Can the electric field be made to penetrate the bulk of a metal? When an external electric field $\textbf{E}_\textrm{ext}$ is externally applied to a metal, the free electrons move opposite to the direction of the field inside the metal and create an internal field $\textbf{E}_\textrm{int}$ opposite to $\textbf{E}_\textrm{ext}$ until they balance so that the net field is zero inside. 
When all the free electrons have moved to crowd one side of the metal the internal field cannot increase any longer, what will happen to if the external field is further increased? 
Will now the electric field start to penetrate the metal and will have a non-vanishing value inside?
 A: A metal has such a high conduction electron density $n∼ 10^{22} cm^{-3}$ that in a metal with dimensions on the order of centimeters you will not be able to accumulate all of them on one surface with any practically achievable outside electric field. Maximally applicable outside electric fields are on the order of $10^7 V/cm$ because of the onset of appreciable field electron emission (Fowler-Nordheim tunneling). See https://en.wikipedia.org/wiki/Field_electron_emission . This means that the maximum surface charge density $\eta$ of electrons you can achieve is on the order of $$\eta=E\epsilon_0=10^7 Vcm^{-1} ·8.85·10^{-14} Fcm^{-1}=8.85·10^{-7} Ccm^{-2}=5.5·10^{12} qcm^{-2}$$ where the electron charge is $q=1.6· 10^{-19}C$. Thus, practically, you can only accumulate about $5.5·10^{12}cm^{-2}$ electrons on the surface, which means that you can accumulate all conduction electron on the surface only in an extremely thin metal sheet with thickness $d=5.5·10^{-10} cm$ which is on the order of an atomic radius. 
Thus you will practically never be able to achieve a penetration of a static electric field into the bulk of a metal. It will always be shielded.
A: Let place a metallic conductor into an external electrostatic field. The electrostatic field will act on all the charges of the conductor, as a result all negative charges will be displaced in the direction against the field. This current will continue until a certain charge distribution sets in, at witch the electric field at all point inside the conductor vanished. Now we can suppose that the external electric field is so intensive that all the conduction band electrons have streamed in one sides of the conductor. So now it remains a material with only non conducting electrons and uniformly positive charged(except in the external surface) since all the conductive electrons have left the inside of the metal concentrating on one side of the surface. In this new situation E inside the material is no more equal to zero, in fact there is a uniform positive charge inside it(gauss theorem). The metal bond is broken.
