Dislocation density based modeling of three-stage work hardening behaviour of type 316LN SS with varying nitrogen content and its finite element implementation for different notch radii

Journal Pre-proof Dislocation density based modeling of three-stage work hardening behaviour of type 316LN SS with varying nitrogen content and its finite element implementation for different notch radii

C. Praveen, J. Christopher, V. Ganesan, G.V. Prasad Reddy, B.K. Choudhary PII:

S0254-0584(19)31157-5

DOI:

https://doi.org/10.1016/j.matchemphys.2019.122342

Reference:

MAC 122342

To appear in:

Materials Chemistry and Physics

Received Date:

26 September 2018

Accepted Date:

19 October 2019

Please cite this article as: C. Praveen, J. Christopher, V. Ganesan, G.V. Prasad Reddy, B.K. Choudhary, Dislocation density based modeling of three-stage work hardening behaviour of type 316LN SS with varying nitrogen content and its finite element implementation for different notch radii, Materials Chemistry and Physics (2019), https://doi.org/10.1016/j.matchemphys.2019.122342

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Journal Pre-proof

Journal Pre-proof Dislocation density based modeling of three-stage work hardening behaviour of type 316LN SS with varying nitrogen content and its finite element implementation for different notch radii C. Praveen, J. Christopher*, V. Ganesan, G.V. Prasad Reddy, B.K. Choudhary Materials Development and Technology Division Indira Gandhi Centre for Atomic Research, HBNI, Kalpakkam – 603 102, Tamil Nadu, India (*E-mail: [email protected]) Abstract Dislocation density based modified Kocks-Mecking approach has been employed to understand the influence of nitrogen on kinetics of dislocation storage and annihilation processes during tensile deformation of type 316LN stainless steel at 300 K. As compared to original Kocks-Mecking approach, the modified version based on the evolution of forest dislocation density and mean free path of dislocations provides better description of threestage hardening behaviour of the steel at all the nitrogen content. The effect of increase in nitrogen on work hardening behaviour is clearly reflected in systematic increase in flow stress, mobile and forest dislocation densities and decrease in mean free path at a given plastic strain. The predicted decrease in mean free path with increasing nitrogen has not only influenced the enhancement of dislocation-dislocation interactions leading to higher storage of dislocation but also provides a large driving force for higher dynamic recovery in the steel. The material model is implemented into finite element code via user-defined material subroutine using implicit algorithm to study the influence of notch radius on tensile properties of type 316LN stainless steel for different nitrogen content. Detailed analysis showed that notch-strengthening effect increases with increase in nitrogen and decrease in notch radius in the steel. Keywords: Type 316LN SS, Nitrogen, Mean free path, Dynamic recovery, Notch radius

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Journal Pre-proof 1. Introduction Type 316LN austenitic stainless steel containing 0.06 to 0.08 wt.% nitrogen is widely used as a structural material for primary components (main vessel, inner vessel, intermediate heat exchanger, etc.) of sodium cooled fast breeder reactors. The basis for the choice of 316LN SS as the structural material in primary side components of fast breeder nuclear reactors is mainly due to its good high temperature mechanical properties, improved resistance to stress corrosion cracking, compatibility with liquid sodium coolant and adequate weldability [1,2]. Addition of controlled nitrogen in the range of 0.06-0.08 wt.% facilitates tensile, creep and low cycle fatigue properties comparable or higher than that of normal grade type 316 stainless steel [3]. Presently, fast breeder reactors are designed for 40 years lifetime, and there is a strong desire to increase the design life to 60 years and more for improved nuclear energy economy [4]. As a part of the efforts to develop high strength 316LN SS suitable for longer design life, the influence of nitrogen with concentration higher than 0.08 wt.% of 316LN SS on high temperature mechanical properties is being investigated extensively [5,6,7,8,9]. Nitrogen is a strong austenite stabilizer and effective solid solution strengthener [10,11,12,13]. The beneficial influence of high nitrogen content on tensile [7,10,11], creep [8,14] and low cycle fatigue properties [9,15] and fatigue crack growth resistance [16] has been demonstrated in type 316LN austenitic stainless steel. It has been suggested that the increase in strength with increasing nitrogen in 316L SS is derived either by one or a combination of several mechanisms such as (i) solid solution strengthening, (ii) decrease in stacking fault energy, (iii) precipitation hardening, (iv) formation of interstitial-solute complexes and (v) order strengthening [13,17,18,19,20]. Most of the above strengthening mechanisms essentially depend on the interactions with dislocations. However, the investigations related to influence of nitrogen on dislocation storage and dynamic recovery 2

