. Ther yeld functon was presented as a generalzaton of several classcal crtera, and t also gves an approxmaton of the Gurson crteron. Trllat and Pastor [86][87] analyzed Gurson s model, and proposed a yeld crteron for porous meda wth sphercal vods to satsfy Gurson crteron condtons. Enakoutsa and Leblond [64] dd some mprovements n the numercal aspect and algorthm for the assessment of a phenomenologcal nonlocal varant of Gurson s model suggested by Leblond et al. [] Models based on contnuum damage mechancs Lematre & Chaboche [][3][4][5][9] There s no doubt that contnuum damage mechancs (CM) s connected to the names of Lematre and Chaboche. Secton 3.3 has gven a detaled descrpton of the thermodynamcs of sotropc damage founded by Lematre and Chaboche, from the development of potental to the defnton of threshold and crtcal damage n the framework of rreversble thermodynamcs. The formulaton of Lematre damage model can be found there. The multaxal ductle plastc damage stran model s as followng: Tong WU 57

70 Thess: Experment and Numercal Smulaton of Weldng Induced amage In dfferental form: p p p eq H R c + + = ) 3( ) ( 3 ν ν (3-9) In ntegrated form: + + eq H R c p p p p ) 3( ) ( 3 ν ν (3-93) In the represented form, besdes materal parameter ν (Posson s rato), Lematre model contans 3 materal damage parameters (see as Fgure 3. 3): ),, ( c R p p P = (3-94) Identfcaton of ths model conssts of the quanttatve evaluaton of the three coeffcents characterstcs by mean of tensle tests. Further, Lematre and Chaboche [4] ntroduced the damage varable nto the vscoplastc model. The rate of creep damage can be wrtten as: ) ( ) ( ) ( χ χ k r A = (3-95) where, ) ( χ s equvalent stress n creep fracture. A, r and are three characterstc creep damage coeffcent for the materal. k Bonora [8][9][] Bonora [8][9] proposed a damage model n the framework of CM. Ths formulaton s materal ndependent, allowng the descrpton of dfferent damage evoluton laws wth plastc stran wthout the need to change the choce of the damage dsspaton potental. In Bonora s work [8][], the followng expresson for the damage dsspaton potental was proposed: n n c p S S Y F / ) ( / ) ( ) ( + = α α (3-96) The assocated damage varable Y s expressed by: = eq H eq f E Y ) ( (3-97) 58

71 Chapter 3: amage Mechancs and Weldng amage The traxalty functon f ) s defed by Equ ( H eq If the von Mses stress of a dutcle materal can be guded by Ramberg-Osgood power law [59]as: eq ( ) = Kp n (3-98) where K s a materal constant, we have: F Y eq = ( ) H f eq ES ( ( c ) (+ n) / p α ) / α n (3-99) Fnally, damage rate s gven n dfferental form: α F p ( c ) H ( α ) / α = λ = α ( + ν ) + 3( ν ) ( c ) (3- ) Y ln( pr / p) 3 eq p Gven the assumpton of proportonal loadng, the damage evoluton s derved as: α ln( p p = + ) ( H ( c ) f ) (3- ) ln( pr p) eq where α s the damage exponent, and present respectvely the ntal and crtcal damage. The rupture stran s p. The p s the threshold stran. Here no dstncton s made between total strans anf plastc strans. R c Compared wth Lematre model, the consttutve equaton of damage proposed by Bonora contans two addtonal parameters: andα. It ncludes 5 materal parameters: P = ( ε, ε,,, α ) (3- ) th f c Saanoun [35][67][68] Saanoun proposed a fully coupled sotropc and sothermal formulaton by usng a sngle yeld surface both for plastcty and damage. The dual (or force) varables (, X, R, Y ) are derved from the state potental defned as the Helmholtz free energy whle the flux varables (, α, r, ) are derved from an approprated dsspaton potental F (, X, R, Y, ) together wth a sngle yeld functon f (, X, R, ) n the framework of non-assocatve theory. The complete sets of state consttutve equatons are gven as followng: Helmholtz free energy Tong WU 59

72 Thess: Experment and Numercal Smulaton of Weldng Induced amage Plastc potental C 3 Q e e ρ Ψ = ( ) ε : (µ + λe ) : ε + ( ) α : α + ( ) (3-3) r γ + 3 a b A Y F f X : X R (3-4) = + 4 C ( ) + Q ( ) + ( ) β γ + A Yeld functon n the stress space f (, X, R) = X R S y < (3-5) Knetc law of damage evoluton F Y λ Y ( ) A γ = λ = ( ) (3-6) β where ρ s the materal densty n the current undamaged confguraton. λ and µ are the classcal Lame s constants. Q the scalar sotropc hardenng modulus, C the knematc hardenng modulus, a the knematc hardenng non-lnearty coeffcent, b the sotropc hardenng non-lnearty coeffcent, γ, A and β are the materal coeffcents characterzng the ductle damage evoluton. Saanoun s model s wdely appled n the metal process of formng. [68] gves an exhaustve presentaton of the theoretcal, numercal and geometrcal aspects of the vrtual metal formng wth ductle damage. After the foundaton and development of damage model, another mportant work s the study on assessment and comparson of dfferent knds of models. For example, rabek and Bohm compared Gurson model and Rousseler model [6][6]. Hambl [9] dd comparatve study between Lematre and Gurson damage models n crack growth smulaton. Lammer and Tsamaks [3] compared dfferent CM models wth reference to homogeneous and nhomogeneous deformatons provdng a formulaton generalzaton to fnte deformaton Ansotropc damage Second order damage tensors have been ntroduced e.g. by Murakam and Ohno [4], Betten [6], Krajcnovc [], Chow and Wang [38], Murakam [4], Lu and Chow [3], Bruhns and Schesse [], and Stenmann and Carol [74]. Brung [6] proposed an ansotropc CM model by usng the porosty as a defnton for the damage varable. The fourth rank damage tensors were already ntroduced, for example, n the ansotropc creep damage modellng by Chaboche [3]. The smplest way to model ansotropc damage s to assume that damage wll only occur n the plane perpendcular to the hghest prncpal stress []: 6

73 Chapter 3: amage Mechancs and Weldng amage = (3-7) 3 [ ~ ] where < < 3 Murakam [4] has developed an approach based on geometrcal consderatons wth a second order damage tensor. In hs model, not only the effectve cross secton s decreasng durng loadng, but also the orentaton of that cross secton. Assumng that the shape of the consdered cross secton s not changng (ths leads to the case of orthotropc damage) we can wrte accordng to the sotropc case: ~ ( ) n S n ~ δ = δs (3-8) where s the second order dentty tensor, and s the second order damage tensor. When the prncpal drectons of stress and damage concde, the followng expresson s vald: = (3-9) 3 3 [ ~ ] Concluson In terms of metallc metrals, t s not necessary to use the complcated ansotropc damage under the condton of RVE sze n mesoscopc approach. Thus, the sotropc damage s chosen to develop the damage model. Vscoelastcty s preponderant n polymers and all ther compounds (especally organc matrx compostes), t s of lttle mportance for metals and alloys. Vscoelastcty s less mportant for weldng due to the short tme of weldng process. Therefore, the couplng between damage and elastoplastcty can descrbe the damage behavour nduced by weldng. Tong WU 6

74 Thess: Experment and Numercal Smulaton of Weldng Induced amage 3.5 Weldng damage 3.5. Introducton Macro cracks are observed n heat affected zone (HAZ) or weldng lne of metallc materals under specfc weldng condton. In addton, because nonmetallc nclusons n a metal can be the stes of vod nucleaton, and weldng process deepens on such segregaton of non metallc elements, and causes vods nucleaton, growth and coalesence, at the end leads to ductle fracture of the structure [97]. There are also many small or mcro defects n the weldng component nduced by many factors such as materal ngredent fluctuates, generaton of hydrogen, ncluson of mpurty, etc Heat affected zone (HAZ) of weld Usually, a weldment s made of three man areas (see Fgure 3. 4): The weld metal deposed durng the weldng operaton. The heat affected zone (HAZ) where the mcrostructure of the base metal s affected by local heatng due to the depost of the weld metal n the lqud state (.e. at ts meltng temperature). The base metal. In addton, the HAZ of weldments conssts of varous mcrostructural states (manly three) n a narrow zone. In fact, the weldng procedure nduces local heatng of the base metal at temperatures rangng from 4 C at the fuson lne to 75 C n the so called heat affected zone (HAZ). Mcro structural heterogenety n the HAZ can be related to phase transformatons. The schematc dagram of phase transformaton was gven n Fgure.4. There the ntal and end temperatures of austente transformaton are respectvely noted Ac and Ac3. The three man parts of the HAZ are the followng (see Fgure 3. 4): Coarse graned heat affected zone (CGHAZ): T max >> Ac3, the austente transformaton s complete and austente gran growth s promoted by elevated temperature (Nearest zone from the fuson lne). Fne graned heat affected zone (FGHAZ): T max s just above Ac3, the austente transformaton s nearly complete but austente gran growth s lmted. Inter crtcal heat affected zone (ICHAZ): Ac < T max < Ac3, the martenste s partally transformed nto austente durng the weldng thermal cycle. 6

75 Chapter 3: amage Mechancs and Weldng amage Fgure 3. 4 Schematc dagram of dfferent zones of a weld jont amage and crackng nduced by weldng Basng on the study on senstvty of weldments to cracks, there are dfferent types of crackng that may appear due to the weldng procedure: Soldfcaton crackng: occurs for metals whch have a hgh thermal expanson coeffcent. Hot crackng: occurs where the soldus temperature s reached as a lqud flm may be trapped between the soldfcaton lmt and the columnar gran. When flms are unable to accommodate the tensle contracton strans, crackng occurs. Generally, the austente former elements are sad to ncrease the hot crackng susceptblty and ferrte former elements are sad to lower t. Barnes et al [4] studed the effect of Nb, Mn and weldng condtons on hot crackng. They found that for hgh Nb content, local segregaton may lead to hgh local concentraton of Nb, and a lqud flm whch s an Nb ntermetallc phase very senstve to crackng can formed (f no crack occurs, ths ntermetallc phase s soldfed n δ ferrte whch s retaned durng coolng). On the other hand, Mn ncreases the resstance to hot crackng. From, ther study, Barnes et al also [4] concluded that the weldng parameters have lttle nfluence on the hot crackng susceptblty. Reheat crackng: may occur for two reasons. The prncpal cause s that when heat treatng susceptble steels, the gran nteror becomes strengthened by carbde precptaton, forcng the relaxaton of resdual stresses by creep deformaton at the gran boundares. The second factor of reheat crackng s the segregaton of elements at gran boundares. It s also called reverse temper embrttlement. To nvestgate the weldment sensblty to reheat crackng due to temper embrttlement, a step coolng, whch s a mult stages temperng treatment, s often performed. Hydrogen crackng: s possble f hydrogen s dssolved n the weld metal durng the weldng procedure. Indeed, when the austente martenste transformaton proceeds, the hydrogen solublty strongly decreases and t may leads to the formaton of hydrogen bubbles. However, applyng some procedures of weldng process, such a preheatng process, a post weldng heat Tong WU 63

76 Thess: Experment and Numercal Smulaton of Weldng Induced amage treatment, a TIG weldng nstead of GMAW, could mnmzed the rsk of hydrogen nduced crackng (HIC). In our study, our model focus our attenton on the weldng damage and crackng nduced by weldng restrant, stress of phase transformaton and vscoelastoplastty. The factors of Hydrogen, oxdaton and corroson are neglected. 3.6 Summary In ths chapter, defntons of damage varable and thermodynamcs of sotropc damage are presented n the framework of rreversble thermodynamcs. Varous wdely used models are studed, and these models are foundaton of the coupled-transformaton damage consttutve equatons. Our study heren s based on the sotropc damage and CM. Although ansotropy s closer to realty of most materals, the assumpton of sotropc damage s realstc n many cases for metallc materals, especally under proportonal loadng when the drectons of prncpal stress reman constant. In Chapter and Chapter 3, we ntroduced two most mportant phenomena of our topc studed: phase transformaton and damage, as a revew and concluson of current stuaton of these scentfc felds. The knowledge n these two chapters s a foundaton of further study on damage n multphase metallc materals, whch wll be studed and explaned n Chapter 4. 64

77 Chapter 4: Evoluton amage Multphase Modellng Chapter 4 Evoluton amage Multphase Modellng 4. Introducton The objectve of ths chapter s to develope a multphase damage thermal vscoelastoplastc model. Compared to the consttutve equatons n secton 3.3, an addtonal varable z s ntroduced n ths chapter. Here, volume fracton of the phase z s a functon of temperature because the phase transformaton s supposed to be drven by thermal loadng. Mechancs coupled wth damage depends on phase fracton and phase transformaton. Therefore, consttutve equatons n multphase are coupled wth damage, temperature and metallurgy. Two models wll be developed and ntroduced n the chapter: mesoscopc damage model and two-scale damage model. The secton s lmted to a -phase model, and we make a conventon that represents martenstc phase and refers austente. In terms of martenstc stanless steel, two phase transformatons are ncluded: and. Volume fracton of martenste s z and volume fracton of austente s. z We have: z + z. = Before developng consttutve equatons, several hypotheses should be gven: Hypothess : the materals are ntally sotropc. Hypothess : the damage s sotropc. Hypothess 3: the damage has no nfluence on metallurgy. Hypothess 4: we consder addtve decomposton of total stran n a thermoelastc part and plastc one. There s no uncouplng between elastcty and plastcty. Tong WU 65

78 Thess: Experment and Numercal Smulaton of Weldng Induced amage Hypothess 5: the vscoplastc stran occers only when appled stran s large enough. The threshold surface s gven as followng: f = J X ) R ( y (4- ) Hypothess 6: the plastcty s supposed to be sotropc. Let us recall some elementary notons for hardenng plastcty. The plastc deformaton of a materal almost always causes hardenng. At mcroscopc scale, hardenng s characterzed by an ncrease n the number of dslocatons wthn materal. Wth ncreasng dslocaton densty, they lose ther mobltes.to allow accountng for ths hardenng, one defnes a feld of elastcty and studes ts evoluton accordng to the deformaton. The evoluton of ths feld can be represented by a combnaton of dlaton, rotaton, translaton and dstorton. Isotropc hardenng s characterzed by a homothetc swellng of the yeld surface. In absence of dstorton and on the bass of a crcular surface (sotropy), sotropc hardenng s then represented by the couple of scalars ( r, R ). Knematc hardenng s characterzed by the translaton of the center of the yeld surface represented by a couple of tensors of nternal varables ( α, X ). Fgure 4. shows these two hardenngs when no damage occurs. The representaton of hardenngs wth damage s gven n Fgure 4.. Ω e ntal 3 y + R y R y X X O O ε p Ω e actual Fgure 4. Geometrcal representaton of sotropc and knematc hardenngs Ω e ntal 3 ( ) y + R y ( R ) y X X O O ε p Ω e actual Fgure 4. Geometrcal representaton of sotropc and knematc hardenngs couplng wth damage 66

79 Chapter 4: Evoluton amage Multphase Modellng 4. efnton of damage n multphase Because of assumpton of sotopc and homogeneous damage, the RVE of two phases can be presented n -. We suppose that each phase has ts own damage. We dstngush a mesoscale for each damaged phase from a mcroscale to defned damage n each scale. The damage defnton n two phases s shown n Fgure S S Phase Mcrovod n phase ds ds Phase RVE Mcrovods n phase ds ds Level : macroscale Level : mesoscale Level 3: mcroscale Fgure 4. 3 amage defnton at dfferent scale levels At mesoscopc scale, we neglect the fact the RVE conssts of martenste (phase ) and austente (phase ) and do not dstngush damage n the two phases. The damage varable s defned as the surface densty of mcrovodes and mcrocracks. S = (4- ) S At mcroscopc scale, the local damages can occur n martenste and austente. Accordngly the damage n phase at mcroscale can be defned by: u ds = ( =, ) (4-3) ds The damage varable n Equ. 4- can be dvded nto two parts: damage n phase and damage n phase. S S S S S S Introduce Equ. 4-3 nto Equ. 4-4, we get: S S = = + = ds + ds (4-4) s s µ ds + s S S = µ s ds (4-5) Let us defne Tong WU 67

