Constitutive equations for prediction of the flow behaviour of duplex stainless steels

Materials Science and Engineering A 527 (2010) 4218–4228

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Constitutive equations for prediction of the flow behaviour of duplex stainless steels S. Spigarelli a,∗ , M. El Mehtedi a , P. Ricci a , C. Mapelli b a b

Dipartimento di Meccanica, Università Politecnica delle Marche, via Brecce Bianche 60131, Ancona, Italy Dipartimento di Meccanica, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy

a r t i c l e

i n f o

Article history: Received 16 November 2009 Received in revised form 9 March 2010 Accepted 9 March 2010

Keywords: Duplex stainless steel Hot working Constitutive equations

a b s t r a c t The high temperature workability of a 2205 duplex stainless steel has been investigated by torsion testing between 950 and 1200 ◦ C. The constitutive equations relating peak flow stress, temperature and strain rates were obtained, and the composite model, where austenite and ferrite are the hard and soft phases, respectively, was found to give an excellent description of the experimental data. The peculiar shape of the equivalent stress vs. equivalent strain flow curves has been discussed; a model based on the hypothesis that during torsion of wrought duplex stainless steels the load transfer from ferrite and austenite is delayed to a later stage of the straining process has been developed, and successfully used to describe the experimental data. Thus it was concluded that during the early stage of the tests the deformation is inhomogeneous, since strain accumulates in ferritic regions, while austenite is almost undeformed. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Duplex stainless steels (DSS) constitute a class of highly appreciated materials for their excellent combination of high strength and resistance to corrosion in presence of chloride-containing environments and to stress corrosion cracking, especially when compared with conventional single phase ferritic or austenitic stainless steels grades. Although these alloys have been extensively used in several industrial applications, their relatively poor hot workability is an important drawback that results in the need for a strict control in processing variables. The scarce DSS hot workability is caused by wide regions of low ductility in the processing windows commonly used for shaping conventional austenitic steels. The plastic deformations behaviour of DSS at high temperature strongly depends on their peculiar microstructure, consisting in more or less equivalent portions of ferrite and austenite. From a general point of view, the high temperature workability of ferrite is higher than that of austenite, in terms of lower flow stresses and of higher ductility. The excellent response to high temperature deformation is a consequence of the ability of ferrite to undergo extensive dynamic recovery (DRV), consisting in the annihilation of a high fraction of the total dislocation population and in the arrangement of the remaining dislocations to form subgrain boundaries [1]. The subgrain size attains, for sufficiently high strains, an equilibrium value, marked by the existence of a macroscopic steady state level of the flow stress; this equilibrium subgrain size

∗ Corresponding author. Tel.: +39 071 2204746; fax: +39 071 2204801. E-mail address: [email protected] (S. Spigarelli). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.03.029

increases with decreasing strain rate and/or increasing temperature [2–4]. In addition, grain boundary sliding significantly delays cracking at grain boundary [1]. The hot straining response of austenite is extremely different; reduced DRV, in this case, leads to the onset of dynamic recrystallization (DRX), that, once a critical strain is attained, induces a significant flow softening after a peak in flow stress has been reached; the peak roughly corresponds to a microstructure that for 35% underwent DRX, while the onset of a true steady state value of the flow stress, that follows the softening after the peak, corresponds to a 95% DRX of the deformed microstructure [5–8]. The grains created by DRX have a low resistance than the original deformed structure and above 900 ◦ C, DRX results in a grain boundary migration that reduces crack formation [1]. The observation that both ferrite and austenite have a good workability at high temperature, could lead to the premature conclusion that a similar behaviour should be exhibited also by DSS. Quite on the contrary, the presence of non-negligible amounts of austenite in ferrite significantly decreases ductility, probably due to the dissimilar properties of these two phases [9]. The analysis of the possible reasons for this poor hot workability of DSS pointed the occurrence of several concurring phenomena, for example a reduced and delayed DRX in the austenitic portions of the microstructure [10]. In addition, DRV, consisting in the formation of a complex network of boundaries, characterized by a mix of high- and low-angle walls [11], has been found to be less effective in the ferritic regions of DSS than in single phase ferritic grades [1]. An important factor to be considered in the study of hot workability of the DSS is the strain partitioning between the soft ferrite and the hard austenite; during the early stage of deformation strain

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accumulation in ferrite, while austenite remains almost undeformed, is frequently observed [12–14]. The difference in strength between the two phases is considered the main cause for the shear bands formation in ferrite. In addition, the different orientation of the ferritic and austenitic constituents in as cast and wrought DSS have been observed to result in a significant variation of the shape of the stress vs. strain curve obtained by torsion testing [12,13]. Last, but not least, the occurrence of sliding of the /ı interfaces has been observed; this mechanism consists in the translation of one grain with respect to an adjacent one and is severely hindered when the interface has a coherent or semi-coherent nature, as in the case of the as cast DSS [12,13]. Sliding is thus localised to few interfaces, and this localization results in the early crack formation at the interfaces, one of the most probable cause of the low ductility [1,13]. Sliding is by far easier in wrought (rolled) DSS, due to the incoherent nature of the interfaces, and as a results damage nucleation is in this case delayed [12–14]. Even though the main microstructural mechanisms that influence and determine the hot workability response of DSS steels has been quite extensively investigated, some important features, such as the modelling of the flow stress variation with strain, have been only qualitatively addressed (see, for example, the very interesting analysis of the possible causes of the difference of the stress vs. strain curves in as cast and wrought 2205 DSS tested in torsion and in plane strain compression [12–14]). The aim of the present work was thus to investigate the high temperature workability of a DSS steel and to develop a family of constitutive relationships able to describe the peculiar straining behaviour of this material, in term of quantitative prediction of the relationship between the stress and strain.

