On the significance to use dislocation-density-related constitutive equations to correlate strain hardening with microstructure of metallic alloys: The case of conventional and austempered ductile irons

Journal of Alloys and Compounds 669 (2016) 262e271

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On the significance to use dislocation-density-related constitutive equations to correlate strain hardening with microstructure of metallic alloys: The case of conventional and austempered ductile irons G. Angella a, *, F. Zanardi b, R. Donnini a a b

National Research Council (CNR) e Institute for Energetics and Interphases (IENI), Via R. Cozzi 53, 20125 Milano, Italy Zanardi Fonderie S.p.A., Via Nazionale 3, 37046 Minerbe, VR, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 December 2015 Received in revised form 26 January 2016 Accepted 29 January 2016 Available online 2 February 2016

In order to highlight the importance of selecting the proper constitutive equation of plastic flow in metallic alloys, the strain hardening behaviours of conventional (DI) and austempered ductile irons (ADI) with similar hardness were investigated as study-case. The empirical Hollomon equation and the dislocation-density-related Voce equation were compared. Hollomon equation showed strong limitations in describing the different strain hardening behaviours of DIs and ADIs. Conversely, Voce equation approximated properly all the flow curves, and Voce parameters could describe successfully the different strain hardening behaviours of the two groups of alloys. Voce parameters, rationalised by the KocksMecking-Estrin physical concepts, could also give an insight to the micro-mechanisms that underlie strain hardening, like dislocation density multiplication, dynamic recovery and microstructure features that affect these micro-mechanisms during straining. Through the analysis of Voce parameters it could be highlighted that the DIs strain hardening behaviour was mainly caused by the fine pearlitic structure, consisting of ferritic lamellae with sub-micrometric widths. The improvement of austempered alloys in strength and ductility was related to a significantly lower propensity of ADIs to recover dynamically, which was mainly attributed to the dual phase structure of these alloys. Furthermore, the strain hardening analysis through Voce formalism could identify in ADIs a critical condition of strain hardening rate to be investigated to improve further their ductility, which could not be found with Hollomon equation. The dislocation-density-related Voce equation describes properly the strain hardening behaviour of metallic alloys and give certain correlations between strain hardening behaviour and microstructure. © 2016 Elsevier B.V. All rights reserved.

Keywords: Strain hardening Constitutive equations Microstructure Ausferrite Metals and alloys

1. Introduction Production routes of commercial alloys are often characterized by solid state transformations. Strain hardening is very sensitive to microstructure and, therefore, strain hardening analysis is a powerful tool to study microstructure evolution during production processes. One of the most significant example is the determination of the initiation of dynamic recrystallisation in alloys deformed at high temperatures according to the widely-used method of inflection point of strain hardening rate [1]. In works focussed on

* Corresponding author. E-mail addresses: [email protected] (G. Angella), [email protected] (F. Zanardi), [email protected] (R. Donnini). http://dx.doi.org/10.1016/j.jallcom.2016.01.233 0925-8388/© 2016 Elsevier B.V. All rights reserved.

correlating strain hardening and microstructure, empirical equations like Hollomon and Hollomon-type equations with no physical meaning [2e5] have been often fitted to the experimental flow curves and the trends of the characteristic parameters of these equations are correlated to the process conditions or chemical compositions. However, this is an improper procedure, since strain hardening analysis should be first performed by studying the experimental strain hardening rate (ds/dεp) vs. flow stress (s) with εp the plastic strain, and then an opportune constitutive equation should be selected to fit the flow curves [2,6e9]. In this way, other constitutive equations could appear more adequate than the empirical Hollomon-type equations to describe the plastic behaviour of alloys. There are indeed constitutive equations that have physical meaning, like the dislocation-density-related constitutive

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equations [6e9], where their characteristic parameters can be correlated directly to alloys microstructure and micro-mechanisms of plasticity. Therefore, these equations are potentially more useful to characterize metallic alloys and their microstructure. The Hollomon-type constitutive equations (Hollomon, Swift, Ludwik and Ludwigson) are power law relationships between s and εp. Hollomon equation is the most widely used for its simplicity, that is

s ¼ K$εnp

(1)

