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A b stra c t : In this article discrete spectral approximations to the nonlinear evolutionary partialdi®erential equations that model the dynamics of thermomechanical solid-solid phase transitionsin one-dimensional shape memory alloys with non-convex Landau-Ginzburg potentials are con-structed. By using the theories of analytic semigroups and interpolation spaces and a generalizationof Gronwall's lemma for singular kernels, the convergence of the approximations is proved. For thealloy Au Cu Zn numerical results are shown under di®erent external distributed actions and2 3 3 0 4 7initial conditions.

K e y w o r d s: S h a p e M em o ry A lloy s, n o n -con vex p oten tial, h y steresis, co n servatio n law s,in itial-b o u n d ary va lu e p ro b lem , sp ectral ap p rox im a tio n s.

A M S S u b je c t C la ssi¯ c a tio n s: 35A 35 , 3 5A 40 , 3 5M 05 , 7 3U 05 , 7 3C 3 5.

1 . In tro d u c tio n

In th is a rticle w e co n sid er th e follow in g on e-d im en sion a l n on lin ear in itia l-b o u n d ary va lu e

yThe work of the authors was supported in part by CONICET, Consejo Nacional de Investigaciones

Cient¶³¯cas y T¶ecnicas of Argentina and UNL, Universidad Nacional del Litoral through project CAI+D

94-0016-004-023.

2

p rob lem : ¡ ¢3 5½ u ¡ ¯ ½ u + ° u = f (x ;t) + 2 ® (µ ¡ µ )u ¡ 4® u + 6 ® u ; x 2 (0;1); 0 · t · Ttt x x t x x x x 2 1 x 4 6x x x

(1.1a )2C µ ¡ k µ = g (x ;t) + 2 ® µ u u + ¯ ½ u ; x 2 (0;1); 0 · t · T (1.1b )v t x x 2 x x t x t

u (x ;0 ) = u (x ); u (x ;0) = v (x ); µ (x ;0) = µ (x ); x 2 (0 ;1 ) (1.1 c)0 t 0 0

u (0;t) = u (1;t) = u (0;t) = u (1;t) = 0 ; 0 · t · T (1.1d )x x x x

µ (0;t) = µ (1;t) = 0 ; 0 · t · T (1.1 e)x x

S y stem (1.1 a-e) arises from th e co n servation law s gov ern in g th e th erm om ech an ica l p ro -

cesses in o n e-d im en sio n al S h a p e M em o ry A lloy s (S M A ). T h ese p ro cesses a re ch ara cterized

b y solid -solid p h ase tra n sition s (m a rten sitic tra n sform ation s). E q u a tio n s (1 .1 a) an d (1.1b )

re° ect th e con serva tio n o f lin ea r m om en tu m an d en ergy, resp ectively. T h e fu n ction s an d vari-

ab les p resen t in sy stem (1.1a-e) h ave th e follow in g p h y sical m ean in g: u (x ;t) = tra n sverse

d isp lacem en t, µ (x ;t) = a b so lu te tem p eratu re, C = sp eci c h ea t, k = th erm al co n d u ctiv ityv

co e± cien t, ¯ = v isco sity con sta n t, f (x ;t) = d istrib u ted lo ad s (in p u t), g (x ;t) = d istrib u ted

h eat so u rces (in p u t), T = p rescrib ed ¯ n al tim e, ® , ® , ® , µ , ° are n on n ega tiv e con stan ts2 4 6 1

-d ep en d in g on th e m ateria l b ein g con sid ered - ap p ea rin g in th e free en ergy p o ten tial w h ich

is tak en in th e L a n d au -G in zb u rg formµ ¶µ °2 4 6 2ª (²;² ;µ ) = ¡ C µ ln + C µ + C + ® (µ ¡ µ )² ¡ ® ² + ® ² + ² (1.2)x v v 2 1 4 6 xµ 22

w h ere ² = u is th e lin earized sh ea r strain . T h e con stan ts µ an d µ in (1.2) a re tw o criticalx 1 2

tem p era tu res a n d C rep resen ts a ¯ x ed en erg y referen ce lev el. T h e b o d y is assu m ed to b e a

sim p ly su p p orted u n it-len gth b ea m th erm ally in su la ted at b oth en d s.

T h e P D E 's in (1.1a -b ) are cou p led a n d n on lin ear d u e to th e term s co m in g from th e

p artial d eriva tiv es o f th e free en ergy. T h e ¯ rst eq u a tio n ca n b e regard ed a s a n o n lin ear

h y p erb olic eq u a tio n in u w h ile th e seco n d is a n o n lin ear p ara b olic eq u a tio n in µ (for a

d etailed d erivatio n o f eq u a tio n s (1 .1a -b ) see [28 ]).

A lth o u gh th ere are several rep resen ta tion s for th e free en ergy p oten tial o f p seu d o ela stic

m ateria ls (see for in stan ce [1 2], [1 3], [2 6], [2 7], [3 1]) th e fo rm (1.2) seem s to b e th e sim p lest

3

on e w h ich is a b le to rep ro d u ce several p h en om en a -su ch as h y steresis, sh ap e m em ory a n d

su p erelasticity - ob serv ed in rea l m aterials u n d er d i® eren t ex tern al th erm o m ech a n ical a ction s.

F or va lu es of µ close to µ , ª is a n o n co n v ex fu n ction of ² a n d th e stress-strain law s ob ta in ed1

from (1 .2 ) are stron gly tem p era tu re-d ep en d en t (see F igu re 1). A t low tem p eratu res th ese

cu rves ex h ib it an elasto -p lastic b eh av io r at sm a ll lo ad s an d a seco n d ela stic b ra n ch at large

load s, w h ich p erm its th e b o d y to w ith stan d fo rces b eyo n d th e p la stic y ield , after w h ich ,

su b seq u en t u n load in g p ro d u ces resid u a l d eform ation . In th e in term ed ia te tem p era tu re ra n ge

th e b eh av ior is su p erela stic, a lso called p seu d o elastic. H ere, a p la stic y ield is also fou n d .

H ow ev er, loa d in g b ey on d th is p lastic y ield follow ed b y com p lete u n loa d in g d o es n ot lead to

resid u al d efo rm a tion b eca u se of th e ex isten ce of an in term ed ia te ela stic b ra n ch w h ich th e

b o d y rea ch es b y creep in g b a ck a fter th e loa d fa lls b eyon d a certa in critica l valu e. F in a lly,

in th e h igh tem p era tu re ra n ge th e b eh av ior is alm o st lin early ela stic w ith h ig h er m o d u lu s of

ela sticity for h ig h er tem p era tu res. H y steresis lo o p s are o b served in th e stress-stra in cu rves

at low an d in term ed ia te tem p eratu res (see [28 ] an d th e referen ces th erein ).

D u e to th eir u n iq u e ch ara cteristics S M A h ave a lread y fo u n d a b road sp ectru m o f a p -

p lication s a m o n g w h ich w e ¯ n d orth o d on tic a n d oth er d en ta l d ev ices ([4]), h eat en gin es,

tem p era tu re sw itch es a n d fu ses, p ip e co u p lin g d ev ices ([1 4]), h y b rid co m p osites ([2 4]) a n d

several in terestin g a p p licatio n s in M ed icin e ([9 ], [1 4], [2 5]).

S in ce th e d iscovery o f N iT in ol (a N ikel-T ita n iu m a lloy ) b y B u eh ler ([17 ]) in 196 2 several

m ath em a tical m o d els w ere p ro p o sed an d stu d ied ([1], [2], [3 ], [12], [1 3], [16 ], [18], [1 9], [2 0],

[32]). M o st of th is m o d els, h ow ev er, w ere static an d d id n ot take in to acco u n t th e stro n g

cou p lin g b etw een th e m ech an ical an d th erm al p ro p erties, w h ich is o n e of th e d istin gu ish in g

featu res p ossessed b y S M A . It w as n ot u n til recen t y ea rs th a t m ath em atica l m o d els w ere

ab le to d eal w ith m o st o f th e u n u su a l p rop erties o f S M A an d , a t th e sam e tim e, to a llow

for th e in clu sio n o f b ou n d a ry a n d d istrib u ted ex tern al a ction s th a t ca n b e u sed as con trol

variab les ([21 ], [2 2], [26 ], [2 7], [30], [31 ], [2 8]). T h is article follow s th e ap p roach in tro d u ced

recen tly in [28].