2. Experimental details The material investigated in the present study is the grade 2205-UNS31803, with the following chemical composition (wt.%): C = 0.018, Mn = 1.8, Si = 0.34, Ni = 5.3, Cr = 22, Mo = 2.6, N = 0.15, P = 0.027, S = 6 ppm. The alloy was provided in the form of hot-rolled slab 18 mm thick. Mechanical characterization consisted in torsion tests carried out at temperature ranging from 950 to 1200 ◦ C. Torsion testing was preferred since the sample dimensions during straining remains constant, and the test is not affected by instability phenomena such as necking, or by barrelling. Conventional tensile testing is of limited use at high temperature, since necking prevents sufficiently large strains being attained. Compression test, in principle more suitable for hot workability studies, is not affected by basic instabilities, but the effect of friction has detrimental effects on the quality of the results [15], since it leads to barrelling. Torsion testing is not affected by these problems, although strain, stress and strain rate vary linearly along the radius of the sample; to overcome this problem, the surface values of the strain, strain rate and stress are considered. In parallel, microstructural investigations are focused on the analysis of the surface of the sample. Although the ratio of shear-to-normal stress is 1, i.e. higher in torsion than in the commonly used hot working operation, the equivalent strain to fracture can be considered a useful comparative index for the ductility of the investigated materials. Torsion samples were machined from the slab with the longitudinal axis parallel to the rolling direction. Torsion tests were carried out on a computer controlled torsion machine; the specimens were heated at 1 ◦ C/s by an induction coil and maintained 5 min to stabilize the testing temperature before torsion. Isothermal tests have been performed in air and specimens just after the fracture were rapidly quenched with water jets to avoid microstructure modifications during slow cooling.

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The shear stress and strain were calculated using the relationships [16]: √ 3M (3 + m + n ) (1a) = 2R3 =

2NR L

(1b)

where L and R are the specimen gauge length and radius, respectively, N is the number of revolutions, M is the torque, m = ∂ log M/∂ log N˙ (strain rate sensitivity coefficient at constant strain), and n = ∂ log M/∂ log N (strain hardening coefficient at constant strain rate). At the peak stress, clearly n = 0; in addition, for the sake of simplicity, also the strain sensitivity coefficient was considered = 0. The equivalent stress, , and the equivalent strain, ε, were calculated by the Von Mises criterion; in particular, the equivalent stress was calculated by the equation =



(l 2 + 3 2 )

(2)

where  l is the normal stress, while  is given by Eq. (1). In the case of the torsion sample considered as a whole, the normal stress is zero (no external longitudinal load); the same is not true for the microstructural components of the duplex structure, ferrite and austenite, subject to normal residual stresses of thermal origin. Surface equivalent strain rates ranged from 0.05 to 5 s−1 . Deformed samples were observed by optical microscopy near the surface of the gauge length after polishing and electrolytic etching in a solution of 10% oxalic acid in hydrogen peroxide. 3. Results Fig. 1 illustrates the microstructure of the alloy after rolling; the structure is typical of hot-rolled DSS, consisting in an arrangement of elongated austenite islands within a ferritic matrix. The equivalent flow stress () vs. equivalent strain (εDSS ) curves have the characteristic shape of the wrought DSS tested in torsion (Fig. 2); as observed in [12], these curves exhibit a parabolic strain hardening at low strains, followed by a nearly linear hardening rate stage up to a peak. After the peak, in most cases the flow stress decreases monotonically up to fracture; only in the low strain rate/high temperature regime, after a moderate decrease of the flow stress after the peak, a proper steady state is observed. In all the other experimental conditions, early fracture occurs before the steady state has been attained. The equivalent fracture strain in general does not exceed εDSS = 10, even though a well defined trend

Fig. 1. Microstructure of the as received 2205 DSS.

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Fig. 2. Representative equivalent stress vs. equivalent strain curves obtained by torsion testing.

toward its increase with increasing temperature and decreasing strain rate was observed. Peak stress values ( p ) as a function of strain rate and temperature are shown in Fig. 3. The dependence of  p on T and strain rate was analysed by the relationship n

ε˙ DSS = A[sinh(˛p )] exp

 −Q

DSS

RT



(3)

The stress exponent with ˛ = 0.012 MPa−1 ranged from 3.5 to 4.5 (Fig. 3a); the apparent activation energy (QDSS = 474 kJ/mol) was calculated by the relationship QDSS = 2.3nRS

