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The pseudoelastic force balance and its application to β-FeBe alloys
Materials Science and Engineering, 50 (1981) 109 - 116
109
T h e P s e u d o e l a s t i c F o r c e B a l a n c e a n d its A p p l i c a t i o n t o / 3 - F e - B e A l l o y s
MARTIN L. GREEN
Bell Laboratories, Murray Hill, NJ 07974 (U.S.A.) MORRIS COHEN and G. B. OLSON
Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received January 24, 1981)
SUMMARY
A pseudoelastic force balance equation for a single martensitic plate is introduced and proposed as the mechanical analog of the thermoelastic force balance introduced by Olson and Cohen. The difference between these two balances is the presence o f a mechanical work term in the pseudoelastic case, in addition to the chemical term common to both. The chemical term was both experimentally determined and independently estimated for the complex case of pseudotwinning and pseudoelastic behavior in Feo.slBeo.19, and the upper limits o f these two values are found to be in agreement. Finally, analysis of the pseudoelastic force balance for the Feo.slBeo.19 case shows that the transformation strain energy contributes substantially to the force balance and cannot be ignored.
1. INTRODUCTION
The term "pseudoelasticity" refers to mechanically reversible behavior commonly observed in thermoelastic martensitic alloys [1, 2]. The application of stress to the parent phase of such an alloy can induce a martensitic transformation, giving rise to macroscopic yielding with an apparent plastic strain. The total or partial disappearance of this strain on unloading, due to the reversion of the martensitic phase to the parent phase, is designated as "pseudoelastic behavior". This is the mechanical analog of thermoelastic behavior; once the stress-assisted martensitic phase forms, the same factors which would have enabled the transformation to proceed 0025-5416/81/0000-0000/$02.50
thermoelastically on cooling without an applied stress will determine whether the transformation will proceed pseudoelastically under an applied stress. The purpose of this paper is twofold. First, thermoelastic and pseudoelastic force balances will be compared and contrasted through analogous mathematical developments. Then the pseudoelastic force balance equation will be applied to ~-Fe-Be alloys, whose pseudoelastic behavior has been described fully in an earlier paper [ 3 ].
2. T H E R M O E L A S T I C FORCE BALANCES
AND
PSEUDOELASTIC
Olson and Cohen [4] have analyzed the energy attending the formation of a martensitic plate (oblate spheroid, with a small ratio of thickness to diameter) and have derived a thermoelastic force balance equation. For the ideally reversible case, in which all the transformation strain is accommodated elastically and there is no frictional resistance to interfacial motion, this force balance equation is Agch + 2 Agel = 0
(1)
where Agcb is the chemical free-energy change per unit volume of martensite and Age1 is the transformation (elastic) strain energy per unit volume of martensite. Although the above terms have the units of energy per volume, eqn. (1) is a force balance. This can be demonstrated by deriving the pseudoelastic force balance, in which the equilibrium of a single martensite plate and the surrounding matrix, created by the © ElsevierSequoia/Printed in The Netherlands
110
application of a stress at a constant temperature, will be considered. Adopting the same model used by Olson and Cohen [4], the freeenergy change A G attending the formation of an oblate spheroidal martensitic plate of radius r and semithickness c is AG = 4 ~r2c( Agch - - 1"7T) + 4 ~rc2A + 2~r20
(2) where A(c/r) = Ages is the transformation (elastic) strain energy, o the interfacial energy per unit area, r the resolved shear stress and 7T the transformation shear strain. Equation (2) is eqn. (1) of ref. 4 where the work term 1"7T is introduced negatively because it represents work done on the system by the applied stress r. Therefore --rTT constitutes a mechanical contribution to the free-energy change A G. (When a volume change accompanies the transformation, an additional work term --anen can be incorporated where an and en are the resolved stress and transformation strain normal to the habit plane, as treated by Patel and Cohen [5] .) A force balance can be derived from an energy expression such as eqn. (2) by utilizing the concept of radial and thickness transformational growth forces [6], as was done by Olson and Cohen [4]. On the assumption that the radial growth of a martensitic plate is stopped by some obstacle such as a grain boundary, the plate will continue to thicken in response to the transformational force --a (hG)/ac in the c direction. In the absence of frictional effects, thickening will stop when this force is zero and the plate reaches a mechanical equilibrium under the applied stress: a(AG) -
ac
÷=
0 =
4 ~ ' r 2 ( A g c h - - TTT) + ~8 ' r 2 3
L~
d
o~_
3~
Age!
= 0
C
~ " ~ d ~gch+2Agej=O T=O
1
Ms
POSmONSo,e
I"''THERMOELASTICITY
._,AI
TO
POSITIONSb
(4)
(5)
POSITIONSc
~ c =~c, "~,~b /(z~gch-TYT)+2Aget=0
~il ~ t ~"P' SEUDOELASTI ~ CITY -
Equation (4) is the pseudoelastic force balance; it should be noted that the terms have the units o f force because of the dimensional coefficients of the energy terms. This equation can be simplified to (Agch - - T T T ) + 2 Age1 = 0
w
(a)
'rcA
(3)
or, since Age1 A (c/r),
Equation (5) is a more general statement of eqn. (1). Both equations are force balances which can be interpreted as follows. Thermoelastic-pseudoelastic force equilibria can be achieved when the force generated by the transformational driving free energy is equal in magnitude to the force generated by the stored transformation (elastic) energy. For thermoelasticity the driving force for transformation is Agch , whereas for pseudoelasticity it is A g c h - - T~/T. Thus, unlike a thermoelastic transformation, Agch need not be negative for a pseudoelastic transformation; however, A g c h - - TTT must of course be negative for the pseudoelastic transformation to occur. To clarify further the analogy between the two force balances, a diagram of the type given in ref. 7, Fig. 1, was developed for the pseudoelastic force balance. Figure I shows this type of diagram for thermoelasticity (Fig. l(a)) and pseudoelasticity (Fig. l(b)). Figures l(a) and l(b) are both plots of the volume of a single martensitic plate as a function of driving force, recognizing that strain in the pseudoelastic case is proportional to the volume of transformation product. Whereas the driving force for the thermoelastic transformation is a function of temperature (zero applied stress) and increases with decreasing temperature, the driving force for the pseudo-
I " ~
T =CONST.
"~.
TS Tr ..t~T
(b)
POSITIONSd
%
(c)
Fig. 1. Ideal (no friction) transformation hysteresis loops for (a) thermoelasticity and (b) pseudoelasticity. Microstructures corresponding to various points on both loops are depicted in (c).
