r__ o a L INIHHIIWJIIIHIHHUIHIIIIHlllHllHll'lllllllllfl 129301565 9203 r ‘ LIBRARY Michigan State University This is to certify that the dissertation entitled M4 [QR m1) Theo/(Ltcmeclka MCI/1AA PlVlSJQ- TH \vS\(AC'l}V\—§LT( (9L 1" i)(+U/bbm All 8%» \ACL‘ (Lv v\.€{ A 7.qu U xYCkc er M “M19 un/C; presented by X {aoClA Man} LU M has been accepted towards fulfillment of the requirements for DOCS‘w 4 PLs‘XoSvé‘aflegree in fiwhflk Medmn {64 XML. . T 101' professor DateLM /0,, /677 MS U is an Affirmative Action/Equal Opportunity Institution PLACE II RETUMI BOX to man thb Mom horn your noord. TOAVOID____F—INE8rutumonorbdondm¢uo. DATE DUE DATE DUE DATE DUE OCT 5319190111: 01 -2334 MSU chn Afflrmdtvo ActlaVEqml Opportunity Imam Wanna-m MODELING THERMOMECHANICAL PHASE TRANSFOR- MATIONS BETWEEN AUSTENITE AND A TWO VARIANT MARTENSITE By Xiaochuang Wu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Materials Science and Mechanics 1997 ABSTRACT MODELING THERMOMECHANICAL PHASE TRANSFOR- MATIONS BETWEEN AUSTENITE AND A TWO VARIANT MARTENSITE By Xiaochuang Wu Both austenite/martensite transformations and martensite/martensite variant reorienta- tion are central to shape memory actuation and pseudoelasticity. The approach is to aug- ment conventional continuum mechanical descriptions with internal variables that track fractional partitioning of the material between austenite and the various martensite vari- ants. A three-species model involving austenite and two complementary martensite vari- ants provides sufficient generality to capture the variant distributions that underlie shape memory, and the strain-accommodation associated with pseudoelasticity. Transformations between any of these species can be tracked on the basis of triggering algorithms and kinetic continuation that reflect both transformation hysteresis and the variation of trigger- ing stress and temperature, as given by the Clausius-Clapeyron relation. The particular algorithm that we describe here is for temperature- and stressdependent response. It requires only the following experimental parameters: the four transformation temperatures Mf, Mr A,, Af, the crystallographic transformation strain, the Young’s modulus and the transformation latent heat. The martensite flow and finish stresses are also introduced. As an application of the model, Two Element Thermal Engine (TSTE) is investigated to pre- dict a reciprocal movement upon thermal heating/cooling pulses controlled for example by - an electrical signal. To my parents ACKNOWLEDGMENTS It is with heartfelt thankfulness that I here acknowledge my indebtedness to my advi- sor and friend, Professor Thomas Pence for his instruction and guidance in my doctoral studies, for his patient direction and discussion in the course of my research, and for his constructive suggestion and careful modification which contribute greatly to the comple- tion of this dissertation. I should also extend my gratitude to Professor David Grumman, who has given me inspiration and advice on physical understanding, whose experimental work and results helped in modifying the mathematics model studied in this approach. I am also grateful to Professor Zhengfang Zhou, who has spent time discussing with me the solution of some difficult mathematics problem, who has been kindly supportive wherever needed. I want to say thanks to Professor Ronald Averill, who has given me advice on numerical simula- tions. My thanks also go to my friend Rangfu Chen and all others for their help and sup- port. Last but not the least, I owe my gratitude to my wife Ping Du, who has been taking great care of me during these years of my study here at Michigan State University, who has been understanding and loving, who is always there when help is needed. iv TABLE OF CONTENTS LIST OF TABLES ........................................................................................................... viii LIST OF FIGURES ......... g - ................................. ix 1 INTRODUCTION ....... ....... l 1.1 BackgroundtoThis Work -- -1 1.2 Scape of This Work 6 2 FORMATION OF A TWO VARIANT PHASE DIAGRAM ............................... 11 2.1 Concept of Nuetrality Curves .............. --l 1' 2.2 Special Nuetrality Curves and the X-unfolding - 15 2.2.1 Unfolding the Basic Three Species Phase Diagram 15 2.2.2 Formation of X-unfolding _ - -16 2.2.3 Building Detwinning into the X-unfolding- - - - ............ 19 2.2.4 Isothermal Behavior of X-unfolding ............ 27 2.3 Modified Nuetrality Curves and the pY-unfolding -29 2.3.1 Thermodynamic Considerations for Detwinning and a Primitive Y-unfolding ........................................ 29 2.3.2 Mathematical Description of pY-unfolding -- - -32 2.4 Refined pY-unfolding Associated with Experiments 35 2.4.1 Experimental Justification on Detwinning 35 2 ..4 2 Mathematical Formulation of the Y—unfolding - 38 2. 4. 3 Summary =44 2.5 Analysis of Isothermal Loading/Unloading Processes ______ -- - 45 2.6 Transformation of a Special Variant .......................................................... 52 3 TRANSITION TYPES FOR THE PHASE FRACTION EVOLUTION ............. 55 3.1 Transition Types ........... - - -- --55 . 3.2 Criteria for Determining Transition Type _ 57 3.2.1 Algebraic Description 57 3.2.2 Geometric Illustration 59 3.3 Criteria for Detwinning Transition Types 61 3.4 Example - - - 452 4 THE HYSTERESIS ALGORITHMS 64 4.1 Brief Review of the Previous One-variant study ....................................... 64 4.1.1 One Variant Algorithm ................................................................... 64 4.1.2 Envelope Function _______________ - -_ _ -66 4.2 Two Variant Constitutive Functions _- - - 68 4.2.1 Constitutive Function Extension .................................................... 68 4.2.2 Possible Extension of the Algorithm to the Two Variant Problem 70 4.3 Analysis on the Algorithm Associated with Transition Types ................... 74 4.3.1 Unique Algorithm ................. - _ 74 4. 3. 2 Algorithm Consistency between One and Two Variant Problems .76 INTEGRABILITY AND PATH DEPENDENCE ................................................ 79 5.1 Integration of The Hysteresis Equations for The Phase Fractions ............. 79 5.1.1 Integration of Transition Type ('I'I‘l) - - -- - -- - - 80 5.1.2 Integration of Transition Type ('I'T2) -- - ---81 5.1.3 Integration of Transition Type (TT3) - - -----83 5.1.4 Integration of Transition Type ('I'T4)- - - -- 84 5.1.5 Integration for Detwinning Process; ..... -85 5.1.6 Path Independence of the depleted Species within a Transition Type 36 5.2 Path Dependence and Path Independence within a Transition Type ......... 86 5.2.1 General Path-Independent Condition -_ 86 5 ..2 2 Path-dependent Analysis for the case of A, > M, - - - 87 5. 2. 3 Path-dependent Analysis for the case of M, > A, -_--_-----90 5.3 Example - -- _ - -- _ -- - -----92 5.4 Discussions on the Solutions ..................................................................... 96 BEHAVIOR OF THE MODEL ............................................................................ 98 6.1 Isothermal Behavior ................................................................................... 98 6.1.1 Pseudoelastic Behavior ___________ - - ............. 99 6.1.2 Internal Hysteresis Loops - - ............ w 103 6.1.3 Shape Memory Effect and Isothermal Behavior below A, ........... 109 6.1.4 Load Cycling and Saturation..- --117 6. 2 Differences between Loading-Cooling and Cooling-Loading Paths ....... 120 6.3 Comparison with Other Models.-_- _ - 123 APPLICATION OF THE MODEL TO AN ACTUATOR DESIGN .................. 127 7.1 Analysis on Basic Structure of TSTE- - -- - -127 7.2 Defamation Consistency -- _ - 130 7.3 Heat-le Element(I): The First Stroke _ - 132 7.3.1 Heating Process-u ----- - - 132 7.3.2 Cooling Process 137 7.3.3 An Uniqueness Point for the Solution of the Equation (7.3.2.4) .140 10 7.4 Later Heat-le Strokes .......................................................................... 142 7.5 Limit Analysis on Thermal Cycles ......... 144 7.5.1 Numerical Convergence on Residual Phase Fraction .................. 144 7. 5 .2 Stable Response Loops -------- - - --153 7.6 Discussion -- - -- - -- - - - ---------------- 156 CONCLUSION AND PROPOSED FUTURE WORK ...................................... 158 REFERENCE ...................................................................................................... 162 APPENDIX-- - ........... - - -- - - - - - - -- - ------- -- - -167 REFINEMENT OF THE TWO VARIANT PHASE DIAGRAM _- - ---167 10. l Clausius-Clapeyron Relations - - - - 168 10. 2 Determination of Strain rn Process A (—) M: .................................. . ....... 171 10.3 Determination of Strain in Process A (-) M_ ------------- -- - 174 10.4 Determination of Strain in Process M_ —) M + ................ 177 TABLE 1 TABLE 2 TABLE 3 TABLE 4 TABLE 5 TABLE 6 TABLE 7 TABLE 8 TABLE 9 TABLE 10 TABLE 11 TABLE 12 LIST OF TABLES Transformation Possibilities for Isothermal Loading and Unloading ........ 28 Simulation Parameters ............................................................................... 46 Temperatures Corresponding to Points a, b, c and d in Figure 13 ............. 47 Transition Types with Their Algorithm ..................................................... 78 Path-dependent Category for M, < A, ..................................... . .................. 90 Path-dependent Category for M, >As- - - -- ---------------- 91 Phase Fractions at the End (0, M!) of the Three Paths: ,ll, 12, I3 ................. 96 Phase Fraction Distributions......................-------- - ----- - - - 119 [1A = 4.0*10"4; [1M = 2.0‘10"4 MPa ....................................................... 149 11,, = 4.0‘10"4; PM = 2.8*1W MPa ....................................................... 150 HA = 4.0*10"4; [1,4 = 3.4*10’\4 MPa ....................................................... 151 [LA = 3.0*10"4; IN = 3.0*10"4 MPa ....................................................... 152 Figure 1 Figure 2 Figure 3 Figure 4 LIST OF FIGURES Triple point phase diagram for a transformation that neither admits hysteresis nor stable phase mixtures. Values for the phase fraction triple {§,, Q, Q} are restricted to the three types shown. ----- --16 The terminal nuetrality curves in the X-unfolding given by (2.2.2.7) and (2.2.2.8) for a material with transformation temperatures: A}: 42, A, = 13, Mf = -7, M, = 22 °C; moduli of austenite and martensite: [LA = 50,000MPa, u” = 20,000MPa; transformation strain 1‘ = 0.07; entropy difference: ni-n: = rig—n? = 0.7x106. Since uAatuM these curves are not linear. - ----- -- 19 X-unfolding of the triple point for materials obeying A, < M ,. The S- regions are stable zones in which transitions do not occur so that the triple {§_, §A, 8",} is static on these regions. The W-regions allow for changes in the phase fraction triple as (1:, 7) changes. For example, transitions A —> M + can occur in that portion of and which is between M,., and Mf+' Similarly, M + -) A can occur in the above two regions between A,+ andAf+. - - - - ' - - .-21 X-unfolding of the triple point for the case of A, > M ,. In contrast to the case described in Figure 3, here the region of at between M” and A“, does not admit transformations and so is, formally, a stable zone. For simplicity the a notation is retained and the region is referred to as a dead zone. A similar dead zone exists in Q] . 22 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Transition paths are shown in X-unfolding for A, < M,. Solid lines indicate active transformation while dash or dot lines indicate inactive transformation. - -- - Temperature segments for cases A, < M, (a) and A, > M, (b) in X- unfolding. ------ - ---------- -- _ - - - _---29 pY-unfolding is obtained by modifying the X-unfolding with eight specified nuetrality curves: A“, M", A”, M,“ Af_, M,_, A,” M,-. The former four are “bent” to vertical positions in t S 0 upon encountering M,; and the latter four are similarly “bent” in ‘t 2 0 upon encountering Mf.P Here A, < M,; materials with A, .> M, are treated similarly. -- - - - -- ----- --31 Phase transformation active and inactive zones are shown in the pY- unfolding, which is obtained by modifying the X—unfolding below M,; ande_.-- ------ -- -- - - - 32 Refined pY-unfolding, say, Y-unfolding, improves upon the pY-unfolding by allowing the detwinning flow stress 1:, and finish stress 9 to be specified as additional material properties. This diagram is for the case of . A, < M ,. In the present development, the terminal nuetrality curves undergo abrupt slope changes upon crossing t = 0 and upon meeting the nuetrality curves Mf- and Mf+. - -- - 37 Active and inactive zones in Y—unfolding corresponding to Figure 9. ....... 37 Subdomains R1, R2 and R3 are shown in t > 0 for materials obeying A, <- M,. -40 Figure 12 Figure 13 Figure 14 Figure 15 Intersecting points of nuetrality curves of detwinning flow and finish for k} > k; (a) and for k} < k; (b). - - - - -- - --41 Y-unfolding for 1: > 0 with k2 = 0. Four intersection points a, b, c and d are shown as: a (t,,, T,,), b (1,); T ,1), c (1,, T,,), d (1}, Tfl). - 48 Loading behavior for initial conditions corresponding to initial conditions of maximum austenite, CFAF (equal amount of martensite two variants). For T> M, the initial condition is {§_, Q, §,}={0, 1, 0} and for T< Mf it is {§,, Q, §+}={0.5, 0, 0.5}. For Mf< T < M, the initial condition is a more general {E,,, §, §} with 5,, = Q. The four associated transition paths p1, p2, p3 and p4 go from left to right. On p], so that T > M,. _ segments 01, 12 and 23 indicate austenite elastic, single transformation A —> M, and pure elastic M+ deformations respectively. On p2, so that Tf, < T < M,. segment 01, 12, 23, 34 represent single transformation A —) M,, double transformation A —> M, & M, —> A , detwinning M_ —) M, and elastic M,., deformations. On p3, segments 01, 12, 23 and 34 indicate single transformation A -) M, , two variant martensite elastic, detwinning M _ —> M, and elastic M+ deformations. On p4, segments 01, 12 and 23 represent two variant martensite elastic, detwinning M . -9 M, and elastic M, deformations.----- - -- -- 5--0 These six transition paths p1, p2, p3, p4, p5 and p6 associated with maximum martensite initial conditions, I-IFMF (equal amount of martensite two variants) go from left to right. On p1, segments 01, 12 and 23 indicate two variant martensite elastic, detwinning M , —) M, and right-shear martensite elastic deformations. On p2, segments 01, 12, 23 and 34 represent two variant martensite elastic, double transformation - M.—)A & A—)M,, detwinning M_-)M, and elastic M, deformations. On p3, the first two segments 01 and 12 are the same with Figure 16 Figure 17 Figure 18 Figure 19 those on p2, segment 23 and 34 indicate single transformation A —) M, and elastic M, deformations. On p,, segments 01, 12, 23, 34 and 45 represent two variant martensite elastic, single transformation M _ -> A , double transformation M _ —) A & A —) M, , single transformation A —) M, and elastic M, deformations. On p5, segments 01, 12, 23 and 34 indicate single transformation M . —) A , austenite and M, elastic, single transformation A -) M, and elastic M, deformations. On p6, segments 01, 12 and 23 represent austenite elastic, 'single transformation A -) M, and elastic M, deformations.---- -- ------ - - -- - - -----51 These three unloading paths go from the right to left with initial condition of 100% M,. All the dashed lines on p1, p2 and p3 indicate M, elastic deformations, except the portion to the left of Af, on p1 which represents elastic austenite. All the solid lines on p1 and p2 indicate single transformation M, -) A deformations.------ - - -- ---------52 For 1: > 0, the lowest temperature and largest stress for conducting M_ -)A are Tf, and If, and the highest temperature for conducting M_->M,isTflt.---------- --------- - - - 54 Four open cone areas at a point p in Q, for X- and Y-unfolding show the transition possibilities when a path passes through this point. If the path passing through p proceeds into N (S, W and B), then transition type (1T1)(('I'12),('I'I‘3)and('l'l‘4))isinprcgress - -- - 50 This graph shows that how the transition types occur when one follows a counter-clock wise ellipse in the (t, T )-plane. From point t2 to t3: (TT3) occurs; fromt3toe: ('I'T2)occurs;frompointbtoc: ('I'I‘5)occurs;inall other parts there is no transformation. 63 Figure 20 Figure 21 Figure 22 Figure 23 For the case A, > M ,, three path-dependent zones d2, d3 and d, are separated in OT . g, and a, are path—dependent in d2 if transition type (TI‘2) occurs (S-paths). 5A and §_ are path-dependent in d3 if transition type (TT3) occurs (W-paths). £1, and §, are path-dependent in d, if transition type (1T4) occurs (E-paths) -------- - ---------------- - ----- 89 For the case A, < M,. six path-dependent zones a2, a3, a4, a23, a24 and am are separated in OT . The situations occurring in a2, a3 and a, are the same with those in d2, (1;; and d4 of the case A, > M, shown in Figure 20. In a23, if transition type (T12) is in process then the phase fractions §_ and g, are path-dependent, while if (TT3) occurs then EA and §_ are path- dependent. In a24, the condition is similar to that in a23 under interchange of('I'T3)and(‘I'T4)aswellas iand§,.1na234,§and§,arepath-' dependent if ('I'T2) occurs, §A and §_ are path-dependent if (TT3) occurs, and, §A and §, are path-dependent if ('I'T4) is in process .......................... 92 Three paths 1,, 12 and 13 go from (r, o) = '(o, M,) to (o, M,.) in the path- dependent zone of transition type ('I'T2) with initial condition {§., §A, §,}={0, 1, 0}. Transition type ('l'T2) occurs on all three paths. Path 12 consists of two straight segments which meet at point MEI-EM“ -Mf), %(MJ + M f) ). Path 13 is similar. The phase fractions E. and §, are path dependent while §A is not. The values of the triple {§,, fiA, §,} at the ends of the three paths are listed in TABLE 7 ..................... 95 Pseudoelastic behaviors in both tension and compression conditions at test temperature 1', = 335. In r > o, A (-) M, processes are involved with the loading/unloading, while, in 1: < 0, A (—) M_ processes are involved with the opposite loading/unloading. ------------- -100 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure30 Pseudoelastic behaviors for M, < A, at different temperature levels: T, = 315, 325, 335 °K, all of which are greater than Af= 308 °K. .................. 100 By Falk’s model (1980), austenite transforming to martensite occurs at the highest point on the left ascending branch (top dashed line) upon loading. The reverse transformation, martensite to austenite occurs at the lowest point on the right ascending branch upon unloading. - -- - - - - 102 By Landau-Devonshire’s model, load-deformation diagrams in three different temperatures (T 1 < T2 < T3) show that the heights of the hysteresis loops decrease with the temperature increase. ----- -- -102 A dead zone between the top and bottom bands in the stress-strain diagram is illustrated. This dead zone corresponds to the portion between points 2 and 3 in the phase diagram. Points f and g correspond in stress- strain diagram to points 1 and 4 in the phase diagram. The internal loop formation condition is that unloading has to reach the bottom band and loading has to reach the top band shown as a-d-b-c-h path. .................... 104 The stress-strain trajectory approaches a stable internal loop in the stress- strain diagram with oscillating scope of stress between points a and b (between 1" and 1‘ in stresses) in the phase diagrarn.- - 106 , Cycling loadsareappliedbetweent=1brAMattest temperature T > Afto form internal hysteresis loops. It can be seen that the top and bottom bands are covered inside the cycling range ............... 107 Shape memory effects occur during loading-unloading-heating-cooling processes. - - -- - _109 xiv Figure 31 Figure 32 Figure 33 Figure 34 Two different procedures recover the residual strain. (a) shows the residual strain recovered upon heating, (b) shows that the residual strain can be recovered by further unloading. Here the test temperature is 301 °K during the loading/unloading process and the original phase fraction is §= {0, 1, 0}. ----------- - -------- -- - ----111 The residual strains are recovered by further unloading (a —) b —) c ). The plateau of the transformation A -) M, decreases with the test temperature decrease. In the opposite, the yielding plateau of the reverse transformation M, -) A increases in the negative direction of the t-axis as the test temperature decreases. - ----- - - - -- - - =-112 Ferroelastic behaviors in both tension and compression conditions at test temperature T, = 200. In 1: > 0, M_ ->M, process is involved with the transformation, while, in t < 0, M, -> M_ process is involved with the transformation. ......................................................................................... l 13 Initial conditions are obtained by cooling the temperature from above A, to the test temperature in a stress free circumstance (CFAF). General features of the transfonnation process for loading/unloading were described in Figure 14. In (a) the initial condition is {0, 1, 0}. 1->2: austenite elastic deformation; 2 -> 3: phase transformation A -) M ,; 3->4: M, elastic deformation; 4-)5: M, elastic unloading; 5—>6: partial reverse transformation M, -) A which gives a certain amount of residual strain left at the end of the unloading. In (b) the initial condition is still {0, 1,0} because 270°K>M, (=263 °K). 142->344: conduct the same deformation mechanism as drose segments in (a) correspondingly; 4 -) 5: M, elastic unloading. In (c) the initial condition is {0.0135, 0.973, 0.0135}. 1->2: phase transformation A -)M,; 2->3: combined transformations A-)M, and M_->A; 344: XV Figure 35 - detwinning M_ -) M ,; 4 -) 5: M, elastic deformation; 5 —> 6: M, elastic unloading. Since the phase fraction of M, is small, there is no significant change in segments 1 -) 2 , 2 —> 3 and 3 -) 4. In (d) the initial condition , is {0.222, 0.556, 0.222}. 1 -) 2: phase transformation A -> M ,; 2 -) 3: elastic twinned martensite; 3-94: detwinning M-—)M,; 4->5: M, elastic deformation; 5 -9 6: M, elastic unloading. In (b), (c) and (d) the residual strains are the phase transformation strain 7‘. -115 Initial conditions are obtained by heating the temperature from below Mf to the test temperature in a zero-stress condition (HFMF). General features of the transformation process for loading/unloading were described in Figure 15. In (a) the initial condition is {0.2801, 0.4397, 0.2801}. 142: phase transformation M_ -)A; 2-—)3: elasticity of combined austenite and right-shear martensite; 3 —> 4: phase transformation A ->M,; 4—>5: M, elasticity; 5—)6: elastic M, unloading; 6 —> 7: partial reverse transformation M, -) A upon continuous unloading which gives a certain amount of residual strain left at the end of the unloading. In (b) the initial condition is {0.5, 0, 0.5} which is also the initial conditions for (c), (d) and (e). 1-) 2: fully twinned martensite elasticity; 2 -> 3: phase transformation M _ —) A; 3 —> 4: elasticity of combined austenite and right-shear martensite; 4->5: phase transformation A—rM,; 5—>6: M, elasticity; 6—>7: elastic M, elastic unloading. In (c) 1—> 2: elasticity of twinned martensite; 2 —) 3: combined phase transformation M _ -) A and A-)M,; 3->4: phase transformation A—>M,; 4-95 and 6-97 are similar to 5 —>6 and 6 ->7 in (b) respectively. In (d) 1—>2: elasticity of twinned martensite; 2 -) 3: combined phase transformation M _ -) A and A-rM,; 344: detwinning M,-)M,; 4-)5 and 5-96 are similarto the corresponding sections in (c). The only difference of (c) with (d) is that there is only one section to conduct detwinning M_ -> M ,, which is xvi Figure 36 Figure 37 Figure 38 Figure 39 Figure 40 Figure 41 2 -) 3 . In (b), (c), (d) and (e) the residual strains left are phase transformation strain 7.. -- -- - 116 Isothermal response (b) under cyclic loads (a) at test temperature T = 270 °K (M, < 270 °K < A,). The initial condition is from HFMF with § = {0.5, 0, 0.5}. The outside profile is the same with Figure 35 (c). M_ -) A and A —> M, occur upon loading while elastic relaxations in mixture phase states occur upon unloading. M. phase is roughly consumed out after the fourth cycle. Austenite remains until the tongue-shape stress-strain response reaches the far right lateral straight line of the outside profile. 118 Two groups of driving paths starting at point (0, 325) on T-axis with initial condition {0, 1, 0} go to point (20, 220), (40, 220), (68.9, 220), (110, 220), (150, 220), (160, 220), (170, 220), (180, 220), (190, 220) respectively ............................................................................................... 121 Phase fractions of the martensite variant M, upon driving paths of loading-cooling (upper point plot) and cooling-loading (lower point plot). Here 6' = 68.9, r, =1so and rf= 200MPa. -------------------- -- -123 Structure of the TSTE confined in a fixed frame. .................................... 129 The path segment p52 has three possibilities: I, 2, 3, as shown in the above. Path 3 is the desired situation and approachable for many shape memory alloys, which will be discussed in the following. - 134 A physical interpretation, for the condition that ensures that detwinning occurs during heating in a stroke. - 135 xvii Figure 42 Figure 43 Figure 44 Figure 45 Figure 46 Figure 47 Six (0', T I)-path segments and their connecting points related with the first stroke (heating/cooling element (1)) are schematically presented in the phase diagram for o > 0. .......................................................................... 140 Temperature pulses applied on the two elements. The maximum temperature in heating must be sufficient to fully austenitize the operating element. .................................................................................................... 146 (0’, T )-paths for the first stroke for element (I) (a) and for element (II) (b). The operating element here is element (I) with initial temperature To = Mf, which is equal to the constant temperature of the response element (II) during this process. Here the material properties come from TABLE 2. In particular the moduli are u, = u” = 3.0x 10‘ MPa. .......................... 154 (o, T)-curves in the second stroke are presented in (a) for the element (II) and in (b) for the element (I). The operating element is element (II) with initial temperature To = Mf, which is equal to the constant temperature of element (I) during this process. In the first part of p52 both M- -) A and M, -) A are involved in the phase transformation (refer to Figure 42). 154 This graph shows (a, T)-curves of the operating elements in ten strokes. A stable (0, T)-loop (the darker one) is reached after the first cycle (the lighter one). -- -- - - - - ---155 The corresponding displacement against stress (a) and temperature (b) after ten strokes. The lighter dots correspond to the first stroke. The darker dots correspond to the following nine strokes, which are indistinguishable at this scale. The head down loops correspond to operating element (II) xviii while the head up loops correspond to Operating element (II) in both (a) Figure 48 A given phase diagram as might be determined by experimental measurements. R1 is the region above nueu'ality curve M f , for 1: 2 0; R2 is the region between M f, and M f_ for t > 0; R3 is the region between M f, and Taxis for 1: 2 0. m, E and R3 are corresponding mirror images ole,R2andR3 about T-axis for 1:<0. .................................... 