Energy dissipation and storage in iron under plastic deformation (experimental study and numerical simulation)

Transcription

1 Focussd on: Infrard hrmograhic Analysis of Matrials Enrgy dissiation and storag in iron undr lastic dformation (xrimntal study and numrical simulation) A. Kostina, A. Iziumova, O. Plkhov Institut of Continuous Mdia Mchanics of th Ural Branch of th Russian Acadmy of Scincs, 61413, Ac. Korolva Strt, 1, Prm, Russia ABSRAC. h work is dvotd to th xrimntal and numrical invstigation of thrmodynamic ascts of th lastic dformation in Armco iron. Dissiation and stord nrgis was calculatd from rocssd xrimntal data of th surfac tmratur obtaind by infrard thrmograhy. An original mathmatical modl dscribing th rocss of msoscoic dfcts accumulation was usd for numrical simulation of th quasistatic loading of iron samls and for calculation of thortical valu of th stord nrgy. Exrimntal and modld valus of th stord nrgy ar in a good agrmnt. KEYWORDS. Stord nrgy; Infrard thrmograhy; Msodfct volution; Numrical simulation. INRODUCION h xrimntal and thortical study of nrgy balanc in mtals undr lastic dformation has a long history. h imortanc of this issu for undrstanding of natur of irrvrsibl dformation of mtals and its transition to fractur was originally shown by J. H. Lambrt in 1779 in his statmnt concrning th nrgy similarity of mchanical and thrmal failur rocsss of solids. h rviw of xrimntal works dvotd to th mthods of stord nrgy study in th matrial undr lastic dformation and th culiaritis of this rocss for diffrnt matrial structurs and load conditions is availabl in [1]. o analyz th thrmodynamic charactristics of dformation rocsss in solids, it is ncssary to tak into account th fact that th lastic dformation work is convrtd into two arts: hat nrgy causd by th movmnt and annihilation of dfcts at various structural lvls, and hiddn (stord) nrgy of lastic dformation accumulatd in th lastic filds of dfcts. h first art of nrgy can b dtctd by diffrnt xrimntal tchniqus and givs us information about currnt stat of structur volution. During th last dcads th nhancd ability to dtct tmratur volution during mchanical tsting lads to a grat intrst in th alication of infrard (IR) thrmograhy to study hat dissiation rocss causd by dfct volution undr lastic dformation in mtals. h thortical and xrimntal study of this roblm can b usful for xlanation of many hnomna of mchanics of solids such as cold work volution, study of damag to fractur transition and crack roagation rocss, invstigation of crack ti closur ffct, validation of linar failur mchanics formula to th invstigation of fatigu cracks roagation in mtals undr diffrnt loading conditions and so on. h aformntiond xrimntal rsults indicat th imortanc of furthr dvlomnt of mthods for invstigation of th thrmodynamics of th dformation rocss and dvlo adquat thortical modls of lastic dformation dscribing th nrgy balanc in mtals undr lastic dformation. his work is dvotd to invstigation of nrgy dissiation in Armco iron undr quasistatic tnsion. It was shown xrimntally [2] that from macroscoic oint of viw th dformation rocss of Armco iron can b dividd into homognous and htrognous arts. As a first st of 28

