Analytical Description of Pseudo-Plasticity

Transcription

1 Analytcal Descrpton of Pseudo-Plastcty Andrew Rusnko Óbuda Unversty, Népszínház u. 8, H-8 Budapest, Hungary Abstract: As phase transformatons are fndng ever-wdenng applcatons, the problem of ther modelng upon the applcaton of thermal and/or mechancal loadngs arses. Ths has resulted n a number of analytcal models to descrbe the phenomena resulted from the phase transformaton. Ths paper s amed at presentng the model of PT-nduced deformaton n terms of the synthetc theory. After the basc, general equatons are developed, the case of pseudo-elastcty wll be consdered n more detal. Keywords: Phase transformaton (PT), PT-nduced deformaton, effectve temperature, pseudo-elastcty, synthetc theory Introducton Let us start wth the defnton of phase transton (PT). There exsts a temperature-dapason wthn whch materals undergo so-called phase transformatons. PTs do not occur by the long-range dffuson of atoms but rather by some form of cooperatve, homogeneous movement of many atoms that results n a change n crystal structure. These movements are small, usually less than nteratomc dstances, and the atoms mantan ther relatve arrangemets. The PT temperature nterval s dfferent for dfferent metals and alloys. For example, ron has body-centered cubc structure (BCC) at temperature below 9 С, face centered cubc structure (FCC) for the temperature range from 9 С to 9 С; ron structure above 92 С returns to BCC structure. The phase transformaton n tn starts at the temperature of 8 С; ttanum, whose melt pont s of 66 С, has hexagonal close packed (HPC) structure below 882 С and BCC structure at hgher temperature. Followng [], a low-temperature phase s called martenste or martenstc phase and a hgh-temperature phase austente or austentc phase. Austente martenste transformaton s referred to as drect transformaton and martenste austente transformaton s called the reverse transformaton. Phase transformatons of most metals are not accompaned by notceable deformaton. However, there exst such alloys whose phase transformatons are 267

2 A. Rusnko Analytcal Descrpton of Pseudo Plastcty accompaned by sgnfcant non-elastc deformaton. So-called Shape Memory Alloys (SMAs) belong to them. The shape memory alloy (also known as a smart alloy, memory metal, or muscle wre) s an alloy that "remembers" ts shape, and can be returned to that shape after beng deformed, by applyng heat to the alloy. A typcal representatve of SMA s nckel-ttanum (N-T) alloy (metallde wth approxmately equal N- and T-concentratons). In temperature nterval from -5 to +2 С, ths alloy undergoes phase transformatons, the character and sequence of whch depends on the component-concentraton and expermental detals. In nckel-ttanum alloy N-T (~45% N) only martenste austente transton upon heatng and austente martenste transton upon coolng occur, whereas N-T 55 at.% N alloys undergoes so-called two-step phase transformatons, upon heatng, the martenstc phase frst transforms nto an ntermedate structure and then the austentc phase starts to grow. There s a stuaton such that a heat-nduced ntermedate phase does not nucleate, whereas t arses upon coolng. Another mportant feature of martenste/austente transformatons s ther reversblty. Ths means that the knetcs of atoms and ther rearrangements n drect and reverse transtons are dentcal. Further, durng both coolng and heatng, PT can be termnated at any stage and redrected n the opposte drecton. In constant temperature, the structure of load-free materal remans unchangeable and there s a balance between the martenste/austente phase and parent matrx. 2 Pseudo-Elastcty and Shape Memory Alloys Consder the deformaton of specmen loaded n tenson at the temperature close to the martenste start temperature; ts stress-stran dagram s shown schematcally n Fg.. The ntal porton of the dagram, segment OM, obeys Hooke s law. Beyond pont M, the specmen experences ntensve flow (porton M M 2 ) caused by the formaton of martenste. After the transformaton s completed, begnnng from pont M 2, the materal deforms elastcally agan untl the yeld lmt σ S s acheved (pont M ); the further ncrease n stress nduces an usual plastc deformaton. The unloadng from the pont lyng on the porton M M 2 leads to the complete or partal, so-called pseudo-elastc, recover of deformaton accompaned, as a rule, wth a sgnfcant hysteress. Snce an unloadng s equvalent to the (effectve)temperature ncrease, the martenstc phase, whch has arsen at tensle, transforms nto austentc phase. It s ths fact that explans the reducton of phase-nduced stran n unloadng. The phasenduced stran n loadng s referred to as pseudo-elastcty, whch s represented by the porton M M 2 n Fg.. 268

