Constitutive model for describing flow behaviour of Fe–18Cr–10Mn–0·39N austenitic stainless steel

Introduction

Austenitic stainless steels have found wide spread use in strategic industries due to their excellent formability, weldability and workhardening, high corrosion resistance, and energy absorption capabilities. ¹ Fe–Cr–Ni and Fe–Cr–Mn–N are the main alloying systems of austenitic stainless steels. Most of earlier investigations are focused on the earlier one. However, replacement of Ni and improvement of mechanical and corrosion properties have caused attention of many researchers towards the later system with a rising trend. ² These studies are mainly devoted to high nitrogen stainless steels (N concentration >0·4 wt-) with stable austenite.

Metastable austenitic alloys are prone to transformation from the initial face centred cubic (fcc) (austenite) γ phase to ϵ-martensite phase with hexagonal close packed crystal structure and/or body centred cubic α′-martensite phase through plastic deformation slightly above the martensite start temperature M_s. Earlier studies suggest that α′-martensite nucleates at the intersections of shear bands.^3,4 Shear band is a collective term for the planar defects (such as faults, twins and ϵ-martensite) that form as a result of overlapping of stacking faults on austenite {111) planes during plastic deformation.

It is widely accepted the modified Ludwik equation is suitable for describing flow behaviour of low stacking fault energy (SFE) fcc alloys ⁵ 1 where K₁ and n₁ are the strength and workhardening coefficients respectively. At low strains, the tensile properties of low SFE fcc materials deviate from the power law, and the second term on the right hand side of equation (1) accurately describes the flow behaviour. Despite Fe–Cr–Ni austenitic alloys, a few studies have been devoted to model the stress–strain response of Fe–Cr–Mn–N austenitic alloys, especially the metastable grades. Li et al. ⁶ showed that flow behaviour of high N (0·82–0·96 wt- N) stable austenitic stainless steels is well described by equation (1). Milititsky et al. ⁷ showed that the modified Ludwik equation is appropriate for modelling the flow behaviour of stable Fe–Cr–Mn–N austenitic stainless steels but not metastable grades. In a similar study, Soussan et al. ⁸ reported that the workhardening behaviour of stable 316 L alloys with nitrogen contents up to 0·25 wt- could be successfully modelled using the modified Ludwik relation. Lee at al. ⁹ investigated the effects of nitrogen content on deformation induced martensitic transformation (DIMT) in Fe–18Cr–10Mn–XN alloys and applicability of the modified Ludwik relation for describing their flow behaviour. They observed that during tensile deformation, the σ (true stress)–ϵ (true strain) response was changed at different transient strains [ϵ (), ∼19–22] depending on the nitrogen content (ranging from 0·33 to 0·51 wt-). At the beginning of deformation, a γ→ϵ martensitic transformation occurred, whereas the formation of deformation induced α′-martensite was predominant after transient strain. They showed that the modified Ludwik relation is not suitable for metastable grades, which was related to the occurrence of DIMT. Figure 1 shows the experimental σ–ϵ curve of Fe–17·83Cr–9·73Mn–0·39N–0·03C–0·24Si (wt-) alloy and fitted modified Ludwik curve from the work of Lee et al. ⁹

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Experimental σ–ϵ curve of Fe–17·83Cr–9·73Mn–0·39N–0·03C–0·24Si (wt-) alloy and fitted modified Ludwik curve ⁹

These studies suggest that development of a predictive model for characterising the flow behaviour of metastable Fe–Cr–Mn–N austenitic steels appears to be lacking. In the present study, a basic constitutive model is developed to predict the tensile stress–strain response of metastable nickel free austenitic stainless steels at room temperature using the approach of Narutani et al. ¹⁰ The results of Lee et al. ⁹ study on describing the flow behaviour of metastable Fe–18Cr–10Mn–0·39N alloy (Fig. 1) are used to validate the model as a typical example. The findings can be extended to other chemical compositions. It is believed that development of such predictive models can be useful for designers and manufactures in predicting material response in case of service loads or estimating the required force for material forming.

Experimental

In the present work, a commercial grade of 4340 steel with chemical composition of Fe–0·39C–0·78Mn–0·32Si–0·03Cu–0·71Cr–0·2Mo–1·66Ni–0·001S–0·007P (wt-) was used to determine tensile behaviour of martensite at room temperature. Grain size of the alloy was ∼35 μm. This material was first annealed at 860°C for 30 min and then was directly quenched in industrial oil bath with a temperature of ∼25°C. Fraction of retained austenite was determined to be ∼2. Tension test was performed with an Instron type tensile machine at 25°C with a crosshead speed of 0·5 mm min⁻¹.

Model development

The microstructure of Fe–Cr–Mn–N austenitic stainless steels can be composed of different constituents depending on SFE. Lee at al. ¹¹ studied the correlation between SFE and deformation microstructure of high interstitial alloyed austenitic Fe–18Cr–10Mn–(N or N+C) alloys. They reported that as the content of the interstitial elements increases, the deformation microstructure changes in a sequence DIMT (SFE<15 mJ m⁻²), mixture of martensite and twin (15<SFE<20 mJ m⁻²) and finally deformation twin (20<SFE<30 mJ m⁻²). Spencer et al. ¹² showed that α′-martensite acts as an elastic reinforcing phase as it supports a higher stress than the austenite tensile loading, even though the martensite codeforms plastically with the austenite. Therefore, microstructure of a deformed metastable alloy (SFE<15 mJ m⁻²) can be considered as a composite material in which austenite is the matrix and α′-martensite is the reinforcing constituent. Narutani et al. ¹⁰ derived a constitutive model for predicting the flow behaviour of metastable Fe–Cr–Ni austenitic alloys from the deformation induced transformation kinetics and the flow properties of the two separate phases austenite and martensite. They used model of Olson and Cohn ¹³ to predict the kinetics of the strain induced α′-martensite transformation. In this model, it was presumed that the α′-martensite is nucleated at the shear band intersections, and that the nucleation and growth process of α′-martensite is controlled by two parameters, α and β. As a result, the following equation for the α′-martensite volume fraction f_α′ as a function of ϵ was obtained: 2 where α controls the rate of the shear band formation. It was assumed to be dependent on SFE and strain rate. β is proportional to the probability that the α′-martensite is nucleated at a shear band intersection, and it is dependent on the chemical driving force and temperature. The exponent n describes the rate of the formation of the shear band intersections, and it was found to have the constant value of 4·5. The model predicts a sigmoidal f_α′–ϵ curve.

