Shock and Vibration

Volume 2018, Article ID 2054873, 21 pages

https://doi.org/10.1155/2018/2054873

## A Two-Step Hybrid Approach for Modeling the Nonlinear Dynamic Response of Piezoelectric Energy Harvesters

Correspondence should be addressed to Giuseppe Acciani; ti.abilop@inaicca.eppesuig

Received 30 August 2017; Revised 20 December 2017; Accepted 11 January 2018; Published 26 February 2018

Academic Editor: Paul Cahill

Copyright © 2018 Claudio Maruccio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An effective hybrid computational framework is described here in order to assess the nonlinear dynamic response of piezoelectric energy harvesting devices. The proposed strategy basically consists of two steps. First, fully coupled multiphysics finite element (FE) analyses are performed to evaluate the nonlinear static response of the device. An enhanced reduced-order model is then derived, where the global dynamic response is formulated in the state-space using lumped coefficients enriched with the information derived from the FE simulations. The electromechanical response of piezoelectric beams under forced vibrations is studied by means of the proposed approach, which is also validated by comparing numerical predictions with some experimental results. Such numerical and experimental investigations have been carried out with the main aim of studying the influence of material and geometrical parameters on the global nonlinear response. The advantage of the presented approach is that the overall computational and experimental efforts are significantly reduced while preserving a satisfactory accuracy in the assessment of the global behavior.

#### 1. Introduction

The possibility of using electronic devices that do not require periodic replacement of batteries is nowadays the major challenge in several engineering fields. In particular, vibration-based piezoelectric energy harvesting devices are emerging as a valid technological option to power miniaturized electronic sensors in civil structural health monitoring applications [1–3]. It can be easily understood, therefore, that the implementation of an efficient computational framework for the analysis and design of these devices is of paramount importance in order to foster the future large-scale applications of this technology.

In this perspective, the electromechanical response of piezoelectric devices can be assessed through different numerical and analytical techniques, for example, reduced-order models, finite element (FE) models, and circuit analogy methods [2, 4–9]. Most of the efforts in this field are based on the hypothesis of linear behavior [10–14], but nonlinear phenomena can greatly impact the final performances [15–18]. The causes of the nonlinear response can be traced back to several mechanisms, such as instability phenomena [19–22], nonlinear material constitutive law [23–27], geometric effects [28–30], impacts [31–34], and damping [35]. Additionally, nonlinearities can arise from the electrical circuit (diodes, among others) [36].

As regards the applications in civil engineering, the dynamic response of most structures and infrastructures is characterized by low frequency content. In these cases, the frequency tuning of piezoelectric beams necessitates the use of flexible materials, low thickness-to-length ratios, and/or relatively heavy additional masses [37, 38]. Such expedients make piezoelectric energy harvesting devices more prone to exhibit nonlinear mechanical behaviors.

While FE simulations can be a viable computational strategy for nonlinear static analyses, reduced-order models (ROMs) are more attractive for nonlinear dynamic analyses because they allow saving elaboration time. Moreover, ROMs are useful in order to separate and identify the effects due to material and geometric nonlinearities [35]. With some exceptions [5], however, most reduced-order models assume a linear electromechanical response and the modal superposition principle is extensively adopted to derive the state-space representation of the system.

Therefore, in this paper, we propose an efficient multiscale hybrid approach to model accurately the nonlinear dynamic response of PVDF energy harvesters. The term multiscale refers specifically to the device and system scales. First, the FE method is employed to solve the equations governing the response under static loading (device scale). Hence, a global curve that provides the tip displacement evolution for increasing values of the external load is obtained (i.e., pushover curve). The global FE-based solutions are then used in place of experimental data to identify the values of linear and nonlinear lumped coefficients of the reduced-order model (system scale). Finally, the nonlinear differential equations governing the dynamic behavior are solved to estimate the frequency response functions of tip displacement and output voltage.

