Mechanical Properties and Strain-Hardening Models of Supermartensitic Stainless Steels Alloyed to Nitrogen and Vanadium

Abstract

This work presents a detailed study directed to supermartensitic stainless steels (SMSS) alloyed to nitrogen (UNS S41425) and vanadium (UNS S41427), showing comparison between them by varying their tempering temperature, evaluating the behavior in its mechanical properties, presenting a fractographic and microstructural analysis between these alloys, besides to demonstrate techniques for modeling these strain-hardening results with best coefficients that determine these materials with good predicted values. The Hollomon, Voce and Chaboche models were determined, using an iterative regression method (R²) in order to obtain the best coefficients for each analysis condition. This work concludes by presenting graphic results of the effect of tempering temperature increase on the final mechanical properties, evaluates the differences obtained between these materials and presents an evaluation of the behavior of each coefficient of the models presented separately, showing the best R² fitting results. With the models determined, it is possible to use the results in this work to perform numerical simulations and meet the needs of the engineering on industry.

Keywords:
Supermartensitic stainless steels; heat treatment; slow strain test; mechanical properties; modeling

1. Introduction

Carbon steels are the main materials used in the construction of oil and gas production fields due to their low cost and high availability¹. With the advancement of the sector and the search for greater performance of materials in face of market requirements, alloys with greater resistance to corrosion were developed²,³. In view of this demand, martensitic stainless steel alloys were developed for applications involving environments with the presence of CO₂. In more severe cases, supermartensitic steels (SMSS) are used, which are currently widely used in rigid pipelines, subsea and offshore platforms.

Supermartensitic steels stand out for their reduced carbon content, combined with controlled additions of molybdenum and nickel, in addition to other microalloy elements, such as nitrogen, vanadium and titanium, enabling the achievement of a martensitic microstructure of low hardness when compared to conventional martensitic steels⁴. This alloy was developed to overcome the application limitations of conventional martensitic stainless steels (MSS) used in large scale in oil and gas industry.

In general, SMSS were developed in search of alternatives to replace the duplex and superduplex steels used in the manufacture of pipelines in the oil and gas industry. For this, the SMSS is subjected to austenitization treatments, usually followed by quenching and tempering with controlled time and temperature, in order to obtain the desired mechanical properties in these steels through phase transformations⁵.

The mechanical properties of SMSS will depend on both the chemical composition and the thermal and/or mechanical processing applied. The variations of alloying elements in the composition of these steels have been widely studied, as a way of evaluating their effects on the final properties of the material.

The effect of Nickel was decisive in the development of low carbon martensitic stainless steels, favoring the expansion of the austenite field to the point where the formation of ferrite can be largely avoided⁶. Studies showed that chromium is an alphagen element capable of forming a protective passive oxide layer, being the main alloying element in stainless steels and responsible for its corrosion resistance⁶,⁷. Contents above 12% of Cr produces semi-ferritic microstructures and increases air hardenability⁸. Molybdenum has proven to be critical for increasing corrosion resistance in applications involving transporting oil and gas, reducing the rate of uniform corrosion and increasing corrosion resistance at elevated temperatures⁹-¹¹. Carbon is kept at low concentrations (<0.03% by weight) in order to suppress the precipitation of chromium carbides, maintaining the corrosion resistance properties of steel promoted by this alloying element and favoring the weldability of these steels with less susceptibility to cracking⁹. Titanium shows itself as a ferritizing and stabilizing element, due to its ability to form fine and extremely stable carbides¹². Like titanium, niobium also has a ferritizing characteristic, with affinity to carbon, reducing the precipitation of intergranular chromium carbides, through the preferential formation of niobium carbides and, therefore, improving the resistance to intergranular corrosion¹³.

Nitrogen is a micro-binding element, used to partially replace the hardening effect of carbon in SMSS, generating additional benefits for the material's corrosion resistance¹⁴. Commercial alloys exploiting the addition of nitrogen are found on the market, however, few references and literary information are found. Study conducted by Qi et al.¹⁵ analyzed 5 heat-treated alloys under different nitrogen contents, concluding that the greater the added nitrogen content, the greater the volumetric fraction of austenite retained on the quenched and tempered samples. It was also observed that the increase in nitrogen up to values ​​of 0.05% favors the corrosion resistance of these materials.

Vanadium interacts in the alloy as an element that forms fine carbides. Studies conducted by Everson and Johnson¹⁶ on the alloy with vanadium additions UNS S41427 demonstrated that this element provided good mechanical and corrosion resistance when compared to other commercial alloys without this element. Lee et al.¹⁷ demonstrate in their work that the presence of fine vanadium carbides act as hydrogen trapping sites, being an alternative to target resistance to the embrittlement phenomenon associated with hydrogen.

When different heat treatments are applied, different mechanical properties can be obtained, depending on the microstructural changes arising from the treatment used. Studies reported by Gennari et al.¹⁸ and Calliari et al.¹⁹ demonstrates the importance of how an in-depth study of process variation in isolation can measure its influence on the mechanical properties of the material or on its microstructural behavior, aiding in the development of predictive mathematical models. Thus, it´s important to study the influence of the main applicable heat treatments, in order to estimate and predict the changes obtained on its microstructural and mechanical properties. Other works reported in the literature by different researchers explore the mechanical properties of SMSS under the effects of heat treatments employed⁴,²⁰-²³.

Knowledge of the mechanical properties of SMSS in different heat treatments is essential to predict the behavior of these materials in view of the numerous application possibilities. With a shortage of literature, mathematical models that represent the mechanical behavior of these materials can be used to obtain reliable numerical solutions for different engineering analyses. Therefore, the development of mathematical models that can correctly represent the material for engineering numerical analysis with the support of finite elements in large range use is essential, in order to obtain reliable and representative final answers.