Journal Pre-proof during tensile deformation of type 316LN austenitic steel in the literature are very limited. In view of this, dislocation density based constitutive models have been used to examine tensile flow and work hardening behaviour of type 316LN austenitic steel at 300 K. Among the dislocation based models, Kocks-Mecking formalism has been widely used to understand the deformation behaviour of materials [21]. In Kocks-Mecking model, work hardening behaviour of materials has been described in terms of evolution of one-internal-variable, namely total dislocation density with the plastic strain. The wide applicability of the KocksMecking model for the description of the tensile [22,23] and creep [24,25] deformation behaviour of materials has been demonstrated for different metals and alloys. Kwofie [26] extended the Kocks-Mecking one-internal-variable model for accounting the cyclic hardening and softening behaviour. The predictive capability of the model has also been demonstrated by Kwofie [26] for the cyclic deformation data of annealed polycrystalline copper. Recent investigations suggests that modified Kocks-Mecking model based on the evolution of forest dislocation density and mean free path with plastic strain proposed by Barlat et al. [27] provides better description of work hardening behaviour for fcc metals [28,29]. In the present investigation, modified Kocks-Mecking approach [27] has been employed for describing the work hardening behaviour of type 316LN SS with different nitrogen contents. Applicability of the model has been critically examined with a view to understand the effect of nitrogen on dislocation storage and annihilation mechanisms. The material model is implemented into finite element code to study the influence of notch radius on the tensile properties of 316LN SS for different nitrogen content. 2. Experimental Details Four heats of type 316LN SS with different nitrogen contents were produced using double melting process. Air induction melting was performed as primary melting. In order to attain a good control over the chemical composition, pure raw materials were used for 3

Journal Pre-proof primary melting. The required quantity of nitrided ferrochrome was added during primary melting to achieve the desired nitrogen levels in the four heats. The carbon content was maintained at ~ 0.03 wt % in all the heats. Apart from carbon content, the concentrations of other elements were also controlled to the specified level. Electroslag refining process as a secondary melting was carried out in order to obtain low inclusion content in the steel. The ingots obtained from electroslag refining process were hot forged into slabs followed by hot rolling into plates with 22 mm thickness. The hot rolled plates were given solution annealing at 1373 K followed by water quenching. The chemical composition of the four heats of type 316LN steel with 0.07, 011, 0.14 and 0.22 wt.% nitrogen content is presented in Table 1. Specimen blanks having the dimensions of 10 mm diameter and 60 mm length were machined from the plates in the hot rolled direction. Specimen blanks were solution annealed by soaking at 1373 K for 30 minutes followed by quenching in water. Following solution annealing heat treatment, button head cylindrical specimens of 26 mm gauge length and 4 mm gauge diameter were machined from the specimen blanks. Tensile tests were carried out in an Instron universal testing machine equipped with three-zone-resistance heating furnace (Fig. 1). Tests were performed at the nominal strain rate of 3  103 s1 at 300 K. The verification of specimen alignment with test frame has been carried out as per the ASTM E1012 standard [30] in order to avoid the strain inhomogeneity across the specimen gage dimension. At least two specimens were tested for each nitrogen level. The derived mean load-elongation data obtained from two or more consistent tests has been used for the analysis. From the load-elongation data, engineering stress (S)-engineering strain (e) was estimated at each test condition. The true stress () vs. true strain () data was then calculated for 316LN SS based on the following relationships i.e.  = S (1+e) and  = ln (1+e).

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Journal Pre-proof Optical metallographic examination was performed on solution-annealed specimens. The samples were etched electrolytically in 60% HNO3 solution. The grain size of all the four heats was measured using linear intercept method. In order to examine the formation of strain induced martensite during tensile deformation, ferrite scope measurements on the deformed samples has been performed in the present study. In addition, X-ray diffraction studies on solution-annealed as well as deformed (up to uniform plastic strain) samples have been carried out for the type 316LN SS having 0.07% N and 0.22% N. X-ray diffractometer with Cu-K radiation was used for the phase analysis. For transmission electron microscopy (TEM) studies, thin foils were prepared from the tensile tested specimens deformed up to ultimate tensile strength for 0.11 and 0.22% N. 10 pct perchloric acid in methanol (in 1:9 ratio) was used as electrolyte for twin-jet thinning. The foils were examined with 120 kV TEM ("Tecnai G2" from FEI). 3. Modelling dislocation dynamics and optimisation methodology The relationship between flow stress (f) and dislocation density () according to Taylor [31] is given as

 f   0  M  b 1 2 ,

(1)

where M is the Taylor factor,  is the constant,  is the shear modulus, b is the Burgers vector and 0 is the flow stress component arising from dislocation glide resistance due to lattice friction, grain boundaries, precipitates, solutes and dispersoids. In Eqn. (1), the term Mb1/2 indicates flow stress contribution from the dislocations (d). The differential form of Eqn. (1) with respect to plastic strain is represented as

d f d p



M  b d  . 21 2 d p

(2)

In order to define the work hardening behaviour, the evolution of dislocation density with plastic strain needs to be coupled with the Eqn. (2). According to Kocks-Mecking approach 5

Journal Pre-proof [21], the evolution of total dislocation density with accumulated shear strain () on slip 

systems (



d d d ) is controlled by a dislocation storage rate ( ) and recovery rate ( ) and d d d

this is mathematically represented as 



d d d 1     k2  , d d d bL

(3)

where b is the Burgers vector, L is the mean free path and k2 is the dynamic recovery parameter. In the Kocks-Mecking approach [21], the mean free path of dislocations (L) is proportional to 1/2. Using the interrelationship between accumulated shear strain and plastic strain (p) as  = Mp (where M is the Taylor factor), the evolution relationship for dislocation density with plastic strain can be represented as

 k 1 2  d  1  M  k2    M  1  k2   , d p  bL   b 

(4)

where k1 is the dislocation storage parameter. In Kocks-Mecking model, it was assumed that the total dislocation density ( = m + f) was mainly constituted by forest dislocations i.e., f >> m, and the mobile dislocation density is considered to be constant during deformation. Therefore, d/dp can also be defined as df/dp in Eqn. (4) [27]. In the modified KocksMecking approach proposed by Barlat et al. [27] the evolution of two internal variables such as forest dislocation density and mean free path have been considered and the relationships are given as df

