Abtract hape-memory alloy how eature not preent in material traditionally ued in engineering; a a conequence, they are the bai or innovative applicati

Transcription

1 HAPE-MEMORY ALLOY: macromodelling and numerical imulation o the uperelatic behavior Ferdinando Auricchio Univerita' di Roma "Tor Vergata" Robert L. Taylor Univerity o Caliornia at Berkeley Jacob Lubliner Univerity o Caliornia at Berkeley Publihed on Computer Method Appl. Mech. Engrg, 46 (997) auricchio@unipv.it

2 Abtract hape-memory alloy how eature not preent in material traditionally ued in engineering; a a conequence, they are the bai or innovative application. A review o the available literature how a dearth o computational tool to upport the deign proce o hape-memory-alloy device. A major reaon i that conventional inelatic model do not provide an adequate ramework or repreenting the unuual macrobehavior o hape-memory material. The preent work ocue on a new amily o inelatic model, baed on an internal-variable ormalim and known a generalized platicity. Generalized platicity i adopted herein a ramework or the development o one- and threedimenional contitutive model or hape-memory material. The propoed contitutive model reproduce ome o the baic eature o hape-memory alloy, uch a uperelaticity, dierent material behavior in tenion and compreion, and the ingle-variant-martenite reorientation proce. For iothermal condition the implementation o the model in a nite-element cheme and the orm o the algorithmically conitent tangent are dicued in detail. Numerical imulation o typical tet perormed on hape-memory material (e.g., uniaxial loading, our-point bending and three-point bending tet) are preented and compared with available experimental data. Baed on the overall development, it appear that the propoed approach i a viable bai or the development o an eective computational tool to be ued in the imulation o hape-memory-alloy device.

3 Content Introduction 2 2 uperelatic application 4 3 Eential eature o the hape-memory alloy micromechanic 6 3. Phae tranormation Kinematic Kinetic An example o hape-memory material: Nitinol Toward a continuum mechanic model Contitutive model or hape-memory alloy 4. A impliying aumption: iotropy D- model Control and internal variable Phae tranormation and activation condition Flow rule Tet example D- model Control and internal variable Phae tranormation and activation condition Model review Material-parameter pecialization to uniaxial tate Time-dicrete iothermal model and algorithmic implementation Kinematic and elatic contitutive equation Time-dicrete model Integration algorithm or the time-dicrete model Algorithmic tangent pecialization to linear and exponential ow rule Tet example Finite-element imulation o hape-memory-alloy device Material-parameter characterization Four-point bending tet Three-point bending tet Conideration on the imulation Cloure and uture reearch direction 37

4 hape-memory alloy: uperelatic behavior 2 Introduction In the 960 Buehler and Wiley [7] developed a erie o nickel-titanium alloy, with a compoition o 53 to 57 % nickel by weight, that exhibited an unuual eect: everely deormed pecimen o the alloy, with reidual train o 8-5 %, regained their original hape ater a thermal cycle. Thi eect became known a the hape-memory eect (Figure ), and the alloy exhibiting it were named hape-memory alloy. It wa later ound that not only do other material have the hape-memory property, but that at uciently high temperature uch material alo poe the property o uperelaticity, that i, the recovery o large deormation during mechanical loading-unloading cycle perormed at contant temperature (Figure 2). A a conequence o uperelatic and hape-memory behavior, hape-memory alloy lend themelve to innovative application. Recently propoed deign baed on uch material range rom el-expanding micro-tructure or the treatment o body veel occluion to device or the deployment and control o pace tructure uch a antenna and atellite. A review o the available literature and peronal contact in the indutry, however, how a dearth o computational tool to upport the deign proce. A major reaon i the act that conventional model o inelatic behavior, uch a claical platicity, do not provide an adequate ramework or repreenting uperelatic and hape-memory behavior. Baed on ome pioneering work by Phillip and collaborator [, 37], over the pat two decade a new amily o inelatic model ha been developed [20] that allow the decription o eature not repreentable by claical platicity, though it include claical platicity a a pecial cae; it ha accordingly been called generalized platicity [2]. A ha recently been dicued [3, 4, 23], the numerical implementation o model belonging to thi amily i traightorward. Conequently generalized platicity appear to be a viable and exible environment or the development o contitutive material with complex behavior. The chie objective o thi work i to propoe a plauible initial development o computational tool or deign with hape-memory alloy through an exploration o the applicability o generalized platicity to the repreentation o the uperelatic behavior, and pecically () the development o contitutive model that reproduce the uperelatic behavior, (2) the numerical implementation o uch model in a nite-element etting, and (3) the imulation o application to how the viability o the propoed approach a an eective computational tool or the deign o device baed on hape-memory alloy. The work i organized a ollow. ection 2 explain the reaon or the in- In the ollowing the ollowing abbreviation are ued: MA or hape-memory alloy and E or uperelaticity. FA & RLT & JL CMAME 46 (997) ection

5 hape-memory alloy: uperelatic behavior 3 creaing interet on hape-memory material through a brie urvey o MAbaed application exploiting the uperelatic behavior. ection 3 overview the MA micromechanic and introduce the terminology ued in the tudy. The urvey o the phae tranition occurring in hape-memory alloy allow the interpretation o the macrocopic material behavior and prepare the ground or the development o a continuum material model. ection 4 decribe a major impliying aumption (iotropic behavior) and preent a one-dimenional and a three-dimenional contitutive model, which reproduce the uperelatic behavior. ection 5 decribe a time-dicrete iothermal verion o the three-dimenional model; it algorithmic implementation within a nite-element ramework i careully addreed. Numerical tet perormed to check the algorithm are preented. The model ability o reproducing macrocopic eect, uch a Luder band, i dicued. ection 6 aee the ability to perorm imulation o typical MAbaed application and how the potentialitie o the propoed approach. Finally, ection 7 cloe and comment on the work preented. Extenion and uture development are alo dicued. FA & RLT & JL CMAME 46 (997) ection

