Improved shell finite element for piezothermoelastic analysis of smart fiber reinforced composite structures

https://doi.org/10.1016/j.finel.2010.03.009Get rights and content

Abstract

Present article deals with the development of an improved eight noded layered shell finite element formulation for piezothermoelastic analysis of smart fiber reinforced polymer (FRP) composite shell structures with bonded piezoelectric sensors and actuators. Stress resultant-type Koiter's shell theory has been used and twist curvature component has been incorporated to keep the strain equations complete. The transverse shear effect has also been considered according to Mindlin's hypothesis. Pyroelectric effect has been considered in the formulation. The developed formulation has been observed to give accurate results for both deep and shallow shells and will be useful for analysis of smart FRP shell structures. It is observed that pyroelectric effect has a significant influence on the response of such shells under piezothermoelastic loading. Different types of smart shell panels viz. spherical, ellipsoidal, doubly curved and cylindrical have been analyzed and the coupled thermo-electro-mechanical responses have been presented.

Introduction

With the emergence of health monitoring and vibration and shape control of flexible structures as significantly important areas of research, enormous effort is currently being put on the development of smart or intelligent structures. Piezoelectric materials are probably the most popular active material used in both sensor and actuator applications because of low cost, low power consumption, light weight, quick dynamic response and ease in embedding or bonding over the structure. In this direction, fiber reinforced polymer (FRP) composites embedded with piezoelectric sensors and actuators have varied potential in active vibration control applications especially in aerospace structures and are commonly termed as smart FRP laminated structures. Deployment of such smart FRP laminated structures in actual applications demands complete understanding of the behavior of such structures subjected to loading. Researchers have paid significant attention in recent times for development of efficient finite elements for analysis of such smart structures as modeling and analysis of adaptive piezothermoelastic laminated structures represent high level of sophistication and complexity. Recently, an increasing number of investigators have addressed piezothermoelasticity. Some of the important works in the direction are presented in the following paragraph.

Tauchert et al. [1] examined the response of laminated plate embedded with piezoelectric lamina to stationary thermal and electric loads using the classical lamination theory and assuming temperature to be linearly distributed in the thickness direction. Jonnalagadda et al. [2] developed a formulation to study the static response of graphite/epoxy composite plates with an attached polyvinylidene difluoride (PVDF) layer subjected to mechanical, thermal and electrical loading using first-order shear deformation theory and assuming linear variation of temperatures in the thickness direction. Influences of temperature on piezoelectric sensors and actuators of beam-type precision devices have been studied based on a thin piezothermoelastic solid finite element by Tzou and Ye [3]. Xu and Noor [4] investigated the response of a laminated cylindrical shell to mechanical loading, temperature change and electric potential by three-dimensional analytical solutions.

A new three-dimensional thin hexahedron piezothermoelastic solid finite element with three internal degrees of freedom was formulated using a variational formulation by Tzou and Ye [5]. Distributed sensing equations were derived and pyroelectric and thermal strain effects of the piezoelectric transducers of a laminated plate were investigated. Lee and Saravanos [6] derived a thermo-piezoelectric multilayer beam element using linear shape functions along the beam and linear through the thickness of each layer (layerwise linear). Shang et al. [7] studied the thermal buckling of a laminated piezoelectric plate with uniform temperature. Batra et al. [8], [9] dealt with shape and vibration control of plates for finite deformations, taking into account nonlinear constitutive equations for piezoelectric patches. Coupled thermo-piezoelectric-mechanical models of composite laminates with surface bonded piezoelectric actuators were developed by Chattopadhyay et al. [10] and Jingmei et al. [11]. Blandford et al. [12] developed a hierarchical finite element approximation of the governing equations using reduced material stiffness coefficient to study static response of beams.