3 σ x σ S M M 2 σ M O ε x Fgure. Stress-stran dagram showng pseudo-elastc deformaton (secton M M 2 ); pseudo-elastc deformaton starts arsng at stress σ Shape memory effect. Let a specmen be heated/cooled to such a temperature so that two phases are present. Further, the specmen s subjected to (a) loadng n tenson, (b) unloadng, and (c) heatng n the unloaded state. The stress-stran and stran-temperature dagram of the specmen s shown n Fg. 2. Segment OM gves a pure elastc stran of the specmen. Along the porton M M 2, PT-nduced stran s occurred, pseudo-elastcty; by the PT the martenstc transformaton s meant. The porton of the dagram OM M 2M 4 n Fg. 2 s dentcal to the dagram n Fg.. After the complete unloadng (pont M 4 ) and heatng (along porton M 4 M 5 the stran remans unaltered), the ntensve decrease n stran (porton M 5M 6 ) occurs that s caused by austentc transformaton due to the temperature ncrease. σ x M 2 M ε x M 5 T M 4 M 6 T Fgure 2. Stran-stran dagram n tenson showng pseudo-elastc deformaton (M M 2 ) and effect of the shape memory (M 5 M 6 ) 269

4 A. Rusnko Analytcal Descrpton of Pseudo Plastcty The Modelng of PT-nduced Strans n Terms of the Synthetc Theory The V.A. Lchachev and V.G. Malnn monograph [] should be emphaszed due to some prncpal statements are taken from t to develop the model of phase transformaton nduced deformaton n terms of the synthetc theory. Let a drect (martenstc, dashed arrows) or reverse (austentc, sold arrows) transformaton occurs n a local volume of body (gran); The plot of the phase character functon s shown n Fg.. The martenste formaton rate s determned as ( M s M ') = T, () * f where T* s effectve temperature []: ( D q) T* = T σ. (2) M f A s ' f M M s ' A s A f T Fgure. Phase character functon () vs effectve temperature D are the crystal deformaton components related to crystallographc axes for a complete phase transformaton when the functon changes from zero to unt; σ are stress components n the local volume related to the crystallographc axes; q s the phase transformaton thermal effect. The components D and q are the known physcal characterstcs of phase transformaton for a gven alloy. The martenste s formed at the reducton of T* for the temperature range M f ' T* M s. Therefore, Eq. () holds at T * < as M f ' < T* < M s. The decrease n the martenste fracton, occurrng at the effectve-temperature- range A ' < T* <, s s A f 27

5 ( A f A ') = T *, () s.e. Eq. () s applcable at T * > for As ' T* A f. As M s M f ' = A f As ', Eqs. () and () are dentcal. The effectve temperature (2) depends on stress components actng n the local volume. As well as n the concept of slp, we assume that the state of stress n a local volume s dentcal to the macro-state of stress. Therefore, the components σ n Eq. (2) are known values for a gven macro-stress and, consequently, the effectve temperature T* s known as well. In addton, the martenste fracton can be calculated by Eq. () due to the temperatures M s and M f ' are known physcal constants of materal. Now we proceed to the determnaton of PT-nduced stran components generated n the local volume, ε. In most cases, t s possble to expresses the ε components through the PT-nduced lattce dstorton D and martenstc transformaton rate as []: ε = A D, (4) where A s the factor that takes nto account that the PT-nduced deformaton may not be the same as the deformaton of crystal lattce. However, A = s often the case. If the deformaton of crystal gran s characterzed by the tensor D at the complete transformaton, the average deformaton of the gran at partal transformaton s determned by D. It s ths statement that forms the bass of Eq. (4) gvng ε components both at drect ( > ) and reverse ( < ) transformaton. We take Eq. (4) as the basc relatonshp for the determnaton of PT-nduced stran rate components (further throughout, we wll also use the phase deformaton noton). To calculate by Eq. () the value of, we need to dfferentate Eq. (2) wth respect to tme. As a result, Eq. (4) yelds the form A D D klσkl Dklσ kl ε = T T (5) M s M f ' q q In contrast to plastc deformaton, the phase deformaton s realzed not by shears. At the same tme, n most cases t s not accompaned by volume change. On the other hand, as well known, a deformaton of any knd can be obtaned as shears 27