K is the strength coefficient and n is the strain hardening exponent. The Hollomon parameters K and n can be easily found through plotting the experimental data s e εp in a logelog plot. Dislocation-density related constitutive equations are exponential decay relationships between s and εp [6e9]. Voce equation [10] is the simplest of these equations and is defined as


s ¼ sV þ ðso  sV Þ$exp  εp εc

(2)

sV is the saturation stress that is achieved asymptotically with plastic straining, εc is the characteristic strain that defines the velocity with which sV is achieved and so is the back-extrapolated stress to εp ¼ 0. Actually, the saturation stress is never achieved during tensile tests because of the specimen instability due to necking. Therefore, sV is an ideal value that can be used to compare materials when Voce equation is used to model flow curves. The differential form of Voce equation is a linear relationship between ds/dεp and s, since ds s 1 ¼ V s dεp εc εc

(3)

So, if linear regions in the experimental data strain hardening rate (ds/dεp) vs. flow stress (s) are found, Eq. (3) is used to find the Voce parameters εc and sV, and Voce equation can fit properly the flow curves. There is a significant benefit in using Voce equation, since it can give an insight into the relationship between strain hardening and microstructure. In fact, although Voce equation was originally proposed as an empirical equation in 1948 [10], later Kocks and Mecking [6,7], and Estrin [8,9] gave physical meaning to the differential form of Voce equation in Eq. (3), through relating the Voce parameters εc and sV to the micro-mechanisms underlying plastic deformation, like dislocation density multiplication, dislocation density reduction through dynamic recovery and microstructure features affecting these micro-mechanisms, as reported also in austenitic stainless steels [11,12]. The case of ductile irons is quite significant, since completely different microstructures in these alloys of similar compositions can be obtained through different heat treatments and solid transformations. The results of analysing the plastic behaviours of ductile irons through the Hollomon and the Voce equations are reported to compare the capabilities of these two kinds of equations to describe strain hardening and correlate strain hardening behaviour with microstructure. Conventional Ductile Irons (DI) identify a group of alloys that contains limited percentages of alloying elements (mainly C, Si and minor Cu, Mn and Mg) to provide a microstructure characterized by graphite with nodular shape. The application of DIs in engineering components is widely used for their good combination of fine mechanical properties and economic return of their industrial production over irons and steels [13,14]. As increasing strength and toughness of DIs have been required, chemical compositions and production routes have been widely investigated to obtain different microstructures and good mechanical properties [15e17]. Depending on the conventional ductile iron grade the microstructure can be fully ferritic, ferritic-pearlitic or fully pearlitic. Ferritic

263

matrix with graphite nodules results in good ductility, impact resistance, tensile and yield strength. Pearlitic matrix provides higher strength, better wear resistance with moderate ductility and impact resistance. Ferritic-pearlitic structure presents intermediate mechanical properties with good strength and toughness [18e20]. Isothermed ductile irons (IDI) [21] are characterised by perferritic structure, where ferrite and pearlite are interconnected instead of having the conventional “bulls-eye” shape. This structure is obtained through austenitizing low alloyed ductile irons in the intercritical range of temperatures where only part of the matrix transforms into austenite [22]. The interconnection improves strength and ductility with respect to conventional DIs with similar hardness. Austempered Ductile Irons (ADI) are produced through austempering conventional ductile irons. Austempering consists of austenitizing in the range 850e890  C and isothermally holding the alloys in salt bath at temperatures typically between 250 and 400  C, where the austenite decomposes resulting in a microstructure of acicular a ferritic-bainitic Widmanst€ atten plates, metastable gHC austenite with high carbon content, and graphite nodules dispersed in the a þ gHC matrix [23e25]. These alloys exhibit an outstanding combination of high strength, ductility, toughness and fatigue strength [23,26,27] and due to their superior mechanical properties, many engineering components such as gears and crankshafts have been recently produced in ADIs [28,29]. Investigating plastic behaviour of these materials is necessary to rationalize their general mechanical properties, like wear and low cycle fatigue [30e34]. It has recently underlined [22] that there is a significant gap between the potential and the actual applications of DIs, and this gap has been attributed to a conservative approach regarding the material properties, particularly for perferritic IDIs and upper ausferritic ADIs. In order to fill this gap, the plastic behaviour analysis cannot be limited to determine engineering parameters, as yield stress (YS), ultimate tensile strength (UTS) and elongation to fracture (LF), but the strain hardening behaviour of materials (or the shape of the flow curves) has to be analysed deeply, correlating possibly strain hardening and microstructure. Strain hardening analysis has been used to characterize the products of different austempering processes [35e37]. However, in these works on ADIs the strain hardening analysis has been carried out only by using empirical equations, like Hollomon and Hollomon-type equations [30e37]. In Ref. [38] Hollomon, Ludwigson and also Voce equations were used to fit the tensile flow curves of compacted graphite cast irons at elevated temperatures, but the work was focused on the comparison of the capability of these equations in approximating the experimental flow curves, but no proper analysis of strain hardening based directly on microstructural aspects was carried out. The first part of the present work is focused on the comparison between the procedures of analysis of the plastic behaviours of conventional and austempered ductile irons with Hollomon and Voce equations, focusing on the capabilities of the equations to approximate flow curves and particularly to describe the relevant differences in strain hardening behaviours of DIs and ADIs with similar hardness. In the second part, the physical interpretation of the Voce parameters based on mechanistic equation of strain hardening proposed by Kocks-Mecking-Estrin was used to rationalise the significantly different strain hardening behaviours of conventional and austempered ductile irons. 2. Theoretical background 2.1. Physical meaning of Voce equation In FCC and BCC metals with hard obstacles to dislocation