Based on the works reported, the objective of this work is to evaluate the influence of different tempering heat treatments on the mechanical properties of SMSS UNS S41425 and UNS S41427, characterized by having nitrogen and vanadium as microalloying elements, respectively. Under different heat treatment conditions, tensile tests with low strain rate (SSRT) were carried out. This work concludes presenting the strain-hardening modelling and a detailed comparison regarding its best fitting values that represent these materials in order to be used to obtain reliable numerical solutions for different engineering analyses.

2. Materials and Methods

2.1. Materials and procedules

The materials selected in this work were two supermartensitic stainless steel tubes (SMSS) containing 13% chromium, meeting the specifications of UNS S41425 and UNS S41427 respectively, which chemical compositions are shown in Table 1, in accordance with the certificate granted by manufacturer. S, Al, Ti and Nb are kept at low levels.

Table 1
Chemical composition of the UNS S41425 and UNS S41427 studied (wt.%).

Material Identification	C	Mn	Si	Cr	Ni	Mo	P	Cu	N	V
SMSS “N” *	0.020	0.76	0.32	13.52	4.61	1.66	0.02	0.07	0.07	0.03
SMSS “V” *	0.016	0.35	0.20	12.16	5.40	1.95	0.01	0.08	0.012	0.16

• ^*
  SMSS “N”: UNS S41425; SMSS “V”: UNS S41427

The tubes of both designations exhibit the same geometric characteristics, with an outer diameter (OD) of 203.2 mm, an inner diameter (ID) of 171.45 mm and an overall length of 622 mm. The tubes were received in the as-received condition prior to cutting and machining. All samples were taken with their length in the axial direction to the tube. Assays were conducted in duplicate. It is important to emphasize that two of the three tubes belong to the same run and, therefore, only two of them were used to make the specimens and samples. Both tubes were delivered under different heat treatment conditions, and UNS S41425 was subjected to a solubilization treatment at 932ºC for 6h with air cooling and later tempered at 620ºC for 13h. In the case of UNS S41427, the treatment consisted of a solubilization at 950ºC for 6h with cooling in oil, followed by double tempering at 620ºC for 11h and 550ºC for 7h, both air-cooled. The heat treatment for the samples is performed as described on Table 2.

Table 2
Specimen identification according to the material and heat treatment applied.

Heat Treatment Condition	Material ID.
	UNS S41425	UNS S41427
As Received	NCR	VCR
Austenitized at 1000°C for 1h and quenched in oil	NT1	VT1
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 300°C for 1h and cooled by air	NT1R3	VT1R3
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 400°C for 1h and cooled by air	NT1R4	VT1R4
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 500°C for 1h and cooled by air	NT1R5	VT1R5
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 550°C for 1h and cooled by air	NT1R55	VT1R55
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 575°C for 1h and cooled by air	NT1R575	VT1R575
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 600°C for 1h and cooled by air	NT1R6	VT1R6
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 625°C for 1h and cooled by air	NT1R625	VT1R625
Austenitized at 1000°C for 1h and quenched in oil followed by tempering at 650°C for 1h and cooled by air	NT1R65	VT1R65
Austenitized at 1000°C for 1h and quenched in oil followed by double tempering of 670°C for 1h and 600°C for 2h, both cooled by air	NT1R67-6(2)	VT1R67-6(2)
Austenitized at 1000°C for 1h and quenched in oil followed by double tempering of 670°C for 1h and 600°C for 8h, both cooled by air	NT1R67-6(8)	VT1R67-6(8)

The test was performed in duplicate. In total of forty-eight (48) specimens for slow strain rate test (SSRT) were used for the analysis. The SSRT were performed based on ASTM-E8/E8M recommendations, with samples made with d=4mm² of diameter and L0=18mm of length and the tests performed with a velocity of $1.0\text{*}{10}^{-5}mm.s^{-1}$ on a Cortest® brand tensile test machine, model Cortest's Constant Extension Rate Test with Honeywell load cell in a 45 kgf of preload²⁴. Results were processed using Origin ® 9.0 software. The slow strain performance was applied in order to avoid the influence of the kinematic contribution and the multiplication of dislocations, exerting an influence on the mechanical properties of the material.

The data were treated in a similar way to that reported in the work performed by Chales et al.²⁵. The first step in the treatment of acquisition data was the identification of deviations and residual errors arising from slips and deviations found during the test. These errors were observed and removed from the data obtained for each sample. Then, using the dimensions of the samples, their respective engineering stress x strain curves were raised. At this stage, the effects of the stiffness of the test equipment were removed, correcting the curve obtained according to the actual modulus of elasticity of the material, in accordance of ASM standard²⁶. The SSRT tests were performed in duplicate. From SSRT curves, the respective true stress-strain curves (${\sigma }_r$x${\epsilon }_r$) were estimated, through the equations Equation 1 and Equation 2:

${\epsilon }_r=\text{ln}\left({\epsilon }_c+1\right)$ (1)

${\sigma }_r=\frac{P}{S_0}*\left({\epsilon }_c+1\right)$ (2)

where ${\epsilon }_c$ is the engineering strain, P is the force applied on test and S₀ is the initial section area of the specimens. The curves ${\sigma }_r$x${\epsilon }_r$ were adjusted for their average curves taking into account the Equation 3;

$x_n^m=\frac{{{\displaystyle \sum }}_{i=1}^k{p}_n^i}{k}$ (3)

where n is the number of values obtained during SST and k is three (3), this is, the number of specimens evaluated. Thus, the average curves for each condition were analyzed.