6 A. Rusnko Analytcal Descrpton of Pseudo Plastcty (slps) along several (maxmum 5) slp systems. Ths reasonng allows us to use the synthetc theory for the modelng of phase deformatons. In terms of the synthetc theory [2,], each slp system s represented by tangent plane wth normal vector N n the Ilyushn fve-dmenson space and stresses σ kl s replaced by scalar product S N, where S s the stress vector. We desgnate through e the rreversble (phase) stran vector gvng the PT-stran developed wthn one slp system, e s co-drected wth the plane normal vector N. We propose to determne e, whch s assumed to be co-drected wth N as well. Therefore, n terms of synthetc theory, Eq. (5) can be wrtten as e φ N N φ, (6) = r N = T ( D S N) + D TS N = ( M M ') ( A D), D = D q r where s f, (7) φ N s termed as stran ntensty rate and D s called reduced shear for a complete transformaton as the functon changes from to. The reduced shear s consdered as the known physcal characterstc of alloy whch accounts for the response of crystal grd to the phase transformaton. Although the magntude of the deformaton e s a fnte quantty relatve to the gran sze, ts contrbuton nto macro-deformaton s nfntesmal due to the number of tangent planes (slp systems) s suggested to be nfnte large. The mcro phase-deformaton rate ncrement components [2,] e d ( =,2, ) are ( e ) dv φ m cos λdv de = =, dv = cosβdαdβdλ. (8) N At a pont of body, the macrodeformaton rate components generated n phase transformaton are [2,] e = φ m cos λ cosβdαdβdλ, (9) where αβλ N φ N s gven by Eq. (6). Snce Eqs. (6) and (9), are obtaned va Eqs. (-), they capable of modelng phenomena when () martenstc transformaton nduces deformaton co-drected wth actng stresses (), austentc transformaton gves the recovery of deformaton. 272

7 s f Upon the applcaton of both thermal and mechancal loadngs, local volumes (grans) of the specmen undergo deformatons nduced by phase-transformatons of dfferent ntensty; wthn some of them the phase reactons does not occur at all. In terms of the synthetc model, the local volume of body s represented by tangent plane wth normal vector N. Therefore, frst of all, t s necessary to formulate the general rule that determnes the orentatons of N n whch phase mcro-deformatons proceed at a gven nstant. The reducton of effectve temperature T* n the range M f ' < T* < M s (Fg. ) leads to the martenste phase arses nducng stran co-drected wth the acton of load; the temperature ncrease n the range A ' < T* < A causes the recovery of the stran. Therefore, PTnduced stran develops at a gven nstant and n a gven drecton N f for T * < ( M ' < T* < ) φ > N φ < N, () f M s, () for T * > ( A ' < T* < ) s A f where φ N s determned by Eq. (6) whose rght-hand sde can have dfferent sgns dfferent drectons N. If the sgn of functon φ N from Eq. (6) does not satsfy the condton () or () for some drectons, the phase deformaton does not acqure an ncrement n these drectons at the gven nstant. There are also other dfferences between the determnaton of rreversble stran due to load [2] and phase transtons:. A dslocaton shape change s neglected due to a PT-nduced deformaton occurs under stresses below the yeld lmt. 2. In terms of the synthetc model appled to the descrpton of PTs, there s no concept of yeld/creep surface. Therefore, planes, whch capable of producng shape changes, fll completely the Ilyushn three-dmensonal subspace. Any plane s set by ts normal vector m (by angles α and β) and the dstance between the plane and the orgn of coordnates s characterzed by angle λ; λ = and λ = π 2 gve the plane wth zero and nfntely large dstance, respectvely.. Planes do not move,.e. the requrement that a tangent plane produces rreversble stran f only t moves on the endpont of stress vector s not vald. 4. We do not take nto account creep deformaton ether because loadng, unloadng, and heatng/coolng of body last relatvely short tme. 27

8 A. Rusnko Analytcal Descrpton of Pseudo Plastcty 4 Modellng of Pseudo-Elatc Stran [4] Consder the followng procedure. A load-free fully austentc sample s under temperature T, M s < T. Further, holdng the temperature T constant, the specmen s subjected to any proportonal loadng. Let us consder the case when a thn-walled ppe s subjected to torson ( S = 2τ xz > ) and we wsh to determne the PT-nduced deformaton of the ppe. Eq. (2) gves the effectve temperature T* as ( S N ) T = T D. (2) In torson, we have S N = cos ξ, cos ξ = sn β cos λ. () S Hence, Eqs. (2) and () gve that T = T ( D cosξ) S, (4) meanng that the loadng of body at constant temperature s equvalent to the decrease n effectve temperature, and ths reducton vares from plane to plane wth dfferent values of angles β > and λ. T* s smallest for the plane wth β = π 2 and λ = : T If ( D ) mn = T S. (5) T mn decrease to the value of M s, the martenste phase starts to growth n the plane wth β = π 2 and λ =. Equatng the stress nducng pseudo-elastc stran phase flow lmt T mn to M s, we obtan the equaton for S (σ n Fg. ) as: = M s S. (6) D T As seen from the above formula, the phase flow lmt grows wth the temperature of body T. If T = M s, S = and the dagram n Fg. becomes nonlnear at once from the orgn of coordnates. The further ncrease n component S leads to that the effectve temperature becomes smaller than M s meanng the martenstc phase starts to form. The formaton of martenste nduces stran whose ntensty φ s determned by Eqs. (6) and () as N 274