80 Thess: Experment and Numercal Smulaton of Weldng Induced amage = S S µ ds ( =, ) (4-6) S ξ = ( =, ξ + ξ = ) (4-7) S where represents average damage n phase ; ξ s damaged surface fracton. We get = ξ + (4-8) ξ The damage varable s defned and deduced from surface fracton whereas the phase fracton s defned by volume. We now ntroduce volume fracton n phase transformaton nto damage consttutve equatons. Under the assumpton that the partcle of martenste and austente are sphercal shape (Fgure 4. 4), we have: V R 3 z = = ( ) (4-9) V R S R ξ 3 = = ( ) = z (4- ) S R z = z (4- ) Therefore, we get: ξ = ξ (4- ) z = ξ + ξ = z 3 + ( 3 ) (4-3) The damage mxture law s derven by volume fracton at power /3. R R Phase Phase Fgure 4. 4 Schematc llustraton of two phases wth sphercal shape 68

81 Chapter 4: Evoluton amage Multphase Modellng 4.3 A proposed damage equaton Based on the study of Lematre-Chaboche damage model and Borona s equaton, a damage stran model s proposed n the framework of contnuum damage mechancs n ths secton. The Lematre-Chaboche damage model n secton can be wrtten as followng n the unaxal cases: c = p p (4-4) pr p In the above equaton, the damage s lnear wth plastc stran. But t s not n good agreement wth some materals characterstcs. For example, the stran-damage law does not ft the nonlnearty of 5-5PH tensle test at room temperature (see Fgure 5. for damage expermental results of 5-5PH). Borona s damage equaton adopted the exponent form of plastc stran, and the damage exponent α was successfully dentfed through expermental method [8]. However hs model s coupled wth an elastoplastc Ramberg-Osgood power law whch s not very adapted to the materal lke 5-5PH studed here. We propose now an extenson of Lematre-Chaboche damage model wth multaxal stran. The damage dsspaton potental s chosen as: s+ Y κ κ p p F = (4-5) s + S where S, s andκ are materal parameters whch depend on temperature. s F Y κ = λ = κ p p p wth λ = p (4-6) Y S wth Y = eq E( ) H f eq (4-7) H f eq = 3 ( ν ) + 3( ν ) H eq (4-8) The ductle damage occurs only when the stran-hardenng s saturated and materal s then consdered perfectly plastc. The expresson of the plastc crteron ( ) R k = shows that: eq eq = ~ = constant = K eq (4-9) Tong WU 69

82 Thess: Experment and Numercal Smulaton of Weldng Induced amage Then K = ES H f ( ) κ eq s p p κ p (4- ) Integratng from t to t ( p = p t, p t = p, t = ), we have: K H = f ( ) ES eq s p p κ (4- ) At tme tc ( p t p c t = ), we get the crtcal damage as: = R, c c c K H = f ( ) ES eq s p R p κ (4- ) In proportonal loadng cases, s a constant. Consequently, from the above two equatons, we get: wth If we take the ntal damage p H p eq p κ = c (4-3) pr p s / κ κ K H p = f ( ) R c (4-4) ES eq of materal nto account, the proposed equatons wrte as: In unaxal loadng: In multaxal loadng: c κ = p p κ (4-5) ( pr p ) = ( p p ) R c p ( + ν ) + 3( ν ) κ 3 H eq p κ (4-6) The damage exponent κ s a parameter of materal dependng on the temperature. In fact, f we neglect the ntal damage of materal and κ s equal to, the proposed model s smplfed nto Lematre s damage model. Compared wth Lematre model, the consttutve equaton of the proposed damage model contans two addtonal parameters: parameters ( p pr, κ, c andκ. It ncludes 5 materal P =,, ) (4-7) 7

83 Chapter 4: Evoluton amage Multphase Modellng Identfcaton of ths model conssts of the quanttatve evaluaton of the fve coeffcents characterstcs by mean of tensle tests. 4.4 Consttutve equatons of mesoscopc model n multphase 4.4. State potental In ths secton, we ntroduce a mesoscopc model to descrbe damage under phase transformaton. Ths damage s defned at mesoscale (Fgure 4. 3, Equ.4-4),.e. there s no dfference between the damage n austente and n martenste. The state potental of each phase s ntroduced through the method of local state n the thermodynamcs of rreversble process. The total nternal energy or the Helmholtz free energy s the sum of each phase s energy based on the lnear mxture of dfferent phases. In our study, t s restrcted n sothermal process, and the totally nternal energy nternal energy conssts of elastc and thermometallurgc, vscoplastc (ncludng sotropc and knematc hardenng) nternal vp vpc energy e and nternal energy nduced by phase transformaton e tr. The followng equatons show these hypotheses: wth: where e th e th met e e e e tot tot e e e th met e vp vpc e tr ( ε, T, p, α,, z ) = e ( ε, T,, z ) + e ( ε, T, p, α,, z ) + e ( T, z) (4-8) e th met vp vpc ( e ρψ e th e met e ε, T,, z ) = ( ε, T,, z ) + ρ ψ ( ε, T, z ) (4-9) =, ρψ vp =, =, ρψ e vpk ( ε, T, p, α,, z ) = ( p, T,, z ) + ( α, T,, z ) (4-3) tr tr e ( T, z) = ρψ ( T, z) (4-3) ρ ψ : thermoelastc energy of phase met ρψ : metallurgc energy durng transformaton vp ρ ψ : blocked energy nduced by sotropc hardenng of phase vpk ρ ψ : blocked energy nduced by knematc hardenng of phase tr ρ ψ : phase-transformaton energy And ρψ e th can be defned by: ρ ψ Tref Tref [ α ( T )( T T ) α ( T )( T T )] e e e e ε, T,, z ) = z ( ) A ( T ) ε : ε ref ref A ( T ) I : ε ( T z C T ) (4-3) e th ( Tong WU 7

84 Thess: Experment and Numercal Smulaton of Weldng Induced amage ρ ψ : met e e ( ε, T, z ) = ( z z) ε A ( T ) I ε (4-33) The blocked energy by the mechansm of the sotropc hardenng coupled wth the temperature s selected n exponental form: vp ρ ψ ( p, T,, z ) = zc ( ) p + exp( γ p) (4-34) γ where the blocked energy by the mechansm of knematc hardenng coupled wth the temperature s descrbed a quadratc form of α. vpk ρ ψ ( α, T,, z ) = z g( T ) b ( T )( ) α : α (4-35) 3 tr We propose to take the ρψ ( T, z) as a smple expresson: z tr ρψ ( T, z) = ( zγ ) L ( T ) (4-36) z where L represents the latent heat of phase transformaton whch can, per assumpton, be taken as constant wth the temperature. The partal dfferentals of the free energy provde the followng state s laws: tot e = e ε = z =, e Tref Tref {( ) A ( T ) ε [ α ( T )( T T ) α ( T )( T T )] A ( T I} ref ref ) + A ε (4-37) ( T )( z z) e R = p tot = z c ( T )( ) [ exp( γ p) ] (4-38) =, X tot e = = α z =, g( T ) b ( ) α 3 (4-39) Y e = z A ( T ) + ε g( T ) α : α + c ( T ) p (4-4) =, 3 tot e s = (4-4) T k = =, e z tot (4-4) 7

85 Chapter 4: Evoluton amage Multphase Modellng 4.4. sspaton potental The yeld functon of each phase can be wrtten as: f = J ( X ) R y f(, R, X, ) = (4-43) then a b S ( T ) Y s ( T ) + F = f + J ( X ) + R + [ ] (4-44) ( ) ( ) [ s ( T ) + ]( ) S ( T ) where y : the yeld stress of phase The pseudo-potental of dsspaton ϕ s the functon of all the dual varables: e ϕ = ϕ(, A, gradt, Y, k; ε, V, T,, z) (4-45) k The total potental of dsspaton s wrtten by the sum of that of each phase: vp ϕ [ k vp ptr th tr d ϕ = ϕ + ϕ + ϕ + ϕ + ϕ ] (4-46) =, : correspond to the classcal vscoplastc potental, whch s defned: u vp + n ϕ = z f (4-47) + n ptr ϕ : potental of phase-transformaton, whch s an only functon of the stress because, at the temperatures of phase change, stran hardenng effects are neglgble. tr ϕ ptr 3 ' ϕ = : F ( z ) z (4-48) 4 : potental metallurgcal transformaton, whch s a transformaton by dffuson, a temporal varaton of the temperature nfluences the speed of dffuson consderably, by consequence the knetcs of transformaton. It s defned by: wth ϕ z (4-49) tr p = G ( T, ε, z) p G( T, ε, z) : a functon to be determned phenomenologcally. Tong WU 73

86 Thess: Experment and Numercal Smulaton of Weldng Induced amage To ensure the postvty of ths potental, we propose to take the functon to be a Heavsde step functon 3 : d ϕ the damage potental whch s defned as: H ( k) = k (4-5) d = zs Y s + ϕ ( ) (4-5) ( s + )( ) S th ϕ the thermal potental s wrtten: z K T grad( T ) T th ϕ = ( )( ) (4-5) The whole of the selected varables and the potentals prevously defned lead to a thermodynamcally acceptable model whose general formulaton s presented here. The vscoplastc flow: p ϕ 3 p X 3 ' ε = = + F ( z ) z (4-53) ( ) J ( X) =, The evoluton of the nternal varable assocated wth sotropc hardenng: ϕ n p = = zu F (4-54) R =, The evoluton of the nternal varable assocated wth sotropc hardenng: The evoluton of the damage varable: ϕ 3 p X n α = = u F X (4-55) X ( ) J ( X =, ) zs Y s ϕ = = ( ) (4-56) Y =, ( ) S The knetcs of phase transformaton: tr ϕ z = (4-57) k In concluson, the mesoscopc model takes nto account, vscoplastcty wth sotropc and knematc hardenng, temperature, phase transformaton and damage. The phase transformaton s ntroduced n the model by phase fracton depended on temperature. The totally nternal energy s the energy n each phase (by lnear mxture of volume fracton) plus one nduced by transformaton plastcty. 3 The Heavsde step functon, H, also called unt step functon, s a dscontnuous functon whose value s zero for negatve argument and one for postve argument. 74

87 Chapter 4: Evoluton amage Multphase Modellng 4.5 Consttutve equatons of two-scale multphase model 4.5. Introducton The two-scale model, whch we ntroduce heren, s developed through a method of localzatonhomogenzaton. The homogenzng procedure used s the Taylor s approxmaton [83], whch assumes homogeneous deformatons n a heterogeneous medum wth nonlnear behavor. Ths law provdes the closest possble match wth Leblond s theoretcal case for elastoplastc phases. Such approach, called mcro-macro, conssts of starng from the behavor of each phase and workng back to the macroscopc behavor of the materal. After localzaton, the behavors of each phase can be treated respectvely, wthout couplng. Thus, the model can provde the freedom to choose the behavor type of each phase. Snce each phase has ts own mechancal behavour, and the damage varable two phases s dfferent:. The damage mechancs of phase s depended on the materal characterstc, whch s related wth temperature. The followng hypotheses are ntroduced nto the two-scale model. Hypothess : the rate of total macroscopc stran can be dvded nto two parts: the rate of mcroscopc stran of the phases and the transformaton plastcty rate. Hypothess : there s no drect couplng between elastoplastcty and transformaton plastcty. Hypothess 3: the mcroscopc stran rate of martenste s equal to the austente stran rate n mcro scale. Hypothess 4: there s no nteracton between two phases, e.g. no damage or stran transfers from one phase to another. Hypothess 5: the homogenzed macroscopc stress s obtaned by the mxture of mcroscopc stress of each phase accordng to the volume fracton of the phases. Hypothess 6: we neglect the latent heat nduced by phase transformaton and the temperature n each phase keeps the same. n 4.5. Stran localzaton The approach of localzaton s based on the Taylor s approxmaton wth equal repartton of stran rates n all phases of the multphase compostes. The classc stran rate at macroscale s equal to total stran rate of sngle phase at mcroscale. Thus, we get: c E = ε ( =, ) (4-58) Accordng to the prncple of localzaton mentoned above, we splt the total stran rato nto two parts, one comng from the total mcroscopc stran rate of the phases, and the other representng the plastc transformaton stran rate. E + tot c pt = E E (4-59) Tong WU 75

88 Thess: Experment and Numercal Smulaton of Weldng Induced amage tot pt E = ε + E wth ε = ε (4-6) e ε = ε + ε + ε (4-6) thm vp The transformaton plastcty stran rate s guded by Leblond s transformaton plastcty model, whch was ntroduced n secton.6.. A smplfed form s as followng equaton: E pt th 3 ε γ y γ = α S ( ln z) z f f z.3 z >.3 (4-6) After localzaton of stran, the problem of multphase becomes the sngle phase behavor Mechancs n martenste In terms of martenste, we choose the elastoplastc model wth sotropc hardenng. The consttutve equatons are coupled wth damage varable of martenste. The free energy of martenstc phase wth the plastc behavor: tot e conssts of two parts when the elastc behavor s uncouplng e tot ( e e p = e T, ε, ) + e ( T, p, ) (4-63) The blocked energy by the mechansm of sotropc hardenng coupled wth the temperature s gven as: a e p ( T, p, ) = p ( ) + b p (4-64) The thermoelastc free energy s: e ε (4-65) e e e e e ( T,, ) = ( ) ε : A ( T ) : ε + B ( T ) : ε + C ( T )( T T ) Therefore, the equatons lnk between state varables and assocated varables are: tot e e = = ( ) A ( T ) : ε + B ( T ) (4-66) e ε e R = ( + b (4-67) tot = ap ) p tot e p Y = Y + Y e = (4-68) = (4-69) e e Y A ( T ) ε 76

89 Chapter 4: Evoluton amage Multphase Modellng a Y p = (4-7) p The pseudo-potental of dsspaton s decoupled nto the thermal dsspaton and plastc dsspaton, whch s couplng wth damage dsspaton: p th d ϕ = ϕ, R,, T ) + ϕ ( gradt, T ) + ϕ (, Y, ) (4-7) ( T The thermal dsspaton can be wrtten as: th gradt gradt ϕ ( gradt, T ) = k( T ) (4-7) T T The damage dsspaton of martenste can be chosen as: ϕ ( d Y κ, Y, T ) = κ p p (4-73) S where S and κ are materal damage parameters of phase dependng on temperature. s threshold plastcty to damage. p The yeld functon coupled wth damage s gven by: f J ( ) b R y, R, ) = ( ) (4-74) ( T The evoluton laws are obtaned from the generalzed normalty law: p 3 p = ϕ ε λ = (4-75) J ( ) ϕ b p = λ = λ (4-76) R Y ϕ λ (4-77) κ = = κ p p p Y S Mechancs n austente For the austente, the elastoplastc model wth lnear knematc work hardenng s employed. The consttutve equatons are coupled wth damage varable of austente. The free energy of austente phase uncouplng wth the plastc behavor: tot e s dvded nto two parts when the elastc behavor s e tot ( e e p = e T, ε, ) + e ( T, α, ) (4-78) Tong WU 77