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movement, the flow stress is given by Refs. [39e41].

s ¼ sf þ sOr þ Mao Gbr1=2

 (4)

sf is the friction stress because of solid solution and PeierlsNabarro stress (negligible in FCC metals), whilst sOr is the stress to overcome hard phases through bowing according to the Orowan mechanism. The third term is related to dislocation reciprocal interactions and depends on the total dislocation density r. M is the Taylor factor (z3.0 in polycrystalline FCC and BCC metals), ao the dislocationedislocation interaction strength (z0.5 [8]), G the elastic shear modulus and b the length of the Burgers vector. During plastic deformation the dislocation density increases continuously, so the stress required to produce slip raises according to Eq. (4). This phenomenon is called strain hardening. Kocks-Mecking [7,8] and Estrin [9,10] gave physical meaning to the differential form of Voce equation in Eq. (3). Strain hardening consists of two competing micro-mechanisms. The first positive term (sV/εc) of the right side in Eq. (3) is related to hardening because of dislocation multiplication and storage. This term is inversely proportional to mean free path of mobile dislocations that is correlated to microstructural features like, for instance, dislocation interactions Lo, grain size D and ferritic lamellae width l in pearlite [41], according to the relationship [7e10].

sV 1 1 1 f þ þ εc Lo D l

(5)

Large values of (sV/εc) indicate high ability to strain-harden because of dislocation multiplication. Therefore, fine structure (small D and l) induces high strain hardening rates. The negative term (s/εc) in Eq. (3) is related to softening that occurs during straining because of annihilation of dislocations with opposite Burgers vectors through cross-slip and formation of low energy dislocation structures [7e10]. All together these two phenomena of dislocation reduction are called dynamic recovery. The term 1/εc is related to dislocation properties that, in turn, depends on the crystallographic lattice where mobile dislocations move producing plastic strain [8]. Small 1/εc indicates low ability to recover dynamically. The overall strain hardening behaviour of materials depends on the competition between these two micromechanisms. Large (sV/εc) and large 1/εc cause significant strain hardening rates, whilst small (sV/εc) and small 1/εc cause low strain hardening rates. The saturation stress sV is achieved when dislocation multiplication and dynamic recovery balance, resulting from the ratio between (sV/εc) and 1/εc. 2.2. Uniform strain The uniform deformation εu is defined as the true strain corresponding to the beginning of geometrical instability in tensile re's criterion. testing, which occurs at ds/dεp ¼ s, that is the Conside During tensile testing some local damage occurs because of the heterogeneous microstructure of these alloys. However, the flow curves before geometrical instabilities are related to the average plastic behaviour that is mainly affected by the interactions between mobile dislocations and glide obstacles. εu describes the ability of materials to deform uniformly, so it is used as formability parameter in ductile materials [5,42]. Theoretical uniform strain εTh u can be calculated from constitutive equations by imposing the re's criterion. Comparison between experimental εu and Conside theoretical εTh u can be used to evaluate the ability of constitutive equations in approximating the experimental flow curves [4]. In Th Hollomon formalism εTh u is straightforward, as in εu ¼ n, that is the strain hardening coefficient [2e4]. The theoretical uniform plastic strain calculated with the Voce formalism is