M M  k2  f d p  bL

 , 

(5)

dL   K L  L  Ls  , d p

(6)

where KL is the rate parameter with which mean free path approaches from initial (LI) to saturation (Ls) value. In modified Kocks-Mecking approach [27], the interdependency 6

Journal Pre-proof between the incremental mobile dislocation density and incremental mean free path is given as

d m 

M b

 d ln L    .  Ls   K L  

(7)

The evolution of mobile dislocation density with plastic strain can be expressed as

d m M  1   d p b  L  Ls

 dL .  d   p

(8)

Eqn. (10) implies that the evolution of mean free path can be used to predict the evolution of dislocation density with plastic strain. The coupled differential equations such as Eqns. (2), (5) and (6) have been integrated by fourth-order Runge-Kutta method and the parameter set (LI ,LS ,KL ,k2) associated with the modified Kocks-Mecking approach have been numerically optimised. The initial values for mobile and forest dislocation densities were fixed as 1  1010 m2 and 1  1011 m2, respectively. In order to evaluate the instantaneous work hardening rate () and rate of evolution of forest dislocation density (df/dp) from the d - p and f - p data, the central difference formulation has been used in the present analysis. Using the formulation of Mbf1/2, the predicted flow stress contribution from the dislocations (d) has been computed from the dislocation density data. The experimental d is obtained using

 d     y ,

p  0.001

 M  b 01/2 , where  y ,  p 0.001 is the experimental yield strength at 0.001

plastic strain and 0 is the initial forest dislocation density. 4. Results and discussion 4.1. Microstructure Fig. 2 shows the optical microstructure of the material in the solution treated condition. Equiaxed grains free of carbide precipitates were observed in all the heats. The grain sizes of 316LN SS with four different nitrogen content 0.07 % N, 0.11 % N, 0.14 % N 7

Journal Pre-proof and 0.22% N are measured as 87 μm, 96 μm, 78 μm and 87 μm, respectively. Ferrite scope measurements on the deformed samples revealed that there is no evidence for the formation of martensite during the deformation of type 316LN SS at all the nitrogen content. XRD pattern also confirmed that both the solution annealed and deformed samples of type 316LN SS having 0.07% N and 0.22% N exhibit austenitic phase as shown by Fig. 3. The peaks corresponding to the (110) and (200) planes of martensitic structure which used to lie within the 2 range of 40˚ to 80˚ have not been noticed in the XRD pattern. The plastic deformation accompanying the martensitic transformation is connected with martensite start temperatures or Md (30/50) C. Md refers to the temperature at which 50% of the martensite is produced after 30% true deformation under tensile condition. The estimated Md temperatures for type 316LN SS having 0.07% N, 0.11% N, 0.14% N and 0.22 % N are 49.7 C (223.3 K), 73.8 C (199.2 K), 82.0 C (191 K) and 122.3 C (150.7 K), respectively. The derived Md temperatures for the steel are much lower than the test temperature 300 K. The relationship proposed by Angel et al. [32] has been used for the estimation of Md (30/50) C temperatures in the present investigation and it is given as Md (30/50) C = 413  462 (%C+%N)  9.2 (%Si)  8.1 (%Mn)  13.7 (%Cr)  9.5 (%Ni)  18.5 (%Mo). TEM micrographs of solution annealed samples shows the dislocation structures consisting of a few randomly distributed dislocations and some array of dislocations (Figs. 4 (a) and (b)). For the deformed condition, high dense dislocation networks have been observed for both the nitrogen contents (Figs. 4 (c) and (d)). TEM investigations on the deformed samples of type 316LN SS having 0.11% N and 0.22% N clearly indicated that dislocation-dislocation interactions are predominant during the deformation of type 316LN SS. All the above observations clearly indicated that type 316LN steel with four different nitrogen contents have stable austenitic structure even during tensile deformation at 300 K.

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Journal Pre-proof 4.2. Applicability of modified Kocks-Mecking approach The variations in flow stress (σf) with true plastic strain (εp) for 316LN SS with four different nitrogen contents of 0.07, 0.11, 0.14 and 0.22 %N are presented as double logarithmic plots in Fig. 5. The predicted true stress vs. true plastic strain data using KocksMecking and modified Kocks-Mecking models have been superimposed as broken and full lines respectively in order to compare the prediction capabilities between two approaches. For all the nitrogen contents, Kocks-Mecking approach exhibited considerable positive deviations at low strain values whereas the prediction using modified Kocks-Mecking model describes the experimental data more accurately. This is more discernible from observed higher 2 values for Kocks-Mecking approach than for modified Kocks-Mecking model for all the nitrogen contents (Table 2). The low value of reduced 2 for any given curve fitting denotes the minimal deviation between the experimental and calculated stress values. The absence of strain induced plasticity effects and the observed predominance of dislocationdislocation interactions during deformation of type 316LN SS suggested that the choice of modified Kocks-Mecking model based on the evolution rate of dislocation density and mean free path is appropriate for the steel. The analysis in terms of θσd vs. σd has been extensively used to understand the three-stage work hardening behaviour of type 316LN SS as shown in Fig. 6 for 0.11% N [33]. Experimental θσd vs. σd data showed a gradual increase in θσd at low σd values (Transient stage) followed by a linear increase in θσd with σd in stage-II and an inverted parabolic hardening at high σd values (Stage-III). The predicted θσd vs. σd using Kocks-Mecking approach fails to account for the initial transient stage but it follows the stage-II and stage-III behaviour to a reasonable extent (Fig. 6). For modified Kocks-Mecking approach, a good match between experimental and predicted θσd vs. σd depicting all the work hardening stages can be seen from Fig. 6. These observations strongly implies that modified