III
elastic transformation is a function of applied stress (at a given temperature) and increases with increasing stress. Otherwise, the two figures are analogous, and comparable microstructures (Fig. l(c)) are generated at corresponding points in the respective diagrams. Thus rs, the applied stress at which the pseudoelastic transformation starts, is analogous to Ms; r0, the applied stress at which the transformation would have started in the absence of any nucleation barrier, is analogous to To; r~, the unloading stress at which the plate completely reverts, is analogous to A~. rr can be determined by considering the radial force --a(AG)/ar obtained from eqn. (2). Similar to the thermoelastic case treated in ref. 4, a radial instability is found to occur when
2(Aotl/2
T r - - - T O ---- - -
~
(6)
7T\ r / The value of r0 is simply given by r0 -
Agch
tion, as evidenced by the fact that M s < To and r s > r0. Figure 2 introduces the effect of frictional resistance and also illustrates the relationship between the ideal and real pseudoelastic loops. Figure 2(a) is the same ideal case as Fig. l(b) but rotated 90 ° clockwise to give the usual representation of a r-~/ curve. Figure 2(b) illustrates the role of frictional resistance, in a manner analogous to ref. 7, Fig. 2. The stress rs at which transformation starts on loading is increased, and the applied stress must drop by 2r~ on unloading before the plate can begin to revert. Consequently, the magnitude of the term Tf"/T can control the degree of hysteresis between the forward and reverse transformations. Figure 2(c) shows an actual pseudoelastic loop observed for a ~-Fe-Be sample [3]. A m e t h o d of determining the value of rf from such a loop [7, 8] is also shown. The hysteresis loop of Fig. 2(c) results from multiple-interface transformation (i.e. multiple plates), with concomitant work hardening and imperfect
(7)
7T
Equations (1) and (5) represent ideal force balances in plate growth and reversal, i.e. the transformation strain is elastically accommodated and there is no frictional resistance on the transforming interface. A frictional resistance term can be introduced into eqns. (1) and (5) by taking this resistance as a reverse stress, or a drag stress r~, acting to oppose the interface motion. Thus the net driving force for transformation in both cases is effectively reduced by the term r~TT. For the pseudoelastic force balance,
Y
(a)
loading
(Agch - - ~'TT) + 2 Agel =
~
"fr Tf'YT
(8)
unloading
For the thermoelastic case, Olson and Cohen [4 ] had previously derived
(b)
cooling
Agch + 2 Age1 =
~
heating
rfTT
(9)
(Olson and Cohen [4, 7] used ro for r~). The frictional resistance term takes a negative value when the plate is thickening and a positive value when it is reverting. The reversion condition of eqn. (6) would be modified similarly by friction. Figures l(a) and l(b) are idealized (no friction) hysteresis loops. In these instances the hysteresis results from obstacles to nuclea-
Y
(c) Fig. 2. Pseudoelastic transformation hysteresis loops, illustrating (a) ideal case (no friction rf ffi 0), (b) addition of friction (1"f > 0) and (c) real case (multiple° interface transformation in Feo.slBeo.19 ).
112
elastic accommodation of the transformation strain [3]. (Another difference between the model of Figs. 2(a) and 2(b) and the experiment of Fig. 2(c) lies in the nature of the constraints imposed during testing. The transformation on cooling (Fig. l(a)) takes place under controlled driving free energy Ageh. The analogy developed for the pseudoelastic case (Figs. l ( b ) , 2(a) and 2(b)) would correspond to transformation under a fixed applied stress r as in "dead-weight" loading, while the experimental curve of Fig. 2(c) was determined under the condition of a constant imposed strain rate. This will influence the detailed shape of the r - 7 curve; such effects of stress-assisted martensitic transformation under constant imposed strain rate have been treated elsewhere [9] .) None the less, the similarity between Fig. 2(c) and Fig. 2(b) is evident.
3. APPLICATION TO ~ - F e - B e ALLOYS
3.1. Pseudoelastic behavior in ~-Fe-Fe alloys The concept of pseudoelasticity, originally defined as a reversible deformation process arising from a stress-assisted martensitic transformation in a thermoelastic alloy, can be extended to include identical mechanical phenomena in fl-Fe-Be alloys, which deform b y a b.c.c, twinning mode. Twinning can be regarded as a special case of the stress-assisted martensitic transformation for which the chemical term of the driving force is zero. Pseudotwinning is defined as a "twinning" process for which the chemical term is not zero because of some structural variable; in ~ - F e - B e alloys containing a b o u t 25 at.% Be a two-phase a(b.c.c.) plus B2(CsC1) microstructure exists which gives rise to this p h e n o m e n o n [3]. In Section 3.2 the pseudoelastic force balance will be applied to Fe0.sl Be0.19, which can exhibit significant pseudoelastic behavior if a modulated coherent a plus B2 microstructure, illustrated in ref. 3, Fig. 5(c), is present. All experimental data used in Section 3.2 and in Appendix A were obtained on typical samples heat treated to produce the required microstructure. This heat treatment was as follows: 24 h at 1125 °C plus a water quench, followed by 1 h at 375 °C plus a water quench. Samples thus treated recover more
than 95% of the pseudoelastic strain on release of the applied stress. 3.2. Pseudoelastic force balance in {3-Fe-Be alloys Equation (8) can be written in a more specific form for the thickening of a pseudotwin in ~ - F e - B e alloys; (Agch - - r T t w ) + 2 Agel = - - r f T t w
(10)
where 7tw is the b.c.c, twinning shear, which remains operative although twinning in the usual crystallographic sense does not occur because the structure of the B2 phase undergoes a change [ 3 ]. In this section the various terms in eqn. (10) will be calculated, for the purpose of analyzing the pseudoelastic force balance. The value of Agch is difficult to calculate independently, partly because the appropriate data are n o t adequately known for ~-Fe-Be alloys and partly because the terms themselves have not been derived rigorously. An attempt at this calculation is carried o u t in Appendix A; in this section, Agch will be determined by difference from eqn. (10). The transformation strain energy can be calculated in the following manner, for the simple shear of a thin plate [10] : Agel -
Ac r
~(2 --v) -
-
-
8(1
--v)
p7 2
c --
(11)
r
where v is Poisson's ratio, p the shear modulus and 7 the transformation shear. Taking v = !3 and ~ = 7tw = 21/2/2 for b.c.c, twinning, we obtain c Agel ~, ~1 - p -
(12)
r
Transmission microscopy on residual pseudotwins (those that remain after release of the applied stress; see ref. 3, Fig. 8(a)), indicated that c = 0.4 × 10 -6 m, as determined by observing the pseudotwins in their edge-on orientation, i.e. with the beam direction in the plane of K1, the twinning plane. The average grain diameter of the sample was 2.0 × 10 -3 m; on the assumption that the pseudotwins propagated until they were stopped at the grain boundaries, r = 1.0 X
113
10 - 3 m. c/r is thus 4 × 10 -4, typical of twins in b.c.c, alloys. The shear modulus of Fe0.slBe0.19 is not known, b u t as a first approximation the value for a similar ~ phase alloy, Fe0.ssAlo.i9 , can be used, giving u = 1.25 × 1 0 il J m -3 [11]. Substituting into eqn. (12) we obtain &gel /> 2.5 × 107 J m - 3
(13)
The inequality is introduced because Agel thus c o m p u t e d represents a lower limit to the pseudoelastic force balance value, since c/r could only be measured on remnant pseudotwins. The fact that these pseudotwins remained after the release of the stress implies that they had reduced their elastic transformation strain energy through some plastic accommodation, which then prevented their complete reversion. The stresses r and rf can be measured from stress-strain curves such as those in Fig. 2(c). For the purposes of the following calculations, r is chosen as the resolved macroscopic yield stress corresponding to the first plates to form, where multiple-plate interactions will be minimal. Since these events will occur in the most favorably oriented grains, a Schmidt factor of 0.5 (the maximum value) is assumed. The friction stress rf is taken as the resolved shear stress equal to one-half of the vertical width of the stress-strain hysteresis loop [7, 8 ] , as is illustrated in Fig. 2(c). These values are found to be r = 340 MPa and Tf = 150 MPa. Therefore ~'Ttw = 2.4 × 10 s J m - 3
(14)
and TfTtw = 1.1 X 10 s J m-3
(15)
Substituting eqns. (13) - (15) into eqn. (10), we obtain Agch < 8.0 X107 J m - 3
(16)
As was mentioned earlier in this section, the chemical free-energy change accompanying the pseudotwinning transformation in Feo.slBeo.19 is difficult to calculate independently. However, as it is a complex quantity because of the unusual microstructural and crystallographic features of the alloy, an independent calculation of Agch is undertaken in Appendix A, to illustrate these complexities.
