Question $2$: For triaxial loading conditions and for the case of a non$-$associative flow rule, derive the expression for the loading index $($:$L$:$)$ in terms of strain increments. Your final result should be:

Start by showing that $d{}_p=d{}_p^e+L\frac{\text{d}\text{e}\text{l}\text{g}}{\text{d}\text{e}\text{l}\text{p}},d{}_q=d{}_q^e+L\frac{\text{d}\text{e}\text{l}\text{g}}{\text{d}\text{e}\text{l}\text{q}}$ and hence that $d{p}^{\text{'}}=K^{\text{'}}(d{}_p-L\frac{\text{d}\text{e}\text{l}\text{g}}{del{p}^{\text{'}}}),dq=3G(d{}_q-L\frac{\text{d}\text{e}\text{l}\text{g}}{\text{d}\text{e}\text{l}\text{q}})$ and then apply the consistency condition, $df=0,$ to solve for $L.$