A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis

Abstract

The aim of this paper was to obtain a new model for the bending-stretching of an anisotropic heterogeneous linearly piezoelectric cantilever rod when the electric potential is applied on the both ends. The process is assumed to be static, and the piezoelectric material is monoclinic of class 2. To derive the model, we start with the corresponding threedimensional problem, introduce a change of variable together with a scaling of the unknowns and then we use a passage to the limit procedure, based on arguments of asymptotic analysis taking the diameter of the cross-section as small parameter. Finally, we prove a result of strong convergence that justifies both the method and the one-dimensional model obtained. One of the most relevant features of this one-dimensional model is that the stretching is coupled with the electric potential, while the bendings are not.Mathematics Subject Classification (2010). Primary 74K10; Secondary 41A60 · 35C20 · 35Q60.

Mathematical and numerical analysis of an obstacle problem for elastic piezoelectric beams

“…As a result, it may come into contact with a conductive deformable foundation. Based on our previous works [17, 18, 31, 32], we associate the following mathematical model with this physical setting.…”

“…See, in particular, Castillero et al [9], Ghergu et al [10] and Miara et al [11] for homogenization of piezoelectric plates and shells and Bellis and Imperiale [5], Collard and Miara [12], Sabu [13], Weller and Licht [14] and Figueiredo and Stadler [15] (and references therein) for formal asymptotic analysis to justify lower dimensional constitutive laws for piezoelectric plates, shells and rods. Recently, pioneering work for modelling of thin linearly piezoelectric beams including convergence results has been presented [16–18].…”

A Contact Model for Piezoelectric Beams

“…As a result, it may enter in contact with a conductive deformable foundation. Based on our previous works [9,13,16], we associate to this physical setting the following mathematical model.…”

“…M > 0 being a positive constant that depends on the characteristic length of the system. Equations (2.2)-(2.3) are the piezoelectric equations for elastic beams presented in [13], and (2.4)-(2.6) represent the boundary conditions, in which q 0 and q L denote the electric potentials applied on both ends of the beam.…”

“…Rodríguez-Arós was supported by the project "Modelización y simulación numérica de sólidos y fluidos en dominios con pequeñas dimensiones. Aplicaciones en estructuras, biomecánica y aguas someras", MTM2012-36452-C02-01 financed by the Spanish Ministry of Economía y Competitividad with the participation of FEDER of thin linearly isotropic piezoelectric beams was performed in [1,5,8,[13][14][15]17]. Models for elastic beams in contact with a foundation have been justified in [7,9,16], based on the ideas in [12].…”