  Initially stressed plates are widely used in modern fabrication techniques,
such as additive manufacturing and UV lithography, for their tunable morphology
by application of external stimuli. In this work, we propose a formal
asymptotic derivation of the F\"{o}ppl-von K\'{a}rm\'{a}n equations for an
elastic plate with initial stresses, using the constitutive theory of nonlinear
elastic solids with initial stresses under the assumptions of incompressibility
and material isotropy. Compared to existing works, our approach allows to
determine the morphological transitions of the elastic plate without
prescribing the underlying target metric of the unstressed state of the elastic
body. We explicitly solve the derived FvK equations in some physical problems
of engineering interest, discussing how the initial stress distribution drives
the emergence of spontaneous curvatures within the deformed plate. The proposed
mathematical framework can be used to tailor shape on demand, with applications
in several engineering fields ranging from soft robotics to 4D printing.