168 xix 1 INTRODUCTION 1.1 Background to This Work As materials that have many active and interesting properties, shape memory alloys (SMAs) have received more and more attention both theoretically and practically in recent decades. Their many employments in various sensitive areas, which include: the driving force in heat cycle engines (Banks and Weres, 1976), orthopedic devices for securing frac- tured bones (Wayman, 1980), integrated actuator/sensor fibers in special composite sys- tems for active control of dynamic and structural behavior (Rogers et al., 1989), blood clot filters and displacement sensors (Takeda er al., 1986), and many more all attribute greatly to their enchanting qualities. Although many inventions have been made, potentials of shape memory alloys still hold great chances for more implementation and further devel- ' opment. Shape memory alloys associated with martensitic phase transformation are char- acterized by first-order solid to solid transformation without atomic diffusion. Their most important properties are the shape memory effect (SME) and pseudoelastic effect which are responsible for many innovative applications. Thus it has become necessary to have an accurate understanding of the therrnomechanical behavior of the shape memory alloys. There are plenty of literature on modeling the behavior of the shape memory alloys, and what we are going to-discuss in the following are the most representative approaches that we have reached. Based on Landau-Devonshire’s theory, Falk (1980) illustrated the 2 stable phase transition between austenite and two martensite variants in a single crystals by minimizing non-convex type of Helmholtz free energy, which can either describe stress induced or thermal induced phase transformations strategically. In Falk’s model, charac- terized by the equality of the phase transformation stress and the Maxwell stress, basic features of SMAs, such as pseudoelasticity, lattice softening and shape memory effect (SME) presented, are qualitatively agreeable with experimental works in some sense. Achenbach et al. ( 1986) derived rate-dependent types of model, from statistical mechanics and thermodynamics point of views, to rehearse the plane-strain responses of a polycrystalline object under biaxiel loading. In this publication they took the polycrystal- line body as a three-phase configuration of austenite and martensite twins, and operated the fractions of the three phases as internal variables which were parametrized by the assumed continuous distribution of orientations of lattice layers for any instant. By incorporating uses of Helmholtz free energy and dissipation potential, Tanaka (1986) came up with a rate-type constitutive and an evolutionary equations which could represent the pseudoelasticity and shape memory effect again under the condition of one- dimensional tensile only for stress induced martensite transformations in a polycrystal, which the new nucleation and the growth of the martensite may be understood to be fully governed by macroscopic transformation kinetics. The internal variable that depends on - stress and temperature is the phase fraction of martensite. Liang and Rogers (1990), based on the integrated form of Tanaka’s equation, presented a thermomechanical constitutive model. In their study, a rate-independent type of equation was proposed to fit the marten- site fraction and temperature relations in order to predict pseudoelasticity and shape mem: - cry behavior. The results from this model had good coincidence with experimental observations made by them. 3 As far as pseudoelasticity is concerned, Muller and his colleagues made an extensive investigation. For example, with the consideration of non-monotone load-deformation curves generated from non-convex energy forms which are the type deliberate in Falk’s model (1980), Muller and Xu (1991) developed Falk’s model by taking into account the dissipative effects from interfaces and coherency. They gave a thorough description of pesudoelasticity loops for a single crystal, in which a width of a hysteresis loop of stress- strain relation depends on interfacial energy of phases rather than depends on the tempera- ture, and the hysteresis loop maintains metastable states which loose their stability on a line defining phase equilibrium. Based on the mixture approach of the work done by Muller (1979), and Muller and Xu (1991), Fedelich and Zanzotto (1991) extrapolated the isothermal hypotheses to nonisotherrnal conditions by contemplating the hardening obser- vation of stress-strain relation with increasing temperature. Only for two-phase situations where the stress and temperature are high enough, they found out, that the elongation rates have a significant influence on phase transformations in a bar, and the slope of the slanted sides of deformation-temperature 100ps is rate dependent, which was given a reasonable explanation. In Leo er al ’s macroscopic descriptions of two phase system (1993), they assumed the austenite to martensite transition stress to be equal to the Maxwell stress plus an additional constant stress equal to half the height of the isothermal hysteresis 100p which accounts for dissipative processes associated with the phase transformation. They also incorporated the temperature and Stress dependence of energy with heat transfer associated with austen- ite/martensite single interface. Mth this model they explained a phenomenon that the hys- teretic stress-strain curves depend strongly on the strain rates at which the wire is extended. 4 Brinson (1993) modified the works by Tanaka & Iwasaki (1985); Tanaka (1986); Liang & Rogers (1990) to describe the SME below a martensite start temperature by split- ting the martensite phase fraction into temperature-induced and stress-induced parts. This subdivision is justified by the micromechanical behavior of SMAs and is effected such that the stress-induced martensite fraction represents the extent of transformations of the material into a single martensite variant. This model can well capture the main conducts of either phase transformation between austenite and over all martensite variants in higher temperature ranges .(pseudoelasticity), or conversion of all martensite variants in lower temperature ranges (SME), with the suggested equilibrium phase diagram. Latter on, in the study by Bekker and Brinson (1994), more detailed discussions on the this phase dia- gram were made to generalize the problem. In all, energy consideration has been the most concentration point in these works men- tioned above. From plasticity point of view, Bondaryev and Wayman (1988) combined the plasticity flow theory and the change of Gibbs free energy to narrate phase transformations in the biaxial case. The idea is cast terms of the change of Gibbs free energy of phase tran- sition or detwinning processes to determine the threshold stress for the changes and then using the plastic flow rule to set up the stress-strain relations during the two processes. This model can also recite the phenomena of pseudoelasticity at high temperature and fer- roelasticity at low temperature. The work or Graesser and Cozzarelli (1991) utilized Bonc-Wen type’s model of rate independence. It employs the dynamic analysis (Wen, 1976), to generate one-dimensional evolutionary equations of plasticity to modify the macroscopic stress-strain character of SMAs. Later on, they extended the one-dimensional model to three-dimensional cases by means of the agreement of volume preserved between the plastic flow and martensitic 5 phase transformations. This model was able to offer some helpful theoretical results for different loading conditions with properly choosing parameters to modify the plastic flow theory and then to describe the behavior of SMAs. All the modeling investigations mentioned above can reflect the. most basic character- istics of SMAs to some degree. However, most of them in general are focused on some specific aspects. When more features of SMAs are needed to be described, it follows that the constitutive equations are usually needed to be changed. For instance, Achenbach er al’s work (1986) can hardly be used to describe plane stress problems, neither is it easily applicable in engineering; Tanaka and his colleaques’s works (or Liang and Rogers’s work) are well suitable for the isothermal stress-induced martensitic transformations or reverse transformations, but they are short of enough descriptions of adiabatic, convective, as well as therrnal—induced situations of martensite transformations. Muller and other’s studies (or Fedelich and Zanzotto’s work) have their most concentration on pseudoelastic- ity but little focus on other properties. Some of the above investigations, such as plastic type of modeling, are in some way or the other in a difficult condition with respect to ' parameter choosing, physical understanding, or practical engineering. Thus, a more com- plete model which is easily applied, as well as supported by a accurate theoretical resources, is the motivation of the current study. Recently, on the basis of the work done by {Coleman & Hodgdon (1986 & 1987), Ivshin and Pence furthered the Duhem-Madelung Model for magnetic hysteresis to the constitutive relations of shape memory alloys (Ivshin & Pence, 1994 a,b) to describe the macroscopic behavior of SMAs from the hysteresis, thennodynamics and continuum mechanics point of view. In the study of Ivshin & Pence (1994 a), a model for rate inde- pendent hysteresis was examined. The evolution equations for phase fractions of austenite 6 and one martensite variant was derived by considering temperature as the only driving forces. Several restrictions on the hysteresis envelopes, which are the maximum and mini- mum values of austenite phase fraction during phase transformations, were given to ensure monotonicity, containment and orientation requirements. Later in the approach made by Ivshin & Pence (1994 b), evolution equations were developed to govern the time history of ‘ shape memory alloys under changes in stress, strain and temperature for one martensitic variant problems. In addition to that, Ivshin and Pence also presented simulations of the relations of stress and strain in either isothermal or adiabatic conditions and internal hys- teresis transition loops. Comparison, made by Brinson and Huang (Brinson and Huang, 1996), between this model and Tanaka’s model, yields nearly identical results if both mod- els are used with the same kinetic law. However, Ivshin and Pence’s model gives more flexibility in a variety situations varying from isothermal to adiabatic conditions. To extend one variant model to two variant model, a basic analysis for two variant martensite problems was given in the work by Pence er al. (1994). They began from the discussion of “triple point organizational kernel”, and then explored the idea to the mixture of coexist- ence and hysteresis unfoldings. Lateral pseudoelasticity associated with phase transforma- tions was obtained under cyclic loading in high temperature regions. 1.2 Scope of This Work The present study will give a relatively complete description to the two variant marten- site model by refining and extending the one-dimensional work of Pence et al. (1994). We consider a two-phase material which consists of a high temperature austenite phase A and two symmetric variants of a low temperature martensite phases M, and M _ which could 7 be treated as twin related. The first of these two variants M, is favored in positive stress (1: > 0) and the second M _ is favored in negative stress (1: < 0 ). We assume that, the for- ward (austenite to martensiti:) transition temperatures M, and M f, and backward (marten- site to austenite) transition temperatures A , and A f, could be measured by decreasing and increasing the temperature at zero-load. In the present study we are going to employ a phase fraction triple {§., fiA, §,} as internal variables, while temperature T and stress 1 are driving forces. Here §A indicates the phase fraction of austenite, while g, and §_ indicate the phase fractions of the two martensite variants. We treat the situation in which the temperature and stress vary with time in some prescribed fashion. This generates paths in a (1:, T )-plane, which will be referred to as the state plane. Pairs (t, T) will be referred to as states, and continuous paths (1:(t), T(t)) in the state plane will be referred to as state paths. The main purpose of this research is to study, refine and apply an algorithm for phase transformations between the three species A, M, and M _ as the state of the system varies. Thus we seek to determine the values of Q, E, and g, on state paths. Hysteresis in these transformations indicate that Q, §, and §_ will not in general be state functions, that is, the instantaneous value of (t, T) does not determine the values of 5,, g, and §_ . Rather we study an algorithm which determine these values on the basis of known initial values for Q, g, and E, , and knowledge of the subsequent state path. At each instant of time, the three phase fractions satisfy the following balance relation 6A0) +§,(t) + 5-0) = 1 (1.2.0.1) To achieve the purpose, various kinds of state plane partitions into M,. A, M, regions (X-unfolding, pY-unfolding, Y-unfolding) are developed from a thermodynamic restric- tion which ensures that Gibbs free energy satisfies the second law of thermodynamics. 8 These partitions are different unfoldings of a standard triple point phase diagram. Charac- teristic curves in the state plane are obtained which govern processes A <-> M, and A <—) M _ respectively, in a way that is similar to (Pence et al. 1994). Descriptions of the M, <--) M , process are then developed by modifying the two austenite/martensite pro— cesses. The characteristic curves act as nuetrality curves which classify both austenite] martensite phase transformations and detwining processes (M, <—) M _) by regions in the (t, T)—plane. They also enter the evolution equations for the phase fraction triple {§_, §A, Q} as an internal variable. Internal variable descriptions for thermodynamic behavior are common, for example, Coleman and Gurtin (1967) employed thermomechanical and internal variables together to generalize dissipation problems. In section 2, we begin by extending the triple point phase diagram (Pence et al., 1994) to an X-unfolding by taking thermodynamic considerations on the three species M., A and M,. The nuetrality curves, describing phase transformations between austenite/martensite at high temperature, are modified as detwinning nuetrality curves, when they enter into non-austenite regions. That means the Clausius-Clapeyron relation for austenitelmarten- site is used to describe the detwinning process, which is not exactly the case. To justify this point, the entropy of austenite is replaced by that of martensite in the Clausius-Clap- eyron relation so as to generate constant stress nuetrality curves for detwinning, in which the detwinning flow and finish stresses are predicted as a consequence of this procedure. Experimentally, it is often the case that the detwinning flow and finish stresses are smaller that the ones predicted by the X—unfolding., If the detwinning flow and finish stresses are instead treated as material constants, a Y-unfolding is suggested. In Section 3, we first define all transition possibilities (transition types) when tempera- ture and stress trigger the transformations. Then criteria for different transition types are 9 derived based on the nuetrality curves, which are useful for us to determine what kinds of transition types occur at any point on a given path in (t, T )-plane. These criteria have an explicit geometric illustration. An example of how to use these criteria is given at the end of this section. Chapter 4 opens with a brief review of one variant model (Ivshin and Pence, 1994, a, b). To extend the model to two martensite variants M. and M,, we introduce a concept of constitutive functions of austenite which is similar to that of the one variant case. We then work on the extension of algorithms from one variant to two variant. Numerous possible algorithms in the two variant case make the problem difficult, but with the symmetry of two martensite variants and the coherence of phase transition possibilities, we arrive at a unique algorithm for each transition possibility. The extension is verified by considering the self-accommodated process in the two models. In chapter 5, based on the algorithm arrived in the previous section, the solutions of phase fractions for each transition type are calculated. This permits further analysis of the stress-strain relation. The phase fraction evolution depends on the state and the state-path orientation in the (1:, T)-plane. Conditions that would ensure path independent algorithms for the various transition possibilities are derived mathematically. However analysis of the path-independent conditions shows that path independence within a transition type is - often not the case. In chapter 6, several numerical results in stress-strain—temperature relations are given. Pseudoelasticity, shape memory and the features of the associated internal hysteresis loops are thoroughly studied. Some isothermal behaviors between the temperatures Af and Mf are also conducted for different initial conditions. Finally, comparisons with other models are made at the end of this section. 10 As an application of this model, a prediction on reciprocal motions of two element devices made of shape memory alloys is analyzed in chapter 7. The analysis focuses on the prediction of phase fraction changes inside the device due to temperature pulses that alternate between the two elements. The result show that a stable cyclic linear motion can be reached after several repititions of the temperature pulses. Conclusions and possible ideas for future work are presented in Chapter 8. An appendix is also given for the inverse issue of Chapter 2. In the previous study, it begins with considering the second law of thermodynamics to create the phase diagram. Linear stress-strain relations of each phase are assumed during this process. The predic- tion for the phase diagram is usually different with experimental observations. Therefore, it naturally arises an inverse problem to determine the stress-strain relation under the same thermodynamic regard and a given phase diagram by experimental measurements. In the appendix, a problem related with the above consideration is well posed. Nonlinear stress- strain relations are obtained corresponding to the given phase diagram. The solution also shows that the two issues are coincident under the same phase diagram. 2 FORMATION OF A TWO VARIANT PHASE DIAGRAM In this study we begin with assuming that phase transformation triggered by tempera- ture and stress satisfy the second law of thermodynamics. According to this point, charac- teristics of a differential equation, obtained from increments of Gibbs free energy of the three species system obeying the Maxwell relation, are derived in association with both the process A (—) M, and A <-) _M_ . Those characteristics are interpreted as nuetrality curves in the state plane and govern as a “modifier” transformation between austenite and the martensite variants within the system. At low temperatures they are reinterpreted in terms of a direct M _ <—) M, process. Then considerations of consistency between these new process interpretation and the Gibbs/Maxwell argument, modify the low temperature characteristic. This then adjusts the mathematical treatment of the M _ H M, detwining process. 2.1 Concept of Nuetrality Curves First of all, we recall some thermodynamic views introduced in the work by Pence et at. (1994) to describe the phase transformations. Entropy and strain of the fine mixture of the three species system are extensive variable counterparts to the temperature T and shear stress 1:. High temperature and low stress corresponds to a situation of high entropy and low strain (austenite favored); low temperature and high stress is consistent with a condi- ll 12 tion of low entropy and high strain (martensite favored). Under a rule of mixtures, these extensive variables are defined by n = Erna + in. + 5-11- . (2.1.0.1) 7 = §A7A+§+Y++§-Y-- (2.1.0.2) Here, nub and 7,“, are entropies and strains of pure phases, and are assumed to be state functions of the driving forces (1:, T). The increments of Gibbs free energy G of the mix- ture of the three species are required to obey the relation d0 = - ndT — 7dr. (2.1.0.3) Here (11, T) and (y, 1:) each form a conjugate pair of thermodynamic variables. The math- ematical development is general enough to include both the case where (y, t) are regarded as shear variables and are regarded as normal variables. Of course in making contract with experiment, the sense in which (7, t) are to be regarded must be specified. To avoid any violation with the second law of thermodynamics, Gibbs free energy G must be a state function of temperature and stress, so that the overall entropy and strain satisfy the Maxwell relation (2. l .0.4) $3” 37: it Under the assumption that entropies and strains of pure phases are state functions, which means the Maxwell relation is satisfied in each pure phase, and abides by the law (1.2.0.1) of the balance of the three species, the above relation gives 13 gm), -n,,) +§annn> = 3%(Yt-Yn) +3510- ’YA) . (24-0-5) which is a thermomechanical restriction on the system. To interpret the physical meanings of the equation, we are going to consider the phase transformations among the three spe- cies. Changes of temperature and stress in (t, T)-plane drive transformations between the three phases M _ , A and M, . Let us first classify the phase transformations into two courses A <—> M, and A H M,. There are two transition directions in each course. The classification remains potentially incomplete until a process is given for describing M, (—-) M _. For the present we only contemplate the first two processes and later on Sec- tion 2.2.3 the third course M, (—> M, will be introduced in terms of the work by Wasilewski (a,b,c, 1971). In process A <—) M ,, beginning at any initial point in the (t, T)-plane it is assumed that certain paths away from this point will favor the transformation of A to M, (A —) M ,). Paths in the opposite direction would then favor the transformation of M, into A (M, —> A). It is assumed that there exists exactly one neutral direction (including its opposite direction) away from the initial point in which neither an A -> M, nor an M, —> A transformation is favored. The curve traced out as one travels in this neutral direction is an A/M, nuetrality curve which will be discussed next. In the above we assumed that the overall phase transitions obey the Maxwell relation (2.1.0.5). In particular, this equation must hold for A H M, process in which the phase fraction 5, does not change. For such a virtual process the thermomechanical restriction (2.1.0.5) on the whole system specializes to 14 mi. 3§+ 3.1:. (11+..nA) = BTW+_YA) . (2.1.0.6) The characteristic equation associated with this differential equation is the following dT 7+ " YA dt n,—nA’ (2. 1.0.7) which, in turn, gives the g, nuetrality curves for the process A H M ,. This is familiar as the Clausius-Clapeyron equation for process A H M, . Following a similar procedure one finds that the Clausius-Clapeyron equation for process A H M _ is dT 'Y- " YA — = — . 2010008 ‘11: IL ’ TIA ( ) Integration of the Clausius-Clapeyron equations (2.1.0.7) and (2.1.0.8) yields a param- etrization of nuetrality curves in the form of B:(1:, T) = C + and 6;,(1, T) = C ' for the two processes. Here C+ and C' are integration constants which locate the individual curves that make up the two families for A H M, and A H M _ respectively. The develop- ment so far has made no assumption as to the symmetry of the variants M, and M -. For simplicity we consider a situation in which these two variants are symmetric with respect to stress, that is, the effect of a particular value to in triggering M, is identical to the effect of -ta in triggering M_ (all other factors being equal). This requires that y,(t, T) = -'y,(—t, T) and 11,(t, T) = 111—1, T) . This symmetry assumption implies ten- sion/compression synunetry where (y, 1:) are regarded as normal components. On a finer crystallographic scale where the shear interpretation may be useful, this implies that M, and M, represent variants with equal and opposite lattice shears. 15 These symmetries give B:('t, T) = B;(-t, T) and implies nuetrality curves that are symmetric about t = 0 . It is often convenient to choose this parametrization so that {32(0, T) = T = 0;,(0, T) and we shall choose this convention. Then B: and {3; play the role of a generalized temperature. Therefore, we let increasing C+ and C- correspond to M, -) A and M_ -> A transformations respectively, and decreasing C+ and C' corre- spond to A -> M, and A -) M _ transformations respectively. Setting [3:01; T) and 3;,(1, T) equal to the four transition temperatures Mf M,, A, and Af, gives eight canonical nuetrality transition curves. We name these eight curves the terminal nuetrality curves in the rest of this study. 2.2 Special Nuetrality Curves and the X-unfolding 2.2.1 Unfolding the Basic Three Species Phase Diagram Before we find the nuetrality curves it is helpful to recall the unfolding of the standard triple point phase diagram introduced in the approach by Pence et al. (1994). Figure 1 is the standard triple point diagram which most simply categorizes the system of the three species of austenite and martensite two variants. In this phase diagram, all phase transfor- mations of austenite/martensite and reorientations of the two variants are abrupt so that either {§_, EA, fi,}={ 1, 0,0}, {0, 1, 0} or {0, 0, 1}; these correspond respectively to pure M_ martensite, pure austenite, or pure M, martensite (Figure 1). If a continuous path (t(t), T(t)) is prescribed, then the austenite/martensite phase transformations or the mar- tensite/martensite reorientations take place whenever the path either crosses one of the dual species transformation curves, or else crosses the triple point (0, T. ). The same 16 transformation curves operate in each process direction, so that these processes are also not hysteretic. To build a natural hysteresis and coexistence of SMAs in the model, we may unfold this triple point phase diagram by the thermodynamic derivations obtained in the last section following Pence et al. (1994). A<—>M, ‘T transition curve A <->M, transition curve SALE-({0, or 1}) SM.({19 0, 0}) (0,T*) >t transition curve Figure 1. Triple point phase diagram for a transformation that neither admits hysteresis nor stable phase mixtures. Values for the phase fraction triple {§_, SA: 5,,} are restricted to the three types shown. 2.2.2 Formation of X-unfolding To unfold the triple point phase diagram, it is necessary to determine the form of the characteristics or those nuetrality curves from the Clausius-Clapeyron equations. Integra- tion of equations (2.1.0.7) and (2.1.0.8) gives l7 (1. T) 1 1530. 775 , 0 j (n.-n.)dT+(vr-r.)dt (2.2.2.1) "A " 11410.70) 1 (T. T) B;(t. 1) a o a j (11,, — n,)dT + (y, -y,)dt (2.2.2.2) “A - Ti. (0’ To) where, To is a fixed reference temperature, 11? = n,(0, T0) are zero-stress reference entropies at this temperature for i = -, A, +. In particular if the stress-strain behavior is lin- early elastic and temperature independent in each pure phase with moduli It, (Ill. = I1. = “111) ie., 7.. = W... i. = ‘r/lwv’. Y- = min-r". (2.2.2.3) and if the entropies of the three species are simply expressed as Tl.- = C ln(T/ T o) + 11?. (2.2.2.4) where C is a common specific heat for the all three species with 11: = 11?, then the same results as in the approach by Pence et al. (1994) gives 2 ( - )1 . Blane“ 01 0[ "‘2‘“ :1“ -y t], (2.2.2.5) 2 - ' ( " )1 e B,(t.D.--T+ 01 0[ "g" t‘ +71). (2.2.2.6) Note that, in each phase/species (M, A, M,) that (2.2.2.3) and (2.2.2.4) are consistent with 18 the Maxwell relation (2.1.0.4). Here 7. is the zero-load transformation strain. For exam- ple, a system with: moduli of austenite and martensite, u, = 50, 000 , it“ = 20, 000 MPa, the difference of the reference entropies between austenite and martensite, 112—112 = 112-1]? = 0.7x 10‘5 j/(m3 °K), and transformation strain is assumed 7‘ = 0.07 , gives the nuetrality curve parametrization (331,1) -_.- T- 1 x 10"r—2.14 x 10’5t2, (2.2.2.7) B;(r, T) = T + 1 x 10“r — 2.14 x 10‘5r2. (2.2.2.8) Continuing with this example, suppose A f = 42, A, = 13 , M f = —7 and M, = 22 °C. Then, each of equations (2.2.2.7) and (2.2.2.8) gives 4 curves passing through those 4 tem- 3+9 3+, peratures, and so that eight terminal nuetrality curves: A f ,, A M f ,, M A f_, A,-, M f_, M ,_ are obtained, which, together, generate an “X-shape” in the (1:, T )-plane (Figure 2). This formally unfolds the triple point phase diagram by separating both the A H M, and A H M _ phase transformation curves. Compared to the triple point phase diagram in Figure 1, this unfolding bifurcates the triple point (0, T‘) into (0, M,), (0, A ,), (0, M ,) and (0, Af), and extends A H M, and A H M , processes into negative and positive stress areas respectively. We temporarily ignore any unfolding of the M, H M _ transformation curve and instead focus on the formal extensions of the terminal nuetrality curves back- wards into the low temperature region of phase diagram. This, rather questionable, scheme will be called the X-unfolding. We will discuss this unfolding in its own right, and then show how an additional modification can be used to account for an unfolding of the M, H M . curve associated with martensite variant reorientation processes. 19 Z .. T M. J. at 5" .: . 2: \M" .s,‘ Figure 2. The terminal nuetrality curves in the X—unfolding given by (2.2.2.7) and (2.2.2.8) for a material with transformation temperatures: Af = 42, A, =13, Mf = -7, M, = 22 °C; moduli ofaustenite and martensite: “A = 50,000 MPa, ”M = 20,000 MPa; transformation ~ strain 7 = 0.07 ; entropy difference: 112—112: 112-1]? = 0.7x106.Since "A #uM thesecurvesarenotlinear. 2.2.3 Building Detwinning into the X-unfolding It is convenient to define eleven zones in (1:, T)-plane associated with this X-unfold- ing, which are indicated in Figure 3 and Figure 4 for the separate cases of material obeying A, > M, and A, < M, respectively. Formally these regions are given by 20 QT={(I. T)|M,sB;(t,r)5Af,MfsB;(r,nsAf}, 9.} ={ (T, T)IB:(T. 71M). MISBJLDSAf }. 9? ={(1.T)|B;(1.D>A,,M,s B:(t, DSA, }, a; ={(r, r)|pj;(t, nA,.B;,(t. T)>Af}. Spam. T)|B;(r. n > A ,. 63o. T) A ,1. Those regions defined in the above way are shown in the following two graphs for materi- als obeying eitherA, < M, (Figure 3) or A, > M, (Figure 4). 21 \\\tM 1. \\ Figure 3. X-unfolding of the triple point for materials obeying A, < M ,. The S-regions are stable zones in which transitions do not occur so that the triple {§., §A, §,} is static on these regions. The D-regions allow for changes in the phase fraction triple as (1:, T) changes. For example, transitions A —> M, can occur in that portion of Q: and QT which is between M,, and Mf,. Similarly, M, -> A can occur in the above two regions between A,, and A,,. 