90 Thess: Experment and Numercal Smulaton of Weldng Induced amage The blocked energy by the mechansm of knematc hardenng coupled wth damage varable s descrbed by a quadratc form of : α : ) )( ( 3 ),, ( α α α T g T e p = (4-79) The thermoelastc free energy s: ) )( ( : ) ( : ) ( : ) ( ),, ( T T T C T B T A T e e e e e e + + = ε ε ε ε (4-8) Therefore, the equatons lnk between state varables and assocated varables are: ) ( : ) ( ) ( T B T A e e e tot + = = ε ε (4-8) ) )( ( 3 α α T g e X tot = = (4-8) p e tot Y Y e Y + = = (4-83) e e T A Y ) ( ε = (4-84) : ) ( 3 α α T g Y p = (4-85) The pseudo-potental of dsspaton s decoupled nto the thermal dsspaton and plastc dsspaton, whch s couplng wth damage dsspaton: ),, ( ), ( ),,, ( T Y T gradt T X d th p ϕ ϕ ϕ ϕ + + = (4-86) The thermal dsspaton can be wrtten as: T gradt T gradt T k T gradt th = ) ( ), ( ϕ (4-87) The damage dsspaton of austente chosen s the exponental functon: d ϕ ) ( ) )( ( ),, ( + + = s d S Y s S T Y ϕ (4-88) where and are materal damage parameters of phase dependng on temperature. s S The yeld functon of austente can be wrtten: ) ( ) ( ),, ( T X J X f y = (4-89) 78

91 Chapter 4: Evoluton amage Multphase Modellng The potentals prevously defned lead to a thermodynamcal model. The evoluton laws are obtaned from the generalzed normalty law: 3 X p p = ϕ ε λ = (4-9) J ( X ) ϕ 3 p X α = λ = (4-9) X J ( X ) s = λ = ( ) (4-9) ϕ Y λ Y S Memory effect durng phase change The memory effect descrbes that nternal varables n daughter phase nhert form mother phase when one phase (mother phase) s dsappeared and the other phase (daughter phase) occurs. Memory coeffcent determnes how many percent values of nternal varables transfer from an old phase to a new phase. Heren, η s memory coeffcent of damage varable, µ s memory coeffcent of nternal varable A. The followng equatons gve the relatonshp between nternal varables n daughter phase and n mother phase: daughter mother =η (4-93) A daughter = µ A (4-94) mother In the equatons, the memory effect s null f η = or µ = and full f η = or µ = Stress and damage homogenzaton The homogenzed macroscopc stress s obtaned by a lnear law of mxture guded by the volume fracton of each phase. Ths smple homogenzaton could take stress at mcroscale back to macroscopc stress, whch s nvolved n structure analyses. = z (4-95) =, The damage varable s defned and deduced from the surface fracton whereas the phase fracton s defned by the volume. The damage homogenzaton s gven by the followng equaton: z = ξ + ξ = z 3 + ( 3 ) (4-96) Ths two-scale model takes only two phases nto account. It s not dffcult to extend to stuaton wth more phases. Compared to two-phase stuaton, multphase has the same localzaton-homogenzaton laws. The dfference les n where the model for multphase needs more dentfcaton for phases. Tong WU 79

92 Thess: Experment and Numercal Smulaton of Weldng Induced amage An example n one dmenson In ths secton, a smple example n one dmenson s gven to llustrate the two-scale model presented prevously. The smulaton was mplemented n Matlab 6.5. We ntroduce a smlar Satoh 4 type test that a unaxal bar s clamped on the top and bottom and smultaneously suffer thermal loadng. The test bar s supposed to be cooled homogeneously from hgh temperature to room temperature. Thus longtudnal dsplacements of bar are restraned durng coolng. Thus may explan the same or smlar phenomena as that can be observed n weldng heat-affected zone (HAZ). The test contans several complex coupled phenomena, and they are thermal (temperature), metallurgcal (phase-transformaton) and mechancal (stran-stress, damage) behavors. We supposed the materal propertes as followng: C Thermal stran dfference between the two phases: ε α γ =. Thermal expanson coeffcent of the austentc phase ( T ref = C ): α γ = Thermal expanson coeffcent of the martenstc phase: T = α T ( T 35 C ); α = 5 ( 35 C < T < 7 C ) α The begnnng temperature of martenstc transformaton: α Ms = 4 C T ( C) E (GPa) ym (MPa) ya (MPa) m=martenste, a=austente Table 4. Materal propertes depended on temperature We suppose that the bar has the elastcty and perfect plastcty and that the Young s modulus and yeld stress depend on temperature, whch s gven n Table 4.. The damage s supposed only n martenste whereas there s no damage n austente. The damage model was presented n Equ. 4- and Equ.4-3 n Secton 4.3. Parameters of the damage model are supposed as: R =., s = 6, p = and κ =. The temperature loadng of the bar s from C to C wth T = -9.8 C/s. The phase-transformaton model adopts Kostnen and Marburger model n Equ. -3 n Secton.3.4 (see Fgure 4. 5). 4 The test was performed by nducton heatng of a tensle specmen wth both ends clamped. urng the test, the reacton force-versus-temperature curves were recorded. The orgnal Satoh test was desgned for two purposes. Frst, for a gven heat treatment, the Satoh test can be used for characterzaton of materals. Secondly, for a gven materal, the test can be used to study the effect of heat treatment. 8

93 Chapter 4: Evoluton amage Multphase Modellng Temperature - Proporton.8 Proporton of M Proporton of A Proporton.6.4. Stran Temperature ( C) Fgure 4. 5 Proporton of each phase Temperature - Stran Elastc, A Elastc, M TRIP wthout damage Metallurgcal Thermal, A Thermal, M TRIP wth damage Temperature ( C) 4 Fgure 4. 6 Varous strans vs. temperature Temperature - Stress Stress (MPa) 8 6 component M, wthout dhomogenzed, wthout dcomponent A component M, wth damage homogenzed, wth d Temperature ( C) Fgure 4. 7 Comparng stress wth/wthout damage Tong WU 8

94 Thess: Experment and Numercal Smulaton of Weldng Induced amage.6. component A component M homogenzed amage evoluton amage Tme (s) Fgure 4. 8 amage varables Stran varables are shown n Fgure 4. 6, n whch the TRIP stran s domnatng comparng the others. Such bg value s due to the large macroscopc stress at low temperature. Furthermore, the TRIP stran that comes from the damage-mechancal model s smaller a lttle than one ganed from the usual stran-stress model (wthout damage) because of the declne of stress n damage condton. The thermal stran has observable effect to the stress n Satoh test, but the metallurgcal stran from austente to martenste weakens ths effect because ts volume ncreases whereas the thermal stran s negatve. From the Fgure 4. 7, t s observed that both austente and martenste yeld at the begnnng (martenste, near to 4 C). Generally, the austentc component of stress s much lower than the martenste s because of the bg dfference of yeld stresses. It s evdent that damage decreases the effectve stress, especally for martenstc component. Before martenstc phase-transformaton, all varables of martenste are null and the mechancal behavor s entrely determned by austentc phase. Wth the development of phase transformaton, the martenste plays more and more role not only n evoluton of stress but also to the damage s growth. At the end of phase-transformaton stage, the macroscopc stress ncreases slowly or even decreases because of damage s effect. amage evoluton s shown n Fgure Concluson Two mechancal models coupled damage, thermal and metallurgy were presented n the chapter. The mesoscopc model was developed based on the lnear mxture of dsspaton n framework of thermodynamcs and defnton of damage varable at mesoscale. The beneft of mesoscopc model s smple calbraton of damage parameters: neglectng the dfference between each phase. The two-scale connects behavors of multphase and sngle behavors through localzaton-homogenzaton laws. Such a modelng brngs a great flexblty of calculaton. On the one hand, each phase can have ts own 8

95 Chapter 4: Evoluton amage Multphase Modellng behavor (elastoplastc, vscoplastc ). On the other hand, one keeps a great freedom on the modelng of the plastcty of transformaton. Tong WU 83

96 Thess: Experment and Numercal Smulaton of Weldng Induced amage 84

97 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models Chapter 5 Expermental Study and Identfcaton of amage and Phase Transformaton Models 5. Introducton The consttutve equatons were proposed n the Chapter 4. The parameters of transformatonplastcty model and damage model wll be calbrated through expermental study n ths chapter. On the one hand, phase transformaton and transformaton plastcty can be dentfed through tests of round bar wth temperature loadng. On the other hand, cyclcal load-unload tests of round bar help for measurement of damage. In addton, we wll ntroduce tensle tests of flat notched specmen wth the help of dgtal mage correlaton system, n order to analyze and study stran localzaton and damage evoluton on surface of specmen. In terms of flat notched specmens, we wll not only compare that three notch szes of specmen nfluence on stran localzaton and damage dstrbuton, but also study the stuaton at dfferent temperatures. At last, the mcroscopc observaton wll be mplemented to study metallurgcal and mechancal behavours of the mcro scale. 5. esgn of specmens Two categores of specmen are used n ths study: round bar (RB) and flat notched specmen (FNS). The geometres of specmens are depcted n Fgure 5.. RB and FNS are adopted for the followng purposes: Tests of round bar provde stran-stress curve, dlatometry curves, mechancal propertes, thermal behavour, and damage varable by measurement of the varaton of elastc modulus. Tong WU 85

98 Thess: Experment and Numercal Smulaton of Weldng Induced amage Tests of flat notched specmen lead to the verfcaton of damage model and stran localzaton due to the fact that varous notch rad nduce dfferent traxal stress rato profles n the md secton. a) Round bar (RB) b) Flat notched specmen (FNS) Fgure 5. Specmen geometry and dmenson n mllmeter (RB and FNS) As far as tests of flat notched specmens are concerned, we desgned three specmens wth dfferent rad of notch wth R=mm (Case A), R=.5 mm (Case B) and R=4 mm (Case C) because the notch geometry generates stress and stran gradents from the specmen centre to the notch edge. The notch nfluence on traxalty rato can be calculated wth fnte element method. Fgure 5. gves an example of traxalty dstrbuton on surface of specmen wth mnmum dameter under elastc tenson. One can clearly see traxalty s maxmum n the centre n Case C and near the notch for the Case A. 86

99 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models.6 Traxalty rato of three cases.5 Traxalty rato.4.3. Case A Case B Case C X (mm) Fgure 5. Notches (Case A, Case B and Case C) nfluence on traxalty (strbuton along mnmum secton under elastc tenson) 5.3 Expermental devces The tests are performed on a MTS SCHENCK servo-electro-hydraulc tenson-compresson machne wth maxmum capacty of 5KN (Fgure 5. 3a). Heatng s generated by electromagnetc nducton, and power s suppled by a 3KW Celes generator (Fgure 5. 3d). Water-coolng setup provdes fast coolng rate. An extensometer wth the precson of mcron s used to measure the stran. It conssts of two stcks, connectng components, fxng component and conductng wres. The two stcks are placed n parallel and the gauged dstance s 5 mllmeters. The thermal couples connect the Converter TEPI (Model BEP34) for sgnal amplfcaton and converson; these sgnals are then collected by the FlexTest control system (Fgure 5. 3c). The automated confguraton ncludes the controller, a PC, and the model 793. system software bundle. In the tests of flat notched specmen, two C.C.. cameras (Fgure 5. 3b) are used to record dgtal mages of the sample s surface. 3 dgtal mage correlaton (LIMESS measurement technque and software) are utlzed to process dgtal mages and analyze the stran localsatons. The applcaton of C.C. cameras and mage correlaton makes t possble to montor stran localzaton and observe ductle damage evoluton. Tong WU 87

100 Thess: Experment and Numercal Smulaton of Weldng Induced amage Load cell Inductor col Upper grp Thermal couples Extensometer Water coolng Lower grp a) MTS SCHENCK b) General vew TEPI converter Flex Test c) TEPI and FlexTest d) Celes generator Fgure 5. 3 Expermental setup 88

101 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models 5.4 Measurement 5.4. Temperature The temperature measurements are performed usng thermocouples (type K) mcrowelded on the surface of the specmen (see Fgure 5. 4). Four thermocouples (T, T, T3 and T4) are used to montor the temperature n the useful zone. One thermocouple (T) s connected to the Heat nductor CELES for temperature feedback and control. 4 X 5mm T,T,T,T3,T4 Thermal couples Fgure 5. 4 Thermal couples welded on surface of specmen 5.4. Force and dsplacement Global force and dsplacement are respectvely measured by the loadng system: load cell and LVT (Lnear Varable splacement Transducers) sensor of tensle test machne are recorded n Controller System (PC component) Stress The nomnal stress * s ganed from force F and ntal cross-sectonal area S by the formula: F S * = (5- ) Consderng the cross-secton wll shrnk, the ntal cross-sectonal area should be replaced by the effectve one. It s supposed the materal s plastc ncompressble and the volume keep the same n the dfferent force loadng condton. And then the nomnal stress leads to true stress guded by: F F L * L + L * * = = = ( ) = ( + ε ) S S L L (5- ) Stran The local stran s measured by the extensometer (Fgure 5. 5). Such recorded stran s called nomnal stran. In order to obtan the stran (Hencky stran), the followng formula s appled: l dl L L + L * ε = = ln( ) = ln( ) = ln( + ε ) (5-3) l l L L Tong WU 89

102 Thess: Experment and Numercal Smulaton of Weldng Induced amage where L : the ntal length of measured part L : the elongaton * ε : nomnal stran ε : true stran L =5mmm LVT Fgure 5. 5 Extensometer amage amage s not easy to be measured drectly. Its quanttatve evaluaton, lke any physcal value, s lnked to the defnton of the varable chosen to represent the phenomenon. We can measure damage by measurng the modfcaton of the mechancal propertes of elastcty. The method of varaton of the modulus of elastcty, whch s a non-drect measurement, s preferred because of ts smple prncple and great precson (see Fgure 3.). Recall effectve stress n the case of unaxal tenson [4] : ~ = = Eε e (5-4) ~ E( ) = E can be nterpreted as the elastc modulus of the damaged materal. amage varable can be wrtten: = E ~ / E (5-5) ~ where E s Young s modulus at moment of undamaged materal. E represents Young s modulus at moment of damaged materal. espte ts apparent smplcty, ths measurement s rather trcky for the followng reasons: ) the measurement of modulus of elastcty requres precse measurement of very small strans; ) damage s usually localzed, thus requres a very small extensometry base (about.5 to 5mm); 9

103 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models 3) the best straght lne n the stran-stress graph representng elastc loadng or unloadng s dffcult to defne; 4) at hgh temperature, metal becomes weak and stran-stress curve shows very strong nonlnearty. Therefore, an adequate procedure s chosen to optmze the measure of Young s modulus. The stage of unloadng ramp s used to measure Young s modulus. The evaluaton of the elastcty modulus durng elastc unloadng must avod the zones of strong nonlnearty. Ths s acheved by performng the evaluaton n a range of stress defned as follows: < < (5-6) mn max where =.5 ( ε ), and : s the upper stress lmt of lnear part. mn max In the stran-stress regon defned above, we ntroduced the error rectangle (Fgure 5. 6), whch conssts of stran error and stress error, to ensure that the useful expermental data are wthn n the error tolerance. Stran error s due to the errors of varous devces nvolved n measurement of strans. Smlarly, stress error s related to force measurng nstrument. The resultng computed error rectangle wdth s: for stan error =.5 and for stress error =.5 MPa. stress error Stress stran error Stran Fgure 5. 6 Error rectangle and lnear measurement of stran-stress curve 5.5 gtal mage correlaton (IC) The dgtal mage correlaton was nvolved n our expermental study. An mage processng gves the space-tme evoluton of varous knematc varables (dsplacement, velocty, stran, stran-rate, etc.) on the sample surface. The dgtal mage correlaton technque was orgnally ntroduced n the early 8s by researchers from the Unversty of South Carolna [5]. The dea behnd the method s to nfer the dsplacement of the materal under test by trackng the deformaton of a random speckle pattern appled to the component's surface n dgtal mages acqured durng the loadng. Mathematcally, ths s accomplshed by fndng the regon n a deformed mage that maxmzes the normalzed cross- Tong WU 9

