An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non‐linearities

Abstract

SUMMARYAn incremental formulation to model geometric non-linearities in piezoelectric solids is presented. First, Lagrangian and Eulerian measures for the electric field and the electric displacement are discussed together with traditional mechanical measures. Using the Lagrangian energetic conjugated measures, Hamilton's principle is used to derive a consistent variational incremental description for the finite movement of a piezoelectric body. Those equations are approximated using the finite element method, and the resulting equations are solved using the Newton-Raphson method and arc-length constraints. Examples with non-linear behaviour such as stress stiffening, large rotations and limit points are presented.

A beam element for piezoelectric planar frames under small strains but large rotations using the total Lagrangian description

“…Here 's and 's refer to the components of the second Piola-Kirchhoff stress tensor and electric displacement vector; ϵ's and ã's refer to the components of the Green strain tensor; and , and are the elastic, piezoelectric and dielectric coefficients. It is timely to say that the adopted measure for the electric displacement and electric field vectors, like the second Piola-Kirchhoff stress and Green strain tensors, refer to the initial configuration and are unaffected by rigid-body motion (Cardoso and Fonseca, 2004). This latter feature justifies the use of the relations (18), identical to those of the corresponding linear theory, as explained in Bathe (1996).…”

“…The principle of virtual displacements for the piezoelectric beam, shown schematically in Figure 3, takes the form (Cardoso and Fonseca, 2004):…”

“…Smart structures has been used in aerospace, hydrospace, nuclear, defense, automotive and civil structural applications. A vast literature is already available mainly focused on the linear analysis of beams, plates and shells, whereas less attention has been given to the more sophisticated nonlinear analysis (Cardoso and Fonseca, 2004;Chróścielewski et al, 2019). However, consideration of geometric nonlinearities can be crucial because external excitations can cause large rotations of smart structures, still in the range of small strains, due to their inherent flexibility (Zhang et al, 2015).…”

“…Nonlinear models of piezoelectric beams subjected to large rotations with some restriction of magnitude (Mukherjee and Chaudhuri, 2002;2005;Kapuria and Alam, 2004) are more common in the literature than those with rotations free of any restrictions (Cardoso and Fonseca, 2004). In this paper, the work of Lucena Neto et al (2019) is modified to account for piezoelectric patches attached to the beam surface, assuming electric potential with linear variation through each patch thickness (Marinković et al, 2009).…”

A geometrically nonlinear structural formulation for analysis of beams with a new set of generalized displacements considering piezoelectric effects

“…Pushing forward the geometric nonlinearities in structures possessing piezoelectric layers, Cardoso and Fonseca 18 propose an incremental Lagrangian formulation for piezoelectric bodies, and assess the nonlinear geometric electro‐mechanical coupling and its implications in the behavior of a frame. Olympio et al 19 present a study of composite beams with embedded piezoelectric layers for application in morphing aerostructures, investigating the sensitivity of the element to the stacking sequence.…”