2 thortical considration of th hnomna w trid to simulat th nrgy storag rocss in Armco iron undr homognous lastic dformation. h simulation was basd on th original statistical modl of dfct volution and includs th saration of lastic dformation into dissiativ and thrmodynamic (structural) arts. h nonlinar constitutiv quation for structural snsitiv aramtrs allows us to obtain th qualitativ agrmnt btwn xrimntal and thortical rsults. MAERIAL AND EXPERIMENAL CONDIIONS E xrimntal study was carrid out on th Armco iron scimns. Chmical comosition of th xamind matrial is rsntd in ab. 1. Fig 1. shows gomtry of th scimns. С Mn Si S P Ni Сr Mo abl 1: Armco iron chmical comosition (wt%). Figur 1: Gomtry of scimns. All sizs ar in millimtrs. Aftr tooling, scimns wr annald in non-oxygn atmoshr with 173 K tmratur during 8 hours. Non-oxygn atmoshr was cratd by co-annaling of mtal and cor samls with dvlod surfac for a bttr oxidation. Mchanical tst was carrid out on srvo hydraulic machin INSRON 85. Bfor starting of xrimnts th surfac of th scimns was mchanically olishd (at th last stag of th olishing th diamond susnsion with a charactristic grit siz of 3 microns was usd). Aftr olishing th surfac was covrd of matt black aint. It was don to obtain missivity of th scimn s surfac clos to 1 for bttr conditions of infrard monitoring. Scimns wr loadd in uniaxial strtching mod with strain rat sc -1. Infrard camra CEDIP Jad III was usd to rcord th tmratur volution on th surfac during mchanical tst. h maximum fram siz is ixls; th satial rsolution is 1-4 mtrs. h tmratur snsitivity is from 25 mk to 3 K. ENERGY CALCULAION BASED ON HE PROCESSED EXPERIMENAL DAA I nfrard monitoring of th surfac of xamind scimn during quasistatic tnsion taks ossibility to obsrv changs in tmratur fild in loading rocss. h Fig. 2 rsnts th strss strain rlation and tmratur volution of th gag art of th scimn for invstigatd matrials. W can divid th dformation rocss into thr arts. At th bginning if th rocss th lastic dformation has a htrognous natur. W can obsrv th roagation of Ludrs bands (lastic dformation wavs) on th scimn surfac. his rocss is accomanid by th tmratur wav roagation. h dtail study of this art of lastic dformation in Armco iron can b found in [3]. 29

3 3 38, MPa mratur, K 5 31 () 3 () Figur 2: Strss strain volution of Armco iron scimn (solid lin) and local tmratur volution (dash lin); strain rat s -1. h lastic wav roagation is finishd by th transition to hardning rocss accomanid by an aroximatly homognous lastic dformation of th saml (th scond art of lastic dformation). h last art of th dformation rocss is charactrizd by ncking htrognous tmratur distribution of th scimn surfac. h scond art of th rocss can b considrd as a homognous at macroscoic lvl. h nrgy balanc in scimn at this art of dformation will b simulatd in th nxt art of this ar. All othr arts of th rocss rqusts th considration th nonlocal ffcts in dfct nsmbl and lads to th mor comlx modl. Hat sourcs calculation h hat sourcs volution can b calculatd basd on th following quation [4] 2 2 sx (, yt, ) c k 2 2 t x y whr is th tmratur, ρ is th dnsity (787 kg/m 3 ), c is th hat caacity (444 J/(kg K)), k is th hat conductivity (76.2 W/(m K)), s is th unknown scific owr of hat sourc (W/m 3 ) and τ is th constant dscribing th hat xchang rocss with th surroundings (16 sc) [5]. h dfinition of hat sourcs volution rqusts th calculation of th sac and tim drivation of noisy tmratur fild. It lads to th situation whn rcordd xrimntal data of th tmratur is not nough suitabl to calculat th hat dissiation rat. In this work w us tim and sac filtr for xrimntal data. o imlmnt filtration rocdur it was alid discrt Fourir transform with Gaussian krnl. h culiaritis of filtring rocdur ar dscribd in [6]. his rocdur allows us to smooth satial and tim htrognity. Fig. 3 rsnts fild of tmratur rang bfor and aftr rocssing. (1) a) b) Figur 3: mratur rang on th surfac of scimn bfor data rocssing (a) and obtaind fild of tmratur rang with zon of intrst (b). 3

4 Stord nrgy calculation Fig. 4 rsnts fild of hat dissiation rat on th small ara of scimn surfac limitd by black rctangl shown in Fig 3b. h small ara for calculation was slctd du to saml sha changing during dformation rocss. h fild of hat sourcs is a littl bit htrognous vn during th scond art of lastic dformation. Figur 4: Exrimntal data of th hat dissiation rat (W/m 3 ) at th bginning and nd of intrst tim rang. h intgral hat dissiation in zon of intrst can b calculatd as x 2 y 2 Q() t s( x, y,) t dxdy (2) x1 y1 whr x1, x2, y1, y2ar th rctangular coordinats of th intrst zon (Fig. 3b). W suos that som of th irrvrsibl lastic work contributs to hat gnration, whil th rst is stord as th nrgy of crystal dfcts accomanying lastic dformation, traditionally known as th stord nrgy of cold work [7,8]. h goal of this work is to calculat valu of th stord nrgy rat. Plastic work snd on dformation of th scimn has to b known for calculation of th nrgy stord in mtals during dformation. W obtaind lastic work rsntd in xrssion (3) as a function of strain rat V and loading forc F(t). W( t) F() t V (3) W calculat stord nrgy of cold work as diffrnc btwn lastic work snt on dformation and intgral hat dissiation (W (t) Q(t)) [9]. im dndnc of ths valus is shown in Fig. 5. Q() t Ratio is usually usd for dtrmination of th aramtr β charactrizd xtnt of matrial damag [1]. h stord W( t) nrgy ratio can b calculatd as follow W() t Q() t 1 (4) W () t In Fig. 6 valu of 1 β dnding on strain is rsntd. Valu of stord nrgy dcrass in th nd of dformation rocss that mans most of lastic work convrtd into hat and matrials rard to th failur. Rd lin indicats art from sc to 28.9 sc (surfac of hat dissiation rat in th bginning and nd of this rang is rsntd in Fig.4) dscribd th scond art of lastic dformation. his art was chosn for numrical modling of th lastic dformation rocss in Armco iron. 31