4. DISCUSSION
In thermoelastic martensitic transformations, Agch and Age1 must have opposite signs. Agcn , the driving force, has to be negative before the transformation will proceed, while Age~ , the transformation strain energy, opposes the transformation. Hence on cooling, as Agch becomes more negative, Agel can become more positive, thereby allowing the transformation to proceed while maintaining the thermoelastic force balance described in eqn. (1). The pseudoelastic force balance represents a more complicated case in that a mechanical as well as a chemical driving force, Ageh -TTT , is operative. Agch -- rTT must be negative in order for the pseudoelastic transformation to proceed; however, Agch itself does not have to be negative for this type of transformation. Indeed, the calculations performed here indicate that Agch is actually a positive quantity for pseudotwinning in Feo.slBe0.i9. If Agch = 0 during a pseudoelastic transformation, then eqn. (5) describes the force balance for twinning with elastic accommodation of the twin. In contrast, the usual elastic twinning reported in the literature [12, 13] typically involves some crystallographic feature which can give rise to a finite Agch term during the operation of the twinning shear and hence is not true twinning in the classical crystallographic sense [14]. Recent analyses of the thermodynamics of thermoelastic martensitic transformations have given rise to some question regarding the role of transformation strain energy in the attainment of a thermoelastic force balance. The Olson-Cohen approach, which is the basis of this paper, includes this term in an essential way. However, Tong and Wayman [15, 16] consider the transformation strain energy to be zero for the first plate to form, although they introduce a frictional term to oppose the formation of the growing plate. From eqns. (13) and {15), it is possible for us to calculate the following ratio for our Fe0.siBeo.19 alloy: 2/Xg~l >/ 0.45 (17) Tf'Ytw
-
-
which can be interpreted to mean that the force opposing the growing plate due to
114
elastically stored transformation strain energy is at least 45% of the frictional force opposing the growing plate. Therefore there is no justification for omitting the transformation strain energy term. Consistent with this conclusion is the experimental determination that the strain energy term in Cu-Ni-A1 martensites can be as large as 15% o f the magnitude of the corresponding chemical enthalpy change [17]. Agreement is found between the upper limits of Agch calculated by difference from eqn. (10) and independently in appendix A: Agch ~< 8.0 × 1 0 7 J m - s (by difference from eqn. (10)); Agch ~< 5.9 × 107 J m -3 (Appendix A). The agreement suggests that the observed pseudotwinned plates were close to meeting the elastic a c c o m m o d a t i o n condition of the pseudoelastic force balance. This assessment is consistent with the observation of reversible mechanical behavior in these samples [ 3 ] . Evidently, accommodation defects which may be created because of imperfect elastic accommodation contribute primarily to r~, the frictional resistance stress. Whereas eqn. (8) states that a force balance can still exist in the presence o f these defects, a state o f true thermodynamic equilibrium cannot exist if a frictional term, arising from any mechanism, is involved.
5. CONCLUSIONS A pseudoelastic force balance equation was proposed and analyzed as the mechanical analog of the thermoelastic force balance introduced by Olson and Cohen [4, 7 ] . Whereas for the thermoelastic force balance the driving free energy has only a chemical component, the pseudoelastic force balance has both chemical and applied work components. The force balance equation was applied to the complex pseudotwinning and pseudoelastic behavior observed in Fe0.slBe0.19 [3]. The force opposing the growth of the plate due to the elastically stored transformation strain energy is found to be at least half as large as the frictional force, and it cannot be ignored, as has been suggested previously [15, 16]. Upper limits to the chemical component of the driving force for pseudoelastic transformation were calculated by t w o
methods. The agreement between the two results suggests that the pseudotwins in Fe0.slBeo.19 are elastically accommodated to a substantial extent.
ACKNOWLEDGMENTS The authors are grateful to Professor S. M. Allen and Dr. J. W. Cahn for their very helpful contributions to the ideas and calculations presented in this paper. One of the authors (M.L.G.) .also wishes to express his appreciation to the AMAX Foundation for a Doctoral Fellowship which made his graduate work possible. This research is part of a long-range program sponsored at Massachusetts Institute of Technology by the National Science Foundation under Contract DMR79-15196.
REFERENCES
1 H. Warlimont and L. Delaey, Prog. Mater. Sei., 18 (1974). 2 R. V. Krishnan, L. Delaey, H. Tas and H. Warlimont, J. Mater. Sei., 9 (1974) 1536. 3 M. L. Green and M. Cohen, Aeta Metall., 27
(1979) 1523. 4 G. B. Olson and M. Cohen, Scr. Metall., 9 (1975) 1247. 5 J. R. Patel and M. Cohen, Acta Metall., 1 (1953) 531. 6 V. Raghavan and M. Cohen, Acta Metall., 20 (1972) 779. 7 G. B. Olson and M. Cohen, Scr. Metall., 11 (1977) 345. 8 K. Otsuka, C. M. Wayman, K. Nakai, H. Sakamoto and K. Shimizu, Acta MetaU., 24
(1976) 207. 9 G.B. Olson and M. Cohen, Proc. U.S.-Japan Semin. on the Mechanical Behavior Associated with Displacive Transformations, Rensselaer Polytechnic Institute, Troy, N Y , June 12 - 15,1979, p. 7. 10 J. D. Eshelby, Proc. R. Soc. London, Set. A, 241
(1957)376. 11 H. J. Leamy, E. D. Gibson and F. X. Kayser, Aeta Metall., 15 (1967) 1827. 12 M. V. Klassen-Neklyudova, Mechanical Twinning o f Crystals, Consultants Bureau, New York, 1964. 13 J. Y. Guedou, M. Paliard and J. Rieu, Scr. Metall., 10 (1976) 631. 14 J. W. Cahn, Aeta Metall., 25 (1977) 1021. 15 H. C. Tong and C. M. Wayman, Acta Metall., 22 (1974) 887. 16 C. M. Wayrnan and H. C. Tong, Ser. MetaU., 11 (1977) 341. 17 R.J. Salzbrenner and M. Cohen, Acta Metall., 27 (1979) 739.
