Just check your math. Correct derivation is

$$\begin{eqnarray} {f \over \epsilon_{11}} & = & {S_{11} \over \epsilon_{11}}=(2\mu+\lambda)+\lambda {\epsilon_{22}+\epsilon_{33} \over \epsilon_{11}} = \\ & = & (2\mu+\lambda)-{\lambda^2 \over \lambda + \mu} = {(2\mu + \lambda)(\lambda + \mu) - \lambda^2 \over \lambda + \mu}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu} \end{eqnarray}$$

Just check your math. Correct derivation is

$$\begin{eqnarray} {f \over \epsilon_{11}} & = & {S_{11} \over \epsilon_{11}}=(2\mu+\lambda)+\lambda {\epsilon_{22}+\epsilon_{33} \over \epsilon_{11}} = \\ & = & (2\mu+\lambda)-{\lambda^2 \over \lambda + \mu} = {(2\mu + \lambda)(\lambda + \mu) - \lambda^2 \over \lambda + \mu}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu} \end{eqnarray}$$

And this is precisely what Young modulus is defined to be (see e.g. here).