A quantitative understanding on deformation twins generation in single-phase austenitic stainless steels

Abstract

Single-phase austenitic stainless steels (316L) have attracted widespread attention from scientists because of their duplex microstructure. In this paper, to have a quantitative understanding on the microstructure deformation of 316L, a physical model based on dislocation theory and strain gradient theory is established to find out the critical conditions when deformation twins generate. The twinning stress and the stress caused by strain gradients are two factors affecting the deformation twinning process. Numerical simulation results reveal that the twinning stress decreases with the increase of twin spacing and the decreases of volume fraction of twins and the orientation of external shear stress; the stress caused by strain gradients increases with the decrease of matrix grain size.

Keywords

Single-phase austenitic stainless steel plastic deformation mechanism deformation twin twinning stress

Introduction

As a functional and modern structural material, the single-phase austenitic stainless steel (AISI 316L) has played an important role in engineering field because of its high work hardening, excellent resistance to corrosion and oxidation, and good formability over a wide range of temperature [1 –3]. Many scholars have done research on it. Leicht et al. [4] characterised the microstructure of 316L produced by additive manufacturing; Man et al. [5] studied influences of microstructures on its corrosion behaviour; and Inoue et al. [6] carried out an in-situ electron irradiation experiment to investigate the influence of annealing on the microstructure of 316L. From experimental phenomena in these literatures, we can clearly see that 316L is a material with a special microstructure.

It has a novel type of duplex microstructure yet single phase. The micro-size statically recrystallised (SRX) austenitic matrix grains, which are plastically compliant, and the coarse non-recrystallised grains, which with strengthening inclusions – nanotwinned austenitic (nt-γ) structures form the bimodal grain size distribution, as shown in Figure 1(a) and (b) [7]. This new synthesised stainless steel has been proven to be an effective compromise between ductility and strength. Micro-size matrix provides strength; coarse grains (CGs) provide ductility and nt-γ structures also contribute to material strength improvement through the recrystallised/nanotwinned duplex microstructure. Moreover, according to the experiment by Yan et al. [7], deformation twins (DTs) generated in 316L during plastic deformation because of the strain gradients between nanotwinned grains and their conjunctional SRX grains caused by stretching, and this phenomenon had a great influence on the subsequent deformation of this metal. So, it is necessary to do research on the generation of DTs in 316L.

Figure 1

(a) and (b) A scanning electron microscope-electron channelling contrast image of 316L [7].

Although a large number of experiments have studied the microstructures of 316L prepared by different methods [4,8 –10], and massive examinations including experimentations and molecular dynamics (MD) simulations have been done on the deformation twinning in ultrafine-grained (UFG)/nanocrystalline (NC) metals [11 –14], the theoretical research on the microstructure deformation of this neoteric stainless steel is still lacking. The purpose of this paper is to find out the critical conditions when DTs generate in 316L under tension through theoretical calculation.

Zhu et al. [15] explained the nucleation and growth of DTs in NC Al by presenting an analytical model based on classical dislocation theory. To evaluate the mechanical property of bimodal NC materials, Liu et al. [16] developed a composite constitutive model consisting of CGs and NC matrix in consideration of strain gradient. Zhu et al. [17] set up a dislocation density-based constitutive model of bimodal nanotwinned metals, which have structures similar to 316L, to study their mechanical behaviours. With the help of the above theoretical research, we establish a physical model to describe the generation of DTs. The effects of the twin spacing, the volume fraction of twins and the orientation of external shear stress on twinning stress, and the influence of the size of SRX grains on the stress caused by strain gradients are investigated.

Theoretical model

A model for the bimodal microstructure

To achieve the aforementioned goals, a composite model with a bimodal distribution of microstructures is set up for 316L, as shown in Figure 2. It can be divided into two parts: recrystallised micro-size austenitic grains as bulk matrix and non-recrystallised CGs embedded with nt-γ structures as strengthening inclusions. In contrast to some traditional alloys with hard second phases or structures, the two parts of 316L have the same elastic modulus and no phase boundaries occur between them [7]. So, phase boundaries are not taken into account in this model. It applies only to single-phase metals with a bimodal distribution and is not rigorous for dual or multiphase metals.

