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The analytical approach and the thermomechanical behavior of a Cr-Ni-Mo-Mn-N austenitic stainless steel were characterized based on the parameters of work hardening (
*h*), dynamic recovery
*(r*) and dynamic recrystallization (
*n*,
*t*
_{0.5}), considering constitutive equations (
*σ*,
*ε*) and deformation conditions expressed according to the Zener-Hollomon parameter (
*Z*). The results indicated that the curves were affected by the deformation conditions and that the stress levels increased with
*Z* under high work hardening rates. The
*σ*
_{c}/
*σ*
_{p} ratio was relatively high in the first part of the curves, indicating that softening was promoted by intense dynamic recovery (DRV). This was corroborated by the high values of r and average stacking fault energy,
*γ _{s}*

_{fe}= 66.86 mJ/m

^{2}, which facilitated the thermally activated mechanisms, increasing the effectiveness of DRV and delaying the onset of dynamic recrystallization (DRX). The second part of the curves indicates that there was a delay in the kinetics of dynamic softening, with a higher value of t

_{0.5}and lower values of the Avrami exponent (n) due to the competing DRV-DRX mechanisms, and steady state stress (

*σ*

_{ss}) was achieved under higher rates of deformation.

Stainless steels combine good mechanical properties, corrosion resistance and good weldability, which are characteristics required in many applications in the chemical, petrochemical and energy industries [

The literature describes the influence of alloying elements (Mo, Ni, N, Mn) on the chemical composition and stacking fault energy (SFE) of stainless steels, and therefore, on thermally activated phenomena [^{2}, while that of AISI 305 steel is about 34 mJ/m^{2} and that of AISI 316LVM steel is 78 mJ/m^{2} [

During thermomechanical processing, stainless steels with austenitic structure and low SFE tend to soften by DRX, after reaching a certain degree of deformation [_{c}), corresponds to the minimum strain required for the onset of DRX, at which the stress is called critical stress (σ_{c}) [

where θ represents the work hardening rate. The values of θ can be obtained by calculating the derivative of the experimentally measured σ vs. ε curve, Equation (2):

In the physical simulation of thermomechanical processing, the occurrence of DRX in materials with low SFE leads to the phenomena of work hardening and dynamic softening, represented on the curve as a stress peak, with a decrease in the stress level during hot deformation [_{0.5}), which can be used to optimize the parameters in various deformation conditions [

The characterization of the thermomechanical behavior of stainless steels based on constitutive equations provides an excellent database for modeling of industrial proce- sses such as rolling and forging. According to the literature, on the first part of the curve, before the peak, the main formalism applied is the Estrin-Mecking-Bergstrom model [

where the first term on the right side represents the contribution of work hardening or the work hardening rate (h) and the second term represents the contribution of DRV or the DRV rate (r), considering that hardening is constant relative to deformation and that DRV follows a first order reaction [

After a period of increasing stress deformation, the stress tends towards a hypothetical value of saturation (σ_{sat}), which can be attributed to the equilibrium between dislocation storage and annihilation rates, with dρ/dε = 0. With additional algebraic manipulation (differentiation) [

which is the equation of the tangent curve of the θσ vs. σ² graph with m = −0.5r. It should be noted that there are no studies that portray the profile of curves based on these coefficients of hardening (h) and DRV (r) up to the stress peak.

The characterization of the second part of the curve, after the peak, can be expressed following the Avrami formalism, where the equation describes DRX based on the evolution of the recrystallized fraction along the applied deformation, Equation (6) [

where n is the Avrami exponent associated with nucleation sites, t_{0.5} represents the time required for 50% of recrystallized fraction, σ_{sat} is the saturation stress when only DRV occurs, and σ_{ss} is the steady state stress, with all the parameters expressed as power laws with the Zener-Hollomon parameter (Z) [

The objective of this research is to characterize the thermomechanical behavior of a 16.6Cr-12Ni-2.1Mo-1.4Mn-0.08N-0.025C austenitic stainless steel based on the parameters of work hardening (h), dynamic recovery (r) and dynamic recrystallization (n, t_{0.5}), considering constitutive equations (σ, ɛ) with deformation conditions expressed according to the Zener-Hollomon parameter (Z) and also the width (w) and depth (d) of the dip generated in the curve of the work hardening rate (θ) vs. equivalent stress (σ).

The material was obtained in the form of 3/4" diameter rolled bars, solubilized at 1030˚C for 60 min and water cooled, with an average initial grain size of 47 μm.

