Layerwise partial mixed finite element analysis of magneto-electro-elastic plates

Abstract: Lagrangian dynamics of mechanical systems 1.1 Introduction 1.2 Kinetic state functions 1.3 Generalized coordinates, kinematic constraints 1.4 The principle of virtual work 1.5 D'Alembert's principle 1.6 Hamilton's principle 1.7 Lagrange's equations 1.8 Lagrange's equations with constraints 1.9 Conservation laws 1.10 More on continuous systems 1.11 References 2 Dynamics of electrical networks 2.1 Introduction 2.2 Constitutive equations for circuit elements 2.3 Kirchhoff's laws 2.4 Hamilton's principle for electrical networks 2.5 Lagrange's equations 2.6 References 3 Electromechanical Systems 3.1 Introduction 3.2 Constitutive relations for transducers 3.3 Hamilton's Principle 3.4 Lagrange's equations 3.5 Examples 3.6 General electromechanical transducer 3.7 References 4 Piezoelectric Systems 4.1 Introduction 4.2 Piezoelectric transducer 4.3 Constitutive relations of a discrete transducer 4.4 Structure with a discrete piezoelectric transducer 4.5 Multiple transducer systems 4.6 General piezoelectric structure 4.7 Piezoelectric material 4.8 Hamilton's principle 4.9 Rosen's piezoelectric transformer 4. 10 References 5 Piezoelectric laminates 5.1 Piezoelectric beam actuator 5.2 Laminar sensor 5.3 Spatial modal filters 5.4 Active beam with collocated actuator-sensor 5.5 Piezoelectric laminates 5.6 References 6 Active and Passive Damping with Piezoelectric Transducers 6.1 Introduction 6.2 Active strut, open-loop FRF 6.3 Active damping via 1FF 6.4 Admittance of the piezoelectric transducer 6.5 Damping via resistive shunting 6.6 Inductive shunting 6.7 Decentralized control 6.8 General piezoelectric structure 6.9 Self-sensing 6.10 Other active damping strategies 6.11 Remark 6.12 References Bibliography Index

Abstract: An approximate solution for the free vibration problem of two-dimensional magneto-electro-elastic laminates is presented to determine their fundamental behavior. The laminates are composed of linear homogeneous elastic, piezoelectric, or magnetostrictive layers with perfect bonding between each interface. The solution for the elastic displacements, electric potential, and magnetic potential is obtained by combining a discrete layer approach with the Ritz method. The model developed here is not dependent on specific boundary conditions, and it is presented as an alternative to the exact or analytical approaches which are limited to a very specific set of edge conditions. The natural frequencies and through-thickness modal behavior are computed for simply supported and cantilever laminates. Solutions for the simply supported case are compared with the known exact solution for piezoelectric laminates, and excellent agreement is obtained. The present approach is also validated by comparing the natural frequencies of a two-layer cantilever plate with known analytical solution and with results obtained using commercial finite element software.

Abstract: This paper discusses the different electromagnetic boundary conditions on the crack-faces in magnetoelectroelastic materials, which possess coupled piezoelectric, piezomagnetic and magnetoelectric effects. A notch of finite thickness in these materials containing air (or vacuum) is also addressed. Four ideal crack-face electromagnetic boundary condition assumptions, that is, (a) electrically and magnetically impermeable crack, (b) electrically impermeable and magnetically permeable crack, (c) electrically permeable and magnetically impermeable crack and (d) electrically and magnetically permeable crack, are investigated separately. The influence of notch thickness on the field intensity factors at notch tips and the electromagnetic field inside the notch are obtained in closed-form. The results are compared with the ideal crack solutions. Applicability of crack-face electromagnetic boundary condition assumptions is discussed.

