Boundary friction for thermoelastic martensitic transformations

Acta metall, mater. Vol. 39, No. 8, pp. 1995-1999, 1991 Printed in Great Britain

0956-7151/91 $3.00 + 0.00 Pergamon Press pie

BOUNDARY FRICTION FOR THERMOELASTIC MARTENSITIC TRANSFORMATIONSf Y. D E N G ~ and G. S. A N S E L L 2

t University of Science and Technology Beijing, Beijing 100083, China and 2Colorado School of Mines, Golden, CO 80401, U.S.A. (Received 28 April 1990; in revised form 25 September 1990)

Abstract--This paper is mainly to discuss the free energy, phase boundary friction and reversibility for thermoelastic martensitic transformations. A method is suggested to obtain, quantitatively, the energy consumed for boundary friction. The base for this calculation is the phenomenological theory suggested by authors before. Applications and results of the method for different data are studied. Comparison of this method with that of Ortin and Planes is discussed. The basic viewpoint in this paper is that the energy consumed for boundary friction converts totally into irreversible heat. This is different from the "entropy argument" of Ortin and Planes. The base of this argument is also reviewed. R r s u m ~ e t article est principalement consacr6 fi une discussion sur l'rnergie libre, le frottement fi une limite de phase et la rrversibilit6 des tranformations martensitiques thermorlastiques. On suggrre une m6thode pour obtenir quantitativement l'rnergie dissipre par le frottement interfacial. Le calcul est bas6 sur la throrie ph6nomrnologique proposre auparavant par les auteurs. Les applications et les rrsultats de cette mrthode sont 6tudires pour diffe~entes donnres exprrimentales. On compare cette m&hode avec celle d'Orin et Planes. Le point de vue fondamental de cet article rrside dans le fait que l'rnergie dissip6e par le frottement interfacial est totalement convertie en chaleur irrrversible. Ceci est diff6rent de 1' "argument d'entropie" d'Ortin et Planes. On consid/~re aussi les fondements de cet argument. Zusammenfassung--Diese Arbeit behandelt im wesentlichen freie Energie, Reibung an der Phasengrenze

und Reversibilit/it thermoelastischer martensitischer Umwandlungen. Eine Methode wird vorgeschlagen, mit der die bei der Reibung aufgewendete Energie quantitativ ermittelt werden kann. Grundlage dieser Rechnung ist die von den Autoren vorgeschlagene ph/inomenologische Theorie. Anwendungen und Ergebnisse der Methode werden fiir verschiedenen Datensfitze untersucht. Die hier entwickelte Methode wird mit der von Ortin und Planes verglichen. Der grundlegende Gedanke in dieser Arbeit ist, dab die bei der Phasengrenzreibung aufgewandte Energie vollst/indig in irreversible W/irme umgewandelt wird, im Gegensatz zu dem "Entropie-Argument" von Ortin und Planes. Der Hintergrund fiir dieses Argument wird ebenfalls kurz dargestellt,

1. INTRODUCTION According to the extents of deviation from ideal case, thermoelastic martensitic transformations can be classified into four types, as shown in Fig. 1. Figure l(a) shows an absolute ideal case, i.e. during the transformation, there is no boundary friction and therefore no hysteresis. Furthermore, no elastic strain energy and interfacial energy resist the transformation. Thus, the martensitic transformation and the reverse transformation conduct at the same temperature, i.e. the same vertical line. This transformation is too ideal to be practised in any experiments. Figure l(b) shows a transformation without boundary friction, therefore without hysteresis. But the transformation causes elastic strain energy and, thus, the transformation path is an incline line. However, the elastic energy is a reversible energy and the transformation is a reversible process. This is also an ideal case which has little possibility to be realized. tDue to circumstances beyond the Publisher's control, this article appears in print without author corrections.

Figure l(c) shows a transformation with boundary friction but without elastic energy. A carefully designed and conducted experiment could realize this type of transformation. Salzbrenner and Cohen [1] investigated transformations for single crystals with single interface, and the obtained results belong to this type. Figure l(d) shows transformation with both elastic energy and boundary friction. Quite a few investigators obtained this type of transformations [2-6]. Furthermore, even a transformation loop of a single plate martensite has been obtained [7]. In general, for a thermoelastic martensitic transformation loop, its slope results mainly from eleastic strain energy and its gap between heating and cooling segments comes from boundary friction. Based on this viewpoint, it is possible to separate following terms from each other: chemical free energy, elastic energy and interfacial energy, and the energy consumed for boundary friction. This separation is essential for understanding the nature of the transformation. This paper only deals with the friction, and a following paper in preparation will discuss the elastic energy and interfacial energy.

