^{1}

^{2}

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Deformation in the model high entropy alloy CoCrFeMnNi is assessed using an internal state variable constitutive model. A remarkable property of these alloys is the extraordinarily high strain hardening rates they experience in the plastic region of the stress strain curve. Published stress-strain measurements over a range of temperatures are analyzed. Dislocation obstacle interactions and the observed high rate of strain hardening are characterized in terms of state variables and their evolution. A model that combines a short-range obstacle and a long-range obstacle is shown to match experimental measurements over a wide range of temperatures and grain sizes. The long-range obstacle is thought to represent interactions of dislocations with regions of incomplete mixing or partial segregation. Dynamic strain aging also is observed at higher temperatures. Comparisons with measurements in austenitic stainless steel show some common trends.

In recent years there has been increased interest in the materials engineering community in a class of metals known as High-Entropic Alloys. These metallic alloys contain five or more primary elements, approximately equimolar in concentration. The term high-entropy alloy refers to the high entropies of mixing present in the material. These alloys often concurrently possess high strength and good ductility [

The objective of the work presented herein is to analyze the temperature dependent stress-strain behavior in a model high-entropy system. An internal-state variable constitutive model is applied and possible internal state variables identified and related to observations of deformation microstructures using electron microscopy analysis provided that accompanies published mechanical property measurements [

Several scientific papers have been published with regards to a specific alloy, CoCrFeMnNi. This report will focus upon a specific high-entropy system CoCr FeMnNi studied by Otto et al. [

Stress strain data is provided for these systems at differing grain sizes and temperatures. In each system, these curves and the yield stress tests were run according to ASTM 1876-01 at a strain rate of 10^{−3} s^{−1} for the Otto et al. measurements [^{−4} s^{−1} for the Licavoli et al. measurements [

The model begins with an analysis of the dependence of the yield stress of the materials with temperature, strain rate (although strain rates were held constant), and grain size. The model calculations require knowledge of the strain- rate, burgers vector (b), and the temperature dependent shear modulus. For this material_{0} (the shear modulus at absolute zero) = 85 GPa [

where _{1} and s_{2} are functions (defined from zero to unity) that describe the temperature and strain rate dependence of the internal variable strength contributions [_{1}, s_{2}, and s_{ε} are defined as [

where k is the Boltzmann constant and g_{0,j} is the normalized activation energy for the specified interaction.

The model for this system was selected by varying the g_{0,j} and _{1} is non-present, or is overwhelmed by the impurity strength contribution s_{2}, and thus s_{1} assumes a value of zero. The values selected for these systems are presented in

The CG/MG/FG (Coarse Grain/Medium Grain/Fine Grain) systems are the systems defined in Otto et al. [_{0,1} and _{0,2} and _{o} values could be produced by interactions of mobile dislocations with larger clusters with inhomogeneous chemistry, perhaps due to incomplete mixing or the initial stages of segregation.

Material | g_{o}_{,1} | g_{o}_{,2} | g_{o}_{,ε} | |||
---|---|---|---|---|---|---|

CG/MG/FG | 0.16 | 0.0045 | 3.0 | 0.0016 | 1.6 | 0 |

HEA 1 | 0.4 | 0 | 4.0 | 0.0078 | 1.6 | 0.001 |

Material | D (μm) | σ_{a} (MPa) | |
---|---|---|---|

FG | 4.4 | 180 | 0.477 |

MG | 50 | 55 | 0.141 |

CG | 155 | 29 | 0.080 |

HEA1 | 75 | 43.6 | 0.155 |

The athermal stress contribution ^{1/2}. Recall that for these analyses the model parameters in