115
APPENDIX A For ~-Fe-Be alloys, Agch can be expressed as the sum of three terms: A g c h = ~-a~ch^ rfpstw + "a~chA rr(G + B2)coh + ~'aSchA~(G + B2)sur
(A1) where ^~pstw is the chemical free-energy d i f ,.a6ch ference between the ~ plus B2 microstructure and the pseudotwinned microstructure (per volume of pseudotwin), Ag ~+ B2)coh is the change in the coherency strain energy of the plus B2 microstructure as a result of pseudotwinning (per volume of pseudotwin) and Ag(~ +s2)sur is the change in surface energy of the a plus B2 microstructure as a result of pseudotwinning (per volume of pseudotwin). A~P~tW arises from the inability of the $ch b.c.c, twinning shear to twin the B2 phase in the ~ plus B2 microstructure. The application of this shear to the B2 superlattice brings about the transformation P m 2 m ( B 2 ) -~ Cmmm, and the accompanying change A,p~tw in chemical free energy [A1] ~eh A,(~ + 82)coh arises from the tendency of the Sch coherent modulated ~ plus B2 microstructure (ref. A1, Fig. 5(c)) to be preserved during pseudotwinning (ref. A1, Fig. 8(c)), thereby causing a change in the state of the coherency strain energy of the two phases. Finally, /~,~(~ 5 c h + B2)sur arises because the surface area of the B2 particles is increased when they are sheared during pseudotwinning.
energy change; the a phase, being a disordered b.c.c, solid solution, is structurally unchanged by the twinning shear. X-ray diffraction experiments on single crystals of Feo.slBe0.z9 enable us to measure an average order parameter SAV2 defined as SAv 2 = SB22fB2
(A3)
SAV corresponds to the order parameter Ss2 of the B2 phase averaged over the total volume of the alloy. Therefore substituting eqn. (A3) into eqn. (A2) we find that A~cPstw _ SAV 2 /X V ~-~Dch
(A4)
Vm
A ground state calculation for h U can be made using the following relationship [A2] : U = --
No 2
(A5)
~-,Viqi
where U is tile energy of a configuration of atoms relative to the pure components (per mole), No is Avogadro's number, Vi is the ith nearest-neighbor interaction energy (per atom) and q~ is the number of ith nearestneighbor unlike bonds (per atom). Knowing the structure of the B2 and C m m m phases [A1], we can construct Table A1 which gives q values for first- and second-nearest neighbors. From this table A U = N o ( V z ~ V2)
(A6)
and Agpl~tw - S A v 2 N ° ( V l - - V 2 )
(A7)
Vm
A . 1 . C a l c u l a t i o n o f Agp~tw
The chemical flee-energy change attending the pseudotwinning of the B2 phase of the plus B2 microstructure, P m 3 m ( B 2 ) -* C m m m , can be expressed as A ecpstw 6ch
- SB22fB2 A V
Vm
T A B L E A1 q values for first- and second-nearest neighbors Phase
q i=1
i=2
4 2
0 2
(A2)
where Ss2 is the long-range order parameter of the B2 phase, fB2 the volume fraction of the B2 phase, A U the internal energy difference between the B2 and C m m m phases (per mole of pseudotwin) and Vm the molar volume of Fe0.siBe0.z9. The pseudotwin is defined as the region of the a plus B2 microstructure that has been sheared. However, only the volume fraction that was originally the B2 phase contributes to the chemical free-
Pm3m(B2) Cmmm
The entropy change neglected in the derivation of eqn. (A7) consists of configurational, vibrational and electronic terms. For a pseudotwinning transformation the configurational entropy change is zero because there is a one-to-one correspondence between atoms in the P m 3 m and C m m m phases [A3].
116
Although it cannot be demonstrated at present that the other entropy terms are unimportant, that assumption will be made here. V1 and V2 for ~ - F e - B e alloys have recently been calculated, using a pairwise interaction model [ A 4 ] . With our sign convention for the configurational energy in eqn. (A5), the values of V1 and V2 yield
Ag(~ +B2)c°h ~ 2.8 X 107 J m - s
(A12)
A.3. Calculation of Ag(c~+S2)sur We can write Aoc hp(ce+ B2)sur = g v AA 7s
(A13)
For Fe0.s1Be0.19 , SAV2 = 0.037 (from our X-ray diffraction measurements) and Vm = 6.7 × 10 - e m 3 tool -1. Therefore
where Nv is the number of B2 particles per unit volume, AA the difference in surface area between a Cmmm and a B2 particle and % the a - B 2 interfacial energy. The number o f B2 particles per unit volume is given by
A,pstw 5oh = 2.8 X 107 J m -3
Nv = fa2/~ur 4 3
V1 -- V2 = 8.37 × 10 -21 J atom -1
(AS)
(A9)
A.2. Calculation of A g ~ +B2)coh To evaluate Ag(c~+B2)coh, w e must obtain the coherency strain energies of both the parent and the pseudotwinned microstructures. By invoking the assumptions of the Crum theorem [A5] and taking the coherent B2 particles to be spheres, we can calculate the coherency strain energy of the parent microstructure. However, the particles no longer have a cubic symmetry in the pseudotwin and so the Crum theorem does not hold; furthermore, the elastic constants of the Cmmm phase are unknown. We shall assume that the coherency strain energy of the parent microstructure is an upper limit to the difference between the coherency strain energies of the parent and pseudotwinned microstructures, i.e. A.(~+ B2)coh ~ b~ce+ B2 5ch J-~coh
where r is the averageradius of the B2 particles. The change in surface area on pseudotwinning can be approximated from knowledge of the principal twinning strains [A6] :
AA ~ ~r2/4
(A15)
% is estimated to be 0.20 J m - 2 , before and after shearing of the B2 particles; this value, 200 erg cm -2, is probably an upper limit for the coherent interfaces involved here. Finally, it was determined from X-ray diffraction linewidths that r ~ 30 × 10 - l ° m for Feo.slBeoa9. Therefore A..,(~+B2)sur ~< 0.3 × 107 J m - s 6ch
(A16)
Adding eqns. (A9), (A12) and (A16), we obtain Agch ~< 5.9 X107 J m-a
(A17)
(AI0)
The coherency strain energy v~+ ~'~cohB2 of the plus B2 parent microstructure can be expressed as [A5] EC~+ B2 2 l + r (_~)2 ¢oh - ~ ~ - - ~ /~ fB2
(AI4)
(All)
where A V / V is the volumetric strain between the a and B2 phases. As in the calculation of eqns. (11) and (12), p = 1.25 × 1011 J m - 3 and v = ~. From transmission electron microscopy observtions, fB2 = 1 . Also, from the metastable phase diagram of ~ - F e - B e and the variation in lattice parameter with composition (ref. A1, Figs. 10 and 11) A V / V can be estimated as 4.5 X 1 0 - 2 . Therefore
R E F E R E N C E S F O R APPENDIX A
A1 M. L. Green and M. Cohen, Acta Metall., 27 (1979) 1523. A2 S. M. Allen and J. W. Cahn, Acts Metall., 20 (1972) 423. A3 G. F. Boiling and R. H. Richman, Acta Metall., 13 (1965) 709. A4 H. Ino, Acts 'MetaU., 26 (1978) 827. A5 J. W. Christian, The Theory o f Transformations in Metals and Alloys, Pergamon, Oxford, 1975, pp. 457 - 475. A6 J. W. Christian, The Theory o f Transformations in Metals and Alloys, Pergamon, Oxford, 1975, p. 52.