1995

1996

DENG and ANSELL: THERMOELASTIC MARTENSITIC TRANSFORMATIONS (o)

b)

where Gc is chemical free energy change, Go is elastic strain energy and G i is interfacial energy. For a mole material with m % transformed martensite, the free energy of the system can be written as

M t°

o%[

(5)

G = Go + gch m + gelm + gin m

where Go is the free energy for 100% parent phase, and g~h, g~ and gin are respectively chemical free energy change, elastic energy and interracial energy for 100% martensite in 1 mol material. Obviously, both standing for free energy of 1 mol material, Fn and G are the same physics parameter, thus they must be identically equivalent to each other

T

c)

(d)

100"/ M

Fn -= G. 0%

(6)

Both (a : N) and Go stand for the free energy during 100% parent phase for the material, thus

T

a : N -= Go. M = martensite percent

T = temperature

(7)

From the data of authors [3]

Fig. 1. Four types of thermoelastic martensitic transformations. (a) No friction and no elastic energy. (b) No friction and some elastic energy. (c) Some friction but no elastic energy. (d) Some friction and some elastic energy. Ortin and Planes [8] suggested a method to separate the above terms from calorimetry data based on a hypothesis "entropy argument". This argument will be discussed later and its base seems rather weak. In this paper, a method to obtain the energy consumed for boundary motion friction is suggested. The applications and results for different authors' data [1, 2] are studied.

gch = 1.612T -- 409.5 J/tool.

(8)

When external stress equals zero, and from (I)-(8), then [(0.06451T - 15.33)m + 0.5 m 2] -N - ( 1 . 6 1 2 T - 409.5)m + (gel + gin) m-

(9)

Then, compare the term of ( T ' m ) in both sides and obtain N = 24.99 J/mol.

(10)

Thus F~ = G0+(1.612T - 383.1)m + 12.49 m 2

2. FREE ENERGY FUNCTION

+(1.847 x l O - 3 T - O . 4 8 5 8 ) a m

A phenomenological theory for thermoelastic martensitic transformation has been suggested [3], in which the free energy (density) function of a system is

where m is the percent of transformed martensite, a is external applied stress, and a, b, c, e are constants for argument m (but can vary with temperature). For a C u - 2 9 % Z n - 3 % A I alloy, these constants are [3] a = a(T)

")

)

b = 0.06451 T - 15.33 c=0.5 ". (2) e = 7.393 x 10 -5 T - 1.944 x 10-2(MPa) -t The F function is dimensionless free energy. Now, let us introduce a unit converting factor N to convert F into an ordinary unit system. Then F n = N . F.

(3)

The parameter N can be determined as following. According to classictheory [9],the freeenergy change caused by martensitic transformation is AG = Gc + G, + G i

(4)

(11)

This is the free energy function of the system for one mole material in ordinary unit system. 3. BOUNDARY FRICTION

(1)

F = a + bm + cm 2 +eam

J/mol.

According to the phenomenological theory [3], the boundary motion friction can be expressed as a dimensionless friction function F ' = n ' -- T - S '

(12)

where H ' is friction quasi-enthalpy and S ' is friction quasi-entropy. By using the data in [3] and the factor N, it is obtained that Hr=(-96.84+0.4081a)m

J/mol

Sr=(0.4875 = 1.847 x 10-3a)m Fr=Hr-TS

"~

J/mol.KI

(13)

r

Then Fr, H, and S~ are all in ordinary unit system, and Fr is the energy consumed for the boundary friction. It is noteworthy that T in Fr function is the temperature in middle point between the same m % of cooling and heating processes. From (13) and noticing the linear relationship between m and T in the range of 20-70% martensite, it is easy to calculate

DENG and ANSELL: THERMOELASTIC MARTENSITIC TRANSFORMATIONS 100

1997

Lc=m +0.1666T-48.99=0

l.

L h = m + 0.09999T - 31.79 = 0

J

80

(21)

The dimensionless free energy function and friction function are respectively [3]

60

F = 0.5 mZ+(0.1333T - 40.39)m

(22)

F ' = (0.03333T -- 8.599)m.