6 hape-memory alloy: uperelatic behavior 4 2 uperelatic application hape-memory alloy have unique propertie which are not preent in many material traditionally ued in engineering application. Accordingly, their ue introduce new deign capabilitie, which make it poible to improve device perormance a well a to propoe innovative olution. The preent ection review application exploiting the uperelatic behavior; the goal i to ue practical example to how the MA range o potentialitie and to explain the reaon or the growing interet in MA-baed device. In general, uperelaticity-baed application take advantage o one o the ollowing eature: () the poibility o recovering large deormation (up to train o 8-5%) (2) the exitence o a tranormation tre plateau, which guarantee contant tre over non-negligible train interval. Medical guidewire A guidewire i a long, thin, metallic wire, paed into the body through a natural opening or a mall inciion. It erve a a guide or the ae introduction o variou therapeutic and diagnotic device. The ue o uperelatic alloy may (a) reduce the complication o the guidewire taking a permanent kink, which may be dicult to remove rom the patient without injury; (b) increae teerability, that i, the ability to tranmit a twit at one end o the guidewire into a rotation o identical degree at the other end [43]. tent tent i the technical word indicating el-expanding micro-tructure, which are currently invetigated or the treatment o hollow-organ or duct-ytem occluion. The tent i initially tretched out to reach a mall prole, which acilitate a ae, atraumatic inertion o the tent. Ater being releaed rom the delivery ytem, the tent el-expand to over twice it compreed diameter and exert a nearly contant, gentle, radial orce on the veel wall [32]. Orthodontic During orthodontic therapy tooth movement i obtained through a bone remodeling proce, reulting rom orce applied to the dentition. The optimal tooth movement i achieved by applying orce that are low in magnitude and continuou in time. Recently, coil pring uing uperelatic material have been deigned. A proved experimentally [40], they produce excellent reult due to the contant tre that hape-memory alloy are able to exert during a ubtantial part o the tranormation. FA & RLT & JL CMAME 46 (997) ection 2

7 hape-memory alloy: uperelatic behavior 5 The Homer Mammalok The Homer Mammalok i a needle wire localizer deigned to meet an important and demanding need in the medical eld. It i oten recommended that any woman over the age o orty ha mammogram every other year. Thi procedure requently identie mall leion which mut be removed and examined microcopically. The procedure or removing the leion i made dicult becaue there are no landmark to guide the urgeon and becaue the leion cannot be eaily ditinguihed in any obviou way rom the urrounding tiue. At the ame time, a little a poible exce tiue hould be removed. The method currently ued i to place a needle in the breat at the ite o the leion a a guide to the urgeon during the removal. To prevent accidental advancement or withdrawal during the tranit to the operating room, the end o the needle ha a barb; due to the harp barb hape, tiue damage are poible and the repoitioning the needle i not poible. uperelatic alloy can be ued to have el-deploying, rounded hook, which do not move, due to their hape, and which can be eaily repoitioned without cauing tiue damage [36]. Eyegla rame By uing the uperelaticity, eyegla rame component can withtand extreme deormation and pring back completely [9]. Concluion The urvey preented above clearly how that hape-memory material have a wide range o application. Recently, international conerence on uperelatic and hape-memory technologie have alo been organized, tetiying to an increaing interet on the ubject. However, a review o the available literature and peronal contact in indutry how a lack o computational deign tool. Accordingly, the aim o the next ection i to propoe a plauible initial development o uch deign tool. To reach thi goal, we rt review the phae tranormation occurring in hape-memory material. Thi allow an interpretation o the uperelatic eect rom a micromechanical point o view. A thermomechanical ramework already preented in literature i then adopted or the development o MA contitutive model. Finally, ater dicuing a nite-element implementation o the model, ome ample problem are tudied. FA & RLT & JL CMAME 46 (997) ection 2

8 hape-memory alloy: uperelatic behavior 6 3 Eential eature o the hape-memory alloy micromechanic The preent ection preent a urvey o the hape-memory alloy micromechanic. It then ocue on Nitinol, which i probably the hape-memory material mot commonly ued in application. The main objective are to help the undertanding o the material macrocopic behavior and to prepare the ground or the development o contitutive model. 3. Phae tranormation hape-memory alloy belong to a cla o material that can undergo reverible change in the crytallographic ymmetry-point-group. uch change can be interpreted a martenitic phae tranormation, that i, a olid-olid diplacivediuionle (or lattice-ditortive) phae tranormation between a crytallographically more-ordered phae, called autenite or parent phae, and a crytallographically le-ordered phae, called martenite. Typically, the autenite i table at high temperature and low value o the tre, while the martenite i table at low temperature and high value o the tre. For a tre-ree tate we indicate with T A the temperature above which only the autenite i table and with T AM the temperature below which only the martenite i table. The phae tranormation between autenite and martenite are the key to explain the uperelaticity eect. For the imple cae o uniaxial tenile tre a brie explanation ollow (Figure 2). Conider a pecimen in the autenitic tate and at a temperature greater than T A ; accordingly, at zero tre only the autenite i table. I the pecimen i loaded, while keeping the temperature contant, the material preent a nonlinear behavior (ABC) due to a treinduced converion o autenite into martenite. Upon unloading, while again keeping the temperature contant, a revere tranormation rom martenite to autenite occur (CDA) a a reult o the intability o the martenite at zero tre. At the end o the loading-unloading proce no permanent train are preent and the tre-train path i a cloed hyterei loop. The tudy o martenitic tranormation can be approached rom dierent point o view. In the ollowing we briey review ome apect o the kinematic and the kinetic o the martenitic phae tranormation occurring in hapememory alloy. 3.. Kinematic By the kinematic o a martenitic phae tranormation we mean the change o atomic tructure occurring during the phae tranormation. Formation o a ingle martenite crytal FA & RLT & JL CMAME 46 (997) ection 3

