Kinetics of interstitial segregation in Cottrell atmospheres and grain boundaries

Svoboda, Jiřı́; Zickler, Gerald A.; Kozeschnik, Ernst; Fischer, Franz Dieter

doi:10.1080/09500839.2015.1087652

Cited by 21 publications

(12 citation statements)

References 34 publications

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“…The rate of C segregating from the undisturbed crystal into the stress field of dislocations has been derived using the Thermodynamic Extremal Principle, [ 15 ] yielding

{\overset{y}{true}}_{normalT} = \frac{2 D π ρ}{f 4extrue(\ln 4extrue(\sqrt{\frac{1}{f}} 4extrue) - \frac{3}{4} 4extrue)} y_{normalL} (\ln \frac{y_{normalL} false(1 - y_{normalT} false)}{y_{normalT} false(1 - y_{normalL} false)} + \frac{Δ E}{R_{normalg} T})

where D is the bulk diffusion coefficient of C, ρ the dislocation density, f the volume fraction of dislocation trap sites, y L the site fraction of C uniformly distributed in the matrix, y T the site fraction of C in dislocation traps, Δ E the trapping energy for C in dislocations, R g the gas constant, and T the temperature. The volume fraction of dislocation trap sites, f , is given by

f = Z c_{dc}

where c dc is the volume fraction of atoms located in the dislocation cores and Z the coordination number

c_{dc} = 2 π {r_{c}}^{2} ρ

Z = \frac{A}{T^{2}}

…”

Section: Modelingmentioning

confidence: 99%

“…The main input parameters for the evolution of C-trapping are temperature, C-content, and dislocation density. The Gibbs energy for trapping can be derived as [15] G…”

Section: Cottrell Atmosphere Formationmentioning

confidence: 99%

“…A similar treatment has been reported by Baluffi et al [12] and De Cooman and coworkers. [13,14] Recently, Svoboda et al [15] have presented a model for interstitial segregation into Cottrell atmospheres and grain boundaries based on the thermodynamic extremal principle. [16] Among others, [17,18] Wepner [19] has comprehensively studied the full range of yield strength increase during strain aging on an experimental basis.…”

Section: Introductionmentioning

confidence: 99%

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Modeling of Bake Hardening Kinetics and Carbon Redistribution in Dual‐Phase Steels

Shan

Soliman

Palkowski

et al. 2020

steel research int.

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The bake hardening response (BHR) of a dual‐phase steel with a martensite volume fraction of 0.22 is studied by means of mechanical testing, microstructural characterization, and computational simulation. After austenitization and quenching, the material is exposed to temperatures between 100 and 220 °C for times up to 10 000 min. The increase in yield strength during bake hardening (BH) follows the well‐known two‐stage characteristics. Herein, the kinetics of BH is analyzed on the basis of computational models for the formation of Cottrell atmospheres and precipitation, which represent the cause of strength increase, as well as the long‐range diffusion of C, which is partitioning between the martensite and ferrite phases during aging. The simulated yield strength evolution is in good agreement with the experiments. The model clearly describes the shape of the experimentally observed BH curves as a function of temperature, time, and degree of prestrain. In contrast to previous simulation attempts, all metallurgical phenomena are treated as coupled processes and modeled in an integrated framework based on a single set of input and state parameters.

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“…The rate of C segregating from the undisturbed crystal into the stress field of dislocations has been derived using the Thermodynamic Extremal Principle, [ 15 ] yielding

{\overset{y}{true}}_{normalT} = \frac{2 D π ρ}{f 4extrue(\ln 4extrue(\sqrt{\frac{1}{f}} 4extrue) - \frac{3}{4} 4extrue)} y_{normalL} (\ln \frac{y_{normalL} false(1 - y_{normalT} false)}{y_{normalT} false(1 - y_{normalL} false)} + \frac{Δ E}{R_{normalg} T})

f = Z c_{dc}

where c dc is the volume fraction of atoms located in the dislocation cores and Z the coordination number

c_{dc} = 2 π {r_{c}}^{2} ρ

Z = \frac{A}{T^{2}}

…”

Section: Modelingmentioning

confidence: 99%

“…The main input parameters for the evolution of C-trapping are temperature, C-content, and dislocation density. The Gibbs energy for trapping can be derived as [15] G…”

Section: Cottrell Atmosphere Formationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modeling of Bake Hardening Kinetics and Carbon Redistribution in Dual‐Phase Steels

Shan

Soliman

Palkowski

et al. 2020

steel research int.