The Hooke’s law is applied for the SMSS alloy 13% Cr considering well-defined linear behavior typical for these steels and is represented by the Equation 4;

$\sigma =E\text{*ε}$ (4)

Where the σ is the stress, E is the modulus of elasticity of the material and ε its specific deformation. However, the distortion of the sample during the tensile test is accompanied by the distortion of the machine itself. In this way, the modulus of elasticity obtained from a curve recorded directly by the traction machine is considerably different from that characteristic of the material, presenting a false decrease in its stiffness. The relationship between the stiffness of the machine, the specimens tested and the equivalent stiffness can be expressed according the Equation 6

$\frac{1}{kt}=\frac{1}{E_{exp}}\text{*}\frac{1}{E_r}$ (5)

Where E_exp represents the modulus of elasticity obtained by testing and E_r the real modulus of elasticity for the material. The E_r can be obtained by strain gages glued in contact with the specimens tested. In order to remove the deviations reported by the test apparatus, the actual modulus of elasticity was obtained directly from the manufacturer, which presents the value of 200GPa of modulus of elasticity for both materials UNS S41425 and UNS S41427, in accordance with the ASM International standard²⁶.

Once the elastic limit of the material has been reached, the plastic deformation occurs followed by a hardening where the stress must be continuously increased in order to continue producing plastic deformation. The nonlinear models of plasticity approached by strain-hardening models are used to represent mathematically the hardening of the material up to its ultimate strength point.

2.2. Strain hardening models

Based on an isotropic hardening criteria, Hollomon's model is one of the first simplified mathematical models that can accurately describe the hardening behavior of the material obtained at the macroscopic level through tensile tests²⁷-²⁹. Through several studies, a good relationship was clearly seen between this model and the stress hardening behavior in metals. Hollomon's equation, which is a function related to true stress x true strain, is given by Equation 6 where σ is the true stress, ${\epsilon }_p$ is the true strain, n is the material hardening exponent and k is the force coefficient of the material.

$\text{σ}=\text{k*}{\epsilon }_p{}^n$ (6)

Using formulation based on exponential form para models based in an isotropic hardening behavior, the Voce equation was developed. This model has a good predicted response, being suitable for materials with little hardening and high plasticity, as the super martensitic steels on heat treatment conditions reported in this work²⁹-³². The equation can be described by the Equation 7.

$\sigma ={\sigma }_S+R_{sat}\left(1-e^{-b\text{*}{\epsilon }_p}\right)$ (7)

where ${\sigma }_S$ is the elasticity limit, ${\epsilon }_p$ is material’s plastic strain and parameters that defines the material hardening, represented by $R_{sat}$ and b coefficients.

In cyclic loading applications, the material exhibits a kinematic hardening response. When a sample is carried past the yield point in one direction (and then undergoes hardening), there is a decrease in the yield point in the opposite direction³³. This phenomenon is known as the Bauschinger effect and can be modeled mathematically by kinematic hardening, being evident in cyclic loadings. Thus, Chaboche presents an equation which represents the phenomena and describes the strain hardening curves with a good correlation. Although it results in good estimates of the Bauschinger effect, the Armstrong Frederick model results in overestimates of the ratchetting effects. Thus, Chaboche proposed the generalization of the Armstrong-Frederick model, according to Equation 8, in which i is the desired number of terms in the sum³⁴.

$\sigma ={\sigma }_S+{\displaystyle \sum }_{i=1}^n\left[\left(\frac{{\text{C}}_i}{{\gamma }_i}\right)\left(1-e^{-{\gamma }_i\text{*}{\epsilon }_p}\right)\right]$ (8)

where ${\sigma }_S$ is the elasticity limit, ${\epsilon }_p$ is material’s plastic strain and parameters that defines the material hardening, represented by Ci and γi. The Chaboche model allows greater flexibility in adjusting the material parameters, maintaining the advantages of the Armstrong-Frederick model, including non-linearity. Thus, the effects overestimated by Armstrong-Frederick can be adequately calculated by this model³⁵.

2.3. Fitting techniques

To determine the coefficients that best fit the results, it is necessary to use an iterative process. The generalized reduced gradient (GRG) algorithm is employed to solve nonlinear optimization problems in which the objective function can have nonlinearities of any nature³⁰,³⁶,³⁷. This method was used to solve the convergence models by finding the best approach parameters. Through the Excel software, it was possible to use the implementation of this method to vary the coefficients in order to obtain the best fitting, with a correlation index close to unit. The correlation index, identified as R squared, indicates how well our data fits the model of regression, given by Equation 9.

$R^2={\left(\frac{n\left(\sum xy\right)-\left(\sum x\right)\text{*}\left(\sum y\right)}{\sqrt{\left[n\sum x^2-{\left(\sum x\right)}^2\right]\text{*}\left[n\sum y^2-{\left(\sum y\right)}^2\right]}}\right)}^2$ (9)

where x and y are two variables for which we want to determine for any linear or non-linear correlation. The value of R squared shall indicate that if there is correlation between the two variables, a change in value of the independent variable will likely result to a change in the dependent variable.

3. Results and Discussion

3.1. Microstructural analysis

Figure 1 shows the microstructures of NCR and VCR samples, revealed using the Vilella Reagent. Both the NCR and VCR samples exhibited microstructural characteristics typical of a supermartensitic stainless steel, with the lath-shaped martensitic structure clustered within the previous austenite grains. However, despite the similarities, it is possible to qualitatively note that the NCR sample has a more refined structure when compared to the VCR.

Figure 1
Microstructure of NCR and VCR samples. Attack performed with Vilella reagent.

For the quenched samples, the microstructure analysis revealed the existence of martensitic microstructure in slats in the form of blocks, as can be seen in Figure 2. These quenched samples on tempering conditions -R1 -R3 and -R4 exhibited some similarity, with slight qualitative variation in the number of carbonitrides dispersed in different regions of the matrix, revealing little influence of microstructural variations between tempering temperatures of 300 and 400°C.

Figure 2
Microstructure of NT1 and VT1 samples. Attack performed with Vilella reagent.

On the other hand, Figure 3 shows the microstructures of samples NT1R5 and NT1R55. Under these conditions, a greater refinement of the martensite packet size is observed, in addition to clearer previous austenite contours. It can be seen that tempering at 550°C resulted in an appreciable distinction in the size of precipitates. With the increase of temper, it is observed in samples NT1R575 and NT1R6 a more refined structure of packages and more intense precipitation, preferentially in the contours between packages and of previous austenite. Increasing the tempering up to 650°C, it is possible to identify by samples NT1R625 and NT1R65 an increase in the packages and refinement of the martensite blocks.

