Mechanical properties of a Cr–Ni–Mo–Al–Ti maraging steel in the process of martensitic transformation

Materials Science and Engineering A308 (2001) 25 – 37 www.elsevier.com/locate/msea

Mechanical properties of a Cr–Ni–Mo–Al–Ti maraging steel in the process of martensitic transformation K. Nagayama a, T. Terasaki a, K. Tanaka a,*, F.D. Fischer b, T. Antretter b, G. Cailletaud c, F. Azzouz c a

Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6 -6, J-191 -0065, Hino/Tokyo, Japan b Institute of Mechanics, Montanuni6ersita¨t Leoben, Leoben, Austria c Centre des Mate´riaux, Ecole Nationale Supe´rieure des Mines de Paris, E6ry Cedex, France Received 29 September 2000; received in revised form 8 December 2000

Abstract Mechanical properties of a Cr–Ni–Mo–Al–Ti maraging steel are studied experimentally in the process of martensitic transformation. The transformation-start and -finish temperatures are determined from the dilatometric curves under a tensile, compressive or shear hold stress. An anomalous temperature-dependence of the yield stress is investigated. By checking the closing of the dilatometric loop in a full heating–annealing– cooling thermal cycle under the applied hold stress, a back stress is confirmed to exist in both the axial and radial directions in the thin-walled tubular specimen, and its initial value is identified. The annealing condition is proved to have a marked effect on the initial value of the back stress. The back stress is shown to evolve in the progress of transformation. The evolution of the TRIP strain is evaluated under the hold stress in both the axial and shear direction. Especially when the hold stress is removed in the process of transformation, a backflow, due to the Magee effect, is observed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: TRIP; Martensitic transformation; Martensite-start line; Yield stress; Back stress; Dilatometric loop; Greenwood– Johnson effect; Magee effect

1. Introduction The thermomechanical response of materials associated with the martensitic transformation has for long been one of the challenging targets of research in the field of continuum mechanics [1 – 5]. And great efforts still continue, aiming to construct theoretical frameworks of describing, for example, the alloy response in the hardening processes, the transformation-induced plasticity (TRIP), the shape memory alloy performance under thermomechanical loads, the transformation toughening in ceramics [6 – 10]. The theories are expected to have a high impact on the material design with respect to selecting an appropriate thermomechanical treatment for the optimal design of microstructures in the materials. * Corresponding author. Tel./fax: +81-42-5858654. E-mail address: [email protected] (K. Tanaka).

Restricting the present discussion to the issues in TRIP [11 –14], the classical TRIP theory [2–5] explains merely the Greenwood –Johnson effect, the elastic and plastic accommodation due to transformation strain (the eigen strain) induced in the newly transformed martensite variants. The Magee effect or the orientation effect, representing the orientation process due to formation of the preferred martensite variants, plays, however, an important role in many practical thermomechanical loading situations [12,13]. For example, Cailletaud et al. [15 –17] have investigated the TRIP performance of a Cr –Ni –Mo –Al maraging steel under full and partial unloading in the process of martensitic transformation during cooling. The phenomenon cannot be explained by the classical TRIP theory based on the Greenwood –Johnson effect. The TRIP theory, capable of describing both effects, should urgently be established to meet growing engineering demands.

0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 9 9 9 - 7

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

26 Table 1 Chemical composition (wt.%) Cr

Ni

Mo

Al

Ti

C

Si

Mn

P

S

N

Fe

12.15

9.05

2.03

0.70

0.35

B0.01

0.05

0.03

0.009

B0.002

0.0045

Bal.

In fact, some attempts proved promising [18 –22]. Based on their micromechanical modeling, Fischer and co-workers [23,24] have successfully carried out FEM simulations to show that the Magee effect may have a dominant influence on the evolution of TRIP strain under the thermomechanical conditions which Cailletaud et al. [15 – 17] had investigated. They also pointed out that the back stress, similar to the one in plasticity [25], has to be introduced in transformation thermomechanics when formulating the constitutive equation of the TRIP strain. In order to establish a full theoretical framework describing macroscopically or microscopically the material behavior in the process of martensitic transformation and plastification, the steady accumulation of sound data under broad thermomechanical conditions is still highly necessary. In this paper the TRIP behavior in the same steel, Cailletaud et al. [15 – 17] have investigated, is studied experimentally under tensile, compressive and shear load. The dilatometric loops in the heating – annealing – cooling process are carefully monitored with or without hold stress. The stress – dependence of the martensitestart and -finish temperatures, which are determined from the dilatometric loops, is an issue to be studied. The yield stresses identified from the isothermal stress – strain curves exhibit an anomalous temperature-dependence. The issue is discussed in view of the martensitic transformation progressing in the material during the test. The dilatometric loops under hold stress conclusively prove the existence of the back stress in both the axial and radial direction of thin-walled tubular specimens. The effect of annealing is checked by evaluating, the value of the jump in strain during transformation by means of the data pertaining to the dilatometric loops. Furthermore, the evolution of the back stress in the process of transformation is qualitatively proven.

