One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil-structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.