Figure 3
Microstructure of samples NT1R5, NT1R55, NT1R575, NT1R6, NT1R625 and NT1R65. Reagent: Villella.

Similarly, as reported by Figure 3, these conditions were also observed for the “V” alloy. Comparing the samples of alloy “N” and “V” in the tempering conditions at 300 and 400°C, it was noticed that samples VT1R3 and VT1R4 exhibited more refined packages, larger blocks and smaller number of precipitates. For the VT1R5 samples, the number of precipitates scattered in the matrix is qualitatively higher than that reported for the “N” alloy. With tempering at 650°C, the “V” alloy promoted greater microstructural refinement than at the other temperatures, manifesting itself in a more accentuated way when compared to the “N” alloy.

Figure 4 shows the microstructures of samples NT1R67-6(2) and NT1R67-6(8). For the double tempering, the presence of a coarser substructure of packages and blocks was observed, as well as more intense previous austenite contours. Compared to the single tempered samples, these showed a more refined substructure, with some regions of more defined contour, as a result of a more intense precipitation. Analogously, the results were similarly reported for the “V” alloy, but with slightly more refined blocks when compared to the “N” alloy.

Figure 4
Microstructure of samples NT1R75-6(2) and NT1R75-6(8). Reagent: Villella.

3.2. Mechanical properties results

In this section, the results of SSRT are presented for both specimens of the UNS S41425, identified as “N” material and UNS S41427, identified as “V” material.

The results presented are the mean curves for the duplicate testing specimens. Correction in elastic and plastic fields were performed eliminating distortions and slips of the test apparatus, preloading and the rigidity of the machine²⁵.

Using the described equations Equation 1 to Equation 5 the specimens results were treated, obtaining their answers and their work hardening curves through the graphs of stress x plastic deformation as shown in Figures 5 and 6. Figure 5 shows the mean strain hardening curves provided by SSRT after data processing for material S41425, identified by “N”. Figure 6 shows the mean strain hardening curves provided by SSRT after data treatment for material S41427, identified by “V” and with the same heat treatment used.

Figure 5
Strain-hardening curves of S41425 (“N”) material plotted on true stress x plastic strain graph under different heat treatment conditions.

Figure 6
Strain-hardening curves of S41427 (“V”) material plotted on true stress x plastic strain graph under different heat treatment conditions.

From Figure 5, it is possible to identify that a sample in the NCR condition had the lowest ultimate stress, as well as the lowest elasticity limit, but with the highest hardening, configuring a saturation Rsat (value between the ultimate stress and the material yield limit) approximated at 290 MPa. From Figure 6, it is possible to identify that a VCR condition presents similarity with the results reported to the NCR, both values ​​close to YS, σrup and Rsat. In fact, it is identified that the micro alloyed effect of the element nitrogen present in UNS S41425 and the vanadium present in UNS S41427 do not confer changes in the "as received" condition, but the changes in mechanical properties are accentuated when subjected to different heat treatment conditions employed in this work.

In general, it is observed that with the increase of tempering in the samples, the material tends to present a lower ultimate stress, associated with an increase in its final elongation. This effect is more noticeable in “N”, being in “V” the closest values ​​observed during each treatment. The -T1R5 condition, in both materials, had the highest value on ultimate stress, while, similarly, a condition without the tempering -T1 had the lowest elongation values. As noted, tempering favored the toughness of the material, achieving the best results at 500 °C, in both conditions. For tempering temperatures above 500 °C, the material tends to return its properties to the “as received” condition. The -T1R67-6 (2) condition was similar to that reported in the -T1R67-6 (8) condition, with no relevant differences with the tempering exposure time and with the values ​​closer to the initial -CR condition.

Table 3 presents the mechanical properties taken from the SST for each condition. In general, the heat treatments used in "N" have higher values ​​of strength and toughness when compared to "V". As observed, duplicate tests have low deviations, with VT1R3 and NCR conditions having the highest deviations obtained. With the results provided by the Table 3, its presented the Figure 7 which contains a graph comparison between the stress results for both materials and Figure 8 which presents a comparison with elongation and toughness property results from "V" and "N" specimens tested. From Figure 7, it is possible to visualize the superiority of the material “N” in the ultimate tension in all the heat treatments used. The smallest Rsat is observed between the conditions -T1R55 for “N” and -T1R6 for “V”.

Table 3
Mechanical properties taken from SST for the specimens of SMSS “N” and “V” materials.