Figure 2

Schematic drawing of 316L with the assumption of the composite model.

From a published literature [7], during the tensile process of 316L, the twinning stress σ_twin and the stress caused by strain gradients σ_SG will be produced in CGs and SRX grains, respectively. When σ_twin and σ_SG satisfy Equation (1) for the first time, DTs generate. (1)

A model for nanotwinned grains

According to some experiments [18,19] and MD simulations [20,21], near grain boundaries (GBs) and twin boundaries (TBs), there exist obvious strain gradients, which can be attributed to the accumulation of dislocations along GBs and TBs during plastic deformation. So, as shown in Figure 3, the configuration of a nanotwinned crystal can be made up of a grain boundary effect zone (GBEZ) represented by the octagonal frame with blue background near GBs, twin boundary effect zones (TBEZs) represented by polygons filled with yellow profile lines near TBs, and interior crystal represented by white polygons filled with black profile lines. The thicknesses of GBEZ and TBEZ are d_GBEZ and d_TBEZ, respectively. λ is twin spacing and d_c is the grain size of the nanotwinned crystal.

Figure 3

Schematic drawing of nanotwinned metals with grain size d_c, twin spacing λ, thickness of grain boundary effect zone d_GBEZ and thickness of twin boundary effect zone d_TBEZ.

Twinning stress

Figure 4 is an analytical model of twin lamellae based on the classical dislocation theory. This model is used to predict the critical conditions when DTs nucleate in face-centred cubic (fcc) metals, and it is consistent with experimental observations [11,12]. Because stacking fault (SF) is the source of deformation twinning, we suppose that there is an SF in twin lamellae. The twin lamellae are subjected to an external shear stress τ , which is at an angle of α with line EF. b ₁ and b ₂ are leading and trailing Shockley partial dislocations, respectively. Under the stress τ , b ₁ emits from the TB AB and stops at the two points, A and B. This causes partial dislocation lines AE and BF to deposit on TBs. Likewise, b ₂ also emits from the TB AB and generates a dislocation line ACDB. Apparently, SF separates the partial dislocation lines CD and EF. Then, the reaction between the two partial dislocations generates AC and BD, the two full-edge dislocation parts [15]. If the partial dislocation line EF moves up a distance Δs, the external shear stress τ performs the work given by (2) where τ and b₁ are absolute values of τ and b ₁, respectively; β is an angle between the leading Shockley partial b ₁ and the dislocation line EF. It equals to 30° [15].

Figure 4

Schematic illustration of dislocation model in twin lamellae.

Given the motion of EF, the partial dislocation lines CE and DF lengthen 2Δs. According to a validated algorithm based on dislocation theory [15], the dislocation line energy increase as follows: (3) where μ is shear modulus; ν is Poisson's ratio; R is roughly equivalent to the value of twin spacing λ and r₀ can be estimated as the magnitude of the lattice dislocation b. This metal is fcc, thus , . a is the lattice parameter.

Moreover, overcoming the energy barriers from energy path related to twinning is necessary. Accordingly, the variation of generalised planar fault energy (GPFE) can be described as [22] (4) where γ_UT is the energy barrier to generate a twin fault and γ_US is the unstable stacking fault energy (SFE), which is the energy barrier to form an intrinsic stable SF.

Given the motion of b ₁, the external shear stress τ must work to surmount the increase of dislocation energy caused by lengthening segments AE, BF and the energy about twinning energy path GPFE. Thus (5) By substituting Equations (2), (3) and (4) into Equation (5) then simplifying, and using the relation σ_twin=M · τ_twin, we can acquire the twinning stress as (6) where M is Taylor factor. Equation (6) elaborates the relationship between the twinning stress σ_twin and the twin spacing λ.