9 hape-memory alloy: uperelatic behavior 7 In general, ingle martenite crytal have a platelet hape. From experimental obervation occurring during the ormation o a martenite ingle crytal rom a parent ingle crytal pecimen, it can be deduced that the plate ormation i accompanied by a macrocopic deormation, called hape change or hape deormation. Moreover, a well-dened interace or contact plane between the autenite and the martenite, the o-called habit plane, i oberved. From a crytallographic point o view, we jut recall that in general MAparent phae have uper-lattice BCC tructure and are claied a -phae alloy. The martenite crytal have periodic tacking order tructure. ince in the martenite atom o dierent radii are packed without any ymmetry, the uper-lattice tructure tend to deorm lightly, reulting in a typical monoclinic conguration. el-accommodation and martenite variant A wa recognized by Bain [5], the ormation o martenite plate cannot imply ollow the ingle-crytal mechanim dicued above. In act, thi mechanim introduce a major accommodation problem, due to the mit between the martenite and the urrounding autenite. I there i no preerred direction or the occurrence o the tranormation, the martenite take advantage o the exitence o dierent poible habit plane, orming a erie o crytallographically equivalent variant. The product phae i then termed multiple-variant martenite and i characterized by a twinned tructure, which minimize the mit between the martenite and the urrounding autenite (Figure 3). On the other hand, i there i a preerred direction or the occurrence o the tranormation (oten aociated with a tate o tre), all the martenite crytal tend to be ormed on the mot avorable habit plane. The product phae i then termed ingle-variant martenite and i characterized by a detwinned tructure, which again minimize the mit between the martenite and the urrounding autenite (Figure 4). According to the exitence o dierent type o ingle-variant martenitic pecie, the converion o each ingle-variant martenite into dierent ingle variant i poible. uch proce, known a reorientation proce, can be interpreted a a amily o martenitic phae tranormation and i aociated with change in the parameter governing the ingle-variant martenite production (hence, it i oten aociated to non-proportional change o tree). Martenite-martenite tranormation Depending on the alloy, dierent martenitic tructure may progreively be ormed rom the autenite during the cooling proce. They all have tackingorder tructure with the ame baal plane and dier jut in the tacking equence. The proce o production o thee dierent martenitic tructure can alo FA & RLT & JL CMAME 46 (997) ection 3

10 hape-memory alloy: uperelatic behavior 8 be interpreted a martenitic phae tranormation Kinetic By the kinetic o a phae tranormation we mean the phae tranormation evolution proce. The martenitic tranormation occur by nucleation and growth. Depending on the accommodation repone between the parent and the product phae (either platic or elatic), the martenitic phae tranormation are claied a non-thermoelatic (uch a in Fe-Ni) or thermoelatic (in typical - phae alloy) 2 [46]. For thermoelatic tranormation the accommodation o the martenite plate within the parent phae i eentially elatic, with no dilocation generation; the interace remain gliile, capable o backward movement, and the revere tranormation proceed through hrinkage o martenite plate. The magnitude o the hyterei, which i related to the tranormation driving orce, i in general mall. hape-memory alloy uually preent thermoelatic martenitic tranormation, guaranteeing the reveribility o the tranormation. Once the condition or the nucleation are atied, depending on the alloy, the martenitic tranormation may proceed in three dierent mode: the athermal mode, the iothermal mode and the burt mode [38]. In general, hape-memory alloy undergo the athermal (or achronic) mode, that i, the amount o martenite ormed i a unction only o the temperature and not o the length o time at which the alloy i held at that temperature. Athermal tranormation tart at well-dened temperature, which are uually inenitive to rate-eect. The phae-tranormation kinetic i in general trongly inuenced by a complex combination o internal and external parameter. Internal parameter are: the alloy ytem, the compoition and the lattice tructure including deect. External parameter are: the thermomechanical treatment and the training. Thereore, or each alloy a careul macrocopic characterization o the material kinetic through experimental invetigation i needed. 3.2 An example o hape-memory material: Nitinol Having laid down the general eature o the martenitic tranormation, we now pecialize the dicuion to nickel-titanium alloy, which are probably the hape-memory material mot requently ued in commercial application. hape-memory propertie or nickel (Ni) titanium (Ti) alloy were dicovered in the 960, at the Naval Ordnance Laboratory (NOL) [7]; hence, the acronym NiTi-NOL or Nitinol, which i commonly ued when reerring to Ni-Ti baed hape-memory alloy. tarting rom the 970, Ni-Ti ha been widely invetigated due to it requent ue in application [0, 6, 48]. 2 The non-thermoelatic phae tranormation are alo called trong, the thermoelatic phae tranormation are alo called weak. FA & RLT & JL CMAME 46 (997) ection 3