from 793 to 828 K for 4 h and was air-cooled. The grains are equiaxed shape in all directions after a series of heat treatment. The average grain size at RT is about 10 mm. The material was chosen due to its excellent quenching capability and the absence of diffusional transformations. The martensite-start temperature is lower than 470 K, and the martensitic transformation is fully completed during cooling down to room temperature (RT). The reverse transformation from the martensite phase to the austenite phase takes place during heating, so that the austenite microstructures can fully be restored after the annealing process explained in Section 4. This allowed us to use the same specimen dozens of times. The chemical composition (wt.%) and mechanical properties of the material are listed in Table 1 and Table 2. The thin-walled tubular specimen in Fig. 1, with an internal diameter of 16.5 mm, an outer diameter of 19.5 and 12 mm gauge length, was machined from the as-forged bars. Tests were performed on a multiaxial servo-hydraulic testing machine (SHIMADZU, EHF-ED5/TD05-10L) equipped with a high frequency induction heater. The axial displacement and the rotation angle were measured with a multiaxial DTF extensometer, and the temperature by means of a platinum –platinum rhodium thermocouple spot-welded at the center in the gauge length of the specimen. The specimen was heated up to 1113 K, with the heating rate 0.5 K s − 1, and held there for 30 min to Table 2 Mechanical properties Young’s modulus Shear modulus E [GPa] G [GPa]

Yield stress |0.2 [MPa]

RT 194

RT 900

473 K 179

RT 73.5

2. Experimental procedure and fundamental material performance

2.1. Alloy steel and experimental procedure The material tested is a maraging TRIP steel with structural hardening, Marval X12 provided by Aubert et Duval [15,16]. The material, initially forged, was heat-treated at 1113 K and cooled in air. Subsequently the material was maraged at the temperature ranging

Fig. 1. Test specimen.

473 K 171

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

27

Fig. 2. Thermomechanical loading path.

ensure full austenitization. The effect of the annealing (1113 K× 30 min) is discussed later. The specimen was then cooled down to RT in air with the cooling rate 2 K s − 1, by controlling both the power input and the amount of the pressurized air blown to the outer and inner surfaces (cf. Fig. 2a). The tests were fully controlled by a personal computer, and the stress, temperature and the strain responses of the specimens were stored in the PC for later analysis. In the stress holding tests (cf. Fig. 2a and b), the tensile, compressive or shear stress was applied during cooling, with the (von Mises equivalent) stress rate of 80 MPa s − 1, at 473 K before the start of the martensitic transformation. The applied stress was then held constant in the subsequent cooling down to RT. The dilatometric loops drawn from the strain and temperature output in the whole thermomechanical loading process provide the information about the martensitestart condition and the back stress. The evolution of the TRIP strain is also a target of investigation in this test. Isothermal loading tests were performed at the (von Mises equivalent) strain rate of 5×10 − 6 s − 1: The specimen was isothermally loaded in tension/compression or shear direction at a test temperature in the process of cooling (Fig. 2d). The yield stresses |0.005, ~*0.05 and |0.05 were determined as the 0.05 and 0.005% proof stresses, respectively, from the stress –strain curves.

Fig. 2(a) and (c) illustrates the thermal and mechanical load paths in the unloading test; the specimen is loaded up to 80 MPa. The martensitic transformation starts in the subsequent cooling process under the applied stress. The stress applied was then removed partially in the subsequent process of martensitic transformation. The change in the evolution of the TRIP strain was studied in order to evaluate the Magee effect in addition to the classical Greenwood –Johnson effect. The evolution equation of the back stress was also estimated qualitatively. In this study, the von Mises equivalent stress and strain ~*= 3~,

k*= k/ 3,

are employed to compare the shear stress ~ and the shear strain k with their axial counterparts.

2.2. Fundamental material performance In a successive heating and cooling process with no applied stress (|h = 0 MPa), a dilatometric loop labeled by |h = 0 MPa in Fig. 3 was obtained in the axial direction. The loop clearly shows that the specimen has fully transformed to the martensite phase at RT, and the austenitization starts at about 900 K during heating, exhibiting a transformation shrinkage. The martensitic transformation starts at about 420 K in the subsequent cooling process, showing a marked trans-

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formation expansion. The loop is not closed, and a negative amount of strain, mrat has remained after a full thermal cycle. The martensite-start and -finish temperatures,

Fig. 7. Effect of hold stress on ratchet strain.

marked with the hollow circle and box, respectively, on the loop, are identified by Fig. 3. Dilatometric loop under applied stress.

