Microstructure evolution of Al–Si alloys under shear loading

Abstract

In this paper, an attempt is made to predict the microstructure evolution in Al–Si alloy two-dimensional (2D) system under shear loading conditions. The importance of damage accumulation events in delamination wear is studied. The conducted molecular dynamics (MD) simulations are based on the Modified Embedded Atom Method (MEAM). As a result a cohesive zone type of model relating the shear stress and the shear displacement has been suggested.

Appendix A

The forces acting on the atoms are readily determined

$$ f_{k} = - \frac{{\partial E}} {{\partial \ifmmode\expandafter\vec\else\expandafter\vec\fi{r}_{k} }} = - {\sum\limits_{j \ne k} {{\phi }\ifmmode{'}\else$'$\fi{\left( {r_{{kj}} } \right)}\frac{{\ifmmode\expandafter\vec\else\expandafter\vec\fi{r}_{{kj}} }} {{r_{{kj}} }}} } - {\sum\limits_{i = 1}^{N_{m} } {A_{i} F_{c} {\left( {\log \hat{\rho }_{i} + 1} \right)}\frac{{\partial \hat{\rho }_{i} }} {{\partial \ifmmode\expandafter\vec\else\expandafter\vec\fi{r}_{k} }}} } $$

(A1)

The second term in Eq. A1 can be reorganized

$$ \begin{aligned}{} & {\sum\limits_{i = 1}^{N_{m} } {{\left( {\log \hat{\rho }_{i} + 1} \right)}\frac{{\partial \hat{\rho }_{i} }} {{\partial \ifmmode\expandafter\vec\else\expandafter\vec\fi{r}_{k} }} = {\sum\limits_{i \ne j} {{\left( {\log \hat{\rho }_{i} + 1} \right)}\hat{\rho }^{\prime}_{i} {\left( {r_{{ij}} } \right)}{\left( {\delta _{{ik}} - \delta _{{jk}} } \right)}\ifmmode\expandafter\vec\else\expandafter\vec\fi{r}_{{ij}} } }} } \\ & \quad \quad\quad\quad \quad = {\sum\limits_{j \ne k} {\hat{\rho }^{\prime}_{i} {\left( {r_{{ik}} } \right)}{\left( {\log \rho _{k} + \log \rho _{j} + 2} \right)}\ifmmode\expandafter\vec\else\expandafter\vec\fi{r}_{{kj}} } } \\ \end{aligned} $$

(A2)

Using the definition of the relative electron density, $\hat{\rho }_{i}$ (Eq. 3), $\hat{\rho }^{\prime }_{i}$ is obtained

$$ \hat{\rho }^{\prime}_{i} = \frac{{2e^{{ - {\sum\limits_{s = 1}^3 {t^{{(s)}} {\left( {\frac{{\rho ^{{(s)}}_{i} }} {{\rho ^{{(s)}}_{0} }}} \right)}^{2} } }}} }} {{{\left( {1 + e^{{ - {\sum\limits_{s = 1}^3 {t^{{(s)}} {\left( {\frac{{\rho ^{{(s)}}_{i} }} {{\rho ^{{(s)}}_{0} }}} \right)}^{2} } }}} } \right)}^{2} }}{\left( { - {\sum\limits_{s = 1}^3 {t^{{(s)}} \frac{d} {{dr_{{ij}} }}{\left( {\frac{{\rho ^{{(s)}}_{i} }} {{\rho ^{{(s)}}_{0} }}} \right)}^{2} } }} \right)}. $$

(A3)

The necessary derivatives of the partial electron densities are given as

$$ \frac{d} {{dr_{{ij}} }}{\left( {\frac{{\rho ^{{(s)}}_{i} }} {{\rho ^{{(s)}}_{0} }}} \right)}^{2} = \frac{{{\left( {\rho ^{{(s)}}_{0} } \right)}^{2} \frac{{d{\left( {\rho ^{{(s)}}_{i} } \right)}^{2} }} {{dr_{{ij}} }} - {\left( {\rho ^{{(s)}}_{i} } \right)}^{2} \frac{{d{\left( {\rho ^{{(s)}}_{0} } \right)}^{2} }} {{dr_{{ij}} }}}} {{{\left( {\rho ^{{(s)}}_{0} } \right)}^{4} }}, $$

(A4)

$$ \begin{aligned}{} & \frac{{d{\left( {\rho ^{{(s)}}_{i} } \right)}^{2} }} {{dr_{{ij}} }} = 2{\sum\limits_{j,k \ne i} {{\left( {\rho ^{{a(s)}}_{i} (r_{{ij}} )} \right)}^{\prime} \rho ^{{a(s)}}_{j} (r_{{ik}} )L^{{(s)}} {\left( {\cos \theta _{{jik}} } \right)}} } \\ & + {\sum\limits_{j,k \ne i} {\rho ^{{a(s)}}_{i} (r_{{ij}} )\rho ^{{a(s)}}_{j} (r_{{ik}} ){\left( {L^{{(s)}} {\left( {\cos \theta _{{jik}} } \right)}} \right)}^{\prime} } } \\ \end{aligned} $$

(A5)

Finally the differentiation of the electron densities yields

$$ {\left( {\rho ^{{a(s)}}_{i} (r_{{ij}} )} \right)}^{\prime} = S_{{ij}} f_{c} {\left( {r_{{ij}} } \right)}f^{0}_{i} e^{{ - \beta ^{s}_{i} {\left( {\frac{{r_{{ij}} }} {{r_{0} }} - 1} \right)}}} {\left( { - \frac{{\beta ^{s}_{i} }} {{r_{0} }}} \right)}. $$

(A6)

The first term in Eq. A1 represents the forces due to the pair interactions. The magnitude of $\phi^{\prime} (r_{kj})$ can be obtain from Eq. A3

$$ {\phi }\ifmmode{'}\else$'$\fi{\left( {r_{{kj}} } \right)} = \frac{2} {Z}{\left( {\frac{{dE^{u} (r_{{ij}} )}} {{dr_{{ij}} }} - \frac{{dF_{i} {\left( {\ifmmode\expandafter\bar\else\expandafter\=\fi{\rho }_{i} {\left( {r_{{ij}} } \right)}} \right)}}} {{dr_{{ij}} }}} \right)}, $$

(A7)

where

$$ \frac{{dE^{u} (r_{{ij}} )}} {{dr_{{ij}} }} = E_{c} \frac{{\alpha ^{2} }} {{r_{0} }}{\left( {\frac{{r_{{ij}} }} {{r_{0} }} - 1} \right)}e^{{ - \alpha {\left( {\frac{{r_{{ij}} }} {{r_{0} }} - 1} \right)}}} . $$

(A8)