(4)

where S is the average slope of the straight line obtained by plotting, in semi-logarithmic coordinates, sinh(˛) vs. 1/T. Fig. 3b plots sinh(˛) as a function of the Zener–Hollomon parameter (Z = ε˙ exp(QDSS /RT )). All the experimental data align on the same line, with slope close to 0.25, giving n = 4. An analysis of the data available in the literature for 2205 DSS shows values of the activation energy for high temperature deformation ranging from 394 [1] to 569–578 kJ/mol [12,17], with stress exponents from 3.8 [1] to 6.5 [17]. It must be mentioned that the magnitude of the stress exponent is severely influenced by the selection of the ˛-value; in addition, also the temperature range and the type of test, e.g. plane strain compression or torsion as in [12], can significantly affect the magnitude of the constitutive parameters. Cabrera et al. [17], for example, tested in compression the 2205 DSS at temperatures ranging from 600 to 1100 ◦ C, while Duprez et al. [18] tested a similar material in torsion between 950 and 1200 ◦ C, obtaining

Fig. 3. (a) Peak stress vs. strain rate and (b) peak flow stress (solid symbols) and equivalent strain to fracture (open symbols) as a function of the Zener–Hollomon parameter (Z).

QDSS = 425 kJ/mol. Although the value of the activation energy for high temperature deformation of DSS has thus been observed to present significant variations in the various investigations, in most cases it is higher than the activation energy in austenitic and ferritic steels. A universally accepted explanation for this high value of the activation energy is the increase in volume fraction of soft ferrite at expenses of the hard austenite with increasing temperature [1,12]; this effect was observed also in the present study, and is illustrated in Figs. 4 and 5; the latter, in particular, shows the marked decrease of the volume fraction of austenite (f ) with temperature (the error bars for the various measurements were omitted for the sake of clarity). Fig. 4 plots few representative examples of the microstructure after torsion; the reduction of f with increasing temperature is indeed accompanied by a marked fragmentation of the original massive austenite islands, an indication that this phase was more severely deformed at the highest temperatures. Damage, in form of cavitation, is frequently observed in the samples deformed at low temperature and high strain rates. The simple analysis of Fig. 4 does not give unambiguous indications on the mechanism of dam-

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Fig. 4. Microstructure of the torsioned samples: (a) 950 ◦ C–0.05 s−1 ; (b) 1200 ◦ C–0.05 s−1 ; (c) 950 ◦ C–5 s−1 ; (d) 1000 ◦ C–5 s−1 ; (e) 1200 ◦ C–5 s−1 .

age nucleation, even though crack formation at the interface and its propagation in the adjacent phases is likely to be the origin of the early fracture. 4. Discussion 4.1. Preliminary remarks: effects of thermal residual stresses on the tensional state in the temperature regime typical of hot working The presence of residual stresses in austenite and ferrite in duplex steels has been clearly assessed, and described at length by Werner and co-workers [19–24]. These authors modelled the irreversible differential deformation of austenite and ferrite in forged duplex stainless steels exhibiting a microstructure consisting of equivalent volume fractions of the constituent phases, whose grains form axially elongated interwoven network (not dissimilar to the microstructure of the steel of the present study). In these

duplex steels, as in any other material consisting of two phases with different thermal expansion coefficients, internal stresses are generated in austenite and ferrite by a variation in temperature [19]. In synthesis, as long as they do not exceed the yield stress of the phase, the internal stresses introduced in ferrite and austenite, under the assumption of an extended, homogenized material description (see Ref. [19] for the detailed analysis of the micromechanical model) can be calculated as: ˛,l =

,l =

f E˛ E EDSS,l f˛ E˛ E EDSS,l



T

(˛ − ˛˛ )dT

(5a)

(˛˛ − ˛ )dT

(5b)

T0



T

T0

where fx , Ex , ˛x and  x,l are the volume fraction, the elastic modulus, the thermal expansion coefficient and the internal stress acting in longitudinal direction for the phase x, i.e. fer-

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Fig. 5. Variation of the austenite volume fraction as a function of testing temperature.

rite (˛), austenite (), or for the duplex stainless steel (DSS), respectively. If the internal stress in one of the constituents approaches the magnitude of the yield stress of that phase, plastic deformation occurs even in absence of any external load. In this case, the thermal expansion response of the duplex material is strongly affected by the accommodation of the thermal mismatch through plastic deformation [19]. Siegmund et al. obtained an estimate of the plastic deformation produced by the difference in thermal expansions of ferrite and austenite; the input data of their analysis, i.e. the variation of the yield stress, of the thermal expansion coefficient and of the Young modulus with temperature, are replotted in Fig. 6. Fig. 7 plots the irreversible plastic deformation calculated by these authors, for a duplex steel with equivalent volume fractions of ferrite and austenite. Even more interesting is the quantification of the residual stresses (Fig. 8) generated in ferrite and austenite by the strains illustrated in Fig. 7. In the initial wrought condition, the constituent phases are loaded by a residual stress (tensile in austenite and compressive in ferrite) generated during the cooling from the working temperature (rolling temperature in the case of the steel considered in the present study). On heating up to the testing temperature, the phases are progressively unloaded from their stress state, until, when temperature exceeds a certain level, austenite is loaded in compression, and ferrite in tension. In parallel, the yield strength of the two phases decreases with increasing temperature (Fig. 6), and, as result, yielding occurs. Since the model assumes for the constituents an elastic-ideal plastic behaviour, after yielding both phases are loaded by an internal stress, either in tension or compression, equivalent to the yield stress of that phase, at that specific temperature and under a certain strain rate. The calculated variation of the internal stresses with temperature is thus a direct consequence of the considered yield stress dependence on T. On the other hand, mechanical properties, in particular yield and peak stresses, in the high temperature regime strongly depend on the strain rate. An approximate estimation of the order of magnitude of the instantaneous strain rate can be given by the expression of the irreversible thermomechanical longitudinal strain rate increment (ε˙ tm,l ), calculated when either ferrite or austenite is plastified [19]:



ε˙ tm,l ≈

˛ − ˛˛ f˛

E˛ EDSS,l

− ˛ f

E EDSS,l

Fig. 6. Variation of Young modulus and thermal expansion coefficient (a) and of the yield strength (b) for austenite and ferrite, used as input data for the thermomechanical model [19].



