Inverse load identification in vibrating nanoplates

In this paper, we consider the uniqueness issue for the inverse problem of load identification in a nanoplate by dynamic measurements. Working in the framework of the strain gradient linear elasticity theory, we first deduce a Kirchhoff-Love nanoplate model, and we analyze the well-posedness of the equilibrium problem, clarifying the correct Neumann conditions on curved portions of the boundary. Our uniqueness result states that, given a transverse dynamic load n-ary sumation m=1Mgm(t)fm(x)$$ {\sum}_{m equal to 1} circumflex M{g}_m(t){f}_m(x) $$, where M >= 1$$ M\ge 1 $$ and {gm(t)}m=1M$$ {\left\{{g}_m(t)\right\}}_{m equal to 1} circumflex M $$ are known time-dependent functions, if the transverse displacement of the nanoplate is known in an open subset of its domain for any interval of time, then the spatial components {fm(x)}m=1M$$ {\left\{{f}_m(x)\right\}}_{m equal to 1} circumflex M $$ can be determined uniquely from the data. The proof is based on the spherical means method. The uniqueness result suggests a reconstruction technique to approximate the loads, as confirmed by a series of numerical simulations performed on a rectangular clamped nanoplate.