εTh u ¼ εc $ln

εc þ 1 sV  so $ εc sV

 (6)

In Eq. (6) εTh u is related to the ability of the material to strainharden through the parameters εc, so and sV that have physical meanings. Therefore, conversely to the Hollomon approach, in Voce formalism εTh u can be correlated to the micro-mechanisms causing strain hardening behaviour and εTh u can be used also to rationalize the ductility of the investigated ductile irons. 3. Experimental 3.1. Materials The materials were selected in order to compare the plastic behaviour of conventional and austempered ductile irons with similar hardness. Materials Brinell hardness (HB) was measured complying to ASTM E10-12. In this work the materials were divided into two groups according to their microstructures. Group DI consisted of three ductile irons with ferritic-pearlitic microstructures and different hardness: 1. ductile iron DIFP 700 e HB 240; 2. isothermed ductile iron IDI 800 e HB 290; 3. isothermed ductile iron IDI 1000 e HB 330. Group ADI consisted of three austempered ductile irons with different hardness: 1. ADI 800 e HB 290; 2. ADI 1050 e HB 330; 3. ADI 1050 e HB 350. The alloys microstructures were observed through a high resolution Scanning Electron Microscope (SEM) SU-70 by Hitachi after conventional metallographic polishing procedure and final etching with Nital 2.5%. 3.2. Tensile tests and strain hardening analysis Tensile tests at room temperature and strain rate 104 s1 were performed on round specimens with initial gage diameter do of 12.5 mm and gage length lo of 50 mm complying to ASTM E8-8M. In Fig. 1 the engineering tensile flow curves S vs. e are reported, where S ¼ F/Ao and e ¼ (l  lo)/lo. S and e are the engineering stress and strain. F, Ao and l are the instantaneous load, the initial sectional area and instantaneous gage length, respectively. The shapes of the flow curves of the two Groups of alloys appear very different. DI materials yield at lower stresses, strain-harden rapidly and achieve the Ultimate Tensile Stress (SUTS) at small strains, so the DI flow curves have smooth transitions from the elastic region to the plastic one. In ADIs the yield stresses are higher than in DIs with similar hardness, but present sharp transitions from the elastic region to the plastic one, where they strain-harden slowly and almost linearly from yielding up to failure. The elongations to rupture for ADIs are significantly larger than in DIs with similar hardness, showing a considerably higher ductility that is typical of ADIs [23,26,27]. For plastic behaviour investigations the true stress s e true strain ε flow curves were considered, with s ¼ S$(1 þ e) and ε ¼ ln(1 þ e). The constitutive equations were fitted on the plastic components of flow curves only, by subtracting the elastic strain εe ¼ s/E (E ¼ Young modulus) to the total strain ε, that was εp ¼ ε  εe. The Young modulus E was calculated in each test by determining the slope of the initial elastic portion of the flow curve. Only the true

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265

Fig. 1. Engineering flow curves S vs. e of alloys (a) Group DI; (b) Group ADI.

re's criterion (ds/dεp ¼ s) was plastic flow curves before the Conside investigated, since beyond this condition the flow curves are not representative of materials behaviour. The parameters of Hollomon equations were found through fitting the experimental flow data lns vs. lnεp by using strains εp > 0.01. An example of the fitting procedure to find the Hollomon parameters and the Hollomon curve is reported for IDI 800 e HB 290 in Fig. 2a and b, respectively. Hollomon equation seems to approximate quite well the flow curve at small and intermediate strains. However, it does not match the curve at high strains and predicts a continuously increasing flow stress with increasing strain. The theoretical uniform strain according to Hollomon is εTh u ¼ n ¼ 0.142 that is well beyond the experimental uniform strain εu ¼ 0.091 in IDI 800. Strain hardening analysis to find the Voce parameters was performed through the examination of the experimental strain hardening data ds/dεp vs. s. Regions of data linearity in the plots ds/ dεp vs. s were found both for DIs and ADIs, so Eq. (3) could fit the strain hardening data linearity to find 1/εc and sV/εc. so was ultimately worked out by fitting Voce equation to the experimental flow curves (with 1/εc and sV from the previous strain hardening analysis). so is not the yield stress, even if it is close to that [43], and it incorporates the initial dislocation density, the stress because of

