The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first-order system in velocity and stress. We demonstrate how to use this condition for the same equation written as a second-order system in displacement. This facilitates use in the context of numerical schemes based upon such a system, e.g. the finite-element method, the spectral-element method and some finite-difference methods. We illustrate the efficiency of this second-order perfectly matched layer based upon 2-D benchmarks with body and surface waves.