Furthermore, during deformation, except for the obstacle of TBs to dislocations, twins providing an additional source of dislocations in the metal represent an extra obstacle to the motion of dislocations [23]. Here, we introduce a parameter – the average thickness of twins t. It can be described by the following formula [23]: (7) where f_nt-γ is the volume fraction of twins. The value of t is approximately 23 nm [7]. By deforming Equation (7), we obtain (8)

By substituting Equation (8) into Equation (6) and simplifying, we can acquire the twinning stress as (9) Equation (9) elaborates the relationship between the twinning stress σ_twin and the volume fraction of twins f_nt-γ.

A model for SRX austenitic matrix

The matrix consists of recrystallised austenitic grains, which are plastically compliant. With strain increasing, partial dislocations generating from CGs/SRX grains interfaces penetrate into twin lamellae successively. Then, because of the inhomogeneous deformation occurring in SRX grains around CGs, strain gradients gradually develop. Thus DTs emerge, and the grains with DTs grow into large crystals by merging with the adjacent grains. An experiment conducted by Yan et al. [7] can approve this process, and the plastic co-deformation mechanisms between matrix and CGs are roughly exhibited in Figure 5(a–d).

Figure 5

(a–d) Schematic illustration interpreting the process of DTs generation in 316L.

The stress caused by strain gradients

Strain gradient is a critical factor resulting in the generation of DTs. By combining the early strain gradient formula based on the Taylor's theory [24] and the relation σ = M · τ, we can acquire the stress caused by strain gradients (hereinafter referred to as strain gradient stress): (10) where ρ_T represents the total dislocation density. Here, ρ_T = ρ_NI + ρ_GBEZ. ρ_NI is the density of intragrain dislocations of the matrix part and ρ_GBEZ is the density of dislocations in GBEZ.

ρ_NI lies on athermal storage and annihilation of dislocations, which are two dynamic processes during the deformation process dominated by dislocation mechanism [25]. For a given overall strain ϵ_p, we can obtain (11) where and are densities of dislocations of athermal storage and annihilation, respectively. Based on the statistical approach proposed by Kocks and Mecking [26], Capolungo et al. presented two models explaining the athermal storage and the annihilation of dislocations [27]. On this basis, the relationship between and ϵ_p is (12) where ϵ_p is plastic strain; d_n represents the size of SRX grains and its average value is 2.2 μm [7]; ψ is the proportionality factor. The dynamic recovery process governed by annihilation is expressed by (13) where k₂₀ and are both dynamic recovery constants, and n₀ varies inversely with temperature T. Thus the evolvement of the density of dislocations with the rise of ϵ_p can be described by adding Equations (12) and (13), i.e. (14)

ρ_GBEZ is given by [28] (15) Here, (16) (17) where φ_GB is a geometrical factor; ϵ_GB denotes the plastic strain formed by dislocations in GBEZ and ϵ_GB can be expressed as (18) where n_GB represents the number of dislocations near GBs. By combining Equations (15)–(18) and simplifying, we can obtain the final expression about ρ_GBEZ: (19)

Results and discussion

Our aim is to determine the critical conditions when DTs generate in 316L. Parameters used in all calculations [17] are summarised in Table 1. For graphic beauty, some coordinate axes are logarithmic.

Table 1

Descriptions, symbols and magnitudes for different parameters in the model.

Parameter	Symbol	Magnitude
Shear modulus (GPa)	μ	86
Burgers vector (nm)	b	0.26
Poisson's ratio	ν	0.29
Taylor factor	M	3.06
Taylor constant	α	0.3
Grain size (μm)	d _n	2.2
Proportionality factor	ψ	0.2
Dynamic recovery constant	k ₂₀	18.5
Reference strain rate (s⁻¹)		1.75
Dynamic recovery constant	n ₀	12.25
Geometric factor	φ _GB	0.5–1.5
Thickness of GBEZ (nm)	d _GBEZ	3.6

