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We investigate non-Hermitian degeneracies, also known as exceptional points, in continous elastic media, and their potential application to the detection of mass and stiffness perturbations. Degenerate states are induced by enforcing parity-time symmetry through tailored balanced gain and loss, introduced in the form of complex stiffnesses and may be implemented through piezoelectric transducers. Breaking of this symmetry caused by external perturbations leads to a splitting of the eigenvalues, which is explored as a sentitive approach to detection of such perturbations. Numerical simulations on one-dimensional waveguides illustrate the presence of several exceptional points in their vibrational spectrum, and conceptually demonstrate their sensitivity to point mass inclusions. Second order exceptional points are shown to exhibit a frequency shift in the spectrum with a square root dependence on the perturbed mass, which is confirmed by a perturbation approach and by frequency response predictions. Elastic domains supporting guided waves are then investigated, where exceptional points are formed by the hybridization of Lamb wave modes. After illustrating a similar sensitivity to point mass inclusions, we also show how these concepts can be applied to surface wave modes for sensing crack-type defects. The presented results describe fundamental vibrational properties of PT-symmetric elastic media supporting exceptional points, whose sensitivity to perturbations goes beyond the linear dependency commonly encountered in Hermitian systems. The findings are thus promising for applications involving sensing of perturbations such as added masses, stiffness discontinuities and surface cracks.