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This paper deals with the asymptotic behavior of the first Bloch eigenvaluein a heterogeneous medium with a high contrast $\ep Y$-periodic conductivity.When the conductivity is bounded in $L^1$ and the constant of thePoincar\'e-Wirtinger weighted by the conductivity is very small with respect to$\ep^{-2}$, the first Bloch eigenvalue converges as $\ep\to 0$ to a limit whichpreserves the second-order expansion with respect to the Bloch parameter. Indimension two the expansion of the limit can be improved until the fourth-orderunder the same hypotheses. On the contrary, in dimension three a fibersreinforced medium combined with a $L^1$-unbounded conductivity leads us to adiscontinuity of the limit first Bloch eigenvalue as the Bloch parameter tendsto zero but remains not orthogonal to the direction of the fibers. Therefore,the high contrast conductivity of the microstructure induces an anomalouseffect, since for a given low-contrast conductivity the first Bloch eigenvalueis known to be analytic with respect to the Bloch parameter around zero.

First Bloch eigenvalue in high contrast media