104 Thess: Experment and Numercal Smulaton of Weldng Induced amage correlaton score wth regard to a small subset of the mage taken whle no load was appled. By repeatng ths process for a large number of subsets, full-feld deformaton data can be obtaned. Helm et al. extended the IC method to use multple cameras, permttng the measurement of threedmensonal shape as well as the measurement of the three-dmensonal deformaton [93]. The threedmensonal technque requres the use of at least two synchronzed cameras acqurng mages of the loaded specmen from dfferent vewng angles. By determnng correspondng mage locatons across vews from the dfferent cameras and trackng the movement throughout the loadng cycle, the shape and deformaton can be reconstructed based on a smple camera calbraton. Correlaton technques on dgtal mages have been used to measure n-plane dsplacement components on plane samples by many researchers [37][4] and to study neckng of sheet metal n the laboratory of contacts and sold mechancs of INSA-Lyon n France [3][4][4][85].. It s necessary for the acquston of the qualfed dgtal mages. In order to obtan good dgtal mage of sample durng tensle test, we should pay attenton to llumnaton, speckle pattern, choce of spray pant, calbraton of two cameras and pant movement wth the surface. Reflectons and shadows should be absolutely avoded. Small speckles loads to hgh spatal resoluton. Calbraton s necessary for 3- IC, but no need for - IC. In our test, we used black and whter spray pants to get proper speckle pattern and to coordnate the contrast gradent of mage. However, t s dffcult to fnd a spray pant wth resstance to hgh temperature. The FNS wth speckle pattern and FNS wth mage correlaton are respectvely shown n Fgure 5. 7a and Fgure 5. 7b. Q M O N Y X P 5mm a) FNS wth speckle pattern b) FNS wth mage correlaton Fgure 5. 7 Flat notched specmen n tensle test 9

105 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models 5.6 Expermental programs 5.6. Round bar tests In terms of the tests of RB, three categores are desgned: Type : Free expanson-contracton tests, whch am to dentfy the thermal parameters, such as coeffcent of heat expanson, Ms, Ac, Ac3 and the dfference of thermal stran dfference between two phases. Type : Expanson-contracton tests wth appled tensle load, whch are used to study the phase transformaton nduced plastcty. Type 3: Cyclc load-unload tests, whch not only provde stran-stress curve and mechanc propertes, such as Young s modulus, yeld stran, yeld stress and hardenng behavour, but also permt to dentfy the damage varable. For the test Type, one specmen wth varous thermal hstores s mplemented. The specmens are loaded wth heatng rate of 4-5 C/s and coolng rate of -3 C/s. It s mposed four thermal cycles: the frst three cycles reach maxmum temperature of 86 C, and the maxmum temperature of last one s 5 C (see Fgure 5. ). The temperature hstory of the tests Type (Fgure 5. 8) s smlar wth that of the test Type. Specfc experments are desgned to measure to dentfy the phase transformaton nduced plastcty through loadng pressure at coolng stage. Loadng pressures wth value of 37.5MPa, 5MPa, 75MPa and 5.5MPa (see Fgure 5. 8), are appled before Ms temperature durng the coolng stage, and last to the completon of martenste transformaton. Temperature ( C) Four appled forces 5 Ms Force = KN KN KN KN Force 5 tme (s) F Stress = MPa MPa MPa MPa 53 6 tme (s) Fgure 5. 8 Temperature and force loads of TRIP tests (Test type ) Tong WU 93

106 Thess: Experment and Numercal Smulaton of Weldng Induced amage Condton P P P3 P4 P5 P6 P7 P8 P9 P P Test T ( C) Max T ( C) Phase M M M M A A A M M A A Note: T presents temperature; M means martenste phase; A s austente phase; heatng rate: 4-5 C/s; coolng rate (average): -3 C/s. Table 5. Temperature loadng condton of RB tests Condton P P P3 P4 P5 P6 P7 P8 P9 P P Load Unload sp. ncrement Tme (s) sp. ncrement Tme (s) Total Cycle Note: unt of dsplacement s mm. Table 5. splacement loadng condton of RB tests Temperature ( C) 5 6 splacement 5 4 tme (s) tme (s) Fgure 5. 9 Temperature and dsplacement loads of P6 (at 6 C, Austente) As far as test Type 3 s concerned, eleven specmens (namely P, =,,, ) are used wth dfferent temperature hstory and stretched at several constant temperatures. There s no hold tme at maxmum temperature. The temperature heatng rate s 4-5 C/s and the coolng one s about - C/s. The dea s to compare the stress stran response and the damage dentfcaton for a specmen at the 94

107 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models same temperature havng had a dfferent temperature hstory. Table 5. s a summary of the performed experments: one can observe that test P and P8 should produce the same materal characterstc f the temperature hstory had no nfluence on materal response. The tensle tests are dsplacement controlled. The dsplacement loadng hstory conssts of several cycles (loadng and unloadng), and up to rupture of specmens except the free expanson specmens. The detaled nformaton of dsplacement loadng s n Table 5.. For example, the temperature and dsplacement loads of P6 are gven n Fgure Tensle tests of flat notched specmen The tests of flat notched specmens wth varous notches were performed at varous temperatures. The fve condtons of test (namely F, =,,, 5) were gven n Table In every condton, three samples (Case A, B, C) were tested. Consequence, ffteen specmens are used n ths knd of test (namely Fj, =,,, Case A). 5, j = A, B, C, e.g. F A represents the specmen wth thermal condton F and notch Condton F F F3 F4 F5 Test T ( C) 3 Heatng rate ( C/s) Coolng rate ( C/s) Max T ( C) Phase M M M MIX A Table 5. 3 FNS test condtons 5.7 Expermental results and dentfcaton of parameters 5.7. Thermal and metallurgcal data The loaded temperature hstores are measured through thermocouples welded on surface of specmen. For example, the temperatures of specmen P are recorded by thermocouples and the temperatures vs. tme curves at four postons of specmen surface are shown n Fgure 5.. It shows the temperatures at mddle of specmen have a good homogenety, and the maxmum dfference T = 4 C max. Tong WU 95

108 Thess: Experment and Numercal Smulaton of Weldng Induced amage Temperature loadng Temperature ( C) T T T3 T4 S5-H Tme (s) Fgure 5. The loaded ttemperature hstores of P (measured ponts: T, T, T3, T4) Experment results obtaned from the tensle tests of round bar, such as coeffcent of expanson, metallurgcal characterstcs are lsted n the Appendx B. These basc data are necessary to smulate materal behavor durng heatng and coolng (temperature hstory from C to 5 C) by lnear nterpolaton between the ndcated ponts. In Fgure 5., four cycles are mposed to the specmen. The frst cycle s for the purpose of elmnatng, at least partly, the nternal stresses dependent on the means of rollng of sheet. The second and the thrd cycle are used for the thermo-metallurgcal characterstcs. The parameters of phase transformaton are drectly dentfed through fttng free expanson curves (see Table 5. 4)..6 Free expanson.4.. Cyc No. (Max 86 C) Cyc No. (Max 86 C) Cyc No.3 (Max 86 C) Cyc No.4 (Max5 C).8 Stran Temeprature ( C) Fgure 5. Free dlatometers wth four cycles thermal loadng (the frst three cycles reach maxmum temperature of 86 C, whereas the maxmum temperature of last one s 5 C) 96

109 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models Parameters Ms (5) Ms (86) Ac Ac 3 α α α γ th εα γ β Value 5 C C 76 C 8 C.5E-5.9E E-3. Note: Ms (5)/Ms (86) means that the maxmum temperature of the thermal cycle s 5/86 C. Table 5. 4 Parameters of phase transformaton from experments It s notced that the fst three cycles have the same martenste start temperature (Ms) whereas t s dfferent from that of the last cycle. The explanaton s based on the materal metallurgy. Martenste s formed based on the gan of austente when the temperature s cooled to Ms. Consequence, martenste transformaton s related to the gran sze of austente. Generally, the hgher temperature leads larger gran sze of austente, n whch the martenste nucleaton s more dffcult and needs more energy or condensate depresson. Thus, the larger gran sze of austente makes the lower Ms. In addton, the precptaton nfluences the transformaton. At hgh temperature, the second phase s precptated among gran boundares, and lowers the martenste start temperature. Therefore, the gran sze of austente affects the martenste transformaton knematcs. It should be taken nto account n the smulaton of hot processng Mechencal data The mechancal behavors of 5-5PH are dentfed by tensle test of round bar at varous temperatures. The detals of mechancal characterstcs, such as Young s modulus, yeld strength and ultmate strength, are gven n the Appendx B. The Fgure 5. and Fgure 5. 3 gve the tensle curves of martenste and austente at dfferent temperatures. These data are necessary for weldng smulaton of 5-5PH. An mportant phenomenon s observed n the tensle curves: the thermal hstores have a large nfluence on mechancal characterstcs. The maxmum temperature has to be taken nto account n the materal studed here. In Fgure 5., stran-stress curve for P, P8 and P9 are plotted at room temperature; the curves have some largely mportant dfferences. P and P6 n Fgure 5. 3 are tested at 6 C, however P has much more strength than P6. An mportant factor may explan the dfference: the larger the gran sze of a crystallne materal, the weaker t s. The gran boundary dslocaton of small gran sze ncreases the resstng force of plastc flowng. For example, the yeld stress follows the Hall Petch relaton (see Equ. -) for a number of steels. Tong WU 97

110 Thess: Experment and Numercal Smulaton of Weldng Induced amage Tensle curves at dfferent temperatures (M) Stress (MPa) 8 6 P, C P, C P3, 6 C P4, 7 C P8, C P9, C Stran (%) Fgure 5. Tensle curves of martenste at dfferent temperatures P5, 85 C, max(5 C) P6, 6 C, max(5 C) P7, C, max(5 C) P, 3 C, max( 86 C) P, 6 C, max( 86 C) Tensle curves at dfferent temperatures (A) 5 Stress (MPa) Stran (%) Fgure 5. 3 Tensle curves of austente at dfferent temperatures P Martenste C Force (KN) Rupture splacement (mm) Fgure 5. 4 Force vs. dsplacement curves of cyclcal load-unload tests at C, martenste state (P) 98

111 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models 4 P5 Austente 85 C Rupture Force (KN) splacement (mm) Fgure 5. 5 Force vs. dsplacement curves of cyclcal load-unload tests at 85 C, austente state (P5) 4 P Martenste C Stress (MPa) Stran (mm/mm) Fgure 5. 6 Stress vs. stran curves of cyclcal load-unload tests at C, martenste state (P) P5 Austente 85 C Stress (MPa) Stran (mm/mm) Fgure 5. 7 Stress vs. stran curves of cyclcal load-unload tests at 85 C, austente state (P5) Tong WU 99

112 Thess: Experment and Numercal Smulaton of Weldng Induced amage Eleven fgures wth load-unload curves were acqured from the correspondng tests (from P to P). Heren, only two examples of cyclcal load-unload tests (P and P5) are reported here (Fgure 5. 4 and Fgure 5. 5), and others are gven n Appendx B. The correspondng stran-stress curves are shown n Fgure 5. 6 and Fgure Other tensle curves at dfferent temperatures are gven n the Appendx B. The test of 5-5PH at room temperature (martenste state) was performed and generated the force-dsplacement curve shown n Fgure It shows that the materal deterorated and force decreased wth the ncreasng of dsplacement after the force reached ultmate strength. At last, materal was broken sharply. In terms of 5-5PH at 85 C (austente state, see as Fgure 5. 5), t s much weaker than that n room temperature, and the behavor of full falure s less of sudden because materal s more ductle. The cyclcal load-unload leads to varaton of Young s modulus, then cause the measurement of damage varable, the detaled analyses wll be gven n Secton Identfcaton of parameters of transformaton plastcty model In terms of phase-transformaton model (Magee model), there are materal parameters ncluded: M s and β. M s can measure drectly from the curve of free expanson, and parameter β should do fttng process to expermental results. The results are gven n Table As far as transformaton nduced plastcty model (Leblond model) consderaton, t contans th y th parameters: ε and. can be measured drectly from the curve of free expanson, and ts γ α γ ε γ α result s lsted n Tab.. The parameter y γ should be dentfed from expermental data. The Lelond s transformaton plastcty model (see Secton.6.) provdes ncrement of transformaton nduced plastcty. In order to dentfy the parameter γ y, the ntegral form s gven as: th 3 ε pt γ α ε = S z dz y ln (5-7) γ where, S = n one dmensonal tensle case. eq 3 ε ε γ ] (5-8) th pt pt α z ( z) ε () = eq[ z(ln( z) ) y γ ε ε th pt pt γ α ( ) ε () = y eq (5-9) γ y Through fttng the data n the Table 5. 5, the parameter γ was dentfed to be: (see Fgure 5. 9). y = 7 ±.5MPa (5- ) γ

113 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models P (MPa) pt ε (%) Table 5. 5 Phase-transformaton nduced plastcty under stress loads (expermental results) Transformaton plastcty Stran SIG=37,5MPa SIG=5MPa SIG= MPa SIG=75MPa -.6 Appled pressure SIG=5,5MPa Temperature ( C) Fgure 5. 8 TRIP from the experment wth heatng and coolng loadng (only thermal and plastc stan heren, elastcty s moved). Identfcaton of transformaton plastcty.8 exp. data Lnéare (exp. data) y =.88x -.85 R =.9644 Plastcty (%) Loaded strength (MPa) Fgure 5. 9 Identfcaton of parameters of transformaton plastcty model Tong WU

114 Thess: Experment and Numercal Smulaton of Weldng Induced amage Identfcaton of parameters of damage mdoel The damage model (Lematre model and the proposed model n Chapter 4) requres several parameters to be dentfed [4][8]. These constants and ther explanaton are as follows: : ntal damage. The ntal damage s dffcult to determne because t s strctly related to the nclusons dstrbuton n the vrgn materal mcrostructure. A scannng electron mcroscope (SEM) nvestgaton can provde an dea of the ntal partcle dstrbuton and amount of damage n the straned area. As usual, ths parameter could be assumed to be zero for a vrgn materal or at the begnnng stage of the damage calculaton. For example, we can measure the damage area n Fgure 5. 3a. A rough value of ntal damage s estmated to be:.. p : threshold damage plastc stran. Plastcty damage starts to occur only after a specfc stran level. Below ths stran level the materal mcrostructure behaves as a contnuum. Mcrovods can nucleate ether by debondng of the ncluded partcles from the ductle matrx or by partcle breakng. p, R : rupture plastc stran and crtcal damage. When ths crtcal stran s reached, falure c occurs. p R s drectly related to the crtcal amount of damage that crtcally reduces the load carryng capablty of the effectve resstng secton. In theory, when falure occurs the crtcal damage varable c should be equal to. In fact, Expermental observatons of almost metals show that falure occurs before =. κ : damage exponent. Ths constant carres nformaton gudng the type of damage evoluton and can be determned from tensle tests. The value gves the degree of nonlnearty of the damage evoluton law. For a gven p,, and, a dscrmnatng κ value exsts whch pr determnes the convexty of the damage evoluton as a functon of plastc stran. If s assumed to be equal to, Equ.4-5 wrtes: c = ( p R c p ) κ p p κ (5- ) The exponent κ s determned as the slope of the best fttng lne of the expermental damage measurements gven by: ( ) ln( ) p p ln = c + κ ln (5- ) pr p And then we have: wth ( ) = ln ( p p ) C ln κ (5-3)