5 Hat dissiation Plastic work Stord nrgy 1.8 W, Q and W -Q, J im, sc Figur 5: im dndnc of lastic work, hat dissiation nrgy and stord nrgy Figur 6: Strain dndncs of stord nrgy ratio (1-β). MAHEMAICAL MODEL OF ENERGY BALANCE IN MEALS UNDER HOMOGENEOUS PLASIC DEFORMAION I t is now wll known that ral mtals hav comlx structur, which is a hirarchy of diffrnt lvls. Undr dformation rocss th structural volution obsrvd at all scal lvls and lads to irrvrsibl dformation and dstruction. o dvlo a modl of dfct volution undr lastic dformation w hav to choos th basic hysical lvl of dscrition of th matrial microstructur and to dscrib th gomtry of th lmntary dfcts. On of th ossibl dscritions of dfct kintics is th statistical modl of dfct nsaml. his modl has to tak into account th stochastically rortis of dfct initiation, thir nonlinar intgration and link btwn microlasticity and damag accumulation rortis. Problm statmnt yical msoscoic dfcts (msoshifts) ar dscribd as th symmtrical tnsor s, which has th following form [11,12]: 1 s S ( nl ln ) 2 whr l is a unit normal to th shar lan, b is a unit vctor in th shar dirction, S is a shift intnsity. Dfct dnsity tnsor that coincids with th strain causd by dfcts is dfind as: ns whr n is a dfct dnsity. h distribution function of dfcts can b rrsntd in th following form: W Z 1 x( E/ ) whr E is th nrgy of th dfct, Z is th normalizing factor, θ is th ffctiv tmratur factor rsonsibl for th systm susctibility. Avraging rocdur lts us to obtain th slf-consistncy quation btwn micro and macro aramtrs in th following form: NsW ( s) ds (5) h solution of th Eq. (5) was roosd in [12]. Basd on ths rsults w can introduc a hnomnological modl of th quasistatic rocss. Full strain rat can b rrsntd as th sum of thr comonnts: lastic strain rat, lastic strain rat and strain rat causd by dfcts : 32

6 A. Kostina t alii, Frattura d Intgrità Struttural, 27 (214) 28-37; DOI: /IGF-ESIS.27.4 (6) Elastic strains ar dfind by linar Hook s law: K (7) d 2G d (8) whr K - isotroic lastic modulus, G - lastic shar modulus, - shrical art of th strss tnsor, d - dviator art of th strss, - shrical art of th lastic strain tnsor, d - dviator art of th lastic strain. Dissiation function for a mdium with dfcts can b rrsntd in th following form [13]: : ( ): q (9) whr is a fr nrgy, - dnsity, q - hat flux vctor, - tmratur. Dividing th thrmal and mchanical roblms and basing on th Onsagr rincil, w can obtain from (9) constitutiv quations for calculating kintics of lastic and structural strains: ( ) (1) ( ) (11) h kintic cofficints, and hav th following form: 1 1 S 1 Ex c a1 1 1 H(,,, c, c) S y 1Ex a 2 1 whr,, - charactristic rlaxation tims, - strss intnsity tnsor, S c a 1, a 2 - matrial constants, S y - yild strss, - intnsity of, c, c - scaling factors, H(,,, c) 2 c ( f1) c c - matrial function ( it can b considrd as dgr of systm nonquilibrium ). It is suosd that thrmodynamic forc can b writtn as: 1 c f 1 2Gc c c c whr f ( ) dnots a owr function for modling of nonlinar hardning rocss: (12) f k c c a k is a scaling factor, a is th xonnt. Eqs. (6)-(8) and (1)-(13) rrsnt a closd systm for a lastically dformd solid with nonlinar hardning. From th first thrmodynamic law, w can obtain th xrssion for calculation th aramtr in th following form: (13) 33