Figure 2 schematically shows a composite microstructure of austenite/α′-martensite. Strength of this composite material can be expressed as follows, based on the assumption that the strain in both phases is equal to the macroscopic plastic strain 3 or 4 where σ_mixture is the strength of mixture; σ_γ and σ_α′ are the strength of austenite and α′-martensite respectively; and f_γ and f_α′ are the volume fraction of austenite and α′-martensite respectively. Since the right hand variables of equation (4) depend on strain, it is possible to determine the strength of the alloy as a function of strain by determining the correlation between these variables and strain.

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Schematic representation of composite microstructure of austenite/α′-martensite

In equation (4), it is assumed that the effective strain in austenite and martensite is the same and is equal to the macroscopic strain. However, Narutani et al. ¹⁰ showed that this assumption might not be valid since some of the latter strain arises from a bias of the martensitic transformation shape change, which does not correspond to the working of either phase. The state of plastic strain of the phases would then correspond to the total strain minus a transformation strain ϵ_T. This strain is proportional to the amount of transformation (ϵ_T = αf). They carried out tensile testing of metastable austenitic steel at −196°C at which stress assisted transformation took place. It was found that the slope of the strain versus α′-martensite volume fraction curve (α) had a constant value of 0·12, which corresponds to the transformation strain of 0·12 when the alloy is fully transformed to martensite. This is an upper limit estimate of the coefficient α. Accordingly; equation (4) can be corrected as follows 5

Results and discussion

Figure 3 represents the change in volume fraction of α′-martensite (f_α′) as a function of true strain for the selected alloy using the data provided by Lee et al. ¹¹ It is observed that at strains up to ∼0·1, f_α′ is smaller than ∼2. Therefore, one can assume the alloy microstructure to be single austenitic phase at this strain range and its flow behaviour is described using Ludwik type equation, as follows 6

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Formation of deformation induced α′-martensite as function of strain

Several models have been developed to describe the kinetics of deformation induced α′-martensitic transformation. In this present work, we use the Shin et al. ¹⁴ model. They applied the inelastic deformation theory to determine the volume fraction of deformation induced α′-martensite f_α′ as a function of strain 7 where β characterises the retained austenite stability, n denotes the deformation mode, f_s is the saturation value of transformed martensite volume fraction and ϵ₀ is the inelastic strain required for onset of martensitic transformation.

In Fig. 3, the predicted curve by Shin et al. model is shown for comparison with experimental data. It is observed that there is relatively a good agreement between the experimental and predicted values of deformation induced martensite. f_α′ can be determined in relation to strain using equation (7), as follows 8

Now, the only remaining unknown parameter of equation (5) is . It has been reported that N and C at concentrations of <0·6 wt- (2·85 at.-) affect the tetragonality ratio (c/a) of martensite in a similar way. ¹⁵ Therefore, a conventional Fe–C based alloy with equivalent concentration of C may be used to determine strength of Fe–N martensite in the studied alloy. Accordingly, σ_α′ is determined from a power law fit to the homogenous plastic deformation part in the σ–ϵ curve of 4340 steel (undergoing <4 plastic strain before necking): 9

Figure 4 represents the flow behaviour of the austenite, martensite and their mixture in comparison with experimental data of the alloy. It is observed that at the beginning of deformation, the austenite phase provides the main contribution to the mixture strength. With further deformation, beyond a transient strain, the martensite phase affects the mixture strength more strongly. This is related to the increase in deformation induced α′-martensite volume fraction with deformation. In addition, it can be seen that with increase in deformation that the alloy flows at evidently lower stress, which is attributed to the dynamic softening contribution of martensitic transformation Δσ_ds. This is related to the strain produced by the operation of phase transformation as a competing deformation mechanism. In terms of thermodynamics, the dynamic softening effect may be understood as the external load needed to deform the material is facilitated by the chemical driving force of the ϵ-martensite transformation. This contribution can be taken into account as follows: 10

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Flow behaviour of austenite, martensite and their mixture in comparison with experimental data of alloy

Figure 5 shows the variation of Δσ_ds as a function of f_α′ predicted by equation (8). It is observed that there is a linear relationship between these parameters (R² = 0·99): 11

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Variation of Δσ_ds as function of f_α′ predicted by Eq. (8)

Combining equations (5)–(11) provides the complete model for describing the flow behaviour of the alloy. Figure 6 shows a good agreement between the experimental and model σ–ϵ curves of the alloy. Comparison of Figs. 1 and 6 reveal that composite model developed in this present work is more appropriate than the modified Ludwik relation for describing flow behaviour of metastable Fe–Cr–Mn–N austenitic alloys.

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Experimental and model σ–ϵ curves of Fe–17·83Cr–9·73Mn–0·39N–0·03C–0·24Si alloy

Conclusions

In this present work, a constitutive model is developed for describing flow behaviour of a metastable Fe–18Cr–10Mn–0·39N austenitic stainless steel, assuming alloy microstructure as a composite mixture of austenite/martensite. Results of previous studies were used to validate the model, and the following were found.