11 hape-memory alloy: uperelatic behavior 9 Metallurgical propertie A detailed dicuion o crytallographic apect, including poible el-accommodation mechanim, can be ound in Reerence [6, 24, 25, 26, 27, 33, 34]. The phae tranormation occur in temperature range which trongly depend on the material compoition. To obtain a deired phae-tranormation temperature range, the required compoition accuracy i motly higher than the one obtainable by uual error in chemical analyi. Accordingly, in mot cae the tranormation temperature are directly meaured a quality control, rather than the chemical compoition. Depending on the compoition, T AM range rom -00 o C to 00 o C. Thermomechanical propertie The amount o heat recoverable memory train and the ize o the hyterei loop trongly depend on alloy compoition, thermomechanical proceing, teting direction and deormation mode (that i, i the material i in imple tenion, imple compreion or hear) [, 3]. I train larger than the recoverable train are induced, then the reverible martenitic procee and the dilocation reulting rom platic ow interact, reulting in a reduction o the memory train. For the ull autenite-martenite phae tranormation the recoverable memory train i o the order o 8%, while the hyterei width i typically o o C. For uniaxial tate o tre and in the uual range o application the tretemperature region in which the phae tranormation may occur are delimited with good approximation by traight line with lope ranging rom 2.5 MPa/ o C to over 5 MPa/ o C. Experimental evidence how that: Phae tranormation do not exhibit preure dependence in the cae o long-aged Ni-Ti; or hort-aged Ni-Ti the R-phae (B2-R) tranition i unaected by preure, while the martenitic (R-B9 0 ) tranormation i preure dependent [7]. Phae tranormation are inenitive to temperature rate and to tre rate [8]. 3.3 Toward a continuum mechanic model The preented urvey on the MA micromechanic and in particular the dicuion on the Nitinol highlight the variety and the complexity o the phenomena occurring at the material microcale. Accordingly, the contruction o a hapememory-alloy contitutive model baed directly on the phae-tranition micromechanic eem to be an overwhelming tak. Thu, we reort to a thermomechanical continuum theory baed on an internal-variable ormalim. The pecic material micromechanic i then tied in through the choice o appropriate impliying aumption, internal variable, and evolution equation or the internal variable. FA & RLT & JL CMAME 46 (997) ection 3

12 hape-memory alloy: uperelatic behavior 0 From the previouly dicued kinematic o the MA martenitic phae tranormation, we may deduce ome indication on the tructure o the internal variable. In act, depending on the level o decription to be reached, the internal variable hould decribe the preence and the eature o the dierent phae; thu, they hould be related to the phae raction and orientation. On the other hand, rom the previouly dicued kinetic o the MA martenitic phae tranormation, we may deduce ome general eature to take into account in the contitutive model through the choice o the internal-variable evolutionary equation. For example, the model hould be able to take into conideration: Dierent initial and nal condition or each phae tranormation to reproduce the hyteretic behavior. Dependence o the phae-tranition initial condition on tre and temperature. Preure dependence o ome phae tranition in term o initial condition and evolutionary equation. Poibility o completely revering the phae-tranormation eect through appropriate thermomechanical cycle. Finally, ince experimental evidence how that the phae tranormation occurring in everal hape-memory alloy are rate-independent, the goal o the next ection i to develop inelatic rate-independent model able to reproduce the uperelatic behavior. FA & RLT & JL CMAME 46 (997) ection 3

13 hape-memory alloy: uperelatic behavior 4 Contitutive model or hape-memory alloy A general inelatic theory well uited or the development o contitutive model or material undergoing olid-olid phae tranition ha been preented in Reerence [2, 22]. uch a theory i pecialized herein to a pecic cla o material undergoing olid-olid phae tranition, the hape-memory alloy. Ater dicuing a major impliying aumption, a one-dimenional model (D- model) and a three-dimenional model (3D- model) are preented. The D- model ha been already been preented in literature [22]; however, due to the novelty o the inelatic rame adopted 3, it i reviewed here with a didactic purpoe. Moreover, ince or proportional loading the 3D- model reduce to the D- model, the latter can alo be interpreted a the bai or the development o the more complex three-dimenional model. 4. A impliying aumption: iotropy The urvey on the kinematic o the MA martenitic phae tranormation preented in ection 3.. pointed out the exitence o everal type o martenite, each one preent with it own amily o variant. To limit the dimenion o the problem to handle, in the ollowing we conider only one type o martenite. Moreover, we aume the material to be iotropic; accordingly, we do not ditinguih between the dierent ingle-variant pecie. Hence, we propoe to work with only two phae: the autenite (A) and the ingle-variant martenite (). Taking into account only two phae, the number o poible phae tranormation dratically decreae. However, to properly model the three-dimenional macrobehavior o hape-memory material, we need to take into account the ingle-variant-martenite reorientation proce. Thereore, in the mot general cae we conider three phae tranormation: converion o autenite into ingle-variant martenite (A! ) converion o ingle-variant martenite into autenite (! A) ingle-variant martenite reorientation (! ) Finally, we recall that all the propoed model are rate-independent; thi aumption, with good approximation, i appropriate or application uing Nitinol (reer to ection 3.2) a well a other hape-memory alloy requently ued. 3 everal contitutive model or hape-memory alloy have been propoed and are available in the literature. Without even attempting to give an exhautive lit, ew example can be ound in Reerence [2, 3, 35, 5, 39, 44, 45, 47]. FA & RLT & JL CMAME 46 (997) ection 4

14 hape-memory alloy: uperelatic behavior D- model The D- model reproduce the uperelatic eect or one-dimenional tate o tre (or equivalently, or proportional loading condition) Control and internal variable A control variable, we aume the uniaxial tre,, and the relative temperature, T, ubequently reerred to a the temperature. A internal variable, we may chooe either the ingle-variant martenite raction, 4, or the autenite raction, A, which are aumed to atiy the ollowing relation at each time: + A = () Accordingly, there i only one independent internal variable, choen herein to be. From equation we may alo get a relation between the rate o the raction: _ + _ A = 0 (2) For implicity, during the contruction o the model we preer to deal with both parameter, enuring that the raction evolution alway atiy equation 2; a a conequence, at each time equation i alo atied. Finally, or each phae tranormation we nd it more convenient to etablih rt the evolutionary equation aociated with the reducing raction, deriving the evolution o the other raction by enorcement o equation Phae tranormation and activation condition To reproduce the uperelatic behavior in a uniaxial tate o tre, only two phae tranormation can be conidered: the converion o autenite into inglevariant martenite (A! ) and the converion o ingle-variant martenite into autenite (! A). A howed experimentally, or uniaxial tre-temperature change and in the uual range o application, the region in which phae tranormation may occur are delimited with good approximation by traight line [4, 3]. Aigning a raction change to each proce we et 5 : _ = _ A + _ A (3) _ A = _ A A + _ A A (4) 4 By convention, the capital letter and A ued a indice reer to pecic raction ( = ingle-variant martenite, A = autenite). Moreover, i = 0 (i = ; A) indicate the abence o the correponding phae in the material, while i = indicate that the material i completely in uch a phae. 5 By convention, the upercript reer to pecic evolution procee; accordingly, the upercript A reer to the converion o autenite into ingle-variant martenite, while the upercript A reer to the converion o ingle-variant martenite into autenite. FA & RLT & JL CMAME 46 (997) ection 4