115 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models ( p p ) ln( ) C = κ ln (5-4) R c where the constant C s the ntersecton of the fttng lne wth the ordnate axs. Usually the falure stran and crtcal damage can be measured pretty well, but the threshold stran easy to determne as a result of the expermental scatter. p s not always so Once the slope s found, t s possble to have a good estmaton of the form of prevous expresson as c [ ( p p ) C] = exp κ ln (5-5) R Through fttng the expermental results, t s not dffcult to dentfy the parameters of damage model (see Fgure 5. ). Snce the parameters are calbrated (Table 5. 6), the evolutons of damage provded by consttutve equatons are obtaned. For example, Fgure 5. dsplays the comparson between the fttng curve and expermental results of damage evoluton n martenste state at C. Fgure 5. shows the materal s more dffcult to be damaged (damage evoluton curves lower) whle temperature ncreases. Comparng damage evoluton curves of P (maxmum temperature: C), P8(maxmum temperature: 5 C), P9(maxmum temperature: 86 C) at room temperature, hgher maxmum temperature experenced by materal make t to have better ductlty. It may be explaned by the change of materal characterstc: there reman small quanttes of austente after martenstc transformaton n P8 and P9, the hgher maxmum temperature leads gran sze coarsenng of austente. amage evoluton laws strongly depend on temperature hstory f the transformaton temperature s exceeded. Fgure 5. gves the damage fttng of P and the fttng of other specmen can be found n Appendx B. The damage parameters dentfed by experment of round bar are lmted n the martenste state. In the tests of austente, the decrease of Young s modulus s not observed wthn regon of % stran. The measurements of damage over 6 C by cyclcal load-unload tests becomes very dffcult because the very hgh temperature enhance the ductlty of the materal and also because the occurrence of sgnfcatve vscous behavour. Temperature p p R c C κ P, C P, C P3, 6 C P8, C P9, C Table 5. 6 Identfed damage parameters of 5-5PH (martenstc phase) Tong WU 3

116 Thess: Experment and Numercal Smulaton of Weldng Induced amage amage P exp. P fttng P exp. P fttng P3 exp. P3 fttng P6 exp. P8 fttng P9 exp. P9 fttng amage evoluton of martenste Plastc stran (mm/mm) Fgure 5. Fttng of damage evoluton at martenstc state, at C (P, P8, P9), C (P), 6 C (P3) -.6 amage fttng of P Ln () y =.455x -.46 R =.9638 κ =.455 C = Ln(p p ) Ln (EPS-EPS) Fgure 5. Identfcaton of damage exponent of damage model (P) 4

117 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models Comparatve analyss of flat notched specmen The evolutons of stran localzaton on surface wth speckle pattern of flat notched specmen (FNS) are obtaned by dgtal mage correlaton of tensle experments of FNS. Three cases (Case A, B, C) wth dfferent notches and fve thermal condtons are analyzed n order to compare study the stran on surface of FNS. Fgure 5. 6a and Fgure 5. 3a show that the notch radus has a large nfluence onto the axal stran (EPYY) along lne MN (md secton, see Fgure 5. 7a). However, t has less nfluence on transversal stran (EPXX) (Fgure 5. 6a). Largest stran always les n edge of notch what ever ts radus. Fgure 5. 6b and Fgure 5. 3b ndcate that materal state nfluences dstrbuton of stran on surface of specmen because dfferent phases have dfferent ductlty. Temperature does not seem to have an mportant effect on stran dstrbuton. Along lne PQ (axal lne, see Fgure 5. 7a), largest stran occurs near md plane of the specmen whereas t s near edge of notch along lne MN. Fgure 5. 8a shows that radus of notch has a small nfluence on the overall tensle curve (dsplacement vs. force). However, temperature and materal phase state play an mportant role n tensle test (Fgure 5. 8b). In Fgure 5. 9, the dsplacement s measured between pont M and pont N and t s ncreasng wth the force loadng. Stran comparng of Case A,B,C at C.5.4 Stran Cas e A, EPXX Cas e A, EPYY Case B, EPXX Case B, EPYY Case C, EPXX Case C, EPYY X (m m ) a) Case A, B, C at C Stran comparng of Case B at dfferent temperatures.5 Stran EPXX, C, M, C EPYY, C, M, C EPXX, C, M, C3 EPYY, C, M, C3 EPXX, C, M, C EPYY, C, M, C EPXX, C, MIX, C4 EPYY, C, MIX, C4 EPXX, 3 C, A, C5 EPYY, 3 C, A, C X (mm ) b) Case B at,, 3 C Fgure 5. Comparson of stran dstrbuton on lne MN on surface when dsplacement s equal to.5 mm (d=.5mm) Tong WU 5

118 Thess: Experment and Numercal Smulaton of Weldng Induced amage Stran (EPYY) comparng of Case A,B,C, at C Case A Case B Case C Stran Y (mm) a) Case A, B, C at C Stran (EPYY) comparng of Case B at dfferent temperatures.3.5. C, M, F C, M, F3 C, M, F C, MIX, F4 3 C, A, F5 Stran Y (mm) b) Case B at,, 3 C Fgure 5. 3 Comparng stran dstrbuton (EPYY) on lne PQ on surface when dsplacement s equal to.5 mm (d=.5mm) sp-force of Case A, B,C at C Case A Case B Case C 3 Force (KN) splacement (mm) a) Case A, B, C at C 6

119 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models sp-force of Case B at dfferent temperatures Force (KN) C, M, F C, M, F3 C, M, F C, MIX, F4 3 C, A, F splacement (mm) b) Case B at,, 3 C Fgure 5. 4 splacement vs. force of tensle tests of FNS Tensle curves at C Case A Case B Case C 3 Force (NK) splacement MN (mm) Fgure 5. 5 splacement (between pont M and pont N, md secton) vs. force of tensle tests of FNS at C Fgure 5. 6 shows that thckness reductons are qute dfferent n varous flat specmens dependng on the notch rad: the maxmum thckness reducton s near the notch edges for Case A, somewhere nsde for case B and at the center for case C. Fgure 5. 7 shows that the prncpal strans (n longtudnal drecton, EPYY) of three cases have the smlar dstrbuton along lne MN: peak value of stran always occurs at the notch edges. The reducton of thckness could be used as rough damage evoluton estmaton, and we could consder that maxmum damage les n the place where mnmum secton appears. Therefore, the concluson resultng from the comparson of Fgure 5. 6 and Fgure 5. 7 s that the largest stran s not at the same locaton as maxmum thckness reductons. The strans are rather large near falure (>%). The measured strans are manly elastoplastc because of ncompressblty. It s an ndcaton that damage s developng. The elastoplastc computatons of Case A to Case C show that the peak values of traxal stress rato move from edges to center, thus s n agreement wth the observatons mentoned above. The expermental results leaded to the concluson that the stress traxalty plays a domnatng role n the evoluton of damage. Tong WU 7

120 Thess: Experment and Numercal Smulaton of Weldng Induced amage a) Case A R= mm (FA) b) Case B R=.5 mm (FB) c) Case C R=4 mm (FC) Fgure dstrbuton of vertcal dsplacement (UZ) on notched samples (FA, FB, and FC)when loadng dsplacement s equal to.5 mm (d =.5 mm) n unaxal tensle test at room temperature by usng dgtal mage correlaton. UZ EPYY % a) Case A R= mm (FA) b) Case B R=.5 mm (FB) c) Case C R=4 mm (FC) Fgure 5. 7 strbuton of longtudnal stran (EPYY) on notched samples (FA, FB, and FC) when loadng dsplacement s equal to.5 mm (d =.5 mm) n unaxal tensle test at room temperature by dgtal mage correlaton. Fgure 5. 8 and Fgure 5. 9 show that the dsplacement and stran are smlarly dstrbuted on surface of FNS at dfferent temperatures because the notch geometry decdes stran gradents from the specmen center to the notch edge. However, there s a quanttatve dfference: dsplacement Z or stran EPYY at martenstc phase state (Fgure 5. 8a, b and Fgure 5. 9a, b) have larger level than ther n austentc phase (Fgure 5. 8c and Fgure 5. 9c) or at mxture state (Fgure 5. 8d and Fgure 5. 9d). In other words, severe neckng can be observed n martenstc state, but not n austentc state when the dsplacement s loaded up to.5 mllmeters. 8

121 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models UZ a) C, FB, M b) C, FB, M c) 3 C, F5B, A d) C, F4B, Mx Fgure dstrbuton of dsplacement of Case B wth dfferent temperature hstores (FB, FB, F5B and F4B) when loadng dsplacement s equal to.5 mm (d =.5 mm) by usng dgtal mage correlaton. (A s austente; M means martenste; Mx ndcates n mxed phase state.) EPYY % a) C, FB, M b) C, FB, M c) 3 C, F5B, A d) C, F4B, Mx Fgure 5. 9 strbuton of stran n longtude drecton (EPYY) of Case B wth dfferent temperature hstores (FB, FB, F5B and F4B) when loadng dsplacement s equal to.5 mm (d =.5 mm) by usng dgtal mage correlaton. a) Case A (FA) b) Case B (FB) c) Case C (FC) Fgure 5. 3 Macroscopc fracture observaton of FNS wth varous notches at room temperature at the moment of rupture (Case A, Case B and Case C) Tong WU 9

122 Thess: Experment and Numercal Smulaton of Weldng Induced amage The mages at moment of rupture are traced by dgtal camera (see Fgure 5. 3). Fgure 5. 3c shows the ntal crack occurs at center and then extends to both edges for Case C. Ths s n good agreement wth the dstrbuton of maxmum damage level of Case C. For Case A and Case B, the ntal crack could not be captured because the specmens break nto two parts too quckly. 5.8 Mcrostructure characterzaton of 5-5PH Mcroscopc observaton s performed due to study damage mechansm at mcro scale and to provde an ndcaton for the choce of an approprate damage models. Mcrostructure characterzaton can be qute dfferent due to the varous methods of mechancal process as well as the dfferences of thermal hstores appled. In ths secton, characterzaton of 5-PH martenstc stanless at room temperature wll be dscussed n detal. a) Optcal mage of topographc observaton X Y um b) Optcal mcro observaton of rupture zone

123 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models y x μm c) SEM mage of rupture zone Smooth undulatng surface mplng surface d) SEM mage of near rupture zone Fgure 5. 3 Mcro observaton of 5-5PH by optcal mcroscopy and scannng electron mcroscopy (SEM) Fgure 5. 3a gves a topographc observaton of 5-5PH before tensle tests are performed. It shows that the mcrostructure s n form of lathng martenste at room temperature and dameter of gran szes are between and mcron. Fgure 5. 3b and Fgure 5. 3c show the fracture surface (longtudnal cross-secton) of RB after tensle test, and the rupture zone wth delamnaton are respectvely observed by usng optcal mcroscopy and SEM. Fgure 5. 3b show the edge of rupture presents sawtooth-lke shape. The fracture s torn not at conventonal 45 but at axal drecton, and then extended to.5 mm deep. A typcal broken secton wth characterstc dmensons s presented n Fgure 5. 3b, X s the length of horzontal segment, perpendcular to the strength drecton. The length of vertcal segment parallel to the strength loadng s denoted by Y. They were measured from the Fgure 5. 3b: X = 6um, Y = 3um and X / Y =. In order to construct a model of mesomechansm, t s mportant to know the dstrbuton of and Y all along the entre cross-secton of a target, and provdes element dmensons of mesh of mesoscopc model. X x and n Fgure 5. 3c y Tong WU

124 Thess: Experment and Numercal Smulaton of Weldng Induced amage can be a reference for dmensons of mcroscopc model. For example, heren has x = um, y = um and x / y =. 55. Compared the data of Fgure 5. 3b, t s found that the effect of dmenson s obvous: X / Y > x / y,.e. the fractography s dfferent between at mesoscale and at mcroscale. In fact, t s necessary to do statstcal analyss of all segments along the broken secton n order to get an average value. Vods or cavtes wth dameter of - mcrons can be observed near the edge of rupture n Fgure 5. 3c. The hgh magnfcaton vew of the fracture surface n Fgure 5. 3d clearly shows the smooth delamnaton surface as well as the dmpled ductle fracture area. The smooth undulatng surface suggests some type of decoheson of the gran boundares. Mcro cracks can be observed n the smooth delamnaton zone. Mcrovods have hgher densty and larger dmensons near the broken secton accordng to hgher plastc stran. a) Plastc stran p =, 6 X b) Plastc stran p = %, 6 X

125 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models c) Plastc stran p = %, 6 X d) Plastc stran p = %, 3 X Fgure 5. 3 Scannng electron mcroscope (SEM) mages of 5-5PH specmens when varous strans are loaded by usng SEM at room temperature (thermal condton C). Fgure d) s a zoom of the square area marked n Fgure c). The crcle shows mcrovods observed and the crcle shows a mcrocrack. Fgure 5. 3 shows mcrostructures of 5-5PH under varous deformatons. Before deformaton s performed (Fgure 5. 3a), the materal mcrostructure s qute smooth except a few mcrovods and mcrocracks that can be regarded as ntal damage. Wth the loadng of deformaton, grans are twsted and surface of mcrostructure are n concavo-convex shape to some extent (Fgure 5. 3b). Wth hgher deformaton, dmples can be observed n the surface n Fgure 5. 3 and dmpled ductle damage happen. Fgure 5. 3d shows a larger vod wth coalescence because of the applcaton of hgher magnfcaton. Tong WU 3

126 Thess: Experment and Numercal Smulaton of Weldng Induced amage The expermental observaton of mcrostructure provdes basc nformaton of damage mechansm under tensle condton. Ths nvestgaton helps to choose or develop a damage model n order to smulate damage and resdual stress. The typcal mcrovods and mcrocracks, whch were observed n mcroscopc experment, are basc concept of damage Kachanov n RVE. The observed dstrbuton and evoluton of mcrovods verfy the ductle damage nduced by plastc stran, whch s used n Lematre & Chaboche model. 5.9 Concluson In ths Chapter, we dscussed the dentfcaton of the parameters of damage model, macro cracks of tensle tests and mcrostructure of 5-5PH. In addton, mechancal propertes at hgh temperature or wth dfferent temperature hstores were ganed from tests of RB. The followng conclusons can be drawn based on the nterpretatons presented n ths paper: (). amage varables can be measured by varaton of Young s modulus resultng from crcularly load-unloadng tests of round bar. The careful processng of data leads to the dentfcaton of the parameters of damage models, whch we have chosen (Lematre s model and Borona s model). The processed results ndcate that 5-5PH n martenstc state at low temperature are easy to damage. When the same specmen has been heated a hgh temperature the damage rate s less mportant. In a seres of tests, the damage n martenstc state of 5-5PH can be observed and measured whereas the damage n austentc state are not obvous wthn the appled plastc stran less than%. Ths mples the austente have large value of the threshold stran to cause damage. However, ths does not mean the threshold stran s over %. A SEM nvestgaton should fll n the absence of damage parameters of austente phase. It s reasonable to dsregard the damage n austente phase n smulaton of multphase n lmted deformaton condton, typcal of weldng stuaton. (). Expanson-contracton tests wth appled stress allowed successful dentfcaton of phase transformaton (Kostnen and Marburger) and fttng of TRIP (Leblond model). (3). amage and rupture n macroscopc scale are observed n the tensle tests of FNS. The stran localzaton s analyzed usng dgtal mage correlaton. The results show that largest damage does not always occur n the place of the largest stran because damage s a compettve result between plastcty and traxal stress rato. (4). Comparson of tests wth varous temperature hstores shows that the mechancal behavors (e.g. stran-stress curve) ncludng damage are affected by hstores of thermal loadng. The damages are strongly affected by hgh temperature. Therefore, the effect of temperature hstory should be taken nto account n modelng. (5). Gran sze, rupture zone, mcrovods and mcrocracks are characterzed n the mcroscopc measurement. Mcrovods grow wth ncreasng of deformaton. mples are observed to be the domnant features and were formed as a result of the ntaton and coalescence of 4

127 Chapter 5: Expermental Study and Identfcaton of amage and Phase Transformaton Models mcrovods. Thus provdes a justfcaton of the choce of Lematre and Borona damage models assocated to mcrovods and mcrocracks based on contnuum damage mechancs. The nformaton s also necessary to understand better the damage behavors at macroscale. Snce the parameters of phase transformaton, transformaton plastcty, and damage models are successfully calbrated through expermental methods, the numercal smulaton can be mplemented n next chapter. Tong WU 5