4

2 . S ta te -S p a c e F o rm u la tio n a n d P re lim in a rie s

In th is section w e sh all fo rm u late th e in itial-b ou n d a ry valu e p rob lem (1.1a-e) as an

ab stract sem ilin ear C a u ch y p rob lem in an ap p ro p ria te H ilb ert sp a ce a n d b rie° y reca ll so m e

p relim in a ries w h ich w ill b e n eed ed la ter on . © ª: 9W e d e¯ n e th e ad m issib le p ara m eter set Q = q = (½ ;k ;C ;¯ ;® ;® ;® ;µ ;° )jq 2 IR ,v 2 4 6 +

1 2 2 2an d for q 2 Q th e sta te sp ace Z as th e H ilb ert sp ace H (0 ;1) \ H (0 ;1) £ L (0;1 ) £ L (0 ;1)q 0

w ith th e in n er p ro d u ct*Ã ! Ã !+ Z Z Z1 1 1~uu C: v00 00 ~v ~v; = ° u (x )~u (x ) d x + ½ v (x )~v (x ) d x + µ (x )µ (x ) d x :~ k0 0 0µ µ

q

N ex t, for q 2 Q , th e op erator A o n Z is d e¯ n ed b yq q¯(Ã ! )4 00 00u 2 H (0;1 ); u (0) = u (1 ) = u (0 ) = u (1 ) = 0u ¯ 1 2vD (A ) = 2 Z v 2 H (0;1) \ H (0 ;1)¯q q 0¯ 2 0 0µ µ 2 H (0 ;1);µ (0 ) = µ (1) = 0Ã !uvan d for z = 2 D (A ),q

µ 0 1Ã ! Ã !0 I 0u u° 4 2: ¡ D ¯ D 0@ Av vA = ½qk 2µ µ0 0 DC v

n: @nw h ere D = .n@ x

W e a ssu m e th at th e fu n ction s f (x ;t), g (x ;t) satisfy th e follow in g h y p oth esis.

2(H 1 ). F o r ea ch ¯ x ed t ¸ 0 , th e fu n ctio n s f (x ;t), g (x ;t) a re in L (0;1) a n d th ere ex ist

2n o n n eg a tiv e fu n ctio n s K (x ), K (x ) 2 L (0;1) su ch th a tf g

jf (x ;t ) ¡ f (x ;t )j · K (x )jt ¡ t j; jg (x ;t ) ¡ g (x ;t )j · K (x )jt ¡ t j1 2 f 1 2 1 2 g 1 2

fo r a ll x 2 (0;1), t ;t 2 [0 ;T ].1 2 Ã !u (x )0

+v (x )W e a lso d e¯ n e z (x ) = a n d F (q;t;z ) : Q £ IR £ Z ! Z b y00 q q0

µ (x )0 Ã !0

f (q;t;z )F (q ;t;z ) = ;2

f (q;t;z )3

5

w h ere

¡ ¢3 5½ f (q;t;z )(x ) = f (x ;t) + 2® (µ ¡ µ )u ¡ 4® u + 6® u ;2 2 1 x 4 6x x x

2C f (q;t;z )(x ) = g (x ;t) + 2® µ u u + ¯ ½ u :v 3 2 x x t x t

W ith th e ab ove n ota tion , th e IB V P (1 .1 a-e) can b e fo rm a lly w ritten a s th e follow in g sem i-

lin ear C au ch y p ro b lem in th e H ilb ert sp a ce Z :q½d z (t) = A z (t) + F (q ;t;z ); 0 · t · Tqd t(P ) (2.1)z (0) = z 0

T h e follow in g resu lts ca n b e ea sily d eriv ed from th eorem s 3 .7 an d 3.11 in [2 8] w ith o n ly

slig h t m o d i ca tio n s in o rd er to tak e in to accou n t for th e slig h tly d i® eren t b o u n d ary con d itio n s

b ein g co n sid ered h ere. S in ce th e m o d i catio n s n eed ed a re triv ia l an d n ot im p orta n t for th e

go als p u rsu ed b y th is article, w e d o n ot g iv e d eta ils h ere.

T h e o re m 2 .1 . ([28]) L et q 2 Q , A : D (A ) ½ Z ! Z a s p rev io u sly d e¯ n ed . T h enq q q q

i) A is d issip a tiv e;q

¤ ¤ii) T h e a d jo in t A o f A is a lso d issip a tiv e a n d is g iv en b y D (A ) = D (A ),q qq q0 1Ã ! Ã !0 ¡ I 0u u° 4 2: D ¯ D 0@ Av vA = ½qk 2µ µ0 0 DC v

iii) T h e o p era to r A h a s p u re p o in t sp ectru m ¾ (A ) g iv en b yq p q

© ª © ª1 1 1+ ¡¾ (A ) = ¸ [ ¸ [ f ® g ;p q nn n n = 0n = 1 n = 1

w h ere ³ ´p kp+ ;¡ 2 22¸ = ¹ ¡ r(q ) § r (q ) ¡ 1 ; ® = ¡ n ¼n nnC v

a n d p4 4 ¯ ½° n ¼

¹ = ; r(q ) = :pn½ 2 °

T h e co rresp o n d in g n o rm a lized eig en v ecto rs in Z a re, resp ectiv ely,qà ! à ! à !e (x ) k e (x ) 0n n n+ ¡ 0; ; ;¸ e (x ) k ¸ e (x )n n nn n

 (x )0 0 nn = 1 ;2 ;¢¢¢ n = 1 ;2 ;¢¢¢ n = 0 ;1 ;¢¢¢

6

w h ere

· 1 = 2+ 2¹ + j j 2n n2k = ; e (x ) = sin (¼ n x ); n = 1 ;2;¢¢¢ ;nn ¡ 2 + 2¹ + j j ½ (¹ + j j )n nn n

µ ¶ µ ¶1 = 2 1 = 2k 2k

 (x ) = ;  (x ) = cos(¼ n x ); n = 1 ;2 ;¢¢¢ ;0 nC Cv v

iv ) T h e o p era to r A g en era tes a n a n a ly tic sem ig ro u p o f co n tra ctio n s T (t) o n Z .q q q

T h e o re m 2 .2 . ([2 8]) (L o ca l ex isten ce o f so lu tio n s) L et q 2 Q a n d A a s d e¯ n ed a b ov e.q

T h en fo r a n y in itia l d a ta z 2 D (A ) th ere ex ists t = t (z ) su ch th a t th e IV P (P ) h a s a0 q 1 1 0

1u n iq u e cla ssica l so lu tio n z (t) 2 C ([0;t ) : Z ) \ C ((0;t ) : Z ).q 1 q 1 q

It w ill b e u sefu l to in tro d u ce som e n o ta tio n fo r certa in in terp olation sp aces. If X is a

pB an ach sp a ce an d p ¸ 1 , L (X ) w ill d en o te th e B an ach sp a ce of a ll B o ch n er m easu ra b le¤ R: 1p p d tm ap p in gs u : [0;1 ) ! X su ch th at ku k = k u (t)k < 1 . If X , X are tw op 0 1XL (X ) 0 t¤

B an ach sp aces w ith X co n tin u ou sly a n d d en sely em b ed d ed in X , p ¸ 1 a n d º 2 (0;1 ), w e0 1

d en ote w ith (X ;X ) th e sp a ce o f averag es, -or \ rea l" in terp o la tion sp ace-0 1 º ;p ¯½ ¾¡ º p¯: 9 u : [0 ;1 ) ! X ;i = 0 ;1; t u 2 L (X );i i 0 0¤¯(X ;X ) = x 2 X :0 1 1 1¡ º pº ;p t u 2 L (X ) an d x = u (t) + u (t) a.e.1 1 0 1¤