Material ID.	YS	σrup	ε	Resilience	Toughness
-	MPa	MPa	%	J/m3	J/m3
NT1	998.75 ± 7.97	1170.52 ± 2.6	13.57 ± 0.38	4.35 ± 0.1	129.16 ± 3.8
VT1	889.56 ± 4.84	984.33 ± 1	14.19 ± 0.66	3.71 ± 0.01	111.18 ± 6.25
NT1R3	968.54 ± 9.91	1153.57 ± 4.21	18.09 ± 0.17	4.16 ± 0.08	177.66 ± 0.06
VT1R3	782.22 ± 61.24	974.78 ± 6.17	17.72 ± 1.17	3.23 ± 0.03	142.9 ± 11.3
NT1R4	1005.74 ± 1.11	1220.47 ± 1.22	19.19 ± 0.54	4.42 ± 0.02	205.34 ± 6.11
VT1R4	855.99 ± 16.35	981.72 ± 5.9	15.75 ± 0.47	3.46 ± 0.09	128.12 ± 4.11
NT1R5	1030.01 ± 22.11	1223.82 ± 7.22	21.34 ± 1.15	4.65 ± 0.15	230.8 ± 16.12
VT1R5	864.29 ± 4.34	1011.06 ± 2.1	19.49 ± 0.37	3.51 ± 0.08	168.4 ± 3.26
NT1R55	950.72 ± 29.09	1080.43 ± 8.8	19.17 ± 0.16	4.13 ± 0.28	178.62 ± 0.09
VT1R55	866.26 ± 5.25	977.39 ± 11.13	20.67 ± 0.71	3.6 ± 0.07	173.28 ± 8.44
NT1R575	905.62 ± 4.03	979.53 ± 1.42	18.61 ± 0.18	3.87 ± 0.01	161.75 ± 2.28
VT1R575	864.98 ± 14.32	958.19 ± 11.8	19.49 ± 1.24	3.6 ± 0.16	158.98 ± 12.78
NT1R6	867.69 ± 12.17	939.46 ± 0.96	19.3 ± 0.04	3.65 ± 0.07	161.92 ± 0.3
VT1R6	824.48 ± 2.15	868.17 ± 0.39	19.67 ± 0.89	3.41 ± 0.05	145.49 ± 8.11
NT1R625	817.01 ± 1.55	899.28 ± 0.14	19.63 ± 0.48	3.38 ± 0.02	160.41 ± 3.96
VT1R625	805.55 ± 1.86	847.33 ± 0.64	20.7 ± 0.78	3.33 ± 0.03	149.84 ± 6.68
NT1R65	706.78 ± 4.06	877.26 ± 7.14	19.8 ± 1.65	77.19 ± 105.45	159.13 ± 10.43
VT1R65	777.49 ± 18.52	830.92 ± 7.6	21.23 ± 0.37	3.1 ± 0.05	151.29 ± 4.37
NT1R67-6(2)	614.63 ± 6.45	866.36 ± 0.34	21.95 ± 1.1	2.12 ± 0.04	169.3 ± 9.93
VT1R67-6(2)	630.81 ± 25.27	803.44 ± 11.33	23.22 ± 1.39	2.27 ± 0.2	160.21 ± 11.5
NT1R67-6(8)	614.68 ± 10.13	861.44 ± 5.4	21.92 ± 0.97	2.22 ± 0.05	167.37 ± 9.3
VT1R67-6(8)	653.71 ± 29.57	799.06 ± 6.32	24.03 ± 1.18	8.48 ± 8.41	165.47 ± 11.27
NCR	555.98 ± 35.3	851.53 ± 23.42	23.25 ± 0.39	1.88 ± 0.17	170.66 ± 12.97
VCR	545.79 ± 10.99	831.37 ± 26.97	22.79 ± 0.61	1.81 ± 0.03	163.72 ± 14.51

Figure 7
Yield Strength and ultimate stress results from S41425 (“N”) and S41427 (“V”) materials upon different heat treatment conditions.

Figure 8
Elongation and toughness results from S41425 (“N”) and S41427 (“V”) materials upon different heat treatment conditions.

From Figure 8, it is observed the superiority of the "N" specimens compared to the toughness generated in all treatment conditions. The difference in toughness becomes very significant on tempering range between 300°C and 500°C. It is important to highlight the influence of nitrogen-rich M₂X carbonitrides, which are one of the main factors responsible for the increase in hardness and, according to the results shown in this section, in the mechanical strength of the “N” alloy.

For tempering higher than 500°C, is observed a sharp drop in the toughness of the "N" specimens, being close to the results observed for "V" specimens. It is also observed an inversion of the superiority of the "N" specimens on the elongation, being presented greater elongations for the "V" specimens on tempering range between 550°C and 670°C. It’s also noted that for tempering range 575°C to 670°C, the "N" specimens present a level of constant toughness, while its elongation gives constant increments with the increase of the temperature, consistent with the constant decrease in the ultimate tension of this material observed by Figure 7. These results indicate that the lower content of fine precipitates in the "V" alloy results in a microstructure with lower mechanical strength when compared to the "N" alloy, but with a smaller decrease in this property when tempered at a higher temperature. This is partly due to the lower content of interstitial elements and the presence of higher levels of vanadium, whose effect impacts the reduction of the kinetics of diffusional reactions, as well as the growth of packages and the reversion of austenite.

3.3. Fractography analysis

The Figures 9 to 16 presents the scanning electron micrograph (SEM) of the fracture surface that was deformed in uniaxial tension, showing the overall morphology of failure under the most relevant analysis conditions

Figure 9
Scanning electron micrograph of the fracture surface features from VCR specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 10
Scanning electron micrograph of the fracture surface features from NCR specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 11
Scanning electron micrograph of the fracture surface features from NT1 specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 12
Scanning electron micrograph of the fracture surface features from VT1 specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 13
Scanning electron micrograph of the fracture surface features from NT1R5 specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 14
Scanning electron micrograph of the fracture surface features from VT1R5 specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 15
Scanning electron micrograph of the fracture surface features from NT1R67-6(8) specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

Figure 16
Scanning electron micrograph of the fracture surface features from VT1R67-6(8) specimen that was deformed in uniaxial tension, showing the overall morphology of failure with presence of dimples and voids covering the fracture surface.

From Figure 9, the VCR specimen presents a ductile fracture with the presence of dimples and voids covering the fracture surface. From (a) it is possible to observe a distinct transition from the radial zone to the fibrous zone. In (b) shows a core with high dimple formation, with the coalescing of the microcavities generated by the increase of load in the material.

From Figure 10 presents the NCR, in comparison with the VCR condition illustrated by Figure 9. It’s possible to identify the necking, characteristic of a ductile fracture. In (a) note the transition from the radial to the fibrous zone. In (b) observe the formation of microcavities (“dimples”) resulting from normal rupture, with a visual cup and cone appearance.

In greater comparison with VCR, it is possible to identify that the NCR sample has a fibrous zone, with a shorter radial zone.

In general, even with different aspects regarding the fault zones, both presented similarity to the observed mechanics, with approximate elongations of 23% and Rsat close to 290Mpa.