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Journal Pre-proof Kocks-Mecking approach

is more appropriate for modeling tensile work hardening

behaviour of type 316LN steel with different nitrogen content.

4.3. Influence of nitrogen on three-stage work hardening behaviour using d vs. d plots In order to understand the influence of nitrogen on three-stage work hardening behaviour of type 316LN SS, d vs. d analysis is carried out. The predicted d vs. d plots obtained for four different nitrogen contents using modified Kocks-Mecking approach clearly demonstrated the increase in flow tress and work hardening rate with increasing nitrogen as shown in Fig. 7. Since θσd is directly proportional to df /dp, the equivalence between θσd vs. σd and df / dp vs. σd can be expected [34]. At all the nitrogen content, the variations in df / dp as a function of σd also displayed a similar variations shown by the θσd vs. σd (Fig. 8). For a given d values, a marginal difference in the values of d with increasing nitrogen can be seen in the transient stage followed by a continued widening in d values with increasing nitrogen in stage-II and stage-III. With increasing nitrogen, increase in the values of d and d for the onset of stage-II and stage-III can be seen in Fig. 7. The values of d and d for onset of stage-II and stage-III as well as the range of d (d) and d (d) for stage-II are given in Table 3. During transient stage, gradual increase in θσd with σd is due to the fact that deformation is predominantly controlled by planar slip mechanisms at low stress levels and the main substructural features are observed to be planar or single slip in most of the grains with homogeneous distribution of dislocations [33]. Increase in values of θσd or dρf/dεp with nitrogen in transient stage indicates that addition of nitrogen induces the pronounced planar slip in type 316LN stainless steel (Figs. 7 and 8). The formation of effective slip obstacles through strong interactions between the dislocations lying in the

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Journal Pre-proof primary and secondary slip systems leads to rapid increase in d vs.d values during stage-II in fcc metals [35,36,37,38]. In austenitic stainless steel, it has been shown that the activation of multiple slip during stage-II induces the formation of heterogeneous substructures such as tangles, hexagonal network, walls and cells [28,33]. Enhanced linear stage-II work hardening rate observed from d vs. d (Fig. 7) and dρf/dεp vs. d (Fig. 8) plots clearly suggest dislocation-dislocation interactions increase with increase in nitrogen leading to accumulation of dislocations at rapid rate for high nitrogen steels. The onset of stage-III hardening characterized by the deviations from the linear θσd vs. σd plot indicates the predominance of onset of the dynamic recovery over storage rate (Fig. 7). In general, dynamic recovery favor annihilation of dislocations results from the cross-slip of screw dislocation at low and intermediate temperatures and climb of dislocations at high temperatures [21,39]. At room temperature, dominance of cross-slip is expected to govern stage-III hardening in type 316LN stainless steel. The observed increase in the stress to onset of stage-III hardening with increasing nitrogen clearly suggests that nitrogen retards onset of dynamic recovery in stainless steel (Fig. 7). However the inverted parabolic work hardening of stage-III increases with increase in nitrogen content. The continuous widening in work hardening rate with increasing nitrogen content at any given strain during stage-II and stage-III can be explained in terms of the increase in flow stress with nitrogen for different plastic strain levels ranging from 0.002 to 0.25 as presented in Fig. 9a. At low strain levels (p < 0.02), flow stress (σf) vs. wt. % N plots for different strain levels appear to be parallel. Contrary to this, flow stress vs. nitrogen plots displayed a systematic increase in slope with increasing plastic strain beyond 0.02. This clearly suggests significant increase in flow stress at higher strains due to increase in nitrogen, particularly for nitrogen content greater than 0.11 wt. %. The variations of slope obtained from the best fit plots of flow stress vs. nitrogen (Fig. 9a) with true plastic strain are