E n d ow ed w ith th e n orm ¯( )¡º pt u 2 L (X );0 0¤: ¯¡º 1¡ º 1¡º pp pk x k = in f k t u k + k t u k t u 2 L (X ) a n d ;¯0 1(X ;X ) 1 1L (X ) L (X )0 1 ¤0 1º;p ¤ ¤

x = u (t) + u (t) a.e.0 1

(X ;X ) is a B an a ch sp ace. In th e p a rticu la r case w h en p = 2 an d X , X are H ilb ert0 1 0 1º ;p

sp a ces, w e sh a ll d en ote (X ;X ) = [X ;X ] (see [5]).0 1 0 1º ;2 º

If B is th e in ¯ n itesim al gen erator o f a n an a ly tic sem igro u p S (t) o n a B a n ach sp ace X

±su ch th a t 0 b elon gs to th e resolven t set of B , ½ (B ), th en th e fraction al ±-p ow ers (¡ B ) are

w ell d e¯ n ed , closed , lin ear, in vertib le o p erato rs for an y ± > 0 (see [23 , p p . 6 9-7 5]). M o reov er,° °¡ ¢ :± ±° °D (¡ B ) en d ow ed w ith th e top ology o f th e grap h n orm k x k = (¡ B ) x is a B an a ch±

sp a ce. T h e fo llow in g resu lt can b e fou n d in [5 ]

7

T h e o re m 2 .3 . ([5 ]) L et X b e a H ilb ert sp a ce a n d B th e in ¯ n itesim a l g en era to r o f a n

a n a ly tic sem ig ro u p o n X su ch th a t 0 2 ½ (B ). T h en , fo r a n y ± 2 (0;1 ), th e H ilb ert S p a ce¡ ¡ ¢ ¢±D (¡ B ) ;k ¢ k is iso m o rp h ic to th e in terp o la tio n sp a ce [D (B );X ] . in th e sen se o f a n± 1¡±

iso m o rp h ism .

F rom T h eorem 2.1, it follow s th at f ¸ 2 C : R e(¸ ) > 0g ½ ½ (A ). H en ce th e fraction alq

p ow ers o f I ¡ A are w ell d e¯ n ed , closed , lin ea r, in v ertib le o p erators an d , for an y ± 2 (0;1),q ° °¡ ¢ :± ±° °D (I ¡ A ) en d ow ed w ith th e n o rm k z k = (I ¡ A ) z is a B an a ch sp ace, w h ich w eq ± q q

d en ote w ith Z . T h is sp ace co in cid es w ith th e in terp olation sp ace [D (A );Z ] . T h eq ;± q q 1¡±

follow in g resu lt co n cern in g th e reg u la rity of F w as p roved in [29].

1T h e o re m 2 .4 . ([2 9]) A ssu m e (H 1 ) h o ld s. L et q 2 Q , 0 < ² < a n d U a b o u n d ed su b set4³ ´

3 + ²4o f [0;T ] £ D (I ¡ A ) . T h en th ere ex ists a co n sta n t L > 0 d ep en d in g o n U , ² a n d q,q

su ch th a t ³ ´3k F (q;t ;z ) ¡ F (q;t ;z )k · L jt ¡ t j + k z ¡ z k1 1 2 2 q 1 2 1 2 + ²4

fo r ev ery (t ;z ); (t ;z ) 2 U . M o reov er, th e co n sta n t L ca n b e ch o sen in d ep en d en t o f q o n1 1 2 2

co m p a ct su b sets o f Q .

O b se rv a tio n . T h e op era tor I ¡ A ab ove can b e rep la ced b y ´ I ¡ A for an y ´ > 0 w ith ou tq q

ch a n gin g an y of th e a ssertio n s. T h e ch oice ´ = 1 h as n o p a rticu lar m ean in g.

3 . S p e c tra l A p p ro x im a tio n s

In th is sectio n ¯ n ite-d im en sion a l ap p rox im a tin g so lu tion s to p ro b lem (P ) are d e¯ n ed

an d th eir con v erg en ce to th e ex act solu tion is sh ow n .

In th e seq u el th e p aram eter q 2 Q w ill b e ¯ x ed , so, w h erever it is clear fro m th e con tex t,

w e sh a ll su p p ress it fro m th e n ota tion .

F or ¯ x ed N 2 IN letà ! à ! à !sin ¼ n x sin ¼ n x 0

: : :N + N ¡ N¸ sin ¼ n x ¸ sin ¼ n x 0¯ (x ) = ; ¯ (x ) = ; ¯ (x ) = ;n nn N + n 2 N + n0 0 cos ¼ (n ¡ 1 )x

8

+ ;¡ Nfor n = 1 ;2;¢¢¢ ;N , w h ere ¸ are a s in T h eorem 2.1, a n d let u s d e¯ n e Z to b e th e sp ann1[© ª3 N: NN^of ¯ = ¯ (x ) en d ow ed w ith th e Z -n orm . T h en Z is d en se in Z an d , sin ce th eN n n = 1

N = 1N N N¯ 's are eig en vectors o f A , it fo llow s th at Z is in varian t u n d er A . N ote a lso th at Z isn

itself a H ilb ert sp ace. ¡ ¢N NN ex t, w e d e¯ n e th e ¯ n ite-d im en sio n al ap p rox im a tin g p ro b lem P in Z , a s fo llow s.

½ d N N N N N¡ ¢ z (t) = A z (t) + F (t;z (t)); 0 · t · TN d tPN Nz (0 ) = P z 0

N N N Nw h ere P : Z ! Z is th e orth o gon al p ro jectio n of Z on to Z , A is th e restriction of A to1[: NN N NZ a n d F (t;z ) = P F (t;z ). T h e d en sity o f Z in Z im p lies th e stro n g co n vergen ce of

N = 1NP to th e id en tity. M o reov er, a straigh tforw ard calcu lation u sin g th e sp ectral d eco m p osition

1[ ° °N N° °of A sh ow s th at Z is also d en se in Z an d P z ¡ z ! 0 , 8 z 2 Z .q ;± q ;±±

N = 1N N NS in ce Z h as ¯ n ite d im en sion , th e o p erator A o n Z is b ou n d ed an d lin ea r, an d a

Nfortiori, it gen erates a u n iform ly co n tin u ou s sem ig rou p of b o u n d ed lin ea r o p erato rs T (t)

Non Z . ¡ ¢NW e h av e th e follow in g resu lt o n lo ca l ex isten ce of solu tio n s o f p rob lem P .

NT h e o re m 3 .1 . L et z 2 Z . T h en , fo r a n y p o sitiv e in teg er N , th ere ex ists t > 0 su ch th a t0 1¡ ¢N NP h a s a u n iq u e so lu tio n o n [0 ;t ).1

3P roof. L et ± 2 ( ;1), z 2 Z a n d N 2 IN b e ¯ x ed . B y v irtu e of T h eorem 2.4, th ere ex ists a04

0 0con sta n t L (r;t ) su ch th at fo r an y r > 0 an d t > 0

¡ ¢0 ±k F (t;z ) ¡ F (s;w )k · L (r;t ) jt ¡ sj + k (I ¡ A ) (z ¡ w )kZ Z

¡ ¢0 ±for every t;s 2 [0 ;t ] an d z ;w 2 D (I ¡ A ) w ith k z k · r, k w k · r. T h en , fo r ev ery± ±

9

0 Nt;s 2 [0;t ], an d z ;w 2 Z w ith k z k · r , kw k · r w e h av e± ±

° °N N N° °k F (t;z ) ¡ F (s;w )k N = P (F (t;z ) ¡ F (s;w ))Z Z

· k F (t;z ) ¡ F (s;w )k Z¡ ¢0 ±· L (r;t ) jt ¡ sj + k (I ¡ A ) (z ¡ w )k Z

0= L (r;t ) (jt ¡ sj + k z ¡ w k )±

0· L (r;t )C (±;N ) (jt ¡ sj+ k z ¡ w k N )Z

Nw h ere th e con sta n t C (±;N ) a p p ears b eca u se of th e eq u ivalen ce of th e n orm s in Z .