T˙

(6)

As a matter of fact, an estimate of the inelastic strain rate can be directly obtained from Fig. 7 by simply observing that a lon-

Fig. 7. Evolution of the value of the longitudinal inelastic strain in ferrite and austenite, and resulting irreversible strain for the duplex steel, as calculated in Ref. [19], for a heating cycle 20 ◦ C–900 ◦ C–20 ◦ C.

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ferrite [25], are fully consistent with the use of low heating rates (1 ◦ C s−1 ), i.e. lower strain rates. The calculated values of the tensile internal stresses in ferrite are overestimated by the model of Siegmund et al., probably as an effect of the lack of input experimental data on the variation of the yield strength between 700 and 900 ◦ C (Fig. 6). The numerical results shown in Figs. 7 and 8, based on the model described in [19] are thus markedly affected by a number of factors, including:

Fig. 8. Evolution of the value of the longitudinal stress in ferrite and austenite, as calculated in Ref. [19], for a heating cycle 20 ◦ C–900 ◦ C–20 ◦ C.

gitudinal thermomechanical strain of εl = 0.001 is accumulated in ferrite during 300 or 15 s for heating rates of 1 and 20 ◦ C s−1 , giving ε˙ ≈ 3 × 10−6 and 7 × 10−5 s−1 , respectively. A similar calculation gives ε˙ ≈ 9 × 10−7 and 2 × 10−5 s−1 for the compressive inelastic strain accumulated in austenite. The decrease of the yield strength with temperature is thus responsible for the non-monotonic variation of the internal stress, that reaches a maximum; in any case, at high temperature, plastic deformation of the softer phase (ferrite) mainly accommodates the thermal mismatch in thermal expansion of the two constituents. Fig. 9 plots some of the calculated values of the internal stresses of Fig. 8, few additional experimental results obtained by Kamachi et al. [25], and the peak stress values for different strain rates [26,27] (the equations used for this estimation will be discussed at length in the following). As above mentioned, the internal stress in ferrite and austenite in principle is equivalent to the yield strength on the assumption of an elastic and perfect-plastic material, but, in any case, under the more realistic assumption of materials undergoing strain hardening, the internal stress cannot exceed the value of the peak stress for those given strain rate and temperature. Fig. 9 can thus used to have a rough estimate of the possible magnitude of the internal stresses in ferrite and austenite for temperatures between 800 and 1200 ◦ C; the analysis of the figure suggests that the magnitude of internal longitudinal stresses in austenite, as given by the model in [19], is consistent with the use of a high heating rate (20 ◦ C s−1 or similar); on the other hand, the values of the residual stresses in austenite, measured in [25] and reported in [19], are considerably lower. The experimental values of the residual stresses in

i. the choice of the proper yield strength data, that, as previously noted, at high temperature are strongly strain rate dependent [19]; taking into account the order of magnitude of irreversible thermomechanical longitudinal strain rate, the values of the internal stress reported in Fig. 9 (see “Section 4” for further details) seems exaggerated for a heating rate as low as 1 ◦ C s−1 ; ii. the distribution of constituent phases, even when relatively simple cases of phase arrangements are considered [22]; iii. the possible occurrence of interphase grain boundary sliding [12,13]; if the two phases can slide, the effect of the differences in thermal expansion could be substantially reduced since grain boundary sliding can accommodate them. The above mentioned factors partly explain the overestimation of the irreversible strains as given by the micromechanical model (the predicted strain is about twice the experimental one), and, as a result, of the internal stresses. 4.2. The composite model: peak stress dependence on strain rate and temperature The analysis of the high temperature response of duplex stainless steels clearly demonstrated that the hot workability of these materials is critically influenced by their peculiar microstructure, composed by a harder phase (austenite) that coexists with a softer one (ferrite). A possible consequence of this observation is that the composite model, described in detail by Cho and Gurland [28], could be successfully applied to duplex stainless steel not only at room temperature, as in [28] (where austenite is the softer phase), but also in the high temperature regime typical of hot working operations. The composite model, in its simplest form, is based on equations that assume the forms: DSS = f  + (1 − f )ı

(7a)

and εDSS = f ε + (1 − f )εı

(7b)

where   ,  ı , ε and εı are the stress and the strain acting in austenite and ferrite, respectively. Differentiation of Eq. (7b), in absence of any significant variation of the austenite volume fraction with time, leads to ε˙ = f ε˙  + (1 − f )ε˙ ı

Fig. 9. Evolution of the longitudinal stress in ferrite and austenite between 700 and 900 ◦ C, as calculated in Ref. [19], for a heating cycle 20 ◦ C–900 ◦ C–20 ◦ C, in comparison with the experimental data obtained by Kamachi et al. [25], and the curves of the peak stress in austenite and ferrite for different strain rate as given by Eqs. (10a) and (10b) [26,27].