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Journal Pre-proof shown in Fig. 9b. These observations indicate strong influence of plastic deformation due to nitrogen addition resulting in the enhanced work hardening at high strains in type 316L SS. 4.4. Variations in model parameters with nitrogen content Understanding the variations in model parameters with nitrogen content provides better understanding towards the influence of nitrogen on the evolution of internal variable during tensile deformation of 316LN austenitic stainless steel. The variations in parameters related to the evolution of initial (LI) and final mean free path (LS), and rate parameter (KL) with respect to nitrogen are shown in Fig. 10a. The initial (LI) and final mean free path (LS), and rate parameter (KL) decrease with increase in nitrogen content. Reverse ‘S’ shaped curves (of L vs. p) obtained for all the nitrogen content are shown in Fig. 10b. A gradual decrease in mean free path in transient stage followed by a rapid decrease at stage-II and approaching towards saturation in Stage-III have been observed for all the nitrogen contents. Further, a decrease in the values of mean free path with increasing nitrogen has been observed at all plastic strain as shown in Fig. 10b. The decrease in rate parameter (KL) with reduced mean free path values signifies increase in dislocation storage rate associated with increase in nitrogen content. The evolution of mean free path with respect to strain is also clearly reflected in the evolution of forest, mobile and total dislocation densities predicted by modified Kocks-Mecking model. A rapid evolution of forest, mobile and total dislocation densities at low plastic strains followed by saturation at high strains in stage-III were noticed for all the nitrogen contents as shown in Fig. 11a for 0.11% N steel. In general, faster evolution of mobile over forest dislocation density at low strains followed by an increase at high strains has been noticed. The predicted evolution of forest dislocation density with plastic strain has been presented in Fig. 11b for all the nitrogen contents. For a given plastic

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Journal Pre-proof strain, the observed systematic increase in forest dislocation density with increasing nitrogen contributes to higher work hardening rate. Other parameters initial stress and dynamic recovery parameter increase with increase in nitrogen in type 316LN SS (Fig. 12). The obtained true yield stress values at 0.001 plastic strain are superimposed and good agreement between the predicted initial stress and experimental yield stress values can be seen in Fig. 12. The increase in k2 with increasing nitrogen demonstrated that nitrogen influences the stage-III work hardening behaviour of the steel. For 0.07% N, θσd exhibiting a nearly constant value at high σd during stage-III indicate low contribution of dynamic recovery (Fig. 7). On the Contrary, the increasing influence of dynamic recovery with increasing nitrogen is clearly revealed by increase in the inverted parabolic hardening for nitrogen ranging from 0.11 to 0.22% N (Fig. 7). It is known that lower mean free path of dislocations enhances the rate of dislocation generation and its accumulation inside grains [40,41]. In general, rather than formation of bands of tangled dislocations, the planar structure i.e. pileup of dislocations along their slip planes were often reported in nitrogen added type 316 L austenitic stainless steel [42]. Increasing nitrogen content promotes more planar slip by reducing stacking fault energy in austenitic stainless steels [18,43,44,45]. It is reported that stainless steel having Fe-Cr-Ni alloy system exhibited a marginal decrease in stacking fault energy with increase in nitrogen content up to 0.25% N [46,47,48]. Contrary to this, stainless steel fitting in to the class of Fe-Cr-Mn and Fe-Cr-MnNi showed an increasing trend in stacking fault energy with increase in nitrogen content [20,49]. Since type 316L(N) austenitic stainless steel belongs to the Fe-Cr-Ni system, the decreasing in stacking fault energy value with increase in nitrogen content is expected. Recently, the influence of interstitial content on SFE in 316LN SS was reported by Santhosh et al [50] based on the thermodynamic analysis. The calculations showed that SFE decreases from 27 mJ/m2 to 17 mJ/m2 with an increase in nitrogen content from 0.07 to 0.22 (wt%.) at 13

Journal Pre-proof room temperature for the steel. The results obtained from the thermodynamic analysis were further validated using the XRD line profile analysis by Santhosh et al. [50]. Apart from reduction in stacking fault energy, the increase in nitrogen content would also play a role in the formation of short-range ordering by pairing with Cr or Mo atoms [51,52,53,54,55]. It was observed that the average separation distance between the adjacent pairs is roughly eight times of burgers vector [53]. Dislocations moving through the lattice would have to destroy the strong bonding between these pairs and this would act as strong obstacle for the dislocation motion. The above observations suggested that the decrease in stacking fault energy and increase in formation of short-range ordering are the most important factors for the formation of enhanced planar dislocation structures with increase in nitrogen content. However, influence of individual contributions of these mechanisms i.e. reduction in stacking fault energy and formation of short-range ordering on the work hardening behaviour of austenitic stainless by the addition of nitrogen content is yet to be studied. Although contradictory effects of nitrogen on the stacking fault energy have been reported in the literature [54], there is a consensus on its effect on dislocation glide motion when present in the range 0.02-0.25 wt.% [56]. This is clearly reflected in enhanced d and df/dp i.e. higher dislocation storage rate with increase in nitrogen from 0.07 to 0.22 wt.% N at transient stage and stage-II work hardening behaviour (Figs. 7 and 8). The observed higher dense dislocation networks for the steel having 0.22% N over 0.11% N suggested that increase in nitrogen enhanced the dislocation storage rate as well as work hardening rate in austenitic stainless steel. The predicted lower final mean free path for high nitrogen steels (0.22% N) is in well agreement with the observed high dislocation density population in its deformed microstructure. The resulting high dislocation density associated with the higher internal energy provides a large driving force for dynamic recovery [40,41]. The observed decrease in mean free path of dislocations with increasing nitrogen not only influences the enhancement