Influence of various factors on twinning stress and strain gradient stress

Figure 6 presents the relationship between twin spacing λ and twinning stress σ_twin with different orientations of external shear stress α. For the angles not exceeding 90°, twinning stress decreases with the increase of twin spacing. Such a tendency is consistent with the work by Zhu et al [29]. Figure 6 also indicates that the impact of orientation of external shear stress on twinning stress is mainly reflected in the maximum value of twinning stress. As the angle increases, the maximum value of twinning stress increases significantly. Figure 7 depicts the twinning stress σ_twin as a function of volume fraction of twins f_nt-γ with different orientations of external shear stress α. The trend of these curves is contrary to that of the curves in Figure 6. Twinning stress rises as the volume fraction of twins increases, i.e. the more twins existing, the higher twinning stress. As for the influence of the orientation of external shear stress, it is similar to that in Figure 6.

Figure 6

Effect of twin spacing λ on twinning stress σ_twin with different orientations of external shear stress α.

Figure 7

Effect of volume fraction of twins f_nt-γ on twinning stress σ_twin with different orientations of external shear stress α.

Figure 8 shows strain gradient stress–strain curves with different sizes of SRX grains. As the sizes of SRX grains d_n decrease, the strain gradient stress σ_SG increases, and the gap between the two adjacent curves is more and more significant. It also shows that the contribution of strain gradient to the stress increases with the decrease of grain size. Because strain distribution is related to the distances between grains and CGs/SRX grains interfaces. Their specific relationship is as follows [30]: (20) where τ_∞ denotes an externally applied equilibrated stress; i represents two bonded phases; y indicates the distance; μ_i is shear moduli; c_i is gradient coefficient and (21)

Figure 8

Stress–strain curves for strain gradient stress σ_SG with different sizes of SRX grains d_n.

From the above two formulas, it can be seen that the strain of every SRX grain in matrix depends on their grain sizes, and the larger the grain sizes are, the smaller the strain is.

The generation of DTs

For twins with an average thickness t of 23 nm, the corresponding volume fraction of twins f_nt-γ can be obtained from a figure derived from an experiment by Soulami et al. [23]. In addition, the orientation of external shear stress α equals to 90°, and the sizes of CGs d_c are 50 μm. By substituting these values in Equation (9), the twinning stress is 732.546 MPa. When the sizes of SRX grains d_n are 2.2 μm, the strain gradient stress–strain curve is shown in Figure 9. Make a straight line parallel to the x-axis with y = 732.546 MPa, and the drawing at its lower right corner is a partial enlargement. When the strain gradient stress σ_SG exceeds the twinning stress σ_twin for the first time, the strain is approximately 22.5%. This result is consistent with an experimental result [7]. Therefore, at a strain of 22.5%, DTs generate in the SRX grains around CGs in 316L.

Figure 9

The critical strain for DTs generation in SRX grains adjacent to coarse grains.

Conclusion

This paper presents a quantitative understanding on the microstructure deformation of 316L. In other words, the generation of DTs in 316L during tensile process is studied by theoretical research. Our main findings are as follows.

The volume fraction of twins f_nt-γ and the orientation of external shear stress α have positive effects on the twinning stress σ_twin, i.e. the greater the values of f_nt-γ and α, the higher σ_twin. While, the effect of twin spacing λ on σ_twin is negative. The larger the twin spacing, the smaller the twinning stress.

The size of SRX grains d_n is the only factor affecting the strain gradient stress σ_SG, and the contribution of strain gradient to σ_SG increases with the decrease of d_n. It is easier for DTs to generate if σ_SG is greater. So, deformation twinning can be suppressed by grain refinement.

When strain reaches 22.5%, the strain gradient stress σ_SG in SRX grains exceeds the twinning stress σ_twin in CGs for the first time, and DTs generate in the SRX grains adjacent to CGs. Consequently, obvious microstructure deformation occurs in 316L.

Geolocation information

School of Mechanical and Power Engineering, Nanjing Tech University, 30 Puzhu South Road, Pukou District, Nanjing City, Jiangsu Province, China

Footnotes

Disclosure statement

No potential conflict of interest was reported by the authors.

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