(8)

Resolution of Eqs. (7a) and (8) is particularly easy in two extreme cases [12]; the first case is characterized by the equivalence of the stress acting in soft and hard zones (constant stress model). The obvious consequence of the constancy of the applied stress in all the structure is that, when hard and soft zones are in series, the softer zone will deform preferentially. The other extreme case occurs when the hard and soft zones deform in parallel with the same strain rate. At the beginning of the test it could be in principle supposed that, in absence of significant internal stresses generated by the difference in thermal expansion coefficients, the applied stress is equivalent in hard and soft zones. In the case of the wrought DSS

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this assumption does not take into account the proved existence of longitudinal stresses, introducing an error that is important only at the early beginning of the test. In any case, even considering the residual stresses, the softer zone (ferrite, already loaded in tension by residual stresses) tends to deform preferentially, but this event is impeded by the action of the harder zone, less prone to straining, that exerts an opposing stress. A similar stress, acting in opposite direction (forward stress), is exerted by reaction by the softer phase on the harder one. Thus the net deforming force acting in soft and hard zones will be different in magnitude, even though the strain rate in the two regions is (approximately) the same. In this context, the possibility of applying the composite model to the description of the experimental data obtained by testing duplex stainless steels at high temperature is very tempting. A good description of the peak stress dependence on applied strain rate has been obtained by applying the constant strain rate model in Ref. [12]. Iza-Mendia and co-workers [12,13], on the other hand, acutely observed that the peculiar sigmoidal shape obtained in the earlier stages of torsion of wrought DSS, is a clear indication that at the beginning of the test a non-homogeneous strain repartition between ferrite and austenite takes place. It can be thus concluded that the softer ferrite is the strain-controlling phase at low strains, due to the stress distribution during the early stage of torsion testing [12,18]. Since austenite plates in wrought materials are in general parallel to the longitudinal axis of the sample, at the beginning of the test they are aligned at 45◦ with respect to the direction of the principal stress. Only in a later stage of the test, the plate structure aligns parallel to the direction of principal stress. As a consequence, after the early stages of the test, an effective load transfer from ferrite to austenite progressively occurs, and the two phases deform with similar strain rates [12]. It is worthwhile to mention that Ref. [13] illustrates a detailed investigation on the mechanisms of strain accommodation in torsion of as cast and wrought 2205 DSS steel; the authors concluded that the incoherent /ı interface allows sliding of the two phases during torsional straining. Such a sliding, that accommodates the different strains in ferritic and austenitic regions, is responsible for the larger strain to fracture observed in the wrought material. In the as cast alloy an opposite situation takes place, since partially coherent interfaces inhibit sliding and lead to shearing of ferrite to maintain the continuity of the structure [13]. On the other hand, the curvature of the scratches (used to quantify the deformation on microstructural scale) inside ferritic region observed by these authors (see Figs. 7 and 8 in Ref. [13]) is consistent with the presence of a backward stress, resulting in the partial obstruction of the straining of the ferritic regions laying close the /ı interface, while in the centre of the ferritic zones the material can be more easily deformed. Although, as above mentioned, the peak stress dependence on strain rate and temperature has been successfully described by the composite model according to the constant strain rate configuration, very recently other authors [26] applied a modified form of Eq. (7a), rewritten as ε˙ = P ε˙  + (1 − P)ε˙ ı

(9)

where P was considered a “strain interaction coefficient”, unrelated to the volume fraction of hard phase. Farnoush et al. [26] analysed the chemical composition of both ferrite and austenite in the 2205 DSS, and prepared two model alloys with similar composition (Table 1), to be tested at high temperature to obtain the constitutive equation for each of the two phases. Substitution of these equations (which parameters are summarised in Table 2) into Eq. (8) gave a good description of the experimental data, provided that an implication, that was apparently ignored, is accepted. Following this approach, based on the assumption of a constant stress acting in both phases, the authors obtained that the austenite, i.e. the harder phase, deforms preferentially, while the softer phase, i.e.

Table 1 Chemical composition (wt.%) of some austenitic and ferritic alloys (see text). Composition and source of the data

C

Mn

Si

Cr

Mo

Ni

Ferrite [26] 434C [30] Austenite [26] 304 [29] 316 [29]