Abstract: In this article, static analysis of functionally graded, anisotropic and linear magneto-electro-elastic plates have been carried out by semi-analytical finite element method. A series solution is assumed in the plane of the plate and finite element procedure is adopted across the thickness of the plate such a way that the three-dimensional character of the solution is preserved. The finite element model is derived based on constitutive equation of piezomagnetic material accounting for coupling between elasticity, electric and magnetic effect. The present finite element is modeled with displacement components, electric potential and magnetic potential as nodal degree of freedom. The other fields are calculated by post-computation through constitutive equation. The functionally graded material is assumed to be exponential in the thickness direction. The numerical results obtained by the present model are in good agreement with available functionally graded three-dimensional exact benchmark solutions given by Pan and Han [Pan, E., Han, F., in press. Green’s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J. Solids Struct.]. Numerical study includes the influence of the different exponential factor, magneto-electro-elastic properties and effect of mechanical and electric type of loading on induced magneto-electro-elastic fields. In addition further study has been carried out on non-homogeneous transversely isotropic FGM magneto-electro-elastic plate available in the literature [Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates].

Abstract: In this article, free vibration studies on Functionally Graded Materials (FGM) magneto-electro-elastic cylindrical shells have been carried out. A series solution is assumed in circumferential and axial direction such that the three-dimensional character of the solution is preserved. The finite element model is derived based on constitutive equation of magneto-electro-elastic material accounting for coupling between elasticity, electric and magnetic effect. Frequency behavior with the variation of power law index is analyzed and explained on the basis of constitutive material. One of the aims of the study is to evaluate the influence of magnetic and piezoelectric effect on the structural frequency. In addition influence of length to radius ratio and radius to thickness ratio on the frequency behaviour of such shells have been attempted.

Abstract: The use of composite materials in engineering structures continues to increase dramatically, and there have been equally significant advances in modeling for general and composite materials and structures in particular. To reflect these developments, renowned author, educator, and researcher J.N. Reddy created an enhanced second edit

Abstract: Equations of Anisotropic Elasticity, Virtual Work Principles, and Variational Methods Fiber-Reinforced Composite Materials Mathematical Preliminaries Equations of Anisotropic Entropy Virtual Work Principles Variational Methods Summary Introduction to Composite Materials Basic Concepts and Terminology Constitutive Equations of a Lamina Transformation of Stresses and Strains Plan Stress Constitutive Relations Classical and First-Order Theories of Laminated Composite Plates Introduction An Overview of Laminated Plate Theories The Classical Laminated Plate Theory The First-Order Laminated Plate Theory Laminate Stiffnesses for Selected Laminates One-Dimensional Analysis of Laminated Composite Plates Introduction Analysis of Laminated Beams Using CLPT Analysis of Laminated Beams Using FSDT Cylindrical Bending Using CLPT Cylindrical Bending Using FSDT Vibration Suppression in Beams Closing Remarks Analysis of Specially Orthotropic Laminates Using CLPT Introduction Bending of Simply Supported Rectangular Plates Bending of Plates with Two Opposite Edges Simply Supported Bending of Rectangular Plates with Various Boundary Conditions Buckling of Simply Supported Plates Under Compressive Loads Buckling of Rectangular Plates Under In-Plane Shear Load Vibration of Simply Supported Plates Buckling and Vibration of Plates with Two Parallel Edges Simply Supported Transient Analysis Closure Analytical Solutions of Rectangular Laminated Plates Using CLPT Governing Equations in Terms of Displacements Admissible Boundary Conditions for the Navier Solutions Navier Solutions of Antisymmetric Cross-Ply Laminates Navier Solutions of Antisymmetric Angle-Ply Laminates The Levy Solutions Analysis of Midplane Symmetric Laminates Transient Analysis Summary Analytical Solutions of Rectangular Laminated Plates Using FSDT Introduction Simply Supported Antisymmetric Cross-Ply Laminated Plates Simply Supported Antisymmetric Angle-Ply Laminated Plates Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported Transient Solutions Vibration Control of Laminated Plates Summary Theory and Analysis of Laminated Shells Introduction Governing Equations Theory of Doubly-Curved Shell Vibration and Buckling of Cross-Ply Laminated Circular Cylindrical Shells Linear Finite Element Analysis of Composite Plates and Shells Introduction Finite Element Models of the Classical Plate Theory (CLPT) Finite Element Models of Shear Deformation Plate Theory (FSDT) Finite Element Analysis of Shells Summary Nonlinear Analysis of Composite Plates and Shells Introduction Classical Plate Theory First-Order Shear Deformation Plate Theory Time Approximation and the Newton-Raphson Method Numerical Examples of Plates Functionally Graded Plates Finite Element Models of Laminated Shell Theory Continuum Shell Finite Element Postbuckling Response and Progressive Failure of Composite Panels in Compression Closure Third-Order Theory of Laminated Composite Plates and Shells Introduction A Third-Order Plate Theory Higher-Order Laminate Stiffness Characteristics The Navier Solutions Levy Solutions of Cross-Ply Laminates Finite Element Model of Plates Equations of Motion of the Third-Order Theory of Doubly-Curved Shells Layerwise Theory and Variable Kinematic Model Introduction Development of the Theory Finite Element Model Variable Kinematic Formulations Application to Adaptive Structures Layerwise Theory of Cylindrical Shell Closure Subject Index