lattice friction (Peierls-Nabarro stress) particularly significant in BCC materials like ferrite, solid solution and the stress to overcome hard undeformable obstacles to dislocation motion [40,41]. An example of the fitting procedure of Voce equation is reported in Fig. 3 for IDI 800 e HB 290. In Fig. 3a, (sV/εc) and 1/εc are found from the strain hardening analysis; in Fig. 3b so is finally established from fitting Voce equation to the flow curve. Voce equation does not match the curve at small strains, whilst fits excellently the flow curves at intermediate and high strains. A saturation stress is predicted in the Voce equation: however, geometrical instabilities occur during mechanical testing, so sV could be defined as the maximum theoretical strength achievable. The theoretical uniform strain according to Voce formalism in Eq. (4) is εTh u ¼ 0.087 that is in excellent agreement with the experimental uniform strain εu ¼ 0.091 in IDI 800.

4. Results 4.1. Flow curves and strain hardening analysis The best Hollomon fits of DIs and ADIs flow curves are reported in Fig. 4a and b, respectively. Hollomon equation fits well ID 1000 e

Fig. 2. Fitting procedure of Hollomon equation for IDI 800 e HB 290: (a) lns vs. lnεp to find the Hollomon parameters n ¼ 0.142 and K ¼ 1408.1 MPa; (b) final Hollomon equation fitting the flow curve.

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Fig. 3. Fitting procedure of Voce equation for IDI 800 e HB 290: (a) analysis of strain hardening data ds/dεp vs. s to find the Voce parameters; (b) final Voce equation fitting the flow curve.

Fig. 4. Experimental true stressestrain curves and Hollomon fits (a) Group DI; (b) Group ADI.

HB 330 and IDI 800 e HB 290 at intermediate strains, fails at high strains, whilst approximates very badly the softer DIFP 700 e HB 240. On the other hand, the Hollomon equation approximates always badly the ADIs flow curves. In Table 1 the parameters of Hollomon equations for DIs and ADIs are summarised. The main result is that the strength coefficients K are higher in ADIs than in DIs with similar hardness. However, in both alloy Groups, the strain hardening exponent n decreases with increasing hardness (unless ADI 1050 e HB 350), whilst the strength coefficients K increase. Unfortunately, it is not possible to relate these parameters to the significantly different shapes of flow curves, with DIs having pronounced concavities with high strain hardening rates and short elongations to rupture, and ADIs slight concavities with slow strain

hardening rates and large elongations to rupture. The results of differential data analysis are reported in Fig. 5 for the fitting procedure of Voce equation. The lines through the origins represent the geometrical instability condition in tensile re's criterion ds/dεp ¼ s. Besides transients testing, i.e. the Conside at small stresses soon after yielding, for all materials linear data regions were found at higher stresses, so the Voce parameters could be determined. In Table 2 the results of strain hardening analysis through Voce equation are summarised. The plastic behaviour of DIs in Fig. 5a is quite consistent, since the slopes 1/εc and the intercepts (sV/εc) of the best linear fits increase consistently with increasing hardness, with 1/εc ranging from about 25.1 to 34.1, and (sV/εc) from 2.15  104 to 4.00  104 MPa. In Fig. 5b ADIs have strain

Table 1 Hollomon parameters from the best fits of plastic flow curves with εp > 0.01. Material Group DI

Group ADI

DIFP 700 IDI 800 IDI 1000 ADI 800 ADI 1050 ADI 1050

Hardness Brinell

n

K (MPa)

240 290 330 270 330 350

0.221 0.142 0.103 0.144 0.121 0.131

1389.1 1412.5 1516.4 1337.3 1633.9 1738.4

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re's criterion ds/dεp ¼ s. Fig. 5. Results of strain hardening analysis of (a) Group DI, (b) Group ADI. The lines through the origin represent the Conside