0.020 0.066 0.03 0.062 0.010

0.40 0.58 2.8 1.72 1.87

0.38 0.47 0.24 0.47 0.62

25.9 16.6 20.0 18.3 16.4

2.8 0.96 2.3 0.28 2.73

3.3 0.21 8.8 8.3 12.1

ferrite, is responsible for a mere 10% or less of the total strain rate. Since this conclusion is unrealistic, an indirect confirmation that, in correspondence of the peak stress, the DSS behaves like a composite which constituents deform with similar strain rate is obtained. On this basis, the same model has been used to describe the peak stress dependence on strain rate obtained in the present study. On the other hand, application of Eq. (7) or (8) requires the knowledge of the constitutive equations relating strain rate, stress and temperature, for the austenitic and ferritic phases. Table 1 summarises few examples of the chemical compositions of austenitic and ferritic grades for which the values of the constitutive equations (Eq. (3)) are available in the literature (Table 2, Fig. 10). Analysis of Fig. 10 demonstrated that the strain rates obtained by using the different equations differ significantly, in some cases of more than two orders of magnitudes. In particular, for ferritic alloys, an intermediate description is given by the model proposed by Farnoush et al. [26], while the same authors proposed a constitutive relationship that seems to strongly overestimate the austenite strength. In this case, an intermediate description is obtained by using the constitutive equation proposed for AISI 304 by Sellars et al. [27], and already used in [12], while the AISI 304 and 316 tested by Cingara and Mc Queen seem to be softer. In order to maintain the description as simple and general as possible, in the present study the constitutive equations giving intermediate descriptions of the ferrite and austenite dependence on strain rate were selected, i.e.: ε˙  = 2.8 × 1014 s−1 [sinh(0.012p )]4.0 exp ε˙ ı = 6.3 × 1012 s−1 [sinh(0.0103pı )]3.6 exp

 −400 kJ/mol  RT

 −310 kJ/mol  RT

(10a) (10b)

Eqs. (10a) and (10b) were used also to estimate the peak stress values compared, in Fig. 9, with the numerical estimation of the internal stresses. Substitution of the estimated equivalent stress acting is soft and hard zones into Eq. (7a), with the value of f typical of each temperature from Fig. 5, should thus result in model curves to be compared with the experimental value of the peak stress. In the presence of thermally generated residual axial stresses ( l ), the calculation of the tensional state in the two phases is more Table 2 Values of the constitutive parameters for austenitic and ferritic alloys; except were expressly stated, the values here included were presented in tabular or equation forms in the original sources. Composition and source of the data

A (s−1 )

˛ (MPa−1 )

n

Q (kJ mol−1 )

Ferrite [26] Ferrite [28]a 434C [30] Austenite [26] 304 [30]a 304 [29]b 316 [29]b

6.32 × 1012 3.52 × 1011 5.0 × 1014 1.5 × 1015 1.6 × 1015 2.8 × 1014 6.4 × 1015

0.0103 0.0115 0.0172 0.0066 0.008 0.012 0.012

3.6 3.3 3.8 4.6 4.8 4.0 5.3

310 261 397 454 410 400 430

a

The values of the constitutive parameters are those reported in Ref. [12]. The values included in the table were obtained by the authors of the present study by interpolating the original experimental data. b

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Fig. 11. Description of the peak stress dependence on strain rate by the composite model (constant strain rate configuration).

into Eq. (7a), are compared with experimental data into Fig. 11. The analysis of the figure again unambiguously demonstrates that this approach gives an excellent description of the peak stress experimental data, except that at the lowest temperature, for which an underestimation of the strength of the alloy is obtained. 4.3. The stress vs. strain curves: a model for repartition of strain in ferrite and austenite The evidence above illustrated seems to support the idea that, during torsion testing of wrought DSS, although almost exclusively ferrite deforms at the early stage of the test, load transfer to the harder phase occurs as, at a later stage, austenite progressively starts to deform. Thus it can be reasonably supposed that, at the beginning of the straining process, ε˙  ≈ 0 and ε˙ ı = K ε˙ DSS

(11)

According to the composite model, this gives the maximum strain rate ferrite could undergo in absence of any significant deformation of austenite, namely ε˙ ımax = Fig. 10. Constitutive equations for deformation of austenitic and ferritic alloys (the value of the different parameters are summarised in Table 2).

complex, since it should be based on Eq. (2): =



(l 2 + 3 2 )

where  is the strain rate- and temperature-dependent component introduced in torsion. The maximum value of the internal stresses can be observed at 950 ◦ C, and should be equivalent, for a heating rate of 1 s−1 , to the yield stress at that temperature for a strain rate close to 3 × 10−6 s−1 in ferrite, and 9 × 10−7 in austenite; this gives a residual stress, approximately calculated by Eq. (10), lower than 5 MPa for ferrite, and close to 11 MPa for austenite. It is thus reasonable to conclude that the error introduced by neglecting into Eq. (2) the residual stresses in ferrite and austenite is negligible. On these bases the term  l was omitted in the calculation of the equivalent stress; the resulting curves, obtained by substituting Eq. (10)

ε˙ DSS 1 − f

(12)

Almost no information is available on the effective amount of straining in ferrite during testing, except some measurements carried out in Ref. [13] on an as cast material after an equivalent strain of εDSS = 0.16, giving εı = 0.2–0.3 (average value εı = 0.25). Taking f = 0.37 (Fig. 5), one obtain that ε˙ ı is close to 98% of ε˙ ımax , as expressed by Eq. (12); on this basis, it has been here calculated that at the early beginning of the test, ε˙ ı = 0.98ε˙ ımax , i.e. the austenitic component undergoes a very small but non-zero strain. A second assumption concerns the modality through which the strain rate in ferrite declines toward the “steady state” value, i.e. the testing strain rate (condition of equivalent strain rate, valid at the peak). In the present instance, it was arbitrarily supposed that the variation of K from its maximum value, i.e. 0.98/(1 − f ) to its minimum (K = 1) assumed a sigmoidal shape, described by the following equation:



K = 1+

0.98(1/(1 − f)) − 1 1 + (εDSS /0.5εt )p



(13)