N NH en ce, th e m ap p in g (t;z ) ! A z + F (t;z ) is lo cally L ip sch itz co n tin u ou s from [0 ;T ]£¡ ¢N N N NZ in to Z an d th erefore th ere m u st ex ist t > 0 su ch th at p ro b lem P h as a u n iq u e1

Nso lu tion on [0;t ). ¥1

NT h e fo llow in g resu lt rela tes th e sem igro u p s T (t) an d T (t).

N N NL e m m a 3 .2 . L et T (t), T (t), A , A , Z a n d Z b e a s a b o v e a n d let R (¸ ;A ) d en o te th e

: ¡ 1reso lv en t o f A a t ¸ , R (¸ ;A ) = (¸ I ¡ A ) . T h en

Ni) fo r ev ery ¸ 2 ½ (A ), th e sp a ce Z is in va ria n t u n d er R (¸ ;A );

N Nii) th e restrictio n o f T (t) to Z co in cid es w ith T (t) fo r ev ery t ¸ 0 , i.e.

NT (t)j N = T (t) 8 t ¸ 0 :Z

:N^P roof. i) L et ¸ 2 ½ (A ) a n d » b e an elem en t o f th e b a sis ¯ of Z an d d e¯ n e z = R (¸ ;A )».N

T h en z is a n eigen vector of A corresp o n d in g to th e sam e eigen valu e ¾ o f ». In fact, (¸ I ¡

A )A z = A » = ¾ » , w h ich im p lies A z = R (¸ ;A )¾ » = ¾ z . S in ce all th e eigen va lu es of A are

Nsim p le, z m u st b e a co n stan t m u ltip le o f » an d th erefo re z 2 Z . P a rt i) th en follow s b y

th e lin earity o f R (¸ ;A ).

N N N~ii) S in ce Z is in varian t u n d er A , th e op era tor A , th e p a rt of A in Z , d e¯ n ed b y

© ª:N N N~D (A ) = z 2 D (A ) \ Z : A z 2 Z³ ´:N N~ ~A z = A z ; z 2 D A ;

1 0

N N N~coin cid es w ith A , th e restrictio n o f A to Z . H en ce A gen erates a u n ifo rm ly con tin u o u s

N Nsem igro u p o n Z a n d b y p art i), Z is in varian t u n d er R (¸ ;A ) for ev ery ¸ w ith R e ¸ > 0.

N N~B y T h eo rem 4.5.5 in [23 ] it fo llow s th a t A = A is th e in ¯ n itesim al gen era tor of th e

Nrestriction o f T (t), th e sem igrou p gen erated b y A , to Z . ¥

W e sh all n eed th e fo llow in g gen era lizatio n o f G ron w all's L em m a for sin g u la r k ern els

w h o se p ro of can b e fou n d in [1 5, L em m a 7.1.1].

L e m m a 3 .3 . ([1 5]) L et a (t) b e a n o n n eg a tiv e, lo ca lly in teg ra b le fu n ctio n o n 0 · t · T ,

L ¸ 0 a n d 0 < ± < 1. T h en , th ere ex ists a co n sta n t K = K (±) su ch th a t ev ery fu n ctio n u

sa tisfy in g Z t 1u (t) · a (t) + L u (s) d s

±(t ¡ s)0

o n 0 · t · T , a lso sa tis¯ es

Z t a (s)u (t) · a (t) + K L d s; fo r 0 · t < T :

±(t ¡ s)0

:W e d e¯ n e th e o p erato r A = A ¡ I w ith D (A ) = D (A ). F ro m th e p rop erties of A itI I

follow s ea sily th a t A gen era tes a n ex p o n en tially stab le a n aly tic sem ig rou p T (t). M o reov er,I I

¡ t ±T (t) = e T (t). A lso , sin ce 0 2 ½ (A ), th e fra ction al p ow ers (¡ A ) a re w ell d e¯ n ed fo r a n yI I I

± 2 (0 ;1).

W e n ow p ro ceed to state an d p rove ou r m ain resu lt ab ou t th e con v erg en ce of th e a p -

p rox im atin g solu tio n s.

¡ ¢ ¡ ¢3 ± NT h e o re m 3 .4 . L et ± 2 ;1 , z 2 D (¡ A ) a n d su p p o se z (t), z (t) a re so lu tio n s o f0 I4¡ ¢

NP a n d (P ), resp ectiv ely, a n d let [0 ;t ) b e th e m a x im a l in terva l o f ex isten ce o f z (t). T h en ,1

0 N 0fo r a n y t < t th ere ex ists a co n sta n t N su ch th a t z (t) ex ists o n [0;t ] fo r ev ery N ¸ N1 0 01 1

N 0a n d z (t) co n v erg es to z (t) in th e n o rm o f Z fo r ev ery t 2 [0;t ]. M o reov er, th e co n v erg en ce1

±h o ld s in th e n o rm o f th e g ra p h o f (¡ A ) .I

¡ ¢3 0 N NP roof. L et ± 2 ;1 , t < t an d fo r each N 2 IN let t > 0 b e su ch th a t z (t) ex ists on11 14

1 1

N 0 N[0;t ). T h en , for t 2 [0;m in f t ;t g) an d N 2 IN1 1 1 Z tN N N N N Nz (t) = T (t)P z + T (t ¡ s)P F (s;z (s)) d s;0

0Z t

z (t) = T (t)z + T (t ¡ s)F (s;z (s)) d s:00

T h erefore

° °¡ ¢N ± N° °k z (t) ¡ z (t)k = (¡ A ) z (t) ¡ z (t)± I° °¡ ¢° °± N N· (¡ A ) T (t)P z ¡ T (t)z° °I 0 0Z ° °t £ ¤° °± N N N+ (¡ A ) T (t ¡ s)P F (s;z (s)) ¡ T (t ¡ s)F (s;z (s)) d s° °I

0: N N= ½ (t) + ½ (t):1 2

±S in ce T (t) co m m u tes w ith (¡ A ) ,I

° °¡ ¢N ± N° °½ (t) = (¡ A ) T (t) P z ¡ zI 0 01 Z° °¡ ¢

± N° °· k T (t)k (¡ A ) P z ¡ zL(Z ) I 0 0 Z° °N° °· P z ¡ z :0 0 ±

NS im ilarly, for th e in tegran d d e¯ n in g ½ (t) w e h av e2

° °£ ¤± N N N° °(¡ A ) T (t ¡ s) P F (s;z (s)) ¡ T (t ¡ s)F (s;z (s))I Z° °¡ ¢

± N N° °= (¡ A ) T (t ¡ s) P F (s;z (s)) ¡ F (s;z (s))I Z° ° ° °± N N° ° ° °· (¡ A ) T (t ¡ s) P F (s;z (s)) ¡ F (s;z (s))I L(Z ) Z° °C0 ±t N N° °1· e P F (s;z (s)) ¡ F (s;z (s)) : (3.1)

± Z(t ¡ s)

B u t

° ° ° ° ° °£ ¤ ¡ ¢N N N N N° ° ° ° ° °P F (s;z (s)) ¡ F (s;z (s)) · P F (s;z (s)) ¡ F (s;z (s)) + P ¡ I F (s;z (s))

Z Z Z° ° ° °¡ ¢N N° ° ° °· F (s;z (s)) ¡ F (s;z (s)) + P ¡ I F (s;z (s)) :

Z Z(3 .2 )

1 2

H en ce, fro m (3.1) a n d (3.2) it fo llow s th at

Z t ° °10N t N° °1½ (t) · C e F (s;z (s)) ¡ F (s;z (s)) d s±2 Z±(t ¡ s)0Z t ° °10t N° °1+ C e (P ¡ I )F (s;z (s)) d s:± ± Z(t ¡ s)0

2T h e in tegra n d of th e seco n d term on th e R H S ab ove is b ou n d ed b y k F (s;z (s))k 2Z±(t ¡ s)

1L (0;T ), u n iform ly in N , an d co n v erg es to zero as N ten d s to in ¯ n ity. B y th e D o m in ated

C o n v erg en ce T h eorem th e seco n d term o f th e last in eq u ality ten d s to 0 as N g o es to in ¯ n ity.