Fig. 4. Definition of martensite-start temperature.

Ms0 = 424 K,

Mf0 = 352 K,

where the subscript 0 denotes the value for the closed dilatometric loop. The martensite-start temperature is defined as the temperature Ms2 in Fig. 4, at which the strain-temperature curve starts departing from the thermal contraction line of the parent phase [26]. The volume expansion due to martensitic transformation, l, is identified by means of the closed dilatometric curve to be l/3=64× 10 − 4. Detailed discussion will be given in Section 3. The coefficients of thermal expansion in the parent and martensite phases were almost constant in the temperature range considered, and were given by hA = 20× 10 − 6 1/K, and hM = 12× 10 − 6 1/K, respectively.

3. Back stress

Fig. 5. Ratchet deformation during thermal cycling under no applied stress.

Fig. 6. Change in ratchet strain during thermal cycling under no applied stress.

Since a certain negative amount of residual strain is observed after a heating/cooling thermal cycle on the axial dilatometric loop under no applied stress (see Fig. 3), a ratchet deformation may progress in the negative direction, if the thermal load is cycled. The ratchet deformation was, in fact, observed during thermal cycling under no applied stress, as shown in Fig. 5. Fig. 6 clearly reveals that the ratchet strain mrat per cycle, defined in the figure, converges to a limit value of 22× 10 − 4 after almost four cycles of thermal loading. The progress of the ratcheting becomes constant in the subsequent thermal cycling. The ratchet strain mrat stands for the value after four thermal cycles in this study. The value of the ratchet strain depends on the hold stress |h applied in the cooling process, as demonstrated in Fig. 3. The relation, illustrated in Fig. 7 with solid boxes, suggests that the dilatometric loop closes (mrat = 0) under a certain value of the applied hold stress. The situation is fully confirmed also by the

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

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The constitutive equation of the TRIP strain can be expressed by means of an effective stress defined by |eff = | −X0.

Fig. 8. Effect of hold stress on dilatometric loop in shear direction.

consideration below; The total axial strain m(T) corresponding to the temperature T in the process of martensitic transformation during cooling may be written as m(T)= mMs +(l/3)x − (Ms −T)[hA(1 − x) +hMx] + mTRIP,

(1)

where mMs =m(Ms) and mTRIP stand for the strain at the martensite-start temperature Ms and the TRIP strain, respectively. The volume fraction of the martensite is denoted by x, and the transformation strain, in other words, the eigen strain due to transformation, by l/3, where l represents the transformation expansion. At the instant when the transformation finishes, i.e. x =1, T= Mf and mMf =m(Mf), Eq. (1) reduces to m max TRIP +l/3=mMf − mMs +(Ms −Mf)hM,

(2)

where m means the maximum value of the TRIP = mTRIP(Mf). Since the value of the rightstrain, m hand side in Eq. (2) can be evaluated from each dilatometric loop, one can draw a relation between the value m max TRIP +l/3 and the applied hold stress |h. The result is plotted in Fig. 7 with hollow circles. The following conclusions can be drawn from Fig. 7: If the applied hold stress is 28.4 MPa (X0z, say), the dilatometric loop fully closes. In other words, the back stress tensor X0 exists in the specimen, and its axial component is X0z. Since no TRIP strain should be observed under the hold stress X0z, and mrat =0 holds in this case, the value of the transformation strain can be evaluated as l/3= 64× 10 − 4. max TRIP max TRIP

(3)

In each test leading to the results plotted in Fig. 7, the shear strain k* was almost unchanged as shown in Fig. 8. Since neither a transformation expansion nor a thermal shrinkage is observed in the shear direction, the shear strain k* measured in Fig. 8 is just the shear TRIP strain k*TRIP. The material behavior given in the figure proves, therefore, that the back stress tensor X0 has no shear component. In order to estimate more precisely the anisotropy of the back stress tensor, the dilatometric performance was identified under no applied stress with tiny specimens, 2 mm in diameter and 12 mm in length, cut out in the axial, radial and circumferential directions of the forged bar [16,17]. The ratchet strains determined from the dilatometric loops in each specimen are tabulated, together with other data, in Table 3, showing a strong orientation dependence. The dilatometric loops were open, and the ratchet deformation was observed in both the axial and circumferential specimens, whereas the dilatometric loops were almost closed, exhibiting no ratchet deformation, in the radial specimens. The tests suggest that the back stress tensor X0 has a non-zero component X0q in the circumferential direction, the value of which is roughly the same as X0z = 28.4 MPa. Its radial component, however, turns out to be zero; X0r = 0 MPa. It should be emphasized here that no marked anisotropy was observed in this material with respect to both the coefficient of thermal expansion in the martensite and austenite phases, and the martensite-start temperature. The back stress, which may be understood as an internal variable in the theory of transformation thermomechanics [23,24], must evolve in the transformation process, just like the back stress in plasticity changes its value in the process of plastic deformation [25]. So the back stress tensor X0 is denoted here with the subscript 0, emphasizing that only its initial value is identified in the present section. The evolution of the back stress will be discussed qualitatively later in Section 7.