Abstract: This paper makes a first attempt to survey and discuss the advances and trends in the formulations and applications of the finite element modeling of adaptive structural elements. For most contributions, the specific assumptions, in particular those of electrical type, and the characteristics of the elements are precised. The informations are illustrated in tables and figures for helpful use by the researchers as well as the designers interested in this growing field of smart materials and structures. Focus is put on the development of adaptive piezoelectric finite elements only. However, papers on other applications and active systems are also listed for completeness purpose. In total, more than 100 papers were found in the open literature. Taking this number as a measure of research activity, trends and ideas for future research are identified and outlined. 7 2000 Elsevier Science Ltd. All rights reserved.

Abstract: Exact solutions are derived for three-dimensional, anisotropic, linearly magneto-electroelastic, simply-supported, and multilayered rectangular plates under static loadings. While the homogeneous solutions are obtained in terms of a new and simple formalism that resemble the Stroh formalism, solutions for multilayered plates are expressed in terms of the propagator matrix. The present solutions include all the previous solutions, such as piezoelectric, piezomagnetic, purely elastic solutions, as special cases, and can therefore serve as benchmarks to check various thick plate theories and numerical methods used for the modeling of layered composite structures. Typical numerical examples are presented and discussed for layered piezoelectric/piezomagnetic plates under surface and internal loads.

Abstract: This review article is devoted to the use of the Reissner Mixed Variational Theorem ~RMVT! forward two-dimensional modeling of flat and curved, multilayer structures. A thorough review of the literature involving the use in the modeling of multilayered plates and shells using RMVT is also presented. In the first part, the paper overviews relevant key points that should be taken into account for an accurate description of strain and stress fields in multilayered plate and shell analysis. It is then shown that RMVT has been originated in view of the fulfillment of such key points, herein referred to as C-Requirements ~zig-zag form of the displacement fields in the thickness direction and continuity of transverse normal and shear stresses at each layer interface!. Classical variational statements are used to introduce Reissner’s Theorem. In the second part, the paper presents various ways in which RMVT can be used to develop plate and shell theories in a systematic manner. The so called layer-wise and equivalent single layer variable description are considered. Both strong and weak ~finite element! forms of governing equations have been derived. A Weak Form of Hooke’s Law ~WFHL!, is also discussed as an idea to eliminate transverse stress variables leading to standard classical models with only displacement unknowns. Two appendices display details of governing equations related to multilayered doubly curved shells and to finite element matrices of multilayered plates. A third part reviews the works that have appeared in literature which make use of RMVT. Mainly papers on multilayered plate and shell modelings have been addressed. The final part of the paper is devoted to giving an overview with selected results of numerical performance that can be acquired by RMVT applications; extensive comparison to elasticity solutions and to other significant analyses, based on classical and refined approaches, are given. It is concluded that Reissner’s Mixed Theorem should be considered as a natural tool for multilayered structure analyses; it plays a similar role to that of the Principle of Virtual Displacement in the analysis of isotropic single-layer structures. This review article includes 119 references. @DOI: 10.1115/1.1385512#