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The effect of the presence of internal stresses due to thermal expansion, not considered in the above calculation, deserves a further comment. As above mentioned, due to the very low value of the strain (heating) rate, the magnitude of the longitudinal internal stress is quite low in the investigated range of testing temperatures (Fig. 9). As a result, it does not significantly influence the shape of the flow curves, except in the very early stage (in practice corresponding to the first point of the experimental curve), when the shear component of the equivalent stress is very low. 4.5. Validation of the model

Fig. 12. Variation of the strain rate repartition ratio, as a function of total strain.

where εt is the total strain required to produce the transition to a constant strain rate configuration, and p = 6.37 (Fig. 12). Fig. 12 shows the variation of the “strain rate repartition ratio” (expressed as the ratio between the strain rate acting in ferrite and austenite, respectively) as a function of the total strain in the DSS. The curves depend on temperature though the variation of the volume fraction of the austenite, while a weak dependence on strain rate should occur through the variation of εt (see below). 4.4. The modelling of the stress vs. strain curves for austenite and ferrite The previous paragraph has led to the introduction of the K parameter that expresses the strain repartition between the softer and the harder phases, and its evolution with the overall strain of the composite. The following step requires the determination ˙ T ) for each of the constituent phases. This process of  = (ε, ε, is relatively simple in the case of austenitic stainless steels, since an equation has been developed by Cingara and McQueen, staring from data obtained in torsion [31,32], in the form:  = p



ε εp





exp

1−

ε εp

 c

(14)

where εp and  p are the peak strain and stress, respectively and c∼ = 0.24 (average of the values reported in [31] for AISI 304 between 900 and 1100 ◦ C, for strain rates of 0.1 and 1 s−1 ). The peak strain can be estimated by the following relationship: εp = 0.0033p + 0.24

(15)

Combination of Eqs. (10a), (14) and (15), for a given strain rate, permits to draw the flow curve up to the peak; since for strain exceeding εp , Eq. (14) is not longer valid, for the sake of simplicity it was here assumed that the flow stress remains constant after the peak. This assumption, as obvious, does not permit to take into account the effect of dynamic recrystallization, i.e. the softening after the peak so typical of austenitic stainless steels. On the other hand, it is interesting to note that, other investigators observed that dynamic recrystallization of austenite is inhibited [12] or delayed [10] in DSS; as a result, one can reasonably wonder if the constancy of the stress after the peak is a condition that is at least in part fulfilled in austenitic regions of the duplex structure. The determination of a relationship, similar to Eq. (14), for ferrite poses a major problem; again, for the sake of calculation, it was here assumed that the same Eq. (14) applies also for ferrite, with c = 0.1 and εp = 0.1, irrespective of the straining conditions. Substitution of Eq. (10b) into Eq. (14) then gave a series of curves that were considered a reasonable approximation of the real behaviour of ferrite.

The constitutive relationships presented in the previous paragraphs can be summarised giving a series of equations that, properly solved, can be used to create model curves illustrating a modelled version of the stress vs. strain curves of the composite:



ε˙ ı = 1 + ε˙  =

0.98(1/(1 − f )) − 1 1 + (εDSS /0.5εt )6.37

ε˙ DSS − (1 − f )ε˙ ı f



p =

1 arcsin h 0.012

ε˙ DSS



ε˙  exp(400, 000/RT )



ε˙ ı exp(310, 000/RT ) 6.32 × 1012

ε˙ DSS εı ≈ ε˙ ı t ∼ = ε˙ ı εDSS



ε εp



ε 1− εp

εp = 0.0033p + 0.24 ı = pı



εpı = 0.1

εı εpı





exp

 1/3.6 (16d)

(16f)



exp

(16c)

(16e)

εDSS − (1 − f )εı f

 = p

 1/4

2.8 × 1014

1 = arcsin h 0.0103

ε =

(16a)

(16b)



pı



1−

 0.24 (16g) (16h)

εı εpı

 0.1 (16i) (16j)

For each arbitrary data point at a given testing time t, corresponding to a total strain εDSS = t ε˙ DSS , the value of the strain rate in ferrite and austenite (Eqs. (16a) and (16b)), and the correspondent values of the peak stresses (Eqs. (16c) and (16d)) have been calculated; the values of the strain in the two phases has been then estimated by Eqs. (16d) and (16f), and the flow curve for each constituent (Eqs. (16g) and (16h) and (16i) and (16j), respectively) was thus calculated. The simple application of Eq. (5a) constituted the last step, giving the model curve for the DSS. The only parameter that required an adjustment to obtain the best possible fit of the experimental data was εt . Fig. 13 gives an example of the variation of the different parameters with total strain. The figure illustrates the behaviour described in the previous sections: after an initial stage during which the strain is localised almost exclusively in ferrite, the local strain rate in the softer phase decreases; as a result, the flow curve for ferrite exhibits a drop, that does not depend on any microstructural softening mechanism, but is only an effect of strain redistribution. By contrast, the strain in austenite increases in the intermediate transition region, until, in stage 3, the strain rate acting in soft and hard zones becomes equivalent. The model suggests that when early fracture due to cavitation occurs, i.e. mostly in the high-Z regime (Fig. 3), the ratio between the strain accumulated in ferritic and austenitic regions is relatively high; in the case