Table 3 Anisotrophy of material properties Orientation

Longitudinal Radial Circumferential a

Ratchet strain mrat×10−4

14.0 (22.0)a 1.35 11.0

Coefficient of thermal expansion

Volume expansion

hM

hA×10−6 [K−1]

×10−4

11.6 11.8 12.1

18.3 19.3 19.3

42.86 64.25 50

The value identified by means of the large specimen in Fig. 1.

Ms0 [K]

429 429 421

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

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Ta, held there for ta min, and then cooled down at the cooling rate of around 99 K min − 1. No stress is applied in the process of cooling. The dilatometric curves in the whole thermal process, with no applied stress, are the object to be investigated. The effect of the following annealing conditions (Ta × ta) was studied:

Fig. 9. Effect of annealing condition on dilatometric loop.

*1113 K× 30 min

*1113 K ×400 min

*1173 K× 30 min

*1173 K ×60 min

*1173 K× 120 min *1208 K× 30 min *1273 K× 30 min*1373 K×400 min *1373 K× 30 min *1473 K× 30 min

Fig. 10. Evolution of axial drift strain.

Fig. 11. Effect of annealing condition on strain jump during transformation.

4. Annealing condition In order to evaluate the effect of the annealing condition on the thermomechanical performance of the material, the specimens in Fig. 1 were heated at a heating rate of 0.5 K s − 1 to an annealing temperature Ta, held there for ta min, and then cooled down with a cooling rate of 2 K s − 1. For comparison similar tests were performed by means of tiny specimens, a diameter of 2 and 25 mm gauge length. Specimens, sampled from the bar material in the axial (abbreviated by A) and radial (abbreviated by R) directions, were heated at a heating rate of 5 K min − 1 to the annealing temperature

Some axial dilatometric loops for the large specimen are altogether illustrated together in Fig. 9. A certain amount of the drift strain mdrift was observed in each test during annealing, which is explicitly shown in Fig. 10. The martensite-start/finish temperatures, marked on the dilatometric curves in Fig. 9, exhibit no systematic effect of the annealing conditions, and the coefficients of thermal expansion of both martensite and parent phases are also not influenced by the annealing conditions. The effect of the annealing condition on the back stress is represented implicitly in Fig. 11, which plots the transformation jump mMf − mMs measured from the dilatometric curves. It should be noted that the data of both large and tiny specimens exhibit no appreciable discrepancy. The result clearly reveals that the material, exhibiting a notable anisotropy even after the annealing at the low temperature, becomes almost isotropic after annealing at 1473 K for 30 min. The initial texture built in during the forging process can, therefore, be erased by the annealing of 1473 K× 30 min. The figure also shows that the effect of the annealing time is small compared to the effect of the annealing temperature. The 30 min of annealing time is enough to acquire a stable material response. Substituting the converged value mMf − mMs =56× 10 − 4 at 1473 K in Fig. 11 to Fig. 2, and taking into account the fact that m max TRIP = 0 in the present situation, the transformation strain is evaluated to be l/3=66× 10 − 4, which is reasonably agree with the value 64× 10 − 4 determined in Section 2 from the closed dilatometric curves. According to the discussion in Section 3, concluding that the closed dilatometric curve under stress-free state means no back stress in the specimen, the specimens are free from the back stress after the annealing of 1473 K×30 min. A precise metallurgical investigation of the material response during annealing process, the change in the

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

texture structure due to forging and its influence on the macroscopic performance of the material, will be reported elsewhere. In this study, the annealing conditions are fixed at 1173 K×30 min. The initial back stress X0 is, therefore, regarded as an intrinsic material property.