15 hape-memory alloy: uperelatic behavior 3 We now dicu the two production procee in more detail. ingle-variant martenite production To decribe the region correponding to the ingle-variant martenite production proce, we introduce the unction: with: F A =? C A T (5) F A = F A? R A (6) F A = F A? R A (7) R A = C A T A (8) R A = C A T A (9) where C A, T A and T A are material parameter, with the latter two repreenting initial and nal temperature at which the tranormation may occur at zero tre (Figure 5). The region in which the tranormation may take place i decribed by: F A > 0 ; F A < 0 ) F A F A < 0 (0) Moreover, to activate the production o ingle-variant martenite, a tre increae, a temperature decreae or a proper combination o thee action hould occur. Hence, we require: F_ A > 0 () Taking into account equation 0-, we expre the raction evolutionary equation relative to the A! phae tranormation in the orm: _ A A = K A <? F A =? _ A A _ A F A >< F_ A > where K A i a calar unction o the tate variable (control and internal variable) and < > i the Macaulay bracket, dened a <x> = (x + jxj)=2 where j j i the abolute value. Autenite production imilarly to the above, we introduce the unction: with: F A =? C A T (2) F A = F A? R A (3) F A = F A? R A (4) R A = C A T A (5) R A = C A T A (6) FA & RLT & JL CMAME 46 (997) ection 4

16 hape-memory alloy: uperelatic behavior 4 where C A, T A and T A are material parameter, with the latter two repreenting initial and the nal temperature at which the tranormation may occur at zero tre (Figure 6). The region in which the tranormation may take place i decribed by: F A < 0 ; F A > 0 ) F A F A < 0 (7) Moreover, to activate the production o autenite a tre decreae, a temperature increae or a proper combination o thee action hould occur. Hence, we require: _ F A < 0 (8) Taking into account equation 7 and 8, we expre the raction evolutionary equation relative to the! A phae tranormation in the orm: _ A = K A <? F A _ A A =? _ A F A where K A i a calar unction o the tate variable. ><? F_ A > Remark 4. Becaue o the general ramework in which the model i developed, there i no limitation to the relative poition o the two phae-tranition zone; hence, they may interect or they may be dijoint, ince neither cae would be problematic or the contitutive model Flow rule To complete the pecication o the rate equation, we need to chooe K A and K A. We retrict attention to very imple rt-order orm. A poible choice i an evolution o exponential type, uch a: _ A A _ A =? _ A =? _ A <? F =? A A A jf A <? F A =? A A jf A F A > F A j F A > F A j < _ F A > (F A ) 2 (9) <? _ F A > (F A ) 2 (20) where A and A are calar contant, meauring the rate at which the tranormation proceed. A econd poible choice i an evolution o power type, uch a: _ A A _ A =? _ <? F A = + A A A jf A =? _ <? F A A =? A A jf A F A > F A j F A > F A j < F _ A > F A <? F_ A > F A (2) (22) FA & RLT & JL CMAME 46 (997) ection 4

17 hape-memory alloy: uperelatic behavior 5 In the ollowing we limit the dicuion o the power-type evolution to A = A = ; hence, the evolution equation i termed a linear type. A dicuion o the propoed evolutionary equation can be ound in Reerence [2]. Clearly, other choice or K A and K A are poible [29]; however, rom experimental reult a rt-order kinetic rule eem to be adequate [8]. ince the model ha only one independent internal variable (choen to be ), we need only one evolutionary equation. Recalling equation -3, we have: _ = _ A + _ A = A (? ) <? F A jf A F A > F A <? F? A A F A F A j jf A j > < F_ A > (23) (F A ) 2 <? F_ A > (F A ) 2 or the exponential model and: _ = _ A + _ A =?(? ) <? F A jf A F A > F A <? F A F A? jf A F A j j > < _ F A > F A <? _ F A > F A (24) or the linear model. The Macaulay bracket manage the choice o the active evolution proce. We may expre the ow rule alo in the ollowing equivalent orm: _ = H A A (? ) or the exponential model and: _ F A F A 2 + H A A _ F A F A 2 (25) _ =?H A (? ) _ F A F A + H A _ F A F A (26) or the linear model. The calar parameter H A and H A embed the phaetranormation activation condition and are dened by the relation: H A = H A = ( i F A > 0 ; F A < 0 ; F_ A > 0 0 otherwie ( i F A < 0 ; F A > 0 ; F_ A < 0 0 otherwie (27) (28) Remark 4.2 The ow equation can be integrated in cloed orm, a decribed in Reerence [2]. FA & RLT & JL CMAME 46 (997) ection 4