Compared to Figure 9, from Figure 11 is presented the VT1 specimens, with the effect of the temper used and its overall morphology of failure. In (a) we see the transition from the radial zone to the core. It is possible to observe small points of microcavities already in the radial zone. In the core, the presence of dimples with a fibrous aspect, typical of a ductile fracture, is observed. When compared to VCR, there is a smaller number of dimples, with less fibrous appearance, and larger grain boundaries. This aspect results in a lower toughness of the material, with approximately 40% less in its elongation and 65% less in Rsat.

From Figure 12, it’s presented the NT1 specimens showing a reduction in core size when compared to NCR, with a larger radial zone area. The heat treatment applied to the NT1 sample gave a notably less rough appearance to the core compared to the sample in the “as received” condition. (a) presents microcavity coalescence mechanisms defined as normal rupture, typical of ductile fracture. (b) shows an approximation of the fibrous zone, showing a region of coalescence of the microcavities with subsequent onset of rupture. Compared with the NCR sample, the NT1 sample showed an approximate reduction of 60% in elongation and 40% in Rsat.

The tempering temperature at 500°C favored the VT1R5 samples with greater mechanical strength, combined with good ductility, shown in Figure 13. From SEM, this VT1R5 condition presented a visual cup and cone appearance, with balanced areas between the fibrous and radial zones. (b) the radial zone is observed with punctual presence of dimples in its transition to the nucleus, with (b) a higher concentration of dimples and defined alveoli in the nucleus.

From Figure 14, supported by Figures 7 and 8, the NT1R5 specimens also proved to have the highest mechanical strength, combined with good material ductility. Compared to the “as received” condition, there is a balance of areas between the fibrous and radial zones in the sample, with approximately 8% less elongation and 35% less Rsat. In (a) the radial zone with the formation of refined and shallow alveoli is shown. In (b) there is a rough and dimpled core, with occasional cracks (arrows) coming from the coalescence of these microcavities.

Compared to the VT1R5 condition, the NT1R5 condition differs with the presence of deep cracks in its core and more refined alveoli in its radial zone, presenting a resistance 20% greater and an elongation 10% greater, approximately.

Comparatively, it is possible to observe that in this same condition, both “V” and “N” specimens presents an aspect of ductile, fibrous fracture and with microcavities in “cup and cone” model. The material “V” presents a greater necking, with a greater reduction of observed area. In turn, “N” has a larger nucleus and smaller radial zone when compared to “V”.

The VT1R67-6(8) condition is characterized by a high formation of dimples, presenting in (a) a radial zone with the presence of dimples increasing as it enters the nucleus and in (b) a nucleus with a high concentration of alveoli with depth, marked by the high presence of coalescence of these microcavities, typical of a ductile fracture.

In general, the fractography of the VT1R67-6(8) condition is similar to the reported VCR “as received” condition. Such similarity is also observed in the NT1R67-6(8) condition compared to its "as received" condition. Analyzing the SEM for all reported conditions, it is observed the formation of precipitates interfering with the plastic deformation in its vicinity, corroborating the appearance of cracks (arrows indicated), more attenuating for the steels alloyed to nitrogen, mainly on NT1R67-6(8) and NCR specimens. The appearance of cracks can be explained by the material's difficulty in plastically deforming in well-located regions, where cracking and expansion begins with the evolution of loading³⁸.

3.4. Plastic models curve fitting

Using the GRG through Excel software, it's obtained the coefficients that determine the models for each condition analyzed using Equation 6 until Equation 8. The results consider the average curves corrected and plotted in a stress x plastic strain graph, until the formation of the necking, as illustrated by Figures 5 and 6. The fitting was performed using the best correlation index results, as described by Equation 9. The Figures 17 to 19 shows the curve fitting employed for Hollomon, Voce and Chaboche models to predict the "V" and "N" materials with the effect of different heat treatment conditions aborded in this work. The Table 4 to 6 show the best fitting parameters of Hollomon, Voce and Chaboche 1°Term models for both "N" and "V" specimens and Table 7 presents the general comparison of fitting results by using the correlation index value R².

Figure 17
Behavior of Hollomon model parameters for “V” and “N” material under different heat treatments.

Figure 18
Behavior of Voce model parameters for “N” and “V” materials under different heat treatment conditions.

Figure 19
Behavior of Chaboche 1°Term model parameters for “N” and “V” materials under different heat treatment conditions.

Table 4
Best fitting parameters of Hollomon for “N” and “V” specimens under different heat treatment conditions.

Material ID	K		n		R2
	"N"	"V"	"N"	"V"	"N"	"V"
-T1	1651.2	1245.1	0.0767	0.0511	0.944	0.953
-T1R3	1469.7	1239.4	0.0604	0.0585	0.967	0.945
-T1R4	1559.2	1230.1	0.0624	0.0538	0.967	0.969
-T1R5	1545.7	1265.7	0.0595	0.0561	0.990	0.984
-T1R55	1326.9	1195.4	0.0503	0.0492	0.993	0.995
-T1R575	1184.2	1154.9	0.0464	0.0448	0.957	0.994
-T1R6	1149.8	1014.6	0.0510	0.0360	0.940	0.937
-T1R625	1127.1	999.9	0.0598	0.0390	0.936	0.917
-T1R65	1205.0	999.2	0.0942	0.0454	0.983	0.923
-T1R67-6(2)	1316.1	1129.6	0.1364	0.1036	0.983	0.980
-T1R67-6(8)	1307.6	1106.1	0.1364	0.0986	0.981	0.972
-CR	1363.5	1331.8	0.1564	0.1562	0.988	0.988

Table 5
Best fitting parameters of Voce for “N” and “V” specimens under different heat treatment conditions.