5. Kinetics A closed dilatometric loop, obtained under the hold stress |h = 29 MPa ( :X0z ), is given in Fig. 3. It should be noted that in this case mTRIP =0 in Eq. (1); m(T)=mMs +(l/3) x− (Ms0 −T) [hA(1 − x) +hMx], (4) l/3= mMf −mMs +(Ms0 −Mf0) hM, where the martensite-start and -finish temperatures are denoted as Ms0 and Mf0, respectively, in order to emphasize that the process is carried out under the effective stress-free condition |eff z =|h −X0z = 0. The fraction of martensite x is calculated at each temperature T by means of Eq. (4) and the strain data for the closed dilatometric loop in Fig. 3, if the material data of l/3, Ms0, hA, and hM are used. The x–T relation evaluated for two specimens is plotted in Fig. 12, in which the best fit of the Koistinen – Marburger kinetics [27]

x=1− exp[− a(Ms0 − T)],

31

a= 6.3× 10 − 3 1/K, (5)

is represented by the solid line. The Koistinen –Marburger kinetics fails to describe the initial stage of transformation in which the transformation rate gradually increases starting from zero. Leaving the exact identification of the volume fraction x under general thermomechanical processes [28,29] as an issue for the next studies, the extent of transformation is measured in this study by the undercooling Ms0 − T or Ms −T.

6. Martensite-start line and yield stress In the stress holding tests (cf. Fig. 2 a and b) with the axial or shear hold stress, the martensite-start and -finish temperatures, Ms and Mf, were determined from the dilatometric curves, m− T or k*− T, as a ‘proportional limit’ (by means of the definition Ms2 in Fig. 4). The results are plotted in Fig. 13, in which thevertical axis represents the effective stresses defined by; |eff z = |h − X0z |eff rq = ~*h

in the axial direction, and

in the shear direction.

It should also be noted that in the shear stress holding tests the axial hold stress |h = X0z and the shear hold stress ~*h are applied at the same time in order to realize a full stress-free state in the specimen in the axial direction. The following material characteristics should especially be emphasized: The martensite-start line (Ms-line) is represented by the Clausius –Clapeyron like linear relation which depends on the direction of the hold stress. The lines are characterized by their slopes; 0 c+ = 3.7 MPa/K M

in tension,

0 c− M = − 14.9 MPa/K

c M = 8.9 MPa/K Fig. 12. Transformation kinetics under no hold stress.

Fig. 13. Transformation-start and -finish lines.

in compression,

in shear.

The dependence of the martensite-start line on the stress direction has widely been recognized in shape memory alloys [30,31], and has been discussed theoretically by evaluating the contribution of the mechanical 0 driving force [24]. The order of the slopes, c + M Bc M B −0 c M , is observed in shape memory alloys as well. * The martensite-finish lines are almost independent of the direction of the hold stress as opposed to the results in SMAs [30 –32], and need to be investigated more precisely. The stress –strain curves, determined in the straincontrolled isothermal loading tests (cf. Fig. 2d), are illustrated in Fig. 14 and Fig. 15 for the tensile tests. In the high temperature range, around 460 K and higher,

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K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

Fig. 14. Stress–strain curves.

Fig. 15. Stress–strain curves.

Fig. 16. Comparison of two sets of stress–strain curves.

no martensitic transformation took place during loading. The curves exhibit a normal gradual decrease of the work-hardening with the progress of deformation. In the intermediate temperature range, from about 440 to 460 K, the transformation expansion and the TRIP strain were induced at the start of the martensitic transformation during loading, resulting in a stress drop on the stress – strain curves since the tests were controlled by the total strain. The high yield stress and the high work-hardening in the low temperature range (B430 K) are attributed to the ‘composite effect;’ in this temperature range, specimens consist of the harder thermally-induced martensite and the softer retained austenite phases at the start of the test. The amount of the martensite phase is larger at the lower temperature, and it increases during loading due to stress-induced martensitic transformation.