S. Spigarelli et al. / Materials Science and Engineering A 527 (2010) 4218–4228

Fig. 13. (a) Model curve obtained by Eqs. (5a) and (16); the figure also shows the variation of the stress acting in austenite and ferrite, and the experimental curve; (b) variation of the strain and strain rate in ferrite and austenite, calculated by Eq. (16).

of a sample deformed at 950 ◦ C–0.05 s−1 (Fig. 4a), for a total strain εDSS = 5, i.e. just before the fracture, the model predicts that the strain accumulated in ferrite is only 1.12 times higher than that in ferrite. When the strain rate increases at 5 s−1 , the strain to fracture decreases due to extensive cavitation, a damage mechanism well documented in Fig. 4c; for εDSS = 1.3, again close to the fracture, the strain in ferrite is twice that accumulated in austenite. On this basis, due to early damage propagation, it can be concluded that the microstructural analysis will not reveal significant traces of deformation in austenite in all those cases in which the fracture occurs before or near εt . Fig. 14 shows a comparison between the model curves and the experimental data; the agreement between the model and the experiments is very good; the only “free” parameter, εt , was found to increase from 0.8 to 1.1 with increasing strain rate from 0.05 to 5 s−1 . The analysis of the figure demonstrates that the model is indeed able to “capture” the crucial feature of the mechanical behaviour in torsion of wrought DSS, i.e. the sigmoidal shape of the first part of the stress vs. strain curve. 5. Final remarks The model presented in the present study is based on the hypothesis that during torsion of wrought DSS, the load transfer mechanism is delayed to a later stage of the straining process. The stress redistribution through load transfer is necessary, in absence of any other strain accommodation mechanism, to preserve the continuity of adjacent phases characterized by different attitudes to straining. On the other hand, sliding of the austenite/ferrite incoherent interfaces in rolled DSS has been observed to significantly contribute to the accommodation of the high strains accumulated

4227

Fig. 14. Comparison between the experimental data and the model curves: (a) curves obtained under a strain rate of 0.5 s−1 ; (b) curves obtained at 1100 ◦ C. Only the part before the peak is illustrated, since the model does not take into account the subsequent softening.

in the soft ferrite, while austenite undergoes little deformation. It can be thus reasonably concluded that is the possibility of sliding that delays the onset of load transfer, causing the sigmoidal shape of the stress strain curve. If this conclusion is true, the absence of sliding, due to the semi-coherent nature of the ferrite/austenite interface in as cast DSS, should correspond to the early occurrence of load transfer and stress redistribution. Fig. 15a plots the model curves calculated at 1000 ◦ C–1 s−1 , with εt = 0.05 and 1, respectively. Provided that the model in its present form cannot describe the softening after the peak, the shape of the first part of curves, if not the peak values, that are different, closely resembles that of the as cast and wrought DSS tested in torsion by Iza-Mendia et al. [12]. Fig. 15b plots other curves, calculated at different temperatures under the same strain rate with εt = 0.05, and a marked similitude with the initial part of the curves obtained in [12] by testing in torsion the as cast alloy is again observed. This analysis is a confirmation that the absence of sliding leads to the early occurrence of load transfer through the semi-coherent interfaces, an effect that can be described by a low value of εt , in contrast with the case of incoherent interfaces that do not inhibit sliding thus determining a late transfer of the load from ferrite to austenite (high εt ). The use of Eq. (5b) for calculation of the strain acting in the duplex structure deserves a last mention; Cho and Gurland [28] showed that the validity of Eq. (5) is restricted to small deformations; for true strains Eq. (5b) should be rewritten in the form [33]



εDSS = ln f ε + (1 − f )εı



(17)

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an excellent description of the experimental data; this observation suggested that, in correspondence with the peak stress, both phases deform with similar strain rate; 2. in excellent agreement with the findings of other authors, the equivalent stress vs. equivalent strain curves were found to exhibit a sigmoidal shape. This effect was attributed to the strain partitioning between the soft ferrite and the hard austenite; during the early stage of deformation strain accumulates in ferrite, while austenite remains almost undeformed. A simple mathematical model describing the transition from this initial configuration to the final condition, where both phases deform with similar strain rate was then proposed, and used to successfully describe the experimental flow curves. 3. The different shape of the flow curve in the as cast material was explained by the early occurrence of load transfer, leading to an early onset of a constant strain rate configuration in both phases. References

Fig. 15. (a) Comparison between the shapes of the calculated curves with εt = 1 (delayed load transfer due to easy interfacial sliding) and εt = 0.05 (early onset of load transfer due to inhibited sliding) at 1000 ◦ C–1 s−1 with experimental data for as cast and wrought 2205 DSS [12] and (b) curves calculated at 1 s−1 under different temperatures with εt = 0.05, and experimental data for as cast DSS [12].

that reduces to Eq. (5b) only for low strains. The error introduced in the model, on the other hand, is relatively negligible when one considers that, for high strains, the flow stresses are considered to be constant and equivalent to the peak stress. 6. Conclusion The high temperature workability of the 2205 duplex stainless steels has been investigated by torsion testing in a wide range of experimental conditions, drawing the following conclusions: 1. The dependence of peak flow stress on strain rate and temperature has been described by the composite model, where austenite is the hard phase and ferrite in the soft one, obtaining

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