7 2 : Q 1 W :( ) whr Q - inlastic contribution to th hat, W - lastic work. Undr isontroy conditions it has th form: : Q 1 W :( ) Rsults of modling Fig. 7 rsnts th xrimntal rsults for tnsion of Armco iron samls. o simulat th homognous art of lastic dformation, w roos th xistnc of two rocss sts. h first st is corrsonding to th loading of th saml u to yild strss, Ludrs bunds roagation and unloading. h scond st corrsonds to th loading of rloadd scimn. During th scond st th scimn xhibits th homognous lastic dformation, arabolic hardning and dcrasing of stord nrgy ratio. his rocss can b simulatd by dfinition of nonzro initial dformation. h schmatic of th rocss is rsntd in Fig. 7. Figur 7: Exrimntal strss-strain curv for Armco iron. Numrical simulation of th considrd rocss was carrid out in th finit lmnt ackag Simulia Abaqus Fig. 8 shows th finit-lmnt msh of th scimn. Eight-nod linar brick lmnts wr usd for th simulation. h abov xlaind modl was alid for th matrial bhavior dscrition using th subroutin UMA. Arrays of matrial constants, strain, strain incrmnts and th tim st assd as inut data to th rocdur. Incrmnts of th strss tnsor comonnts and incrmnts of th dfct dnsity tnsor comonnts ar dtrmind from th systm of constitutiv quations. Valus of ths comonnts at th nxt tim st ar dfind as th sum of th valus on th rvious st and th aroriat incrmnt. Fig. 9 rsnts th strss-strain curvs obtaind from xrimntal data (dash dot lin) and rsults of numrical simulation (solid lin). Rsults calculatd by th modl ar in a good agrmnt with xrimnt. Figur 8: Finit-lmnt msh of th scimn. 34

8 25 2 numrical rsults xrimntal rsults 15, MPa Figur 9: Numrical and xrimntal strss-strain curvs for Armco iron. Fig. 1 dmonstrats th simulation rsults of th 1 β valu. h modling rsults coincid to th xrimntal on th homognous stag of th lastic dformation. Dash lin corrsonds to th rd lin in Fig. 6. h numrical rsults xhibit incrasing branch of stord nrgy ratio and coincid to th xrimntal rsults aftr som dformation valu. his fact can b xlaind by th initial conditions for structural snsitiv aramtrs of th modl dfind in th simulation rocss. h initial condition corrsonds to th annald matrials with zro dfct inducd dformation. At th bginning of dformation rocss this matrial xhibits incrasing rat of stord nrgy ratio corrsonding to th incrasing of dfct dnsity [1]. In contrast to this, th xrimntal rsult corrsonds to initially dformd matrial which has cratd ffctiv dfct structur for th dissiation of dformation nrgy. h xtnsion of th ara of alicability of th modl rqusts th simulation of nonlocal rocss of lastic dformation..8.7 numrical rsults xrimntal rsults Figur 1: Valu of 1-β dnding on strain. Simulation rsults. CONCLUSION h infrard tchniqu has bn alid to invstigat th ffct of hat dissiation undr quasistatic loading of Armco iron. A tchniqu for xrimntal data tratmnt has bn dvlod for dtrmination of th stord nrgy rat. h hat dissiation fild and avrag stord nrgy ration wr calculatd for whol dformation rocss including homognous and htrognous ats. h xrimntal rsults stimulat us to dvlo a hnomnological modl dscribing th nrgy balanc in mtals undr lastic dformation. Extnding th rvious rsults of th rsarch grou in Prm, w roos a statistical dscrition of th msodfct nsmbl and introduc th thrmodynamic otntial of solid with dfct. h ky oint of th modl is dfinition of dfct inducd strain which can b considrd as a thrmodynamic variabl. It allowd us to roos a thr dimnsional thrmodynamic intrnal variabl modl of hat dissiation in mtals and dscrib th nrgy storag in Armco iron undr homognous lastic dformation. 35

