Effect of Hydrogen on Dislocation Nucleation and Motion: Nanoindentation Experiment and Discrete Dislocation Dynamics Simulation

Abstract

The hydrogen effect on the nucleation and motion of dislocations in single-crystal bcc Fe with (110) surface was investigated by both nanoindentation experiments and discrete dislocation dynamics (DDD) simulation. The results of nanoindentation experiments showed that the pop-in load decreased evidently for the electrochemical hydrogen charging specimen, indicating that the dislocation nucleation strength might be reduced by hydrogen. In addition, the decrease of hardness due to hydrogen charging was also captured, implying that the dislocation motion might be promoted by hydrogen. By incorporating the effect of hydrogen on dislocation core energy, a DDD model was specifically proposed to investigate the influence of hydrogen on dislocation nucleation and motion. The results of DDD simulation revealed that under the effect of hydrogen, the dislocation nucleation strength is decreased and the motion of dislocation is promoted.

Appendix A

The interaction between two dislocation segments with different core widths a and b is derived here. In the non-singular dislocation theory, the distribution of the Burgers vector is isotropic, which is formulated as [61]:

$$\begin{aligned} {\varvec{b}}=\int {{\varvec{g}}\left( {{\varvec{x}}} \right) } \mathrm{d^{3}}{\varvec{x}} \end{aligned}$$

(A1)

where ${\varvec{g}}\left( {{\varvec{x}}} \right) ={\varvec{b}}\tilde{{w}}_{a} \left( {{\varvec{x}}} \right) ={\varvec{b}}\tilde{{w}}_{a} \left( r \right)$ is the Burgers vector density function, and $\tilde{{w}}\left( r \right)$ is the Burgers vector distribution function within the dislocation core width a. $w_{a} \left( x \right)$ is defined as the convolution of $\tilde{{w}}_{a} \left( {{\varvec{x}}} \right)$ with itself. Under the assumption that $R_{a} \left( {{\varvec{x}}} \right) =R\left( {{\varvec{x}}} \right) *w_{a} \left( {{\varvec{x}}} \right) =\sqrt{R\left( {{\varvec{x}}} \right) ^{2}+a^{2}}$, $w_{a} \left( {{\varvec{x}}} \right)$ can be derived as:

$$\begin{aligned} w_{a} \left( {{\varvec{x}}} \right) =w_{a} \left( r \right) =\frac{15}{8\pi a^{3}\left[ {\left( {r/a} \right) ^{2}+1} \right] ^{7/2}} \end{aligned}$$

(A2)

where $R\left( {{\varvec{x}}} \right) =\sqrt{x^{2}+y^{2}+z^{2}}$. For a dislocation loop with the isotropic distribution $g\left( {{\varvec{x}}} \right)$, its stress fields can be expressed as:

$$\begin{aligned} \mathbf {\tilde{{\sigma }}}_{\alpha \beta } \left( {{\varvec{x}}} \right) =\oint _C {{\varvec{A}}}_{\alpha \beta ijklm} \partial _{i} \partial _{j} \partial _{k} \left[ {R\left( {{\varvec{x}}-{\varvec{x}}''} \right) g_{m} \left( {{\varvec{x}}''{ \mathbf{-}}{\varvec{x}}'} \right) \mathrm{d}^{3}{\varvec{x}}''} \right] \mathrm{d}{\varvec{x}}_{l}^{'} \end{aligned}$$

(A3)

where ${\varvec{A}}_{\alpha \beta ijklm}$ is the elastic tensor. For a dislocation with Burgers vector distribution $\tilde{{w}}\left( r \right)$, the stress acting on another dislocation with the same Burgers vector distribution is:

$$\begin{aligned} \mathbf {\sigma }_{\alpha \beta } \left( {{\varvec{x}}} \right)= & {} \mathbf {\tilde{{\sigma }}}_{\alpha \beta } \left( {{\varvec{x}}} \right) *\tilde{{w}}\left( {{\varvec{x}}} \right) \nonumber \\= & {} \frac{\mu }{8\pi }\oint _C {\partial _{i} \partial _{p} } \partial _{p} R_{a} \left[ {{\varvec{b}}_{m} \mathbf {\varepsilon }_{im\alpha } \mathrm{d}{\varvec{x}}_{\beta }^{'} +{\varvec{b}}_{m} \mathbf {\varepsilon }_{im\beta } \mathrm{d}{\varvec{x}}_{\alpha }^{'} } \right] \nonumber \\&+\frac{\mu }{4\pi \left( {1-\nu } \right) }\oint _C {{\varvec{b}}_{m} \mathbf {\varepsilon }_{imk} \left( {\partial _{i} \partial _{\alpha } \partial _{\beta } R_{a} -\delta _{\alpha \beta } \partial _{i} \partial _{p} \partial _{p} R_{a} } \right) \mathrm{d}{\varvec{x}}_{k}^{'} } \end{aligned}$$