In the continuous isothermal hot torsion tests, cylindrical specimens (L = 11 mm; φ = 8.0 mm) were heated to a soak temperature of 1200˚C in an infrared radiation furnace coupled to a G-III hot torsion testing machine, left there for 300 s to homogenize the microconstituents, and then cooled at a rate of 3.3˚C/s to the deformation temperature, which were done at 1000˚C, 1050˚C, 1100˚C and 1150˚C. The specimens remained at these temperatures for 30 seconds to eliminate temperature gradients, and then deformed at strain rates of 0.1, 0.5, 1.0 and 5.0 s^{−1} to a deformation of 3.5. Immediately after deformation, the test specimens were quenched in water for metallurgical analysis.

The equivalent stress and strain values (σ_{eq}, ɛ_{eq}) were calculated by means of the Von Mises distributions, based on the method of Fields and Backofen [

where R is the radius and L is the length of the useful part of the specimen. The coefficients n and m are related to the sensitivity to hardening and deformation rates.

For application of the method and characterization, the curves were initially analyzed using the Origin 9.0 mathematical manipulation program to eliminate irregularities, noise and fluctuations on the experimental curves. The values of peak stress (σ_{p}) and steady state stress (σ_{ss}) were determined directly from the experimental curves. The values of critical stress (σ_{c}) and saturation stress (σ_{sat}) were determined by means of analytical methods, establishing the curves of work hardening rate (θ) as a function of applied stress (σ) up to the peak stress, with a 3rd order polynomial fit, followed by second order differentiation to determine the minimum value of the derivative that represents the inflection point on the curve and the onset of DRX (σ_{c}).

To characterize the first part, the curves of the product of the hardening rate were built based on the stress (θσ) vs. square of the stress (σ^{2}) to determine the curvature (m = −0.5r), where r is the coefficient of DRV. The values of the strain hardening parameter (h) were determined according to the relation [

The second part of the curve, after the peak stress, was characterized according to the values of the Avrami exponent (n) and the time required to attain to 50% of DRX (t_{0.5}), based on the fraction of dynamic softening (X_{s}) along the applied deformation. These parameters were determined from the rearrangement of Equation (6),

The torsion samples were sectioned 250 µm from the outer surface parallel to the axis of the rolling direction in the center in order to avoid the oxidized area for microstructural examination carried out by optical microscopy (OM), with electrochemical etching using 65% nitric acid (HNO_{3}), at a current density of 1A/cm² and variable time. The average grain size was measured according to the ASTM E112 standard [

_{p}). This behavior is characteristic of materials that undergo work hardening and dynamic recovery. After the peak, there is a region of dynamic softening where the stress decreases to an intermediate level between the stress at the onset of plastic flow (σ_{o}) and the peak stress (σ_{p}). This behavior is typical of materials that recrystallize dynamically, competing with the mechanism of dynamic recovery, which may or not reach steady state stress (σ_{ss}).

In the first part of the curves, note the visible decrease in the level of stress in response to increasing temperature and decreasing strain rate. Moreover, in low strain rate and high temperature conditions, the hardening rate decreases rapidly until the peak stress is reached; in these conditions, note that the curvature is also more pronounced. On the other hand, in high strain rate and low temperature conditions, there is a high level of work hardening up to the peak. Immediately after the peak, note the subtle thresholds, which are probably formed due to the effect of DRV with average SFE, the presence of second phase particles responsible for the marked kinetics of DRV, competing with DRX and work hardening. Later, after the peak, intense softening by DRV occurred, with a marked reduction in the level of stress (∆σ~40 MPa) to the steady state, with retardation of the process, particularly in the conditions of low temperature and high strain rate, which hindered the operation of the thermally activated mechanisms, establishing a dynamic imbalance between the DRV and DRX rates. Moreover, the higher the strain rate the greater the difference between peak stress and steady state stress, due to the strong action of the hardening mechanism, leaving the material with low ductility, as reported in the literature [

As can be seen, the level of stress in any condition depends on the deformation conditions, i.e., temperature and strain rate. Taking peak stress as a representative point of the material’s behavior, it is clear that the peak stress (σ_{p}) increases in response to the imposed strain rate and decreases with temperature, as shown in

where _{p} is the peak stress (MPa). These are constants that depend on the variables A, _{def}) at a constant strain rate [

A computational method derived from the methodology proposed by Uvira and Jonas [_{def}) for the hot deformation of the steel under study. Using this method, a value of α was determined that best

fitted Equation (10). After substituting the σ_{p} values for all conditions of strain rate and temperature in Equation 10, linear regressions were used to determine the value of n for a variation in the values of α (0.002 <α < 0.052). The value of α used here is the value that generated the lowest standard deviation for the values of n at all the tested temperatures.

