Problem $4$: Discrete Pile$-$Up

Consider a pile$-$up of three screw dislocations in which the leading dislocation is fixed at $(0,0).$

The other dislocations have coordinates $(x_1,0)$ and $(x_2,0)$ which are to be determined by

applying the equilibrium equations presented in class $($as shown below$).$ Assume that the

externally applied stress is constant $($denoted by $$ and the frictional stress $k=0).$

${\displaystyle \underset{\{\begin{array}{c}\text{:}[i=1]\\ ij\end{array}}{\overset{n}{}}}\frac{A}{x_j-x_i}+{}_{\text{a}\text{p}\text{p}}(x_j)-k=0$

a$.)$ Write down the two equilibrium equations for the two free dislocations $($remember for

screw dislocations, $A=\frac{b}{2}).$ Use dimensionless stress units in which $=f.$

b$.)$ Solve these two coupled equations and show that the equilibrium positions are given by:

$x_1=\frac{b}{4f}(-3+\sqrt[\phantom{2}]{3})$ and $x_2=\frac{b}{4f}(-3-\sqrt[\phantom{2}]{3}).$