#### 2. Numerical Modeling

##### 2.1. A Short Review on Nonlinear Electroelasticity

Piezoelectric polymers like PVDF [39] represent a valid solution for the development of flexible energy harvesting devices [2, 40]. The main advantage of PVDF with respect to other piezoelectric materials (in particular piezoceramics) is the possibility of sustaining large displacement without failure or drastic reduction of the piezoelectric efficiency [41–43]. Throughout this paper, therefore, it will be assumed that the piezoelectric layers are made of PVDF.

As soon as a piezoelectric solid undergoes large deformations and rotations, the classical small strain electromechanical constitutive equations lead to incorrect results. For the sake of completeness, we briefly review hereafter the equations for the continuum mechanical description of a piezoelectric solid under large strains. The interested reader can refer to [44] for a more complete discussion.

The reference and deformed configurations are denoted by and , respectively, where . When the electromechanical body deforms, the nonlinear mapping function at time instant maps the material point onto :The displacement vector is obtained as the difference between the positions vectors of the current and initial configuration:whereas the deformation gradient can be defined as a function of the displacement gradient :whereAccording to Faraday’s law,where is the electric field vector in the current configuration. Consequently, it is possible to define as the gradient of a scalar electric potential :Velocity and acceleration of a material point with respect to the reference configuration are defined, respectively, by the following material time derivatives:In the current configuration, the balance of momentum and Gauss’s law state thatwhere represents the mechanical Cauchy stress tensor, denotes the electric displacement, is the mechanical density, is free electric charge density, and indicates the volume force (in the current configuration). The balance of mass implies that , where is the density in the initial configuration and . Local balance of angular momentum guarantees that . Moreover, if is the electric polarization vector in the current configuration and is the vacuum permittivity, thenThe transformation between volume elements , and electric charges , in the current and reference configurations, respectively, is based on the following equations:Furthermore, the condition ensures that the tensor is not singular and, as a consequence, the deformation process will be smooth.

In the initial configuration, the local balance of momentum given by (8) can be recast with respect to different stress and strain measures:where and are the total first and second Piola–Kirchoff stress tensors, respectively. Moreover, represents the body force in the initial configuration. Constitutive equations satisfying the material objectivity principle are also required. To this end, it is assumed that a strain energy density function exists for the electromechanical body that, in general, can be defined with respect to different kinematics tensors, namely, , , and and the electric field vector . All the quantities refer to the initial configuration. Here, indicates the Green–Lagrange strain tensor that can be expressed as a function of the deformation gradient tensor bywhereis the right Cauchy–Green tensor. Objectivity requires that . A compressible Neo–Hookean type material model with a total energy density is used in this work:where and are the Lame constants, and are further material constants to be calibrated, and , , and are computed for a transversely isotropic material according to [45, 46] and they are equal to where is the identity tensor. In this study, it is assumed that the interaction between electric fields and matter is mainly confined within the finite space occupied by the matter. Once is defined, it is possible to derive the following constitutive equations:Here, is the dielectric displacement vector computed in the initial configuration while the total first Piola–Kirchoff stress tensor and the total Cauchy stress tensor are equal torespectively. The transformation from the material to the current configuration is possible by means of the following relationships:The Dirichlet and Neumann boundary conditions for the mechanical field arewhere and are prescribed mechanical displacement and surface traction vectors in the reference configuration, respectively. The boundary of the domain is ( and are its Dirichlet and Neumann portions, resp.), with and . Moreover, is the outward unit normal to . The boundary conditions for the electric field arewhere and are prescribed values of electric potential on and electric charge flux on , respectively. Moreover, , .

A cantilever-type configuration is here assumed for the energy harvesting device (see Figure 1). It is made of a piezoelectric layer and a substrate layer used for the deposition of the polymeric mixture during the fabrication process before the polarization of dipoles. This second material layer is described here by a compressible Neo–Hookean type material with total energy densities :