The value of α for the steel was found to be 0.011 MPa^{−1}. The ln(ε) vs. ln[sinh(ασ_{p})] graph in

The value of n is associated with the active deformation mechanism, and the literature reports that thermally activated dynamic systems have values of n between 3 < n < 6 [_{def}) of 347 kJ/mol was found in these deformation conditions, which is consistent with results reported in the literature for similar steels [

Using the log graph Z vs. log[sinh(ασ_{p})], not shown here, taking sinh(ασ_{p}) = 1, and making a linear regression analysis, an A value equal to 5.51 × 10^{11} s^{−1} was found, with a good correlation coefficient of 0.95, showing a good fit of the experimental data. This constitutive equation can be used to predict the stress levels in other hot deformation conditions of this alloy in industrial settings or even in 3D computer simulations of thermomechanical processes, leading to a reduction in the costs of production, design and analysis.

_{c}, σ_{p} stresses and the Z parameter, expressed as power laws. The values of the exponent n, which represents the sensitivity to

strain rate, are very similar to the values within the range reported in the literature, 0.07 < n < 0.18 [

The effect of the processing conditions on the stresses, considering the activation energy (Q_{def}) expressed by the Zener-Hollomon parameter (Z), reveals a marked increase in stress with Z, and at high values of Z (10^{15}), the stresses tend to increase markedly in response to minor variations in Z. The value of the exponent of Z (n = 0.14) in response to peak stress reflects the sensitivity of the material to the deformation conditions, with elevation of the work hardening rate occurring more slowly, generating a delay in dynamic softening with strong DRV.

^{2} curve, where the m slope is equal to −0.5r. The value of the parameter of DRV (r), which specifies the inflection on the first part of the curve,

presented variations between 6.2 < r < 9.4, with values within the range reported in the literature for stainless steels, 6 < r < 10 [_{p}), where the DRV is less effective with less influential diffusion mechanisms, increasing the σ_{c}/σ_{p} ratio and requiring higher strains to trigger DRX. This behavior is physically reasonable, since the DRV rate accelerates as the temperature increases and the strain rate decreases.

The determination of the work hardening parameter, h, depicted in

It was also found that the parameter r, which specifies the curvature of the DRV curve, showed higher values in the tests performed at low strain rates and high temperatures, indicating that DRV occurs more rapidly at low values of Z. On the other hand, higher values of r caused the material to reach saturation stress (σ_{sat}) more rapidly with strong DRV, as shown in ^{2} [

alloying elements, which strongly influence the stability of the austenitic phase, also contribute to this effect.

The dependence of the strain hardening parameter (h), in power law, on the Z parameter shown in Equation (15), presented an upward trend:

The work hardening coefficient, h, contributed to increase the level of peak stress during deformation on the first part of the curve, modifying its profile, particularly at low temperatures and high strain rates. In this stage, immovable dislocations are formed and increase in density, accumulating and interacting randomly, becoming entrapped and forming less mobile subgrains, thus requiring a higher level of stress to cause greater plastic deformation [

_{s}) as a function of deformation time, expressed by the sigmoidal curve, using the Avrami formalism. Note that the softening rate accelerates as the temperature and strain rate increase. At a given temperature, the t_{0.5} decreases with increasing strain rate; e.g., in the condition of 1000˚C/0.1 s^{−1}, with t_{0.5} of 7.3 s dropping to 0.38 s at the same temperature when the strain rate increases to 5.0 s^{−1}. At a given strain rate, t_{0.5} also decreases with increasing temperature. It should be noted that dynamically recrystallized fraction (XDRX) differs from the dynamic softening fraction (X_{s}) resulting from the action of DRV, which can be expressed according to the ratio σ_{sat}/σ_{ss} = 1.75. On the other hand, the high value of Xs corresponds to an accelerated behavior in the region of DRX after the peak, with intragrain softening and high driving force, as well as a greater concentration of defects.

Thus, the greater the variation in the level of Δσ = σ_{sat} − σ_{ss} stress the greater the driving force for DRX.

To estimate the Avrami exponent (n), Equation (6) was rearranged by plotting the

In this set of isothermal hot torsion tests, the ratio that best represented the time dependence to attain 50% of softening (t_{0.5}), with the deformation conditions set forth in Equation (16), (see

The dependence of the Avrami exponent (n) on the parameter Z, according to the deformation conditions that best fit the experimental data according to Equation (17) and expressed in

The relationship between t_{0.5} and the depth (d) and width (w) of the dip of the curve generated in the softening region (negative strain hardening rate) is linear, i.e., the longer the t_{0.5} the greater the depth (d) and width (w) of the dip. On the other hand, the r value decreased, accelerating the softening kinetics. The simplicity of this method stems from the small number of experiments it requires, even when numerous parameters are involved. It is worth noting that the kinetics of dynamic softening determined with this methodology encompasses all the active softening mechanisms. In other words, not only the effects of DRV but also those of DRX are computed in the calculated values of t_{0.5}, showing a correlation coefficient of 0.99, as indicated in

The experimental and simulated curves are presented in ^{−1}. The simulated curves present a higher share in the DRV rate (high r). The relative shift of

the curves to the left with respect to peak deformation, in response to higher work hardening, tends to generate differences in the real and simulated curves.