The stress –strain curves at 428 and 453 K, marked with thick lines in Fig. 14, are good examples to clarify the phenomenon: * In the test at 428 K (: Ms0), the martensite phase starts being induced at the early stage of loading, producing the transformation strain and the TRIP strain, and resulting in a low initial slope of the stress –strain curve. The material exhibits a high work-hardening which can be explained by the composite effect. * On the contrary, in the test at 453 K (\Ms0), the non-linear response appears only when the load path crosses the Ms-line at higher stress level. A sudden increase of the strain due to transformation causes a stress drop on the stress –strain curve since the test is controlled by the total strain. Thus the two stress –strain curves discussed above cross when the load increases. Another set of stress –strain curves, marked by ‘Full martensite specimen’ in Fig. 16, is obtained as follows: After a normal heating –cooling process down to RT under no hold stress, the specimens were heated again to a test temperature, also under no hold stress, and finally loaded in the axial direction isothermally. It should be noted that the specimens were in a full martensitic state at the start of the loading tests. The data labeled by ‘At cooling stage’ are the same ones as in Fig. 14 and Fig. 15. The stress –strain curves of the full martensite specimen are reasonably higher, exhibiting higher work-hardening than the specimens composed of the martensite and parent phases. The same phenomenon can be detected in Fig. 15 as well in the temperature range lower than about 430 K. The lower the test temperature is, the higher work-hardening exhibits the stress –strain curve, since the amount of the martensite phase becomes larger, approaching to the situation of the ‘Full martensite specimen’ in Fig. 16. Some tests in Fig. 16, denoted by TR40, were carried out with the higher strain rate 10 − 4 s − 1. A significant effect of strain rate is observed. The viscosity of the austenite phase might play some role. The yield stresses of the fast tests are higher, and no significant stress drop is observed. This is due to the fact that the tests were controlled by the total strain. In both the compression and shear tests a similar behavior was observed in Figs. 14–16. Yield stresses |0.05, ~*0.05 and |0.005, identified in the isothermal loading tests, are summarized in Fig. 17 for the axial tests, and in Fig. 18 for the shear teststogether with the martensitestart temperatures determined from the dilatometric curves (cf. Fig. 13). It is worth noting that in the shear tests in Fig. 18 the axial hold stress | =X0z (= 28.4 MPa) was always applied in addition to the shear stress ~*= 3~. Therefore, the equivalent effective shear stress ~*eff = X 20z + 3~ 2 ,

X0z = 28.4 MPa

(6)

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

is employed to compare the data in the axial and shear tests. The temperature-dependence of the yield stress, explained referring to Fig. 14 and Fig. 15, can be clearly reviewed in Fig. 17 and Fig. 18. In the high temperature range, the plot represents the yield stress in the parent phase. No significant dependence on the direction of the applied stress and the definition of the yield stress is observed. The latter observation reflects the fact that the stress – strain curves are almost flat in this temperature range, exhibiting only the temperature-dependence (see Fig. 15). In the low temperature range, a strong composite effect is observed. A significant difference in |0.05 and |0.005 once again proves that the stress –strain curves exhibit large work-hardening in this temperature range as obviously illustrated

Fig. 17. Yield stresses and martensite-start line; axial direction.

Fig. 18. Shear yield stresses and martensite-start line.

Fig. 19. Change in martensite-start temperature.

33

in Fig. 15. The influence of the direction of the applied stress is remarkable, i.e. the morphology of the martensite phase, more precisely, the size and direction of the martensite variants selected in the thermal and mechanical processes, has a strong effect on the strength of the specimen. This issue should be studied in further detail in the context of the material characterization of stronger TRIP steels by selecting an appropriate ‘thermomechanical treatment’ during cooling. In the intermediate temperature range, the material response, showing an anomalous temperaturedependence, is the result of the competition between the composite effect (which raises the yield stress) and the contribution of the strains due to transformation (which lowers the yield stress). The martensite-start line determined from the dilatometric curves is reproduced partly by the data of the tests with low strain rate [29]. Fig. 19 represents the following results: In the process of cooling after normal annealing, specimens were isothermally loaded in compression up to 0.2% at a test temperature. After unloading, the specimens are subsequently cooled down under free-stress to RT to measure the martensite-start temperature, by means of the dilatometric curves, which are plotted in the figure versus the test temperature of loading. At high test temperatures, the specimen yields plastically but no transformation occurs. The measured Ms temperature represents, therefore, nothing else than the constant martensite-start temperature under stress-free conditions, Ms0. Below about 450 K, however, the martensitic transformation takes place during isothermal loading. The amount of martensite phase x induced by the mechanical loading is larger at the lower test temperature. The martensite-start temperatures observed in the subsequent cooling process are observed on the so-called ‘subsequent martensite-start line [33]’ or the iso-x line in the stress-temperature plane. It is worth noting that the iso-x line is just the martensite-finish line (Mf-line) when x= 1. The present data reveal that, as in the case of the Fe-based SMA [33], the subsequent martensite-start line shifts from the ‘initial’ martensite-start line to the lower temperature side as the fraction of the martensite induced in the pre-loading becomes large. Identification of the iso-x lines, between the Ms-line (x= 0) and the Mf-line, should be an issue of further investigation. This subject is closely related to the formulation of the transformation kinetics. The strain induced during martensitic transformation in the subsequent cooling process, m max TRIP +l/3= mMf − mMs + (Ms − Mf) hM, is also plotted in the figure. The strain is reasonably smaller at the lower test temperature, since the specimen consists of less parent phase at the start of the martensitic transformation during subsequent cooling. The data in Fig. 19 show

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7. Evolution of TRIP strain and back stress

Fig. 20. Evolution of axial strain during cooling.