109
T h e P s e u d o e l a s t i c F o r c e B a l a n c e a n d its A p p l i c a t i o n t o / 3 - F e - B e A l l o y s
MARTIN L. GREEN
Bell Laboratories, Murray Hill, NJ 07974 (U.S.A.) MORRIS COHEN and G. B. OLSON
Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received January 24, 1981)
SUMMARY
A pseudoelastic force balance equation for a single martensitic plate is introduced and proposed as the mechanical analog of the thermoelastic force balance introduced by Olson and Cohen. The difference between these two balances is the presence o f a mechanical work term in the pseudoelastic case, in addition to the chemical term common to both. The chemical term was both experimentally determined and independently estimated for the complex case of pseudotwinning and pseudoelastic behavior in Feo.slBeo.19, and the upper limits o f these two values are found to be in agreement. Finally, analysis of the pseudoelastic force balance for the Feo.slBeo.19 case shows that the transformation strain energy contributes substantially to the force balance and cannot be ignored.
1. INTRODUCTION
The term "pseudoelasticity" refers to mechanically reversible behavior commonly observed in thermoelastic martensitic alloys [1, 2]. The application of stress to the parent phase of such an alloy can induce a martensitic transformation, giving rise to macroscopic yielding with an apparent plastic strain. The total or partial disappearance of this strain on unloading, due to the reversion of the martensitic phase to the parent phase, is designated as "pseudoelastic behavior". This is the mechanical analog of thermoelastic behavior; once the stress-assisted martensitic phase forms, the same factors which would have enabled the transformation to proceed 0025-5416/81/0000-0000/$02.50
thermoelastically on cooling without an applied stress will determine whether the transformation will proceed pseudoelastically under an applied stress. The purpose of this paper is twofold. First, thermoelastic and pseudoelastic force balances will be compared and contrasted through analogous mathematical developments. Then the pseudoelastic force balance equation will be applied to ~-Fe-Be alloys, whose pseudoelastic behavior has been described fully in an earlier paper [ 3 ].
2. T H E R M O E L A S T I C FORCE BALANCES
AND
PSEUDOELASTIC
Olson and Cohen [4] have analyzed the energy attending the formation of a martensitic plate (oblate spheroid, with a small ratio of thickness to diameter) and have derived a thermoelastic force balance equation. For the ideally reversible case, in which all the transformation strain is accommodated elastically and there is no frictional resistance to interfacial motion, this force balance equation is Agch + 2 Agel = 0
(1)
where Agcb is the chemical free-energy change per unit volume of martensite and Age1 is the transformation (elastic) strain energy per unit volume of martensite. Although the above terms have the units of energy per volume, eqn. (1) is a force balance. This can be demonstrated by deriving the pseudoelastic force balance, in which the equilibrium of a single martensite plate and the surrounding matrix, created by the © ElsevierSequoia/Printed in The Netherlands
110
application of a stress at a constant temperature, will be considered. Adopting the same model used by Olson and Cohen [4], the freeenergy change A G attending the formation of an oblate spheroidal martensitic plate of radius r and semithickness c is AG = 4 ~r2c( Agch - - 1"7T) + 4 ~rc2A + 2~r20
(2) where A(c/r) = Ages is the transformation (elastic) strain energy, o the interfacial energy per unit area, r the resolved shear stress and 7T the transformation shear strain. Equation (2) is eqn. (1) of ref. 4 where the work term 1"7T is introduced negatively because it represents work done on the system by the applied stress r. Therefore --rTT constitutes a mechanical contribution to the free-energy change A G. (When a volume change accompanies the transformation, an additional work term --anen can be incorporated where an and en are the resolved stress and transformation strain normal to the habit plane, as treated by Patel and Cohen [5] .) A force balance can be derived from an energy expression such as eqn. (2) by utilizing the concept of radial and thickness transformational growth forces [6], as was done by Olson and Cohen [4]. On the assumption that the radial growth of a martensitic plate is stopped by some obstacle such as a grain boundary, the plate will continue to thicken in response to the transformational force --a (hG)/ac in the c direction. In the absence of frictional effects, thickening will stop when this force is zero and the plate reaches a mechanical equilibrium under the applied stress: a(AG) -
ac
÷=
0 =
4 ~ ' r 2 ( A g c h - - TTT) + ~8 ' r 2 3
L~
d
o~_
3~
Age!
= 0
C
~ " ~ d ~gch+2Agej=O T=O
1
Ms
POSmONSo,e
I"''THERMOELASTICITY
._,AI
TO
POSITIONSb
(4)
(5)
POSITIONSc
~ c =~c, "~,~b /(z~gch-TYT)+2Aget=0
~il ~ t ~"P' SEUDOELASTI ~ CITY -
Equation (4) is the pseudoelastic force balance; it should be noted that the terms have the units o f force because of the dimensional coefficients of the energy terms. This equation can be simplified to (Agch - - T T T ) + 2 Age1 = 0
w
(a)
'rcA
(3)
or, since Age1 A (c/r),
Equation (5) is a more general statement of eqn. (1). Both equations are force balances which can be interpreted as follows. Thermoelastic-pseudoelastic force equilibria can be achieved when the force generated by the transformational driving free energy is equal in magnitude to the force generated by the stored transformation (elastic) energy. For thermoelasticity the driving force for transformation is Agch , whereas for pseudoelasticity it is A g c h - - T~/T. Thus, unlike a thermoelastic transformation, Agch need not be negative for a pseudoelastic transformation; however, A g c h - - TTT must of course be negative for the pseudoelastic transformation to occur. To clarify further the analogy between the two force balances, a diagram of the type given in ref. 7, Fig. 1, was developed for the pseudoelastic force balance. Figure I shows this type of diagram for thermoelasticity (Fig. l(a)) and pseudoelasticity (Fig. l(b)). Figures l(a) and l(b) are both plots of the volume of a single martensitic plate as a function of driving force, recognizing that strain in the pseudoelastic case is proportional to the volume of transformation product. Whereas the driving force for the thermoelastic transformation is a function of temperature (zero applied stress) and increases with decreasing temperature, the driving force for the pseudo-
I " ~
T =CONST.
"~.
TS Tr ..t~T
(b)
POSITIONSd
%
(c)
Fig. 1. Ideal (no friction) transformation hysteresis loops for (a) thermoelasticity and (b) pseudoelasticity. Microstructures corresponding to various points on both loops are depicted in (c).
