Structure Evolution in Austenitic Stainless Steels — A State Variable Model Assessment

Strain hardening in austenitic stainless steels is modeled according to an internal state variable constitutive model. Derivation of model constants from published stress-strain curves over a range of test temperatures and strain rates is reviewed. Model constants for this material system published previously are revised to make them more consistent with model constants in other material systems.


Introduction
The constitutive behavior of annealed, austenitic stainless steels was recently analyzed according to an internal state variable model [1] [2].In this model, which has been described in detail by Follansbee [2], the temperature and strain-rate dependent yield stress, σ, of annealed material (with a low initial dislocation density) is modeled as where σ a is an athermal stress (e.g., due to the strengthening contribution of grain boundaries), ˆi σ is an internal state variable characterizing the strengthening contribution of solute element additions, and ˆN σ is an internal state variable characterizing the strengthening contribution due to nitrogen, μ is the temperature-dependent shear modulus, μ o is the shear modulus at 0 K, and s i and s N are functions (varying from zero to unity) that describe the temperature (T) and strain rate ( ) ε dependence of the two strength contributions.The explicit nitrogen-de- pendent term in Equation ( 1) evolved from analysis of two extensive data sets documenting the effect of the nitrogen content on the temperature-dependent yield stress in austenitic stainless steels [3] [4].
The addition of strain-hardening is modeled by adding another internal state variable to Equation (1): , where ˆε σ is the internal state variable characterizing interactions of mobile dislocations with stored (or immo- bile) dislocations and s ε defines the temperature and strain-rate dependence of these interactions.The analysis of temperature and strain-rate dependent yield stress measurements in a variety of austenitic stainless steels led to the following definitions of s i , s N , and s ε , where k is Boltzmann's constant and b is the Burgers vector: ( ) ( ) ( ) ( ) Consistent with an internal-state variable formulation, the strain-dependence of ˆε σ is defined by the diffe- rential ( ) where θ II is the stage two hardening rate (e.g., of a single crystal), κ is a constant, and ˆs ε σ is the temperature and strain-rate dependence saturation threshold stress.When κ equals unity, Equation (3) becomes the Voce Law.According to Equation (6) the rate of strain hardening begins at θ II and approaches zero as ˆε σ approaches ˆs ε σ .
Finally, the temperature and strain rate dependence of ˆs ε σ is described using a dynamic recovery model [5]   3 ˆln ln ln where ˆso ε σ is the value of ˆs ε σ at 0 K, and so ε ε and so g ε are constants.Values of the model constants in Equations ( 6) and (7) were listed in [1] and [2], but the analysis used to generate these constants was omitted.The purpose of this paper is to document this detail and to report updated values of these constants that are more in line with the constants for the other metals and alloys included in [2].

Evaluating the Evolution Equation
The temperature and strain-rate dependence of evolution (strain hardening) is evaluated by analyzing stressstrain curves measured at various temperatures and strain rates.Rewriting Equation ( 2 A key premise of the internal-state variable model applied here is that evolution does not alter the parameters on the right-hand side of Equation ( 8)-except of course for σ(ε).This premise was shown to be approximately valid by Follansbee and Kocks, through extensive measurements of the evolution of the internal state variable in pure copper [6].In applying Equation ( 8) to stress-strain curves measured in an annealed austenitic stainless steel, introduction of correct values of a σ , ˆi σ , and ˆN σ should give an initial value of ˆε σ equal to zero, and the increase of ˆε σ with strain should follow Equation ( 6). Figure 1 shows the result of this analysis on a stress-strain curve reported by Albertini and Montagnani [7] in annealed 316 L stainless steel measured at 295 K and a strain rate of 0.004 s −1 .For this calculation, 50 MPa , and ˆ243 MPa N σ = . As expected ˆε σ starts close to zero and increases uniformly with strain.Application of Equation ( 8) to a more ex- tensive data set is described in the next section.

