Analysis, Dynamics and Applications Seminar

Large Deflections of Inextensible Beams and Plates

Motivated by piezoelectric energy harvesting applications, this talk focuses on the large deflections of cantilevered beams and plates. In the first half of the talk, we discuss a recent nonlinear evolutionary PDE model that derives the equations of motion for an inextensible beam via Hamilton's principle with Lagrange multipliers. The theoretical results are centered around the existence, uniqueness, and decay of strong solutions for the derived equations. The first is achieved through a spectral Galerkin procedure, but the identification of weak limits requires additional compactness, and thus higher order energy estimates must be obtained. Uniqueness follows from a novel decomposition of the dynamics for sufficiently smooth solutions. Following this, we move to a two dimensional rectangular domain to see how the cantilever's mixed boundary conditions impose challenges for the analysis of plate strong solutions, even in the entirely linear case. Lastly, in an analogous manner with the beam, we derive the equations of motion for an inextensible cantilevered plate and discuss various modeling hypotheses that can be employed during this process.