Material ID	σS		Rsat		b		R2
	"N"	"V"	"N"	"V"	"N"	"V"	"N"	"V"
-T1	855.7	813.7	342.0	192.3	284.59	273.99	0.997	0.997
-T1R3	939.5	776.7	270.6	235.2	108.21	152.85	0.969	0.975
-T1R4	998.5	818.7	298.4	199.0	83.43	141.45	0.952	0.977
-T1R5	1060.8	865.1	262.1	205.9	49.35	74.86	0.954	0.961
-T1R55	963.6	881.2	189.8	167.2	55.70	48.42	0.959	0.966
-T1R575	913.0	875.8	178.7	147.5	22.11	48.90	0.995	0.967
-T1R6	868.8	828.7	196.4	136.6	18.53	19.93	0.998	0.994
-T1R625	813.1	805.9	218.6	155.0	17.53	16.25	0.998	0.997
-T1R65	705.5	778.6	286.5	173.7	25.25	15.75	0.999	0.998
-T1R67-6(2)	608.8	627.5	389.8	283.6	23.30	25.15	0.999	0.999
-T1R67-6(8)	608.1	639.3	389.3	277.7	22.41	21.64	0.999	0.999
-CR	547.4	535.6	422.9	412.6	26.97	26.92	0.999	0.999

Table 6
Best fitting parameters of Chaboche 1°Term for “N” and “V” specimens under different heat treatment conditions.

Material ID	σS		C1		𝛾1		R2
	"N"	"V"	"N"	"V"	"N"	"V"	"N"	"V"
-T1	855.7	813.7	97316.2	52687.7	284.58	274.00	0.997	0.997
-T1R3	939.5	776.7	29286.0	35944.9	108.21	152.84	0.969	0.975
-T1R4	998.5	818.7	24892.3	28144.7	83.43	141.43	0.952	0.977
-T1R5	1060.8	865.1	12933.3	15416.3	49.35	74.87	0.954	0.961
-T1R55	963.6	881.2	10574.3	8092.1	55.70	48.41	0.959	0.966
-T1R575	913.0	875.8	3950.7	7213.0	22.11	48.90	0.995	0.967
-T1R6	868.8	828.7	3640.4	2721.6	18.53	19.93	0.998	0.994
-T1R625	813.1	805.9	3831.6	2519.4	17.53	16.25	0.998	0.997
-T1R65	705.6	778.6	7234.5	2736.9	25.25	15.75	0.999	0.998
-T1R67-6(2)	608.8	627.5	9082.3	7133.0	23.30	25.15	0.999	0.999
-T1R67-6(8)	608.2	639.3	8723.4	6010.3	22.41	21.64	0.999	0.999
-CR	547.4	535.6	11402.6	11105.9	26.96	26.92	0.999	0.999

Table 7
General comparison results of correlation coefficient index R² for all the models analyzed.

Material ID	Hollomon		Voce		Chaboche 1°Term
	"N"	"V"	"N"	"V"	"N"	"V"
-T1	0.9440	0.9529	0.9967	0.9966	0.9967	0.9966
-T1R3	0.9667	0.9450	0.9690	0.9746	0.9690	0.9746
-T1R4	0.9672	0.9691	0.9515	0.9767	0.9515	0.9767
-T1R5	0.9897	0.9841	0.9541	0.9615	0.9541	0.9615
-T1R55	0.9931	0.9953	0.9594	0.9659	0.9594	0.9659
-T1R575	0.9569	0.9940	0.9955	0.9672	0.9955	0.9672
-T1R6	0.9400	0.9369	0.9982	0.9943	0.9982	0.9943
-T1R625	0.9357	0.9175	0.9985	0.9972	0.9985	0.9972
-T1R65	0.9827	0.9228	0.9991	0.9975	0.9991	0.9975
-T1R67-6(2)	0.9827	0.9802	0.9992	0.9992	0.9992	0.9992
-T1R67-6(8)	0.9805	0.9716	0.9994	0.9991	0.9994	0.9991
-CR	0.9882	0.9879	0.9992	0.9993	0.9992	0.9993

3.4.1. Hollomon Fitting

The fitting analysis, using the methods described on 2.3, it’s performed in order to represent the strain hardening behavior of the materials performed on different heat treatments by using the Hollomon model described on Equation 6. From the equation, the K coefficient indicates the strength of the material. Higher values correspond to higher material strength. The n coefficient is the material hardening exponent that increases directly as the hardening occurs during loading. The values of each coefficient that best represent the results and a general comparison between “N” and “V” material are presented on Figure 17 and Table 4.

Through the Figure 17, supported by Table 4, it can be seen that both materials had higher values in the force coefficient k, with a range between 999.2 (VT1R65) to 1651.2 (NT1). In general, as seen in Figure 7, the "N" specimens present greater resistance under all conditions, with superiority on K coefficient values as observed by Figure 17.

It is noted for both conditions that when considered the tempering, the temperature increase on values higher than 600°C provides an also increase on the material hardening exponent n, allied with the increase on its elongation as observed by Figure 17.

In order to generate the models and to fit the curves with the best parameters, its necessary an indicator of how well our data fits the model of regression, so the regression method of R squared is used. The Table 4 presents the results of R² as the coefficients for each condition analyzed for both materials in comparison. Using the Hollomon model, the fitting presented a good correlation index R², with values in a range of 0.936 for NT1R625 and 0.995 for VT1R55.

3.4.2. Voce Fitting

The fitting analysis was also performed for Voce model, using the equation described by Equation 7. Different for Hollomon, this model was developed using a nonlinear term constructed by an exponential form, with a total of 03 coefficients for adjustments. The σS coefficient indicates the point where the beginning of the plastic strain of the material occurs, treated as the elasto-plastic transition point. The Rsat coefficient relates the saturation point of the hardening material. This coefficient measures the difference between rupture and yield stress values. Through the Figure 18, it is possible to verify the magnitude of this coefficient in each proposed condition. The coefficient b the asymptote of the material hardening curve, indicating the way it suffers hardening during loading. The values ​​of each coefficient that best represent the results and a general comparison between “N” and “V” material are presented on Figure 6 in isolation. The values ​​used in Figure 18 were taken from Table 5.

