Inverse load identification in vibrating nanoplates

Abstract

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 ∑ M m=1 g m (t)𝑓 m (x), where M ≥ 1 and {g m (t)} M m=1 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 {𝑓 m (x)} M m=1 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.

Size estimates for nanoplates

“…where M 1 is a positive constant. Within the kinematic framework of the Kirchhoff-Love theory in infinitesimal deformation, the statical equilibrium problem of the nanoplate loaded at the boundary and under vanishing body forces is described by the following Neumann boundary value problem [21]:…”

“…Here, we shall adopt the simplified strain gradient theory proposed by Lam et al [23] to model the mechanical behavior of the material in infinitesimal deformation. Under the kinematic assumptions of Kirchhoff-Love's plate theory, the statical equilibrium problem of the nanoplate loaded at the boundary and under vanishing body forces is described by the following Neumann boundary value problem [21]…”

A sixth-order elliptic equation for nanoplates. Neumann problem and Unique Continuation in the interior

“…Under the kinematic framework of the Kirchhoff-Love theory, and for infinitesimal deformation, the statical equilibrium problem of the nanoplate loaded at the boundary and under vanishing body forces is described by the following Neumann boundary value problem [21]:…”

“…The plate typology, although less common than nanobeams, has some inherent mechanical advantages that include robustness, which is a relevant feature for fabrication and functionalisation, and higher stiffness, which results in higher frequencies and small free vibration energy dissipation in both fluid and gaseous environments [42,7]. Albeit the detection of added mass is one of the most popular issues in applications [18,29,12], other notable inverse problems for nanoplates involve force or pressure sensing from dynamic data [21]. In addition, there has recently been growing interest in the development of diagnostic techniques for assessing the presence of defects in nanoplates, thus paving the way for the extension of methods hitherto designed for large-scale mechanical systems to the nanodimensional size as well [48].…”

“…In what follows we refer to the mechanical Kirchhoff-Love's nanoplate model proposed in [21] within the SSGT elasticity, in which the Neumann conditions are correctly derived for smooth boundary (see problem (3.2)-(3.5)). In Section 3 we propose an alternative, although equivalent, determination of these boundary conditions (Lemma 3.2) and we develop the variational formulation of the Neumann problem.…”