S u m m arizin g, w e h av e

Z t ~ ° °CN N N° °k z (t) ¡ z (t)k · ² (t) + F (s;z (s)) ¡ F (s;z (s)) d s (3.3)± ±(t ¡ s)0

0 N Nw h ere, fo r t 2 [0 ;t ], ² (t) · C for a ll N 2 IN an d ² (t) ! 0 a s N ! 1 . In p a rticu lar1Z 0t1

N² (t) d t ! 0 a s N ! 1 .0

: N~ ~L et K = K (± ) b e as in L em m a 3.3 a n d let u s d e¯ n e K = C + C C K . S in ce z (0 ) =° ° :N N N N~° °P z th ere ex ists ± > 0 su ch th a t z (t) · M + 2K for all t 2 [0;± ], w h ere M =0 ±

:su p k z (t)k . L et L b e th e L ip sch itz co n stan t fo r F corresp on d in g to th e set U =0 ±0· t· t

1n o0 ~[0;t ] £ k z k · M + 2 K . T h en , from (3 .3 )1 ±

Z t° ° ° °1N N N N° ° ~ ° °z (t) ¡ z (t) · ² (t) + C L z (s) ¡ z (s) d s 8 t 2 [0 ;± ];± ±±(t ¡ s)0

an d , fro m L em m a 3.3

° °N N N° °z (t) ¡ z (t) · f (t); 8 t 2 [0 ;± ]; (3.4)

±

R N: t ² (s )N N 0~w h ere f (t) = ² (t) + C K L d s, for t 2 [0 ;t ].± 10 (t¡s )

N 0~W e sh a ll n ow sh ow th at th ere ex ists N 2 IN su ch th at f (t) · K 8 t 2 [0 ;t ], 8 N ¸ N .0 01

NA s w e sh all later see th is w ill im p ly n o t o n ly th e ex isten ce o f z (t) on th e w h ole in terval

0 N 0~[0;t ], 8 N ¸ N , b u t also th e b ou n d k z (t)k · M + 2K , 8 t 2 [0;t ], 8 N ¸ N .0 ± 01 1

1 3

In fa ct, ob serv e th at Z Zt tN² (t) Cd s · d s

± ±(t ¡ s) (t ¡ s)0 0Z t 1· C d s

±s0

C 1¡±= t :1 ¡ ±

1 ¡ ±1¡±C h o osin g ´ = ´ (L ) > 0 su ± cien tly sm all so th a t t · for every t 2 [0 ;´ ], it follow s

2Lth at Z t N² (t) C

· for every t 2 [0;´ ]: (3.5)±(t ¡ s) 2 L0

0O n th e o th er h an d , if ´ < t · t1Z Zt tN N² (t) ² (t ¡ s)d s = d s

± ±(t ¡ s) s0 0Z Z´ tN N² (t ¡ s) ² (t ¡ s)= d s + d s

± ±s s0 ´Z t NC ² (t ¡ s)1¡ ±· ´ + d s±1 ¡ ± ´´Z tC 1 N· + ² (t ¡ s) d s

±2 L ´ ´Z 0t1C 1 N· + ² (s) d s:

±2 L ´ 0Z 0t1

NT h en , sin ce ² (s) d s ! 0, th en th ere ex ists N su ch th at00 Z t N² (t) C 0d s · 8 t 2 [ ;t ] an d N ¸ N : (3.6)01±(t ¡ s) L0

F rom (3.5) an d (3 .6 ) it follow s th at

N 0~ ~f (t) · C + C C K = K 8 t 2 [0;t ] an d N ¸ N ; (3.7)01

as w an ted .

C on seq u en tly, from (3 .4 ) an d (3.7)

° °N N° ° ~z (t) ¡ z (t) · K 8 N ¸ N an d t 2 [0;± ];0±

1 4

w h ich im p lies ° °N N° ° ~z (t) · M + K 8 N ¸ N an d t 2 [0 ;± ]:0± ° °

N 0 0 N° °N ow , let N ¸ N b e ¯ x ed . T h en z (t) ex ists o n [0 ;t ] an d for t 2 [0 ;t ], z (t) · M +0 1 1 ±° °¤ 0 N ¤~ ° ° ~2K . In fact, su p p ose, o n th e co n trary, th at th ere ex ists t < t su ch th at z (t ) = M + 2K1 ±° ° ° °

N ¤ N N° ° ~ ° ° ~an d z (t) < M + 2K fo r 0 · t < t . T h en , from (3.4) z (t) ¡ z (t) · f (t) · K± ±° °

¤ N ¤ N° ° ~on [0;t ), an d th erefo re, z (t) · M + K on [0 ;t ). B y th e con tin u ity of z (t), w e m u st±° ° ° °

N ¤ N ¤° ° ~ ° ° ~h av e z (t ) · M + K , w h ich co n trad icts z (t ) = M + 2K .± ±

N 0~A t th is p oin t w e h av e sh ow n th at 8 N ¸ N , kz (t)k · M + 2K , 8 t 2 [0 ;t ]. T h erefo re,0 ± 1

N 0 0for N ¸ N , ± can b e ch o sen strictly greater th a n t . H en ce (3.4 ) h o ld s in [0 ;t ], i.e.0 1 1

° °N N 0° °z (t) ¡ z (t) · f (t); 8 t 2 [0 ;t ]:1±

N 0F in ally, sin ce b y v irtu e o f th e D om in ated C on vergen ce T h eorem , f (t) ! 0 8 t 2 [0;t ]1

as N ! 1 , th e th eorem follow s. ¥

4 . N u m e ric a l R e su lts

In th is section n u m erical resu lts ob ta in ed u sin g th e ¯ n ite d im en sion al ap p rox im atin g

sch em e in tro d u ced in th e p rev iou s section a re p resen ted . W e sh all m ak e u se th e p a ram eter

¡ 3 ¡1valu es rep orted b y F . F alk in [12 ] for th e alloy A u C u Z n : ® = 24 J cm K , ® =2 3 3 0 4 7 2 4

5 ¡ 3 6 ¡ 3 ¡1 ¡ 3 ¡11:5 £ 10 J cm , ® = 7:5 £ 10 J cm K , µ = 208 K , C = 2:9 J cm K , k =6 1 v

¡1 ¡ 1 ¡3 ¡1 2 ¡11:9 w cm K , ½ = 11 :1 g cm , ¯ = 1, ° = 1 0 J cm . F igu re 1 sh ow s th e stress-strain

cu rves o b tain ed from th e p oten tia l (1 .2 ) for th ese va lu es o f th e p aram eters. T h e d oted

lin es in d icate th e u n sta b le p arts of th e cu rves, w h ile th e h o rizon tal lin es in d ica te p o ssib le

h y steresis lo op s.

In th e cou rse of o u r calcu la tion s w e ¯ rst u sed an ex p licit fou rth ord er R u n g e-K u tta

m eth o d . D u e to th e stru ctu re of th e n o n lin earities of th e sy stem th is m eth o d w a s fo u n d

to b e very u n stab le an d u n e± cien t. T h e e± cien cy o f th e n u m erical alg orith m w as g rea tly

im p rov ed u sin g a h y b rid im p licit-ex p licit E u ler m eth o d w h ich en su res stab ility for m u ch

1 5

larger step sizes. R ou gh ly sp ea k in g, th is m eth o d con sists of a p p rox im atin g th e lin ear p art

dof z (t) in a n im p licit w ay w h ile an ex p licit form is u sed for th e n on lin ear p art. F or th ed t

¡ 5n u m erica l resu lts p resen ted b elow w e u sed th is h y b rid m eth o d w ith N = 3 2 an d ¢ t = 10 .