Fig. 21. Evolution of shear strain during cooling.

Fig. 22. Effect of shear hold stress on axial dilatometric loop.

Fig. 23. Evolution of axial TRIP strain.

that the speed of loading has no remarkable effect on the material performance. A material response similar to the one given in Fig. 19 was observed in both tension and shear loading.

The evolution of the strain observed in the stress holding tests (Fig. 2a and b) is summarized in Fig. 20 under the axial hold stress, and in Fig. 21 under the shear holdstress. The martensite-start and -finish temperatures are marked on each curve in the figures. It should be noted that in the shear stress holding tests the axial stress equal to X0z was applied in addition to the shear hold stress in order to realize the stress-free state in the axial direction. Fig. 20 clearly reveals that the total axial strain consists, as is formulated in Eq. (1), of the thermal contraction, transformation expansion, and the TRIP strain. In the case of a compressive hold stress, the total strain is a result of the competition between the transformation expansion developing in positive direction and the TRIP strain and thermal contraction developing in negative direction. In particular it should be noted in Fig. 21 that no shear strain evolves under ~*= 0 MPa. This is a direct proof that h the back stress tensor X0 has no shear component (see the discussion in Section 3). Since no axial TRIP strain evolves when the hold stress is equal to the back stress X0z, the total strain measured represents the evolution of the transformation expansion strain (see Eq. (4)) if the thermal strain is subtracted. This situation has already been studied in Section 3 and Section 5, relating to the closed dilatometric loop labeled by |h = 29 MPa in Fig. 3. The dilatometric response in the axial direction is almost not influenced by the shear hold stress as illustrated in Fig. 22. Although the martensite variants are induced by both the applied shear hold stress ~*h and the thermal load (by cooling), the deformation response of the specimen in the axial direction is merely the transformation expansion and additionally the thermal contraction, without any development of the TRIP strain. The evolution of the axial TRIP strain under an axial hold stress |h is identified by normalizing its dilatometric curve with respect to the dilatometric curve under the hold stress X0z, meaning that the TRIP strain curve for the hold stress |h is obtained by subtracting the axial strain curve for |h = X0z from the axial strain curve for |h. The result is summarized in Fig. 23. The horizontal axis in the figure stands for the undercooling Ms – T which is a measure of the extent of transformation. Once the transformation kinetics discussed in Section 5 is identified for this material, by taking account the effect of both the thermal and mechanical loads, the undercooling can be replaced by the volume fraction of martensite x. Since neither thermal expansion nor transformation expansion is observed in the shear direction, the total shear strain k* plotted in Fig. 21 is nothing else but the shear TRIP strain k*TRIP, which is re-drawn in Fig. 24 versus the undercooling.

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

The maximum values of the TRIP strain, max m max TRIP and k* TRIP, observed at the end of the transformation, are plotted in Fig. 25 versus the hold stress. The effective hold stresses |h −X0z and ~*h are employed as the horizontal axis to compare the results in the axial and shear directions. The Greenwood and Johnson linear relation [34] holds when the hold stress is less than about 940 MPa. In this range, no remarkable dependence of the stress direction is observed. When the absolute value of the hold stress is larger than 40 MPa, the relation becomes non-linear, and the obvious influence of the stress direction is observed [35]. Due to the presence of the axial back stress X0z, the curve under the axial hold stress is asymmetric with respect to the vertical axis [15,16]. A constitutive relation in rate form [23,24] m; TRIP =A f(x)(| − Xz)x: ,

(7)

Fig. 24. Evolution of shear TRIP strain.

Fig. 25. Relation between maximum TRIP strain and hold stress.

Fig. 26. Evolution of axial strain during unloading.