128 Thess: Experment and Numercal Smulaton of Weldng Induced amage 6

129 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons 6 Chapter 6 Numercal Smulaton and Implementaton of Consttutve Equatons 6. Introducton In ths chapter, we ntroduce numercal analyss of structures under thermo mechancal loadng wth damage and phase transformaton. On the one hand, the numercal calculaton of notched specmen s done and compared wth the expermental results. On the other hand, an example of dsk weldng s gven n order to llustrate the partcularty of numercal smulaton of weldng through the two-scale model, whch s proposed n the prevous sectons. We focus our study on the metallurgy and damage n mechancal calculaton. It s supposed that the temperature feld s already calculated, and proportons of phase are calculated from temperature feld, usng standard methods of these metallurgcal computatons (Kostnen and Marburger s emprcal law, Leblond s model ). In addton, n order to apply to the weldng numercal predcton, the consttutve equatons have to be mplemented n a fnte element program. The proposed model s mplemented n computng software CEA CAST3M fnte element code, whch ntegrates not only the computng processes themselves but also the functons of constructon of the model (pre-processor) and the functons of processng of the results (post processng). The CAST3M s a proper numercal platform, whch provdes a large choce of mechancal models or laws. Tong WU 7

130 Thess: Experment and Numercal Smulaton of Weldng Induced amage 6. Metallurgcal calculaton 6.. Phase trasnformaton calculaton The calculaton of austente phase transformaton (austentzaton) can use the phenomenologcal model (Equ.-4) proposed by Leblond, whch was ntroduced n Secton.3.4. In the case of weldng smulaton of damage and stresses, we can use ts smple form at equlbrum state. For a slow heatng rate, there s enough tme to make the austente fracton to reach equlbrum state for each temperature, and the proporton of austente s approxmately lnear wth the temperature durng phase transformaton (see Fgure.8). For martenstc phase transformaton, Kostnen and Marburger s emprcal law (Equ.-3) can be appled to calculate the proporton of martenste. In the law, proporton of martenste s a functon of temperature and austente s proporton. z = z( e β < Ms T > ) (6- ) where s volume fracton of martenste and z represents volume fracton of austente; Ms s z martenste start temperature; β s coeffcent depend on materal; T represents temperature. In martenstc transformaton, on one hand, there s no dfferental equaton to ntegrate. On the other hand, t s necessary to take nto account the rreversblty of the transformaton,.e. martenste can not be reversed when temperature s below ( ' t z. Practcally, the Equ. 6- can be wrtten as: Ac t + t) = max( z ( t + t), z ( )) (6- ) wth [ exp( M T( t + ) )] z t + t) = z ( t + t) β s (6-3) ' ( t z t + t) = z ( ) (6-4) ( t In the martenstc phase computaton, the nput varables nclude: z ( ), z ( ) and T( t + t). The calculaton of metallurgcal transformaton s performed at Gauss pont of FEM mesh. t t Therefore, the metallurgcal calculaton can be summarzed n the followng: 8

131 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons Phase transformaton calculaton If ( T and T Ac ) or ( T and T Ac3 ) No phase transformaton If T and Ac 3 > T > Ac Austentzaton If ( T < and T > M s ) or ( T < and T < M f ) No phase transformaton If T < and T > M s Martenstc transformaton In addton, the parameters of phase transformaton of 5-5PH are gven n the Chapter Gran sze calculaton The growth of gran sze can be calculated through conventonal equaton proposed by Alberry, Ikawa, and Leblond (see Secton..4.). The followng flow chart summarses the method: Gran sze calculaton If z > d dt a ( G ) Q RT z z a = C exp( ) G (6-5) d a Q or ( zg ) = z C exp( ) (6-6) dt RT If z d dt ( G α ) = C exp( Q RT ) (6-7) where G : gran sze T : absolute temperature R : constant for gases Tong WU 9

132 Thess: Experment and Numercal Smulaton of Weldng Induced amage 4 a, C, Q are postve constants. (Generally, a = 4, C =.4948 mm 4 / sec, Q R = 639 gves satsfactory results.) The Equ. 6-5 can be wrtten as followng form: () d dt where t represents other varables except G. V ( G) = f [ G() t, V () t ] (6-8) The mplct dfferental equaton s ntegrated by Runge-Kutta method. The austente gran sze at tme t + dt s calculated by: G dt ( t dt) = G( t) + [ G + ( G + G ) + G ] (6-9) 6 wth G f [ G() t, V () t ] (6- ) = dt dt G = f G() t + G, V t + (6- ) dt dt G = 3 f G() t + G, V t + (6- ) G = f [ G() t + G dt, V ( t + dt) ] (6-3) Calculaton of transformaton plastcty In the case of numercal smulaton of resdual stresses and damages nduced by weldng, only martenstc transformaton plastcty s consdered because austente transformaton occurs at hgh temperature when the materal has more ductlty and less stress concentraton. The martenstc transformaton ( ) plastcty can be calculated by Leblond s model presented n Secton.6.. The smple expresson s wrtten as: Its ncremental form s: E pt eq 3 ε Σ S h ln y Σ Tref = y ( z ) z [.3, ] z (6-4) E E pt pt Tref eq 3 ε Σ ( t) = S( t) h y y Σ [ ln z ( t) ] [ z ( t + t) z ( ) t ] (6-5) pt pt ( t + t) = E ( t) + E ( t) (6-6)

133 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons T The experments presented n the Chapter 5 have gven the temperature-ndependent parameters: ε ref, eq y. The parameter h( Σ / Σ ), whch translates the non-lnearty of the transformaton plastcty y accordng to the equvalent macroscopc stress appled, has been ntroduced n Secton.6. and Secton.7. n detal. 6.3 Numercal mplementaton of the two-scale model 6.3. Introducton As far as the numercal analyss s concerned, n our smulaton of the two-scale model proposed n Chapter 4, the frst step s to compute the coupled transent temperature and metallurgcal phase feld. The second one s stress, stran and damage feld computaton and then the states of each phase (stresses, strans, damages, nternal varables...) are output. At the end, we can homogenze the strans and stresses, dsplacements and damages. The calculaton of temperature feld s not dscussed n our study, and the calculaton metallurgcal phase feld was gven n the prevous secton. Here, we focus our study on mechancal analyss n two-scale model Algorthm The algorthm n small dsplacements s gven n the followng part (Fgure 6. ). The model presented prevously relates only to the ntegraton of the consttutve law of multphase materal. If the ncremental theory s appled to solve the problems of multphase behavors, the ncrement of macroscopc stress Σ between step p and step p + can be calculated from the followng equatons. p Here, and represent respectvely martenste and austente as the prevous conventon. Σ( p) = Σ(p+ ) Σ(p) = z (p+ ) (p+ ) z = z =, = z =, = z =, (p+ ) (p+ (p+ ) H H H ε e (p+ ) (p+ ) (p+ ) (p+ ) =, z (p) =, H ε e (p) (p) (p) =, (p) e e [ ε ε ] + [ z H z H ] (p+ ) (p) =, (p+ ) (p+ ) (p) (p) ε e (p) tot th p [ ε ε ε ] + [ z H z H ] (p) (p) (p) =, (p+ ) (p+ ) = th z(p+ ) H(p+ ) E(p) z(p+ ) H(p+ ) ε(p) z =, =, =, + =, [ z H z H ] (p+ ) (p+ ) (p) (p) H (p+ ) (p) (p+ ) (p) H (p+ ) (p) ε ε e (p) p (p) The load ncrement s: = T pt T F p) B z(p+ ) H(p+ ) ( E(p) Ep+ ) B z =, =, ( (p+ ) H(p+ ) ε th (p) Tong WU

134 Thess: Experment and Numercal Smulaton of Weldng Induced amage T + ε p T B z(p+ ) H(p+ ) (p) B (z =, =, (p+ ) H (p+ ) z (p) H (p) )H (p) (p) where s defned by: B BΣ = tr[ ΣE(U )] dω Ω U T. Then, t can be wrtten as: wth K th e p pt ( p+ ) U = F (p) (p) + F(p) F(p) + F(p) + F(p) th = T F p) B z =, ( (p+ ) H(p+ ) e = T F p) B (z =, ε th (p) z ( (p+ ) H(p+ ) (p) H(p) ) H(p) (p) = T B = H, (p+ ) (z (p+ ) H (p) (p) z (p) H (p+ ) (p) pt T pt F( p) = B z(p+ ) H(p+ ) E(p) =, If the multphase s coupled wth damage, stffness matrx H s a functon of damage varable and can be wrtten as. H() In the two scale model, the partton of the strans at the mcroscopc level s preserved, and ntal transformaton plastcty s supposed to be null. Intalzaton: E tp ) th ( = ; ε () = ; ε = ε α th Tref () γ The ncrement of thermal stran of each phase : th ε(p) = α(p+ ) T) T(p + ) ( α (T) T The accumulaton of thermal stran of each phase : (p) (p) ε th (q) = ε th (p) + ε th p (p) ε tot = E E pt The accumulaton of thermal stress of each phase : th (p) = H ε th (p) (p) The mxture of thermal force: th T F ( p) = zb =, The resdual force: th (p) R + ext nt th ( p) = F(p) F(p) F(p)

135 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons The followng algorthm gves the evoluton of resdual balance between teraton j and teraton j +. If an equlbrum pont exsts and the steps are not too large, the resdual dsplacement can decrease untl they reach a very small number ξ through teraton. Calculaton of the ncrement and accumulaton of dsplacement: U ( j) = K (p) R th (p) U ( j) ( j) ( j) ( p+ ) = U(p) + U The lnkng force s: j) ( j) F ln( = K (p+ ) U(p+ ) The total stran: wth T K B z H B and ( p+ ) = (p+ ) (p+ ) U Σ= [ Σ ] Ω =, T B Ω tr E(U) d tot( j) ( j) E ( p+ ) = B U(p+ ) The total stran of each phase excludng transformaton plastcty: ε tot( j) tot( j) pt (p+ ) = E (p+ ) E(p+ ) The mechancal stran: ε mec( j) e( j) p( j) tot( j) th (p+ ) = ε(p+ ) + ε(p+ ) = ε(p+ ) ε(p+ ) The ncrement and accumulaton of stress of each phase: ( j) e( j) e( j) (p+ ) = H (p+ ) ε(p+ ) H(p+ ) ε(p+ ) j) j) j) [ ε ] [ ] mec( ε p( H ε mec( ε ( j) = H(p+ ) (p+ ) (p+ ) The nternal force: ( j) ( j) ( j) (p+ ) = (p+ ) + (p+ ) = Σ = (p) j) T ( j) T ( j) F nt( B B z (p+ ) =, ( j) ext( j) nt( j) R = F F + F ln( j) (p) p (p) TEST ( j) R ξ IF NOT U = ( j) ( j) ( p) K (p+ ) R U = + ( j+ ) ( j) ( j+ ) ( p) U(p) U(p) Tong WU 3

136 Thess: Experment and Numercal Smulaton of Weldng Induced amage IF YES Calculaton of the transformaton plastcty at next step: tp E (p) and E tp tp tp ( p+ ) = E(p) + E(p) Fnsh equlbrum teraton. Intalzaton Loadng Ud, Fd, T, z Stran ε th,e YES TRIP E pt NO Calculate E pt E pt = Localzatonn tot pt ε = E E Load step Component s behavors e ε... Homogenzaton = =, z No Equlbrum YES Next step Fgure 6. Flow chart of multphase calculaton 4

137 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons 6.4 Numercal verfcaton of the models Snce the consttutve equatons of model are gven n the Chapter 4, the numercal calculaton are mplemented to verfy the models and the calculated results are compared to expermental results, whch were accomplshed n Chapter 5. The parameters of model dentfed n Chapter 5 provde the basc nput data for the smulaton. All the calculatons are mplemented n Cast3M software Phase transformaton and transformaton plastcty verfcaton The frst case s devoted to calculate phase transformaton and transformaton plastcty n one element n 3- (QUA8, 8 nodes). The algorthms of phase transformaton and transformaton plastc models are respectvely gven n secton 6.. and secton It s supposed that the austentc transformaton s guded by lnear law as we mentoned n secton 6... We use the parameters of martenstc phase transformaton model (Kostnen and Marburger): Ms =, Ac= 7, Ac = 8; y th and the parameters of transformaton plastcty model (Leblond): γ = 7MPa, ε = 9.58E 3. The thermal loadng s lnear heatng and coolng wth tme, and the hghest temperature reaches C (see Fgure 6. ). α γ Fgure 6. Fracton of phase evoluton under temperature loadng The calculated phase fracton s gven n Fgure 6.. It s n a good agreement wth the expermental results n Chapter 5. The transformaton plastcty s calculated n the stuaton of varous appled stresses. Fgure 6. 3 gves the plastc stran nduced by transformaton. Total strans ncludng Tong WU 5

138 Thess: Experment and Numercal Smulaton of Weldng Induced amage transformaton plastcty n experment and smulaton are shown n Fgure The numercal results ft the expermental data. The expermental stran ncreases slower than the smulated from 6 C to 8 C durng heatng because the precptaton hardenng occurs n ths temperture regon and ths factor s not taken nto account. The martenste start temperature (Ms) n smulaton s also dfferent from that n experment. The expermental study shows the martenste start temperature s affected by the hghest temperature (see Secton 5.7..). In our calculaton, Ms s supposed a constant not a functon of hghest temperature for smplcty and also by lack of suffcent nformaton. Stran (X.E-3 mm/mm) Mpa 75. MPa 5. MPa 37.5 MPa MPa Temperature (deg_c) Fgure 6. 3 Transformaton plastcty.6 Transformaton plastcty..8 Stran SIG=37,5MPa SIG=5MPa SIG= MPa SIG=75MPa SIG=5,5MPa Temperature ( C) a) Calculated results b) Expermental results Fgure 6. 4 Total stran ncludng transformaton plastcty (calculated data vs. expermental data) 6

139 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons 6.4. Numercal verfcaton of flat notched bars In ths secton, the study s devoted to stran, traxal stress and damage of notched specmen. The frst example s the comparson of calculated results of three cases wth varous notch rad (Case A R=.mm, Case B R=.5mm, Case C R= 4.mm), whch are tensle tests at room temperature. And then these numercal results wll compare to expermental data (FA, FB, FC). Only one fourth of the flat specmen s taken nto account because of the symmetry n - calculaton (Fgure 6. 5). The mesh conssted of QUA4-type elements. The boundary condton and geometry are shown n Fgure In smulaton of traxal stress state and damage, the method of plane stress, fnte stran and dsplacement s preformed. In 3- calculaton, one eghth of whole specmen (flat part) s modelled wth 6 CUB8-type (8-node volume) elements. Ud Q 7 mm mm Y O N X 5 mm a) Case A n b) Case B n c) Case C n d) Case B n 3 Fgure 6. 5 Meshes of notched specmens Tong WU 7

140 Thess: Experment and Numercal Smulaton of Weldng Induced amage Snce there are expermental result of tensle test of notched specmens (CaseA )The traxal stress state of three cases at mnmum secton of specmens (lne ON) s dsplayed n Fgure It s shown numercal result of longtudnal stran (EPYY) s n a good agreement wth expermental data n the fgure: maxmum stran s at edge of notch, and the specmen wth large notch has flat stran dstrbuton at mnmum secton. Fgure 6. 7 gves stran EPYY at longtudnal secton (lne OQ) wth both expermental and smulated results. EPYY (mm/mm) CaseA cal. CaseB cal. CaseC cal. CaseA exp. CaseB exp. CaseC exp. EPYY at mnmum secton X (mm) Fgure 6. 6 Stran (EPYY) at mnmum secton (ON) when dsplacement loadng s equal to. mm EPYY (mm/mm) EPYY at longtudnal secton CaseA cal. CaseB cal. CaseC cal. CaseA exp. CaseB exp. CaseC exp Y (mm) Fgure 6. 7 Stran (EPYY) at longtudnal secton (OQ) when dsplacement loadng s equal to.mm Stress traxalty dstrbuton across the mnmum secton specmen does not only depend on appled dsplacement but also on notch radus. Fgure 6. 8 shows that the notch has mportant an nfluence the dstrbuton of traxal stress: the maxmum traxal stress of Case B and Case C s at centre whereas t 8