Strain hardening―also referred to as structure evolution―is added to the model by introducing the current rate of change of

where

dence saturation threshold stress [

where

As shown in

where

Material | T (K) | ε_{offset} | |||
---|---|---|---|---|---|

FG | 77 | +60 | 0 | 3300 | 2800 |

FG | 293 | +10 | 0 | 2500 | 2500 |

FG | 473 | +15 | 0 | 2300 | 2500 |

FG | 673 | +20 | 0 | 3300 | 2500 |

FG | 873 | +15 | 0 | 2500 | 2500 |

CG | 77 | −25 | 0 | 2300 | 2500 |

CG | 293 | −40 | 0 | 2200 | 2500 |

CG | 473 | −38 | 0 | 2100 | 2500 |

CG | 673 | −35 | 0 | 2800 | 2500 |

CG | 873 | −25 | 0 | 3500 | 2500 |

CG | 1073 | −10 | 0 | 3000 | 2500 |

HEA | 298 | 0 | +0.035 | 2400 | 2200 |

HEA | 523 | 0 | +0.045 | 1800 | 2200 |

HEA | 773 | +60 | +0.045 | 1500 | 2300 |

HEA | 873 | +60 | +0.045 | 1200 | 2300 |

both the HEA 1 and FG and CG materials) fall along the line represented by Equation (7), but other data points (open triangles) fall well off this line. The line in this figure is characterized by

Deviations from the linear model depicted in

The observation of DSA both in the stress strain curve [

where C_{c} is the carbon concentration. The correlation expressed by Equation (8), which is common for solution hardening, shows that a possible source of the increased strength is the increase in the solute concentration in the vicinity of the dislocation core.

Two defect populations have been postulated for this high entropy alloy system. One is a short range obstacle (characterized by g_{0,1} = 0.16 for the FG and CG materials). It seems highly unlikely that the long range obstacle population could contribute to DSA; rather it must be the short range obstacle population. However, at a temperature of 673 K, which is a temperature where the FG material exhibits DSA, Equation (2) predicts an s_{1} value of 0. The implication of this is that this obstacle population is ineffective at this high a temperature because the added stress would be _{1} = 0. It is interesting in _{0,1} = 0.4. At 773 K, where Licavoli et al. observed serrated yielding, Equation (2) predicts s_{1} = 0.069. It must be that the g_{0,1} value for the FG and CG materials in _{0} value in the range noted for the HEA materials must be added to Equation (1).

The behavior observed in the systems presented herein has been noted as similar to other highly entropic FCC metals, e.g. austenitic stainless steel. Application of the model approach used in this report has been studied in the stainless steel systems and described by Follansbee [

Deformation kinetics is controlled by interactions of dislocations with grain boundaries and short range obstacles (g_{0,1} = 0.16 in CG/MG/FG and g_{0,1} = 0.4 in HEA 1). However, the suggestion of a long range obstacle population (g_{0,2} = 3.0 in CG/MG/FG and g_{0,2} = 4.0 in HEA 1) is new and suggests interactions of dislocations with clusters of solute atoms, perhaps resulting from incomplete mixing or a tendency to reduce energy with segregation. It would take quantitative microscopy to identify these clusters. This may offer a glance to the source of deviations in model parameters―particularly

Dynamic strain aging is observed at the higher test temperatures in these high entropy alloys. Evidence for this is seen in the high rate of structure evolution that deviates from model behavior expressed by Equation (7). With a low value of g_{0,1} for the short-range obstacle system, an inconsistency arises in that this obstacle population is not predicted to be effective at the temperature where DSA is observed. With a two-obstacle model (actually three-obstacle if the stored dislocation density obstacle population is included) one is unable to select model parameters (see _{0,1 }value in the range of 0.4 required to render the obstacle effective at the temperature where DSA is observed. Either a third obstacle population must be added, or the lowest temperature measurements (77 K) in the Otto et al. [

The authors wish to express appreciation to Saint Vincent College for the institution’s support of this undergraduate research project.

Stein, A. and Follansbee, P.S. (2017) Analysis of Deformation in a High Entropy Alloy Using an Internal State Variable Model. Materials Sciences and Applications, 8, 484-492. https://doi.org/10.4236/msa.2017.86033