Table 2 Voce parameters from strain hardening analysis. Material Group DI

Group ADI

Hardness Brinell DIFP 700 IDI 800 IDI 1000 ADI 800 ADI 1050 ADI 1050

240 290 330 270 330 350

hardening parameters significantly lower than DIs, since the slopes 1/εc are from 6.0 to 7.3 and the intercepts (sV/εc) from 0.76  104 to 1.17  104 MPa. Also in ADIs Voce parameters increases consistently with increasing hardness. The resulting saturation stresses sV are considerably higher in ADIs than in DIs. Therefore, Voce equation fittings identify clearly parameters of different values that can be related to the different strain hardening behaviours of two Groups of alloys. The best Voce fits of DIs and ADIs flow curves are reported in Fig. 6a and b, respectively. Besides small transients at yielding, Voce equation fits very well the flow curves for both DIs and ADIs regardless the significantly different shapes. Voce equation cannot describe the flow curves at yielding, which has been commonly reported in metallic materials [8,12,43] and also in compacted graphite cast irons [38]. This discrepancy at small strains has been attributed to transitional evolution of the dislocation structures

sV/εc (MPa) 2.15 2.77 4.00 0.76 0.99 1.17

     

4

10 104 104 104 104 104

1/εc

sV (MPa)

25.1 27.0 34.1 6.0 6.3 7.3

854.6 1024.2 1173.5 1271.9 1577.5 1608.6

[43] and it does not affect the validity of physical interpretation of Voce equation at large strains. It is interesting that through the strain hardening analysis, DIs flow curves were found to achieve always the geometrical instability conditions, that is ds/dεp ¼ s, whilst ADIs never attained it, since ruptures occurred at about constant values ds/dεp ¼ 1.5$s. In Table 3, the theoretical uniform strains εTh u according to Hollomon and Voce equations are reported and compared with the experimental uniform strains εu, whilst εr are plastic strains to rupture, and εc are the characteristic strains. In DIs, where experimental re's conditions were met, there was an excellent match Conside between theoretical Voce values εTh u and experimental uniform strains εu, whilst Hollomon values εTh failed dramatically at u matching εu. In ADIs ruptures occurred at strains εr well before necking, since all ADIs broke at about ds/dεp ¼ 1.5$s. Hollomon

Fig. 6. Experimental true stressestrain curves and Voce fits (a) Group DI; (b) Group ADI.

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Table 3 Comparison between theoretical uniform strains from Hollomon and Voce equations and experimental values: εTh u ¼ theoretical uniform plastic strain; εu ¼ experimental uniform plastic strain; εr ¼ plastic strain to rupture; εc ¼ characteristic strain. Material Group DI

Group ADI

DIFP 700 IDI 800 IDI 1000 ADI 800 ADI 1050 ADI 1050

Hardness Brinell

Hollomon εTh u (¼ n)

Voce εTh u

εu

εr

εc

240 290 330 270 330 350

0.221 0.142 0.103 0.144 0.121 0.131

0.105 0.087 0.065 0.189 0.169 0.165

0.104 0.091 0.067 e e e

0.110 0.106 0.073 0.147 0.129 0.123

0.040 0.037 0.029 0.167 0.159 0.137

values εTh u are smaller than εr in ADI 800 and ADI 1050 e HB 330, which disagrees with experimental findings, whilst in ADIs Voce equations predict that necking should start well beyond the rupture strains εr, which is in agreement with experimental findings. Thus, Voce equation seems to describe correctly the overall plastic behaviour of both DIs and ADIs, whilst Hollomon equation shows strong limitations, confirming the goodness of Voce equation to describe flow curves and strain hardening behaviour of ductile irons.

path of mobile dislocations cannot be easily found as for pearlitic structure. However, the microstructure in ADI 800 is considerably coarser than IDI 800 with similar hardness (Fig. 7b), so in ADI 800 both the widths of ferrite (l(a) in Fig. 8b) and austenite (l(gHC) in Fig. 8b) are between about 0.5 and 1.0 mm. Assuming the Orowan mechanism controlling dislocation motion, it can be inferred that the average mean free path lmean of the ADI 800 structure is in the range of about 0.5e1.0 mm. 5. Discussion