18 hape-memory alloy: uperelatic behavior Tet example We now tet the ability o the model to reproduce the uperelatic behavior under multiple tre cycle. Retricting the dicuion to a mall-deormation regime, we aume an additive decompoition o the total train : = e + L (29) where e i the elatic train and L i the maximum reidual train 6. The elatic train i aumed to be linearly related to the tre: = E e (30) with E the elatic modulu. We limit the tet example to the exponential ow rule and chooe the ollowing material parameter: E = 000 MPa ; C A = C A = MPa/ o C L = 0 ; A = A = 0 MPa T A = 0 o C ; T A = 70 o C ; T A = 90 o C ; T A = 30 o C We aume to tart alway with a pecimen in the autenitic phae ( = 0) and to keep the temperature contant (T = 60 o C > T A ). In the rt imulation (Figure 7) we induce a complete et o tranormation (that i, a complete A! and a complete! A tranormation), ollowed by partial unloading-reloading cycle. A partial reloading implie an incomplete direct tranormation (A! ), while a partial unloading implie an incomplete revere tranormation (! A). Note that the model decribe a erie o loop, which are internal to the complete loading-unloading cycle; uch internal loop preent a ratcheting eect which tabilize ater a ew cycle. A tet example, we alo conider the cae o partial loading with complete unloading, the cae o partial unloading with complete loading and the cae o partial loading and partial unloading (Figure 8-0). Comparing thee reult with experimental data available in the literature [9, 30, 35], it i poible to conclude that the propoed model ha an appropriate qualitative behavior. 6 The maximum reidual train L, regarded a a material contant, i a meaure o the maximum deormation obtainable by multiple-variant martenite detwinning, hence, it i a meaure o the maximum deormation obtainable aligning all the ingle-variant martenite in one direction. Within a mall-deormation regime the aumed additive train decompoition i conitent with the experimental obervation in which or zero tre the reidual train i aociated only with the amount o ingle-variant martenite. FA & RLT & JL CMAME 46 (997) ection 4

19 hape-memory alloy: uperelatic behavior D- model The 3D- model reproduce the uperelatic behavior or three-dimenional tate o tree; hence, it i a three-dimenional generalization o the D- model Control and internal variable A control variable, we aume the tre,, and the temperature, T. To capture the phae tranormation eect, we introduce the tranormation train tr, given by: tr = L u (3) where u i the caled tranormation train and L i a calar parameter repreenting the maximum deormation obtainable only by detwinning o the multiplevariant martenite. Accordingly, a internal variable, we aume u and the ingle-variant martenite raction, ; thu, the model ha one tenorial and one calar internal variable Phae tranormation and activation condition We conider three phae tranormation: converion o autenite into ingle-variant martenite (A! ), converion o ingle-variant martenite into autenite (! A), reorientation o the ingle-variant martenite (! ). Aigning a change o u and to each procee, we et: _u = _u A + _u A + _u (32) _ = _ A + _ A (33) Note that the reorientation proce occur at contant value o the martenite raction, conequently, there i no correponding contribution in the evolution o the ingle-variant martenite raction, that i, _ = 0. Each o the three phae tranormation i aumed to occur in a pecic region o the control-variable hyperplane -T, a dicued in the ollowing. Converion o autenite into ingle-variant martenite (A! ) A mentioned in ection 3, ome phae tranormation are preure-dependent. To model uch an eect, we introduce a Drucker-Prager-type loading unction: F A ( ; T ) = ktk + 3 p? C A T where t i the deviatoric part o the tre (dened a: t =?tr( )=3), p i the preure, C A and are material parameter and k k indicate the Euclidean FA & RLT & JL CMAME 46 (997) ection 4

20 hape-memory alloy: uperelatic behavior 8 norm. Generalizing the one-dimenional model (equation 5-9), the initial and nal tranormation unction can be expreed a: F A = F A? R A F A = F A? R A with: R A = R A = 2 4 A 2 4 A A? C A T A 3 + A? C A T A where A, A, T A and T A are all material parameter. We aume that the condition or the converion o autenite into ingle-variant martenite are: F A > 0 ; F A < 0 ; _ F A > 0 (34) and that the caled tranormation train correponding to the A! phae tranormation evolve a ollow: where: N A = M A km A k _u A = _ A N A (35), M A (36) Recalling again the dicuion on the D- model, or the evolution o the inglevariant martenite raction we may aume either an exponential orm: _ A = H A A (? ) _ F A 2 F A with A a material parameter, meauring the peed o the tranormation, or a linear orm: F _ A =?H A (? ) _ A F A The calar parameter H A embed the condition or the activation o the phae tranormation (equation 34) and it i dened by the relation: H A = ( i F A > 0 ; F A < 0 ; F_ A > 0 0 otherwie (37) FA & RLT & JL CMAME 46 (997) ection 4

21 hape-memory alloy: uperelatic behavior 9 Converion o ingle-variant martenite into autenite (! A) imilarly to the above, we introduce a Drucker-Prager-type loading unction: F A ( ; T ) = ktk + 3 p? C A T where C A i a material parameter. Generalizing the D- model (equation 2-6), the initial and nal tranormation unction can be expreed a: with: R A = R A = F A = F A? R A F A = F A? R A 2 4 A 2 4 A A? C A T A 3 + A? C A T A where A, A, T A, T A are all material parameter. We aume that the condition or the converion o ingle-variant martenite into autenite are: F A < 0 ; F A > 0 ; _ F A < 0 (38) and that the caled tranormation train correponding to the! A phae tranormation evolve a ollow: _u A = _ A N A (39) where: N A = u (40) kuk Recalling again the dicuion on the D- model, or the evolution o the inglevariant martenite raction we may aume either an exponential orm: _ A = H A A _ F A F A 2 (4) with A a material parameter, meauring the peed o the tranormation, or a linear orm: F A A = H A (42) F A The calar parameter H A embed the condition or the phae tranormation (equation 38) and it i dened by the relation: ( i F H A A < 0 ; F = A > 0 ; F_ A < 0 (43) 0 otherwie FA & RLT & JL CMAME 46 (997) ection 4

