The conductance and the transmittance phase shifts of a two-terminal Aharonov-Bohm (AB) ring are analyzed in the presence of mechanical displacements due to coupling to an external can- tilever. We show that phase rigidity is broken, even in the linear response regime, by means of inelastic scattering due to phonons. Our device provides a way of observing continuous variation of the transmission phase through a two-terminal nano-electro-mechanical system (NEMS). We also propose measurements of

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... hase shifts as a way to determine the strength of the electron-phonon coupling in NEMS.