(23)

According to Tong and Wayman [10] To = 0.5(M s + M r ) = 306 K 184

200

216

252

248

264

(24)

thus the entropy change is

Temperoture (K)

A S = A H / T o = - 1.683

Fig. 2. Transformation loop for Cu-29% Zn-3% AI alloy. the energy consumed for the boundary motion friction in this range. That is AF~ = 8.49

J/tool.

(14)

Now the energy consumed for boundary friction for a transformation cycle can be obtained based on the assumption that the energy is proportional to the area of transformation loop. First, Fig. 5 in [3] can be rewritten as Fig. 2, and then the area of the portion between 20% M and 70% M can be measured as Ar(20-70%) =415% K.

A A = AF~/A r (20-70 %)

= 2.047 x 10 -2

(J/tool.)/(% K).

(16)

From Fig. 2, the area of the transformation loop has been measured as Ar(100%) = 1034% K.

(17)

Thus, the energy consumed for boundary friction in one transformation cycle is F~= AA x A~(100%)=21.2

J/mol.

(18)

In the next section, some other auhors' data [1, 2] will be used to exemplify this calculation method.

N = 12.63

Mf = 288 K ,

As = 3 1 8 K ,

Af=308K

(19)

and AH in [1] will be referred AH = - 515

J/mol.

(26)

Fn = 6.313 m2+(1.683T + 509.9)m Fr = ( 0 . 4 2 0 8 T - 108.6)m

J/mol (27)

J/mol.

(28)

Through the same procedure in Section 3, the energy consumed for boundary friction is Ef = 19 J/mol.

(29)

This result is obtained from Fig. 2(d) of [1], i,e. for single-crystal with multiple-interface. By measuring the area of Fig. l(a) in [I], it is easy to obtain the energy consumed for single-crystal single-interface friction as E s = 23

J/mol.

(30)

4.2. Cornelis and Wayman [2], Cu-39.5 at.% Zn

Referred to Fig. 1 of [2], following data have been obtained: linear relationship exists between approximately 37-70% martensite; during cooling the 37% M corresponds to 188 K and 70% M to 178 K; during heating the 37% M corresponds to 201 K and 70% M to 187 K; and To - 243 K. Then L c = m + 0 . 0 3 3 0 T - 6.574 = 0 1 Lh = rn + 0.0236T -- 5.108 = 0 )

Figure 2(d) in [1] will be referred and data from this loop were measured approximately as follows Ms = 294 K,

J/mol.

Thus Fn and Fr in ordinary unit system are respectively

4. SOME EXAMPLES

4.1. Salzbrenner and Cohen [1], C u - 1 4 % A I - 2 % Ni alloy

(20)

From these data, it is obtained that cooling and heating states are respectively [3]

(25)

Through the same procedure in Section 2, the converting factor N for this alloy is

(15)

Thus, from (14) and (15), the energy consumed per unit area is

J/mol.K.

(31)

and F = 0.5 m2+(0.02830T - 5.841)m "] F ' = (4.700 x 10 3T - 0.7330)m

J~ .

(32)

From [11], AH of an alloy C u - 4 0 a t . % Zn is -199.3 J/tool. Although Ms is very sensitive to Zn content for Cu-Zn alloys and C u - Z n - X alloys (X, the third element), AH is not very sensitive for small change of Zn content, of. [3] Table 1. Thus as the first order approximation, the AH in [10] could be used for the alloy in [2]. Then AS = - 199.3/243 = - 0 . 8 2

J/mol-K.

(33)

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DENG and ANSELL: THERMOELASTIC MARTENSITIC TRANSFORMATIONS

Through the procedure in Section 2 N =28.98

J/mol

]

Fn = 14.49 m 2 + ( 0 . 8 2 0 T - 169.3)m F r = ( 0 . 1 3 6 2 T - 21.24)m

J / m o l i ~ (34)

J/mol

and through the procedure in Section 3 E f = 5.4

J/tool.

(35)

Other data, for instance [4-7], could be used for the friction calculation as soon as the relationship between M % and T could be obtained. 5. DISCUSSION First, let us compare the results obtained by different methods but using the same source data [1], as shown in Table 1. It can be seen that, although there are some differences, the results are in the same order. Considering the measure errors from figures, the results could be thought reasonably close to each other. Two points may be noteworthy. First, the value 26 J/mol was almost directly obtained from experiment with the least assumptions and it might be the most reliable. Secondly, in [1] the loop area of Fig. 2(a) is larger than that of Fig. l(d), it is reasonable that the energy consumed for former is more than that for latter in proportion to the area difference. Although the results are close, the bases to obtain them are different. The base for Ortin and Planes [8] is "entropy argument", i.e. "the condition of thermoelastic equilibrium is equivalent to a null change in the entropy of the universe". The authors of [8] stated that "a parent-to-martensite thermoelastic transformation is not reversible in the thermodynamic sense". We agree with this statement. Because of existence of hysteresis, the themoelastic martensitic transformation is not a reversible process in thermodynamics sense, in other words, it is an irreversible process. According to [12], the irreversibility of a process is

I = T.Sp

(36)

where Sp is the "entropy production" (the change of entropy of the associated universe). For a thermoelastic martensitic transformation, as a thermodynamics irreversible process, I > 0, thus Sp > 0.