22 hape-memory alloy: uperelatic behavior 20 For later development, we note that the converion o martenite into autenite i aociated only with a recaling o the tranormation train u. ingle-variant martenite reorientation proce (! ) To model the ingle-variant martenite reorientation proce or non-proportional change o tre (change o direction or rotation), we et: with: R = F = ktk + 3p? C T F = F? R A? C T 5 (44) where C, and T are material parameter. We aume that the condition or the activation o the reorientation proce are: where: F > 0 ; _N 6= 0 (45) N = M km k, M and that the caled tranormation train correponding to the! phae tranormation evolve a _u = H _N (46) where: H = ( i F > 0 0 otherwie (47) Additional conideration From a phyical point o view it eem plauible that whenever the material ha enough energy to induce a converion o autenite into ingle-variant martenite, then it ha alo enough energy to reorient the martenite raction already preent. Hence, we et C A = C and R A = R recalling the expreion or F A and F, we have:, uch that F A = F. Moreover, M A = M = t + = M (48) ktk! N A = N t = + 3 ktk + = N (49) We can now integrate the rate equation or the caled tranormation train, obtaining the ollowing equation: u = ^N (50) FA & RLT & JL CMAME 46 (997) ection 4

23 hape-memory alloy: uperelatic behavior 2 where ^N i the current value o N i H =, otherwie it i equal to the lat value o N attained when H wa equal to. Conitency o equation 50 with the rate orm can be checked computing it time derivative and conidering the two cae H = and H = 0: I H =, then ^N = N and, by equation 50, u i in the direction o N. From equation 40 we may conclude that N A = N. Hence, the rate orm o equation 50 return: _u = _ N + _N = _ A + _ A N + _N which i exactly equal to the um o the three tranormation train rate equation 35, 39 and 46. I H = 0, then H A = 0 and only H A can be non zero. Moreover, ince H = 0, ^N i xed. Hence, the rate orm o equation 50 return: _u = _ A ^N Recalling the denition o ^N and that the! A phae tranormation i a recaling o u, we have that ^N = u=kuk, rom which we obtain equation 39. Remark 4.3 Due to the particular orm o N, the condition N _ 6= 0 required or the reorientation proce to be active (equation 46) reduce to: [I? n n] _ t 6= 0 where n = t=ktk and I i the ourth-order identity tenor. Hence, the reorientation proce occur only or non-radial change o the deviatoric tre Model review Due to the Drucker-Prager-type loading unction, we plit the tranormation train u into the volumetric and the deviatoric component: u = v + w 3 ; w = tr(u) (5) with tr() being the trace operator. The contitutive equation 50 or the tranormation train u can alo be plit into volumetric and deviatoric part: w = 3 (52) v = m (53) FA & RLT & JL CMAME 46 (997) ection 4

24 hape-memory alloy: uperelatic behavior 22 where m i the deviatoric part o ^N. The martenite raction evolution repreent the only part o the model till in rate orm: For the exponential cae we have: _ A _ = _ A = H A A (? ) _ A while or the linear cae we have: = H A A A + _ A (54) F A F A =?H A (? ) A = +H A _ F A F A _ F A F A 2 (55) 2 (56) F A F A with the uual denition or H A and H A (equation 37 and 43) Material-parameter pecialization to uniaxial tate The model preented o ar i in a three-dimenional context. Accordingly, the material parameter involved in the contitutive equation are relative to a threedimenional etting. I the model i pecialized to the cae o uniaxial tate o tre and train, then the contitutive equation hould involve the correponding one-dimenional quantitie, which are indicated herein with a uperpoed tilde and which may be eaily computed rom the correponding three-dimenional parameter through the relation: A = ~ A A = ~ A L = ~ L = A A A (57) (58) The material parameter ued or the numerical example are alway expreed in term o the one-dimenional quantitie ince they have a more clear experimental interpretation. FA & RLT & JL CMAME 46 (997) ection 4

25 hape-memory alloy: uperelatic behavior 23 5 Time-dicrete iothermal model and algorithmic implementation Thi ection decribe a time-dicrete iothermal verion o the three-dimenional 3D- model preented in ection 4.3. The algorithmic implementation o the model within a nite-element ramework and the orm o the conitent tangent tenor are dicued in detail. The dicuion i limited only to a mall deormation regime. While during the development o the time-continuou model (ection 4.3) we aumed tre and temperature a control variable, during the development o the iothermal time-dicrete model we aume the train a the only control variable. Thi choice i conitent with the act that, rom the point o view o integration cheme, the time-dicrete problem i conidered train-driven. Accordingly, we conider two time value, ay t n and t n+ > t n, uch that t n+ i the rt time value o interet ater t n. Then, knowing the train at time t n+ and the olution at time t n, we hould compute the new olution at time t n+. To minimize the appearance o ubcript (and to make the equation more readable), we introduce the convention: a n = a(t n ) ; a = a(t n+ ) where a i any generic quantity. Thereore, the ubcript n indicate a quantity evaluated at time t n, while no ubcript indicate a quantity evaluated at time t n+. We now conider a time-dicrete iothermal verion o the 3D- model under the aumption o mall deormation. 5. Kinematic and elatic contitutive equation Limiting the dicuion to a mall deormation etting, we additively decompoe the total train into an elatic part, e, and a tranormation part, tr : where, recalling equation 3: = e + tr (59) tr = L u (60) We chooe a ree-energy unction quadratic in the elatic train: = ( e ) = 2 e D e e (6) where D e i the ourth-order rank elatic modulu tenor. Accordingly, the tre i given = De e = D e (? L u) (62) FA & RLT & JL CMAME 46 (997) ection 5