22 Figure 4. X-unfolding of the triple point for the case of A, > M,. In contrast to the case described in Figure 3, here the region of of between M,, and A,, does not admit transformations and so is, formally, a stable zone. For simplicity the Q notation is retained and the region is referred to as a dead zone. A similar dead zone exists in Q} . The X-unfolding, preserves the three stable phase zones SA, 5”,, and SM_ in which the phase fraction triple is {0, l, 0}, {0, 0, 1} and {1, 0, 0} respectively. Since this develop- ment treats A H M, and A H M - nuetrality curves, but neglects a direct M, H M _ pro- cess, one identifies the t0p half of the “X” with the unfolded A H M, and A H M _ transition curves of Figure 1. Actually, A H M, and A H M _ may be active in Q: and (2’, respectively while in 9;” both processes may be active. However the bottom half of the “X” does not, at present, permit an obvious correspondence with the M, H M - trans- formation curve of Figure 1. One goal is to naturally develop this correspondence. To do so we identify the region between the two lower branches of the “X” as a region of stable 23 MJM, mixture, which would likely exist in a crystallographically twinned state. For rea- sons outlined below, this region naturally extends to the regions 5”,, and S M-+- The auste- nite phase is not present in the zones S M _, , S M, and S M ,_ , and so that the phase fraction triple must be of the form {§_, 0, §,}. The values fi, and E, are determined by how a state path enters into this combined area. The values 5,, and §_ are then static so long as the driv- ing force path (t(t), T(t)) does not exit S M _, U S M,, U S M ,_ . We now consider transforma- tions between the six stable regions: SA, 3”,, SM, 5M0, 51“,, and SM,,, due to driving force paths (t(t), T(t)) which connect them via the five active regions: (2}, (2:, (IT, (2'2 , 52;. We first examine connection paths that avoid the central region 0;". Consider eight special paths: p1, p2, p3, p4, p-f, p5, p3, p3 (Figure 5), which are organized here into pairs of opposite direction (e.g. (p1, p2), ..., (p3, pz)). They are also taken to be orthogonal to the various nuetrality curves, and so are referred to as transfonnation flux paths; consequently, they will have the greatest transformation gradient. The general case of an arbitrary path can be approximated by infinitesimal path segments that then alternate between the local nuetrality curve direction and the local transformation flux direction. 24 [11: M- -> A , . «a I’ 92: A '> M- p4: A -> M,\\ 155* 1" , . ' i .- M" ‘5: r W" I \ SM- SM-l- pgzM,->(A)=>M_ f2, /nM-+ \\ PFM.->(A)=°M+ \\\\“ ‘ i .a\‘\ \ SM+~ pr (A)-> (M,) " Figure 5. Transition paths are shown in X-unfolding for A, < M ,. Solid lines indicate active transformation while dash or dot lines indicate inactive transformation. Consider the path p1 going from SM_ to 8,, corresponding to transition direction M _ -9 A. Note that the portion between nuetrality curves Mf, and A,, (dash line) is inac- tive while the portion between A,, and Af, is active (solid line). The opposite path p2 run- ning from S A to SM, indicates transition direction A —) M - in which the portion between Af, and M ,, (dash line) is inactive while between nuetrality curves M ,_ and M,; (solid line) is active. Both of the two paths could be active in the area between nuetrality curves A,_ and M,_. Flux paths p3 and p, in Q; are similar to that of paths p1 and p2, where now M, replaces M,. It should be noted that processes A H M, and A H M _ may operate simul- taneously in OT , which implies the possibility of phase fraction triples {§_, 9, 5,,} in (If with0<§,§A,§,<1.Incontrast§,=0in (2'1 andE,=0in 0:. 25 Paths: p1, p2, p3, p4, operate above both Mf, and Mf,. For states (I, T) underneath ' either M,, or M,,, the associated transition of the form A -> M, or A -) M- has already gone to completion, so that there is no austenite in the corresponding area. Any austenite that would be predicted by a formal algorithm immediately transforms to M, if 1: > 0 (A = M,) and immediately transforms to M, ift < 0 (A = M,). This will be referred to as an “instability transformation”. It is reminiscent of Wasilewski, who introduced assumptions on the reorientation of martensite variants in non-austenite regions (Wasilewski, a, b, c, 1971), such that the austenite state serves as an instantaneously inter- mediary status to “switch” the reorientations of martensite variants. In the present study this suggests that, transition direction M _ -> A indicated by flux path p1- between A ,, and Af, (solid line in Figure 5) should be replaced by M_ —) (A) =9 M, , which is M_ —) M, . We will refer here to direct martensite/martensite reorientation transformations from mixed martensite variant fraction {§,, 0, §,} to one that involves only one variant as a “detwinning process”. Since path p1- (dot line) was originally inactive between M,; and A ,, , no transition occurs on p1 in Sm.-. On the other hand, path p: for transition direction A —-) M . (dot line) is completely inactive because of the absence of austenite below Mf,. Similarly, the active part of path p3 (solid line), indicating transition direction M, -) A between A,, and A,,, should be replaced by M, —> (A) :9 M- or detwinning process M, -> M _, while path p5 (dot line) is completely inactive between Mf, and A,,. Path p3, like path p5, is inactive, because of the absence of austenite. Consequently, the terminal nuetrality curves A ,_ and A , , become detwinning flow curves at sufficiently low tempera- ture. Likewise, the terminal nuetrality curves A f, and A f , become detwinning finish curves at sufficiently low temperature. Thus far, two new transition directions have been set up in low temperature regions (2; and Q; , in which detwinning processes will be trig- 26 gered by increasing or decreasing the stress and temperature. Further, since paths p-f is inactive between M,; and A ,, and path p; is inactive between Mf, and A,,, we verify the earlier claim that the stable mixed martensite variant zone SM, extends to regions 51",, and SM, In conclusion, in (21 = Q] U at U Q: , the X—unfolding retains the original features of the previous unfolding by Pence et al. (1994). These include the coexistence of the phase triple {§_, fiA, §,}away from 1: = 0 in OT, and the triggering of the transition direction M . —> A , for example, by increasing the stress in t > 0 at a constant tempera- ture. In addition to that, in Q; and 9'2, detwinning process M, H M_ is modified by M, -> A and M -> A austenite/martensite phase transformations, and the two variant martensites would be coexistent in S M (= S M ,, U S M, U S M, ), Q; and Q; . In the latter two areas detwining processes are acknowledged to occur. It is obvious that the detwin- nin g process in this X-unfolding come from allowing M _ -) A and M, -) A to extend into the areas where there is no austenite, therefore, the corresponding detwinning flow and finish can be obtained directly from the phase transformation flow and finish in the non- austenite areas (I; and 0'2. However, in Q; and 0'2 , one should note that the X-unfold- ing has the property that increasing temperature at a constant stress can trigger the detwin- ning process. This is caused by the specified driving path crossing what are now detwinning nuetrality curves that continue to depend on both temperatures and stresses inherited from the original austenite/martensite transformation curves. This dubious phe- nomenon will be amended naturally in following further investigations on the unfoldings at the beginning in section 2.3. 27 2.2.4 Isothermal Behavior of X-unfolding At the present stage, strategically, one can contemplate the transformation behavior under the X-unfolding. We here consider only the isothermal behaviors in view of the detwining difficulty mentioned above. Variable temperature and stress paths will be addressed in chapter 6. To see the problem clearly we define the following temperature ranges shown in Figure 6, 1p, {T>Af}, 1,, = {TffsTsAf}, T srn {Tf,ST A, or M, < A, in TABLE 1. 28 TABLE 1. Transformation Possibilities for Isothermal Loading and Unloading TRANSFORMATION TRANSFORMATION TEMPERATURE POSSIBILITIES POSSIBILITIES DURING THERMOMECHANICAL RANGES DURING LOADING UNLOADING BEHAVIOR Te Tpd A->M,. M,—)A. Mdoelasticity " M_)Al, '4 M+—)A8, ifM,>A,,thenshapememory ' effect with residual strain associ- 2 9 ated with some more M,,and less T5 T3}; A_)M+ - A—’M- - M.lefi;ifM,A,,thenshapememory ' effect with residual strain associ- 4 ‘1 atedwith someM,andM, leftfor TE Tsrn A'9M+ . A—’M- - T>A,and100%M,leftforT< A,; if M, < A,, then with M, M _) M,_s, 100% left only. *3 A .9 M+6, shape memory effect with 100% T E T.“ M, left. M_ —) M ,7. TE Tu M_—)M+, samewiththeabove. ‘1. HM, >A,, then 1 occurs first when T> M,, and l, 2 occur simultaneously when T < M,. IfM, < A,, then 1 always occurs before 2. ’2. If M, > A,, then 3 and 4 occur simultaneously when T > A,, 4 occurs before 3 when T < A,, and 5 is the last one to occur. If M, < A,, then 4 always occurs before3,and5isthelast. NOTES ‘3. 6 occurs always before 7. '"4. HM, > A,, then 8 would not complete, and 9 is active only when T < M,. IfM, < A,, then 8 is active only when T > A, and would not complete. 9 is inactive. For both the two cases, 8 occurs always before 9 in the sense of8 supports 9 by creating A. ‘5. IfM,>A,, then 10and 11 are active when T>A, with 10 supporting 11 by cre- atingA. IfM, O > O \ , T )k \4, T (a) (b) Figure 6. Temperature segments for cases M, > A, (a) and A, > M, (b) in X-unfolding. 2.3 Modified Nuetrality Curves and the pY-unfolding 2.3.1 Thermodynamic Considerations for Detwinning and a Primitive Y- unfolding In the approach of the previous section, the X-unfolding naturally gives rise to detwin- ning process by first allowing martensite/austenite phase transformations to enter non-aus- tenite areas, and second transforming the obtained austenite into martensite variants immediately in terms of Wasilewski’s assumptions involving an instability transformation of austenite. An issue that must now be addressed is that the slope of these detwinning process nuetrality curves was determined on the basis of a Clausius-Clapeyron relation for austenite/martensite transformations. However, in Q; and (2'2 the processes are now asso- ciated with martensite detwinning. Thus, the Clausius-Clapeyron argument requires modi- ' fication. Based on (2.1.0.5), now the Clausius-Clapeyron equation for the detwinning 30 . + - process m (22 and (22 becomes 2L: “’7', (2.3.1.1) In particular, if entropies of martensite variants are assumed identical, so that n, - TL = 0 , then the slopes of the nuetrality curves pararnetrizing detwinning processes lie parallel to the T—axis. The condition 11, — 11- = 0 is met for the example material char- acterized by (2.2.2.3) and (2.2.2.4) with n: = nf’. Note that this condition need not be met for more generalized models, such as those with temperature dependent elastic mod- uli, since then the individual phase Maxwell relation (2.1.0.4) would in general give a stress dependence on the martensite variant entropy functions 11, and TL , which would, in turn, break the relation 11, = TL. For the realistic special case with 11 , = TL , the above analysis implies that the detwin- ning flow and finish lines are independent of the temperature, which is agreeable to the approach by Krishnan, et. al (1971) and Brinson (1993). Under this scheme, the detwin- ning flow stress, say 1,, could be naturally determined by identifying the stress of the inter- section point between nuetrality curves Mf, and A ,_. In the same manner the detwinning finish stress, named 1,; is naturally resolved by identifying the stress of the intersection . point of nuetrality curve Mf, and with nuetrality curve Af_. These two special stresses are illustrated in Figure 7. This gives a primitive Y-unfolding, referred to as the pY-unfolding; it is shown in Figure 7 and Figure 8 for the case of a material obeying M, > A,. In this sit- uation, zones SA and Q, are the same in the X- and pY-unfoldings, however, zones 0; , 0'2 , S M , and S M. in the X-unfolding change to zones OE, Q; , SM, and 3’”. in the pY- unfolding (Figure 8) respectively while S M turns to SM. We no longer display the trans- 31 formation nuetrality curves for A -) M, and A -) M - in O; and Q} in view of the previ- ous discussion where it was shown that these two processes are not active in the area below 91. '1 ‘ T J: by: 0'. bsx (a. \ / $9 + I Is Tr r Figure 7 pY-unfolding is obtained by modifying the X-unfolding with eight specified nuetrality curves: A,,, M,,, A,,, M,,, A,-_, M,_, A,_, Mf_. The former four are “bent” to vertical positions in 1 S 0 upon encountering M,; and the latter four are similarly “bent” in 1 2 0 upon encountering Mf,. Here A, < M,; materials with A, > M, are treated similarly. 32 Figure 8. Phase transformation active and inactive zones are shown in the pY-unfolding, which is obtained by modifying the X-unfolding below M,, and M,,. 2.3.2 Mathematical Description of pY-unfolding With the above discussion we now describe the modified nuetrality curves below Mf, and M,,. The primary nuetrality curves [33“, r) and B;(r,1), remain suitable in a, (above Mf, and M,,), Below Mf, and Mf, these curves are modified so as to be parallel to the T-axis under the thermodynamic assumption 1] , = 1']. taken in the last section. For example, for any temperature C’ obeying M f < C- < A f , one can always find the intersec- tion point (1, f (1)) of nuetrality curve B:(1, T) = C+ with 5;,(1, T) = M f. In particular, the intersection point value of 1, corresponding to C" = A,, is the detwinning flow -1, in this pY-unfolding. The detwinning finish stress -1, comes from this same procedure with C" = A1: The area between the nuetrality curves 1 = -1, and 1 = 4,: is renamed (2'2 . Since 33 the nuetrality curves are vertically described. Substituting 1 = f(t) into [3;(t, 1) gives the modified nuetrality curves {3;(1, 1) = [3:0, f(1)) below M,. with 1 < 0, which is active only in Q} . Thus, the nuetrality curves of process A H M, in Q, and M, —> M _ below Mf, with 1 < 0 can be written as 1320.1) for 13:11.1) 2M, BT17) = + _ . B,(I.f(1:)) for Ban, 1) 0, the nuetrality curves are found as BAH) for 1331,1121", 1311.1) = - + . 841.3(1)) for Ban, 1) < M f (2.3.2.2) Here (1, g(1)) is the intersection point of nuetrality curves B;(1, T) = C- with B;(t, 1) = M I. Let (2.3.2.1) and (2.3.2.2) be equal to the four transition temperatures respectively, eight nuetrality curves are obtained. Four of these are associated with A, and A f in Q; and (2.2 , and will be naturally interpreted as detwinning flow and finish lines respectively. The other four associated with M, and M f below M,, and Mf, are not significant for any phase transformation process. The parts of the eight nuetrality curves in 91 remain the same functions with those in the X-unfolding. Example Under the material description assumptions given by (2.2.2.3) and (2.2.2.4), the modi- 34 fied forms of (2.2.2.5) and (2.2.2.6) for this pY-unfolding can be obtained by following the above procedures. This evaluates (2.3.2.1) and (2.3.2.2) into the particular forms r 2 1 (”M ' "A” t - T + - 1 ,(1, T) 2 M 5+“, 1) = t 1191- all 21‘4”!" Y ] for B f , (2.3.2.3) ,Mf-(zthni-nint for B; 0, we assume that the detwinning flow stress 1, and finish stress 1,. (both of them are positive) are known for M _ —> M ,. Consequently, in 1 < 0, these two are given by 1, = —1, and 1,. = —1f for M, ->M_ under the symmetry assumptions of the material, so as to determine the two corresponding nuetrality curves vertically (Figure 9). Both the active detwinning nuetrality curves and the inactive nuetrality curves will be vertically modified by the flow stress 1, and the finish stress 1, to satisfy the condition of the instability transformation of austenite and its associated Clausius-Clapeyron relation (2.3.1.1). Recall also that detwinning process M . —) M, is obtained by modifying the pro- cess M_ —-) A below Mf,, this naturally requires that\the two corresponding nuetrality curves connect to each other on Mf,. A similar requirement is made for processes 36 M, -9 M , ' and M, -—) A on Mf_. We call this kind of requirement a consistency condition, namely that: (1) the nuetrality curve Af, continues from 1 > 0 to the curve 1 = -1f below Mf_; (2) the nuetrality curve A ,, continues from 1 > 0 to the curve 1 = -1, below Mf,; (3) the nuetrality curve Af_ continues from 1 < 0 to the curve 1 = 1f below M,,; (4) the nuetrality curve A,, continues from 1 < 0 to the curve 1 = 1, below Mf,. There are many ways to accomplish these procedures, the method taken here is to “bend” each of the nuetrality curves Af,, A ,,, Af, and A,_ twice. As an example, Af, is bent once at point (0, Af) on the T- axis, and again at its intersection point with Mf_. This allows A,,: to remain as it was in the X-unfolding for 1 > 0, to be given by 1 = —1f below M,;, and to have a straight line connec- tion on the resulting boundary of QT . The remaining three nuetrality curves can be modi- fied similarly. Once the above procedures are completed, all the other nuetrality curves must then be modified accordingly. In this refined pY-unfolding, called simply the Y- unfolding in what follows, we still use the notations of the pY—unfolding for a definition of zones (Figure 10). 37 A T A,, ’ T" M, - 4s. a Mr. 9' . . ANG r > 1 F15 Ts "Tf > \ Tf Figure 9. Refined pY-unfolding, say, Y-unfolding, improves upon the pY-unfolding by allowing the detwinning flow stress 1, and finish stress 1f to be specified as additional material properties. This diagram is for the case of M, > A ,. In the present development, the terminal nuetrality curves undergo abrupt slope changes upon crossing 1 = 0 and upon meeting the nuetrality curves M,, and Mf,. 4f: 1004 T . ‘4' S if} \ 80‘?- A k 50» " A11: \ Q; , - Q: ./;~:’r 3,, +- ‘ - Q S 7 ~ 1 M4- 400 I’ll.- '7 0'2 M , ,,, Figure 10. Active and inactive zones in Y-unfolding corresponding to Figure 9. 38 2.4.2 Mathematical Formulation of the Y-unfolding In general, for constructing modified nuetrality curves of the Y-unfolding suitable to meet the temperature independent detwinning flow and finish stresses, one must match the two lines 1 = 1, and 1 = 1,- with the two stipulated nuetrality curves B'(1, T) = A, and B'(1, T) = A f on the terminal nuetrality curve Mf,. The nuetrality curves inside Q; could be simulated by linear interpolation between A f and A, associated with the detwin- ning flow and finish stresses. Nuetrality curves, trivial for M_ -> M, in SM, and SM (1>0), are all vertical lines corresponding to any temperatures in T S A f and M f < T < A, respectively. Based on the consistency condition introduced in section 2.4.1, each such nuetrality curves should connect with a nuetrality curve of M , H A at the point on M f , corresponding the same temperature parameter. This naturally continues B'(1, T) beneath the terminal nuetrality curve M f , in 1 > 0. As discussed above, since B'(1, T) remains unchanged in 1 S 0 , the consistency con- dition restricts the connection between the two nuetrality curve sets of M _ H A for 1 S 0 and M_ —) M, for 1> 0 to describe process M- H A for 1 > 0. At present stage, we ignore the thermodynamic consideration connected with this part, ie. process M , H A for 1 > 0. Appropriate thermodynamic refines to the problem are discussed in the Appendix. Geometrically, instead, there are many ways to link the two parts. The obvious way men- tioned in the previous section involves using straight lines to continue those nuetrality curves with C0 smooth assumptions (Figure 9, Figure 10), which gives an approximate description for process M , H A . Note that this modifies the slopes of the nuetrality curves in region (If for 1 > 0. Because of this, it is convenient to decompose the portion of Q: in 1 > 0 into subte- 39 gions R1 and R2. The region R2 is that portion of Q:— above A,, and the region R1 is that portion of QT below A,,. In addition, let R3 be the region above A,, and Mf, in 1 > 0. Symmetry determines the corresponding regions in 1 < 0. Mathematically, the three regions are expressed as the following shown in Figure 11, R1 = {(1, T)|1>0, B'(1, T) s A,, B+(1, T) ZMf} . R2 = {(1, T)|1>0, A,< B'(1, T) 0, B'(1, T)2Af,B+(1,T)2Mf}. We now discuss this particular construction for our standard model involving pure spe- cies strains as given by (2.2.2.3) and the pure species entropy as given by (2.2.2.4). The straight line slope of Af, in 1 > 0 above Mf,, which satisfies the consistency condition, is then found to be 2 , A —M -k 1 —k 1 kf= f f I" 21(<0). (2.4.2.1) I Here, It, and k2 are given by, e (11 -l'l‘ ) k, = 7 «113—113) (> 0), k2 = A M (>0). (2.4.2.2) 21511151013” 71:) The straight line slope of A ,_ in 1 > 0 above M,,, which satisfies the consistency condition, is then found to be i (< 0), (2.4.2.3) 40 for the same It] and k2. Thus, A,_ and A,, are given by T - k'ft = A f for 1 > 0, (2.4.2.4) T - k,1 = A, for 1 > 0, (2.4.2.5) respectively above Mf,. Figure 11. Subdomains R1, R2 and R3 are shown in 1 > 0 for materials obeying M, > A,. Since detwinning flow and finish stresses 1, and 1fare now regarded as determined by experimental measurements, It, and k} are fully settled for this standard model. Three possibilities might be encountered, ie., either k} > k, , It} = k, or k} < k,. In the first case, the straight line extensions to Af, and A ,, that bound R2 would intersect at a point 111 obeying 1 < 0. In the last case, this intersection would occur at a point 112 obeying 1 > 0 41 (Figure 12). The coordinates for this intersection point are found to be 1 _ E _, 1 =Atk,-A,k'[. , - _ _, , _ _ (2.4.2.6) k,-kf k,-kf Note that k} and k, are negative, so that condition It} > k, makes (2.4.2.6) correspond to a point 111 and condition k} < k, makes (2.4.2.6) correspond to a point 7% The intermedi- ate case k} = It, gives parallel lines and so corresponds to an intersection point at infinity. Figure 12. Intersecting points of nuetrality curves of detwinning flow and finish for k} > k, (a) and for k} < k, (b). The intersection point it , which is the intersection of the extension of nuetrality curves Af, and A,_, can now be used to organize all the remaining nuetrality curves 3' in R1, R2 and R3 corresponding to the process A H M ,. The nuetrality curves, trivial for A H M - in the region R3, are assumed to be straight with slope value It} . Similarly, the nuetrality curves for A —) M- in the region R] (A -) M_ may be active in a part ofR] ifA, > M,) are 42 assumed to be straight with slope value k, . In the intermediate region R2, it is natural to assume that the nuetrality curves are determined by linear interpolation between the M _ -) A nuetrality curves Af, and A,,. Therefore, taken together, the nuetrality curves can be expressed as WI. T) = T - 16,1 in R3. (2.4.2.7) 531.1) = T — k;1 in R1. (2.4.2.8) respectively. And, the nuetrality curves in R2 can be obtained as T(Af — A,) + 1(A,k'f - Afk,) in 13;,(1. 7) = _ _ (Af-A,)+r(k,—k,) R2 , (2.4.2.9) by being assumed to be straight lines with slopes determined by linear interpolation between E, and k,. This algebraic construction corresponds geometrically to one in which straight rays are drawn from 111 if k} > k, or it; if k} < k, which then fill in section R2. Finally, the nuetrality curves for detwinning in 1 > 0 below Mf, are given as 53,,(1. 1) = M , +(k1- k'f)t +k212 in 3”,, (2.4.2.10) 139,411) = Mf+(k1-k,)1 +k212 in 8M, (2.4.2.11) 2 - . A—A M+k1+k1 +Ak-Ak 1 .. (f ‘X f ‘ 2_) _( ‘f ") inn; (2.4.2.12) BUtsfct’ T) = 43 Thus, taking together all of the above results, the characteristics for phase transforma- tion M _ H A and detwinning process M _ —> M, are summarized as r 0;,(1, T), V 1 s O B}(1, T). V (1, T) e R1 B,f(1, T). V (1, T) e R2 Wt. 1) = t B;(t,1). , v (1, T) e R3 . (2.4.2.13) B;,f(1, T). V (1, T) 5 SM, Bane. 7). v (t. T) e 02' Pan“. T). V (1'. T) 6 Sun: > 0) If we use R1, R2 and R3 to indicate the reflection of the areas of R1, R2 and R3 with respect to the T-axis and follow the same analysis for formulating B'(1, T) , then the char- acteristics for phase transformation M, H A and detwinning process M, -> M, can be written as ’13:“. T). V 1 > O B)“, T). V (1, T) e R, BIAT. 7). v (1, T) 6 it, 3+“, 77 = * BI“, 73. V (1. De R3 . (2.4.2.14) BEAT. 7). v (1, T) 5 SM, 651.10.?) v (1, 1) 6 Q; the. 73. V (r, T) e 3,,(150). Here, all the components of fl+(1, T) are decided as the same fashion as those of B'(1, T) 44 but associated with equation (2.2.2.6) and the slopes k} = -k',, k: = -k,. (2.4.2.15) 2.4.3 Summary For sufficiently low temperatures, the pY—unfolding and Y-unfolding involve vertical (temperature independent) detwinning flow and detwinning finish terminal nuetrality curves. These are obtained by assuming 1| ,(1, T) = n,(1, T) for (1, T) values that trigger detwinning under all circumstances. From the above discussions, this pY—unfolding, being another extension of the triple point phase diagram, not only takes over all the features of X-unfolding in Q1 but also reflects the detwinning process between the two martensite variants. In the description of detwinning, since the detwinning nuetrality curves are now all vertical, temperature increase at a constant stress does not traverse nuetrality curves, so that temperature changes no longer trigger reorientation. However, the pY-unfolding com- pletely determines the detwinning flow and finish lines by naturally extending the high temperature phase transformation nuetrality curves. In general, the detwinning flow stress of the pY—unfolding is smaller than that of the X-unfolding, both of which are larger than those determined by experimental measurements (Hou and Grummon, 1995; Miyazaki et I al. 1991). This motivated the further modification of the pY-unfolding into the Y-unfold- ing where the detwinning flow stress and detwinning finish stress are taken as material properties. It is important to point out that, in general, the detwinning flow stress is found to obey a mild temperature dependence. This phenomenon was, for example, observed in a tension 45 experiment of a bulk (the size of the specimens were 1 mm x 1.5 mm x 15 mm at 50.0, 50.5 51.0, 51.5, and 52.0 at%Ni) by Miyazaki, et al. (1986, 1991), in which the slope of detwinning flow line is -0.385 °KIMPa. This kind of phenomenon was explained in terms of a thermally activated effect (Miyazaki and Otsuka, 1984). Also, in the tension test of NiTi thin film by Hou & Grummon (1995), the slope of detwinning flow line is measured -5.41 °K/MPa. Such a tiny temperature influence on the detwinning flow is also observed in the work by Dye (1990). Furthermore, in two other experimental studies (wire speci- mens with diameters 0.9 mm and 1.0 mm at T150Ni47Fe3 and Tr-49.8 at. pct Ni (or Ti-50.6 at pct Ni) respectively) by Miyazaki and Otsuka (1984, 1986), they also found that the slopes of detwinning flow lines changed slightly with temperatures. Actually, for most shape memory alloys, the detwinning flow stress may increase when temperature decreases (Miyazaki, Kohiyama and Otsuka, 1991). The explanation in terms of a thermal activation effect is that at lower temperatures the molecules of the material are less active than at higher temperatures, which makes the detwinning process more difficult at lower temperatures. 2.5 Analysis of Isothermal Loading/Unloading Processes With this Y-unfolding so far, we can illustrate various isothermal behaviors of shape memory alloys, such as pseudoelasticity at high temperatures, shape memory effect at low temperatures, or some properties between the two. Since the phase diagram depends upon parameters: phase transformation temperatures A}; A,, M, and Mr; detwinning flow stress 1, and finish stress 1f; shear moduli 11A and 11M; reference entropy difference between aus- tenite and martensite A110, phase transformation strain 7' and density p. Here, these are 46 determined with reference to work of Hou and Grummon (1995), which are given in the following table. TABLE 2 Simulation Parameters A] A: Mr M] "A “M A71 P I: 1f 7 308 295 263 235 3'10‘ 3*10‘ 414695 6.5‘103 150.0 200.0 0.06 °K°K°K.°KMPaMPaJ/m3°KKg/m3MPaMPa Here, A1] is determined by relation A11=AHITo where AH=17.8 Hg and To=279 °K. An Y- unfolding for 1 > 0 is obtained as shown in Figure 13. It has straight nuetrality curves for the parameters given above, since the condition 11A = 11M annihilates the 12 terms in the various B—functions (see (2.2.2.5), (2.2.2.6)). Recall that T fir and Tf, were previously defined for the X-unfolding. For convenience, we now define two more special temperatures T,, and T,,: This gives four intersection tem- peratures that distinguish transformation areas in 1 > 0, which are expressed in the follow- ing, - - 2 _k -k + k -k +4k A -—M Tss=As+k s 1 J( s 1) 2( s s), , 2,2 (2.5.0.1) . 2 2k2 ’ TSf = A] + k} (2.5.0.2) T f, = A,+k,1,, (2.5.0.3) 47 T f = A f + k}1f . (2.5.0.4) Here, the first subscripts are ad0pted from the first subscripts of nuetrality curves M ,, and M,, and the second are come from the first subscripts of nuetrality curves A ,, and Af_. The values It, and k2 were in general defined previously in (2.4.2.2), and, k} and k, are given by (2.4.2.1) and (2.4.2.3) respectively. These four temperatures are in general dependent on the shear moduli of austenite and martensite in virtue of k2. The corresponding coordi- nates on 1-axis are 1,,, 1,1, 1, and 1,: In the present situation, the four temperatures are found as in TABLE 3, TABLE 3. Temperatures Corresponding to Points a, b, c and d in Figure 13 T, , T,,. T,, T17- 274.6 °K 280.8 °K 256.7 °K 263.9 °K Comparing with the four transformation temperatures, shows that Af>A,>T,f>T,,>Tff>M,>Tf,>Mf, (2.5.0.5) for these particular values of the material parameters. 