Stress-Strain Measurements in AISI 304 and AISI 316 Stainless Steels
Table 1 lists the source of 18 measurements of stress-strain curves in AISI 304 and AISI 316 stainless steels (and variations of these alloys).The data set was selected because of the wide range of temperatures and strain rates investigated, which is necessary for evaluation of the constants in Equation (7).Included in Table 1 are the grain sizes and the nitrogen contents (when specified).The nitrogen contents are listed because of the correlation of the state variable ˆN σ (as well as ˆi σ ) discussed in [1].Each of these stress-strain curves was analyzed ac- cording to Equation ( 8) to derive the variation of ε σˆwith strain.As in Figure 1, the values of a σ and the two state variables were taken as 50 MPa , and ˆ243 MPa N σ = . The slight variation in grain size could result in a σ values slightly greater than (for a smaller grain size) or less than (for a larger grain size) the assumed 50 MPa, but this would be a small effect.Similarly, the variation in nitrogen content could result in ˆi σ and ˆN σ values that differ from the assumed values of 572 MPa and 243 MPa, respectively.The net result of using the assumed values of these parameters on the predicted ˆε σ values is that in softer materials (e.g., AISI 304 versus AISI 316, or AISI 316 LN versus AISI 316), the ˆε σ values would start off negative at zero strain.This can be easily accounted for by adding an "offset" stress so that the ˆε σ values start at zero.The value of the offset used in the calculations is listed in Table 1.Indeed, negative offsets are generally observed in the softer materials, but there some outliers.For instance, there is no reason for the offset stress to differ for tests at different temperatures on the same material.That this is found in a few cases demonstrates the level of experimental scatter in the measurements and analysis.The next step of the analysis is to fit Equation ( 6) to the ˆε σ versus ε curves.In [1] and [2] κ was selected as 3.4, which led to ˆ4000 MPa .While these model parameters enabled close fits with the measurements, they differed from the model parameters published in [2] for a large collection of FCC, BCC, and HCP metals and alloys.In particular the κ-value for these other systems was either κ = 1 or κ = 2. Secondly, the ˆso ε σ value (4000 MPa) was much higher than estimated in all of the other materials.In all of the other systems analyzed, 0.009 0.035 . For the austenitic stainless steels, ˆ0.056 so o ε σ µ = .Finally, the typical value of so ε ε is 10 7 s −1 .Inspection of Equation (6) indicates that κ and ˆso ε σ are not completely independent; a high value of κ along with a high value of ˆso ε σ yields an almost identical stress strain curve over the strain range of interest as a low value of κ along with a low value of ˆso ε σ .Figure 2 shows an example fit of a ˆε σ versus ε curve for the data set given in Figure 1.The two dashed curves are the model fits.
The short-dashed curve is for the model parameters listed above.The long-dashed curve uses κ = 2,   a The final temperatures for tests under adiabatic conditions are listed in parentheses; b The material received a "stress relief anneal"; these treatments are well above the recrystallization temperature of 850˚C and would yield a grain size of 30 μm to 60 μm, depending on the heat treatment time [15]; c The material was reportedly heat treated at 1050˚C for 30 minutes; this is a common solution anneal condition, also well above the recrystallization temperature of 850˚C, that would yield a grain size of 40 μm to 60 μm [15].Each of the measurements listed in Table 1 was analyzed with κ = 2 as described above.Model parameters ( ˆs ε σ and II θ ) that yielded a good match of Equation 6 with the measurements are listed in the last two columns of Table 1.Fits of ˆε σ versus strain for two of these measurements are shown in the deduced values of ˆε σ versus strain; the dashed curves are the model predictions according to Equation 6with the model parameters listed in Table 1.The lower curves are from the Antoun measurements in 304 SS at 344 K and a strain rate of 0.001 s −1 [14].The curves at the higher stress levels are for the measurement by Stout and Follansbee in AISI 304L at 295 K and a strain rate of 100 s −1 [13].
The parameters in the last column of Table 1 suggest a slight strain-rate dependence of θ II .While the extensive measurements by Follansbee and Kocks in copper [6] indicated this strain-rate dependence, one would not conclude this with the limited data set in the stainless steels presented here.The indicated strain-rate dependence is assumed based on the earlier measurements, and the assumed correlation is ( ) where the strain rate ε has the units s −1 .Equation ( 9) is very close to the result published earlier [1] [2], where the constant was 3010 MPa and the multiplier of the logarithmic term was 23 MPa.
The dependence of ˆs ε σ on temperature 2 and strain rate is evaluated using Equation ( 7), shown in Figure 4.
The data points plotted as open triangles fall roughly on a line when ˆso ε σ in Equation ( 7) is set at 2600 MPa and so ε ε from Equation ( 7) is set at 10 7 s −1 .Note that the dashed line passes through the origin, which is consistent with Equation (7).From the slope of the line, the value of so g ε is found to be 0.258.The four open squares in Figure 4 that fall well off the line are for the Albertini and Montagnani data set at 823 K [7] , the Steichen data set at 811 K and a strain rate of 3 × 10 −5 s −1 [8] , the Conway et al data set at 703 K [10], and the Dai et al data set at 623 K [12].It was proposed in [2] that dynamic strain aging becomes active at these high temperatures, which leads to behavior that deviates strongly from that described by Equation (7).A method to include the higher stresses during dynamic strain aging into the constitutive model was introduced in [2].6) to the Stout and Follansbee stress-strain curve in 304L SS at 295 K and a strain rate of 100 s −1 and the Antoun stress-strain curve in 304 SS at 344 K and a strain rate of 0.001 s −1 .