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Journal Pre-proof of stage-II work hardening behaviour but also provide a large driving force for higher dynamic recovery in type 316LN stainless steel (Figs. 10b and 12). 4.5. Finite element implementation The modified Kocks-Mecking model is implemented into commercial finite element software Abaqus 6.16 via user-defined material subroutine (UMAT). The constitutive description provided in the UMAT is used for updating the stresses, state variables and material Jacobian at the end of strain increment (Appendix). Euler backward implicit scheme with radial return algorithm has been employed to update the stress and internal-variables during strain increment. By considering the 2-dimensional axial symmetry along with symmetry plane in the middle of the specimen and the uniaxial loading condition, rectangle geometry with width equal to the specimen radius and height equal to half of the specimen length with only one notch has been chosen for the simulation as shown in Fig. 13. Eightnode quadratic axisymmetric element with reduced integration was used to model the specimen geometry. In order to ensure the size of mesh does not affect the finite element results, mesh convergence check has been carried out and accordingly finer mesh size has been provided near to the notch root. Applicability of the implemented scheme has been examined by simulating the tensile behaviour of type 316LN austenitic stainless steel with different nitrogen content for various U-notch radii as 0.5, 1.25, 2.5 and 5 mm at 300 K. Good agreement between the predicted and experimental results reported by Ganesh Kumar et al. [57] has been obtained for yield and ultimate strength values at 0.07%N content as shown in Fig. 14. The presence of notch in the specimen constraints the deformation in lateral direction resulting in state of tri-axial stress in the notch root and it confines the plastic deformation to the vicinity of notch tip. The difficulty in spreading of yield zone in the presence of tri-axial stresses leads to higher yield strength of the notched specimen than smooth specimen. The constraint effect increases with decrease in notch radius. This implies 15

Journal Pre-proof that the specimen with lower notch radius experiences higher stress triaxiality leading to increase in yield and ultimate tensile strength [58,59]. The plastic strain distribution around the notch root for different notch radius is shown in Fig. 15 at nominal strain of ε = 2% for type 316LN stainless steel with 0.07 wt. % nitrogen. Steep strain gradient is observed along the notch root plane at lower notch radius due to higher stress triaxiality compared to smaller strain gradient at larger notch radius. Similarly the specimen with larger notch root radius shows a much larger yielding or plastic zone than the specimen with smaller radius. In order to study the effect of nitrogen on notch strengthening, the corresponding parameters values associated with modified Kocks-Mecking model for different nitrogen contents (Figs. 10a and 12) have been used in the UMAT code and simulations were performed. With increase in nitrogen higher yield and ultimate tensile strength are observed for all the notch radius as shown in Figs. 16a and 16b respectively. For the given notch radius, the increase in the yield and ultimate tensile strength values suggested notch strengthening effects increases with increase with nitrogen. The distribution of Von-Mises equivalent stress with respect to nitrogen content at the nominal strain of 0.5% is shown in Fig. 17. The peak values of VonMises equivalent stress increase with increase in nitrogen content throughout the notch plane. The combination of increase in constraint effect due to decrease in notch radius and enhanced work hardening rate with increase in nitrogen content contributes to higher notch strengthening in type 316LN austenitic stainless steel at high nitrogen levels. 5. Conclusions Detailed investigation on tensile flow work hardening analysis indicated that modified Kocks-Mecking approach provides better description of three-stage work hardening behaviour of type 316LN austenitic stainless steel over Kocks-Mecking approach for all the nitrogen content. Moreover, the absence of strain induced plasticity effects and the observed predominance of dislocation-dislocation interactions during deformation of type 316LN SS 16

Journal Pre-proof suggested that the choice of modified Kocks-Mecking model based on the evolution rate of forest dislocation density and mean free path is appropriate for the steel. It has been noticed that increase in nitrogen in type 316LN SS not only increases the yield stress but also strongly influences the work hardening behaviour throughout the plastic deformation. This is clearly reflected in systematic increase in flow stress, mobile and forest dislocation densities and decrease in mean free path with increase in nitrogen at a given plastic strain. Enhanced linear work hardening rate in σd vs. σd and dρ/dεp vs. σd plots clearly suggested that dislocation-dislocation interactions increase strongly with increase in nitrogen in Stage-II. The observed higher dense dislocation networks for the steel having 0.22% N over 0.11% N suggested that increase in nitrogen enhanced the dislocation storage rate/work hardening rate, contributing to the lowering of mean free path in type 316LN austenitic stainless steel. The increase in the stress to onset of stage-III hardening with increasing nitrogen has clearly demonstrated that nitrogen retards the onset of dynamic recovery in stainless steel. However, the increase in forest dislocation density with increasing nitrogen provide a large driving force for higher dynamic recovery in type 316LN stainless steel which results in the pronounced inverted parabolic stage-III work hardening for higher nitrogen content. Finite element simulation results showed that notch-strengthening effect increases with increase in nitrogen content and decrease in notch radius in type 316LN austenitic stainless steel at 300 K. Acknowledgement The authors thank Computer division, IGCAR for their support in providing the software Abaqus 6.16. Authors are grateful to Dr. Partha Ghosal, Dr. Veera Babu and Ms. Satabdi Roy, DMRL, Hyderabad for the TEM investigations. Special thanks are due to Dr. K.I. Gnanasekar and Mr. Nair Afijith Ravindranath, IGCAR, Kalpakkam for helping in XRD measurements. 17