48 Figure 13. Y-unfolding for 1 > 0 with k; = 0. Four intersection points a, b, c and d are shown as: a (1,,, T,,), b (1,,, T,,), c (1,, Tf,), d (1f, TI?" The loading/unloading behaviors strongly depend upon the initial phase fraction. To briefly view the problem, we here consider two type initial conditions which were consid- ered by Wasilewski (1971 d). The first one is that the austenite phase fraction is maximum, which is obtained by cooling the material from a temperature that is above Af down to the test temperature (CFAF). The second one is that the martensite (with two symmetric vari- ants) phase fraction is maximum, which is obtained by heating the material from a temper- ature that is below Mf up to the test temperature (I-IFMF). There are four loading types p2 , p2, p3 and p4 (accounted from decreasing temperature direction) in the initial condition case of maximum austenite (CFAF), which are located by the temperature ranges: T > M,, M, > T > Tf,, Tf, > T> Mf, T T) T,,, T,,> T> Tff, A,> T) T,,, A f > T > A, , T > A f , shown in Figure 15. Conversely, there are three unloading types p1, p2 and p3 in both of the two initial condition types, which are located at temperature ranges: T>Af, Af> T>A,, T A when stress increases between temperature range Af and T,,. It is obvious that the temperature range to induce the (stable) stress-assisted austenite of the latter case is wider than that of the former case, ie., the martensite variant M, of HFMF can be initially present up to a temperature T < Af. This kind of transition mechanism, a stable stress-assisted aus- tenite from some special orientations of martensite (special variants), has been suggested by Wasilewski (1971 d). The present model can give detailed predictions about the tem- perature ranges of the specific transformation upon isothermal loading. More general loading behaviors occur for other initial conditions, but the limiting cases addressed here give the general flavor of the model. Finally, the example discussed here and shown in Figure 14 to Figure 16 involved a material with M, < A,. Similar analy- sis can be carried out for a material with A, < M,. 50 - - — - pureelastic single transformation . . . . . — double transformation initial condruons __ detwinning 1M" (0,1,0)- % ..... —>3 p1 d / l (Lillie. {0.0.1, {in §A9 go} ' {0.5, 0, 0.5} § 0 1, If 1' Figure 14. Loading behavior for initial conditions corresponding to initial conditions of maximum austenite, CFAF (equal amount of martensite two variants). For T) M, the initial condition is {§., §A, §,}={0, 1, 0} and'for T< Mfit is {§_, EA, §,}={0.5, 0, 0.5}. For Mf< T < M, the initial condition is a more general {§_, fiA, §,} with i. = §,. The four associated transition paths p1, p2, p3 and p4 go from left to right. On p], so that T> M,, segments 01, 12 and 23 indicate austenite elastic, single transformation A -> M, and pure elastic M, deformations respectively. On p2, so that T,, < T < M,, segment 01, 12, 23, 34 represent single transformation A -) M, , double transformation A —) M, & M _ -) A , detwinning M_ —) M, and elastic M, deformations. On {13, segments 01, 12, 23 and 34 indicate single transformation A -) M, , two variant martensite elastic, detwinning M _ —) M, and elastic M, deformations. On p4, segments 01, 12 and 23 represent two variant martensite elastic, detwinning M _ —) M, and elastic M, deformations. 51 - - - - pureelastic AT — single transformation initial conditions M" — double transformation ‘ 1 . . 2 1M" , {0,1,0} - /_ ..... 9 P6 15.. t... t.)- .......... .>‘ p, {0.5, 0, 0.5} - 2 {0.5, 0. 0.5} . {0.5.0.05} 4 {0 0 1} {0.5, 0, 0.5] o Figure 15. These six transition paths p1, p2, p3, p4, p5 and p5 associated with maximum martensite initial conditions, HFMF (equal amount of martensite two variants) go from left to right. On p1, segments 01, 12 and 23 indicate two variant martensite elastic, detwinning M - —) M, and right-shear martensite elastic deformations. On p2, segments 01, 12, 23 and 34 represent two variant martensite elastic, double transformation M - -) A & A —) M , , detwinning M _ —) M, and elastic M, deformations. On p3, the first two segments 01 and 12 are the same with those on p2, segment 23 and 34 indicate single transformation A -> M, and elastic M, deformations. On p4, segments 01, 12, 23, 34 and 45 represent two variant martensite elastic, single transformation M _ -) A , double transformation M _ —> A & A -> M, , single transformation A -) M, and elastic M, deformations. On p5, segments 01, 12, 23 and 34 indicate single transformation M- —)A, austenite and M, elastic, single transformation A —) M, and elastic M, deformations. On p6, segments 01, 12 and 23 represent austenite elastic, single transformation A -) M, and elastic M, deformations. 52 - - — - pure elastic single transformation initial conditions (0. 1, 0) . {0. :11. i.) {0.0. ll Figure 16. These three unloading paths go from the right to left with initial condition of 100% M,. All the dashed lines on p1, p2 and p3 indicate M, elastic deformations, except the portion to the left of A,, on p1 which represents elastic austenite. All the solid lines on p1 and p2 indicate single transformation M, —) A deformations. 2.6 Transformation of a Special Variant Other than the above isothermal features of the Y-unfolding, it is interesting to look at some different characters of this Y-unfolding. Since the observed detwinning flow stress 1, is, in general, smaller than the extrapolated value from the pY-unfolding (Hou and Grum- mon, 1995), it follows that the nuetrality curve for the reverse transformation M , -) A for t 1 > 0 is below that of the pY-unfolding. Regarding to the continuous assumption between nuetrality curves of M , -> A and M _ —-> M, on the terminal nuetrality curve Mf, (see sec- tion 2.4.2), this implies that the start temperature for M - -—> A is lowered. The above extrapolation reflects a transformation mechanism that the martensite variant M, becomes more unstable in the Y-unfolding for 1 > 0. In other words, the energy barrier for M _ —) A 53 is decreased in 1 > 0. Phenomenally, this decrease tendency of M _ —> A flow coincides with the postulation made by Wasilewski (1971, c, d). Based on an asymmetric isothermal stress-strain relation of a tension-compression experiment conducted below Mf (Wasilewski, 1971 c), Wasilewski concluded that the yield point for a special martensite variant transforming to a transient austenite phase that shifts to a martensite variant simul- taneously is lower than others’s yield point in a different stress circumstance. Protracting the above issue to a test temperature range between Af and M,, where the transformed aus- tenite is no longer a transient phase but stable, (Wasilewski, 1971 d) deduces a similar phenomenon with that from the present model mentioned above (Delaey, et al., 1974). Further, since the transformation M , -> A occurs only in the darker zone R2 of Figure 17 for 1 > 0, one concludes that the lowest temperature and the largest stress for conduct- ing process M _ -) A in 1 > 0 are T,, and 1,— respectively at points c and d. Regarding the detwinning process in a; , the highest temperature and lowest stress for conducting detwinning process M- -) M, in 1 > 0 are Ti and 1, respectively at points (1 and c. The two processes M _ —> A and A —) M, may occur simultaneously inside the zone abcd (which is part of R2 in Figure 17). Another interesting character is that three processes M_ —) A , A -) M, and M, -> M, can be triggered in a same temperature level between T17 and Tf, upon loading. First, process A —> M, proceeds in a small stress range. Second, processes M _ -> A and A —-) M, take place together once the stress increases past the ter- minal nuetrality curve A,_. The third, detwinning M - —) M, occurs once the stress further increases beyond the terminal nuetrality curve Mf,. Those are illustrated in Figure 17. Similarresultsareobtainedfor M,—>A and M,->M_ in1<0. 54 Figure 17. For 1 > 0, the lowest temperature and largest stress for conducting M- -) A are T}, and 1}, and the highest temperature for conducting M _ -) M, is T17" 3 TRANSITION TYPES FOR THE PHASE FRACTION EVOLUTION In chapter 2, phase diagrams were discussed based on certain thermodynamic consid- erations, which gave rise to different phase transition zones. As shown in figures Figure 14 and Figure 15 for isothermal loading, various double transformations can occur in the region QT corresponding to the unfolded triple point of Figure 1. In this section we are going to investigate different single and double transformation possibilities in the different transition zones, as well as the corresponding criteria for determining the associated trans- formation possibilities, which will be organized into transition types. in the following study. 3.1 Wfion Types In general, when temperature and stress trigger the phase transformations in the fine mixture of the three species, austenite and martensite two variants, both of the two trans- formation processes A H M, and A H M _ might occur simultaneously in Q, . Therefore, at each instant of time, it is assumed that there is either a net transformation from A -) M, or else a net transformation from M, -> A . Concurrently, it is assumed at each instant of time that there is either a net transformation from A -—) M , or else a net transformation from M , —-) A . Taken together, in Q1 , they give the follOwing four combination transition 55 56 types: (TTl): M,->A,M-—>A; (TT2): A —>M,,A—)M,; (TT3): M,—>A,A—)M_; (TT4): M-—)A,A—>M,. For relatively high temperatures, transition type ('I‘I‘l) is that which occurs under pure temperature increase; transition type (TT2) is that which occurs for pure temperature decrease. Transition types (TT3) and (TM) are those which occur for pure stress decrease and increase respectively. With these four transition types, pseudoelastic behavior can be simulated when the temperature is relatively high (in Q1) (see Pence et al., 1995). For combined changes in temperature and stress (processes that are not pure), the particular transition type will depend on the local orientation of the state path in the stress-tempera- ture plane. On the other hand, to complete the model when the temperature is relatively low (below Mf, and M14), the detwinning process M, H M _ must also be accounted for. Recall the discussion in section 2.2 to 2.4, this detwinning process is modified by M _ —-) A =9 M, and M, —> A :9 M_ in terms of Wasilewski’s instability transformations when one formally continues A H M - and A H M, transformations in the zones where austenite is unstable. Therefore, for X—unfolding in Q; , we have transition type (1T5): M _ -) M, . and in Q; we have transition type (IT 6): M, —-) M -. For the Y-unfolding and the pY—unfolding, the only modification to this X-unfolding description is that transition types (115) and (U6) operate in (23’ and 1‘); respectively. 57 3.2 Criteria for Determining Transition Type As mentioned above, phase transformation and detwinning are triggered by changes in temperature T and stress 1 as the state path is executed in the (1, T )-plane. The particular transition type that is operating may change at distinct points (1,1) on the state path. These points occur when the state path is instantaneously aligned with one of the two nuetrality curves that pass though the point. In other words, several transition types may occur suc- cessively along a given state path. In Q1, if the local orientation of the state path crosses the AIM, nuetrality curves so as to make dB+ > 0 , then transition M, -) A occurs. Con- versely if path orientation with respect to the Al M , nuetrality curves give (13’ 2 0 , then transition M_ -—) A takes place. Thus, the signs of dB+ and (13' can be used to determine the operative transition type. Recall the discussions in the above, there are totally six transition types introduced in the two variant problems. These transition types hold regionally, i.e., transition types (TT 1) to ('I'T4) apply in Q1, and transition types (TTS) and (1T6) hold away from Q,. Thus, the criteria for the different transition types are going to be treated separately in the two regions. In these treatments, it is convenient to introduce the following notations: Bi=%BT-. B'r=-g%. BI=%%. 51:32: (3.2.0.1) 3.2.1 Algebraic Description The criteria for distinguishing between the four transition types are shown as the fol- lowing based on the nuetrality curves, 58 (TTl) if dB+>0 and dB'>O, (3.2.1.1) (112) if dB+0 and dB’0. (3.2.1.4) To see more clearly the above conditions, they can be expressed by means of the tan- gent of the given state path curve. From simple derivations on (3.2.1.1) and (3.2.1.2), we have that the following transition types occur: £>max -E,-E V d1>0 6"” 6; Br (T'Tl) if 4 + - ; (3.2.1.5) £2< min{-Et-, --B-E} V d1 < 0 dt + - _ is. is. d—T < rnin{-E, -E} V d1 > O ‘1‘ ii; B'r (T12) if < + - ; (3.2.1.6) , g>max{-E§,-E} V dT<0 . Br 37 —B—: < 31 < ..2} v (It > 0_ . Br 1 Br . (TT3) if 4 - + ; (3.2.1.7) _E, < g; < ”a, v d1 < 0 L B [3 - + -E<-d—T<-E Vd1>0 is} d‘ a; (1T4) if 4 B+ B- . (3.2.1.8) 1 (IT 1 --B—+-0andforAHM, in1<0aremodifiedbythe detwinning flow and finish stresses. T 4‘ A k = yt/An A (N) ,- \\\\§ - (E) \\\\\.-, . \ / o 7 (S) \ b 1 Figure 18. Four open cone areas at a point p in Q, for X— and Y-unfolding show the transition possibilities when a path passes through this point. If the path passing through p proceeds into N (S, W and B), then transition type (IT 1) ((112), (TT3) and GT4» is in progress. 61 3.3 Criteria for Detwinning Transition Types Recall the considerations in section 3.1, ('I'T5) comes from M _ -> A and is active in Q; while (’I'T6) comes from M, -—> A and is active in Q; for X-unfolding. Thus, criteria for ('TTS) and (IT 6) could be expressed as (115) if dB'>0in 12;, (3.3.0.1) (TT6) if da” > 0 in (2'2, (3.3.0.2) respectively. Furthermore (3.3.0.1) and (3.3.0.2) give (115) if 2L: >1} v d1>0 in (1;, (3.3.0.3) 131 dT 13“ (1'16) if E < —B—: )1 d1 < 0 in 9'2. (3.3.0.4) 1' One should note in condition (3.3.0.3) that the case d1 < 0 is ignored. Similarly, in (3.3.0.4) the case for d1 > 0 is ignored. The reason is that both (IT 5) and (1T6), associ- ated with detwinning, are modified by transition directions M . —) A amd M, -9 A , which correspond to increasing stress and decreasing stress processes respectively. Therefore, the direction of stress increasing/decreasing should coincide with that of the detwining direc- tion. The other important point that one should note is that criteria (3.3.0.3) and (3.3.0.4) apply only to the X-unfolding. For the pY— and Y-unfoldings, because B; and B} are equal to zero, one can directly use B;-d120 to check (TTS) in Q; and B:-d1201'o check (IT 6) in Q; . This gives 62 (1'15) [3; > 0 V dc > 0 in (2;, (3.3.0.5) (1T6) a; < 0 V (II < 0 in 1°25. (3.3.0.6) In the Y-unfolding, all of the nuetrality curves in Q; and Q} become perpendicular to the 1 -axis. Hence temperature changes at constant stress does not cause detwinning. This contrasts with the X-unfolding, where temperature increase at constant stress causes detwining processes to occur in Q; and Q; . 3.4 Example Assume that B+(‘t, 1) and int, 1) are given as (2.3.2.3) and (2.3.2.4) with equality of moduli of austenite and martensite for pY-unfolding. Here It, (>0) is given (2.4.2.2). If the state-path is an ellipse in the (1, T )plane in Figure 19, we can then determine the transi- tion types on the different parts of the ellipse state-path as follows. With (3.2.0.1), we find 43:43; = 1:1, —B;/B', = -k, in a, , and a; = 0, B“, = 0 a: = —2rtI and B; = 2kl in ft; and 1°13. t1, t2, t3 and t4 indicate the tangent points where the path is tangent to the nuetrality lines respectively. Based on (3.2.1.5) to (3.2.1.8) and the discussions on (TT5) and (1T6) for pY—unfolding, it follows that: from point t2 to t3 the transition type (TT3) occurs in Q1 ; from t3 to e the transition type (1T2) occurs in QI ; in Q} , 8M , SM- and SM, there is no phase transformation; from point b to c the tran- . sition type (TT5) occurs in Q; 63 \\ I \ / \ /\ / A\ / \ ’93» \ /\\ / \\ // \ S C,/ \\ I 3M 14 M / t1 \\ 1’ 3M4» 1 \O - > O; + 3,, 02 Figure 19. This graph shows that how the transition types occur when one follows a counter- clock wise ellipse in the (1, T )-plane. From point t2 to t3: (TT3) occurs; from t3 to e: (112) occurs; from point b to c: ('I'I‘S) occurs; in all other parts there is no transformation. 4 THE HYSTERESIS ALGORITHMS This chapter will extend the hysteresis algorithm of the one-variant martensite case given in the work by Ivshin and Pence (1994 b) to a two-variant martensite case on the basis of the two-variant analysis (Pence et al., 1994). For this reason it is useful to briefly summarize the one-variant model (Ivshin and Pence, 1994 b). The extending work begins from the analysis of the envelope functions and the algorithms presented in (Pence et al., 1994). If only one transition is active then the extension of the algorithm can be expressed by different equivalent forms. However if two transitions are active, then these different forms are no longer equivalent. Thus an examination on combinations of various transition directions from different transformation processes becomes necessary for detemrining the proper extension of the one-variant model into the two-variant regime. 4.1 Brief Review of the Previous One-variant study 4.1.1 One Variant Algorithm The one-variant model studied in the work (Ivshin and Pence 1994 b) involves transi- tion between a high-temperature/low-stress austenite phase A-and a low temperature/high- stress martensite phase M. In the present three species model, this is as if two martensite variants are treated together as one martensite species (Q, = 5,, + §_) and only A H M 64 65 takes place on each state path. At any instant, either there is a net transformation tendency for A —) M (so that fig 5 0) or else a net transformation tendency for M —) A (so that g, 2 0). Note that the nuetral tendency fig = 0 can be regarded as either a trivial A —> M tendency or else a trivial M -) A tendency. Thus the state path (1(t), T(t)) is partitioned into segments on which either the A -) M or the M -—> A transformation tendency occurs. If this partitiOning is known, the transformation is governed by the following equations (see equations (30) and (32) of the work by Ivshin and Pence (1994 b): g?“ = “A 3“ D-%GA_,M(1.T) rorA—aM, (4.1.1.1) _’ r d§ é 27M = I.“ ”(1 no%aM_,A(1,T) forM—rA. (41.1.2) M—rA ’ Here fiA = l-fiM is phase fractions of austenite, orA _,M(1, T) and 01,, _,A(1, T) are constitutive functions; they were given the respective symbols (1,,,,(1, T) and (1,,,-"(1, T) in (Ivshin and Pence, 1994 b), but the symbols used in (4.1.1.1) and (4.1.1.2) will be more convenient for the present discussion. The equations (4.1.1.1) and (4.1.1.2) ensure that the transformation proceeds at a pace that is proportional to the phase that is transforming (A in (4.1.1.1) and M in (4.1.1.2)) and is independent of the phase that is experiencing a net ' increase. In the event that the A -) M transformation begins from pure austenite (fi, = 1), then the transformation governed by (4.1.1.1) gives an fiA(t) that coincides with (I, _, ”(1(t), T(t)). Similarly fiA(t) = (1M _, A(1(t), T(t)) for M —> A transformations that start with pure martensite fi, = 0. 66 4.1.2 Envelope Function As discussed in the approach by Ivshin and Pence, (1994 b), equations (4.1.1.1) and (4.1.1.2) give immediate integrals. There are however practical advantages to operate with the differential equations (4.1.1.1) and (4.1.1.2). The constitutive functions are required to obey (1A _, ”(1, T) 2 01M _, A(1, T) . Both A —) M transformations that start at some initial time to with fiA S a, _, ”(1(t), T(t,)) S 1, and M -9 A transformations that start with fiA 2 (1M _, A(1(t,), T(t,)) 20, are required to generate phase fraction trajectories fiA(t) obeying (1M _, A(1(t), T(t)) S fiA(t) S at, _, ”(1(t), T(t)) . In fact, as shown in (Ivshin and Pence, 1994 b), the above containment requirement are not ensured by (4.1.1.1) and (4.1.1.2) alone, but are guaranteed if the constitutive functions or, _, M and (1M _, A obey certain additional containment restrictions. This justifies the terminology envelope flinc- tions‘ for a, _, M and at” _, A. The special situation involving equality of the envelope functions ensures that fiA(t) coincides with the new single “envelope function” so that fiA(t) is a state function of (1, T); hysteresis is not present in this special situation (A, = M], M, = A,). Envelope functions that are step functions between 0 and 1 give abrupt trans- formations at the locus (1, T) where the step takes place (A, = Af, M, = Mf). Thus, in com- bination, identical envelope functions that involve only a step between 0 and 1 on some (1, T ) curve gives the standard notion of abrupt nonhysteretic phase transitions (Mf= M, = A, = A!" However, unequal envelope functions that smoothly vary between 0 and l gener- ate the hysteretic mixtures which were the focus of the work by Ivshin and Pence (1994 b). Finally it is to be pointed out that the issue of determining whether an A -> M or an M -) A transformation tendency operates is resolved by requiring algorithmic consistency ' in (4.1.1.1) and (4.1.1.2). Narnely at each instant of time (4.1.1.1) and (4.1.1.2) should 67 both give that either fig < 0 or fig > 0 . In the former event then fiA (or fiM) is to be deter- mined from (4.1.1.1) and in the latter event then fi, (or fiM) is to be determined from (4.1.1.2). As shown in (Ivshin and Pence, 1994 b), this requires that at, __,M(1, T) and or” _, A(1, T) display dependence on (1, T ) by means of a characteristic function B(1, T) in the following fashion: 01,... ”(1. 7) = 61......(136. 1)) and «M44171 = (immune. 7)). ' (4.1.2.1) Therefore characteristic functions that describe nuetrality arise naturally in models of the type presently under study. If the state path (1(t), T(t)) ever coincides with the curves B(1, T) = C (constant), then fiA(t) is constant as long as the coincidence is maintained. Here it is to be emphasized that firm, and (1,,,, are functions of a single argument which can be determined by experimental measurements at free-stress circumstance. Thus they obey O S 6‘min(l3) S 61mm) 5 1 flair-"(13) = {(1) ii: :2: 61mm) = {I if B 2 Ms ' . (4.1.2.2) 0 if B s M , 60mm) 2 0 d'mw) 2 0 Additional conditions on rim-,(B) and am,(B) which ensure trajectory containment 68 and trajectory orientation requirements can be. found in the work by Ivshin and Pence (1994 a). Normalization B(0, T) = T gives u,_,,,(0, 1) = own) and aM,,(0, 1) = (1,,,-,(1). (4.1.2.3) 4.2 Two Variant Constitutive Functions To develop the two variant model on the basis of one variant model by Ivshin and Pence (1994 b), there are two things important to be considered, one of them is the exten- sion of the constitutive functions while the other is the extension of the algorithms for phase fraction evolution during transformation. 4.2.1 Constitutive Function Extension In the high temperature two variant study (Pence, et. al., 1994), OtA,(1, T) and at,A(1, T) are used as constitutive functions to describe A -) M, and M, -) A phase transformations, respectively; while aA_(1, T) and a-A(1, T) are used to as constitutive functions to describe A -) M _ and M _ -> A phase transformations respectively. When the AIM, neutrality curves are parametrized by the nuetrality function B+(1, T) introduced in (2.4.2.14) for the Y-unfolding, then the A H M, constitutive functions are of the form: t + aA+(T’ T) 5 (1A _) M,(Ts T) = amax(B (t, ”)9 (4'2'1 '1) a.,(t.1)sa..,_,,(t.1) = 61,...(B+(t.7)). (4.2.1.2) for constitutive functions (im,(x) , armor) of a single variable. More than this, since 69 detwinning process M, —9 M , is considered, 0t,A(1, T) is also the M, —-) M . constitutive function below M,;. Based on the symmetry of the two martensite variants, when the AIM, nuetrality curves are parametrized by the nuetrality function B'(1, T) introduced in (2.4.2.13) for the Y-unfolding. The constitutive functions for A H M _ are defined as 01,111. 7) a a, _, Mir. 7) = 61......(1311. T». (4.2.1.3) 01-),(1. 7) a 01”,, g(T. 7) = 61......(311. 7)). (4.2.1.4) Similarly, a_A(1, T) is also the M _ -> M, constitutive function below Mf,. Under normal- ization B+(0, T) = B'(0, T) = T, we have the following relations, aA,(0, T) :—: aA_(0, T) = dm,,(T) and (1,,,(0, T) a a_A(0, T) = emu). (4.2.1.5) The relations corresponding to (4.1.2.2) in the two variant case are 0 s 01,,(1, T) 501A_(1, T) 31, O s a,A(1, T) S 01A,(1, T) S] 1 if B'ZAf 1 if B'ZM, a_A(‘t, n = 5 “A11: T) = 0 if B' S A. 0 if 13' SM, . (4.2.1.6) 1 if BIZAf 1 if B+2M, a+A(Tr T) = _ 9 “Anita D = 0 if W54, 0 if [3“st 70 4.2.2 Possible Extension of the Algorithm to the Two Variant Problem After the analysis about the characteristics and constitutive functions, it is necessary to introduce a calculation system into the two variant model, which is a group of algorithms to determine the phase fractions associated with different transition directions in two vari- ant problems. ' For purposes of developing the appropriate extensions of the algorithms, the discus- sion is initially confined to phase transformations A H M, and A H M , without the detwinning. Since am“) can be interpreted as the maximum value of austenite phase fraction during phase transformations from 100% austenite to overall martensite, and, B+(1, T) and B'(1, T) parametrize the phase transformation families of A H M, and A H M _ respectively, constitutive functions aA,(1, T) and ctA_(1, T) would be deci- phered as state functions of austenite narrating phase transformations from 100% austenite to individual martensite variants respectively. In a similar fashion, since armor) is the minimum value of austenite phase fraction during phase transformations from 100% over- all martensite ~to austenite, constitutive functions or,A(1, T) and (1,,,(1, T) indicate state functions of austenite phase fraction describing the transformation from 100% individual martensite variant to austenite. Before extending the one variant algorithms (4.1.1.1) and (4.1.1.2), we consider in what follows the observation that each of (4.1.1.1) and (4.1.1.2) can be rewritten in three extra forms with fiA and fiM . These four equations are equiva- lent for either A —-> M or M —) A in the one variant case, but extensions to the two-variant case are sensitive to the form of the one-variant equations that are used before the exten- sion. To get insight into this problem, note since 5,, + fiM = 1, that all of the following 71 forms are equivalent to (4.1.1.1) for governing an A -> M transition direction: 3.?" = “Hinn-%a,_m(t,1), (4.1.1.1)a 4% l-§ d EM: O‘HMf’n-Eothmu), (4.1.1.1)b 1.. dg‘ — g” d (4.1.1.1)c H7 - aA_,M(1,T)'Ea“’M(T’n° A similar four way equivalence holds for (4.1.1.2) with respect to M —> A transition direction. The extra three are expressed as dgA §M -d—t- = 1_aM_,A(1:, T) 'EanACt, 1.), (4.1.1.2)3 dgM I'SA d . _d_t = l—aM Act 1') .EaMfiACt’ T), (4.1.1.2)!) _, 1 (18.4 IT§A d -d-t- = 1_aM_,A(T’n.-J;aM-’A(T’no (4.1.1.2)0 When multiple martensite species (only two in this study) are present, this kind of equivalence no longer holds. For example, suppose that an A —> M, transformation occurs in concert with either an A —-) M_ transformation (TT2) or an M- -) A transformation (1T4). Then fi_ + fi, + fi, = 1 and the A -—> M, transition might arguably be governed by, any one of the four possibilities (see (4.1.1.1)): 72 3% = «___..i: 1) j: 4M“ 73 21-? = 51%5iadr—W“ T) 3% = arr-iii. :i. 2». dgA - l ___—E— d 4A1“: T) 37 - aA,(1, 1') d1 (4.2.2.1) (4.2.2.1)a (4.2.2.1)b (4.2.2.1)c Since in the two variant approach, coefficient 1 - fi, no longer indicates fi,l in process A —> M ,, the last two equations (4.2.2.1)b and (4.2.2.1)c in the set of (4.2.2.1) are not consistent with the viewpoint that this coefficient should give the phase fraction of the pre- cursor phase and so will not be considered in the further study. In a similar fashion, an A —> M_ transition in ('I'T2) and (TT3) might similarly be gov- erned by any one of four equations, only two of which are reasonable, d§- _ _§___g E - 01,,(1 T) dt MA T). dig _ __§_A 27 01,,(1,T) dt _a'wAfiT) (4.2.2.2) (4.2.2.2)a The martensite to austenite transformation admit to similar interpretation. For transi- tion direction M, -) A the possible algorithms for use in (U1) and (TT3) are (see (4.1.1.2)): 73 3‘? = 1 05;“ 1) 54,: _afl,“ 1') (4.2.2.3) %A_ 1 (1:61: T) :11: 4m“ T) , (4.2.2.3)a g. = 1 :31) 1) 57“,,“ 1'), (4.2.2.3)b ‘19-] g, ‘1 (4.2.2.3)c 2? 1- -a,A(1, 1) d: “I’M“ 7) Coefficient 1 — 1;, in equations (4.2.2.3))b and (4.2.2.3)c no longer indicates fi, in pro- cess M, —) A , so that these two expressions will be eliminated since it is again not consis- tent with a coefficient factor that is proportional to the precursor phase. . Following the same manner as the above gives two remaining candidate algorithms for transition direction M _ —) A : 3%-]: 43(1d1).dt—0A(T’ T), . (4.2..24) $.41: ‘41:.“ T) j! _aA“ T) (4.2.2.4)3 The elimination of the “b” and “c” governing equation possibilities for each single process indicates for each two process transition type: (TN) to (TT4) in Q1, there now remain four combination possibilities. For example, the combined A —> M ,, A —) M - pro- cess of transition type (TT2), might possibly be governed by either (4.2.2.1) or (4.2.2.1)a in conjunction with either (4.2.2.2) or (4.2.2.2)a. In addition there are still two possibilities - for each detwinning transition in Q; and Q2 . Here it is interesting to note that the less complete model employed in the work by Pence et al. (1994) only allowed for transition 74 type (‘I'I' l) and (T12) in the notation employed here. There, of the four combination possi- bilities for (TT2), the particular combination ((4.2.2.1), (4.2.2.2)} was employed for A —) M +, A —9 M_. Similarly, the particular combination ((4.2.2.3), (4.2.2.4)} was employed for M + -9 A , M , —) A (transition type (IT 1)). We now inquire into the proper combination possibilities, and in this process arrive at those used in (Pence et al., 1994) for ('I'I‘l) and ('I'I‘2). 