26 hape-memory alloy: uperelatic behavior 24 Recalling the Drucker-Prager orm o the loading unction and the hypothei o iotropy (ee ection 4. or a dicuion) and auming the elatic tenor D to be contant and iotropic, it i convenient to plit equation 59, 60 and 62 into the volumetric and the deviatoric component. Uing the notation o ection 4.3.3, we have: and: = e? 3 (63) e = e e? e 3 (64) e = e e + L m (65) = e + 3 L (66) p = K e = K (? L w) (67) t = 2G e e = 2G (e? L v) (68) where K i the bulk modulu and G i the hear modulu. 5.2 Time-dicrete model Uing a backward-euler integration ormula, the dicrete orm o equation are given by: where: w = 3 (69) v = m (70) = ;n + A + A (7) A = A = Z tn+ tn Z tn+ tn _ A dt (72) _ A dt (73) imilarly, uing a backward-euler cheme to integrate the time-continuou evolutionary equation yield the correponding time-dicrete evolutionary equation. Written in reidual orm and ater clearing raction, the time-dicrete evolutionary equation pecialize to: R A = F A 2 A R A = F A 2 A? H A A (? ) F A? F A n = 0 (74)? H A A F A? F A n = 0 (75) FA & RLT & JL CMAME 46 (997) ection 5

27 hape-memory alloy: uperelatic behavior 25 or the exponential cae (equation 55-56), while or the linear cae (equation 57-58) they pecialize to: R A = F A A + H A (? ) F A? F A n = 0 (76) = 0 (77) R A = F A A? H A F A? F A n The quantitie A and A can be computed expreing F A and F A a unction o A and A and requiring the atiaction o R A and R A. 5.3 Integration algorithm or the time-dicrete model A return-map algorithm i ued a the integration cheme or the time-dicrete model. Initially uggeted by Maenchen and ack [28] and Wilkin [50] or the olution o platicity ormulation, the return map provide an ecient and robut integration cheme baed on a dicrete enorcement o the evolutionary equation [4, 42]. It belong to the amily o elatic-predictor/inelatic-corrector algorithm and, hence, i a two-part algorithm. In the rt part, a purely elatic trial tate i computed; in the econd part, i the trial tate violate the contitutive model, an inelatic correction i computed uing the trial tate a an initial condition. The detail o the algorithm or the time-dicrete model propoed here are:. Trial tate. Aume that no phae tranormation occur (i.e. w = w n, v = v n, = ;n, A = A = 0). Accordingly, compute the trial preure and the trial deviatoric part o the tre: p T R = K(? L w n ) t T R = 2G (e? L v n ) 2. Check reorientation proce (! ). Compute F I F > 0 then et H = update w, m and v recompute p T R and t T R ele et H = 0 end i 3. Check olution with = 0 and =. Compute F A = and F A =0 I F A = > RA then FA & RLT & JL CMAME 46 (997) ection 5

28 hape-memory alloy: uperelatic behavior 26 = i the appropriate olution olution ound et H A = ele = i not the appropriate olution end i I F A < RA then =0 = 0 i the appropriate olution olution ound ele = 0 i not the appropriate olution end i I olution ound then kip to 7 4. Check A! tranormation. Compute F A, F A and Fn A I F A > 0, F A > F A n et H A = ele et H A = 0 end i 5. Check! A tranormation. Compute F A, F A and Fn A I F A < 0, F A < F A n et H A = ele et H A = 0 end i and ;n < then and ;n > 0 then 6. Compute martenite evolution. I H A = or H A = then compute A and A update, v, w, p and t end i 7. Compute algorithmic tangent. I H A = or H A = or H = then compute algorithmic inelatic tangent ele compute elatic tangent end i Remark 5. tep 6 i perormed olving equation or equation FA & RLT & JL CMAME 46 (997) ection 5

29 hape-memory alloy: uperelatic behavior 27 uing a Newton-type iterative algorithm: where: 8 >< >: R? = A k+ A k A 9 >= >; = 8 >< A A A c d and the upercript k indicate the iteration index ? k k 9 8 >= >< 9 k R A >= >;? R? >: k R A >; = " a b #? = " B C D E # (78) (79) 5.4 Algorithmic tangent In the ollowing we addre the contruction o the tangent matrix conitent with the time-dicrete model. The ue o a conitent tangent tenor preerve the quadratic convergence o the Newton method, which we adopt or the incremental olution o the momentum equation rom a nite-element cheme. Linearizing equation 67-7, we get: dp = K(d? L dw) dt = 2G(de? L dv) dw = 3 d A dv = d A + A + d A m + H dn We need to compute dn only or the cae H =. When H =, t, e and v are all in the ame direction (equation 68); hence: In Reerence [2] we how that: where: dn = d t ktk! n = = d t ktk = e kek! e = [I? n n] de kek kek kek = ktk=(2g) + L and I i the ourth-order identity tenor. Uing thee reult in the linearized elatic relation, we get: dp = Kd? 3K L d A + d A " # dt = 2G I? H L (I? n n) de? 2G L d A + d A m kek FA & RLT & JL CMAME 46 (997) ection 5

30 hape-memory alloy: uperelatic behavior 28 The calar quantitie d A and d A are computed rom the linearization o the dicrete-time evolutionary equation (equation or the exponential model. Accord- and equation or the linear model) a unction o, A ingly: and A dr A d A dr A d A which can be written a: R ( d A d A ) d d =? 8 9 A : d A : d For the particular model preented here, we have: : d = 0 : d = 0 (80) A A A @F @F = 2Gn + 3k = 2Gn + 3k Thu, uing the expreion or R? rom equation 79, we can olve the ytem 80 in term o d A and d A : where: d A = T A n + T A 2 : d d A = T A n + T A 2 : d T A = 2G BA A + CA A T A 2 = 3k BA A + CA A T A = 2G DA A + EA A T A 2 = 3k DA A + EA A FA & RLT & JL CMAME 46 (997) ection 5