Journal Pre-proof Appendix Procedure for internal state variable and stress update The constitutive equations developed for the isotropic hardening with rate-independent plasticity [60] have been employed. Note: Bold symbols are used to denote tensor quantities (i) Initialize state variables and strain increment Dislocation density, ρ = ρ , Mean free path, L = LI; Equivalent plastic strain f ,0 f (Δp) = 0 and total strain increment ( Δε ) (ii) Compute trial stress σ tr by assuming total strain increment ( Δε ) as incremental elastic strain









σ tr  2G εte + Δε  Tr εte + Δε I

(A1)

Where εte is the incremental total strain (iii) Check the yield condition f  e   f

 3 tr  tr    σ :σ  2 

  0  M  Gb  f >0

(A2)

Where  e is the equivalent Von-Mises stress and σ tr ' is deviatoric component of trial stress. When yield condition i.e. Eqn. (A2) is satisfied, the iteration would proceed for the estimation of residue and incremental plastic strain as given in the next step. Otherwise, the stress update would be performed using Hooke's law relationship and it is represented as

 

σ  2Gε e  Tr ε e I

(A3)

(iv) Check the size of the residual and estimate the plastic multiplier R =  etr  3Gp   0  M  b 

f

< Tol

(A4)

To achieve the the prescribed tolerance (Tol = 1 × 10–6) Newton Raphson method was employed to update the state variables

18

Journal Pre-proof  etr  3Gp   0  M  b  f d p  M  b  M  3   MK 2    f 2   bL  f

(A5)

(v) Compute plastic strain increment and stress increment

Δε

p

= Δp



f σ

(A6)







p σ  2  εte + Δε  Tr εte + Δε I  2  Δε

(A7)

(iv) Update internal variables Current dislocation density, mean free path and equivalent plastic strain (vi) Update material Jacobian The consistent tangent modulus i.e. material Jacobian needs to be provided in order to obtain a quadratic convergence rate in solving the global force equilibrium in the ABAQUS/Standard.

    σ tr σ tr 2 e e δσ   2  A tr tr  2  tr    K   tr e 3 e e e  

where A 

    3 1 e  tr 2  3   1 e  M b  M Mk     2 f  2  f  bL   

19

          

   II  : δε  

II is the fourth order tensor

(A8)

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24

Journal Pre-proof Figure captions Fig. 1.

Tensile specimen drawing and test set-up.

Fig. 2.

Optical microstructure of 316LN SS having (a) 0.07% N, (b) 0.11% N, (c) 0.14% N, and (d) 0.22% N in solution annealed condition.

Fig. 3.

XRD patterns for the (a) solution annealed and (b) deformed samples of type 316LN SS having 0.07% N and 0.22% N.

Fig. 4.

Dislocation substructure of type 316LN SS in solution annealed condition (a) and (b), and, deformed up to true uniform plastic strain (c) and (d).

Fig. 5.

Variations in flow stress (σf) with true plastic strain (p) for four different nitrogen contents in type 316LN austenitic stainless steel. Predicted σf-p data using KocksMecking and modified Kocks-Mecking approaches are superimposed as dashed lines and full lines, respectively.

Fig. 6.

Variations in σd with σd representing three-stage work hardening behaviour of type 316LN SS for 0.11% N.

Fig. 7.

Variations in d with d for four different nitrogen contents in type 316LN SS. Three stages of work hardening have been marked for 0.22% steel.

Fig. 8.

Variations in df/dp with d for four different nitrogen contents in type 316LN SS. Three stages of work hardening have been marked for 0.22% steel.

Fig. 9.

Variations in (a) flow stress (σf) with nitrogen content for different true plastic d f strain levels ranging from 0.002 to 0.25 and (b) slope (m) i.e. with respect d %N to true plastic strain (p). The slope values were derived from linear fit of flow stress vs. nitrogen content data as shown in Fig. 9a.

Fig. 10.

Variations in (a) initial (LI) and final (LS) mean free path, and rate parameter (KL) with nitrogen content for type 316LN SS and (b) mean free path of dislocations (L) with true plastic strain (p) for four different nitrogen contents in type 316LN austenitic stainless steel.

Fig. 11.

(a) Representative plot of evolution of forest (f), mobile (m) and total () dislocation densities with plastic strain for 0.11% N content and (b) variations in forest dislocation density (f) with plastic strain for four different nitrogen contents in type 316LN SS.

Fig. 12.

Variations in initial stress (I) and dynamic recovery parameter (k2) with nitrogen content for type 316LN SS. Experimental yield stress values obtained for the plastic strain of 0.001 have been superimposed.

25

Journal Pre-proof Fig. 13.

Representative figure of one-quarter of axisymmetric specimen geometry with finite element mesh used for simulation of notch radius of 2.5 mm.

Fig. 14.

Variations in experimental and predicted yield and ultimate tensile strength values for different notch radius of type 316LN SS with 0.07wt % N at 300 K.

Fig. 15.

Plastic strain distribution along notch root plane for different notch radius at nominal strain of 2% for type 316LN SS with 0.07wt % N at 300 K.

Fig. 16.

Variations in (a) yield and (b) ultimate tensile strength values with notch radius for different nitrogen content in type 316LN SS.

Fig. 17.

Variations of Von-Mises equivalent stress along the notch root plane of 0.5 mm notch radius specimen at nominal strain of 0.5 % for different nitrogen content in type 316LN SS.