4.3 Analysis on the Algorithm Associated with Transition Types 4.3.1 Unique Algorithm Discussions in section 4.2.2 lead us to the observation that there remain four possible algorithms for each two process transition type in Q, ('ITl to TM) and two for each detwinning transition in Q; and 9.2 . From many reasonable solicitudes, such as symme- try, basic assumptions on phase transformation, etc., most of them can be excluded as we are going to see next. Here we give two separate lines of argument. The first is based on symmetry and mathematical well-posedness. The second is based on an assumption regarding the transformation process. The first argument begins with a symmetry consideration between M+ and M, in the pair combination governing (1T1) and (112). This immediately reduces the pair combina- tion in the following way: ('I'I‘l): ((4.2.2.3), (4.2.2.4)} or {(4.2.2.3)a, (4.2.2.4)a}. (TT2): ((4.2.2.1), (4.2.2.2)} or ((4.2.2.1)a, (4.2.2.2)a}. The other two transition types (TT3) and (TT4) lose MJM, symmetry and so do not neces- 75 sarily allow for such reductions. Returning now to transition types (TN) and (1T2), we note for (1T2) that the pair combination {(4.2.2.1)a, (4.2.2.2)a} overdetermines Q, and does not determine EA, and 5, individually. Hence one may conclude that (112) is governed by the pair ((4.2.2.1), (4.2.2.2)}. For transition type (IT 1), it can be shown that the pair {(4.2.2.3)a, (4.2.2.4)a} admits an explicit solution for 5,0), §A(t), 8,,(t). This solution is found to have the property that é, depends on the constitutive function (1+A(T, T) and §+ depends on the constitutive function a, A(1, T) . This statement implies that the determina- tion of §, and g, depends on processes M + -9 A and M - -> A respectively, which does not seem reasonable, so that it is preferable to use { (42.2.3), (42.2.4)}. Recall that the basic one variant phase transformation rule cited in the approach by Ivshin and Pence (1994 b) is that phase transformation proceeds at a pace that is propor- . tional to the phase that is transforming and is independent of the phase that is experiencing a net increase. For ('I'I‘l) the A—phase is experiencing a net increase and so this gives an independent and purely physical argument against equations involving S—EA in (TN). However this same assumption for (1T2) would argue against equations involving 3+ 45 and E- in (TT2), which is in contradiction to the pair combination that we have obtained here. This provides a hint that the pair equation governing (TF2) may require some further modification, as we shall show in the next section. For (IT 3), the combined M + —-> A and A —> M_ process, it can be argued that the pos- sible algorithms associated with (4.2.2.2)a should be excluded since the phase fraction of austenite §A should involve dependence on both of the phase fractions 5,, and §,. On the other hand, the algorithm for process M + —> A has ken chosen as (4.2.2.3) in (1T1). Hence for the purpose of consistently using the same algorithm in all transition types, one must select ((4.2.2.3), (4.2.2.2)}. Transition type (TT4), the combined M,—9A and 76 A -)M+ process is similar to (TT3), The selected algorithm for (TT4) is ((42.2.1), (42.2.4)}. Because there is no austenite in the detwinning processes (ITS) and (1T6) in Q; and 9'2 respectively, we do not need to contemplate those algorithms associated with the phase fraction of austenite. Thus, for the coherence considerations between the algorithms of M_ —-)A (M+ ->A) and M_ —)M+ (M+ —> M_), in the whole (I, T)-plane, the algo- rithm selected for ('I'I’S) is (4.2.2.4); the algorithm selected for ('I'l‘6) is (4.2.2.3). 4.3.2 Algorithm Consistency between One and Two Variant Problems So far, we have selected equations for governing the transition types in the two-variant model. However, the extension of the algorithms from one-variant to two-variant models needs further investigations to ensure that the extended algorithms can describe both one- variant and two-variant situations for a self-accommondated process. To examine the con- sistency between one-variant and two-variant models, let us consider a loading path of increasing/decreasing temperature at zero-load in the two variant model, for conditions involving symmetric martensite variants. The corresponding transition types from section 3.1 are (TH) and (TT2) for temperature increase and temperature decrease respectively. We begin with temperature decrease starting at T> Afso that the MM, symmetry in ini- tial conditions keeps M,./M, symmetry for all time. In this case one should note that Q, = §++ §_ = 2E+ = 2§,. For this zero-load case of temperature decrease, algorithm ((4.2.2.1), (4.2.2.2)} for transition type (TT2) can be rewritten in terms of §M as d§M 2§A d E — m'aa DfOTAéM, (4.3.2.1) MA 77 if equation (4.2.1.5) for ('I'I‘2) is used. Similarly, for temperature increase, algorithm {(4223), (42.2.4)} for transition type (“IT 1) can be rewritten as d‘éM _ gu d a _ l-flmin(T(t)) . admina') for M -) A , (4.3.2.2) if equation (4.2.1.5) for ('I'I‘2) is used again. Therefore, comparisons for M -> A between the governing equation (4.1.1.2) of the one-variant and the governing equation (4.3.2.2) of the two-variant models show agree- ment. However comparisons for A -—> M between governing equation (4.1.1.1) of the one- variant and the governing equation (4.3.2.1) of the two-variant models show disagreement by a factor 2. This disagreement induces us to review the explanation of those constitutive functions at the beginning of section 4.2.2. It is noted that constitutive functions aA+(1:, T) and orA-(‘t, T) indicate state functions of austenite phase fractions narrating phase trans- formations from 100% austenite to each individual martensite variant respectively. Fur- thermore, when temperature decreases from a reference either above or below Af at stress- free condition, the austenite phase is always evenly transformed into equal amounts of M. and M+ martensite variants. However in section 4.2.2, this effect was not reflected in the set of (4.2.2.1) and (4.2.2.2). Therefore, to build this reasonable extrapolation into the extended governing algorithms of the two-variant model during the extension, the selected governing equations (4.2.2.1) and (4.2.2.2) for A -) M + and A —) M _ should be modified by d§+ 5.4 d 2; - m ram n M -»M+. (43-2-3) 4%-- E respectively. The equations governing each transition type are summarized in TABLE 4. This algo- rithm, for the six transition types in either X-unfolding, pY-unfolding or Y-unfolding, ensures consistency within the two-variant model, and properly reduces to the one-variant 78 Q model in the zero-load situation. d — —m "d-t'aA_(T, D fOI'A ARI-9 TABLE 4. Transition Types with Their Algorithm AREAS TRANSITION TYPE ALGORITHM (’I'I‘l) M+ -) A, M, -9 A ((4.2.2.3), (4.2.2.4)) (21 (112) A -) M,,, A -+ M_ ((43.2.3), (43.2.4)) ('I'I3) M+ —> A, A —) M_ ((4.2.2.3), (4.3.2.4)) (Tr4) M_ —> A, A -) M+ ((4.2.2.4), (43.2.3)) 9; or (”2; I (’I'I‘S) M_ -+ M+ (4.2.2.4) 9', or (‘2'; I (rrs) M+ -) M_ (4.2.2.3) BALANCE I 5110) +§+(:) +§,(:) = 1 (1.2.0.1) 5 INTEGRABILITY AND PATH DEPENDENCE The phase fraction evolution of the three species, which describes transformations between austenite and martensite or detwinning processes between the two martensite variants, is determined on the basis of the ordinary differential equations listed in the TABLE 4. The overall balance requires that the phase fractions satisfy relation (1.2.0.1). For this process, we assume that T(t) and T(t) are given functions, ie., (1(t), T(t)) forms a state-path in which time t varies from a beginning time t,- to an arbitrary future time I so long as the transition type remains the same. General analytical solutions for the phase fractions can be given in terms of path integrals. These integrals along the state path may, or may not, be evaluatable in terms of the path endpoints. Equivalently, the algorithm within a transition type may, or may not, be path-independent. In this section we investi- gate the solution for each transition type as well as conditions which indicate path-inde- pendence within the transition type. Some of the precursor studies, which guide some aspects of this section’s development, were suggested by Ivshin in unpublished work. 5.1 Integration of The Hysteresis Equations for The Phase Fractions There are totally six transition types in the Y-unfolding: (I'Tl) to (TT 4) which hold in Q, , ('I'I'S) which holds (2; and (”6) which holds in (2.2 . In the following we are going to present a solutions for the phase fraction evolution during a time interval [ti, t], in which 79 80 one of the six transition types occurs. 5.1.1 Integration of Transition Type ('ITl) If in a time interval [1), 1] transition type ('I'I’l) M + —) A, M_ -) A occurs, then the governing equations are (4.2.2.3), (4.2.2.4) and (1.2.0.1) (see TABLE 4). Through drop- ping dt in equations (4.2.2.3) and (4.2.2.4), and then integrating the two, we obtain §+(t) = E(t,-) - (l — or+A(1:(t), T(t))) , (5.1.1.1) §_(t) = D(t,-) - (l - or,A(1:(t), T(t))) , (5.1.1.2) where, the coefficients are determined by _ §+(ti) 50,.) - “a“ (10:), T(ts))' (5.1.1.3) D(t,-) = 5,0,.) (5.1.1.4) 1 - a_A(‘t(ti). T(t.))' Hereupon, the corresponding austenite phase fraction is obtained from the balance equa- tion (1.2.0.1). 5.40) = 1 - E(t.-) - (1 - 01.410). T(t))) - DU.) - ( 1 - CLAW). T(t))) - (5- 1-15) For this situation, 5.4, Q and g are all path-independent in the (t, T)—plane. That is, beginning at known values §A(t,-) , §+(t,-) and §_(ti) at the initial time ti, the values of §A(t) , §+(t) and §-(t) at any future time t depend only upon the current values of the state 81 (1(t), T(t)) and not upon the path connecting (1:01.), T(ti)) to (1(t), T (t) ). Of course here it is required that the path only involves transition type (1T1) M + —> A, M _ -> A . 5.1.2 Integration of Transition Type (TT2) For a time interval {t}, t] in which (’I'I‘2) A -) M +, A -> M- occurs, the governing equations for this transition type are (4.2.2.1), (4.2.2.2) and (1.2.0.1). With a proper deri- vation on the three equations we arrive at EN) = F (1,.) ° Jot/(170). T(t)) - OLA.(T(t). T(t)). (5.1.2.1) where the coefficient F (‘1) is given by i) = €40.) . (5.1.2.2) Fa JaA-(T(t,-). T(t))) - GA+(’€(I;). T(t))) Substituting (5.1.2.1) into (4.2.2.1) and (4.2.2.2) generates upon integration: ‘ —F(t-) a tt,Tt §,(t) = II-TL J“: ((12)) T((t)))) :11: —aA+(r (t), T(t))]dt-I-l; (t), (5.1.2.3) ‘i ' -F t- a r t , T t * §_(:) = II_;_2 ([05:18 T8. 5: _MA( (t), T(t))Id:+§,(t,.). (5.1.2.4) ‘1 Here the derivatives in the above integration are %“A+“(‘)’ T(t)) = VaA+(r(t), T(t))-£10), (5.1.2.5) 82 £42m). m» = Va..-(r(r). T(t)) - $40. (5.1.2-6) where r(t) = 1(:)a,+r(r)e,., (5.1.2.7) Vf(t. T) = aiéfig§ép (5.1.2.8) 3, and a, are unit vectors along 1: and T axes. Result (5.1.2.1) shows that §A is path-independent, Since it depends only upon the ini- tial and the final values of temperature and stress. However expressions (5.1.2.3) and (5.1.2.4) indicate that phase fractions Q. and §_ are, in general, path-dependent. This is because the two integrals cannot generally be evaluated in terms of their endpoints in (1:, T )-plane, unless special restrictions are put on the envelope functions. To see this, assume that §+ and §_ are path-independent, then their expressions (5.1.2.3) and (5.1.2.4) have to satisfy Cauchy-Riemann condition (Apostol, 1962), which give VaA_(r, T) - V01 A((1; T) = Va A +(1:, T) - Va A_(1:, T) = 0 , (5.1.2.9) Here the operator with hyper bar is defined as —— a a Vf(1:, r) = 5%, -391. (5.1.2.10) The scalar form of equation (5.1.2.9) is 83 £301.11. 1) $374.41. 2) = 337a.-(r.1)-aa—TaA.(r. 2). (5.1.2.11) The path-independent condition (5.1.2.9) (or ($12.11)) will be further discussed in sec- tion 5.1.6, where it is shown that this condition can not be satisfied. 5.1.3 Integration of Transition W (113) If transition type (TT3) M + —>A, A -> M_ occurs in a time interval [t,-, t], then, the governing equations are (4.2.2.3), (4.2.2.2) and (1.2.0.1). The solution of equation (4.2.2.3) again gives the result (5.1.1.1), so that Q, is path-independent in (I, T) -plane. With equations (4.2.2.3), (4.2.2.2) and result (5.1.1.1), one can obtain I int) = JaA-(r(r). T(t)) - I ‘1 150,.) Jews). T(t)) dormer“), T(t)) + (5.1.3.1) £1195) . A/ ‘1 A17“ 1‘), T“ 1)) JOLAATO). T(t)) Plugging (5.1.1.1) and (5.1.3.1) into (1.2.0.1) then gives the phase fraction §_. In general, §A and E, are path-dependent in the (1:, T) -plane. We can find the condi- tion for path-independence from the expression (5.1.3.1) by applying Cauchy-Riemann condition to it. This gives the condition VaAjt, T) - Va+A(t, T) = Va+A(1:, T) - VaA_(r, T) = 0 , (5.1.3.2) its scalar form is expressed as 34 a a a a ’ 37011.“. 1) 5701.41. 7) = 3705.11. 73-5-601+A(1. T) . (5.1.3.3) This condition, which is similar to (5.1.2.11), will also be discussed in section 5.1.6. 5.1.4 Integration of Transition Type (TT4) Transition type (TT4) M _ -) A, A -) M + is similar to (1T3) under interchange of §+ and §_ . The governing equations are (4.2.2.1), (4.2.2.4) and (1.2.0.1) for transition type (TT4). The solution of equation (4.2.2.4) again gives (5.1.1.2), which indicates that §, is path-independent. The phase fraction of austenite can be found by solving equations (4.2.2.1) and (4.2.2.4) with the consideration of result (5.1.1.2), that is ‘ B(1,.) :0) = a .(10). 10»- da- (m). T(t))+ " ‘l " Ijamm, T(t)) " (5.1.4.1) 540;) . JaAJTUi), T(t))) $1,141“). T(t)) And by use of (1.2.0.1), (5.1.1.2) and (5.1.4.1), one then obtains §+. The phase fractions of austenite and M+ martensite are in general path-dependent in (1:, T)—plane in general. Applying Cauchy-Riemann condition to (5.1.4.1) gives the Special path-independent condition of the two variables as V01A +(r, T) - Va_A(1:, T) = Va_A(r, T) - VaMfl, T) = 0 , (5.1.4.2) and its scalar form is 85 a 3 a 3 5‘1“”? 0) ' 5741-110: 0) = b—T-GAJT. 6) ~53a,A(T. 0). (5.1.4.3) Again the above condition will also be further considered in section 5.1.6. 5.1.5 Integration for Detwinning Process . All the solutions and path-independent conditions obtained above are suited to the dual process region 91 . In the detwinning zones 0; and Q} , without any austenite, the only - active transition types are (ITS) and (1T6) respectively. For transition type (TTS) M, -) M +, the governing equations are (422.4) and (1.2.0.1) with Q = 0. Thus result (5.1.1.2) is the solution of §_ , which gives :0) = D(ti) - (1 - a_A(‘t(t), T(t))) , (5.1.5.1) §+(t) = 1 — (D(t,-) - (1 — a, A(1.'(t), T(t)))) . (5.1.5.2) For transition type ('IT6), the governing equations are (4.2.2.3) and (1.2.0.1) again with EM = O. The expression (5.1.1.1) gives the solution of §+ as §+(t) = E(t,-) - (1 - or +A(1(t), T(t))) , (5.1.5.3) §,(t) = l — (B(ti) - (l - a+A(r(t), T(t)))) . (5.1.5.4) The phase fractions in both (US) and (1T6) are path-independent. 86 5.1.6 Path Independence of the depleted Species within a 'II'anSition Type For transition types ('1'1‘ 1), (TTS) and (1T6), all of the phase fractions §_(t) , 90) and Q0) are path-independent in the sense that, once the initial conditions are specified, they depend on only the current values of (1:, T). However for the remaining three transition types ('IT2), (TT3) and (TT4), only one of the phase fractions has been shown to display such path-independence, namely: austenite phase fraction EA in (TT2), positively oriented martensite phase fraction Q in (TT3) and negatively oriented martensite phase fraction §_ in (TM). These are precisely the phase species that are being absolutely depleted. For example, transition type (TT3) involves M + -9 A and A -9 M _, thus the M + species is being absolutely depleted, the M - species is being absolutely augmented, and the A spe- cies is in flux (whether it is depleted or augmented depends on which of the two transitions M + -> A and A -) M. is stronger). As regards transition type (1T1), here both M + and M , are being absolutely depleted, and, consistent with the above comments, the associ- ated phase fractions are path independent. The path independence of Q for transition type ('IT 1) can then be regarded as a consequence of the overall balance (1.2.0.1). 5.2 Path Dependence and Path Independence within a 'IIansition Type 5.2.1 General Path-Independent Condition So far we have arrived at three path-independent conditions (5.1.2.9), (5.1.3.2) and (5.1.4.2) in transition types (1T2), (1T3) and (TT4) respectively. However, in view of the fact that the envelope function dependency on (t, T) is mediated by the B-function that describe the nuetrality curves (see (4.2.1.1) - (42.1.4)), condition (5.1.2.9) for ('IT2) can 87 be rewritten as a'....(B*) - a'mtfi') - (131- B;- B: - 3'.) = 0. (5.2.1.1) and (5.1.3.2) for (TT3) and (5.1.4.2) for (TT4) can be rewritten as a’max(fl-) ' a’min(B+) ' (B; ' B; - B: ' 3.1) = O 3 (5.2.1.2) a’mw“) . new) ~ (131- B; - B: - 6}) = 0. (5.2.1.3) respectively. Recall from Section 4.1.2 that 61'”, and d’m-n are equal to zero if (1:, T) are in the zones that are inactive for the process under consideration, that is the transitions either have not began or have gone to completion. Our interest is thus the case when neither fit’mJlr nor fit’min vanishes. The three path-independent conditions then simplify to VB"(r, 1) - VB'(1,1') = 0. (5.2.1.4) It will be Shown in the next section that this condition can never be satisfied by the present Bofunctions. 5.2.2 Path-dependent Analysis for the case of A, > M, In this section all discussion is confined to the pY-unfolding with identical moduli of austenite and martensite, and confined to a material with M s < A 3 . The other case (M 3 > A,) is treated in the next section. Materials obeying M 5 < A 3 involve a dead zone in the temperature driven (zero-load) transformation hysteresis. This gives certain simplifica- 88 tions. It is important to note that all the possibly path—dependent cases occur in certain parts of region QT , and that the associated special condition for path-independence is the requirement (5.2.1.4) on the nuetrality curves. As mentioned above, we are interested in the path-independent conditions in areas where the first derivatives of the two envelopes do not vanish. In transition type (TT2), the domain on which d’mx is nonzero is d2 = {(1, T)|Mf< [3“(1, 1) < M,, Mf< B(1, T) < Ms}. In domain d2 (Figure 20) the path-independent condition of §+ and 5, is given by (5.2.1.4). However, since B; > 0 , B}. > 0 , B; > O and B: < 0 , it follows that 131- E43? B'T>0. (5.2.2.1) in d2. Hence transition type ('IT2) in d2 as determined by equations (4.3.2.3) and (4.3.2.4) generates path-dependent values for E”. and §_ Turning to transition types ('IT3) and (TT4), the areas of nonzero fit’mx and Gt'm-n are given by d. = {(1.T)|M,sB'(r.DSM..A,sB* M,, three path-dependent zones d2, d3 and d4 are separated in of. 2,, and a, are path-dependent in d2 iftransition type (112) occurs (S-paths). g. and a are path-dependent in d3 if transition type (TT3) occurs (W-paths). EA and L, are path- dependent in d4 if transition type ('IT4) occurs (IS-paths). Recall in QT that the determination of transition types correlates with state path directionality in a way that is geometrically associated with compass headings: N, E, S, W. Geometrically we have shown that: §+ and Q are path-dependent in d2 for S-paths, 5A and §_ are path-dependent in d3 for W-paths, Q and Q, are path-dependent in d4 for E- paths. All the above conclusions can be viewed in TABLE 5. TABLE 5. Path-dependent Category for M, < A, PATH-DEPEDENT PATH-DEPENDENT PATH-DEPENDENT ZONES TRANSITION TYPES DIRECTIONS 5.2.3 Path-dependent Analysis for the case of M, > A, The more complicated situation M 3 > A s will be discussed in this section. Again, the pY-unfolding with identical shear moduli of austenite and martensite is considered here. The path~dependent condition is still (5.1.2.4) for transition types (1'12), (113) and (TT4), and the definitions for the three path-dependent zones remain unchanged as in the last sec- tion. The difference for the present circumstance is that the absence of a dead zone means that the three path-dependent zones d2, d3 and d4 overlap each other (Figure 21), which creates certain overlapping areas where either two or three transition types may simulta- neously involve path-dependence. For convenience, we define following six subdomains to further investigate the path- dependent condition, dz = dz-d3’d4, a3 = d3-d2-d4, a4 = d4-d2’d3, 023 = d20d3-d4, 024 = dznd4—d3, 0234 = dznd3nd4, 91 which are shown in Figure 21. Mth a similar discussion as that in section 5.2.2 the path- dependent zones, transition types and corresponding direction cones are listed in TABLE 6. Each transition type can only be triggered by the corresponding transition path, ie., ('ITl) is ignited by N-paths and so on. It is observable from TABLE 6 and Figure 21 that in the relative stability zone for M_, such as d3, the possibility for phase fraction §_ being path-dependent is larger than the others. A similar result can be observed for phase frac- tion 5,, in d4. TABLE 6. Path-dependent Category for M, > A, PATH-DEPEDENT PATH-DHENDENT PATH-DEPENDENT ZONES TRANSITION TYPES DIRECTIONS “2 (TF2) S-paths a3 (1T3) W—paths (14 _(TI' 4) E-paths a23 (TT2), (TT3) S-, W-paths a24 (TT2), (1T4) S-, E-paths (1234 m2), (1T3)! (TT4) 8" W's E’Paths 92 Frgure 21. For the case A, < M,, six path-dependent zones a2, a3, a4, (123, an and am are separatedin Qr.'I‘hesimationsoccurringinaz,a3anda4arethesamewiththoseind2,d3 andd4ofthecaseA,>M, shown in Figure 20. Inaz3, iftransitiontypeCI'I‘2)isinprocess then the phase fractions 5,, and g, are path-dependent, while if (113) occurs then §A and E, are path-dependent. In a24, the condition is similar to that in a23 under interchange of (T13) and (TT4) as well as g and §. In a234, &, and L are path-dependent if (TT2) occurs, 5A and a are path-dependent if (TT3) occurs, and, §A and g, ... path-dependent if ('IT4) is in process. 5.3 Example To illustrate this path dependence, consider a material with M, < A, and contemplate three separate stable paths in d2, all of which involve only transition type ('IT2) through- out. The three paths: ’1 , 12 and 13 are shown in Figure 22. They all start at point (1:, a) = (O, M,) with initial conditions {§,, EM, §}={O, l. 0} and end at point (1:, o) = (O, Mf). The three state paths are the forms of 93 11: {(1,1)11 = 0}; M +M 12: {(1,1)10’ = -ak,r+M,.M,2Tz—‘—2—i); Al+bl (T = nklr+M,,Mfsrs——i‘ 2 )1. M +M 13: {(r, T)|(T = nk,r+M,,M,2Tz’—2—I); M,+Mf respectively. Where, I) is a real number which must be greater that l to stay in d2. It is desired to find the phase fractions {§_, Q, L} at the end point (0, M!) on the three different paths. ' The nuetrality functions that we use here are (2.3.2.3) and (2.3.2.4) for pY-unfolding with equality of the moduli of austenite and martensite, where k1 = Y. = 0.1 m3 0 0 . nA-nM °K/J (Ivshin and Pence, 1994 b). Thus, all the three paths are inside domain d2 for any (2 > 1 and the directions (derivative Z—Z ) of all the points on the three paths are confined to the open cone zone S. The envelope functions used here are linear piecewise defined as the following: 0, v (35114, * - Mf— VM 5 SM 5301 amax(B)"1Ms_-Mfr f B 3 9 (°°°) 1, vpzu, ' 0, v [35A, B-As amin(B) = 4 Af—As, V A‘SBSAI . (5.3.0.2) 1, v (32.4, Based on the definitions (4.2.1.1) to (4.2.1.4), the functions aA+(1:, T) , 01+A('c, T) , aA_(r, T) and 0t_A(1:, T) can be found by substituting (2.3.2.3) and (2.3.2.4) into (5.3.0.1) and (5.3.0.2). It is then desired to determine phase fractions (5..., Q, §+} at the end point (t, T) = (0, Mf) on the three paths, each as a function of Q . A mathematical reduction leads to complicated integral expressions that can formally be integrated with the help of either handbooks or symbolic algebra. Numerically, these phase fractions for different values of Q are found and given in TABLE 7. Since path 12 and path 13 are sym- metric with respeet to T-axis, so that the values of g, and §_ corresponding to path l2 and path (3 switch each other. These results also confirm that EA is path-independent, g, and §_ are path-dependent in transition type (TT2). 95 Figure 22. Three paths I], l; and 13 go from (r, o) = (0, M,) to (0, M,) in the path-dependent zone of transition type ('IT2) with initial condition (g, Q, §}={0, l, 0}. Transition type ('I'I'Z) occurs on an three paths. Path 12 consists of two straight segments which meet at point mafims — M f), gm: + M f) ). Path 13 is similar. The phase fractions 1; and 5,, are path dependent while §A is not. The values of the triple 19,514, §+} at the ends of the three paths are listed in TABLE 7. 96 TABLE 7. Phase Fractions at the End (0, Mf) of the Three Paths: ll, 12, I3 _l___ ,. . ,3 lI—— {0.,5 0,.05} {0,0,1} {1,0,0} 1.01 I {0.50, 0.5} {0.036752, 0, 0.963248} {0.963248, 0, 0.036752} 1.1 I {0.5, 0, 0. 5} {0.120019, 0, 0.879981} {0.879981, 0, 0.120019} 1.5 I {0. 5, 0. 0. 5} {0250921, 0, 0.749079} { 0.749079, 0, 0.250921} 2 I {0. 5, 0, 0. 5} {0.319921, 0, 0.68009} {0.68009, 0, 0.319921} 5 I {0.5.0, 0.5} {0.430292, 0, 0.569708} {0.569708, 0, 0.430292} 20 I {0. 5,0, 0.5} {0.482665, 0, 0.517335} {0.517335, 0, 0.482665} 100 I {0.5, 0, 0. 5} {0.495634, 0, 0.503466} (0.503466, 0, 0.495634} lOOOI {0. 5, 0, 0. 5} {0.499653, 0, 0.500347} {0.500347, 0, 0.499653} 20000 I {0.5, 0, 0.5} {0.499983, 0, 0.500017} {0.500017, 0, 0.499983} no I {0.5, 0, 0.5} {0.5, 0, 0.5} {0.5, 0, 0.5} 5.4 Discussions on the Solutions The phase fraction evolution for different transition types obtained in section 5.1 are suimd to the X-unfolding, pY-unfolding and Y-unfolding. Because different nuetrality curves produce different responses to the envelope functions, the phase fraction evolution 97 in X, pY and Y-unfolding could be different corresponding to the same transition types and the same thermomechanical circumstances. Obviously, in I), both of the X- and pY- unfolding generate the same results for the phase fractions because the nuetrality curves for the two unfolding are identical in (21 . However, for the Y-unfolding, since the detwin- ning flow and finish are independently specified, the description of phase transformations A (951,111 T<0 and A (—> M_ in ‘t>0 are different from the description ofthe other two unfoldings. Predominantly, the Y-unfolding decreases stress ranges of existence of M, in 1.” < 0 and of M _ in r > 0, which is caused by the smaller detwinning flow and finish stresses 1, and “If found in the experimental literature compared to the “natural values” associated with the X- and pY-unfoldings. All phase fraction solutions are determined in an infinitesimal time interval where only one of those transition types occurs. Chasing a given state path in (T, T)-plane, phase frac- tions vary in one transition type for a segment on the path, and then may change to another transition types for the following segment on the path. All the three phase fractions {§,, fiA, §,} are path-independent for transition type (TT 1), ('I'I‘S) or (1T6), which means that they can be uniquely determined at a point in (r, T)-plane provided initial conditions are given. In addition to this, Q in (112), §_ in (1T3) and g, in ('IT 4), which represent con- sumed phases in each of the three transition types, are path-independent. On the other hand, phase fractions of transformed phases U5“. in (T12), §A/§ in (TT3) and §A/§+ in ('IT4) are path-dependent in the areas as shown in Figure 20 and Figure 21. 6 BEHAVIOR OF THE MODEL After a careful discussion on the algorithms and associated studies, we now turn to view the macroscopic thermomechanical behavior of the two variant model by considering some numerical simulations. We emphasize that these simulations are only suited to a case of one dimensional behavior for a material containing only austenite and two martensite variants. This could include a compatible twin structure that is imagined as a symmetric lattice structure in which one lattice is sheared in one direction and the corresponding lat- tice is sheared in the opposite direction. Phase transformation occurs between austenite and martensite variants in (21 , and detwinning occurs between the two symmetric marten- . . .. + a - srte structures In 02 and £22. 6.1 Isothermal Behavior In this section, we are going to confine all simulations for isothermal mechanical pro- cesses. This includes isothermal loading/unloading at high (above Af) and low (below Mf) temperatures, in which pseudoelastic and shape memory behaviors will be generated. More than this, certain loading/unloading processes at some temperatures between Af and Mf are also considered. Further, internal hysteresis loops will be conducted at the end of this section. All the simulation parameters are taken from TABLE 2 of Section 2.5 except 98 99 p, = 4.0 x 104 and ti” = 2.5 x 104 MPa are going to be employed here. In particular, M, < A, for this material. The phase diagram used here is Y-unfolding. The envelope func- tions are the piecewise forms of one argument as given in the following, r .0 I B < Mf - . 