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