(a) µ = 2 00 K (b ) µ = 2 60 K

F ig u re 1 : S tress-S train cu rv es for d i® eren t

tem p eratu res ob ta in ed from (1.2) w ith th e

va lu es of ® , ® , ® a n d µ as in [12]. T h e2 4 6 1

d o tted lin es rep resen t u n sta b le p arts o f th e

cu rves. H orizo n ta l lin es in d icate p o ssib le

h y steresis lo op s.

(c) µ = 600 K

E x p e rim e n t 1 : L ow -tem perature steady-state.

NF or th is ex p erim en t w e to ok f = g ´ 0, µ (x ) ´ 2 00 K , u (x ) = P h (x ), w h ere0 0

½0 :0 5x 0 · x · 0:5

h (x ) =0 :0 5(1 ¡ x ) 0:5 · x · 1;

-0.1 -0.05 0 0.05 0.1 0.15-300

-200

-100

0

100

200

300

strain (%)

stre

ss (

MP

a)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15-400

-300

-200

-100

0

100

200

300

400

strain (%)

stre

ss (

MP

a)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

strain (%)

stre

ss (

MP

a)

1 6

an d v ´ 0. T h u s, th e b eam is in itia lly in th e low tem p era tu re ra n ge co m p osed of tw o0

1segm en ts of m arten sites, n a m ely, m a rten site M on 0 · x < an d m a rten site M on+ ¡2

1 < x · 1 (5% in itia l stra in ). T h e evolu tion of d isp lacem en t an d tem p eratu re ca n b e2

ob serv ed in F ig u res 2a a n d 2b , resp ectiv ely. T h is evo lu tion is d u e to th e fa ct th a t th e in itialà !u (x )0

v (x )con d itio n z (x ) = is n ot a stea d y -state of sy stem (1 .1 a-e). T h e sy stem evo lv es u n til00

µ (x )0 »a stead y -state com p o sed o f tw o sy m m etric seg m en ts of m arten sites M a n d M ( 1 1:25%=+ ¡

»stra in ) an d con sta n t tem p era tu re µ 2 22 K is rea ch ed . F ig u re 2 c sh ow s in m ore d eta il th e=

d isp lacem en t p ro¯ le d u rin g th e ¯ rst 250 m illiseco n d s.

(a) (b )

F ig u re 2 : L ow tem perature steady-state.

E volu tio n o f d isp la cem en t (a, c) an d tem -

p era tu re (b ) from a n u n stead y low tem p er-

a tu re in itia l co n d ition .

(c)

E x p e rim e n t 2 : H igh-tem perature steady-state.

H ere w e to o k µ (x ) ´ 6 00 K an d u , v , f a n d g as in E x p erim en t 1 . T h e evolu tion0 0 0

of d isp la cem en t an d tem p era tu re is sh ow n in F ig u res 3 a an d 3b , resp ectively. T h e b eam

1 7

oscillates u n til th e stea d y -state con sistin g of zero d eform ation an d co n stan t tem p eratu re

»µ 5 05:6 K is rea ch ed . T h is is in agreem en t w ith th e fact th a t ab ove th e au sten ite- n ish=

»tem p era tu re µ = A (in o u r ca se A 2 83 K ) th e stea d y -states sa tisfy u ´ 0, µ ´ co n st. D u e=f f

to th e h ig h -tem p era tu re u n stead y in itial con d itio n th e b eam im m ed ia tely b en d s d ow n w ard

ap p roach in g th e state u ´ 0 w h ile tem p era tu re d ecreases sligh tly, o rig in a tin g th e d am p ed

oscillation s o b served in F igu res 3a a n d 3 b . T h e oscillatio n s of th e m id d le-p o in t of th e b eam

can b e ap p recia ted in F igu re 3 c.

(a) (b )

F ig u re 3 : H igh tem perature steady-state.

(a ) d isp la cem en t p ro ¯ le; (b ) tem p eratu re

p ro ¯ le; (c) m id d le-p o in t d isp lacem en t.

(c)

E x p e rim e n t 3 : P ulse at low tem perature.

In th is ex p erim en t w e stu d ied th e e® ects o f a d istrib u ted fo rce con sistin g o f a p u lse

aro u n d th e m id d le-p oin t o f th e b ea m w h en th e in itia l tem p era tu re is b elow th e m a rten site

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

time (sec)

mid

dle

-po

int

dis

pla

cem

en

t

1 8

»¯ n ish tem p eratu re µ = M 20 8 K . W e to ok u (x ) = v (x ) ´ 0 , µ (x ) ´ 2 00 K , g (x ;t) ´ 0=f 0 0 0

an d ½4 ¡35 £ 10 0 :4 · x · 0 :6 an d 0 < t < 0:5 £ 10

f (x ;t) =0 o th erw ise.

In itially, p oin ts a rou n d th e cen ter m ove u p w a rd w h ile th e e® ect o f th e p u lse p ro p ag ates

to th e en d p oin ts of th e b ea m (F ig u res 4a, 4c). A t ex actly th e sam e tim e th is e® ect rea ch es

th e en d p oin ts, th e m id d le-p oin t d e° ection rea ch es a m a x im u m an d sm all d am p ed o scilla tio n s

b eg in to tak e p la ce (F igu re 4 c) arou n d th e ¯ n al eq u ilib riu m sta te co n sistin g o f tw o sy m m etric

» »segm en ts of m arten sites M , M ( 1 1:05% stra in ) a n d con stan t tem p era tu re µ 22 6 K= =+ ¡

(F ig u re 4b ).

(a) (b )

F ig u re 4 : P ulse at low tem perature. (a),

(c) d isp la cem en t p ro¯ le; (b ) tem p eratu re

p ro ¯ le.

(c)

E x p e rim e n t 4 : P ulse at high tem perature.

In th is ca se w e in vestiga ted th e e® ects of a p u lse aro u n d th e m id d le-p oin t of th e b ea m ,

w h ich w a s set in itially a t a con stan t tem p era tu re a b ov e A . W e to o k µ (x ) ´ 60 0 K a n d u ,f 0 0

1 9

v , f a n d g as in E x p erim en t 3 . A t th e b eg in n in g, th e b eam b en d s u p w ard u n til th e p u lse is0

sw itch ed o® (F igu re 5c). Im m ed iately afterw ard s, d am p ed o scilla tio n s b egin to o ccu r. T h ese

oscillation s take p lace a rou n d th e ¯ n al eq u ilib riu m sta te con sistin g of u ´ 0 an d co n stan t

»tem p era tu re µ 60 2 K (F igu res 5a , 5 b ). R ecall th at ab ove th e au sten ite ¯ n ish tem p eratu re=

th e on ly u n loa d ed stead y -sta te is u ´ 0.

(a) (b )

F ig u re 5 : P ulse at high tem perature. (a),

(c) d isp la cem en t p ro¯ le; (b ) tem p eratu re

p ro ¯ le.

(c)

E x p e rim e n t 5 : W aiting-H eating.

H ere, w e ob serv ed th e e® ects o f h ea tin g th e b ea m w h en it is set in itially at a n eq u ilib riu m

sta te con sistin g of tw o sy m m etric segm en ts of m arten sites M an d M . F o r th is, w e to ok+ ¡

as th e in itial d ata th e ¯ n a l stea d y -state o f E x p erim en t 1 (1 1.25% in itial stra in , µ (x ) ´ 22 20

K ), f (x ;t) ´ 0 an d th e h eat so u rce g (x ;t) con sistin g o f a u n iform ly d istrib u ted h eat p u lse

as follow s ½45 £ 1 0 0:2 < t < 0:25

g (x ;t) =0 oth erw ise.

