  This paper aims at mathematically modeling a new multi-physics conductivity
imaging system incorporating mechanical vibrations simultaneously applied to an
imaging object together with current injections. We perturb the internal
conductivity distribution by applying time-harmonic mechanical vibrations on
the boundary. This enhances the effects of any conductivity discontinuity on
the induced internal current density distribution. Unlike other conductivity
contrast enhancing frameworks, it does not require a prior knowledge of a
reference data. In this paper, we provide a mathematical framework for this
novel imaging modality. As an application of the vibration-assisted impedance
imaging framework, we propose a new breast image reconstruction method in
electrical impedance tomography (EIT). As its another application, we
investigate a conductivity anomaly detection problem and provide an efficient
location search algorithm. We show both analytically and numerically that the
applied mechanical vibration increases the data sensitivity to the conductivity
contrast and enhances the quality of reconstructed images and anomaly detection
results. For numerous applications in impedance imaging, the proposed
multi-physics method opens a new difference imaging area called the
vibration-difference imaging, which can augment the time-difference and also
frequency-difference imaging methods for sensitivity improvements.
