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A Physics-Based Crystallographic Modeling Framework for Describing the Thermal Creep Behavior of Fe-Cr Alloys

Abstract

In this work, a physics-based thermal creep model is developed based on the understanding of the microstructure in Fe-Cr alloys. This model is associated with a transition state theory-based framework that considers the distribution of internal stresses at sub-material point level. The thermally activated dislocation glide and climb mechanisms are coupled in the obstacle-bypass processes for both dislocation and precipitate-type barriers. A kinetic law is proposed to track the dislocation densities evolution in the subgrain interior and in the cell wall. The predicted results show that this model, embedded in the visco-plastic self-consistent framework, captures well the creep behaviors for primary and steady-state stages under various loading conditions. The roles of the mechanisms involved are also discussed.

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Acknowledgment

This work was funded by the US Department of Energy’s Nuclear Energy Advanced Modeling and Simulation (NEAMS). Special thanks go to Professor G.P. Potirniche for providing us with the detailed experimental data.

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Correspondence to W. Wen.

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Manuscript submitted October 31, 2016.

Appendix

Appendix

Previous studies[18,46,52,53,54] show the climb velocity can be expressed as:

$$ v_{c}^{s} = \frac{{I_{v}^{s} \varOmega }}{b} $$
(A1)

To calculate the vacancy current \( I_{v}^{s} \), one needs to analyze the stress and vacancy concentration status around the climbing edge dislocation. A cylindrical control volume around the dislocation line with the radius r is defined. The zone with \( r \le r_{d} \) is considered as the dislocation core region. Therefore the chemical force (Osmotic force) applied on the unit length of edge dislocation segment can be obtained as[46,79]:

$$ f_{\text{os}}^{s} = - \frac{kTb}{\varOmega }\ln \left( {\frac{{C_{v}^{s} (r_{d} )}}{{C_{v}^{0} }}} \right), $$
(A2)

where \( C^{v} ({\text{r}}_{d} ) \) represents the vacancy concentration at \( r = r_{d} \). \( C_{v}^{0} \) is the equilibrium vacancy concentration at a given temperature.

Meanwhile, climb is also affected by the climb component of Peach-Koehler force. The full Peach-Koehler force is defined as \( {\mathbf{f}} = \left( {{\varvec{\upsigma}} \cdot {\mathbf{b}}^{s} } \right) \times {\mathbf{t}}^{s} \) where \( {\mathbf{t}}^{s} \) is the normalized tangent to the dislocation line.[56,57] The climb component of \( {\mathbf{f}} \) for the edge dislocation can be expressed as[58,59,60]:

$$ f_{c}^{s} = {\mathbf{f}}^{{\mathbf{s}}} \cdot {\mathbf{n}}^{s} = \left[ {\left( {{\varvec{\upsigma}} \cdot {\mathbf{b}}^{s} } \right) \times {\mathbf{t}}^{s} } \right] \cdot {\mathbf{n}}^{s} = - \left| {{\mathbf{b}}^{s} } \right|{\varvec{\upsigma}}:\left( {{\mathbf{b}}^{s} \otimes {\mathbf{b}}^{s} } \right) $$
(A3)

When the dislocation is locally in equilibrium state, the total force \( f^{s} = f_{\text{os}}^{s} + f_{c}^{s} \) should be equal to 0. Therefore we get from Eqs. [A2] and [A3]:

$$ C_{v}^{s} (r_{d} ) = C_{{v,{\text{eq}}}}^{s} = C_{v}^{0} \exp \left( {\frac{{ - f_{c}^{s} \varOmega }}{kTb}} \right) $$
(A4)

Notice that the vacancy concentration in the material matrix is assumed to be equal to the equilibrium concentration, \( C_{v}^{{}} (r \ge r_{\infty } ) = C_{v}^{\infty } = C_{v}^{0} \), where \( r_{\infty } \) denotes the radius of the outer boundary for the control volume. Therefore, a vacancy concentration gradient along the radius appears in the control volume which leads to a diffusive flow of vacancies. The dislocation needs to absorb or emit vacancies (climb) to retain the local equilibrium status.

At steady-state the divergence of vacancy diffusion flux \( J \) is null in the absence of defect creation. The associated Laplace equation in the cylindrical coordinate system is:

$$ \nabla^{2} C_{v}^{s} = \frac{1}{r}\frac{\partial }{\partial r}r\frac{{\partial C_{v}^{s} }}{\partial r} = 0 $$
(A5)

with the inner and outer boundary conditions:

$$ \begin{aligned} C_{v}^{s} ({\text{r = r}}_{\infty } ) = C_{v}^{\infty } = C_{v}^{0} \hfill \\ C^{v} ({\text{r = r}}_{d} ) = C_{{v,{\text{eq}}}}^{s} \hfill \\ \end{aligned} $$
(A6)

By solving Eq. [A5], we obtain:

$$ C_{v}^{s} (r) = C_{{v,{\text{eq}}}}^{s} + (C_{v}^{\infty } - C_{{v,{\text{eq}}}}^{s} )\frac{{\ln (r/r_{\infty } )}}{{\ln (r_{\infty } /r_{d} )}} $$
(A7)

Therefore, the net current absorbed or emitted by unit length of dislocation segment is given by:

$$ I_{v}^{s} = 2\pi r \cdot J = 2\pi r\frac{{D_{v} }}{\varOmega }\frac{{\partial C_{v}^{s} (r)}}{\partial r} = \frac{{2\pi D_{v} \left[ {C_{v}^{\infty } - C_{v}^{0} \exp \left( {\frac{{ - f_{c}^{s} \varOmega }}{kTb}} \right)} \right]}}{{\varOmega \ln (r_{\infty } /r_{d} )}}, $$
(A8)

where \( D_{v} \) is the vacancy diffusivity. Then the climb velocity can be expressed as:

$$ v_{c}^{s} = \frac{{I_{v}^{s} \varOmega }}{b} = \frac{{2\pi D_{v} \left[ {C_{v}^{\infty } - C_{v}^{0} \exp \left( {\frac{{ - f_{c}^{s} \varOmega }}{kTb}} \right)} \right]}}{{{\text{b}}\ln (r_{\infty } /r_{d} )}} .$$
(A9)

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Wen, W., Capolungo, L., Patra, A. et al. A Physics-Based Crystallographic Modeling Framework for Describing the Thermal Creep Behavior of Fe-Cr Alloys. Metall Mater Trans A 48, 2603–2617 (2017). https://doi.org/10.1007/s11661-017-4011-3

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Keywords

  • Slip System
  • Creep Rate
  • Creep Test
  • Edge Dislocation
  • Subgrain Boundary