2 0

T h e sy stem rem a in s at th e in itia l state u n til th e h eat p u lse is sw itch ed o n . A t th is tim e

th e tem p era tu re starts to in crease (F ig u re 6b ), th e m arten site cry sta ls are co n v erted in to

au sten ite an d th e b ea m b en d s d ow n w a rd o scilla tin g aro u n d zero d efo rm a tion (F ig u re 6a).

»T h ese oscillation s are q u ick ly d am p ed a n d th e b ea m reach es th e stead y -sta te u ´ 0, µ 33 6=

K .

(a) (b )

F ig u re 6 : W aitin g-H eating. (a) d isp lacem en t an d (b ) tem p era tu re p ro¯ les.

E x p e rim e n t 6 : H eating-W aiting-C ooling

F or th is ex p erim en t w e to ok a ga in as in itial d a ta th e ¯ n al stea d y -state of E x p erim en t

1. W e also to ok f (x ;t) ´ 0 an d th e d istrib u ted h eat so u rce g (x ;t) con sistin g of an in itial

u n iform ly d istrib u ted h eat p u lse w h ich is sw itch ed o® after t = 0 :0 5 sec. A t t = 1 :4 5 sec.

th e o p p osite h eat p u lse is a p p lied u n til t = 1:5 0 sec. w h en it is sw itch ed o ® . M ore p recisely,838 £ 1 0 t < 0 :0 5<

3g (x ;t) = ¡ 8 £ 10 1 :4 5 < t < 1:50:0 o th erw ise.

T h e tem p era tu re raises u n ifo rm ly u p to n early 33 6 K w h ile th e b ea m ap p roach es th e

u n d eform ed state. A fter th e h eat p u lse is sw itch ed o® , tem p eratu re rem ain s at ab ou t 33 6

K w h ile d isp lacem en t sh ow s d a m p ed oscillatio n s a rou n d u ´ 0 d u e to in ertial e® ects. T h e

sa m p le is n ow co m p letely in th e au sten ite p h ase. A t t = 1 :45 , w h en th e op p osite p u lse is

ap p lied , th e tem p era tu re d ecreases u n iform ly a n d rem a in s a t ab ou t 2 22 K , w h ile th e b eam

u n d ergo es a p ro cess o f reverse tran sform ation w h ich ta kes it b ack to th e o rig in a l in itial

con ¯ g u ration sh ow in g th e so -called tw o-w ay sh ap e m em o ry p h en om en on (F igu res 7 a-d ).

2 1

(a) (b )

(c) (d )

F ig u re 7 : H eating-W aiting-C oolin g. (a) d isp lacem en t p ro¯ le; (b ) tem p eratu re

p ro ¯ le; (c) m id d le-p oin t d isp la cem en t; (d ) m id d le-p oin t tem p eratu re.

5 . C o n c lu sio n s

In th is a rticle, d iscrete sp ectra l a p p rox im ation s to th e n o n lin ear p a rtia l d i® eren tial

eq u atio n s th at m o d el th e d y n am ics of th erm o m ech an ica l m arten sitic tra n sform ation s in

on e-d im en sio n al sh ap e m em o ry alloy s w ith n o n -con vex L an d a u -G in zb u rg p oten tials w ere

0 0.5 1 1.5 2 2.5 3180

200

220

240

260

280

300

320

340

time (sec)

mid

dle

-po

int

tem

pe

ratu

re

0 0.5 1 1.5 2 2.5 3-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

time (sec)

mid

dle

-po

int

dis

pla

cem

en

t

2 2

d evelo p ed .

B y u sin g th e th eories of a n aly tic sem igro u p s a n d in terp o la tion sp aces a n d a g en era liza -

tion of G ro n w a ll's lem m a fo r sin g u la r kern els, th e con v erg en ce of th e a p p rox im ation s w a s

sh ow n to h old n ot on ly in th e state-sp a ce n orm b u t a lso in th e stro n ger k ¢ k -n orm .±

T h e n u m erica l ex p erim en ts p erfo rm ed u sin g th is sch em e sh ow th a t u n d er d i® eren t in itial

con d itio n s an d d istrib u ted ex tern al actio n s th e m o d el (1.1 ) is ab le to p ro d u ce so lu tion s w h ose

q u alita tiv e b eh av ior is fou n d to b e in clo se agreem en t w ith la b ora tory ex p erim en ts p erfo rm ed

on S h ap e M em ory A lloy s u n d er sim ilar con d ition s.

F rom a p ra ctical p oin t of v iew it w o u ld b e v ery im p ortan t to ¯ n d th e va lu es of th e

vector p a ram eter q th at \ b est ¯ t" ex p erim en ta l d a ta fo r a g iv en alloy. T h is is called th e

p ara m eter id en ti cation p rob lem ab ou t w h ich n o resu lts are y et k n ow n . In th is reg ard

th e sch em e p resen ted h ere p rov id es a frien d ly m a th em atical fra m ew ork fo r attack in g th is

p rob lem . E ® o rts in th is d irection a re a lread y u n d erw ay a n d resu lts w ill b e p u b lish ed in a

forth com in g a rticle.

R e f e r e n c e s

[1 ] A C H E N B A C H , M . a n d M U L L E R , I., \A M odel fo r S h ape M em ory" , J o u rn a l d e P h y siq u e, C o llo q u e C 4 ,

S u p p l¶em en t a u N o 1 2 , T o m e 4 3 , d ¶ecem b re 1 9 8 2 , p p C 4 1 6 3 -1 6 7 .

[2 ] A C H E N B A C H , M . a n d M U L L E R , I., \ C reep an d Y ield in M arten sitic T ra n sform ation s" , Ig en ieu r-A rch iv , 5 3 , p p

7 3 -8 3 , (1 9 8 3 ).

[3 ] A C H E N B A C H , M . a n d M U L L E R , I., \S ha pe M em ory as a T herm a lly A ctivated P rocess" , P la sticity T o d a y : M o d ellin g

M eth o d s a n d A p p lica tio n s, p p 5 1 5 -5 3 1 , S a w czu k a n d B ia n ch i.

[4 ] A N D R E A S E N , G . F . a n d M O R R O W , R .E ., \L aboratory a n d C lin ical A n alyses o f N iT in o l W ire" , A m . J . O rth o d .,

V o l 7 3 , N o 2 , p p 1 4 2 -1 5 1 .

[5 ] B E N S O U S S A N A ., D A P R A T O G ., D E L F O U R M . a n d M IT T E R S ., \R epresen tation an d C on tro l o f In ¯ n ite D im en -

sio n al S ytem s" , V o l. I B irk h Äa u ser, 1 9 9 2 .

[6 ] B U R N S , J . A . a n d S P IE S , R . D ., \F in ite E lem en t A ppro xim ation o f a S hape M em o ry A llo y" , P ro ceed in g s o f th e

A D P A / A IA A / A S M E / S P IE C o n feren ce o n A ctiv e M a teria ls a n d A d a p tiv e S tru ctu res, A lex a n d ria , V A , 1 9 9 1 .

th[7 ] B U R N S , J . A . a n d S P IE S , R . D ., \M odellin g for C on trol of S ha pe M em ory A lloys" , 3 0 IE E E C o n feren ce o n D ecisio n

a n d C o n tro l, B rig h to n , E n g la n d , 1 9 9 1 .

[8 ] B U R N S , J . A . a n d S P IE S , R . D ., \S en sitivity A n alysis for a D yn am ic M odel of P hase T ra n sition s in M ateria ls w ith

M em o ry" , to a p p ea r in th e P ro ceed in g s o f th e C o n feren ce o n R ecen t A d v a n ces in A d a p tiv e a n d S en so ry M a teria ls a n d

T h eir A p p lica tio n s, V irg in ia P o ly tech n ic In stitu te a n d S ta te U n iv ersity, A p ril 2 7 -2 9 , 1 9 9 2 .

[9 ] C A S T L E M A N , L . S ., M O T Z K IN , S . M ., A L IC A N D R I, F . P . a n d B O N A W IT , V . L ., \ B iocom patibility of N itin ol A llo y

as a n Im plan t M a terial" , J o u rn a l o f B io m ed ica l M a teria ls R es. V o l. 1 0 , p p 6 9 5 -7 3 1 (1 9 7 6 ).

