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Electromagnetic Metamaterials (MTMs) are artificial materials with novel electromagnetic properties not available in nature. MTMs typically consist of a homogeneous host material containing appropriately configured embedded compact inclusions. MTMs have the potential to enable significant improvement on performance of low-profile (i.e. microstrip) and conformal antennas, including reduction of antenna size and antenna coupling. One key limitation for MTM implementation is the possibility of dielectric breakdown from electrical stresses such as high ambient electric fields arising from the transmitter itself, lightning and atmospheric charges, precipitation static (p-static), and electrostatic discharge (ESD). In this paper we investigate dielectric breakdown in the mean- field approximation. Dielectric breakdown is deemed to occur if the electric potential across an insulator exceeds a certain critical value, causing the insulator to become conductive and leading to failure of the insulator. Embedded conductive structures, such as those comprising MTMs, cause electric field enhancement near the metallic inclusion lowering the electric strength. We calculate the detailed electric field distribution within the MTM and compare the peak values to a critical breakdown field. Using the mean-field theory, we replace each inclusion by an equivalent dipole. The effect of remaining dipoles is taken into account by an effective field. The polarizability is determined by a self-consistent solution for the effective field. We determine the detailed field in the vicinity of the inclusion by summing the effective field and the local field due to the inclusion. We find that the presence of inclusions reduces the electric strength of the MTM in comparison to the electric strength of the pure host material. For a dilute MTM, the reduction depends mainly on the geometry of the inclusion. The reduction depends weakly on the concentration of the inclusions and is independent of the perm- - ittivity of the host material. The reduction may be significant even for very dilute MTMs and needs to be taken into account in practical applications. This work is an initial stage in CERDEC's effort to evaluate limitations to MTM parameters given the presence of intrinsic or environmental electromagnetic fields.