4.2. Microstructure As an example, SEM micrographs of the microstructures of IDI 800 and ADI 800 with similar hardness are reported in Figs. 7 and 8, respectively. The microstructures from other DIs and ADIs with similar hardness are comparable to the microstructures here reported. In Fig. 7a IDI 800 shows ferritic-pearlitic structure, where BCC a ferrite is soft and pearlite is strong, consisting of ferritic lamellae confined between hard Fe3C cementite lamellae. The strong component of material that governs the plastic flow is the pearlite that is reported at higher magnifications in Fig. 7b. Because of the chemical etching, ferrite was removed, so the bright lamellae in Fig. 7b are made of cementite, and the spacing between them identifies the ferritic lamellae, that is l(a) in Fig. 7b. Cementite lamellae are obstacles to dislocation motion, so bowing mobile dislocations move in the ferritic lamellae according to the Orowan mechanism, causing plastic strain. Therefore the width of the ferritic lamellae identifies the mean ferritic free path of mobile dislocations [41]. From Fig. 7b it can be deduced that l(a) is in the range of about 100e200 nm. In Fig. 8a ADI 800 shows the typical austempered structure with a high volume fraction (up to ~70%) of €tten acicular laths a (bright areas) and a BCC ferritic Widmansta small volume fraction of high C content metastable FCC austenite gHC (dark areas). In Fig. 8b the microstructure of ADI 800 is reported. In ADIs, both ferrite and austenite deform, so the mean free

From the results of the flow curve fittings and strain hardening analysis, it can be concluded that Hollomon equation fails in approximating the experimental flow curves, particularly in ADIs, and the Hollomon parameters cannot describe the different strain hardening behaviours of DIs and ADIs with similar hardness. The Hollomon parameters seem to be related vaguely to the hardness HB of the alloys. Conversely Voce equation can approximate the flow curves both of DIs and ADIs and the Voce parameters can describe straightforwardly the different strain hardening behaviours of IDIs and ADIs, which suggests that Voce equation should be used to model and to investigate strain hardening behaviour of metallic alloys. The reason of the success of Voce equation is related to the physical meaning of the dislocation-density-related equation. In both Groups of alloys, Voce parameters increase consistently with increasing hardness, which can be easily rationalised through the relationship between microstructure and strain hardening rates of Kocks-Mecking-Estrin model. In Eq. (5) the mean free path l influences considerably the dislocation multiplication term (sV/εc), and in the case of high density of dislocation obstacles like DIs and ADIs where l ≪ Lo and D [7e9], Eq. (5) results in

sV 1 f εc l

(7a)

From Figs. 7 and 8, l(a) in IDI 800 was found to be between 100

Fig. 7. SEM micrographs with secondary electrons of IDI 800 e HB 290: (a) ferritic þ pearlitic microstructure at 5 kX; (b) detail of pearlitic microstructure at 60 kX.

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Fig. 8. SEM micrographs with secondary electrons of ADI 800 e HB 270: ferritic þ austenitic microstructure: (a) at 5 kX; (b) at 20 kX.

and 200 nm, whilst in ADI 800 lmean was inferred to be about 0.5e1.0 mm, that is about 5 times as coarse as in IDI 800: this is in good agreement with the dislocation multiplication terms in Table 2, where sV/εc in IDI 800 is about 3.5 times as big as in ADI 800, according to Eq. (7a). As further example, in Fig. 9 the microstructure of ADI 1050 is reported at the same magnifications of ADI 800, for comparison. In ADI 1050 the microstructure is finer, so lmean is about 0.2e0.5 mm, that is about 2 times the lmean of ADI 800. Furthermore in ADI 1050 sV/εc is about 1.5 times sV/εc of ADI 800, which agrees well with the sizes of the microstructures. The dynamic recovery term 1/εc depends strongly on dislocation properties and, in turn, on the crystallographic lattice where mobile dislocations move. Crystallographic structure, chemical composition and stacking fault energy (SFE) are the parameters that affect mainly dislocation cores and, as a consequence, dynamic recovery rate [8,40]. The deforming matrix in DIs is soft BCC a ferrite, whilst ADIs is a dual phase alloy consisting of hard BCC a ferrite (~70%) and soft FCC gHC austenite (~30%). At this stage, it can be concluded only that the big difference in crystallography and microstructures between DIs and ADIs is consistent to their significantly different dynamic recovery behaviours. However, in order to rationalize in more details the different dynamic recovery behaviours between DIs and ADIs, an deeper strain hardening investigation should be applied through introducing microstructure parameters (l and D) into the model of strain hardening according to Kocks-MeckingEstrin [33e36] and considering the different contribution of a and gHC to the plastic flow of ADIs. Microstructure characterization should support the improved approach, particularly in analysing the dual phase a þ gHC structure in ADIs. So it is important to underline that the aim of the present qualitative analysis is to

rationalise the better performance of Voce equation respect to Hollomon one at describing the strain hardening behaviour of metallic alloys on the bases of the physical meaning of Voce equation. The saturation stresses sV are considerably higher in ADIs than in DIs. The overall strain hardening behaviour of metallic alloys depends on the competition between dislocation multiplication and dynamic recovery. So large (sV/εc) and large 1/εc cause significant strain hardening rates, whilst small (sV/εc) and small 1/εc cause low strain hardening rates. However, the saturation stress sV depends on the balance between dislocation multiplication and dynamic recovery, i.e. the ratio between (sV/εc) and 1/εc, so from Eq. (7a), it results

