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.

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)