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The paper concerns a general variational derivation of all the state equations of a continuum medium with dissipative interfaces in the context of the Strong Discontinuities Approach. The kinematics is based on an enhanced enrichment displacement field, that satisfies a priori the boundary conditions and the continuity requirements. The variational formulation is established for a damaging material under general constitutive hypotheses. The numerical implementation of the strong discontinuity is developed within the Elements with Embedded Discontinuities method. Some considerations about the numerical implementation and the equilibrium condition at the interface are presented.