Summary
Analysis of stress-strain curves reported for annealed austenitic stainless steels has given further evidence of the application of the internal state variable constitutive formulism developed by the author and coworkers.Of particular interest here was the derivation of model parameters describing strain-hardening.A set of model parameters for this alloy system was given in previous publications [1] [2], but the derivation of these parameters was not presented in these earlier publications.
The reanalysis of the literature stress-strain curves presented here demonstrated that the model parameters in Equations ( 6) and ( 7) are somewhat co-dependent.In particular, a high value of κ along with a high value of ˆso ε σ can give almost identical agreement with a specific ˆε σ versus ε data set as a low value of κ along with a low value of ˆso ε σ over the strain range of interest (ε < 1).The proposed value of κ = 2 for the austenitic stainless steels is more in line with model parameters proposed for a wide variety of material systems.The interplay between κ and ˆso ε σ reinforces a conclusion reached in [2] (see Chapter 13) that the empirically based hardening (or structure evolution) model (particularly with κ ≠ 1) lacks a sound mechanistic foundation.Future work on this element of the model would enhance the internal state variable model formulation.

Figure 1 .
Figure 1.Analysis of the Albertini and Montagnani stress-strain curve at 295 K and a strain rate of 0.004 s −1 according to Equation (8) to give ˆε σ versus strain.

Figure 2 . 1 .
Figure 2. Fit of Equation (6) to the deduced values of ˆε σ versus strain curve deduced for the Albertini and Montagnani stress-strain curve at 295 K and a strain rate of 0.004 s −1 .Two sets of model parameters are used to demonstrate the interplay between k and ˆε σ .

Figure 3 .
The solid curves are

Figure 3 .
Figure 3. Fit of Equation (6) to the Stout and Follansbee stress-strain curve in 304L SS at 295 K and a strain rate of 100 s −1 and the Antoun stress-strain curve in 304 SS at 344 K and a strain rate of 0.001 s −1 .

 2
For the three adiabatic tests, the final rather than the initial temperature is plotted in Figure4.

Table 1 .
Stress-strain measurements in annealed AISI 316 and AISI 304 stainless steels (and variations of these alloys) analyzed in this study.