9 = D T cos ξ. (7) rφ N S As seen from (7), the condtons (), φ, s satsfed for < λ < π 2 and < β < π 2. On the other hand, the nequalty T M s requres N S D M T s cos ξ, (8) that s obtaned by equatng T to M s n Eq. (4). Wth account of (6), the above nequalty can be rewrtten as ( S S ) cosξ. Therefore, the ntegraton lmts n (9) are α 2π, β β π 2 and λ λ : ( snβ) snβ snβ cos λ = S S =, sn β = S S. (9) The macro-stran rate vector components nduced by the martenste transformaton ( e ), by makng use of Eq. (9) are 2π S π 2 β λ DT 2 e = dα m sn 2βdβ cos λdλ. (2) 2r Integratng n Eq. (2) gves that e = e and e Z 2πD T = ( x) 2 = e s π 2 λ S 2 2 πd T S sn cos d cos d Z S r β β β λ λ = r S, (2) β 2 + x = arccos x + x x 2x ln. (22) x Further we ntegrate over S n lmts from S to S n (2): 2 e S πdt S πd = T S S = Z ds F, (2) r S r S S 2 arccos x x F( x) = 2 x + x ln. x x Accordng to Eqs. (7) and (6), we rewrte Eq. (2) as 275

10 A. Rusnko Analytcal Descrpton of Pseudo Plastcty ( M ) πaf D T s S S = e F ( M ) s M f '. (24) Ths formula holds f the effectve temperature (4) less than M f ',.e. the martenste transformaton progresses. Otherwse, accordng to Fg. 2, the drect martenste transformaton s fnshed ( = ) at T = M f ' and, consequently, the transformaton-nduced deformaton s termnated. Equatng the rght-hand sde of Eq. (5) to M f ', we obtan the equaton for the boundary value of S, S f, at whch Eq. (4) or (5) s applcable: M f ' S S S f S = f. (25) D T Eq. (2) can be obtaned n another, smpler, way. By ntegratng n Eq. (7) wth respect to tme, we obtan ( S C) rϕ N = DT cos ξ, (26) where C = S cos ξ. Therefore, sn β cos λ rϕ N = DT S. (27) ( S cos ξ S ) = DT sn β If to nsert ths stran ntensty nto Eq. (9) and ntegrate t, we arrve at the result of Eq. (2). Somewhat dfferent approach to the modelng of PT-stran can be found n works [5-7]. Conclusons Utlsng the synthetc theory, the phenomenon of pseudo-elastcty has been modelled. To do ths the PT-nduced lattce dstorton D and martenstc transformaton rate s ntroduced nto the consttutve equtons. Acknowledgement The author expresses thanks to Prof. K. Rusnko for many useful conversatons on the topcs presented n ths paper. References [] Lchachev, V. and Malnn, V. (99). Structure-analytcal theory of strength, (n Russan), St. Peterburg 276

11 [2] Rusnko, A. and Rusnko, K. (29). Synthetc theory of rreversble deformaton n the context of fundamental bases of plastcty, J. Mech. Mater. 4: 6-2 [] Rusynko, K. and Rusnko, A. (2). Plastcty and Creep of Metals, Sprnger, Berln [4] Rusnko, A. Phase transformaton stran n terms of the synthetc theory, Proceedngs of The World Congress on Engneerng and Technology (CET 2), October 28 - November 2, 2, Shangha, Chna [5] Rusynko, K. and Shandrvs'ky, A. (996). Irreversble deformaton n the course of a martenste transformaton, Materal Scence : [6] Holyboroda, I. and Rusnko, K. (996). Dervaton of a unversal relaton between tangental stress and shear stran ntenstes n descrbng reversble martenstc deformaton wthn the framework of a synthetc model, J. Appl. Mech. and Tech. Phys. 7: [7] Holyboroda, I., Rusnko, K. and Tanaka, K. (999). Descrpton of an Febased shape memory alloy thermomechancal behavor n terms of the synthetc model, J. Comp. Mater. Sc. :