141 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons for Case A s near edge of notch. Stress traxalty ncreases slghtly wth the ncreasng of loaded dsplacement n Fgure In Fgure 6., the evoluton of the stress traxalty of Case B wth plastc stran s gven for three locatons: the centre core of the mnmum secton (O), the pont at edge of notch (N) and mddle pont between both (L). Ths curves n Fgure 6. show that stress traxalty n three ponts remans constant at frst because of elastc stage, and then ncrease wth the evoluton of plastcty n Case B and C, after then slghtly drops n correspondence wth the neckng of specmen. However, the stress traxalty at centre (O) s hardly nfluenced by plastc stran..7 Traxal stress state at mnmum secton Traxal stress state Case A Case B Case C X (mm) Fgure 6. 8 Traxal stresses at mnmum secton wth dfferent notches (Case A, B, C) when loaded dsplacement s equal to. mm..7.6 Traxal stress state at mnmum secton (CaseB) Traxal stress state d=. mm d=.5 mm d=. mm X (mm) Fgure 6. 9 Traxal stresses at mnmum secton of Case B wth dfferent appled dsplacement (d =.,.5,. mm) on specmen Tong WU 9

142 Thess: Experment and Numercal Smulaton of Weldng Induced amage Traxal stress rato Locaton N Locaton L.35 Locaton O splacement (mm) Fgure 6. Traxal stress state vs. loaded dsplacement of Case B at dfferent locatons (N, L, O) The damage evoluton across the mnmum secton has been analyzed n order to determne the frst ductle falure locaton amage smulaton n notched geometry has been obtaned analytcally from the consttutve equatons proposed n Chapter 4. The Equaton 4- shows the damage evoluton s a functon of traxal stress and plastc stran. In Fgure 6., damage contour map shows ductle falure evoluton across the mnmum secton. The ntal falure of Case A s near edge of notch whereas the maxmum damage of Case C s at the centre of specmen. Compared wth expermental data obtaned n Chapter 5, the locaton of ntal falure from smulated results s n a good agreement. The damage s manly nfluenced by hydraulc stress. Here, the competton of stress traxalty and plastc stran accumulaton, whch are controlled by the notch effect, determnes falure ntaton locaton (near the notch n Case A). Snce the ntal falure occurred, falure extends across the mnmum secton and fnally entre falure happens. Consequently, the flat notched specmen can be consdered an approprate geometry for the study of damage evoluton under multaxal stress state a) Case A b) Case B c) Case C Fgure 6. amage dstrbuton of Case A, B, C (R =,.5, 4 mm) 3

143 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons 6.5 Smulaton of a dsk heated by laser 6.5. Introducton The ntroducton of weldng jon was ntroduced n Secton 3.5. n detal. In order to smplfy the problem and focus on the metallurgcal and mechancal analyss, an example of dsk heated by laser wll be provded. Compared the weldng jont (see Fgure 3.4), the case of dsk s appled to smulate the HAZ and base metal, and the molten weld part s neglected (see Fgure 6. ). In the secton, we wll smulate the case of dsk heated by laser for purpose of understandng and analyzng the damages and resdual stresses produced durng a weldng operaton. Laser T T melt Ac3 Ac CGHAZ FGHAZ ICHAZ sk HAZ Base metal Fgure 6. Schematc dagram of dsk heated by laser and ts HAZ 5 In the case of smulaton, the nfluences of thermal, metallurgcal and mechancal aspects can be ncluded as: The nfluence of the thermal hstory on the mechancal hstory results smultaneously from varatons of the mechancal propertes (Young's modulus, yeld strength) wth the temperature and from thermal expansons or contractons. The nfluence of the metallurgcal hstory (nseparable from the thermal hstory) on the mechancal hstory results n four factors: ) the metallurgcal structure and the austentc gran sze (proporton of the phases) on the mechancal propertes; ) the expanson and contracton produced by the metallurgcal transformatons; 3) the transformaton plastcty; 4) the phenomenon of restorng nternal varables, such as damage, durng transformatons. The phenomenon of damage memory durng martenstc transformaton can be modelled through ntroducton of a memory coeffcentη. The memory effect s null f η = and full f η =. For ths dsk smulaton, the memory coeffcent can be consdered to be null. 5 Coarse graned heat affected zone (CGHAZ): T max >> Ac3. Fne graned heat affected zone (FGHAZ): T max s just above Ac3. Inter crtcal heat affected zone (ICHAZ): Ac < T max < Ac3. Tong WU 3

144 Thess: Experment and Numercal Smulaton of Weldng Induced amage Ths smulaton s performed n code Cast3M. The structure of smulaton program s shown n Fgure Frstly the temperature fled s generated and then we have gran sze and phase fracton felds. They are nput fles of man program, whch conssts of two subroutnes for calculaton of damage and TRIP. Fnally, the calculated results are transferred to post process. Gran sze TRIP Temperature Localzaton Man Homogenzaton Post process Phase fracton amage Fgure 6. 3 Structure of the program 6.5. Fnte elememt smulaton A dsk made of 5-5PH wth 6 mm n dameter and 5 mm n thckness, was heated at the centre by a spot laser for 7 seconds, then cooled by natural convecton for 73 seconds. The spot laser was modelled by a heat flux whose dstrbuton on the upper sde (see Fgure 6. 4). The lower and lateral sdes were subjected to free convecton. The coeffcent of exchange convecton s: = Wm C. The radaton coeffcent s expressed by = ε, where the emssvty: h c 8 ε =.7 and the Helmotz constant: = The ambent temperature s supposed to C. The mesh conssted of QUA4-type elements and the problem s consdered axsymmetrc (see Fgure 6. 5). h r Q (.E5 W/m) Heat Flux R (mm) Fgure 6. 4 Flux nput on upper surface of dsk 3

145 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons Fgure 6. 5 Mesh and dmensons of dsk Thermal and metallurgcal results The tme for heatng s 7 seconds and the coolng tme s untl to 8 seconds. Temperature evoluton at locaton PA, PB and P s shown n Fgure The temperature fled at the moment of heatng fnsh s observed n Fgure urng the heatng stage, the hghest temperature reaches to 5 C, and the temperature drops to 6 C at the end of coolng. Temperature Temperature ( C) PA PB P Tme (s) Fgure 6. 6 Temperature evoluton on surface of dsk (Locaton PA PB P ) Unte: deg_c Temperature Fgure 6. 7 Temperature feld at the end of heatng (7 s) The smulaton of phase transformaton was performed based on the calculaton method presented n Secton 6... The parameters of phase transformaton were dentfed n Secton The phase Tong WU 33

146 Thess: Experment and Numercal Smulaton of Weldng Induced amage proporton at the end of heatng stage s plotted n Fgure It shows that the phase at the center (CGHAZ and FGHAZ) s austente and t s mxture of martenste and austente at the ICHAZ. The calculaton of gran sze was mplemented accordng to the method presented n Secton 6... Fgure 6. 9 shows the gran sze dstrbuton n dsk at the end of coolng. One may observe a large gradent of gran sze n the heated zone. Austente proporton.5 Martenste proporton. Fgure 6. 8 Phase proporton at the end of heatng (7 s) Unte: mcro Gran sze 4 6 Fgure 6. 9 Gran sze at the end of coolng (8 s) Mechancal results The mechancal smulaton used the prevously calculated results: temperature, phase proporton, and gran sze. The mechancal calculaton s not coupled wth the temperature smulaton, and the calculated temperature fled s an nput date fle. Phase fracton s determned by thermal loadng not by mechancal loadng. Transformaton plastcty s coupled by mechancal condton. The tme dscretzaton can be dfferent between the thermometallurgcal and the mechancal calculatons. The materal propertes are nonlnear wth temperature, and expermental data at some specfc temperatures are nput n the program (see Chapter 5 for materal data). Other values between the specfc temperatures are nterpolated lnearly. 34

147 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons Varous calculatons were performed: the frst one appled macro elastoplastc model wth the mxng law of materal propertes (namely Calcul_). The second calculaton (Calcul_) used the meso elastoplastc model n whch each phase has ts own materal property, but here the transformaton plastcty was not taken nto account. Compared wth Calcul_, the Calcul_3 consdered the transformaton plastcty and appled two-scale model presented n Secton 4.5. The last smulaton (Calcul_4) used the two-scale damage model and ntroduced all factors: phase transformaton, transformaton plastcty, damage, gran sze. The detal condtons of calculaton are gven n Table 6.. All calculatons except the frst one took nto account the affect of gran sze. The used mechancal propertes of 5-5PH are gven n the appendx B. The damage parameters are gven n Secton The temperature fled, phase fracton and gran sze were calculated n Secton As far as the affect of gran sze s concerned, t was carred out through changng the yeld stress usng the Hall-Petch law (see Secton.4.). Ths effect s focused on the center of dsk assocated wth hghest temperature. The zone s not the regon of hgh stress. Therefore, for the case of dsk heated by laser, the nfluence of gran sze s lmted. In Calcul_4, the damage was calculated based on the proposed model proposed n Secton 4.3 and ts parameters were dentfed n Secton In the two-scale model, an mportant parameter should be noted: memory coeffcent η. It ndcates how nternal varables n mother phase are transferred to the daughter phase when phase transformaton occurs. In our calculaton, t s supposed to be zero because t s not easy to choose a specfc value. No. Hardenng Phase change Gran sze TRIP amage Calcul_ Knematc NO NO NO NO Calcul_ Knematc YES YES NO NO Calcul_3 Knematc + sotropc YES YES YES NO Calcul_4 Knematc + sotropc YES YES YES YES Table 6. Comparson of calculatonal condton of four smulatons Fgure 6. shows typcal deformaton (magnfcaton X) at the end of heatng and at the end of coolng: the hghest dsplacement s at the centre of dsk when t s at the end of heatng, whereas the hghest deformaton s at the zone (ICHAZ 6, see Secton 3.5.) where partal transformaton occurs at the end of coolng. The ICHAZ s also a regon where stresses concentrate. a) end of heatng (7 second) (X) b) end of coolng (8 second) (X) Fgure 6. eformed shape of dsk (Calcul_4) 6 Inter crtcal heat affected zone (ICHAZ): Ac < T max < Ac3. Tong WU 35

148 Thess: Experment and Numercal Smulaton of Weldng Induced amage Fgure 6. shows typcal plastc stran n radal drecton (EPRR) and the plastc stran n crcumferental drecton (EPTT) n macro scale. The dsk s stretched n both radal and crcumferental drecton at the centre of dsk, and s compressed near ICHAZ. The strans and stresses are concentrated n the HAZ. % -4. EPRR. a) radal stran b) crcumferental stran Fgure 6. Macro plastc stran dstrbuton of dsk at the end of coolng (Calcul_4) From Fgure 6. to Fgure 6. 5, typcal stresses on the upper surface of dsk are respectvely calculated n four smulaton (from Calcul_ to Calcul_4). Compared to the other three calculatons, the Calcul_ wth macro elastoplastc model has smooth stress dstrbuton on the surface and hgher maxmum stress (see Fgure 6. ). The meso model wth two phases not only changes the stress dstrbuton but also lower maxmum stress n radal and crcumferental drecton. When the transformaton plastcty s taken nto account (Calcul_3), the maxmum stresses are decreased to some extent. The peak stresses dstrbute smlarly on upper surface of dsk n the last three calculatons: the postve peak stress s at the ICHAZ, whereas the negatve les n FGHAZ near ICHAZ sde. Stress (MPa) Radal stress on the upper surface R (mm) Stress (MPa) Hoop stress on upper surface R (mm) Fgure 6. Resdual radal and hoop stress on the upper surface of dsk Macro elastoplastc model (Calcul_) The calculaton result shows that damage has nfluence on resdual stresses. Comparng Calcul_3 and Calcul_4, the damage reduces the maxmum radal stress of about percent, and 8 percent for crcumferental stress. 36

149 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons 8 Radal stress on the upper surface 8 Hoop stress on the upper surface Stress (MPa) Stress (MPa) R (mm) R (mm) Fgure 6. 3 Resdual radal stress and hoop stress on upper surface of dsk Meso elastoplastc model wthout TRIP (Calcul_) 6 Radal stress on the upper surface 8 Hoop stress on upper surface 4 6 Stress (MPa) Stress (MPa) R (mm) R (mm) Fgure 6. 4 Resdual radal stress and hoop stress on upper surface of dsk Two-scale elastoplastc model wth TRIP (Calcul_3) 6 Radal stress on the upper surface 6 Hoop stress on the upper surface 4 4 Stress (MPa) Stress (MPa) R (mm) R (mm) Fgure 6. 5 Resdual radal stress and hoop stress on upper surface of dsk Two-scale elastoplastc model wth TRIP and damage (Calcul_4) Tong WU 37

150 Thess: Experment and Numercal Smulaton of Weldng Induced amage Table 6. shows the comparson of mnmum and maxmum stress among four calculatons. It ndcated that the TRIP and damage decrease the peak stresses n radal and crcumferental drecton. The complete smulaton (Calcul_4) shows the resdual stress of dsk heat by laser n radal drecton reaches up 59 MPa and the peak stress n crcumferental drecton s up to 574 MPa. The damage was plotted at three representatve moments: at 5 seconds when austentc transformaton does not happen (see Fgure 6. 6), at seconds when austente occurs at the centre of dsk (Fgure 6. 7), and at 8 seconds at whch the martenstc transformaton s completed (Fgure 6. 8). The damage has happened before austente occurred because of heatng nduced expanson at centre and restrcton of surroundng cold metal. Even though the value of damage s very small, t ndcated damage occurs frstly at centre. However, the damage was not observed at the centre n Fgure 6. 7 because the central metal was changed nto austente and damage dd not nhert from martenste due to null memory coeffcent η. At that moment, damage appeared on the boundary between martenste and austente. When martenstc phase transformaton was completed wth the decreasng of temperature, damage occurred and accumulated n the whole heat affected zone. No. Model max r mn r max θ mn θ Calcul_ macro Calcul_ meso elastoplastc Calcul_3 scale + TRIP Calcul_4 scale + TRIP + damage Table 6. Comparson of stress on the upper surface of dsk (MPa). 5 s. Fgure 6. 6 amage dstrbuton of dsk at 5 second 38

151 Chapter 6: Numercal Smulaton and Implementaton of Consttutve Equatons. s.4 Fgure 6. 7 amage dstrbuton of dsk at second.7 8 s.4 Fgure 6. 8 amage dstrbuton of dsk at 8 second (end of coolng) 6.6 Concluson Numercal calculaton of the model presented n Chapter 5 was mplemented n software Cast3M n the chapter. Numercal verfcaton of flat notched bars showed a good agreement wth the expermental results n term of stran and damage on the surface. The example of dsk heated by laser was used to smulate the HAZ of weldng. In metallurgc aspect, the Kostnen and Marburger s emprcal law whch does couple mechancal behavour has a good agreement wth the expermental results. Transformaton plastcty behavour guded by Leblond s model, whch couples wth stresses, was quantfed and ntroduced n the calculaton. As plastcty, TRIP nfluenced the damage to some extent and affected resdual stresses. Gran sze has the lmted nfluence resdual stress and damage n the case of heated dsk. Tong WU 39

152 Thess: Experment and Numercal Smulaton of Weldng Induced amage As far as the mechancal calculaton s concerned, the proposed two-scale model brngs the freedom of choosng each materal law. The model supports to trace hstory of each phase s behavours (damage, stress, stran ). In the case of dsk, damage n the martenste decreased about percent of peak resdual stresses on the upper surface. 4

