Received 22 April 2015; accepted 26 May 2015; published 29 May 2015

ABSTRACT

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.

Keywords:

Constitutive Modeling, Internal State Variable, Austenitic Stainless Steel, Strain Hardening

1. 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

(1)

where σ_a is an athermal stress (e.g., due to the strengthening contribution of grain boundaries), is an internal state variable characterizing the strengthening contribution of solute element additions, and 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 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):

(2)

where is the internal state variable characterizing interactions of mobile dislocations with stored (or immobile) 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:

(3)

(4)

(5)

Consistent with an internal-state variable formulation, the strain-dependence of is defined by the differential

(6)

where θ_II is the stage two hardening rate (e.g., of a single crystal), κ is a constant, and 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 approaches. Finally, the temperature and strain rate dependence of is described using a dynamic recovery model [5]

(7)

where is the value of at 0 K, and and 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] .

2. Evaluating the Evolution Equation

The temperature and strain-rate dependence of evolution (strain hardening) is evaluated by analyzing stress- strain curves measured at various temperatures and strain rates. Rewriting Equation (2),

(8)

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, , and should give an initial value of equal to zero, and the increase of 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, , , and. As expected starts close to zero and increases uniformly with strain. Application of Equation (8) to a more extensive data set is described in the next section.

3. 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 (as well as) discussed in [1] . Each of these stress-strain curves was analyzed according to Equation (8) to derive the variation of with strain. As in Figure 1, the values of and the two state variables were taken as, , and. The slight variation in grain size could result in 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 and 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 values is that in softer materials (e.g., AISI 304 versus AISI 316, or AISI 316 LN versus AISI 316), the values would start off negative at zero strain. This can be easily accounted for by adding an “offset” stress so that the 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 versus ε curves. In [1] and [2] κ was selected as 3.4, which led to, , and. 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 value (4000 MPa) was much higher than estimated in all of the other materials. In all of the other systems analyzed,. For the austenitic stainless steels,. Finally, the typical value of is 10⁷ s⁻¹. Inspection of Equation (6) indicates that κ and are not completely independent; a high value of κ along with a high value of 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. Figure 2 shows an example fit of a 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,

^aThe final temperatures for tests under adiabatic conditions are listed in parentheses; ^bThe 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] ; ^cThe 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] .

, , and ¹. It is evident that the two model curves are almost coincident.

Each of the measurements listed in Table 1 was analyzed with κ = 2 as described above. Model parameters (and) that yielded a good match of Equation 6 with the measurements are listed in the last two columns of Table 1. Fits of versus strain for two of these measurements are shown in Figure 3. The solid curves are the deduced values of versus strain; the dashed curves are the model predictions according to Equation 6 with 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

(9)

where the strain rate 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 on temperature² 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 in Equation (7) is set at 2600 MPa and from Equation (7) is set at 10⁷ 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 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] .

4. 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 can give almost identical agreement with a specific versus ε data set as a low value of κ along with a low value of 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 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.

Acknowledgements

The author appreciates the support of Saint Vincent College in the writing of [2] and the compilation of this manuscript.

References

NOTES

¹Only a κ = 2 is selected; the remaining parameters arise from the fit to the full data set listed in Table 1 as described below. Agreement with the data, although, is not as good with κ = 1.

²For the three adiabatic tests, the final rather than the initial temperature is plotted in Figure 4.