35

has recently been proposed for the axial TRIP strain as a generalization of the classical TRIP constitutive equation [3–5] in order to describe both the Greenwood – Johnson effect and the Magee effect in TRIP phenomena. In Eq. (7), A stands for the material constant and f(x) for a material function of the volume fraction of the martensite phase. The relation emphasizes that the TRIP strain depends on the applied stress | and the back stress Xz only through the effective stress | − Xz. The TRIP strain also depends linearly on the transformation rate x: . The evolution of the back stress Xz, staring from the initial value X0z, is essential to catch the Magee effect. The relation is already partly confirmed by the present experiments since no TRIP strain develops when |= Xz. An identification of the material constant and the material function of the present material will be an issue of forthcoming papers. In order to prove qualitatively that the axial component of the back stress Xz actually evolves in the process of transformation from its initial value X0z, unloading tests were performed under the thermomechanical loading path illustrated in Fig. 2a and c. The hold stress |h = 80 MPa was unloaded to the initial value of the back stress X0z = 28.4 MPa just in the process of transformation. Fischer et al. [23] have predicted in their simulations based on micromechanics of martensitic transformation that a backflow due to the Magee effect, which is not predicted by the conventional TRIP theory based on the Greenwood –Johnson effect, should be observed in this thermomechanical situation after unloading. The evolution of the axial strain after unloading is shown in Fig. 26, in which the curves with the label ‘80 MPa’ and ‘X0z ’ represent the development of the axial strain under 80 MPa and X0z of constant hold stress, respectively. The hold stress is lowered from 80 MPa to X0z at the points a, b, c, d and e. The curve starting from each point characterizes the subsequent development of the strain. The evolution of the strain, when the hold stress is lowered at the last stage of transformation (at the point e), is nearly the same as in the case of the 80 MPa constant holding test, i.e. no appreciable subsequent evolution was observed. The constitutive Eq. (7) is verified under this situation since the transformation rate is small. The evolution of the TRIP strain is evaluated, as explained above, by normalizing the response from the curve in the X0z constant holding test, and is shown in Fig. 27. The TRIP strain does not develop at the early stage of transformation (the cases of a and b), whereas a considerable backflow is observed in the middle stage of transformation with a high transformation rate (the cases of c and d). The TRIP strain increment at just the instant of unloading, DmTRIP, is determined as the TRIP strain change due to 1 K temperature change, and is marked in the figure by arrows. The TRIP strain incre-

K. Nagayama et al. / Materials Science and Engineering A308 (2001) 25–37

36

Fig. 27. Evolution of axial TRIP strain during unloading.

Fig. 28. Incremental change of TRIP strain.

Fig. 29. Possible evolution of back stress (Schematic).

ment is plotted in Fig. 28 versus undercooling, to show such a clear tendency that the TRIP strain increment increases in negative direction from initial zero value as the transformation proceeds. Suppose the back stress X0z remains constant in Eq. (7) throughout the entire transformation process. Since |h − X0z =0 after unloading, DmTRIP must vanish at all points of a–e. Fig. 28, therefore, allows the conclusion that the back stress evolves in positive direction from its initial value X0z during the transformation process, and that a qualitative sketch of the back stress evolution illustrated in Fig. 29 is plausible.

8. Concluding remarks Mechanical performance of a Cr – Ni – Mo –Al –Ti

maraging steel is investigated experimentally in the process of martensitic transformation during cooling to provide a database for the establishment of a set of constitutive equations for transformation-induced plasticity (TRIP) under a multiaxial stress state. The transformation-start and -finish temperatures are determined from the dilatometer curves under the hold stress in the tensile, compressive or shear direction. The dependence of the stress direction on the martensitestart line, representing the Clausius –Clapeyron relation, is clearly observed as in the case of shape memory alloys. The yield stresses, identified from the strain-controlled isothermal loading tests, show an anomalous temperature-dependence, which is explained by the competition between the contribution of the harder martensite phase induced thermally and/or mechanically (the composite effect) and the effect of the sudden production of the strains due to transformation. By checking whether the dilatometric loop (the strain-temperature hysteresis loop) closes or not after a full heating –annealing –cooling thermal cycle under the hold stress, the back stress due to the texture built in by the forging process is found in both the axial and radial directions of the thin-walled tubular specimen and its initial value is identified. The annealing conditions, the hold temperature and the hold time, are shown to have a marked effect on the initial value of the back stress. A steady evolution of the TRIP strain is observed under a constant hold stress in the direction of the hold stress. When the hold stress is lowered or removed in the process of transformation, a backflow increases in the subsequent cooling process. The phenomenon is due to the Magee effect, relating the length change to specific martensite variants selected by the current stress state. Therefore the TRIP strain development during unloading also reveals that the back stress evolves in the progress of martensitic transformation. Qualitative evaluation of the evolution of both the TRIP strain and the back stress will be the subject of the next study to be investigated under the multiaxial stress state. Cailletaud et al. have recently shown in [36] a concept how to establish a theoretical framework in transformation thermomechanics consisting of the ‘transformation potential’ and the associated flow rule.

Acknowledgements Part of this work was financially supported by the Special Research Fund/Tokyo Metropolitan Government as well as by the grant-in-aid for Scientific Research (No. 11650095) through the Japan Society for the Promotion of Sciences.

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