III
elastic transformation is a function of applied stress (at a given temperature) and increases with increasing stress. Otherwise, the two figures are analogous, and comparable microstructures (Fig. l(c)) are generated at corresponding points in the respective diagrams. Thus rs, the applied stress at which the pseudoelastic transformation starts, is analogous to Ms; r0, the applied stress at which the transformation would have started in the absence of any nucleation barrier, is analogous to To; r~, the unloading stress at which the plate completely reverts, is analogous to A~. rr can be determined by considering the radial force --a(AG)/ar obtained from eqn. (2). Similar to the thermoelastic case treated in ref. 4, a radial instability is found to occur when
2(Aotl/2
T r - - - T O ---- - -
~
(6)
7T\ r / The value of r0 is simply given by r0 -
Agch
tion, as evidenced by the fact that M s < To and r s > r0. Figure 2 introduces the effect of frictional resistance and also illustrates the relationship between the ideal and real pseudoelastic loops. Figure 2(a) is the same ideal case as Fig. l(b) but rotated 90 ° clockwise to give the usual representation of a r-~/ curve. Figure 2(b) illustrates the role of frictional resistance, in a manner analogous to ref. 7, Fig. 2. The stress rs at which transformation starts on loading is increased, and the applied stress must drop by 2r~ on unloading before the plate can begin to revert. Consequently, the magnitude of the term Tf"/T can control the degree of hysteresis between the forward and reverse transformations. Figure 2(c) shows an actual pseudoelastic loop observed for a ~-Fe-Be sample [3]. A m e t h o d of determining the value of rf from such a loop [7, 8] is also shown. The hysteresis loop of Fig. 2(c) results from multiple-interface transformation (i.e. multiple plates), with concomitant work hardening and imperfect
(7)
7T
Equations (1) and (5) represent ideal force balances in plate growth and reversal, i.e. the transformation strain is elastically accommodated and there is no frictional resistance on the transforming interface. A frictional resistance term can be introduced into eqns. (1) and (5) by taking this resistance as a reverse stress, or a drag stress r~, acting to oppose the interface motion. Thus the net driving force for transformation in both cases is effectively reduced by the term r~TT. For the pseudoelastic force balance,
Y
(a)
loading
(Agch - - ~'TT) + 2 Agel =
~
"fr Tf'YT
(8)
unloading
For the thermoelastic case, Olson and Cohen [4 ] had previously derived
(b)
cooling
Agch + 2 Age1 =
~
heating
rfTT
(9)
(Olson and Cohen [4, 7] used ro for r~). The frictional resistance term takes a negative value when the plate is thickening and a positive value when it is reverting. The reversion condition of eqn. (6) would be modified similarly by friction. Figures l(a) and l(b) are idealized (no friction) hysteresis loops. In these instances the hysteresis results from obstacles to nuclea-
Y
(c) Fig. 2. Pseudoelastic transformation hysteresis loops, illustrating (a) ideal case (no friction rf ffi 0), (b) addition of friction (1"f > 0) and (c) real case (multiple° interface transformation in Feo.slBeo.19 ).
112
elastic accommodation of the transformation strain [3]. (Another difference between the model of Figs. 2(a) and 2(b) and the experiment of Fig. 2(c) lies in the nature of the constraints imposed during testing. The transformation on cooling (Fig. l(a)) takes place under controlled driving free energy Ageh. The analogy developed for the pseudoelastic case (Figs. l ( b ) , 2(a) and 2(b)) would correspond to transformation under a fixed applied stress r as in "dead-weight" loading, while the experimental curve of Fig. 2(c) was determined under the condition of a constant imposed strain rate. This will influence the detailed shape of the r - 7 curve; such effects of stress-assisted martensitic transformation under constant imposed strain rate have been treated elsewhere [9] .) None the less, the similarity between Fig. 2(c) and Fig. 2(b) is evident.
3. APPLICATION TO ~ - F e - B e ALLOYS
3.1. Pseudoelastic behavior in ~-Fe-Fe alloys The concept of pseudoelasticity, originally defined as a reversible deformation process arising from a stress-assisted martensitic transformation in a thermoelastic alloy, can be extended to include identical mechanical phenomena in fl-Fe-Be alloys, which deform b y a b.c.c, twinning mode. Twinning can be regarded as a special case of the stress-assisted martensitic transformation for which the chemical term of the driving force is zero. Pseudotwinning is defined as a "twinning" process for which the chemical term is not zero because of some structural variable; in ~ - F e - B e alloys containing a b o u t 25 at.% Be a two-phase a(b.c.c.) plus B2(CsC1) microstructure exists which gives rise to this p h e n o m e n o n [3]. In Section 3.2 the pseudoelastic force balance will be applied to Fe0.sl Be0.19, which can exhibit significant pseudoelastic behavior if a modulated coherent a plus B2 microstructure, illustrated in ref. 3, Fig. 5(c), is present. All experimental data used in Section 3.2 and in Appendix A were obtained on typical samples heat treated to produce the required microstructure. This heat treatment was as follows: 24 h at 1125 °C plus a water quench, followed by 1 h at 375 °C plus a water quench. Samples thus treated recover more
than 95% of the pseudoelastic strain on release of the applied stress. 3.2. Pseudoelastic force balance in {3-Fe-Be alloys Equation (8) can be written in a more specific form for the thickening of a pseudotwin in ~ - F e - B e alloys; (Agch - - r T t w ) + 2 Agel = - - r f T t w
(10)
where 7tw is the b.c.c, twinning shear, which remains operative although twinning in the usual crystallographic sense does not occur because the structure of the B2 phase undergoes a change [ 3 ]. In this section the various terms in eqn. (10) will be calculated, for the purpose of analyzing the pseudoelastic force balance. The value of Agch is difficult to calculate independently, partly because the appropriate data are n o t adequately known for ~-Fe-Be alloys and partly because the terms themselves have not been derived rigorously. An attempt at this calculation is carried o u t in Appendix A; in this section, Agch will be determined by difference from eqn. (10). The transformation strain energy can be calculated in the following manner, for the simple shear of a thin plate [10] : Agel -
Ac r
~(2 --v) -
-
-
8(1
--v)
p7 2
c --
(11)
r
where v is Poisson's ratio, p the shear modulus and 7 the transformation shear. Taking v = !3 and ~ = 7tw = 21/2/2 for b.c.c, twinning, we obtain c Agel ~, ~1 - p -
(12)
r
Transmission microscopy on residual pseudotwins (those that remain after release of the applied stress; see ref. 3, Fig. 8(a)), indicated that c = 0.4 × 10 -6 m, as determined by observing the pseudotwins in their edge-on orientation, i.e. with the beam direction in the plane of K1, the twinning plane. The average grain diameter of the sample was 2.0 × 10 -3 m; on the assumption that the pseudotwins propagated until they were stopped at the grain boundaries, r = 1.0 X
113
10 - 3 m. c/r is thus 4 × 10 -4, typical of twins in b.c.c, alloys. The shear modulus of Fe0.slBe0.19 is not known, b u t as a first approximation the value for a similar ~ phase alloy, Fe0.ssAlo.i9 , can be used, giving u = 1.25 × 1 0 il J m -3 [11]. Substituting into eqn. (12) we obtain &gel /> 2.5 × 107 J m - 3
(13)
The inequality is introduced because Agel thus c o m p u t e d represents a lower limit to the pseudoelastic force balance value, since c/r could only be measured on remnant pseudotwins. The fact that these pseudotwins remained after the release of the stress implies that they had reduced their elastic transformation strain energy through some plastic accommodation, which then prevented their complete reversion. The stresses r and rf can be measured from stress-strain curves such as those in Fig. 2(c). For the purposes of the following calculations, r is chosen as the resolved macroscopic yield stress corresponding to the first plates to form, where multiple-plate interactions will be minimal. Since these events will occur in the most favorably oriented grains, a Schmidt factor of 0.5 (the maximum value) is assumed. The friction stress rf is taken as the resolved shear stress equal to one-half of the vertical width of the stress-strain hysteresis loop [7, 8 ] , as is illustrated in Fig. 2(c). These values are found to be r = 340 MPa and Tf = 150 MPa. Therefore ~'Ttw = 2.4 × 10 s J m - 3
(14)
and TfTtw = 1.1 X 10 s J m-3
(15)
Substituting eqns. (13) - (15) into eqn. (10), we obtain Agch < 8.0 X107 J m - 3
(16)
As was mentioned earlier in this section, the chemical free-energy change accompanying the pseudotwinning transformation in Feo.slBeo.19 is difficult to calculate independently. However, as it is a complex quantity because of the unusual microstructural and crystallographic features of the alloy, an independent calculation of Agch is undertaken in Appendix A, to illustrate these complexities.
