Plastic Multiplier for Perfectly Plastic Materials

Question

I'm trying to apply the associated flow rule to elastic-perfectly plastic material, i.e.,$$d\epsilon_{ij}^p=\lambda\dfrac{\partial f}{\partial \sigma_{ij}}$$, where $f$ is a yield function with no hardening variable, $\lambda$ is a plastic multiplier. I understand that $\lambda$ can be found through consistency condition ($df=0$), which gives $$\lambda=\dfrac{\dfrac{\partial f}{\partial \sigma_{ij}}C_{ijkl}d\epsilon_{kl}}{\dfrac{\partial f}{\partial \sigma_{ij}}C_{ijkl}\dfrac{\partial f}{\partial \sigma_{kl}}}$$ This seems confusing to me since after yielding is activated for perfectly plastic deformation $d\bf\sigma=0$ and $d\bf\epsilon^e=0$ when $d\bf\epsilon^p>0$. Therefore $d\epsilon_{kl}=d\epsilon^p_{kl}$, substituting this into $\lambda$'s expression renders $\lambda$ indeterminate. Maybe there's something I understand wrong, but if this is the case, how would the plastic deformation be predicted for such the material? Thanks!