1 .521; ' 01M,(B) _ 2(1— cos(M’_Mf)1t) M, s B 514,, (6.1.0.1) L 1 B>Ag ‘ 0 BAf 6.1.1 Pseudoelastic Behavior Pseudoelasticity occurs when loading in the high temperature austenite phase gener- ates biased martensite, subsequent unloading causes complete reversion of the biased mar- tensite to austenite provided T > A]. This kind of behavior has been investigated in many works, such as by Fu, et al. (1993), Ivshin and Pence (1994 b), etc.. Here, as one of many basic features of the two variant model, pseudoelastic behaviors in both tension and com- pression at test temperature T, = 335 °K are given in Figure 23. Three such behaviors at three different test temperature levels: T, = 315 °K, T, = 325 °K and T, = 335 °K are pre- sented (calculated by Mathematica) in Figure 24 for ‘t > 0. In each complete hysteresis loop the lateral slopes are different because of the different moduli of austenite and mar- tensite. 100 0.025 0.05 0375' Y -625 Figure 23. Pseudoelastic behaviors in both tension and compression conditions at test temperature T, = 335. In t > 0, A HM, processes are involved with the loading! unloading, while, in 't < 0, A H M _ processes are involved with the opposite loading/ unloading. I (MPa) t 1: C) 500 315 °K 600 325 °K 600 335 K 500 500 500 100 400 “,0 300 300 300 200 200 200 100 100 100 0.02 0.04 0.06 6.08 0.02 0.04 0.06 0.08 Figure 24. Pseudoelastic behaviors for M, < A, at different temperature levels: T, = 315, 325, 335 9K, all of which are greater than Af= 308 °K. Since M, < A, for the material considered here there is dead zone between the flow stresses of A -> M, and M, -) A (Figure 27). The height A17 of the dead zone at test 101 temperature TI is determined by the following formula, 11‘ = 2(A,-M,) \ Jkl +4k2(T,-M,) + Jk1+4k2(T,—A,) (6.1.1.1) Here (2.4.2.2) gives ’61 = 0.144 rn3°1 0, it follows that the height ATT of the dead zone decreases with test temperature T, increase and vice versa for T, > Af (see Figure 24). The reason for this phenomena in the present model is that ILA it It” through parameter k2 determined by equation (2.4.2.2). In the special case of equal shear moduli (ILA = It” ), one finds k2 = 0, which indicates that ArT = 2(A, - M,)/(2717,). Comparing with the other models, this one is similar to Falk’s model (1980) but is much more general. In Falk’s energy model, forward transformation A —> M happens along the top dashed line when the applied load reaches the maximum of the left ascend- ing branch (elastic loading of austenite) of the load-deformation diagram (Figure 25). The reverse transformation M -) A occurs along the bottom dashed line when the unload reaches the minimum of the right ascending branch of the same diagram. Consequently, a pseudoelastic hysteresis loop is formed between the ascending branches and the two- dashed lines, which indicates the energy dissipation during the process. The present model, which can cover all the above description if one treats the top and bottom dashed lines as the same as top and bottom bands, can generally describe the deformation related to both phase transformation and elasticity simultaneously. In addition, the height ATT decrease with temperature increase could reflect some experimental measurements such as those illustrated in the work by Funakubo, (1987). This trend is also predicted by Landau- Devonshire’s model (Muller and Xu, 1991), for which the load-deformation diagrams in 102 different temperatures are shown in Figure 26. However, in Muller and Xu’s work (1991), an opposite phenomenon was observed experimentally, which shown the loop height slightly increases with the temperature increase. Therefore a corresponding model was suggested to describe this in their approach. ‘P Figure 25. By Falk’s model (1980), austenite transforming to martensite occurs at the highest point on the left ascending branch (top dashed line) upon loading. The reverse transformation, martensite to austenite occurs at the lowest point on the right ascending branch upon unloading. P P P A ‘ 1 T, 7'2 T3 ’ d > d > d o o o . Figure 26. By Landau-Devonshire’s model, load-deformation diagrams in three different temperatures (T, < T; < T3) Show that the heights of the hysteresis loops decrease with the temperature increase. 103 6.12 Internal Hysteresis Loops Internal hysteresis loops are an important behavior. For the specified material with M, < A ,, we will study how the internal hysteresis loop is conducted in the present model. In particular, we retrieve the results similar to those of the work by Ivshin and Pence (1994 b) from their one variant investigation and in so doing provide a mathematical treatment of an intensity filtering phenomenon. For illustration we employ the pseudoelastic behavior at test temperature T, = 335 °K > A, (Figure 27). The A —> M, and M, —. A start stresses are denoted as Na) and rMA(T,) respectively, so that the top band and bottom band of the dead zone are defined by 2 - k, + fl, - 4k2(M, - 1,) TAM(T‘) = 2k 2 (6.1.2.1) ,MAUt) = 2 — k, + Jk, -4k2(A, - 1,) 21:2 (6.1.2.2) respectively. With respect to Figure 27, loading from pure austenite to point a (above the stress I = tAM(T,) but below the stress at which the A -9 M, transformation is complete) gives a mixed state of A and M ,. Then unloading remains elastic until point b on the bot- tom band. The slope value of this unloading portion from a to b is between that of elastic austenite (on the left side) and elastic martensite (on the right side) because the material state is a mixture of austenite and martensite (in general It, > p.” ). Further unloading from point b leads the curve to point C, during which process M, —> A occurs. Point c coalesces with point f if the unloading is large enough to cause M, —) A to go to comple- tion. 104 If reloading from point b, then the curve goes back elastically to point (1 on the Same track as the unloading. This would be followed by A -) M, processes during loading to point e. Point e approaches point g if the loading causes A —> M, to go to completion. This is a behavior different from the plastic hardening in which reloading from point b will return elastically to point a. On the other hand, if reloading from point c, then the behavior is elastic to point h with a mixture state of austenite and martensite inherited from point c. It then travels from h toward g monotonically during which A -> M, occurs. top band bottomband Figure 27. A dead zone between the top and bottom bands in the stress-strain diagram is illustrated. This dead zone corresponds to the portion between points 2 and 3 in the phase diagramPointsfandgcorrespondinstress-straindiagramtopoints 1 and4inthephase diagram. The internal loop formation condition is that unloading has to reach the bottom band and loading has to reach the top band shown as a-d-b-c-h path. It is conspicuous from the above analysis for the material obeying M, < A, that for- ward transformation A —) M ,1 can only occur once the load reaches the top band 1 = 105 TAM(T,). The reverse transformation M, -) A can only occur once the unloading reaches the bottom band 1: = TMA(T,). Any loading/unloading wholly between the two bands is conducted elastically at the modulus associated with the mixture. Consequently, a neces- sary condition for a an internal 100p formation is that a cycling load has to have the maxi- mum load larger than the top band and have the minimum load smaller than the bottom band. This is different with the model made by Muller and his colleagues (1991, 1993), in which the internal transformation is governed by a straight line connecting the martensite start stress and the austenite start Stress in the stress-strain hysteresis loop. Another approach by Tanaka et al. (1994) involves internal transformation that is governed by two curved lines, one of which is connected with the martensite start stress and controls trans- formation A -) M for loading internal paths. The other is connected with the austenite start stress and controls transformation M —-) A for unloading internal paths. The reason for the curved internal transformation lines is that they believe that the formation of the internal loops (subloops) depends upon the prior transformation through the dependence of internal transformation stresses on the transformation history. In the present model, the evolution of internal loops depends on the transformation history but the internal transfor- mation stresses are taken as constants which implies that the phase diagram is fixed. It can be also shown with the present model that, if loading from pure austenite state to a point above the top band, again say, point a, is followed by unloading to any point aboVe point (1, then oscillating the load in a small domain around the selected point will eventu- ally eliminate all austenite phase and end up with pure M+ phase. Similarly, an oscillation of loads in a small domain around a point on or below the bottom band will eliminate all the M, phase and end up with pure austenite phase A. This concludes that the top and bot- tom bands work, in a certain sense like, a phase filter to sift austenite A and M+ variant 106 respectively when the specified oscillating load is applied away from the two bands. Finally oscillating around a point between the bands will give convergence to a mixture state that is independent of initial conditions. This phenomenon is connected with the infinitesimal loop behavior discussed in Ivshin and Pence (1994 a). T; - . . L -b 3250:4114. M, M, o 0.02 0.04 0.05 0.00 7 Figure 28. The stress-strain trajectory approaches a stable internal loop in the Stress-strain diagram with oscillating scope of sness between points a and b (between 1:" and 1‘ in stresses) in the phase diagram. As a Simulation result, an internal saturated hysteresis loop (Figure 28) is obtained at r, = 325 °K under three cycles between Tb = 160 and t' = 560 MPa in stress. It can be observed from Figure 28 that the internal loops drift to the right with respect to the origi- nal one, and quickly approach a firm position.This corresponds to a tendency that strains in a same stress level may initially vary with the cycle number, but then quickly settle to a stable internal loop after several cycles. We are going to prove this point in the following for a stress oscillation Th 5 1: S “It where 1:b < 1"“ and ‘1:l > 1"“ as shown in Figure 29. 107 "I 1‘ A 1 2 3 n A-7M+ ifstressincrease ...- _ ---A 11 ...... A17 cyclinssoope 1M4. --- .. _____ I" - ‘b V 1 2 31 n t O 0 \M,.>Aifstressdecrease Figure29.CyclingloadSareappliedbetweent=1brAMattest temperature T > A, to form internal hysteresis loops. It can be seen that the top and bottom bands are covered inside the cycling range. It should be noted that T >,Af and t > 0 imply that §_ = 0 so that only transformations A H M, are involved in the particular loading/unloading process. Based on equation (4.3.2.3) the right-shear martensite phase fraction is generated for A -) M, when the stress increases between 1: = 1“” and 1: = 1‘ MPa, g 1_ (tom é, —— or ,. W.-.” Here (9)9 = 1 and n (=1, 2, 3,, ...) indicates the point numbers on the lower stress line at (6.1.2.3) 1: = 1:" MPa. Since all the points n-l are on t = 1:" < TMA, it follows that all the valuesof ((1,,,,)n_l = 1 . Based on equation (4.2.2.3) the austenite phase fraction is found as the following for M, -) A when the stress decreases between 1: = 1M0 and 1: = 1:", 108 (to, 5.4 = l_1-(00,A)n (1 " (AA) . (6.1.2.4) here n (=1, 2, 3, ...) indicates the point numbers on the upper stress line at r = 426.6 MPa. Since all the point n are on 1: = 426.6 MPa > TAM, it follows that all of the values of (11M)n = 0. Thus the values of i, at each top peak for 1: = 1." are (§,)n = 1- (§A)n-1 I(01A,)n while the values of Q, at each bottom peak on 1: = 1b are (gnu = 1-(§,)n(l _(a+4)n)' Since 01A, is simply a function of ‘t and T, all the (05“,)u are equal, as are all (00%)“. For convenience, let (11A,)n = u and (“+400 = v (n =1, 2, 3, and n =1, 2, 3, ...). Therefore, the M+ phase fraction at point 11 on 1: = 1‘ can bewrittenas (1%,)n = [Z (uv)"‘I(§,),. (6.1.2.5) .g 1 Since the cycling load/unload conducts an incomplete transformation for both forward and reverse transitions A (-) M, , we have 0 < u, v <1. Thus (6.1.2.5) converges to a limit _ (531 1’4]; - l—uv l—uv’ (5.)... (6.1.2.6) which is larger than (i +)1 . By (6.1.2.4) and (6.1.2.6) the austenite phase fraction on 1: = Tb converges to a limit (9.4)., = 1‘11:J;(1-V)- (6.1.2.7) uv The overall strain expression based on (2.1.0.2) can be rewritten, 109 ‘L' 1: T " = —+ ———+ , 6.1.2.8 r (“M 11.1 r)§. ( ) for this particular loading/unloading condition. Thus the strain converges to fixed values on both top and bottom peaks of the cycling which means that the internal loops in this example move to the right and approach a saturated position. The “right move” tendency corresponds to a situation for phase transformations that more martensite (or twins) is (are) generated during the cyclic load before the saturation is reached. This is similar to the softening feature of materials under high temperature creep. 6.1.3 Shape Memory Effect and Isothermal Behavior below Af Shape memory effects occur when the alloy is deformed into a biassed martensite phase and does not revert to a self-accommodated morphology upon unloading. Its initial shape is recovered when heated into the stable austenite regime, and then cooled back to the original temperature. ‘T(MPa) 200_ Loading\ 150 * . unloading 100 - / SOP 200 0 .‘02 0 .‘04 :E‘ > 7 heating 308 hm Figure 30. Shape memory effects occur during loading-unloading-heating-cooling processes. 110 Here the test temperature of loading/unloading T, = 200 °K which is below Mf (= 235 °K). After unloading the temperature is then increased to A}: The initial phase fraction condition used in the present case is obtained by decreasing the temperature from above Af to the test temperature under a free-stress situation (CFAF), thus it is random martensite E = {0.5, 0, 0.5}. Upon loading (Figure 30) equal elastic deformation occurs in the two mar- tensite variants for stress below the detwinning flow t,=150 MPa. This is followed by a detwinning process M _ —> M, which involves the migration of twinning interfaces within the two martensite variant mixture. This kind of motion of twinning interfaces needs grad- ual increases of external driving forces to overcome the interface resistance to motion. This process is complete at 1: = If. Further loading then involves pure elastic behavior of the fully detwinned martensite (M,,) (the tail part on Figure 30). Unloading from the max- imum sness point to a stress free condition makes the stress/strain relation track on a lin- ear path with the martensite modulus as the slope. This gives a residual strain equal to the phase transformation strain upon unloading to zero stress. Heating the specimen to the temperature A, transforms M, —) A and so eliminates all the residual strain. Cooling again gives random martensite and so leaves the strain unchanged. 111 A 1 (MPa) , 400* loadmg ‘ 400 . 300 . unloading 300 . 200» X 200 100» root 0 a Y 301 0 .‘ 0.1T 0.06 0 .508, b- heatin -100» 308: b 3 T(OK) Figure 31. 'IVvo different procedures recover the residual strain. (a) shows the residual Strain recovered upon heating, (b) shows that the residual strain can be recovered by further _ unloading. Here the test temperature is 301 °K during the loading/unloading process and the original phase fraction is 5 = {0, l, 0}. In general, the residual strain associated with unloading is caused by certain special martensite variants remaining, and it can be recovered upon heating to above Af Discus- sions related with this issue have been conducted by either experimental measurements and theoretical approaches (Funakubo, 1987). In addition, the present model not only gives strain annihilation by heating/cooling (as just shown) but also illustrates that the residual strain can he recovered by “further unloading”. Here the unloading is always with respect to loading, for example, at certain temperature levels, increasing the stress causes A -—) M, to become active, which is the loading process. In the opposite, decreasing the stress causes M, -> A to become active, which is the unloading process. When the stress returns to zero, process M, —) A may or may not be complete. The complete case com- ports to pseudoelasticity while the incomplete case remains at residual strain. Based on the 112 present model, any such remaining strain can be recovered by heating and cooling (a to b in Figure 31. (a)). Conversely, further unloading, which means continuous decrease of the stress to negative values, is an alternative way to either complete or partially complete the process M, -9 A (a to b in Figure 31. (b)). Actually, this further unloading is an opposite loading with respect to the loading of increase Stress, which is favored for the growth of martensite variant M,. Thermodynamically, this corresponds to a situation for which the energy barrier for phase transformation M, —> A is smaller than that of detwinning M, —-) M _ in the temperature range between Af and Tf, (see section 2.6). A similar condi- tion is held for M _ -> A when stress increases to a positive value smaller than “If I (MPa) 1 (MP8) 285 °K ‘°°’ 275 °K 300' A 300: 200’ 200: 100’ 100» . a - - 2 7 0.02 0.404 0 06 1‘ 7 0.02 0.04 06 0.09 4 4—96 -100. (a) (b) -100F" Figure 32. The residual strains are recovered by further unloading (a —> b -) c). The plateau of the transformation A -) M, decreases with the test temperature decrease. In the opposite, the yielding plateau of the reverse transformation M, -) A increases in the negative direction of the T-axis as the test temperature decreases. To demonstrate this phenomenon, consider an example with a initial condition {§_, Q, 5,, }={0, 1, 0} at test temperature T, = 301 °K below A, (=308 °K). Two stress/strain behav- 113 iors are obtained for the residual strain recovery by either heating or by further unloading (Figure 31). If the test temperature is now decreased then the stress/strain profiles move down along ‘t-axis (Figure 32). This Bauschinger efect type of phenomenon can be observed in a certain test temperature range (Figure 32 (a)). which is a concept from clas- sical plastic theory to reflect an experimental observation that, after a certain amount of forward plastic deformation in tension or compression, the material yields at a lower stress when the direction of loading is reversed than for continued forward deformation. Ferroelastic behaviors (Bondaryev and Wayman, 1988) are simulated at test tempera- ture T, = 200 °K with initial condition § = {0.5, 0, 0.5} (Figure 33). AT(MPa) zu—— ——————————————— 4 ————— F r r 2.. O -0 06 -0.04 -0.02 0.02 0.04 0 06 Y -100L ‘225 I— ____________ '2001 Figure 33. Ferroelastic behaviors in both tension and compression conditions at test temperature T, = 200. In 1: > 0, M _ -) M, process is involved with the transformation, while, in 1: < 0, M, -) M , process is involved with the transformation. In addition to pseudoelasticity and shape memory, there are other isothermal behaviors between Af and Mf of interest. Since the initial conditions have a large effect on the ensu- ing stress/strain relation associated with loading and unloading, in the following we will show two groups of graphics regarding the two kinds of initial conditions CFAF and 114 I-IFMF introduced in Section 2.5. Recall that CFAF is obtained by cooling the temperature from above A, to the test temperature, which corresponds to an initial condition of maxi- mum austenite (Figure 34). The other (HFMF) is obtained by heating the temperature from below Mf to the test temperature, which corresponds to an initial condition of maxi- mum equal amount martensite (Figure 35). It is noted that the model allows for transfor- mations M _ -) A to occur in 1: > 0, and also allows for transformation M, —> A to occur in ‘t < 0. Increasing the stress to cause M, -> A in t > 0 could beregarded as further unloading with respect to decreasing the stress (loading) to cause A —) M , in t < 0 at the same temperature level. A similar discussion for M, —> A to occur corresponding to decreasethestressint<0canbearrivedtoo. 115 600» 4 300» OK 4 301°K 3 270 3 400» 200 300» 2 150» 200. 100i 100’ 50 2 6 5 5 A 0.02 0.04 0.06 0.00 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 53 (a) (b) E r I 5 300» 5 100» 4 . 200 200 3 4 3 150» 2 150 100» 100 2 so so 1 6 1 - - 5 ' 0.01 0.02 0.03 0.04 0.05 0.06 0.07 7 ‘ 0.01 0.02 0.03 0.04 0.05 0.06 0.01 7 (C) ((0 Figure 34. Initial conditions are obtained by cooling the temperature from above Afto the test temperature in a stress free circumstance (CFAF). General features of the transformation process for loading/unloading were described in Figure 14. In (a) the initial condition is {0, 1, 0}. l —> 2: austenite elastic deformation: 2 -) 3: phase transformation A ->M,; 3 -)4: M, elastic deformation; 4-35: M, elastic unloading; 5 -)6: partial . reverse transformation M, —) A which gives a certain amount of residual strain left at the end of the unloading. In (b) the initial condition is still {0, l, 0} because 270 °K > M, (=263 °K). l —> 2 -> 3 —> 4: conduct the same deformation mechanism as those segments in (a) correspondingly: 4 —> 5 : M, elastic unloading. In (c) the initial condition is {0.0135, 0.973, 0.0135}. 1 —) 2: phase transformation A -) M ,; 2 -) 3 : combined transformations A -)M, and M,—)A; 3—)4:detwinning M_—)M,; 4-95:M,elasticdeformation; 5 -) 6: M, elastic unloading. Since the phase fraction of M, is small, there is no significant change in segments 1 -) 2 , 2 —) 3 and 3 -) 4 . In ((1) the initial condition is {0.222, 0.556, 0.222}. 1 -) 2: phase transformation A —) M ,; 2 -) 3: elastic twinned martensite; 3 ->4: detwinning M_ ->M,; 4 -)5: M, elastic deformation; 5 -)6: M, elastic unloading. In (b), (c) and (d) the residual strains are the phase transformation strain 7'. 116 (MPa) 6 m, 285°K 5 300' 200. 4 , 3 100 2 1 - 7 0.02 0.04 0.05 0.051 (b) t t 300» 5 300 5 35°. 270°K 4 3,0. 260°K , 4 200. . 3 200 3 150 150 2 2 100’ 100 50 50 l - ._ 6 - 1 6 ‘ 0.01 0.02 0.03 0.04 0.05 0.06 0.01 7 ‘ 0.01 0.02 0.03 0.04 0.05 0.05 0.07 7 (C) (d) r 300 o 4 m 250 K 200 3 150 2 100 50 1 *9 5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Y (c) Figure 35. Initial conditions are obtained by heating the temperature from below M, to the test temperature in a zero-stress condition (I-IFMF). General features of the transformation process for loading/unloading were described in Figure 15. In (a) the initial condition is {0.2801, 0.4397, 0.280] }. l -) 2: phase transformation M. -—> A; 2 —> 3: elasticity of combined austenite and right-shear martensite; 3 —) 4: phase transformation A -> M +; 4 --) 5 : M, elasticity; 5 -> 6: elastic M, unloading; 6 -) 7: partial reverse transformation M , -) A upon continuous unloading which gives a certain amount of residual strain left at the end of the unloading. In (b) the initial condition is {0.5, 0, 0.5} which is also the initial conditions for (c), (d) and (e). 1 -) 2: fully twinned martensite elasticity; 2 —> 3: phase transformation M _ -) A; 3 -) 4: elasticity of combined austenite and right-shear martensite; 4 -5 5: phase transformation A —)M,; 5 —56: M, elasticity; 6—57: elastic M, elastic unloading. In (c) 1 -> 2: elasticity of twinned martensite; 2 -5 3: combined 117 phase transformation M_ -9A and A —)M,; 3 —54: phase transformation A -)M,; 4—)5 and 6—57 are similar to 5-56 and 6—)7 in (b) respectively. In (d) 1—)2: elasticity of twinned martensite; 2 —) 3: combined phase transformation M - -) A and A—5M,; 3-54: detwinning M-—>M,; 4-)5 and 5—)6 are similar to the corresponding sections in (c). The only difference of (e) with (d) is that there is only one section to conduct detwinning M- —> M +, which is 2 -> 3. In (b), (c), (d) and (e) the residual strains left are phase transformation strain 7". 6.1.4 Load Cycling and Saturation Based on the present model, saturated hysteresis loops usually occur in dual direction transformations where two or more phases compensate each other and eventually reach a balanced state of associated phases for the oscillating load process. This behavior, for example, was observed in the process A H M , at high temperature oscillating of loads in Section 6.1.2. A oscillated stress-strain behavior with 0 S 1: S 150 (MPa) is simulated at T = 270 °K < A, (Figure 36). In this process with HFMF initial condition (é = { 0.5, 0, 0.5}), all the three phases M,, A and M, are involved in different periods. Since the oscillated load is operated in t > 0, M_ -) A and A -—) M , occur during loading while only elastic ' relaxation takes place in the mixture state during unloading. M. phase is roughly devoured out after the fourth cycle, while A phase remains until the tongue-shape response reaches the far right lateral straight line (Figure 36 (b)). There is no stable saturated loop because there is no phase compensation in the unloading process. The yielding point on the outside profile (the same with Figure 35 (c)) is higher than the internal yielding points. This is because there is no austenite involved with the transformation at the very first loading so that the yielding does not occur until the path hits the nuetrality curve A,,. This triggers M_ -) A which then supplies austenite for the immediate A —> M , process. Since austen- 118 ite has been generated during the first cycle, the yielding occurs at lower stress in the fol- lowing cycles as shown in Figure 36. The phase fraction distributions at the beginning of each cycle are listed in TABLE 8. The austenite reaches its maximum value right after the firstcycle. T ) T=270°K 1 2 3 4 150 T(MPa) 4 ..4 25 2,0, T=270°K i 200’ t 3: 150} ————— — — - 3.!- —5004» l 2 3 4 -5 =- ----- - - - - la— 01 Y b o 0.01 0:02 0.03 0.04 0.05 0.06 0.0? (b) Figure 36. Isothermal response (b) under cyclic loads (a) at test temperature T= 270 °K (M, < 270 °K < A,). The initial condition is from HFMF with g = {0.5, 0, 0.5}. The outside profile is the same with Figure 35 (c). M. —) A and A —) M, occur upon loading while elastic relaxations in mixture phase states occur upon unloading. M, phase is roughly consumed out after the fourth cycle. Austenite remains until the tongue-shape stress-strain response reaches the far right lateral straight line of the outside profile. 119 TABLE 8. Phase Fraction Distributions 1 {0.5000, 0.0000, 0.5000} 2 {0.1012, 0.3403, 0.5585} 3 {0.0205, 0.2940, 0.6855} 4 {0.0042, 0.2084, 0.7874} 5 {0.0000, 0.1414, 0.8586} 6 {0.0000, 0.0935, 0.9065} 7 {0.0000, 0.0617, 0.9383} 8 {0.0000, 0.0407, 0.9593} co {0.0000, 0.0000, 1.0000} 120 6.2 Differences between Loading-Cooling and Cooling-Loading Paths According to Duerig, et al., 1988, “For reasons which are not entirely clear, plastic defamation will occur below the martensitic yield strength in many materials if one applies the load while cooling through M,” (pp. 183-184). We now show that behavior of this type arises naturally in the present model. The essential feature of this argument were presented in the approach by Wu, Pence and Grumman (1996). In particular we consider two routes for obtaining oriented martensite (M,) at a low temperature and high stress beginning from stress-free austenite at high temperature. Here the high temperature is taken to be T: 325 °K (> A,) and the low temperature is taken as T: 225 °K (< M,). Vari- ous final values of stress 1:,- will be considered. For each such stress value, one of the two routes involves loading at T: 325 °K from 1: = 0 to 1;, followed by cooling at t = 1:,- from T = 325 °K to T: 225 °K. The other route involves cooling at 1: = 0 from T: 325 °K to T: 225 °K, followed by loading at T = 225 °K from 1: = 0 to 1,. We refer to these two routes as loading-cooling and as cooling-loading paths respectively (Figure 37). In all cases the final phase fraction state is martensitic: {§_, Q, E,,} = {§_, 0, §,}. However for a certain range of 1:, namely 1 < “9:, one obtains that the value of §, generated by the loading-cooling path is greater than that generated by the cooling-loading path. The envelope functions to be used in the following are of the forms (6.1.0.1) and (6.1.0.2). All the material properties are given by TABLE 2 of section 2.5. 121 T (°K) 340 - l 320 Mfl" 300 >‘< L/ 280 \ >/ . 5.0- "100‘ A 150‘ H 200 A L 350“ ) Figure 37. No groups of driving paths starting at point (0, 325) on T-axis with initial condition {0, 1, 0} go to point (20, 220), (40, 220), (68.9, 220), (110, 220), (150, 220), (160, 220), (170, 220), (180, 220), (190, 220) respectively. Following the above two kinds of driving paths the model predicts a natural path- dependent phenomenon of shape memory materials. In the loading-cooling paths, the loading portion triggers trivial (TT4), so as to only conduct austenite elastic deformations, and the cooling portion triggers (1T2). Since 1: > 0 transformation A —> M, occurs earlier than A —-> M , upon cooling, therefore, at the end of the driving path the phase fraction 5, is larger than phase fraction E, (and Q, = 0). There is a critical stress 1:. (= 1:3) that is the ' intersection stress between nuetrality curves M,;, and M ,: I. = —(k, -k;)+,/(kl—k;)2+4k2(M,-Mf) 2k2 (6.2.0.1) For the present parameters used here (TABLE 2 of Section 2.5), this critical stress 1'. = 122 68.9 MPa. 11' r, < t‘, then further cooling to temperature 220 °K will generate both of the martensite variants. If the stress surpasses 1:" during loading, then further cooling to tem- perature 220 °K will generate 100% M, martensite ({0, 0, 1}) because transformation A 4—> M , upon cooling can only be conducted in a stress domain [0, t.) in 1: > 0. In the cooling-loading driving paths, (TT2) is triggered by the cooling portion and (TTS) is triggered the loading portion. Cooling from temperature 325 °K to 220 °K along T-axis is a self-accommondated process, so as to produce an equal amount of the two vari- ants of martensite. Then loading at temperature 220 °K will only cause elastic deforma- tions of this equal self-accommodated martensite until the detwinning flow stress 1,. Increasing the stress beyond 1:, then induces the detwinning process M _ -9 M, . The unequal final amounts of the phase fractions of the martensite variants produced by the two different kinds of the driving paths are illustrated in Figure 38. On the other hand, since the loading-cooling paths generate more M, if t < If they give a larger defor- mation than the cooling-loading paths. In other words, the first process of loading-cooling only generates an elastic deformation in the austenite. The following process (cooling) generates a deformation associated with phase transformation A -9 M, which is larger than the elastic deformation. However, following the cooling-loading driving paths, there is no deformation in the self-accommondated cooling process. The loading portion then will only generate an elastic deformation of martensite before the stress reaches the ' detwinning stress 1,. 