4. DISCUSSION
In thermoelastic martensitic transformations, Agch and Age1 must have opposite signs. Agcn , the driving force, has to be negative before the transformation will proceed, while Age~ , the transformation strain energy, opposes the transformation. Hence on cooling, as Agch becomes more negative, Agel can become more positive, thereby allowing the transformation to proceed while maintaining the thermoelastic force balance described in eqn. (1). The pseudoelastic force balance represents a more complicated case in that a mechanical as well as a chemical driving force, Ageh -TTT , is operative. Agch -- rTT must be negative in order for the pseudoelastic transformation to proceed; however, Agch itself does not have to be negative for this type of transformation. Indeed, the calculations performed here indicate that Agch is actually a positive quantity for pseudotwinning in Feo.slBe0.i9. If Agch = 0 during a pseudoelastic transformation, then eqn. (5) describes the force balance for twinning with elastic accommodation of the twin. In contrast, the usual elastic twinning reported in the literature [12, 13] typically involves some crystallographic feature which can give rise to a finite Agch term during the operation of the twinning shear and hence is not true twinning in the classical crystallographic sense [14]. Recent analyses of the thermodynamics of thermoelastic martensitic transformations have given rise to some question regarding the role of transformation strain energy in the attainment of a thermoelastic force balance. The Olson-Cohen approach, which is the basis of this paper, includes this term in an essential way. However, Tong and Wayman [15, 16] consider the transformation strain energy to be zero for the first plate to form, although they introduce a frictional term to oppose the formation of the growing plate. From eqns. (13) and {15), it is possible for us to calculate the following ratio for our Fe0.siBeo.19 alloy: 2/Xg~l >/ 0.45 (17) Tf'Ytw
-
-
which can be interpreted to mean that the force opposing the growing plate due to
114
elastically stored transformation strain energy is at least 45% of the frictional force opposing the growing plate. Therefore there is no justification for omitting the transformation strain energy term. Consistent with this conclusion is the experimental determination that the strain energy term in Cu-Ni-A1 martensites can be as large as 15% o f the magnitude of the corresponding chemical enthalpy change [17]. Agreement is found between the upper limits of Agch calculated by difference from eqn. (10) and independently in appendix A: Agch ~< 8.0 × 1 0 7 J m - s (by difference from eqn. (10)); Agch ~< 5.9 × 107 J m -3 (Appendix A). The agreement suggests that the observed pseudotwinned plates were close to meeting the elastic a c c o m m o d a t i o n condition of the pseudoelastic force balance. This assessment is consistent with the observation of reversible mechanical behavior in these samples [ 3 ] . Evidently, accommodation defects which may be created because of imperfect elastic accommodation contribute primarily to r~, the frictional resistance stress. Whereas eqn. (8) states that a force balance can still exist in the presence o f these defects, a state o f true thermodynamic equilibrium cannot exist if a frictional term, arising from any mechanism, is involved.
5. CONCLUSIONS A pseudoelastic force balance equation was proposed and analyzed as the mechanical analog of the thermoelastic force balance introduced by Olson and Cohen [4, 7 ] . Whereas for the thermoelastic force balance the driving free energy has only a chemical component, the pseudoelastic force balance has both chemical and applied work components. The force balance equation was applied to the complex pseudotwinning and pseudoelastic behavior observed in Fe0.slBe0.19 [3]. The force opposing the growth of the plate due to the elastically stored transformation strain energy is found to be at least half as large as the frictional force, and it cannot be ignored, as has been suggested previously [15, 16]. Upper limits to the chemical component of the driving force for pseudoelastic transformation were calculated by t w o
methods. The agreement between the two results suggests that the pseudotwins in Fe0.slBeo.19 are elastically accommodated to a substantial extent.
ACKNOWLEDGMENTS The authors are grateful to Professor S. M. Allen and Dr. J. W. Cahn for their very helpful contributions to the ideas and calculations presented in this paper. One of the authors (M.L.G.) .also wishes to express his appreciation to the AMAX Foundation for a Doctoral Fellowship which made his graduate work possible. This research is part of a long-range program sponsored at Massachusetts Institute of Technology by the National Science Foundation under Contract DMR79-15196.
REFERENCES
1 H. Warlimont and L. Delaey, Prog. Mater. Sei., 18 (1974). 2 R. V. Krishnan, L. Delaey, H. Tas and H. Warlimont, J. Mater. Sei., 9 (1974) 1536. 3 M. L. Green and M. Cohen, Aeta Metall., 27
(1979) 1523. 4 G. B. Olson and M. Cohen, Scr. Metall., 9 (1975) 1247. 5 J. R. Patel and M. Cohen, Acta Metall., 1 (1953) 531. 6 V. Raghavan and M. Cohen, Acta Metall., 20 (1972) 779. 7 G. B. Olson and M. Cohen, Scr. Metall., 11 (1977) 345. 8 K. Otsuka, C. M. Wayman, K. Nakai, H. Sakamoto and K. Shimizu, Acta MetaU., 24
(1976) 207. 9 G.B. Olson and M. Cohen, Proc. U.S.-Japan Semin. on the Mechanical Behavior Associated with Displacive Transformations, Rensselaer Polytechnic Institute, Troy, N Y , June 12 - 15,1979, p. 7. 10 J. D. Eshelby, Proc. R. Soc. London, Set. A, 241
(1957)376. 11 H. J. Leamy, E. D. Gibson and F. X. Kayser, Aeta Metall., 15 (1967) 1827. 12 M. V. Klassen-Neklyudova, Mechanical Twinning o f Crystals, Consultants Bureau, New York, 1964. 13 J. Y. Guedou, M. Paliard and J. Rieu, Scr. Metall., 10 (1976) 631. 14 J. W. Cahn, Aeta Metall., 25 (1977) 1021. 15 H. C. Tong and C. M. Wayman, Acta Metall., 22 (1974) 887. 16 C. M. Wayrnan and H. C. Tong, Ser. MetaU., 11 (1977) 341. 17 R.J. Salzbrenner and M. Cohen, Acta Metall., 27 (1979) 739.