    1 1 = sV f l εc

(7b)

Although in DIs the microstructure is finer with sub-micrometer

l that is considerably shorter than in ADIs with similar hardness, from Eq. (7b) it can be concluded that the higher stresses sV in ADIs are due to their smaller dynamic recovery terms 1/εc. So, although the coarse microstructure, the better strength of ADIs can be attributed to the significantly lower propensity of these alloys to recover dynamically respect to DIs. In Table 3 the strains to rupture εr are also reported. In ADIs the values εr are between 40 and 60% higher than in DIs with similar hardness, resulting in significant enhancement of ductility that is typical of ADIs [22,23,26,27]. It is important to note that ADIs have theoretical values εTh u between 90 and 120% larger than DIs with similar hardness, so even if the ADI rupture conditions were about

Fig. 9. SEM micrographs with secondary electrons of ADI 1050 e HB 330: ferritic þ austenitic microstructure: (a) at 5 kX; (b) at 20 kX.

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ds/dεp ¼ 1.5$s, ruptures were achieved later at larger strains, resulting in better ductility than DIs. The values εc in ADIs are considerably close to the corresponding εTh u , which means that the large uniform strains εTh u in ADIs are mainly due to the very low tendency to dynamic recovery. This is not the case of DIs, where dynamic recovery terms are larger. It is interesting to highlight that, if also in ADIs the strains to rupture εr could be equal to the theoretical values εTh u , a further improvement of ductility of about 30% could be achieved with significant benefits on the applications of ADIs. However, further investigations on the damage occurring during straining are needed to understand the rupture conditions ds/dεp ¼ 1.5$s.

6. Conclusions In order to highlight the importance of selecting the proper constitutive equation of plastic flow in metallic alloys, the plastic behaviours of conventional and austempered ductile irons with similar hardness were investigated through Hollomon and Voce equations. The ductile irons were subdivided into two groups according to their microstructures. Group DI consisted of ferriticpearlitic ductile irons and Group ADI of austempered ductile irons. Hollomon equation showed strong limitations in approximating the experimental flow curves, particularly for ADIs, and failed dramatically at describing the different strain hardening behaviours of DIs and ADIs. Both DIs and ADIs had linearity regions in the differential strain hardening data ds/dεp vs. s, so the tensile flow curves resulted properly fitted with Voce equations. Voce equation approximated properly the flow curves both of DIs and ADIs, and Voce parameters could describe properly the different strain hardening behaviour of DIs and ADIs with similar hardness. The physical meanings of Voce parameters based on the KocksMecking-Estrin model of strain hardening were used to rationalise the significantly different plastic properties of the considered materials. The following points can be summarised: 1. DIs have high strain hardening rates and achieve rapidly the saturation stresses because of their fine structure and high propensity to recover dynamically; 2. ADIs have low strain hardening rates and achieve slowly the saturation stresses because of their low propensity to recover dynamically; 3. although microstructure is coarser in ADIs than in DIs with similar structure, the improved mechanical properties of ADIs (higher strength and better ductility) are due to the significantly lower propensity to recover dynamically; 4. strain hardening analysis allowed to identify also a critical condition of strain hardening rate for rupture in ADIs that could be investigated in order to improve further their ductility. From the present investigations it can be concluded that dislocation-density-related Voce equation should be used to investigate the plastic behaviours of metallic materials, because of its outstanding capability to describe the strain hardening behaviours and correlate the micro-mechanisms occurring during deformation to the microstructure of metallic materials.

Acknowledgements Mr. E. Veneri, F. Vettore and S. Masaggia from R&D Laboratory Zanardi Fonderie S.p.A. are warmly thanked for their technical support. Mr Della Torre from CNR-IENI Milan is kindly thanked for his help for SEM sample preparation.

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