Nonlinear analysis of functionally graded laminates considering piezoelectric effect

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

Nonlinear dynamic and vibration analysis of piezoelectric eccentrically stiffened FGM plates in thermal environment

This paper concerns arbitrary active constrained layer damping (ACLD) treatments applied to beams. In order to suppress vibration, hybrid active-passive treatments composed of piezoelectric and viscoelastic layers are mounted on the substrate beam structure. These treatments combine the high capacity of passive viscoelastic materials to dissipate vibrational energy at high frequencies with the active capacity of piezoelectric materials at low frequencies. The aim of this research is the development of a generic analytical formulation that can describe these hybrid couplings in an accurate and consistent way. The analytical formulation considers a partial layerwise theory, with an arbitrary number of layers, both viscoelastic and piezoelectric, attached to both surfaces of the beam. A fully coupled electro-mechanical theory for modelling the piezoelectric layers is considered. The equations of motion, electric charge equilibrium and boundary conditions are presented. A one-dimensional finite element (FE) model is developed, with the nodal degrees of freedom being the axial and transverse displacements and the rotation of the centreline of the host beam, the rotations of the individual layers and the electric potentials of each piezoelectric layer. The damping behavior of the viscoelastic layers is modeled by the complex modulus approach. Three frequency response functions were measured experimentally and evaluated numerically: acceleration per unit force, acceleration per unit voltage into the piezoelectric actuator and induced voltage per unit force. The numerical results are presented and compared with experimental results to validate the FE model.

Free vibration of simply supported functionally graded and layered magneto-electro-elastic plates by finite element method

Electromechanical response of thin shell laminated with flexoelectric composite layer

The present paper deals with the nonlinear analysis of the functionally graded piezoelectric (FGP) annular plate with two smart layers as sensor and actuator. The normal pressure is applied on the plate. The geometric nonlinearity is considered in the strain-displacement equations based on Von-Karman assumption. The problem is symmetric due to symmetric loading, boundary conditions and material properties. The radial and transverse displacements are supposed as two dominant components of displacement. The constitutive equations are derived for two sections of the plate, individually. Total energy of the system is evaluated for elastic solid and piezoelectric sections in terms of two components of displacement and electric potential. The response of the system can be obtained using minimization of the energy of system with respect to amplitude of displacements and electric potential. The distribution of all material properties is considered as power function along the thickness direction. Displacement-load and electric potential-load curves verify the nonlinearity nature of the problem. The response of the linear analysis is investigated and compared with those results obtained using the nonlinear analysis. This comparison justifies the necessity of a nonlinear analysis. The distribution of the displacements and electric potential in terms of non homogenous index indicates that these curves converge for small value of piezoelectric thickness with respect to elastic solid thickness.

Piezocomposites are attracting widespread interest since they can offer greater flexibility and better performances in specific applications with respect to traditional piezoelectric wafers. Design of piezocomposites requires accurate homogenisation models for the prediction of the equivalent electro-mechanical properties. In macro-scale models of structures with piezocomposite transducers, these properties are adopted in order to avoid the complexity of the piezocomposite microstructure. In the case of smart structures the accurate modelling of the actuation and the response of the structure is of primary importance. If classical structural finite elements are not sufficiently accurate, higher order or solid elements should be adopted. Thanks to adaptation or mixed-dimensional methods, it is possible to adopt computationally expensive higher order or solid elements only in some sub-domains of the structure. In this work, the modelling of smart structures equipped with thin piezoelectric transducers is considered in a multi-scale framework. Micromechanical homogenisation models are developed and employed for the prediction of the equivalent properties of piezocomposites. A micromechanical model based on the concept of inclusion is proposed to investigate the influence of the shape of the inclusions, of the constituent materials and of the polarisation on the equivalent properties. It has been found that fibre-shaped inclusions should be considered in order to obtain piezocomposites with strong piezoelectric effect and to have at the same time high direction-dependence. The equivalent properties of Macro Fiber Composites are determined via the Asymptotic Homogenisation Method (AHM) with an analytical solution and with a numerical solution via FEM which takes into account the effect of the electrodes. AHM analytical solution is adopted to investigate the effect of the material properties of the matrix on the overall piezocomposite. Results indicate that low values of the Young's modulus and of the Poisson's ratio yield high directional dependence in the piezoelectric properties. A laminated design with anisotropic layers and a piezocomposite layer is investigated via UFM. A configuration with maximum directional dependence in terms of equivalent piezoelectric strain constants is proposed, whereas maximum directional dependence in terms of piezoelectric stress constants is proved to be not achievable with such a design. Hierarchical finite elements for structural analyses based on a Unified Formulation (UF) by Carrera are developed and coupled via the Arlequin method proposed by Ben Dhia. Solid, plate and beam finite elements for mechanical and for piezoelectric problems are presented. Via UF, higher order and piezoelectric elements can be formulated straightforwardly. These elements are combined in variable kinematic solutions in the Arlequin framework. Higher-order elements are adopted locally where the stress field is three-dimensional, whereas the remaining parts of the structure are modelled with computationally cheap lower-order elements. Two electro-mechanical coupling operator for the Arlequin method in the context of piezoelectric analyses are proposed. Results are validated towards monomodel solutions and three-dimensional analytical and numerical reference solutions. Accurate solutions are obtained reducing the computational costs

On the role of higher-order conditions in distortion gradient plasticity

A scaled boundary finite element method for bending analysis of fiber-reinforced piezoelectric laminated composite plates