[1 0 ] D E L A E Y , L . a n d C H A N D R A S E K A R A N , M ., \P roceedin g, In tern a tion al C o n feren ce on M arten sitic T ran sform a-

tion s" , L es E d itio n s d e P h y siq u e, L es U lis, 1 9 8 4 .

2 3

[1 1 ] D E L A E Y , L ., K R IS H N A N , R . V ., T A S , H . a n d W A R L IM O N T , H ., \T herm oelasticity, P seu d oelasticity an d th e

M em o ry E ® ects A ssocia ted w ith M arten sitic T ran sform a tio n s" , J o u rn a l o f M a teria ls S cien ce, V o l. 9 , 1 9 7 4 , p p 1 5 2 1 -

1 5 5 5 .

[1 2 ] F A L K , F ., \M odel F ree E n ergy, M ech an ics a n d T herm odyn am ics o f S hape M em ory A lloys" , A cta M eta llu rg ica , V o l.

2 8 , p p 1 7 7 3 -1 7 8 0 , (1 9 8 0 ).

[1 3 ] F A L K , F ., \O n e D im en sio n al M odel of S hape M em ory A lloys" , A rch . M ech ., 3 5 , p p 6 3 -8 4 , W a rszaw a , (1 9 8 3 ).

[1 4 ] F U N A K U B O , H . (E d .), \S hape M em ory A lloys" , P recisio n M a ch in ery a n d R o b o tics, V o l. 1 , T ra n sla ted fro m th e

J a p a n ese b y J . B . K en n ed y, G o rd o n a n d B rea ch S cien ce P u b lish ers, (1 9 8 7 ).

[1 5 ] H E N R Y , D . \G eom etric T heory of S em ilin ear P arabo lic E qu ation s" , S p rin g er-V erla g , 1 9 8 9 .

[1 6 ] L O H M A N , R . a n d M U L L E R , I., \ A M od el for the Q u alitative D escription of M arten sitic T ran sform a tio n s in M em o ry

A llo ys" , in P h a se T ra n sfo rm a tio n s, A ifa n tis E . a n d G ittu s J ., p p 5 5 -7 5 .

[1 7 ] M A L L O Y , E . C ., \N itin ol P ro vides S hape M em o ry C apabilities" , O n th e S u rfa ce M a g a zin e, 2 2 J u n e 1 9 9 0 .

[1 8 ] M U L L E R , I., \ A M odel fo r a B ody w ith S hape M em ory" , A rch . R a tio n a l M ech a n ics A n a l., 7 0 , p p 6 1 -6 7 , (1 9 7 9 ).

[1 9 ] M U L L E R , I. a n d V IL L A G G IO , P ., \A m od el for an E lastic-P lastic B ody" , A rch . R a tio n a l M ech a n ics A n a l., 6 5 # 1 ,

p p 2 5 -4 6 , (1 9 7 7 ).

[2 0 ] M U L L E R , I. a n d W IL M A N S K I, K ., \A m odel for P ha se T ran sition s in P seu d oela stic B odies" , Il N u o v o C im en to , V o l.

5 7 B , N o 2 , p p 2 8 3 -3 1 8 , (1 9 8 0 ).

[2 1 ] N IE Z G O D K A , M . a n d S P R E K E L S , J ., \E xisten ce o f S olu tion s for a M athem a tical M odel of S tru ctu ral P h ase T ran -

sition s in S hape M em ory A lloys" , M a th em a tica l M eth o d s in th e A p p lied S cien ces, V o l. 1 0 , p p 1 9 7 -2 2 3 , (1 9 8 8 ).

[2 2 ] N IE Z G O D K A , M ., a n d S P R E K E L S , J ., \C on vergen t N u m erica l A pp roxim ation s of the T herm om echan ical P hase

T ran sitio n s in S hape M em ory A lloys" , N u m er. M a th ., V o l. 5 8 , 1 9 9 1 , p p 7 5 9 -7 7 8

[2 3 ] P A Z Y , A ., \S em igrou ps of L in ear O perators an d A pplication s to P a rtia l D i® eren tial E qu a tio n s" , C o rrected S eco n d

P rin tin g , S p rin g er-V erla g , 1 9 8 3 .

[2 4 ] R O G E R S , C ., L IA N G , C . a n d J IA , J ., \ B eha vior of S h ape M em ory A lloy R ein forced C om posites, P art I: M odel F orm u -thlation s an d C on tro l P roblem " , 3 0 A IA A / A S M E / A S C E / A H S / A S C S tru ctu res, S tru ctu ra l D y n a m ics a n d M a teria ls

C o n feren ce, M o b ile, A la b a m a , A p ril 1 9 8 9 .

[2 5 ] S C H M E R L IN G , M . A ., W IL K O V , M . A ., S A N D E R S , A . E . a n d W O O S L E Y , J . E ., \A P roposed M edical A pplication

of the S ha pe M em ory E ® ect: A N iT i H arrin gton R od for th e T rea tm en t of S coliosis" , en \ S h a p e M em o ry E ® ects in

A llo y s" , E d ited b y J e® P erk in s, P len u m P ress, p p 5 6 3 -5 7 4 , (1 9 7 5 ).

[2 6 ] S O N G M U , Z ., \G loba l S olu tion s to the T herm om ech an ica lE qu ation s w ith N on -co n vex L an d au -G in zbu rg F ree E n ergy" ,

J o u rn a l o f A p p lied M a th em a tics a n d P h y sics (Z A M P ), V o l. 4 0 , p p 1 1 1 -1 2 7 , J a n u a ry 1 9 8 9 .

[2 7 ] S O N G M U , Z . a n d S P R E K E L S , J ., \G lobal S olu tion s to the E qu ation s of a G in zbu rg-L an da u T heory for S tru ctu ral

P hase T ran sition s in S ha pe M em ory A llo ys" , P h y sica D , V o l. 3 9 , p p 5 9 -7 6 , (1 9 8 9 ).

[2 8 ] S P IE S , R . D ., \A S tate-S pace A p proa ch to a O n e-D im en sion al M ath em atical M od el fo r the D yn am ics of P hase

T ran sitio n s in P seu doelastic M aterials" , J o u rn a l o f M a th em a tica l A n a ly sis a n d A p p lica tio n s, 1 9 0 , (1 9 9 5 ), p p 5 8 -1 0 0 .

[2 9 ] S P IE S , R . D ., \R esu lts on a M athem atical M od el of T herm om echan ical P hase T ran sition s in S hape M em ory M ateri-

als" , J o u rn a l o f S m a rt M a teria ls a n d S tru ctu res, 3 , (1 9 9 4 ), p p 4 5 9 -4 6 9 .

[3 0 ] S P R E K E L S , J ., \A u tom atic C o n trol of O n e-D im en sion al T h erm om ech an ica l P ha se T ran sitio n s" , in " M a th em a tica l

M o d els fo r P h a se C h a n g e P ro b lem s" , In tern a tio n a l S eries o f N u m erica l M a th em a tics, V o l. 8 8 , B irk h Äa u ser, p p 8 9 -9 8 ,

(1 9 8 9 ).

[3 1 ] S P R E K E L S , J ., \G lobal E xisten ce fo r T herm om echa n ical P rocesses w ith N o n co n vex F ree E n ergies of G in zbu rg-L an d au

F o rm " , J o u rn a l o f M a th em a tica l A n a ly sis a n d A p p lica tio n s, V o l. 1 4 1 , p p 3 3 3 -3 4 8 , (1 9 8 9 ).

[3 2 ] W IL M A N S K Y , K ., \P ropaga tio n of the In terfa ce in S tress-In du ced A u sten ite-M arten site T ra n sform ation " , In g en ieu r-

A rch iv ., V o l. 5 3 , 1 9 8 3 , p p 2 9 1 -3 0 1 .