123 Efi- loading-cooling .. ,1. . ...... 0.9 048’ o 0.7 . . . $coolrng-loadrng 0.6 0.51? 0 IL I O A A 1%) so 100 150 200 250 Figure 38. Phase fractions of the martensite variant M, upon driving paths of loading- cooling (upper point plot) and cooling-loading (lower point plot). Here 1:. = 68.9, 1:, =150 and If: 200 MPa. 6.3 Comparison with Other Models In Brinson’s approach (1993) martensite phase fractions are distinguished as stress- induced martensite Md (favored at a specified stress) and thermo-induced martensite M ', the latter of which is regarded as self-accommondated. This contrasts to the model under consideration here in which the martensite phase fractions are distinguished in terms of variant structures. Note, however, that a connection between these two viewpoints is estab- lished by grouping our minority variant with an equal amount of the majority variant to obtain a self-accommondated martensite structure that can be identified with Bekker and Brinson’s thermo—induced martensite. Then the excess majority variant is identified as purely stress induced martensite. That is, one takes 124 i, = 2min(§.. 6). id = 8;, + §,-§,. (6.3.0.1) To illustrate this transformation, we define two sets of portions of all state paths. The first one P, is a set of all portions of the state paths along which the relation g, > L is held. The other one P, is a set of all portions of the state paths along which the relation 5, < E, is held. If a state path or part of the state path belongs to P,, following it M, is the majority variant, so that g, .-. 25, g, = 5,, - g, (6.3.0.2) or in the reverse forms i=éfl. §=§a+§fl (6.3.0.3) The governing equations corresponding the above situation for all the six transition types, in terms offi, and gel are: 125 (TTl): §r_ ifln Ed 1( 5; )flll.A-l(2§d+§,)ilfr+A dt " 1-00Adr ’dt =21-0t,Adt 21-a,Adt ' (TT2): 3%: ___ _(1 (E: 50%!14 g4, _ (1"zirAigrEA-_(1‘2:::§r)§‘m. (TI’ 3): d_§r _ _(1 ’gd’grflk Ed _ (1 ”gd-grygA-_l(2§d+§l)flx+A dt - or, dt ’dt ' 2a,, dt 2 l-or,A dt ' (TT4): d“ -A dad 1 gr da-A 1" éd gr dam- l-a ..4)— ’dt 41—00925 ( 200,, )tTt (1T5): 1—a§" (TT6): d—gt Z-g, da+A dt=(1-a,A dt ’54 =1 5,, (6.3.0.4) (6.3.0.5) (6.3.0.6) (6.3.0.7) (6.3.0.8) (6.3.0.9) 126 The path-independent issue about the above symbolic description can be obtained by expression (6.3.0.2) and the previous achievement in Section 5.1. Thus Q, .54 and Q are path independent for transition type (1T1), (T15) and ('IT6), whereas only Q is path independent for (TT2). Note now, however, for (TT3), where only §, is path independent, that this is insufficient to give path independence for either Q, Q or Q A similar lack of any path independent component Q, Q and fi, occurs for (TT4). It should be noted that the above equations or their solutions corresponds to a situation that 5,, is rich. Following a continuous state path in the (a, T)-plane, the sovereign phase between 5, and a may exchange each other. If in some portions that §_ is rich, one should use another group of governing equations for the six transition types, which are modified by the following relation based on the rule (6.3.0.1). E: = 2i... 2;; = i - 1;. (6.3.0.10) 7 , APPLICATION OF THE MODEL TO AN ACTUATOR DESIGN As illustrated in the previous chapters, shape memory alloys conduct many interesting and useful behaviors under thermomechanical loads. One of these, shape memory, is pri- marily responsible for many smart device designs. Actually, the shape memory effect is an ability of a material to recover a significant inelastic deformation upon heating (Figure 30). This significant inelastic deformation is caused by either phase transformations between austenite and martensite or martensite variant reorientations. Based on this kind of deformation mechanism, functional thermomechanical devices are designed for various purposes in different aspects. In the following we are going to model the behavior of a potential reciprocal device, Two-Stroke Thermal Engine (TSTE), with the present two variant model. This contraption is designed to carry out a reciprocal movement upon ther- I ma] heating/cooling pulses controlled for example by an electrical signal. These trigger austenite martensite transformations and martensite variant reorientations in the device. 7.1 Analysis on Basic Structure of TSTE The TSTE is made of two identical SMA elements which are confined between fixed frame constraints so that the overall length is constant. We use (I) and (II) to indicate the left and right elements respectively (Figure 39). Normalize the two lengths of the two ele- 127 128 merits by characteristic lengths which are taken to be the length of the element in a state of stress-free austenite, say, 11 for the element (I) and 12 for the element (11). Thermal expan- sion effects are neglected. At any point in the operation, the strain in each element is assumed to be uniform, and is due both to elastic stress and to transformation strains asso— ciated with the phase variant state of the element, 8 = {i Q, £1 } and fin = {a}, a}, :2 }. The phase fractions in each element obey the phase balance condition (1.2.0.1). The reference configuration is obtained by the following three steps: (1) cooling both elements to a temperature To below Mfso as to transfer all austenite phase to random mar- tensite phase, (2) stretching element (I) to a fully detwinned martensite phase (positively oriented) and then releasing the force, (3) connecting the two elements by a thickless rigid thermal isolation to form a interface between the two elements. This generates a stress- free reference configuration with element (I) positively oriented and element (II) fully twinned. The initial phase fractions corresponding to the reference state are El = {0, 0, l} and g“ = (0.5, 0, 0.5}. The stress here is therefore regarded as a normal stress that is associated with normal strain caused by the interface movement due to the thermal heating/coolin g pulses. To emphasize this feature we replace {1, y, u} by {6, e, E}. Therefore, we use 0'} and on (instead of 1,,,.) to indicate the normal stresses in elements (I) and (II) respectively, and e. - (instead of 7') to indicate the corresponding transformation strain. Since there is as yet no external load to be carried, the stresses equilibrate: 61 = on = a. 129 Figure 39. Structure of the TSTE confined in a fixed frame. We will always operate the detwinned element (operating element) by applying a heat- ing/cooling pulse while the other random twin element (response element) is passively driven due to the perfect interface bounding. Heating causes the oriented martensite to transform to austenite so as to annihilates the detwinning strain, which then pulls on the other element causing it to now detwin. Cooling the transformed austenite converts it to random martensite while the other element, now detwinned, undergoes elastic relaxation. Therefore, after a heating/cooling operation on one element, the states of the two elements should exchange with each other. An ideal operation involves a perfect switch between the two elements at the end of each operation. As we will see below, elastic deformation causes a departure from the ideal behavior. The switching of the state of the two elements supplies an initial condition for the following heating/cooling pulse on the other element. Repeatedly inputting a temperature pulse to the current detwinned side will generate some kind of cyclic response. In this study we want to characterize the limiting cyclic output. A similar study has been conducted by Ivshin and Pence (l992), in which a limit cyclic response of two-way shape memory effect was simulated when a laminated sample with 130 residual stresses was considered. That study, however, only involved a one variant marten- site model, and also did not involve alternating reciprocal action. 7 .2 Deformation Consistency The temperatures T; and TH in the two elements are regarded as explicit input vari- ables, while stress 6, along with phase fractions 3 and fin, are treated as output variables. Note since § = {Q Q, Q} that there are totally seven output variables. The phase fraction balance gives two equations: 6 + 53,-1- E: = l , a] + Q: + 51,] = l . There are two evolu- tion equations governing the phase fractions 3 for the element (1), and two governing fin for the element (II). The particular evolution equations to be used for each element will depend upon the transition type in action. So far there are six equations and seven unknowns, which is not a well posed problem. The final equation comes from the con- straint condition for the movement of the perfectly bonded interface which involves all the input and output variables. This equation can be expressed as the following by setting the overall displacement away from stress-free austenite equal to the original displacement due to stress-free transformation strain in element (I), 1,{§{, 5],, :1} - {e_, a, e,} +12{§f‘, $1.61} . {e_, e, e,} = 112‘, (7.2.0.1) where 0' "' 0’ O’ ‘ e_,e ,c,} = ——e ,—,—+e . (7.2.0.2) { A {EM EA EM } The interface displacement from its original location is determined by 131 8 = 0164:1621}. {s-.e,,.e.}). (7.2.0.3) in terms of element (1), or 8 = 12{§E15 g2: :2} ° {8-1 6A5 8+} 9 ‘ (7°2°0-4) in terms of element (11). Determinations of :1 = {£1 Q, g‘, } and :11 = {6, 5,3, €11 } depend upon the state path (a, T) through transition type criteria and the evolution equations, where the temperature T as an input is given, and the stress a needs to obey equation (7.2.0.1). Therefore, the imposed procedure is fully coupled. To solve this problem an initial judgement based upon the input information has to be made for choosing the transition type or the particular evo- lution equation for those phase fractions. This process will be seen in the following sec- tion. Here, we only consider two elements with equal lengths in the stress-free austenite state, ie., I] = 12 = 1. Thus, equation (7.2.0.1) along with equation (7.2.0.2) becomes (i-e‘)(§f+§f‘)+§ —<§l+§l>+(—+e)(§1+§i‘)=e‘. (72-05) EM EM The linear piecewise envelope functions as shown in equations (5.3.0.1) and (5.3.0.2), and the Y-unfolding derived in section 2.4.2 are going to be employed in the forthcoming analysis. 132 7 .3 Heat-le Element(I): The First Stroke In this section we will discuss the stress-temperature relation during the first heating/ cooling (the first stroke) on the element (I) with a the special initial condition £1: {0, 0, l} and :11 = {0.5, 0, 0.5 }. The specialty of this initial condition is that, the operating element . which is under heating/cooling is fully detwinned, and the response element which is in a constant temperature is in an even random martensite state. It will be seen that the initial condition for the following stroke will be 5," = (A, 0, l-A} and :1: {0.5-A, 0, 0.5+A} with A > 0. There are three processes in the heating and three in the cooling, which are distin- guished by either different transformation type or the absence of a phase transformation. Actually, the major displacement of the interface between the two elements is caused in the heating portion. However, a small displacement recovery takes place in the cooling portion. 7.3.1 Heating Process We are now going to determine state paths (0', TI) and (0', Tu) on the phase diagram. The initial conditions of E = {0, 0, l} and €11 = {0.5, 0, 0.5 } give that initially A = 0. Heat- ing element (1) causes the state path (a, T1) for element (I) to increase up the T—axis from the initial point (a, T1) = (0, To =.Mf) to the point (0, A,) while the state path (6, Tu) for element (11) stays at the initial point (0, Tu) = (0, To = M,). In this process for T1 < A, there is no movement of the interface between the two elements because of the absence of phase transformations and the neglect of thermal expansion effects. Heating T1 > A, in an infinitesimal temperature range [A ,, T9] (the corresponding stress range is [0, ai]) initiates ('IT 1) in element (1), which increases the stress 0' in the two 133 elements since the active M, -> A makes element (1) shorter so as to drag the interface toward the left (M _ -> A is inactive since gf = 0 ). An analytical solution for T1 and a can be found for this process segment based on the length constraint (7.2.0.5) and the phase evolution equations ((4.2.2.3), (4.2.2.4)) of (IT 1) for the phase fraction i1 (note the ele- ment (1]) is in elastic with a constant temperature To so that fin = {0.5, 0, 0.5}): 25.101 f-A,)o r, = A,+k,o+k202+ , . (7.3.1.1) EAEMe +(EA-EM)0 Since the first derivative of (7.3.1.1) with respect to stress 0 JT 25215 e‘ A -A 5' = t,+2k,o+ A M ( i ’) (7.3.1.2) e 2 [EAEMe + (E, -EM)o] satisfies the ('I'Tl) criteria (3.2.1.5) for do > 0 in the whole domain [0, 0‘], further heating T1 beyond this infinitesimal temperature range [A ,, Tli] continues to trigger (TT 1) in ele- ment (1), so that expression (7.3.1.1) tracks the response 0 until either the process goes to completion or the stress 0’ reaches the detwinning flow 6,. 134 o * Figure 40. The path segment p52 has three possibilities: I, 2, 3, as shown in the above. Path 3 is the desired situation and approachable for many shape memory alloys, which will be discussed in the following. ' The stress 0 inside the two elements increases, and may or may not reach the detwin- ning flow stress a, before the completion of M, —> A in element (I). This implies that continued T1 increase will cause the state path (a, T1) to follow a path of the form of either path 1, 2 or 3 as shown in Figure 40. Simultaneously, the state path (a, T n) is horizontal in the (a, ”plane since the temperature Tn is held fixed in element (II). Useful device response requires a situation where the M, —) A process in element (I) is sufficient to cause detwinning in element (11). This will happen if the elastic strains are insufficient to relieve the original transformation strain, that is if the following requirement is satisfied e. > a; + 2;, , (7.3.1.3) where, 3 _ as S _ as e, - EA,EM - 511' (7.3.1.4) The physical interpretation of expressions (7.3.1.3) and (7.3.1.4) can be viewed in Fig- ure 41. Generally, the TlNi allay gives 8‘ » 81+ 8;, (Hou and Grumman, 1995), so that continuous heating to trigger process M, —> A will cause the stress 0’ to surpass a, so as to initiate detwinning in element (II). Figure 41. A physical interpretation for the condition that ensures that detwinning occurs during heating in a stroke. Condition (7.3.1.3) indicates that the state path (a, TI) follows a path of the farm of A path 3. Continued heating with T1 > T2 now triggers detwinning in element (11). Since the 136 detwinning provides a softening process, the motion of the interface between the two ele- ments will dramatically increase with this T1> T2 increase, which in turn, causes the state path (a, T1) to veer away from the nuetrality curve A ,, towards the nuetrality curve Af, which it encounters at point (03, T3). This process will be derived in the following discus- sion. The phase evolution equations ((4.2.2.3), (4.2.2.4)) for the phase fraction £1 under (TI‘2), the phase evolution equation (4.2.2.4) for the phase fraction fin under (ITS), along with the length constraint (7.2.0.5) gives the following state path of a and T1 for a, T3 results in the path (a, T1) becoming ver- tical due to stress that is maintained at 0'3 and the element (11) state path (0', Tu) would remain stalled at its endpoint (63, To). Hence any heating beyond T3 does not contribute any additional device actuation. It is also important to note that A“ is unchanged during the subsequent cooling process of the element (I). 7.3.2 Cooling Process _ When cooling T; from T3 the state path (a, T1) starts as a vertical line involving a = 03 since initially there are no transformations to relieve the stress. This persists until the ver- tical element (I) stroke path encounters the nuetrality curve M ,, at point (04, T 4) where 04 = 63. Now further cooling will trigger A -> M, in element (I). This also gives elastic relaxation in element (II) in a infinitesimal temperature range [T4, T11] (the corresponding stress range: [04, 01]). The relation between T1 and a can be found by use of the constraint equation (7.2.0.5) and phase evolution equation (4.3.2.3) for £1 (note that phase fraction §n is unchanged in the form of :11 = {A0 , 0, l-Aa } in this infinitesimal range), 138 ‘ 2 25 o-l-E E e 1-2A TI=Mf+k1o+k2o2+(M,—Mf)[ " A M ( ‘9]. (7.3.2.1) EAEMe‘ + (EA — EM)o Since the first derivative of expression (7.3.2.1) with respect to o ”1 (7.3.2.2) e 3 2(M,-Mf) E ,EMe‘us, + 5,, + 2(5, -EM)Aa] satisfies the (T12) criteria (3.2.1.6) for do < 0 in the whale domain [0, oi], further cooling TI beyond this infinitesimal temperature range [T4, T11] continues to trigger (TT2) (A -> M ,) in element (1). This indicates that the expression (7.3.2.1) applies to the cool- ing procedure until the state path (o, TI) reaches the nuetrality curve M ,, at the point (o5, T5). Further cooling of temperature TI below T5 triggers (TT2) in element (1) involving both A -) M, and A —) M_ processes. Substituting phase fraction expressions (5.1.2.3) and (5.1.2.4) for (TT2) into (7.2.0.5) gives } 2's «his . when _ L. 321- . ad‘s-3:1 ..., . a! 03 57, do a! Go 37, do o, M» A- 2 I II (E +5 )0 (E -E )0 l 1 [l-AQ- 5" A M. ' A Ms (l‘Fs aA+aA-) EAEMe EAEMe , (7.3.2.3) 139 At; = (tbs-(53),, II 5 (abs-($55" = (€474- (é- 1).: = 1-2Aas “1+ = “A,_(O, T1) = (B4165 T])-Mf)/(MS_Mf)’ 01;, = aA,(o, T1) = (BIO: Tr) ’ M 1V (M s ' M!) ’ F5 '3 JaA+(05, T5)/J(IA,(O'5, T5)aA_(65, T5) = l (GA-‘65, T5) = 1 ). Taking the first derivative ta (7.3.2.3) with respect to stress 6 generates ‘17 l r P(o,1‘,)%I = Q(o, T,)+cl -2C2,/aA,aA_, (7.3.2.4) where E -E C1 = 4 t’ C2 = A ‘1’ FSEMe EAEMe P T (1;, 3&1, aA+aaA— C alA- B(1,“... “Ara“;- (0'1" 5'37, Tar-7+2“ T37, “Tim: A+ “A- A- I I ' ' I I l 1 Q“, T) = 2533‘“. - a—AiaaM- C o “_Alaa‘“ + Eta—a” “A- aA‘F (1A, (1A- The state path (o, T1) in this portion is governed by the ODE (7.3.2.4) and the correspond- ing initial condition. T‘la = T,,. The path segment starts at (o, T) = (o5, T5). An '0 05 important issue is the stress associated with cool-down to T1 = M,; In the next section it is shown that o = 0 at T1 = Mfso that this segment p56 has the form shown in Figure 42. 140 Figure 42. Six (0, T1)-path segments and their connecting points related with the first stroke (heating/cooling element (1)) are schematically presented in the phase diagram for o > 0. 7.3.3 An Uniqueness Point for the Solution of the Equation (7 3.2.4) In this section we will focus on the state path p56 as governed by equation (7.3.2.4). It is shown that this path concludes at the point (0, Mf) with o decreasing monotonically from the initial value o = o5. The demonstration holds for the linear envelope functions (5.3.0.1) and (5.3.0.2) and the constitutive functions (4.2.1.1) to (4.2.1.4), and the Y- unfolding derived in section 2.4.2. To show that the solution is monotone, (7 3.2.4) is reformed as do PW. Tl) , (7.3.3.1) 141 the above P and Q functions are 1+Czo B(o, Tr)‘ Mf+ Czo- l B(o, T1)-M, P(6’T1)’M -M +,M -M x f B(o, T,)— M, f B(o, T,)- M, (1+Czo)(k1+2kzo) B(o, TI)- Mf+ k,(Czo- 1) B(o, T1)— M, Q(0.T1)= M -M M -M f 8(6 7”,)- M, f B(o, rp— M, Since it is always true that B(o, Tl) > B+(o, Tr) in the triangle zone enclosed by the T- axis, Mf, and M,,, one has P(o, T1)) 0 for all o > 0. On the other hand, in this triangle zone it is also true that o < 1/C2 = (EAEMe‘yw, - EM) which implies Czo — l < 0. Note that k; < 0 , kl > 0 and k2 > 0 . Therefore, one has Q(o, TI) > 0 for all o > 0. Finally, since as 701,01}, 51, one has C1—2C2,/atk,at;- > 0 (F5: 1). Taken together, it is concluded that the solution of equation (7.3.2.4) (or (7.3.3.1)) is monotone. The solution curve can not penetrate the terminal nuetrality curve M,,. To verify this, assume that the solution curve 71(0) goes toward M,,. It is seen that there is a singularity for the solution of (7.3.2.4) on the terminal nuetrality curve Mf, since «3,, M = 0 . The f+ . . th. llnutof— is do . th - llm (—) = k1 + 2kzo , (7.3.3.2) 0*(0, 1,)» M, d which is equal to the slope of the terminal nuetrality curve M,,. This indicates that the path ps6 can never penetrate M,,. Finally, the path p56 T1(o) definitely goes through the paint (0, M,). To prove this, assume that there is a limit of the solution TI(o): 142 lim T,(o) = L. (7.3.3.3) o—)O+ This limit exists because T1(o) is monotone. If L > M , then the limit of the right hand side of (7.3.2.4) is 0'1“ ' lim P(o, T,)—' = 0, (7.3.3.4) o —) 0+ do which is not equal to the limit of the left hand side of (7.3.2.4): lim [Q(o, 1,) + C1-2C2 a},af,,] = ‘ + C1¢0. (7.3.3.5) 040+ M,;—MI Thus, it must be L = M , which implies that the solution T1(o) definitely goes through the point (0, M,). The proof is thus complete. The above proven conclusion illustrates that the stress between the two elements is erased at the end of the cooling (or when T I reaches the temperature M,). This shows that the elastic deformation of the element (II) of a martensite state upon heating the element (1) can be recovered by subsequently cooling the element (I) to the temperature Mf Since during the whole cooling process the phase fraction of the element (11) remains constant: :11 = {A0 , 0, l-Aa }, the phase fraction of the element (1) could be determined as E = {0.5- ‘ A, , 0, 0.5+Aa } by means of the consistency requirement (7.2.0.5). 7.4 Later Heat-le Strokes The first stroke, as discussed above, is special, since it is the only stroke starting from a fully detwinned state of the element (I) and random martensite state of element (II): :1 = 143 {0, 0, l} and fin = {0.5, 0, 0.5 }. All other strokes involving heating/cooling of the element (1) will start from :1: {A, 0, l-A} amd :11 = {0.5-A, 0, 0.5+A} for some positive A. Simi- larly, all strokes involving heating/cooling of element (11) will start from £1 = {0.5-A, 0, 0.5+A} and :11 = {A, 0, l-A} for some positive A. Let the corresponding sequence in A so generated be: A], A2, A3, A,,... where A1 = A, and add subscripts correspond to end-of- stroke states after heating/cooling of element (1) and even subscripts correspond to end-of- stroke states after heating/cooling of element (II). The analysis of any one of those strokes, taking A,- ta A,,, can be treated in a similar fashion to the initial stroke involving the six path segments: p51, p52, ps3, ps4, ps5, p56, in the particular element that undergoes heating/cooling. Each such segment will involve dif- ferent phase fraction values that lead off each of the segments. Otherwise, the only major qualitative difference involves the analysis of the segment ps2 (ps1 is trivial transformation during heating process so that it is a vertical line along Tlaxis from (0, T0=Mf) to (0, A,)). On segment psz, the initial value é, = Ai in the active element now causes a M _ -) A trans- formation in addition to the M, -) A transformation; this alters the previous description of the p82 segment given in (7.3.1.1). In particular, this path segment is no longer straight even if the elastic moduli of austenite and martensite are equal. This dual transformation in the heated element continues until the path encounters the terminal nuetrality curve A,,. Certain phase diagrams might allow detwinning in the response element before comple- tion of M _ —-> A in the active element. This corresponds to materials that have the follow- ing relation between material parameters A,+k',o, >A,+klo,+ kzof, (7.4.0.1) ie., the temperature on nuetrality curve Af, is larger than that an nuetrality curve A,, at the 144 same stress level o = 68 for o > 0. With the material parameters used here, an opposite condition of (7.4.0.1) is found, so that M _ -> A completes in the active element before detwinning occurs in the response element. Thus, in the simulation results presented below, we find that M _ -> A occurs on the segment p52, which is consequently split into two subpaths ps2a and p52,, The latter path ps2b involves only M, —> A transformation in the heated element, and so is described by the same equations that were used for the treat- ment of the primary ps2 path for the active element (see equation (7.3.1.1)).All subse- quent details are qualitatively the same. Especially, full austentization of the heated element occurs before full detwinning of the constant temperature element, thus giving A,,] > 0. 7.5 Limit Analysis on Thermal Cycles A simulation involving concrete parameters will be conducted in this section. The complication of starting states with nonzero A, and different moduli of austenite and mar- tensite prompt us to consider a fully numerical treatment. The convergence of A after sev- eral thermal strokes applied on the two elements is reached for different values of the austenite and martensite moduli. This in turn generates a stable path-loop in the (t, T)- plane and the (6, T)—plane, where 8, the displacement, is given by either (7.2.0.3) or (7.2.0.4). 7.5.1 Numerical Convergence on Residual Phase Fraction Linear envelope functions: (5.3.0.1) and (5.3.0.2), and the Y—unfolding are taking into 145 account for the present simulation. The pararneters employed here are from the TABLE 2 except for the moduli. The temperature pulse, as an input applied an the two elements, is illustrated in Figure 43 where T,, = M, The ultimate temperature must be sufficient to fully austenitize the operating element under the constraint stresses generated by the process. These ultimate temperatures are different in each stroke but tend to an asymptote. The val- ues of A corresponding to end-af-stroke states for different combination selections of the austenite and martensite moduli are given in TABLE 9 to TABLE 12. These A values can be obtained without considering the state path of full cycles or full strokes. They are deter- mined by the canstraint equation (7.2.0.5) and the fact that the phase fraction of the response element remains unchanged during cooling processes. This is completed by a sample Mathematica program. By doing this, first, general heating state paths are updated by the value of A corresponding to the previous stroke. Second, the intersection point between the updated heating state path and nuetrality curve Af, is found to determine the phase fractions at this point. Finally, by use of the constraint condition (7.2.0.5) to con- clude the A at the end of the current stroke. It is seen in all cases that the values of A rap- idly converge to a stable value A”. 146 TI element ‘ (I) ‘ element (11) stroke 2 stroke 1 stroke 3 stroke 4 , Ar I A! " . A! ' > t b t 0 t1 t2 t3 0 t1 t2 t3 Figure 43. Temperature pulses applied on the two elements. The maximum temperature in heating must be sufficient to fully austenitize the operating element. The following TABLES show that the limit values A,, tend to decrease with an increase of the moduli. This corresponds to a condition that the response element experi- ences greater detwinning because an increased stiffness gives a relatively small elastic deformation so that more of the deformation is related with the phase transformations. Based on the conditions of TABLE 9 and TABLE 12 both of which have the same sum of the moduli of austenite and martensite, it is also observed that the equal moduli situation gives more detwinning for the response element than the uneven moduli situation. Since A is nonzero, the device as described here has not recovered the prevailing state at the very beginning of the operation: :1: {0, 0, 1}, g" = (0.5, 0, 0.5}. Hence the response is not a true “cycle” in the first couple strokes. Continued operation will always involve end-of—stroke states involving same A remnant that is not fully detwinned on the nominally oriented element. Repeatable or true cyclic behavior only occurs if the same A 147 remnant occurs at the end of each two-stroke cycle. In fact, for symmetric device- described here (11 = 12 = I), the same A remnant must occur at the end of every stroke, albeit interchanged between the two elements. The question thus arises as to whether or not there is such a stable cycle, ie. an A, which gives cyclic behavior. Here smallness of A is a measure of the closeness to ideal behavior in terms of stroke distance. That is A,, = 0 gives ideal behavior (maximum stroke), while, at the other extreme A,, = 1/4 gives 8 = §n = {0.25, 0, 0.75} at the end of each stroke, corresponding to zero interface movement. This point can be viewed in the following discussion. To clearly see the problem, we consider the displacement at the end of each stroke. With (7.2.0.3) and (7.2.0.4) employed here, the displacements at the end of each stroke with respect to the reference position are found as the following, 50 = 0: 51 = Kl-ZAOE‘I; 82 = ZIAZE‘; 1(1—2A,)e' if i is odd 21A,.e' if i is even Then the effective stroke distances between two connected strokes are 148 sdl = [81-60] = I(l-2A,)e’; sd2 |82-6,| = 1271—2112—2731); sd ,. [6,-6,_,| = le‘(1-2A,—2A,_,); When enough temperature pulses are operated on the two elements (i —> oo ), our numeri- cal results show that lim A, = .lim A,_l = A... Therefore, the limiting effective stroke 4 -) ’0 l -) 0° distance is obtained as sd, = zc'(l — 4A,). (7.5.1.1) Thus, A,. = 0 represents a maximum stroke situation that the prevailing state: 3 = {0, O, l} and El = {0.5, 0, 0.5} can be recovered at the end of each stroke, while A,, = 1/4 indicates a useless situation. A measure of the ultimate stroke quality is thus Q = l—4A”. Note that Q = 0.5398, 0.6206, 0.6574, 0.5865 for the cases given in TABLE 9 to TABLE 12 respectively. 