153 Chapter 7: Concluson and Perspectves Chapter 7 Conclusons and Perspectves The smulaton of thermomechancal problems wth phase transformaton s a formdable problem whch many research teams dedcated to. When the thermomechancal problems couple damage, the problems become more complcated. The dffculty les not only n the couplng between phase transformaton (almost accompanyng transformaton plastcty) and mechancs but also n that damage develops n the multphase materals. Our contrbuton was devoted to three partcular ponts. The frst objectve was to develop proper consttutve equatons that can correctly predct damage and resdual stresses nduced by severe thermal or mechancal loadng. The consttutve equatons ncluded phase transformaton, transformaton plastcty and damage models. The exsted phase transformaton (Kostnen and Marburger law) and transformaton plastcty models (Leblond s model) whch we chosen were satsfed n the smulaton of materal 5-5PH. A damage equaton was proposed based on the Lematre s ductle damage model for the purpose of extendng applcaton and better fttng our studed materal. The consttutve equatons used a two-scale structure whch provdes a freedom to choose materal law for each phase and where all the phases are represented wth ther own behavour. The second am was to provde an expermental database ncludng thermal, metallurgc and mechancal propertes of the stanless steel 5-5PH. The dentfcaton of phase transformaton model, transformaton plastcty model and damage model, besdes the characterstcs of materal, were performed n the laboratory LaMCoS of INSA n France. The partcular contrbuton was to study damage s mechansm, evoluton and characterzaton. Tensle tests of round specmens were used to dentfy the parameters of damage model as well as mechancal behavours at varous temperatures. Tests of flat notched specmen were desgned to provde the verfcaton of damage model and stran localzaton due to the fact that varous notch rad nduce dfferent traxal stress rato profles n the Tong WU 4

154 Thess: Experment and Numercal Smulaton of Weldng Induced amage md secton. The tests of flat notched specmen appled three dmensonal mage correlaton technologes to trace the stran localzaton and evoluton on surface of specmen. In our expermental study, our tests about phase transformaton and damage were focused on the proportonal loadng n one dmenson and dd not take non proportonal loadng nto account. Many test results showed varable and ratonal, even though the damage parameters at hgh temperature are less accuracy because the measurement of Young s module was dffcult at over 8 C. The expermental database of 5-5PH provded the necessary parameters for the numercal calculaton of resdual stresses ncludng phase transformaton, transformaton plastcty and damage. In addton, the mcroscopc experment (XR, TEM and SEM) were performed to provde mcrostructure characterzaton of 5-5PH and dscover the damage mechansm. The thrd objectve was to mplement numercal calculaton n the code Cast3M. On the one hand, numercal verfcaton of the flat notched bars was performed to compare wth expermental results. The results showed the numercal result has a good agreement wth the expermental data. On the other hand, we used the two-scale model ncludng phase transformaton, transformaton plastcty and damage to smulate the level of resdual stresses of the dsk made of metal 5-5PH heated by laser. The nternal varables, such as stran, stress, damage, were successfully traced n the smulaton of two-scale model. The smulaton results showed the transformaton plastcty changes the level of resdual stresses and s not neglgble; damage decreases about 8 percent peak value of resdual stresses on upper surface of dsk. Even though the nfluence of gran sze s obvous n austentc steel such as 36L, t s small n our case because the coarsenng of austente s concentrated at the centre of dsk where the stress s not hgh. Furthermore, the relatonshp between austente gran sze and the formaton of martenste s not clear. The thermomechancal problems couplng wth damage n the phase-change metal are really dffcult. Our expermental study leaves some problems to be further studed n the future. As far as the damage model and smulaton are concerned, there are also stll some problems whch are open. In expermental aspect: The mcroscopc test of specmen durng phase transformaton s too dffcult to realze because the tme of phase transformaton s extremely short. The mcroscopc observaton at hgh temperature s also a problem because of the oxdaton on the surface of specmen. It s verfed that the martenste start temperature (Ms) s nfluenced by maxmum temperature reached. The accurate relatonshp between both needs more experments to explore t. When we use two cameras to trace the evoluton of tracton of notched specmen, the mage correlaton provdes three dmensonal dsplacement of surface but lacks of stran n thckness drecton because the montorng on only one sde surface can not provdes the thckness data of specmen. Seeng that the three dmensonal stran s mportant to study stran localzaton 4

155 Chapter 7: Concluson and Perspectves and damage, a possble soluton s to use four cameras (settng two on each sde of specmen) to trace both sdes and then generate strans n three drecton through mage correlaton. In modelng aspect: Is there an nteracton between damage n austente and damage n martenste? If there s, how do they affect wth each other? o gran sze and transformaton plastcty nfluence damage or vce versa? oes the memory effect exst durng martenste and austente transfer to each other (memory coeffcentη )? It was known the precptaton s an mportant phenomenon n stanless steel 5-5PH, t seems more reasonable to take the behavour nto our modellng. In the smulaton of resdual stress, another mportant factor s hydrogen dffuson whch may be mportant for damage and crack. In numercal aspect: The model wll be appled to calculate complcated structure or weld jont. Tong WU 43

156 Thess: Experment and Numercal Smulaton of Weldng Induced amage 44

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168 Thess: Experment and Numercal Smulaton of Weldng Induced amage 56

169 Appendx A: Expermental evces Appendx A Expermental evces A.Stran and stress measurements LVT a ) Fgure F A force applquée par un ressort F B pvots L capteur nductf b) agram Fgure A. Extensometer The lnear varable dfferental transformer (LVT) s a type of electrcal transformer used for measurng lnear dsplacement. The transformer has three solenodal cols placed end-to-end around a tube. The centre col s the prmary, and the two outer cols are the secondares. A cylndrcal ferromagnetc core, attached to the object whose poston s to be measured, sldes along the axs of the tube. Tong WU 57

170 Thess: Experment and Numercal Smulaton of Weldng Induced amage Fgure A. LVT SYNOPTIQUE GENERALE U ISPOSITIF ENTREE à 6 voes Amplfcateur Condtonneur Centralp MC33 Modulateur PCM TM 6 K 3 SM Johne + Relhofer Fltres Ant-retours Echantllonneur Bloqueur Multplexeur Convertsseur Analogque gtal Module PCM emultplexage 6 bts parallèles maxmum tratements des données mpresson mse en memore Fgure A. 3 Schematcs of acquston bât d'essas capteur de force servo-valve capteurs de déplacement Armore de régulaton servo-vérn Coffret hydraulque bac à hule M almentaton Centrale hydraulque Fgure A. 4 agram of expermental setup of tensle test 58

171 Appendx A: Expermental evces A. Mcroscopc equpments Fgure A. 5 Optcal mcroscopy (ZEISS AXPHOT) The optcal mcroscopy was used to observer topologc mage of mcrostructures (gran sze, gran boundary) and crack regon of stretched round bar. The specmen was prepared by standard methods nvolvng mechancal grndng (ground usng emery papers of varous grnds from to 5 grt), polshng (slcon grt from 6µm to µm) and vbraton-polshed. Fgure A. 6 Bruker 8 X-ray dffractometer X-Ray ffractometer (XR) was appled n order to quanttatvely assess the phase fractons of specmen through smultaneous X-ray dffracton lne profle analyss of the FCC and BCC. Tong WU 59

172 Thess: Experment and Numercal Smulaton of Weldng Induced amage Fgure A. 7 TEM JEOL FEG Transmsson Electron Mcroscope (TEM) was mplemented to study the mcrostructure of the 5-5PH n the as-receved condton, and analyze the structure of the precptates. The characterstcs of the TEM JEOL FEG s as followng: Acceleratng voltage kv, feld emsson source, pont resoluton. Å, useful lmt of nformaton ~. Å, maxmum specmen tlt capacty +/-3 degrees, absolute mnmum probe sze ~.4 nm, ES spatal resoluton ~ nm, mnmum energy spread ~.65 ev. Fgure A. 8 SEM JEOL 84A LGS Scannng Electron Mcroscope (SEM) was carred out for purpose of observng the mcroscopc characterstcs ncludng mcrovods and mcrocracks. The specmen was prepared wth the same procedure of the optcal electron mcroscopy observaton: mechancal grndng (ground usng emery papers of varous grnds from to 5 grt), polshng (slcon grt from 6µm to µm) and vbraton-polshed for observng surface. 6

173 Appendx B: Expermental Results of 5-5PH n etals Appendx B Expermental Results of 5-5PH n etals B. Test results of round bar B.. Materal propertes Martenste Austente Temperature ( C) ALFA ( -6 / C] Table B. Expanson coeffcents Temperature ( C) E (GPa) Martenste Austente Table B. Young's modulus Temperature ( C) Nonlnear SIG_Y (MPa) Martenste.% SIG_Y (MPa) Ultmate SIG (MPa) Nonlnear SIG_Y (MPa) Austente.% SIG_Y (MPa) Ultmate SIG (MPa) Table B. 3 Yeld strength and ultmate strength Tong WU 6

174 Thess: Experment and Numercal Smulaton of Weldng Induced amage B.. Force vs. dsplacement 9 P Martenste C Force (KN) splacement (mm) 7 Fgure B. Force vs. dsplacement of P at C, martenstc state 8 P Martenste C 7 6 Force (KN) splacement (mm) Fgure B. Force vs. dsplacement of P at C, martenstc state 5 P3 Martenste 6 C 4 3 Force (KN) splacemnt (mm) Fgure B. 3 Force vs. dsplacement of P3 at 6 C, martenstc state 6

175 Appendx B: Expermental Results of 5-5PH n etals 5 P4 Martenste 7 C 5 Force (KN) splacement (mm) Fgure B. 4 Force vs. dsplacement of P4 at 7 C, martenstc state 4 P5 Austente 85 C Force (KN) splacement (mm) Fgure B. 5 Force vs. dsplacement of P5 at 85 C, austentc state 3 P6 Austente 6 C 5 5 Force (N) splacement (mm) Fgure B. 6 Force vs. dsplacement of P6 at 6 C, austentc state Tong WU 63

176 Thess: Experment and Numercal Smulaton of Weldng Induced amage 4 P7 Austente C 35 3 Force (KN) splacement (mm) Fgure B. 7 Force vs. dsplacement of P7 at C, austentc state 9 P8 Martenste C Force (KN) splacement (mm) 6 Fgure B. 8 Force vs. dsplacement of P8 at C, martenstc state P9 Martenste C Force (KN) splacement (mm) Fgure B. 9 Force vs. dsplacement of P9 at C, martenstc state 64

177 Appendx B: Expermental Results of 5-5PH n etals P Austente 3 C Force (KN) splacement (mm) Fgure B. Force vs. dsplacement of P at 3 C, martenstc state 4 P Austente 6 C 35 3 Force (KN) splacement (mm) Fgure B. Force vs. dsplacement of P at 6 C, martenstc state B.. Stress vs. stran 4 P Martenste C Stress (MPa) Stran (mm/mm) Fgure B. Stress vs. stran of P at C, martenstc state Tong WU 65

178 Thess: Experment and Numercal Smulaton of Weldng Induced amage P Martenste C 8 Stress (MPa) Stran (mm/mm) Fgure B. 3 Stress vs. stran of P at C, martenstc state 6 P3 Martenste 6 C 5 4 Stress (MPa) Stran (mm/mm) Fgure B. 4 Stress vs. stran of P3 at 6 C, martenstc state 35 P4 Martenste 7 C 3 5 Stress (MPa) Stran (mm/mm) Fgure B. 5 Stress vs. stran of P4 at 7 C, martenstc state 66

179 Appendx B: Expermental Results of 5-5PH n etals P5 Austente 85 C Stress (MPa) Stran (mm/mm) Fgure B. 6 Stress vs. stran of P5 at 85 C, austentc state 45 P6 Austente 6 C Stress (MPa) Stran (mm/mm) Fgure B. 7 Stress vs. stran of P6 at 6 C, austentc state Stress (MPa) P7 Austente C Stran (mm/mm) Fgure B. 8 Stress vs. stran of P7 at C, austentc state Tong WU 67

180 Thess: Experment and Numercal Smulaton of Weldng Induced amage P8 Martenste C Stress (MPa) Stran (mm/mm) Fgure B. 9 Stress vs. stran of P8 at C, martenstc state P9 Martenste C 8 Stress (MPa) Stran (mm/mm) Fgure B. Stress vs. stran of P9 at C, martenstc state P Austente 3 C Stress (MPa) Stran (mm/mm) Fgure B. Stress vs. stran of P at 3 C, austentc state 68

181 Appendx B: Expermental Results of 5-5PH n etals P Austente 6 C Stress (MPa) Stran (mm/mm) Fgure B. Stress vs. stran of P at 6 C, austentc state B. amage results and fttng exp. data Lnéare (exp. data) amage fttng of P - Ln () y =.455x -.46 R = Ln (EPS-EPS) Fgure B. 3 Identfcaton of damage exponent of damage model (P) Tong WU 69

182 Thess: Experment and Numercal Smulaton of Weldng Induced amage exp. data Lnéare (exp. data) amage fttng of P Ln () y =.79x -.73 R = Ln (EPS-EPS) Fgure B. 4 Identfcaton of damage exponent of damage model (P) exp. data Lnéare (exp. data) amage fttng of P3 Ln () y =.57x R = Ln (EPS-EPS) Fgure B. 5 Identfcaton of damage exponent of damage model (P3) Ln () amage fttng of P exp. data -.4 Lnéare (exp. data) y =.6636x R = Ln (EPS-EPS) Fgure B. 6 Identfcaton of damage exponent of damage model (P8) 7

183 Appendx B: Expermental Results of 5-5PH n etals exp. data Lnéare (exp. data) amage fttng of P9 -. Ln () y =.533x R = Ln (EPS-EPS) Fgure B. 7 Identfcaton of damage exponent of damage model (P9) Plastc stran amage (exp.) amage (fttng) Table B. 4 Expermental and fttng data of plastc stran vs. damage of P Plastc stran amage (exp.) amage (fttng) Table B. 5 Expermental and fttng data of plastc stran vs. damage of P Plastc stran amage (exp.) amage (fttng) Table B. 6 Expermental and fttng data of plastc stran vs. damage of P3 Plastc stran amage (exp.) amage (fttng) Table B. 7 Expermental and fttng data of plastc stran vs. damage of P8 Tong WU 7

184 Thess: Experment and Numercal Smulaton of Weldng Induced amage Plastc stran amage (exp.) amage (fttng) Table B. 8 Expermental and fttng data of plastc stran vs. damage of P9 B.3 Test results of flat notched specmen a) splacement (UZ) b) Stran (EPZZ) Fgure B. 8 splacement and stran dstrbuton of FA (Case A, C, martenste) when loadng dsplacement s equal to.5 mm (d =.5 mm) by usng dgtal mage correlaton. a) splacement (UZ) b) Stran (EPZZ) Fgure B. 9 splacement and stran dstrbuton of F3A (Case A, C, mlartenste) when loadng dsplacement s equal to.5 mm (d =.5 mm) by usng dgtal mage correlaton. 7

185 Appendx B: Expermental Results of 5-5PH n etals a) splacement (UZ) b) Stran (EPZZ) Fgure B. 3 splacement and stran dstrbuton of F3B (Case B, C, martenste) when loadng dsplacement s equal to.5 mm (d =.5 mm) by usng dgtal mage correlaton. a) splacement (UZ) b) Stran (EPZZ) Fgure B. 3 splacement and stran dstrbuton of F3C (Case C, C, martenste) when loadng dsplacement s equal to.5 mm (d =.5 mm) by usng dgtal mage correlaton. a) splacement (UZ) b) Stran (EPZZ) Fgure B. 3 splacement and stran dstrbuton of F4A (Case A, C, mxed phase) when loadng dsplacement s equal to 3 mm (d = 3 mm) by usng dgtal mage correlaton. Tong WU 73

186 Thess: Experment and Numercal Smulaton of Weldng Induced amage a) splacement (UZ) b) Stran (EPZZ) Fgure B. 33 splacement and stran dstrbuton of F5C (Case C, 3 C, austente) when loadng dsplacement s equal to 3 mm (d = 3 mm) by usng dgtal mage correlaton. 74