Raja et al. [13] presented a generalized finite element formulation of a laminated beam with embedded piezoelectric materials as distributed sensors and actuators. In their work, fundamentals of piezothermoelasticity were reviewed first and followed by the development of a new piezothermoelastic triangle composite shell finite element including the temperature effect, extended from the piezoelastic triangular shell element by Ye and Tzou [14]. Among the investigations in piezothermoelasticity, the static and dynamic problems of different structures were discussed by Tauchert et al. [15]. A general solution for dynamic piezothermoelastic problems of transversely isotropic piezoelectric materials was derived by HaoJiang et al. [16]. A higher order temperature (HOT) field theory was developed and implemented in the coupled thermo-piezoelectric-mechanical analysis of composite laminates with surface bonded piezoelectric actuators by Gu et al. [17]. Kim et al. [18] developed the coupled thermo-piezoelectric-mechanical theory, based on layerwise displacement field and higher order electrical and temperature fields, to study dynamic response and control of smart cylindrical composite shells. Altay and Dökmeci [19] formulated the fundamental equations of thermo-piezoelectricity in variational form, and systematically derived the system of one-dimensional (1D) equations for the high-frequency vibrations of a cylindrical rod. Görnandt and Gabbert [20] presented a general finite element concept based on a weak formulation of the equilibrium conditions and the coupled constitutive equations of this thermo-piezoelectric problem to solve such problems numerically. Vel and Batra [21], [22] developed a three-dimensional analytical solution in terms of an infinite series for the thermo-piezoelectric deformations of laminated thick plates with various support edges. Ganesan and Kadoli [23] analyzed the piezoelectric composite cylindrical shells operating in a steady state axisymmetric temperature using a semi-analytical finite element method. The numerical and experimental study of active compensation of thermal deformation of a composite beam using piezoelectric ceramic actuators was considered by Song et al. [24]. Altay and Dökmeci [25] modified Mindlin's equations of thermo-piezoelectricity, by introducing a thermal field vector, and obtained the consistency of the universal gradient equations in thermo-piezoelectricity. Liew et al. [26] investigated the behavior of multilayered composite plates subject to thermo-piezoelectric-mechanical loading using the three-dimensional equations of thermo-piezoelasticity and the differential quadrature (DQ) numerical technique. A new thermo-piezoelectric mixed variational theorem (TMVT) and its corresponding mixed thermo-piezoelectric constitutive equations were proposed for the variational-based modeling of thermo-piezoelectric multilayered composites by Benjeddou and Andrianarison [27]. Heidary and Eslami [28] outlined the equations governing the linear response of piezothermoelastic plate based on Hamilton's principle and finite element methods. Coupled electro-thermo-elastic equations applicable for the analysis of smart structures with piezoelectric patches/layers have been derived from the fundamental principles of mass, linear momentum, angular momentum, energy and charge conservation by Ahmad et al. [29]. Kumar et al. [30] presented the piezothermoelastic model of cylindrical shell using nine noded degenerated shell element. Oh et al. [31] developed an enhanced lower-order shear deformation theory (ELSDT) for the analysis of smart structures under combined thermo-electro-mechanical loading. Tian et al. [32] presented two-dimensional generalized piezothermoelastic problem in terms of Green and Lindsay generalized thermoelastic theory. Jiang and Li [33] recently developed a finite element model for piezothermoelastic composite beam considering higher order displacement field, higher order electrical field and linear temperature field using two nodes Hermitian beam element. Neto et al. [34] developed two finite elements viz. ad hoc smart beam element (ADSBE) based on the first order deformation theory and variational asymptotic smart beam element (VASBE) for the static analysis of smart beams with piezoelectric sensors/actuators.

From the exhaustive literature review, it has been observed that, while a number of works are available in the form of beam and plate finite elements for analysis of smart FRP structures, not many works are available in the form of finite element piezothermoelastic analysis of shell structures. There are few literatures available which also did not consider the complete electro-thermo-mechanical analysis of smart shell structures. However, for actual control applications of such structures, it is important to understand the thermo-electro-mechanical behavior of such structures so that appropriate control system could be designed. In the present work, thus, an attempt has been made to develop an improved shell finite element for coupled piezothermoelastic analysis of smart FRP composite shell structures to study the thermo-electro-mechanical responses of such structures under thermo-mechanical loading considering pyroelectric effect for both deep as well as shallow shells.

Section snippets

Shell finite element for piezothermoelastic analysis

The stress-resultant type Koiter's shell theory [35] has been considered to formulate the present finite element formulation of the smart FRP composite shells. Since Koiter's shell theory is based on the Love–Kirchhoff assumptions, the effect of shear deformation was not considered by Koiter's shell theory. This effect was considered in the formulation of Koiter's shell theory according to Mindlin's hypothesis [36]. It has been established that the assumption of the first order shear

Results and discussions

Based on the finite element formulation discussed in Section 2, a computer code has been developed which is capable of analyzing smart FRP composite shell structures under any kind of possible loading such as thermal, electrical, mechanical or any combination of these with or without considering pyroelectric effect. The developed computer code has been validated with already published results before using the same for analysis and design of smart shell structures.

Conclusions

An eight noded improved layered shell finite element has been developed for piezothermoelastic analysis of smart FRP composite structures with surface bonded PZT sensors and actuators for the analysis of deep as well as shallow shells based on the stress resultant-type Koiter's shell theory including transverse shear effect according to Mindlin's hypothesis. Four different types of smart FRP composite shells viz. spherical, ellipsoidal, doubly curved and cylindrical have been analyzed to

References (44)

  • A. Benjeddou et al.

