_{1}

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.

The constitutive behavior of annealed, austenitic stainless steels was recently analyzed according to an internal state variable model [

where σ_{a} is an athermal stress (e.g., due to the strengthening contribution of grain boundaries), _{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

The addition of strain-hardening is modeled by adding another internal state variable to Equation (1):

where _{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

where θ_{II} is the stage two hardening rate (e.g., of a single crystal), κ is a constant, and _{II} and approaches zero as

where

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),

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 [

steel, introduction of correct values of^{−1}. For this calculation,

The next step of the analysis is to fit Equation (6) to the ^{7} s^{−1}. Inspection of Equation (6) indicates that κ and

Source (Primary Author) | Material Characteristics and Testing Conditions | Analysis Results | ||||||
---|---|---|---|---|---|---|---|---|

Material | Grain Size | Strain Rate (s^{−}^{1}) | Temp (K)^{a } | Nitrogen % | Offset (MPa) | |||

Steichen [ | 304 | ASTM 5 (63 μm) | 3 × 10^{−5} | 811 | 0.052 | +30 | 2200 | 2800 |

100 | 811 (829) | ?60 | 1200 | 3250 | ||||

Albertini [ | 316L | “Virgin” condition | 0.0035 | 823 | ? ? | +50 | 2000 | 2900 |

44 | 295 (381) | +60 | 1600 | 3400 | ||||

0.004 | 295 | 0 | 1850 | 3000 | ||||

Semiatin [ | 304L | ASTM 7.5 (27 μm) | 0.01 | 294 | 0.038 | 0 | 2300 | 2900 |

0.0035 | 673 | 0 | 950 | 3000 | ||||

Conway [ | 316 | ? ? b | 0.004 | 294 | 0.05 | 0 | 2200 | 2885 |

0.004 | 703 | 0 | 2300 | 3000 | ||||

Byun [ | 316 | ? ? c | 0.001 | 294 | 0.031 | 0 | 2100 | 2900 |

0.001 | 437 | 0 | 1750 | 2900 | ||||

Dai [ | 316 LN | ? ? ^{c } | 0.001 | 294 | 0.067 | ?100 | 1800 | 2850 |

0.001 | 523 | ?50 | 1500 | 2900 | ||||

0.001 | 623 | 0 | 2400 | 2850 | ||||

Stout [ | 304L | 40 mm | 0.0002 | 295 | 0.082 | ?25 | 2200 | 3000 |

0.02 | 295 | ?25 | 2300 | 3100 | ||||

100 | 295 (371) | ?100 | 2200 | 3200 | ||||

Antoun [ | 304 | ? ? | 0.0001 | 344 | ? ? | ?80 | 1550 | 2850 |

^{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 [^{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 [

^{1}. It is evident that the two model curves are almost coincident.

Each of the measurements listed in ^{−1} [^{−1} [

The parameters in the last column of _{II}. While the extensive measurements by Follansbee and Kocks in copper [

where the strain rate ^{−1}. Equation (9) is very close to the result published earlier [

The dependence of ^{2} and strain rate is evaluated using Equation (7), shown in ^{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

The four open squares in ^{−5} s^{−1} [

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 [

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

The author appreciates the support of Saint Vincent College in the writing of [