149 TABLE 9. [IA = 4.0*1004; itM = 2.0*10I~4 MPa operating A of response elements at ultimate temperatures elements strokes end-of-stroke (”K) (I) I 1 0.1171521977252063 335.123647800 (II) I 2 0.1 150009266592965 334.622190725 (I) I 3 0.1150510309196698 334.633789633 (II) I 4 0.1150498696891485 334.633520813 (I) I 5 0.1 150498966052308 334633527044 (II) I 6 0.1 150498959813465 334.633526900 (I) I 7 0.1 150498959958073 334.633526900 (II) I 8 0.1 150498959954724 334.633526900 (I) I 9 0.1 150498959954803 334.633526900 (II) I 10 0.1 150498959954798 334.633526900 (I) I 1 l 0.1 150498959954798 334633526900 limit I on A,_ a: 0.1 150499 334.633526900 150 TABLE 10. 11A = 4.0*10"4; int = 2.8*10"4 MPa ‘ operating A of response elements at ultimate temperatures elements strokes end-of-stroke (°K) (I) I l 0.095965644548003 335.433784391 (II) I 2 0.094826994366186 335.117197237 (I) I 3 0.094843410530481 335.121891685 (II) I 4 0.094843174457726 335.121824177 (I) I '5 0.094843177852696 335.121825147 (II) I 6 0.094843177803873 335.121825133 (I) I 7 0.094843177804575 335.121825134 (II) I 8 0.094843177804565 335.121825134 (I) I 9 0.094843177804565 335.121825134 limit I no A,, z 0.09484318 335. 121825134 151 TABLE 11. [IA = 4.0*10"4; um = 3.451004 MPa operating A of response elements at ultimate temperatures elements strokes end-of-stroke (”K) (I) I 1 0.086460055018001 335.588237493 (II) I 2 0.085641 127661279 335327082388 (1) I 3 0.085650372005167 335330032150 (II) I 4 0.085650267840977 335329998912 (1) I 5 0.085650269014702 335329999287 (II) I 6 0.085650269001486 335329999283 (I) I . 7 0.085650269001635 335329999283 (11) I 8 0.085650269001633 335329999283 (1) I 9 0.085650269001633 335.329999283 limit I no A,, z 0.08565027 335329999283 152 TABLE 12. 11A = 3.031004; (1,, = 3.0*10"4 MPa operating A of response elements at ultimate temperatures elements strokes end-of-stroke (° K) (I) I 1 0.104853194379462 335307846732 (II) I 2 0.103344420472474 334914231932 (1) I 3 0.103371280735037 334921227214 (II) I 4 0.103370804176779 334.921103103 (I) I 5 0. 103370812632450 334.921 105305 (II) I 6 0. 103370812482419 334.921 105266 (I) I 7 0.103370812485081 334.921105266 (II) I 8 0. 103370812485033 334.921 105266 (I) I 9 0.103370812485035 334.921105266 (II) I 10 0.103370812485035 334.921105266 limit I oo A,, = 0. 10337081 334.921 105266 153 7.5.2 Stable Response Loops We now consider more detailed numerical simulations of (o, D—paths for a particular case involving equal moduli. The parameters to be used in the following simulation are all listed in TABLE 2. Following the previous analysis of Sections 7.3 and 7.4, the (o, T)- curves corresponding to the first stroke, are presented (by lighter dots) for both the ele- ments in Figure 44. The (o, T).curves for the consequent stroke are given (by darker dots) in Figure 45. Note that the (o, T1)-curves for the response element is always a straight back and forth horizontal line. The difference between the first and the second strokes is that both M_ -> A and M, —) A are involved in the p52 process of the second stroke, while only M, -) A is involved in the ps2 process of the first stroke (refer to Figure 42), since the initial condition of the first stroke involved é, = 0 in the operating element. A combined view of (o, T1)-and (o, Tn)-curves (the lighter dot is the (o, TI)-curve for the first stroke) corresponding to the operating elements under ten cycles are shown in Fig- ure 46. The moduli here are: it, = 3.0 x 104, It” = 3.0 x 104 MPa, so that the simula- tion is that associated with TABLE 12. It is to be noted that, after the first stroke, the various curves are indistinguishable (the darker dot curves), which indicates that the limit cycle corresponding to A,, has been reached. These state paths are generated based on the analysis in sections 7.3 and 7.4. The intercomparable (8, T)- and (5, o)-curves are illus- trated in Figure 47 (a) and (b), which are obtained by use of equations (7.2.0.3) and (7.2.0.4), as well as the derived (o, T) state paths. The lighter dot curves are again the dis- placement against temperature and stress, corresponding to the first stroke, while the darker dot curves correspond to the remaining nine strokes. The effective stroke distances can be found from expression (7.5.1.1) and A,, listed in the above TABLES. For example, 154 the effective stroke distance sd, = 0.5865 Is for this particular case. 3.1 T1(°K) ...1. To (°K) 320. ............. i 2 320. 240 300’ 300. ; 200 .‘ . . - 280. 240 ’ 0 5‘0 100 1 0 200 0 Figure 44. (o, T)-paths for the first stroke for element (1) (a) and for element (II) (b). The operating element here is element (I) with initial temperatme T0 = M,, which is equal to the constant temperature of the response element (II) during this process. Here the material properties come from TABLE 2. In particular the moduli are 11A = 11,, = 3.0x 104 MPa. The value of A at the beginning is A = 0, and at the end is A = 0.104853194379462. 320> 340$ Tn p52 3403 TI b . ps2, : 320- 300> 300» m» 5 280 >‘ ---- 280» 26° c5 2601 2403' 240. o 50 100 110 200 ’ 0 (a) 0' (MPa) 0» “(MW Figure 45. (o, T)—curves in the second stroke are presented in (a) for the element (II) and in (b) for the element (I). The operating element is element ([1) with initial temperature To = M,, which is equal to the constant temperature of element (I) during this process. In the first part of ps2 bath M_ -) A and M, -) A are involved in the phase transformation (refer to Figure 42).The value of A at the beginning is A = 0.104853194379462, and at the end is A = 0. 103344420472474. 155 ‘33 8 8 'c c' 8 Figure 46. This graph shows (0, T)-curves of the operating elements in ten strokes. A stable (o, T)—loop (the darker one) is reached after the first cycle (the lighter one). o I L — E01: 8/1 0.OSLOC~.‘~ :e—‘ -v- o—og‘..— —— _— 0.04» ' 0. 03’ effective stroke distance 0.02» (LIWC—T a. 0—0—0—0- e—e‘...—...— _- 0.01 - 240 260 200 300 (a) ' Figure 47. The corresponding displacement against stress (a) and temperature (b) after ten strokes. The lighter dots correspond to the first stroke. The darker dots correspond to the following nine strokes, which are indistinguishable at this scale. The head down loops correspond to operating element (11) while the head up loops correspond to operating element (II) in both (a) and (b). 1 56 7 .6 Discussion In principle, the deformation caused by thermal expansion is against that caused by phase transformation. The thermal expansion coefficients of austenite and martensite can be taken as 01,, = 11 x 10'6 ll°K and a.” = 6.6 x 10’6 ll°K respectively. To gauge its effect, we choose the bigger one a = 11 x 10'6 l/K° a for both austenite and martensite. The temperature change to actuate a stroke is likely to be on the order of 100 °K based on the above simulation. Then the thermal expansion strain is about 0.0011 « e. = 0.06 . Thus, we conclude that a reasonable approximation allows the thermal-expansion effect to be neglected. Based on the previous achievement, the final cyclic behavior corresponds to a repeat- able limit cycle. Such limit cycles arise naturally in earlier models of SMA behavior given by Ivshin and Pence (1992). Here a limit cycle means that cyclic input, such as repeated alternating temperature pulse excursions TI(t) and Tn(t), generates cyclic output response, here film and §n(t) and stroke 8. For a given cyclic input, the associated limit cycle or cycles (if any exist) may be sought in two general ways. The first involves starting from some initial condition of the output response, here 8(0) and 511(0), followed by perform- ing a large number of input cycles. Limit cycles then arise as the possible large time con- vergence of the functions §I(t) and 90) to a repeatable cycle with the same cyclic ' frequency. Here initial conditions of the output response that happen to be on a limit cycle will generate output response that stays on the cycle, and, ideally, those initial conditions not on a limit cycle will give output response that drifts towards a nearby limit cycle. This was the method pursued above, where the limit cycle was essentially attained on the sec- ond stroke (Flgure 44 to Figure 47). The other general method for determining limit cycles 157 involves setting up a mapping between output response variables sampled at the cyclic fre- quency and seeking fixed points of the mapping. Such fixed points give initial conditions that are on a limit cycle. Although Ivshin and Pence were able to pursue such a direct fixed point treatment in some of their previous studies (Ivshin and Pence, 1992), the more com- plicated nature of the problem under study here (involving more than one martensitic vari- ant, and more than one shape memory element) points to the usefulness of obtaining these limit cycles by numerical simulation procedures. 8 CONCLUSION AND PROPOSED FUTURE WORK The approach is to augment conventional continuum mechanical descriptions with internal variables that track fractional portioning of the material between austenite and the various martensite variants. A three-species model involving austenite and two comple- mentary martensite variants provides sufficient generality to capture the martensite variant distributions that underlie shape memory, and the strain-accommodation associated with pseudoelasticity. Transformations between all of these species can be tracked on the basis of triggering algorithms that reflect both transformation hysteresis and the variations of phase fractions of triggering both stress and temperature. Three phase diagrams are presented based on thermodynamic considerations and experimental measurements. The X-unfolding, as a prototype, is first derived from allow- ing phase transformation nuetrality curves to enter non-austenite areas to describe the detwinning processes. The detwinning in this unfolding strongly depends upon both stress and temperature. Further modification on the X-unfolding, about the entropy of austenite replaced by that of martensite in the non-austenite areas, provides the pY-unfolding. An interesting aspect of this phase diagram is that stresses associated with detwinning start and finish are determined in terms of the other material parameters. Basically, this pY- unfolding requires only the following experimentally determined parameters: the four phase transformation temperatures, the transformation strain, the Young’s moduli of auste- nite and martensite, and the transformation latent heat. Themore sophisticated phase dia- 158 159 gram, Y-unfolding, is generated by specifying the detwinning start and finish stresses independently based on experimental observations. The corresponding nuetrality curves of phase transformations are consequently modified following this change. The comparison of isothermal behaviors, between the theoretical prediction with the Y-unfolding and the experimental measurements on thin films by Hou and Grumman, shows that the model can well reflect the practical situation in some degrees (Grumman and Pence, 1996). Discussions of the path-independent conditions in Chapter 5 reveal some fundamental features of the two variant algorithms of the model. These features are responsible for the hysteresis characteristics of shape memory alloys. The special behavior, discussed in Chapter 6, according to loading/cooling and cooling/loading state paths shows potential for the model to direct designing smart sensors. Pseudoelasticity, shape memory effects and many isothermal behaviors are predicted in Chapter 6. The application of the model in Chapter 7 shows two aspects. One is that the model possesses both theoretical completeness in certain levels and potentiality to guide sample engineering designs. The other is the potential engineering application of the two element device (TSTE). A stable cyclic behavior is reached by performing the cyclic temperature pulse on the device. Further investigation associated with carrying external loads can fol- low up based on the present approach. The interesting inverse problem investigated in the Appendix reflects another view on the model itself. Relations between stress, strain, temperature and entropy for each species are backed out of a three species phase diagram as given by experiment. We believe that there are still certain amounts of work to be done on the thermodynamic issue related with the choice of the particular entropy form of each individual phase. An example shows that there is a consistency between the approach in Chapter 2 and the inverse issue in the 160 Appendix under the same phase diagram. So far in this study, we only focus on l-D isothermal descriptions and some sample behaviors associated with both stress and temperature changes, in which stress and tem- perature are treated as an explicit input. More complicated situations, such as convective and adiabatic processes, can be extended from the present approach by adding certain thermodynamic considerations (Ivshin and Pence, 1994 b). In general, for the convective and adiabatic condition, often the case is that only either stress or temperature, but not both, are given as input and their relation is decided by extra thermodynamic equations (e.g. joule heating, convection, physical constraint). For more general modeling work, TiN i alloys considered usually have either 24 habit plane martensite variants plus austenite, or 12 coherent martensite variants plus austenite, which are distributed in 3-D scopes. Fellows as Patoor et al (1994), Son and Hwang (1993), Boyd and Lagoudas (1994, 1996) contributed certain amount of work on the 3-D modeling work in different aspects. For instance, Patoor et al have developed a constitu- tive model to describe the transformation among all 24 martensite variants and austenite phase based on free energy issue. The model was set up first for the single crystal and then extended to the description of polycrystalline transformation by considering the self-con- sistent micromechanics method. Son and Hwang have acquired a thermodynamic model for both pseudoelasticity and shape memory effects. The transformation criteria of the model is similar to plastic yielding’s in plasticity theories. Later, Boyd and Lagoudas have also developed a microscope constitutive model to phenomenologically narrate the behav- ior of pseudoelasticity and shape memory effects for polycrystalline shape memory alloys. The model using a free energy function and a dissipation potential contains three descrip- tions based on the combination number of internal state variables. 161 Another challenge future work conjunct with the present study is how to extend the present model to the more general case with multiple martensite variants and austenite in 3-D. This extension will include modifying the overall entropy and strain expressions (2.1.0.1) and (2.1.0.2) respectively based on the compatibility theory (Bhattacharya and Kohn, 1996). The corresponding 3-D criteria for phase transformations and reorientations is another open question. The potential way to solve the problem is based on the relation between 3-D stress distributions and orientations of various lattice structures of austenite and martensite variants. 9 REFERENCE 1 Apostol, T., Calculus, Blaisdell Mathematics Series.‘ New York, London. First Edition 1962. 2 Achenbach, M. and I. Muller, Simulations of Material Behavior of Alloys with Shape Memory, Arch. Mech. 37(6), pp. 573-585(1985). 3 Achenbach, M., T. Atanackovic and I. Muller, A Model for Memory Alloys in Plane Strain, Int. J. Solids Structures Vol. 22, No. 2, pp. 171-193(1986). 4 Achenbach, M., A Model for an Alloy with Shape Memory, Int. J. Plast. 5, pp. 371- 395(1989). 5 Banks, R. and O. Weres. 1976. 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Grumman, Model Tracking of Stress and Temperature- Induced Martensitic Transformations for Assessing Superelastic and Shape Memory Actuation, to appear in Symposium of Adv. in Mats for Smart System, Dec. 1996 (Bos- ton). 10 APPENDIX REFINEMENT OF THE TWO VARIANT PHASE DIAGRAM As discussed in the chapter 2, under certain assumptions of entropy and strain of the pure phase species, phase diagrams have been obtained based on thermodynamic consid- erations. The Gibbs free energy is treated as though a state function of temperature and stress in either an individual phase or a mixture phase, by which Clausius-Clapeyron rela- tions are generated to describe transformations between any two species and further to form phase diagrams. The phase diagram gained by this procedure is usually not exactly the same with experimental observations, specifically, in a detwinning process. Thus, the difference between theoretical derivations and experimental measurements supplies a motivation to consider a reverse problem. That is, if a phase diagram is given, it is very interesting to find the distribution forms of strain and entropy of the pure phases that pro- vide the same thermodynamic consistency requirement. This problem is considered here. 167 168 Figure 48. A given phase diagram as might be determined by experimental measurements. R1 is the region above nuetralitycurve M,, for 't 20;R2istheregion between M,, and M,_ for 1:>0; R3 is the region between M,. and T-axis for 1:20.fi, R2andR3-are corresponding mirror images of R1, R2 and R3 about Tlaxis for 1' < 0 . 10.1 Clausius-Clapeyron Relations If a phase diagram is given, shown as Figure 48, the slopes of each neutrality curves are determined for the processes of martensitic transformations and martensite variant reorientations. For convenience, three regions, R1, R2 and R3, are defined in the half- plane 1:20,in whichA HM, and A HM_ occurian and M_—>M, occursinR2and R3. The mirror images of R1, R2 and R3 about t = 0 axis are R1, R7 and R3.. It is sup- posed that the Gibbs free energy is a state function of temperature and stress in both the individual phases and in any mixture state. Thus, the following Clausius-Clapeyron rela- tions have to be satisfied in the half-plane 12 0 if one notes (2.1.0.7), (2.1.0.8) and 169 (2.3.1.1), TIA-11, = —(yA-y,)/f,(‘t) forAHM, ian, (10.1.0.1) TIA-TL = -('yA-‘y,)/f2(1:, T) for AHM_ ian, (10.1.0.2) 11,-n_ = —('y,-y_)/f3(t, T) for M_->M', inRZand R3. (10.1.0.3) Here, fl(1:) , f 2(1, T) and f 3(1, T) are the slopes of each neutrality curve for A 4—) M ,, A H M _ and M _ -> M, respectively, which are regarded as experimentally supplied. Similarly, in the corresponding mirror image regions in the half-plane t < 0 , R1, R2 and R3, following Clausius-Clapeyron relations have to be obeyed, 111-11. = -(yA—'y_)/f1(1:) forAHM_ in iii, (10.1.0.4) nA—n, = -(yA-y,)/f2('t, 7) for AHM, inR_l, (10.1.0.5) 1),-n- = -(y,—y,)/f‘3(t, 7) forM,—>M_ iniz’and f3. (10.1.0.6) Here, 171(1) , f 2(1, T) and f 3(1, T) are the slopes of each neutrality curve for A (-) M- , A (—-) M, and M, -> M _ respectively for r < 0. The mirror image status of '18, E andfi that will hold in the event of symmetry requires the following relations, which are hence- f0rth assumed: f .(t) = -f 1(4). (10.1.0.7) f 2(1. 1) = -f2(-Ts 1). (10.1.0.8) 170 no.1) = -f'3(—t. 7). (10.1.0.9) In this more general phase diagram, the entropy and strain may each depend on bath temperature and stress. The entropies are written as the following in the present study (Ivshin and Pence, 1994 b; Bekker and Brinson, 1994), 1110.77 = Cln(-TT-)+ni(t.1) . (10.1.0.10) 1141.7") = Cln(T£)+nZ(r,1), (10.1.0.11) 7111.1) = ClnITl)+nf’(t.1) . ' (10.1.0.12) Here, the stress is appended in the reference terms of entrapy expressions. Since the heat capacity is defined as T% , it follows that the heat capacities in the three phases are given 0 by C + Tit—2' , so that if I]? is independent of temperature then C is interpreted as actual heat capacity. The mathematical Maxwell relations of Gibbs free energy being a state function in each pure phase gives the following 311,4_3Y,4 3n,__3¥, 3'1-_37- 3; -37, I)? -5,, a? -5,, (10.1.0.13) Assume that the stress-strain relation for austenite satisfies Hooke’s law in all the cases, ie., 171 “(A = 1. (10.1.0.14) ”A which implies that 11; depends upon only temperature based on the Maxwell relation (10.1.0.13). We presume that '1]; is constant for the rest of this Appendix. 10.2 Determination of Strain in A <—> M, Process In R1, equations (10.1.0.1) and (10.1.0.13) give 3 1 a diiflm 0 0201 a—I-‘(YA-Y-r)+mfi(YA-Y+)-—%(YA-Y+) - . (1 ° ' ° ) for A (-—> M, process. The characteristic equation of (10.2.0.1) is 47' _ 2.1. _ flu), (10.2.0.2) Assume that (10.2.0.2) is integrable and its integration form is 111(1, T) = C, (10.2.0.3) with property mm, C) = C, M ,S C SA ,. To solve equation (10.2.0.1) the following boundary condition on 1: = 0 has to be posed for 7,, and y, , YA |T = 0 = 72¢) and 7,], = 0 = 73%) . (10.2.0.4) for M, s g s A ,. Expression (10.1.0.14) implies that 7m) = 0. Therefore, the solution of (10.2.0.1) with boundary condition (10.2.0.4) is found as 172 l +f1(1)o “(I’m 11,, ITO) + ”(HA T» 5 (10.2.0.5) the superscript RI on the right head of 7, indicates the solution in region R1. From ( 10.1.0.1), (10.1.0.10), (10.2.0.5) and (10.1.0.11), the following relation about the entropies is derived, . _ 73mm. 1)) trim, 7) = fr (0) . (10.2.0.6) For A H M - process in RI, combining (10.1.0.4) and (10.1.0.13) follows a %f1(1) 370..- r-)+ WWW-”(1,4, ) (y, -7-) = o , (10.2.0.7) with its characteristic equation 51: flat). (10.2.0.8) Then, the following results are determined in a similar fashion (10.2.0.5) and (10.2.0.6), rffi(r.1)= —+£L(—)‘Yo(fil(1. 7)) (10.2.0.9) 11A fl . . rf’lfira. 7)) -(t, T) = — - , (10.2.0.10) Tl “A f 1(0) if the boundary condition on 1: = 0 is given by 173 7-| = 72¢). (10.2.0.11) 1:0 Here, I-Il(‘t, T) = l; for M f S C S A f is the integration form of (10.2.0.8) with property fil“), C) = C5 For the problem associated with two complementary martensite variants, it is natural to impose the following relation, 713C) = -rf’(§). (10.2.0.12) by a consideration of self-accommodated process. It is easy to prove that (10.2.0.12) is equivalent to the following relation, 7516. T) = 43161. I). (10.2.0.13) by the solutions (10.2.0.5), (10.2.0.9), and the fact of 111(1, 7) = fi1(—1:, 7). (10.2.0.14) Example I For A H M, process in R1, one considers a situation that the boundary condition for y, is constant and equal to the transformation strain, ie., Y,| - ‘y . (10.2.0.15) 1:0 - Assume also that 3.: = fl(1:) = “+3, (10.2.0.16) 174 which implies that rl,(t,7) = T—éAtz-BT. Here, A and B are positive quantities. Solution (10.2.0.5) now gives 7510.7“) = 347'. “M where, 1 A r l — = — +— ”M BY “A Then (10.2.0.6) becomes 113’. = nit-31%- Solving (10.2.0.19) and (10.2.0.20) gives t B: 07 oandA= “A-EMO. TIA - 11+ . “Allufih - 11+) (10.2.0.17) (10.2.0. 18) (10.2.0.19) (10.2.0.20) ( 10.2.0.2] ) Thus, we formally obtain the same results as expressed in (2.2.2.3); and (2.2.2.5). 10.3 Determination of Strain in A H M, Process In R1, combining (10.1.0.2) and (10.1.0.13) gives 175 . $7,,” +, (T min —y,)—:-:——::D(y ,— y) = 0, (10.3.0.1) where, 1120.7) = $1,547). (10.3.0.2) The characteristic equation of (10.3.0.1) is if = f,(r,7), (10.3.0.3) which is assumed integrable. Integration of (10.3.0.3) is taken to have the form 1'12('c, T) = C, (10.3.0.4) with property 112(0, C) = C, M f S C S A ,. Boundary condition for solving this problem is given by (10.2.0.11) and “It -0 = 72(C) = 0. Consequently, the solution of (10.3.0.1) under the above boundary condition is found as R] T fa“! “2(11 T))dl 7' (T’ 7) = 17 ”RIB“, 7))e ° . (10.3.0.5) A Here, 82(15 3(15 C)) «(1.0 = f 2(1. so. Q)’ (10.3.0.6) _ where, T = s(t, C) is a specified neutrality curve parametrized by C for M f S C S A f. 176 On the other hand, rm, 7) can determined by (10.1.0.2), (10.1.0.10), (10.1.0.12) and ( 10.3.0.5), 00. 112(1. Tndt . ., 1311 (15.1))e ° Tl-(t.7)=n,,- 2 f2(T.T) (10.3.0.7) As the same manner, based on (10.1.0.5) and (10.1.0.13) for A H M, process in if, we can obtain the following solutions E T _ 50. fir“. 7))43 r. (1.7) = ,,—+72(I12(7.1))e ° . (10.3.0.8) A [610.020. Tndt o o Via-=12“; ”)3 0 0,6. I) = n - _ (10.3.0.9) " f2(T. 7) where, g = 30.7”). (10.3.0.10) _ 82(1. 5(1. D) an. 1;) = - . (10.3.0.11) . f 2(19 3(it, C)) (10.3.0.12) , _ a _ 82(177') = 'a‘i’fZCttnt and T = 5(1, C) is a specified A HM, nuetrality curve in RI, pararneterized by C, M ,SCSA ,. fi2(t,T) 2C is the integration form of (10.3.0.10) with property 177 1.12“), C) = C It can also be proven that (10.2.0.12) and (10.2.0.13) are equivalent by means of (10.3.0.5), (10.3.0.8), and 112(r, 7) = I'm—t, 7). (10.3.0.13) 10.4 Determination of Strain in M_ —> M, Process First of all, consider the situation in R2, by (10.1.0.3) and (10.1.0.13) the following P.D.E. can be derived a E __1_ 31+,7_83(TsT) _ ' where, 8 83(1. 7) = 37f 3(1. 7). (10.4.0.2) The characteristic equation of (10.4.0.1) can be written as = f3(1:, 7). (10.4.0.3) . Since M, —> M, is modified by M_ —->A , the nuetrality curves of M_ -) M, in R2 is a extension of the nuetrality curves of M_ -—> A in R1. Thus, we are going to continuously employ temperature parameter C (M, S C S A ,) to parameterize the integration curves (nuetrality curves) of (10.4.0.3) which start at M, , . Based on the above consideration, the 178 integration form of (10.4.0.3) is written as 113(1, T) = C, (10.4.0.4) which starts at the intersecting point (11? TC) on M, , . This point can be found by solving equations (10.4.0.4) and M , , (T = M, ,(1)) simultaneously, which gives T; = h(C), T; = T(C) . (10.4.0.5) The boundary conditions for solving this problem are posed as r: = 7511.?) .7? = vil(t.1)| (10.4.0.6) (1, T) e M ,, (t,T)e M,,. Along a specified M_ -> M, nuetrality curve T = s(1, C) that starts at (1;, TC) and goes down to a lower area, boundary conditions ( 10.4.0.6), could be rewritten as 73 = 3.0;). Tf’ = BIC). (10.4.0.7) Therefore, the solution of equation (10.4.0.1) under the above boundary condition is found tobe V(tr "3(10 1.))dt 15%. n-rf'za. 7) = (Bands. 1))—3,0130, 1))). “W . (10.4.0.8) where, 83(15 3(1, 0) f 3(15 3(1, §)) . ' 141.0 = (10.4.0.9) 179 Solution (10.4.0.8) implies that R2 4m 0. nrw’ "’(t’ “W 7, (7.1) = 71(1.I)+B,(H3(7.1))e ’ , (10.4.0.10) f V('r “3(10 ”)4! 46131:. 1)) 75““. T) = 7.6. 7) + B-(H3(t. 7))e (10.4.0.11) Here, 11(1, T) is any function that obeys boundary condition 71(7. DI, T) e 11,, = 0. (10.4.0.12) which is explicit by (10.40.10), (10.4.0.11) and (10.4.0.6) if one assumes that the strains of the two variant martensite satisfy (y,(t, 7), y,(t, 7)) e c“. By (10.1.0.3), (10.1.0.11), (10.1.0.12) and (10.4.0.8), the following relation between the entropies is obtained, . .. 3.0130. 7))-B,(n,(r, 1)) lm,...n,""- 0.0.01.1. - = e . n+ TL f3(15 T) (10.4.0.13) For more about the above specification, a particularly natural assumption is that 71(7. 7) E0. (10.4.0.14) Then, solutions (10.4.0. 10) and (10.4.0.11) become . hm (t. WWI. "311. T))dt 7520. 7) = 3,0130. 7))e ’ , (10.40.15) 180 1 VO. 113(1. TDdt Wu, 7) = B,(n3(t, T))e “"3“” , (10.4.0.16) which would be deciphered as the natural extension of solutions (10.2.0.5) and (10.3.0.5) in R1 into R2, if one notes (10.4.0.6) and (10.4.0.7). For region R3, the only difference from that in region R2 is that the boundary condi- tions are now given along T-axis, which are expressed as (10.2.0.4)2 and (10.2.0.11) for CS M, . Thus, relations, from (10.4.0.1) to (10.4.0.4), still suit the present condition, but, (10.4.0.4) has the property 113(0, C) = C for CS M ,. (10.40.17) Therefore, the difference between the strains in R3 is given by t R3 R3 0 a ‘1'“) 113(1, T))dt 7.. (177—7. (1. 7) = (7,0130. 1))-1.0136. 7)))e ° . (10.4.0.18) Comparing (10.4.0.8) and (10.40.18) will conclude that the two solutions are identical with each other along the boundary M ,_ between R2 and R3 if one notes that 73C) and f(C) are continuous along the T-axis with Co smooth. Solution (10.4.0.18) implies that R3 ‘ a rim, I13(1, T))dr 7,. (T. T) = 72(1. 7) + 7,013“. T))e ° . (10-4-0-19) R3 0 V0. 113(1, T))dr 7- (1.7) = 72(7. 7) +T.(TI3(I. 7))e ° . (10.4.0.20) 181 Here, 72(1, T) is any function in R3. And, on the other hand we have a a 7:013“. T)) - 100130;, 1'))eJ:V(t. 113(1, T))dt - = , 10.40.21 n+ n- f 3(15 T) ( ) in R3. If one assumes that the strains are Co along M ,_ , then, 72(1, T) satisfies 72(7. Blame M,_ = 11(1. 7)] (1’ m M,_. (10.4.0.22) by (10.2.0.5), (10.3.0.5), (10.4.0.6) and (10.4.0.7). Thus, we define an arbitrary C“ func- tion, named 7(1, T) , in R2 U R3 , upon which, 71(1, T) in (10.40.10) and (10.40.11) as well as 72(1, 7) in (10.4.0.19) and (10.40.20) can be replaced by y(t, 7) now. In 7112 and E similar solutions can be obtained as we have illustrated above without any difficulty.