Through the Figure 18, it's observed for -T1 condition, in both "N" and "V" specimens, the b explodes in higher values ​​showing a much tighter work hardening curve, with little elongation generated before necking. With the presence of tempering, the hardening curve tends to raise the elongation, favoring a reduction on b values, which the asymptote of the hardening curve material tends to be smoother. In general, the increase in the tempering temperature favors a smoother strain hardening curve, increasing the elongation of the material, as seen through Figure 8.

Analyzing the σS coefficient, it's also observed the fitting process generate values consistent with those reported by Figure 7. In fact, the fitting performed using the techniques described in chapter 2.3 generated σS values similar to those reported for the "N" and "V" specimens in all heat treatment condition analyzed. This precision is also confirmed by Table 5, which presents an index correlation factor R² close to unit for all specimens. The saturation values, Rsat, reported by Figure 18 can be compared with the difference between YS and σrup presented by Figure 7. It’s noted the coefficient presents a similar behavior along the conditions analyzed, as a result of a good fitting adjustments with R² close to unit.

In general, Voce's model presented the good correlation index R², with greater precision when compared to Hollomon's model, reporting values in a range of 0.952 for NT1R4 and 0.999 for NCR.

3.4.3. Chaboche 1°Term fitting

The fitting analysis was also performed for the Chaboche model, using the equation described by Equation 8. Similarly, this model was developed using a nonlinear term constructed by an exponential form, but the main difference consisting of a final curve from a sum of n partial curves. Thus, the final representative equation allows the attribution of n terms of Chaboche in order to obtain greater precision. The equation described with just 1 term represents the total of 03 coefficients for the adjustments. The σS coefficient, also presents on Voce's equation, indicates the elasto-plastic transition point. The C1 and γ1 coefficients also relates the strength of the material and the asymptote of the material hardening curve, which indicates the elongation the material suffers hardening before necking. The values of each coefficient that best represent the results and a general comparison between “N” and “V” material are presented on Figure 19 in isolation. The values used in Figure 19 were taken from Table 6.

Through Table 5 and Table 6, it is possible to observe the similarity between the results of σS values of the Chaboche and Voce models, determined by the adjustment method described on chapter 2.3. For C1 and γ1, specimens with lower elongation and accentuated hardening behavior until the necking generates higher values on these coefficients. This was observed for both quenched specimens (-T1) showing higher values of C1 and γ1. Higher values on tempering temperature favors an increase on the elongation and smoothing of the strain hardening behavior until the necking point, corresponding to a decrease in the values of these coefficients. Note that the tempering condition at 600° C presents an inflection point on the values, starting to suffer a slight increase with the increase of temperature.

In general, Chaboche's model was able to obtain better correlation index when compared to Hollomon's model. The R² values were similar to those reported by Voce's model. This is explained by the similarity of the asymptotic term applied in its mathematical formulations.

Chaboche manages to stand out from Voce's model when adding more terms to its base, thus increasing its complexity. Based on a kinematic hardening, this model is found in the literature with good accuracy and in material applications involving symmetrical and non-symmetrical loading cycles³⁹-⁴¹. Voce's model, based on isotropic hardening, is used in general applications, usually in directional loads without a cyclic effect²⁵,⁴²,⁴³.

3.4.4. Comparison between R² results

For the fitting method, the approximate ranges of the initial values are chosen on the basis of practical experience and material constant. If the initial values are out of the range or inadequacy, new initial values are chosen to match the equation of that condition. Using the least square method, the fitting will end when the best match coefficients for the lowest normalized deviations are reached. The best match values for Hollomon, Voce and Chaboche models are showed in Table 7.

The results substantiated in Table 7 proved that all the models showed good predictions, with a square R interval between 0.9175 for VT1R625 by Hollomon and reaching a value of up to 0.999 for the -T1R67-6(8) by the Voce model. Chaboche's results were similar to those reported by Voce's model. This is associated with the similar nonlinear exponential term approached between the two models presented. For almost all conditions, the Voce model brings more robustness and precision when compared to the Hollomon model and therefore presents a better convergence of the solution with the best adjustment approach.

4. Conclusions

This paper investigated the effect that the micro alloyed element nitrogen and vanadium have on supermartensitic steels under different applied thermal treatments. From the mechanical properties, it was observed that:

• The "N" alloy presented greater variation on stress-strain behavior as a function of the heat treatment applied and also presented superiority in its ultimate tension in all analyzed conditions.

• The "V" material had lower Rsat values, with subtle variations in its ultimate stresses.

• In both materials, the tempering at 500°C proved to be the treatment which provides the best resistance on the material.

  • From de SEM analysis, it was observed that:

• For all the conditions, the fracture surface showed specimens with the presence of dimples and voids covering the fracture surfaces in all heat treatment observed. In general, its observed rough and dimpled core, with occasional cracks coming from the coalescence of these microcavities and the presence of precipitates interfering with the plastic deformation in its vicinity, corroborating the appearance of cracks in its core, more attenuated on "N" specimens.

From modeling, the techniques were presented to describe the strain-hardening curves from the specimens and it was observed that:

• The curve fitting provided the best coefficients to predict the "N" and "V" specimens for Hollomon, Voce and Chaboche models. The results performed by fitting techniques reached predictive models with good results, provided by higher R² coefficients with values between 0.917 up to 0.999.

• The results proved that both Voce and Chaboche models provide accurate predictions at large strain-hardening behavior until the necking formation.

Finally, the coefficients determined in this paper can be used to perform numerical simulations and meet the needs of the engineering application.

5. Acknowledgments

Acknowledgements to Pro-Defesa IV - CAPES for financial support.

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Publication Dates

• Publication in this collection
  23 Oct 2023
• Date of issue
  2023

History

• Received
  11 Apr 2023
• Reviewed
  29 June 2023
• Accepted
  16 July 2023