A formulation of the response of a system to strain and electric field perturbations in the pseudopotential-based density functional perturbation theory has been proposed by Hamann et al. [D. R. Hamann, X. Wu, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B 71, 035117 (2005)], using an elegant formalism based on the expression of density functional theory (DFT) total energy in reduced coordinates, the key quantity being the metric tensor. We extend this formulation to the projector augmented-wave approach. In this context, we express the full elastic tensor including the clamped-atom tensor, the atomic-relaxation contributions, and the response to electric field change (piezoelectric tensor and effective charges). With this, we are able to compute the elastic tensor for all materials within a fully analytical formulation. The comparison with finite difference calculations on simple systems shows excellent agreement. This formalism has been implemented in the plane-wave-based DFT ABINIT software package. We apply it to the computation of elastic properties of low-symmetry systems of geophysical interest.