Geometric Nonlinear Finite Element and Genetic Algorithm Based Vibration Energy Harvesting from Functionally Graded Nonprismatic Piezolaminated Beams

Energy harvesting technology has the ability to create autonomous, self-powered systems which do not rely on the conventional battery for their operation. The term energy harvesting is the process of converting the ambient energy surrounding a system into some useful electrical energy using certain materials. Among several energy conversion techniques, the conversion of ambient vibration energy to electrical energy using piezoelectric materials has great deal of importance which encompasses electromechanical coupling between mechanical and electrical domains. The energy harvesting systems are designed by incorporating the piezoelectric materials in the host structure located in vibration rich environment. The work presented in this dissertation focuses on upgrading the concept of energy harvesting in order to engender more power than conventional energy harvesting designs. The present work deals with first the finite element (FE) formulation for coupled thermo-electro-mechanical analysis of vibration energy harvesting from an axially functionally graded (FG) non-prismatic piezolaminated cantilever beam. A two noded beam element with two degrees of freedom (DOF) at each node has been used in the FE formulation. The FG material (i.e. non-homogeneity) in the axial direction has been considered which varies (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The various cross section profiles (such as linear, parabolic and cubic) have been modelled using the Euler-Bernoulli beam theory and Hamilton‘s principle is used to solve the governing equation of motion. The simultaneous variation of tapers (both width and height in length directions) is incorporated in the mathematical formulation. The FE formulation developed in the present work has been compared with the analytical solutions subjected to mechanical, electrical, thermal and thermo-electro-mechanical loading. Results obtained from the present work shows that the axially FG nonprismatic beam generates more output power than the conventional energy harvesting systems. Further, the work has been focussed towards the nonlinear vibration energy harvesting from an axially FG non-prismatic piezolaminated cantilever beam. Geometric nonlinear based FE formulation using Newmark method in conjunction with Newton-Raphson method has been formulated to solve the obtained governing equation. Moreover, a real code GA based constrained optimization technique has also been proposed to determine the best possible design variables for optimal power harvesting within the allowable limits of ultimate stress of the beam and voltage of the PZT sensor. It is observed that more output power can be obtained based on the present optimization formulation within the allowable limits of stress and voltage than that of selection of design variables by trial and error in FE modelling

Development of computational efficient shell formulation for analysis of multilayered structures subjected to mechanical, thermal, and electrical loadings

The aim of this work is the development of robust finite shell model suitable for numerical applications in solid mechanics with a remarkable reduction in computational cost. Two-dimensional (2D) structural models, commonly known as plates/shells, are for instance used in many applications to analyze the structural behavior of thin and slender bodies such as panels, domes, pressure vessels, and wing stiffened panels amongst others. These models reduce the three-dimensional 3D problem into a two-dimensional 2D problem, where variables depend on the in-plane axis coordinates. Two-dimensional elements are simpler and computationally more efficient than 3D (solid) models. This feature makes plate/shell theories still very attractive for the static, dynamic response, free vibration, thermo-mechanical and electro-mechanical analysis, despite the approximations which they introduce in the simulation. Nevertheless, analytical solutions for three-dimensional elastic bodies are generally available only for a few particular cases which represent rather coarse simplifications of reality. In most of the practical problems, the solution demands applications of approximated computational methods. The Finite Element Method (FEM) has a predominant role among the computational techniques implemented for the analysis of layered structures. The majority of FEM theories available in the literature are formulated by axiomatic-type theories. In this thesis, attention is focused on weak-form solutions of refined plate/shell theories. In particular, higher-order plate/shell models are developed within the framework of the Unified Formulation by Carrera, according to which the three-dimensional displacement field can be expressed as an arbitrary expansion of the generalized displacements. A robust finite shell element for the analysis of plate and shell structures subjected to mechanical, thermal, and/or electrical loadings is developed. A wide range of problems are considered, including static analysis, free vibration analysis, different boundary conditions and different laminations schemes, distributed pressure loads, localized pressure loads or concentrated loads are taken into account. The high computational costs represent the drawback of refined plate/shell theories or three-dimensional analyses. In recent years considerable improvements have been obtained towards the implementation of innovative solutions for improving the analysis efficiency for a global/local scenario. In this manner, the limited computational resources can be distributed in an optimal manner to study in detail only those parts of the structure that require an accurate analysis. In the second part of the thesis two different methodology are presented to improve the analysis efficiency, and at the same time keeping the finite higher-order plate/shell element accuracy. The two approaches can be collocated in the simultaneous multi-model methodologies. The first is the Mixed ESL/LW variable kinematic method, where the primary variables are described along the shell thickness selecting some plies with an ESL description and others with a LW behaviour by using the Legendre polynomials for both the assembling approaches. The second approach is a new simultaneous multi-model, here presented as Node-Dependent Variable Kinematic method. The shell element with node-dependent capabilities enables one to vary the kinematic assumptions within the same finite element. The expansion order ( along the shell thickness ) of the shell element is, in fact, a property of the FE node in the present approach. Different kinematics can be coupled without the use of any mathematical artifice. The theories developed in this thesis are validated by using some selected results from the literature. The analyses suggest that Unified Formulation furnishes a reliable method to implement refined theories capable of providing almost three-dimensional elasticity solution, and that the two simultaneous multi-theories methods are extremely powerful and versatile when applied to composite or sandwich structures subjected to various mutlifield loadings

On the fundamental equations of electromagnetoelastic media in variational form with an application to shell/laminae equations

AbstractThe fundamental equations of elasticity with extensions to electromagnetic effects are expressed in differential form for a regular region of materials, and the uniqueness of solutions is examined. Alternatively, the fundamental equations are stated as the Euler–Lagrange equations of a unified variational principle, which operates on all the field variables. The variational principle is deduced from a general principle of physics by modifying it through an involutory transformation. Then, a system of two-dimensional shear deformation equations is derived in differential and fully variational forms for the high frequency waves and vibrations of a functionally graded shell. Also, a theorem is given, which states the conditions sufficient for the uniqueness in solutions of the shell equations. On the basis of a discrete layer modeling, the governing equations are obtained for the motions of a curved laminae made of any numbers of functionally graded distinct layers, whenever the displacements and the electric and magnetic potentials of a layer are taken to vary linearly across its thickness. The resulting equations in differential and fully variational, invariant forms account for various types of waves and coupled vibrations of one and two dimensional structural elements as well. The invariant form makes it possible to express the equations in a particular coordinate system most suitable to the geometry of shell (plate) or laminae. The results are shown to be compatible with and to recover some of earlier equations of plane and curved elements for special material, geometry and/or effects