Recent research has identified Adini’s rectangular element as an efficient higher order element for solving second order elliptic partial differential equations such as Poisson’s equation, which governs the steady state heat conduction problem. This type of element includes both the primary field variable and its spatial derivatives as nodal degrees of freedom. Compared to the conventional cubic elements of the Serendipity and Lagrange families, Adini’s element includes the minimum number of nodes per element and has the advantage that the nodal values of the spatial derivatives of the temperature field are directly retrieved from the FEM solution. As a result, the differentiation and averaging procedures that are typically used to obtain the nodal values of the temperature gradients are avoided. In this paper a generalized version of Adini’s element for solving two-dimensional steady state heat transfer problems in non-rectangular geometries is presented. Also, the traditional finite element formulation is modified to allow the application of essential boundary conditions without having to constrain the nodal values of the tangential derivative of the temperature. The resulting higher order element and modified FEM formulation are used to solve an example problem and the accuracy of the solution is compared with solutions obtained using the traditional linear, quadratic, and cubic Serendipity elements to show the efficiency, in terms of accuracy per number of degrees of freedom, of the proposed approach for finding the nodal values of the temperature gradients, which are required to compute the nodal values of the heat flux vector.

The numerical implementation of a gradient-enhanced continuum coupled damage—plasticity model as a constitutive framework to model the nonlocal response of materials is presented. By the introduction of “nonlocal, ” gradient-enhanced measures in the plasticity potential function and yield criterion and in the damage potential function and damage criterion, the proposed model introduces microstructural characteristic material length scales, which allow us to predict the size of localized zones based on material constants, as opposed to local models where the loss of ellipticity causes localized zones to be mesh dependent. As numerical methods are used to compute the gradients of the hardening terms, no additional gradient internal state variables are introduced into the Helmholtz free energy. As opposed to previous theories that only incorporate linear hardening, the gradient model proposed here uses coupled nonassociative plasticity — nonassociative damage and can account for a wide range of material models because of the consistently expanded Laplacian evolution equations. However, due to the nonassociative material model, anisotropic hardening, and the corresponding gradients in plasticity, the model becomes highly complex as additional equations have to be introduced for the evolution of the gradients of plastic normal, plastic strains, damage normal, and damage tensor. The numerical implementation uses a small deformation finite element formulation and includes the displacements, plastic multiplier, and damage multiplier as nodal degrees of freedom, thus allowing the three fields to have different interpolation functions. Higher-order elements are used for the plastic multiplier and damage multiplier to enforce continuity of the second gradients, which requires the introduction of additional boundary conditions.

A new class of materials known as ‘functionally graded materials’ (FGMs) has emerged recently, in which the material properties are graded but continuous particularly along the thickness direction. In an effort to develop the super heat resistant materials, Koizumi [1] first proposed the concept of FGM. These materials are microscopically heterogeneous and are typically made from isotropic components, such as metals and ceramics. In the quest for developing lightweight high performing flexible structures, a concept emerged to develop structures with self-controlling and self-monitoring capabilities. Expediently, these capabilities of a structure were achieved by exploiting the converse and direct piezoelectric effects of the piezoelectric materials as distributed actuators or sensors, which are mounted or embedded in the structure [2, 3]. Such structures having built-in mechanisms are customarily known as ‘smart structures’. The concept of developing smart structures has been extensively used for active control of flexible structures during the past decade [4]. Recently considerable interest has also been focused on investigating the performance of FG plates integrated with piezoelectric actuators. For example, Ootao and Tanigawa [5] theoretically investigated the simply supported FG plate integrated with a piezoelectric plate subjected to transient thermal loading. A 3-D solution for FG plates coupled with a piezoelectric actuator layer was proposed by Reddy and Cheng [6] using transfer matrix and asymptotic expansion techniques. Wang and Noda [7] analyzed a smart FG compositestructure composed of a layer of metal, a layer of piezoelectric and a FG layer in between, while in [8] a finite element model was developed for studying the shape and vibration control of FG plates integrated with piezoelectric sensors and actuators. Yang et al. [9] investigated the nonlinear thermo-electro-mechanical bending response of FG rectangular plates covered with monolithic piezoelectric actuator layers; most recently, Huang and Shen [10] investigated the dynamics of a FG plate coupled with two monolithic piezoelectric layers undergoing nonlinear vibrations in thermal environments. All the aforementioned studies focused on the rectangular-shaped plate structures. However, to the authors’ best knowledge, no researches dealing with the free vibration characteristics of the circular FGM plate integrated with the piezoelectric layers has been reported. Therefore, the present work attempts to solve the problem of providing analytical solution for free vibration of thin circular FG plates with two full size surface-bonded piezoelectric layers on the top and the bottom of the FG plate. The formulations are based

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

Geometrical nonlinear free vibration responses of FG-CNT reinforced composite annular sector plates integrated with piezoelectric layers

A reliable and accurate analytical model is desired for the printed circuit board (PCB) with IC package to predict the system response due to loadings such as shock and vibration simulation or even with thermal effect. This work addresses the procedure of model verification by the adoption of experimental modal analysis (EMA) to validate the finite element (FE) model constructed by FE commercial software. The PCB with one package adhered with the heating pad to emulate the heat effect is first considered for completely free boundary condition. The refined FE model of the PCB consists of detail components, such as the chip, substrate, compound and solder balls. The thermal effect on the PCB is simulated to conduct the temperature field analysis as well as the thermal stress. The modal analysis on the PCB with the heating in steady state is then performed to obtain the structural modal parameters, i.e. natural frequencies and mode shapes. The EMA is also carried out to determine the system modal properties that are used to update the analytical FE model. Through the comparison of frequency response functions and modal parameters between the analytical FE model and the real PCB structure, the refined FE model can be verified for material properties and thermal boundary conditions. The same procedure for model verification is then conducted via both EMA and FEA on the PCB in the fixed boundary that complies with the test fixture for the random vibration test of JEDEC specification. The verified equivalent FE model of the PCB can then be adopted to perform spectrum response analysis accordingly.