115
APPENDIX A For ~-Fe-Be alloys, Agch can be expressed as the sum of three terms: A g c h = ~-a~ch^ rfpstw + "a~chA rr(G + B2)coh + ~'aSchA~(G + B2)sur
(A1) where ^~pstw is the chemical free-energy d i f ,.a6ch ference between the ~ plus B2 microstructure and the pseudotwinned microstructure (per volume of pseudotwin), Ag ~+ B2)coh is the change in the coherency strain energy of the plus B2 microstructure as a result of pseudotwinning (per volume of pseudotwin) and Ag(~ +s2)sur is the change in surface energy of the a plus B2 microstructure as a result of pseudotwinning (per volume of pseudotwin). A~P~tW arises from the inability of the $ch b.c.c, twinning shear to twin the B2 phase in the ~ plus B2 microstructure. The application of this shear to the B2 superlattice brings about the transformation P m 2 m ( B 2 ) -~ Cmmm, and the accompanying change A,p~tw in chemical free energy [A1] ~eh A,(~ + 82)coh arises from the tendency of the Sch coherent modulated ~ plus B2 microstructure (ref. A1, Fig. 5(c)) to be preserved during pseudotwinning (ref. A1, Fig. 8(c)), thereby causing a change in the state of the coherency strain energy of the two phases. Finally, /~,~(~ 5 c h + B2)sur arises because the surface area of the B2 particles is increased when they are sheared during pseudotwinning.
energy change; the a phase, being a disordered b.c.c, solid solution, is structurally unchanged by the twinning shear. X-ray diffraction experiments on single crystals of Feo.slBe0.z9 enable us to measure an average order parameter SAV2 defined as SAv 2 = SB22fB2
(A3)
SAV corresponds to the order parameter Ss2 of the B2 phase averaged over the total volume of the alloy. Therefore substituting eqn. (A3) into eqn. (A2) we find that A~cPstw _ SAV 2 /X V ~-~Dch
(A4)
Vm
A ground state calculation for h U can be made using the following relationship [A2] : U = --
No 2
(A5)
~-,Viqi
where U is tile energy of a configuration of atoms relative to the pure components (per mole), No is Avogadro's number, Vi is the ith nearest-neighbor interaction energy (per atom) and q~ is the number of ith nearestneighbor unlike bonds (per atom). Knowing the structure of the B2 and C m m m phases [A1], we can construct Table A1 which gives q values for first- and second-nearest neighbors. From this table A U = N o ( V z ~ V2)
(A6)
and Agpl~tw - S A v 2 N ° ( V l - - V 2 )
(A7)
Vm
A . 1 . C a l c u l a t i o n o f Agp~tw
The chemical flee-energy change attending the pseudotwinning of the B2 phase of the plus B2 microstructure, P m 3 m ( B 2 ) -* C m m m , can be expressed as A ecpstw 6ch
- SB22fB2 A V
Vm
T A B L E A1 q values for first- and second-nearest neighbors Phase
q i=1
i=2
4 2
0 2
(A2)
where Ss2 is the long-range order parameter of the B2 phase, fB2 the volume fraction of the B2 phase, A U the internal energy difference between the B2 and C m m m phases (per mole of pseudotwin) and Vm the molar volume of Fe0.siBe0.z9. The pseudotwin is defined as the region of the a plus B2 microstructure that has been sheared. However, only the volume fraction that was originally the B2 phase contributes to the chemical free-
Pm3m(B2) Cmmm
The entropy change neglected in the derivation of eqn. (A7) consists of configurational, vibrational and electronic terms. For a pseudotwinning transformation the configurational entropy change is zero because there is a one-to-one correspondence between atoms in the P m 3 m and C m m m phases [A3].
116
Although it cannot be demonstrated at present that the other entropy terms are unimportant, that assumption will be made here. V1 and V2 for ~ - F e - B e alloys have recently been calculated, using a pairwise interaction model [ A 4 ] . With our sign convention for the configurational energy in eqn. (A5), the values of V1 and V2 yield
Ag(~ +B2)c°h ~ 2.8 X 107 J m - s
(A12)
A.3. Calculation of Ag(c~+S2)sur We can write Aoc hp(ce+ B2)sur = g v AA 7s
(A13)
For Fe0.s1Be0.19 , SAV2 = 0.037 (from our X-ray diffraction measurements) and Vm = 6.7 × 10 - e m 3 tool -1. Therefore
where Nv is the number of B2 particles per unit volume, AA the difference in surface area between a Cmmm and a B2 particle and % the a - B 2 interfacial energy. The number o f B2 particles per unit volume is given by
A,pstw 5oh = 2.8 X 107 J m -3
Nv = fa2/~ur 4 3
V1 -- V2 = 8.37 × 10 -21 J atom -1
(AS)
(A9)
A.2. Calculation of A g ~ +B2)coh To evaluate Ag(c~+B2)coh, w e must obtain the coherency strain energies of both the parent and the pseudotwinned microstructures. By invoking the assumptions of the Crum theorem [A5] and taking the coherent B2 particles to be spheres, we can calculate the coherency strain energy of the parent microstructure. However, the particles no longer have a cubic symmetry in the pseudotwin and so the Crum theorem does not hold; furthermore, the elastic constants of the Cmmm phase are unknown. We shall assume that the coherency strain energy of the parent microstructure is an upper limit to the difference between the coherency strain energies of the parent and pseudotwinned microstructures, i.e. A.(~+ B2)coh ~ b~ce+ B2 5ch J-~coh
where r is the averageradius of the B2 particles. The change in surface area on pseudotwinning can be approximated from knowledge of the principal twinning strains [A6] :
AA ~ ~r2/4
(A15)
% is estimated to be 0.20 J m - 2 , before and after shearing of the B2 particles; this value, 200 erg cm -2, is probably an upper limit for the coherent interfaces involved here. Finally, it was determined from X-ray diffraction linewidths that r ~ 30 × 10 - l ° m for Feo.slBeoa9. Therefore A..,(~+B2)sur ~< 0.3 × 107 J m - s 6ch
(A16)
Adding eqns. (A9), (A12) and (A16), we obtain Agch ~< 5.9 X107 J m-a
(A17)
(AI0)
The coherency strain energy v~+ ~'~cohB2 of the plus B2 parent microstructure can be expressed as [A5] EC~+ B2 2 l + r (_~)2 ¢oh - ~ ~ - - ~ /~ fB2
(AI4)
(All)
where A V / V is the volumetric strain between the a and B2 phases. As in the calculation of eqns. (11) and (12), p = 1.25 × 1011 J m - 3 and v = ~. From transmission electron microscopy observtions, fB2 = 1 . Also, from the metastable phase diagram of ~ - F e - B e and the variation in lattice parameter with composition (ref. A1, Figs. 10 and 11) A V / V can be estimated as 4.5 X 1 0 - 2 . Therefore
R E F E R E N C E S F O R APPENDIX A
A1 M. L. Green and M. Cohen, Acta Metall., 27 (1979) 1523. A2 S. M. Allen and J. W. Cahn, Acts Metall., 20 (1972) 423. A3 G. F. Boiling and R. H. Richman, Acta Metall., 13 (1965) 709. A4 H. Ino, Acts 'MetaU., 26 (1978) 827. A5 J. W. Christian, The Theory o f Transformations in Metals and Alloys, Pergamon, Oxford, 1975, pp. 457 - 475. A6 J. W. Christian, The Theory o f Transformations in Metals and Alloys, Pergamon, Oxford, 1975, p. 52.