View full text Add to dashboard Cite

show abstract

“…The smaller k values for the decarburized iron, ball milled iron, and IF iron measurements, as compared to the mild steel value, illustrate the lesser constraining effect produced at grain boundaries by reduction in the carbon content. A recent report has been made on different aspects of carbon segregation at individual dislocations and relating to pile-ups at grain boundaries [16]. Such segregation applies also for high manganese (fcc) austenitic steels [17].…”

Section: Constrained Polycrystalline Plasticitymentioning

confidence: 99%

Crystal Engineering for Mechanical Strength at Nano-Scale Dimensions

Armstrong¹

2017

Crystals

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Abstract:The mechanical strengths of nano-scale individual crystal or nanopolycrystalline metals, and other dimensionally-related materials are increased by an order of magnitude or more as compared to those values measured at conventional crystal or polycrystal grain dimensions. An explanation for the result is attributed to the constraint provided at the surface of the crystals or, more importantly, at interfacial boundaries within or between crystals. The effect is most often described in terms either of two size dependencies: an inverse dependence on crystal size because of single dislocation behavior or, within a polycrystalline material, in terms of a reciprocal square root of grain size dependence, designated as a Hall-Petch relationship for the researchers first pointing to the effect for steel and who provided an enduring dislocation pile-up interpretation for the relationship. The current report provides an updated description of such strength properties for iron and steel materials, and describes applications of the relationship to a wider range of materials, including non-ferrous metals, nano-twinned, polyphase, and composite materials. At limiting small nm grain sizes, there is a generally minor strength reversal that is accompanied by an additional order-of-magnitude elevation of an increased strength dependence on deformation rate, thus giving an important emphasis to the strain rate sensitivity property of materials at nano-scale dimensions.

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“…The problem of interaction between dislocations and interstitial atoms as carbon or hydrogen is of increasing interest with respect to a detailed understanding of Cottrell clouds [1], and their kinetics [2]. To calculate the main physical quantity, i.e., the interaction energy between the stress field of a distinct dislocation and of the interstitial atoms, one can use the superposition of the elastic fields generated by the dislocation and by eigenstrains due to deposition of the atoms in interstitial positions.…”

Section: Introductionmentioning

confidence: 99%

Elastic stress–strain analysis of an infinite cylindrical inclusion with eigenstrain

2017

Self Cite

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The ensemble of interstitial atoms as C attracted to a dislocation is well established as "Cottrell cloud" phenomenon. The deposition of the interstitial atoms in octahedral or tetrahedral positions in a bcc lattice may yield a remarkable internal stress state according to their anisotropic misfit eigenstrains. The stress fields of the atoms may then lead to a significant change in the stress field, e.g., around a dislocation. In such a case, the interstitial atoms are situated near the dislocation core in cylindrical volume elements along the dislocation line. As the occupancy of the sites in each cylinder by interstitial atoms can be considered as constant, also the eigenstrain state tensor is constant in the cylinder. A complete set of analytical expressions for the eigenstress state inside and outside of the cylinder is presented. The resultant stress field is then given by superposition, which allows also the determination of the interaction energy between the Cottrell cloud and the dislocation.

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Kinetics of interstitial segregation in Cottrell atmospheres and grain boundaries

Cited by 21 publications

References 34 publications

Modeling of Bake Hardening Kinetics and Carbon Redistribution in Dual‐Phase Steels

Modeling of Bake Hardening Kinetics and Carbon Redistribution in Dual‐Phase Steels

Crystal Engineering for Mechanical Strength at Nano-Scale Dimensions

Elastic stress–strain analysis of an infinite cylindrical inclusion with eigenstrain

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