(A4)

Since $R_{a} =R\left( {{\varvec{x}}} \right) *\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{a} \left( {{\varvec{x}}} \right)$ is obtained by the second convolution of $R_{a}$ and $\tilde{{w}}\left( {{\varvec{x}}} \right)$, the stress field $\mathbf {\sigma }_{\alpha \beta } \left( {{\varvec{x}}} \right)$ can be obtained analytically and used to calculate the interaction stress between two dislocations with the same core width a. However, because the convolution $R_{ab} =R\left( {{\varvec{x}}} \right) *w{ }_{a}\left( {\mathbf {x}} \right) *w_{b} \left( {{\varvec{x}}} \right)$ does not have an analytical expression, the interaction stress between two dislocations with different core widths a and b cannot be obtained analytically. Therefore, how to obtain the interaction force between two dislocation segments with different core widths a and b is transformed into a mathematical problem to determine the convolution of $\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right)$.

According to the convolution theorem, the Fourier transform of a convolution of two functions is the product of their Fourier transforms:

$$\begin{aligned} F_{\omega } \left( {\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right) } \right)= & {} F_{\omega } \left( {\tilde{{w}}{ }_{a}\left( r \right) } \right) F_{\omega } \left( {\tilde{{w}}_{b} \left( r \right) } \right) =\sqrt{F_{\omega } \left( {w{ }_{a}\left( r \right) } \right) F_{\omega } \left( {w_{b} \left( r \right) } \right) } \nonumber \\= & {} \sqrt{\frac{\left( {a\omega } \right) ^{2}K_{2} \left( {a\omega } \right) }{2}\frac{\left( {b\omega } \right) ^{2}K_{2} \left( {b\omega } \right) }{2}} \end{aligned}$$

(A5)

where $K_{2} \left( {a\omega } \right)$ and $K_{2} \left( {b\omega } \right)$ are the Bessel functions of the second kind. The numerical solution of $\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right)$ can be obtained by performing the inverse Fourier transform in Eq. (A5). However, in the DDD simulation, it is intractable to compute the numerical solution $\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right)$ for every two dislocation segments in every time step, and thus an approximate analytical solution $w_{c} \left( {{\varvec{x}}} \right) \approx \tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right)$ is adopted here. In other words, the convolution of $\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right)$ is replaced by the convolution of $\tilde{{w}}{ }_{c}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{c} \left( {{\varvec{x}}} \right)$. This means the interaction force between two dislocation segments with core widths a and b is replaced by the interaction force between two dislocation segments with the same core width c. The value of c is assumed to be the function $c=f\left( {a,b} \right)$, which is fitted with the numerical solution of $\tilde{{w}}{ }_{a}\left( {{\varvec{x}}} \right) *\tilde{{w}}_{b} \left( {{\varvec{x}}} \right)$ in the range $0.5\le \gamma \le 0.99$ with $\gamma \text{= }\min \left( {a,b} \right) /\max \left( {a,b} \right)$.

Fig. 6

Numerical solution of $\tilde{{w}}{ }_{a}\left( r \right) *\tilde{{w}}_{b} \left( r \right)$ and approximate solution $w_{c} \left( r \right)$

The numerical solution of $\tilde{{w}}{ }_{a}\left( r \right) *\tilde{{w}}_{b} \left( r \right)$ and the approximate analytical solution $w_{c} \left( r \right)$ with $\gamma =0.5$ are compared in Fig. 6, which shows that $w_{c} \left( r \right)$ is in good agreement with the numerical solution of $\tilde{{w}}{ }_{a}\left( r \right) *\tilde{{w}}_{b} \left( r \right)$. According to the method above, the function $c=f\left( {a,b} \right)$ can be obtained as:

$$\begin{aligned} c=\max \left( {a,b} \right) \left( {P_{1} \gamma ^{2}+P_{2} \gamma +P_{3} } \right) \end{aligned}$$

(A6)

where $P_{1} =0.1462$, $P_{2} =0.2191$ and $P_{3} =0.6352$ are the fitting parameters. After c is determined, the interaction stress between two dislocations with different core widths a and b can be calculated by Eq. (A4) with $R_{a}$ replaced by $R_{c}$.