(37)

Therefore the condition of thermoelastic equilibrium is not equivalent to a null change in the entropy of Table 1. The energyconsumed for boundary friction

Authors

Ef (J/mol)

Sample

S and C [1]

26

Single-crystal, single-interface [1]

O and P [8] O and P [8] This paper This paper

51 49 19 23

Single-crystal,single-interface[1] Single-crystal,multiple-interface[1] Single-crystal,multiple-interface[1] Single-crystal,single-interface[1]

the universe. It seemed to us that "entropy argument" needed further clarification and justification. In [3], we suggested to distinguish the equilibrium state, static state and transformation state. The "thermoelastic equilibrium" could be any one of them. The thermodynamics reversible process is and only is the process which always keeps in the equilibrium state. Obviously, the thermoelastic martensitic transformation process is an irreversible process and the irreversibility comes from hysteresis which results from the boundary motion friction. In our opinion, the energy consumed for the boundary friction converts totally into irreversible heat dissipation. This case is very similar to that of the ordinary sliding friction and the boundary behaves as an irreversible thermal engine. The nature of the boundary friction is not clear yet. According to Ortin and Planes [8], the energy or work for driving the boundary motion does not release as irreversible heat, but releases as mechanical wave energy. This is a prerequisite for the entropy argument. However, it seemed that both the statement and the argument needed justification. According to Olson and Cohen [13], the energy for driving the boundary motion is Vtn"g0"~T , where V m is the molar volume, ~'T is transformation shear and z0 is shear stress required to move an interface. The energy could not convert into shear elastic energy which is reversible and can not cause hysteresis. The energy could not convert into internal energy which is part of the chemical free energy. As indicated in [13], the energy is dissipated. The dissipated energy could convert into mechanical wave energy as indicated by Ortin and Planes [8], but it also could convert into irreversible heat as argued by us. 6. CONCLUSIONS 1. A method is suggested to calculate the energy consumed for the boundary friction. The base of this calculation is the phenomenological theory suggested before, and the data needed are the transformation entropy and the transformation loop. The obtained results are close to those of other investigators. 2. The basic viewpoint for the thermoelastic martensitic transformation is that it is an irreversible process and the energy consumed for the boundary friction converts totally into irreversible heat dissipation. This viewpoint is different from that of "entropy argument".

Acknowledgements--The work reported in this paper has been supported by The Office of Naval Research, U.S,A. and is supported by the National Natural Science Foundation of China No. 59071058. REFERENCES

1. R. J. Salzbrenner and M. Cohen, Acta metall. 27, 739 (1979). 2. I. Cornelis and C. M. Wayman, Scripta metall. 10, 359 (1976).

DENG and ANSELL: 3. 4. 5. 6. 7.

THERMOELASTIC MARTENSITIC TRANSFORMATIONS

Y. Deng and G. S. Ansell, Acta metall. 38, 69 (1990). J. Li and G. S. Answell, Metall. Trans. 14A, 1293 (1983). C. M. Friend, Scripta metall. 20, 995 (1986). J. Baram and Rosen, Acta metall. 30, 665 (1982). A. Amengual, F. Garcias, F. Marco, C. Seggi and V. Torra, Acta metall. 36, 2329 (1988). 8. J. Ortin and A. Planes, Acta metall. 36, 1873 (1988). 9. L. Kaufman and M. Cohen, Prog. Metal Phys. 7, 165 (1958).

AM 39/8~8

1999

10. H. C. Tong and C. M. Wayman, Scripta metall. 11,341 (1977). 11. W. Ameodo and M. Ahlers, Acta metall. 22, 1475 (1974). 12. V. M. Faires and C. M. Simmang, Thermodynamics, 6th edn, pp. 116-118. Macmillan, New York (1978). 13. G. B. Olson and M. Cohen, Scripta metall. 9, 1247 (1975).