    A thermopiezoelectric mixed variational theorem for smart multilayered composites

    Computers and Structures

    (2005)
  • F. Heidary et al.

    Piezo-control of forced vibrations of thermoelastic composite plate

    Journal of Composite Structures

    (2006)
  • X. Tian et al.

    Finite element method for generalize piezothermoelastic problems

    International Journal of Solids and Structures

    (2007)
  • J.P. Jiang et al.

    A new finite element model for piezothermoelastic composite beam

    Journal of Sound and Vibration

    (2007)
  • J.N. Reddy et al.

    A higher-order shear deformation theory of laminated elastic shells

    International Journal of Engineering Science

    (1985)
  • J.G. Smits et al.

    The constituent equations of piezoelectric bimorphs

    Sensors and Actuators A

    (1991)
  • D.T. Detwiler et al.

    Finite element analysis of laminated composite structures containing distributed piezoelectric actuators and sensors

    Finite Elements in Analysis and Design

    (1995)
  • P. Bhattacharya et al.

    Finite element analysis and distributed control of laminated composite shells using LQR/IMSC approach

    Aerospace Science and Technology

    (2002)
  • T.R. Tauchert et al.

    Piezothermoelastic behavior of a laminate

    Journal of Thermal Stresses

    (1992)
  • H.S. Tzou et al.

    Piezothermoelasticity and precision control of active piezoelectric laminates

    Journal of Vibration and Acoustics

    (1994)
  • K. Xu et al.

    Three dimensional analytical solutions for coupled thermoelectroelastic response of multilayered cylindrical shells

    AIAA Journal

    (1996)
  • H.J. Lee et al.

    Coupled layerwise analysis of thermo piezoelectric composite beams

    AIAA Journal

    (1996)
  • Cited by (33)

    • Transference of SH waves in a piezoelectric fiber-reinforced composite layered structure employing perfectly matched layer and infinite element techniques coupled with finite elements

      2022, Finite Elements in Analysis and Design
      Citation Excerpt :

      For several years, the precise prediction of the mechanical–electrical properties of PFRCs has remained a dynamic research area. Hence, several noteworthy results on the design of composites by employing analytical, numerical, and experimental techniques such as Method of Cells, Continuum Mechanics, Asymptotic Homogenization method, full-field optical technique, Strength of Materials, Koiter’s shell theory, Series-parallel/parallel-series techniques, Finite element method, Rule of mixtures, Hybridization, etc. can be found from the works of [2–12]. Analysis of wave propagation characteristics in piezoelectric ceramics is a field of extensive research in the modern day and age.

    • Viscoelastic modelling and dynamic characteristics of CNTs-CFRP-2DWF composite shell structures

      2018, Composites Part B: Engineering
      Citation Excerpt :

      The stress-resultant type Koiter's shell theory [52] has been considered in the present finite element (FE) formulation of the CNTs-CFRP woven fabric composite shell structures. The effects of shear deformation is employed based on the Midlin's hypothesis [53–55] in the present FE formulation. The details of shell finite element formulation are presented in the following subsection.

    • Active vibration control of space antenna reflector over wide temperature range

      2015, Composite Structures
      Citation Excerpt :

      The number of smart materials have been investigated and fabricated over the years; some of them are shape memory alloys, piezoelectric materials, optical fibers, electro-rheological fluids, magneto-strictive materials [13]. Among all smart materials, piezoelectric materials are widely used as sensor and actuator because of its numerous advantages like low cost, quick dynamic response, low power consumption, excellent electromechanical coupling, large operating range, light weight and ease in bonding on structure [14–16]. The space antenna reflector needs to be deployed in the harsh thermal environments in earth’s orbit where it experiences wide temperatures ranging from −70 °C to 200 °C [17].

    • Meso-mechanically motivated modeling of layered fiber reinforced composites accounting for delamination

      2015, Composite Structures
      Citation Excerpt :

      Noticeably, the hourglass stabilization is also formulated in such a way, that neither volumetric nor transversal shear locking occur. The accurate determination of stress distributions in thickness direction for laminated layered composites has been recently investigated in [32] using an improved shell formulation and in [33,34] based on the solid-shell concept. For a more elaborate literature overview the interested reader is referred to the review papers [35–37] and the references therein.

    • Numerical analysis of layered fiber composites accounting for the onset of delamination

      2015, Advances in Engineering Software
      Citation Excerpt :

      For laminated layered composites, the accurate determination of the through-the-thickness stress distribution was recently investigated by several authors. For instance, in [27] an improved shell formulation was used for this, whereas in [28] the investigations were based on the solid-shell concept. For a more elaborate literature overview, the reader is referred to the review papers [29–31] and the references therein.

    View all citing articles on Scopus
    View full text