The complete work presented in this thesis has been organized into seven chapters. Chapter 1 is devoted to the literature survey and the proposed objectives of the present work. An overview
Chapter 1
of the contents of the remaining six chapters is presented below.
Chapter 2 presents a coupled 3D piezoelasticity solution for edge effects in the static analysis of Levy-type rectangular hybrid composite and sandwich plates using mixed field extended Kantorovich method (EKM). The effect of adhesive layers for the static response of smart hybrid sandwich plates have also been investigated. The method is derived using Reissner’s variational principle following a mixed formulation approach which takes displacements as well as stresses into account as primary variables. In this method, the boundary and interface conditions are exactly satisfied at all points ensuring the same order of accuracy for all the displacement and stress variables. The accuracy in predicting the order of free edge stress singularity has been verified. Numerical results are presented for thick single layer piezoelectric, smart hybrid composite and sandwich plates subjected to the various type of boundary conditions. The convergence and accuracy of the present method have been substantiated by direct comparison with the 3D exact elasticity solution and 3D FE solution of ABAQUS. It is observed that two/three term in the solution is sufficient enough to predict stresses accurately near the edges.
The 3D EKM developed in Chapter 2 for the static piezoelasticity solution of laminated hybrid composite and sandwich rectangular plates is extended to the free vibration analysis of elastic laminated plates in Chapter 3. In Chapter 3, the 3D EKM has been used to present the free vibration analysis of elastic composite and sandwich plates along with the effect of adhesive characteristics on the free vibration behaviour of composite plates have been investigated. The accuracy and efficacy of the present method have been established by comparing the results with the results from other theories published previously and with the 3D FE results of ABAQUS wherever no other results are available. Benchmark natural frequencies and mode shapes are presented for composite and sandwich plates for different types of boundary conditions.
In Chapter 4, the generalized 3D EKM presented in Chapter 3 for the dynamic analysis of elastic laminated plates has been extended to develop the free vibration solution of bimorph, smart hybrid composite and sandwich plates. The accuracy of the present solution is verified in comparison with the results from other theories and with the 3D FE solutions of ABAQUS. The effect of adhesive layer on the natural frequency of bimorph plate has also been investigated.
Benchmark natural frequencies are presented for bimorph, composite sandwich and smart hybrid sandwich plates. Mode shapes for smart hybrid composite and sandwich plates for various
boundary conditions are presented.
In Chapter 5, the formulation for exact Levy-type free vibration piezoelectric solution is presented for rectangular plates using an efficient zigzag theory (ZIGT) and the third order theory (TOT). The inplane displacements (ux, uy), in this theory, are approximated as a com- bination of third order variation in z across the thickness and a layerwise linear variation of slopes (ux,z, uy,z) at the interface. The field variables in the solution scope are expanded in Fourier series along the normal to the simply supported edge. The mixed formulation approach is followed in deriving the solution process. This solution methodology can be used to obtain the static and free vibration response of elastic/piezoelectric laminated composite plates.
The piezoelasticity exact Levy solution methodology using ZIGT and TOT presented in Chapter 5 is used in Chapter 6 to predict the free vibration of elastic laminated plates as well as static and free vibration response of smart hybrid laminated rectangular plates. The accuracy of the 2D theories i.e. ZIGT and TOT are assessed with the 3D elasticity solution based on EKM and in some cases with the 3D FE solution from ABAQUS.
Finally, the major conclusions of this work and suggestions for future research are summa- rized in Chapter 7.
Chapter 2
3D Piezoelasticity Solution for Static Analysis of Smart Hybrid
Rectangular Plates by Extended Kantorovich Method
2.1 INTRODUCTION
A coupled three-dimensional static piezoelasticity solution [151] for rectangular piezolami- nated plates with Levy-type boundary conditions is presented in this chapter using the mixed- field multi-term extended Kantorovich method (EKM) and Fourier series expansion. The piezo- electric or PFRC layers are embedded or surface bonded on the elastic laminated substrates with poling along the thickness direction. The formulation considers full electromechanical coupling, and the piezoelectric layers can be modeled as both actuators and sensors. The solution con- siders two-way electromechanical coupling, and satisfies all the essential and natural boundary conditions as well as the continuity conditions at the interface exactly at all points.
In Sec. 2.2, modified constitutive relations of piezoelasticity are deduced and subsequently expanded using the strain-displacement and electric field-potential relations. The solution methodology is laid out in Sec. 2.3 following a mixed formulation approach expressing the govern- ing equations with the help of mechanical (displacement and stress) and electrical (potential and displacement) state variables. The state variables and the electromechanical loading functions are expressed in terms of Fourier series in the span coordinate which satisfy the simply-supported boundary condition on the pair of edges along the span direction. The variational principle is implemented along the inplane (x) and thickness (z) directions which furnish 8n ODEs with
constant coefficients and 5n algebraic equations in terms of x. In the first iteration step, the trial function are assumed along thex-direction and are solved for thez-functions which in turn serves as a priori function in the next step to determine the functions in x-direction. The pro- cesses is repeated until a desired convergence is warranted. It is perceived that, in this method, convergence in the results is achieved very fast just in two to three terms of the Fourier expansion in the iteration process.
The objective in this analysis is to accurately characterize the edge effects and the effects of adhesive in finite dimensional piezolaminated plates under mechanical and actuation potential loading, and also to provide a benchmark for assessing the accuracy of the two-dimensional laminate theories. The efficacy and eminence of the present method is ascertained by comparing the present results with other previous results and with the 3D FE solution obtained from ABAQUS and are presented in Sec. 2.4. At first the accuracy in predicting the order of stress singularity is examined taking long dissimilar bi-material strips and then results for rectangular hybrid and sandwich plates are presented. Results of piezoelectric plates with F-F boundary condition under electric potential loading are presented and effect of b/a on the interlaminar stresses has been studied. The effect of electromechanical coupling and the electric boundary conditions on the boundary layer effect is studied providing interesting insight into the behavior of smart plates. The effect of adhesive thickness on the interlaminar stresses are also investigated.