26

Journal Pre-proof Table 1. Chemical composition of the four heats of type 316LN austenitic stainless steel. Elements in wt.% Steel C

N

Mn

Heat 1

0.027

0.07

1.70

Heat 2

0.033

0.11

Heat 3

0.025

Heat 4

0.028

Si

S

P

Mo

Ni

Fe

0.22 0.0055 0.013 17.53

2.49

12.20

Balance

1.78

0.21 0.0055 0.015 17.62

2.51

12.27

Balance

0.14

1.74

0.20 0.0041 0.017 17.57

2.53

12.15

Balance

0.22

1.70

0.20 0.0055 0.018 17.57

2.54

12.36

Balance

27

Cr

Journal Pre-proof Table 2. Reduced 2 values obtained for the Kocks-mecking and modified Kocks-Mecking approaches for different nitrogen contents. Reduced 2 values

Nitrogen content (Wt. %)

Kocks-Mecking

Modified Kocks-Mecking

0.07

29.4

0.6

0.11

12.2

0.5

0.14

19.1

0.4

0.22

23.1

0.3

28

Journal Pre-proof Table 3. The values of d, d, d and d for the onset of stage-II and stage-III for different nitrogen content in type 316LN austenitic stainless steel. Nitrogen content

Onset of Stage-II

Onset of Stage-III

Range for stage-II

(Wt. %)

d

d

d

d

d

d

0.07

2.6 × 105

95.2

4.3 × 105

197.8

1.7 × 105

102.6

0.11

3.1 × 105

117.4

5.3 × 105

225.1

2.3 × 105

107.7

0.14

3.6 × 105

140.7

5.9 × 105

255.4

2.4 × 105

114.7

0.22

5.4 × 105

215.3

8.2 × 105

337.1

2.7 × 105

121.8

29

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Fig. 1.

Tensile specimen drawing and test set-up.

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Fig. 2.

Optical microstructure of 316LN SS having (a) 0.07% N, (b) 0.11% N, (c) 0.14% N, and (d) 0.22% N in solution annealed condition.

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Fig. 3.

XRD patterns for the (a) solution annealed and (b) deformed samples of type 316LN SS having 0.07% N and 0.22% N.

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Fig. 4.

Dislocation substructure of type 316LN SS in solution annealed condition (a) and (b), and, deformed up to true uniform plastic strain (c) and (d).

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Fig. 5.

Variations in flow stress (σf) with true plastic strain (p) for four different nitrogen contents in type 316LN austenitic stainless steel. Predicted σf-p data using KocksMecking and modified Kocks-Mecking approaches are superimposed as dashed lines and full lines, respectively.

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Fig. 6.

Variations in σd with σd representing three-stage work hardening behaviour of type 316LN SS for 0.11% N.

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Fig. 7.

Variations in d with d for four different nitrogen contents in type 316LN SS. Three stages of work hardening have been marked for 0.22% steel.

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Fig. 8.

Variations in df/dp with d for four different nitrogen contents in type 316LN SS. Three stages of work hardening have been marked for 0.22% steel.

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(a)

(b)

Fig. 9.

Variations in (a) flow stress (σf) with nitrogen content for different true plastic d f strain levels ranging from 0.002 to 0.25 and (b) slope (m) i.e. with respect d %N to true plastic strain (p). The slope values were derived from linear fit of flow stress vs. nitrogen content data as shown in Fig. 9a.

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(a)

(b)

Fig. 10.

(a) Variations in initial (LI) and final (LS) mean free path, and rate parameter (KL) with nitrogen content and (b) evolution of mean free path of dislocations (L) with true plastic strain (p) in type 316LN austenitic stainless steel for four different nitrogen contents.

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(a)

(b)

Fig. 11.

(a) Representative plot of evolution of forest (f), mobile (m) and total () dislocation densities with plastic strain for 0.11% N content and (b) variations in forest dislocation density (f) with plastic strain for four different nitrogen contents in type 316LN SS.

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Fig. 12.

Variations in initial stress (I) and dynamic recovery parameter (k2) with nitrogen content for type 316LN SS. Experimental yield stress values obtained for the plastic strain of 0.001 have been superimposed.

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Fig. 13.

Representative figure of one-quarter of axisymmetric specimen geometry with finite element mesh used for simulation of notch radius of 2.5 mm.

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Fig. 14.

Variations in experimental and predicted yield and ultimate tensile strength values for different notch radius of type 316LN SS with 0.07wt % N at 300 K.

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Fig. 15.

Plastic strain distribution along notch root plane for different notch radius at nominal strain of 2% for type 316LN SS with 0.07wt % N at 300 K.

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(a)

(b)

Fig. 16.

Variations in (a) yield and (b) ultimate tensile strength values with notch radius for different nitrogen content in type 316LN SS.

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Fig. 17.

Variations of Von-Mises equivalent stress along the notch root plane of 0.5 mm notch radius specimen at nominal strain of 0.5 % for different nitrogen content in type 316LN SS.

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HIGHLIGHTS  Modified Kocks-Mecking model described the flow behaviour of 316LN with varying N  Varying nitrogen strongly influences the dislocation storage and recovery rates  Lower mean free path with higher dislocation density was observed for 0.22% N steel  Modified Kocks-Mecking model has been implemented into the finite element formulation  Finite element simulations demonstrated the notch-strengthening effect in the steel

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