9 h calculatd valu of stord nrgy rat was comard with xrimntally dtrmind valu. A good agrmnt btwn xrimntal and numrical rsults in th rang of homognous lastic dformation was found. ACKNOWLEDGEMENS h xrimntal invstigation of iron scimn was carrid out at th Laboratory LAMEFIP ENSAM (Bordaux, Franc). h authors thank th rofssors. Palin-Luc and N. Saintir for fruitful discussions and suort of this xrimntal work. his work was suortd by grant of Prsidnt of th Russian Fdration for suort of young Russian scintists and lading scintific schools (MD ) and scintific rojct for young rsarchs and PhD studnts (grant NP-349). NOMENCLAURE b c E F G IR k K l n q Q Q s S S S y V W W Z β unit vctor in th shar dirction hat caacity dfct nrgy loading forc lastic shar modulus infrard hat conductivity isotroic lastic modulus unit normal vctor to th shar lan dfct dnsity msoscoic dfct dnsity tnsor intnsity of hat flux vctor scific nrgy dissiation rat intgratd ovr th coordinats inlastic contribution to th hat microscoic dfct tnsor scific nrgy dissiation rat shift intnsity yild strss tmratur strain rat distribution function of dfcts lastic work normalizing factor variabl charactrizd xtnt of matrial damag lastic strain tnsor shrical art of th lastic strain tnsor dviator art of th lastic strain d lastic strain tnsor ρ dnsity Cauchy strss tnsor strss intnsity tnsor shrical art of th strss tnsor 36

10 d i, i,, τ dviator art of th strss tnsor charactristic rlaxation tims th constant dscribing th hat xchang rocss with th surroundings fr nrgy REFERENCES [1] Bvr, M.B., Holt, D.L., ichnr, A.L., h stord nrgy of cold work, Progrss in Matrials Scinc, 17 (1973) [2] Fdorova, A.Yu., Bannikov, M.V., Plkhov, O.A., A study of th stord nrgy in titanium undr dformation and failur using infrard data, Fractur and Structural Intgrity, 24 (213) [3] Plkhov, O.A., Naimark, O.B., Saintir, N., Palin-Luc,., Elastic-lastic transition in iron: Structural and thrmodynamic faturs, chnical Physics, 54(8) (29) [4] Chrysochoos, A., Louch, H., An infrard imag rocssing to analys th calorific ffcts accomanying strain localization, Intrnational Journal of Enginring Scinc, 38 (2) [5] Chrysochoos, A., Balandraud, X., Wattriss, B., Idntification rocdur using full-fild masurmnts alications in mchanics and structurs, Procssing of CNRS summr school, 213 May 21-25; Montllir, Franc. [6] Fdorova, A.Yu., Bannikov, M.V., Plkhov, O.A., Infrard thrmograhy study of th fatigu crack roagation, Fractur and Structural Intgrity, 21 (212) [7] Hodowany, J., Ravichandran, G., Rosakis, A.J., Rosakis, P., Partition of lastic work into hat and stord nrgy in mtals, Ex. Mch., 4(2) (2) [8] Plkhov, O., Uvarov, S., Naimark, O., hortical and xrimntal invstigation of dissiatd and stord nrgy ratio in iron undr quasi-static and cyclic loading, Strngth of matrials, 1(391) (28) [9] aylor, G.I., Quinny, H., h latnt nrgy rmaining in a mtal aftr cold working, Proc. R. Soc. A, 143 (1934) [1] Rittl, D., Wang, Z.G., Mrzr, M., Adiabatic shar failur and dynamic stord nrgy of cold work, Physical Rviw Lttrs, 96 (26) 7552(1 4). [11] Naimark, O.B., Structural-scaling transitions in msodfct nsmbls as mchanisms of rlaxation and failur in shockd and dynamically loadd matrials (xrimntal and thortical study), J. Phys. IV. Franc, 134 (28) 3 9. [12] Naimark, O.B., Bayandin, Yu.V., Lontiv, V.A., Pantlv, I.A., Plkhov, O.A., Structural-scaling transitions and thrmodynamic and kintic ffcts in submicro-(nano)crystallin bulk matrials, Phys. Msomch., 12(5-6) (29)