As for the kinetics of DRX in the second part of the curves, note that the delay in the process resulted in lower values of the Avrami exponent (n) and the time to attain 50% of dynamic softening (t_{0.5}). The shape of the experimental and simulated curves showed differences in the conditions of 1000˚C and 1050˚C at strain rates of 0.1 and 0.5 s^{−1}. Nevertheless, in modeling the curves, it is clear that the parameters n and t_{0.5} determine the shape of the curves by means of the kinetics of DRX, where higher values of n and lower values for t_{0.5} are associated with the kinetics of accelerated dynamic softening with low Z, as indicated in Figures 11(a)-(d).

^{−1} at strains of 0.4, 1.5 and 3.5. Note that the deformations close to the peak (ɛ = 0.4) show the presence of hardened grains and deformation twinning, and the onset of the formation of small recrystallized grains. With further deformation, after the peak stress (ɛ = 1.5), the presence of elongated grains in the direction of deformation is still visible, with some smaller grains at the boundaries of deformed grains, a sharp drop in the level of stress, and the presence of localized flow at the grain boundaries, indicating plastic instability [_{ss} = 3.5, as indicated by the X-ray analyses in _{p}) to steady state (σ_{ss}) stress caused a

reduction in strength, increasing ductility with the refinement of austenitic grains in the hot working of the steel.

The microstructural features are presented in

It is also worth noting the microstructural evidence of plastic instability with a deformation gradient at the grain boundaries, the presence of deformation bands, and concentration of dynamically recrystallized chain-shaped grains in the conditions of low temperature and high strain rate, which generate heterogeneous deformation with localized flow. On the other hand, in the conditions of high temperature and low strain rate, the kinetics of DRX is accelerated and homogeneous (n − 3), with new grain growth and a significant increase in the recrystallized fraction in response to increasing strain until the steady state is reached.

^{−1} and strain of 1.5. Note the formation of small chain-shaped grains that migrate, induced by deformation, which are formed at deformed grain boundaries and twin boundaries, with evidence of localized flow, indicating instability of the material at some points. Also visible is the presence of precipitates at the grain boundaries, possibly chromium nitride, which can be confirmed by the phase diagram calculated using FSstel database in the FactSage software, as shown in ^{−1} and a strain of 3.5, the grains are fully recrystallized with grain refinement.

In this study, the dynamic softening of austenitic stainless steel was investigated at de- formation temperatures varying from 1000˚C to 1150˚C and strain rates of 0.1 s^{−1} to 5.0 s^{−1}. The following conclusions can be drawn from this investigation.

1) The results indicated that plastic flow curves were significantly influenced by the conditions imposed during thermomechanical processing, and that the stress levels increased in response to the parameter Z at a high work hardening rate.

2) In the first part of the curves, the σ_{c}/σ_{p} ratio was relatively high, indicating that the softening promoted by DRV was intense, and a finding was corroborated by the highest values of r and the average stacking fault energy, γ_{sfe} = 66.86 mJ/m^{2}, which facilitated the thermally activated mechanisms, increasing the effectiveness of DRV and delaying the onset of DRX.

3) The second part of the curves showed that there is a delay in the kinetics of dynamic softening with a higher value of t_{0.5} and lower values of the Avrami exponent (n), owing to the competing DRV and DRX mechanisms, and steady state stress (σ_{ss}) reached at higher strain rates.

4) The microstructure presented work hardened grains with deformation twinning and the formation of small recrystallized grains on the first part of the curve. As deformation continued, after the peak, the presence of elongated grains was visible in the direction of deformation, with the formation of small dynamically recrystallized grains in the shape of chains at triple points, twin boundaries, deformed grain boundaries, the presence of chromium nitride precipitates, and formation of localized flow, indicating plastic instability of the material.

The authors acknowledge the financial support of the Brazilian research funding agencies FAPEMA (Maranhão Foundation for Scientific Research and Development) CNPq and CAPES. We are also grateful for the technical support provided by Prof. Dr. Regina Sousa (UFMA―Federal University of Maranhão) and MSc. José Renato Sucupira.

Ferreira, R.P., Silva, E.S., Nascimento, C.C.F., Rodrigues, S.F., Aranas Jr., C., Leal, V.S. and Reis, G.S. (2016) Thermomechanical Behavior Modeling of a Cr-Ni-Mo-Mn-N Austenitic Stainless Steel. Materials Sciences and Applications, 7, 803-822. http://dx.doi.org/10.4236/msa.2016.712062