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Based on the theories of 3-D elasticity and piezoelectricity and by assuming appropriate boundary functions, the state equation for laminated piezoelectric plate is established. By using the transfer matrix and recursive solution approach, an analytical solution that satisfies all boundary conditions, including the conditions on the top and bottom surfaces, of the laminates is presented. The solution can take into account all the independent elastic and piezoelectric constants for orthotropic and piezoelectric materials and satisfies the continuity conditions between plies of the laminates. Numerical examples are given at the end of the paper to verify the effectiveness of the present method. The results are compared with those of existing analytical and finite element models.

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... Near the clamped/free edges, the stress field changes sharply. For such cases, the general purpose displacement based finite element may not be reliable in the vicinity of the edges [5,6]. The limitations of the general finite element solution for addressing the edge effects has also been pointed out in a recent review article [7]. ...

... The displacement based finite element solution can not capture this behavior due to sharp gradients across the thickness. The results confirm the limitations of the FE solution reported in literature567. The through-thickness variations of stresses ...

A coupled three-dimensional piezoelasticity solution for rectangular piezoelectric laminated plates with Levy-type boundary conditions is presented using the mixed-field multi-term extended Kantorovich method (EKM) and Fourier series expansion. The solution considers 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. The objective is to accurately characterize the edge effects 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 interlaminar shear stress is also accompanied by a singular stress field of the transverse normal (peel) stress near the edge. The numerical study is conducted for both single layer PZT plates for sensory applications and hybrid sandwich plates for smart structural applications. It is shown that the EKM solution predicts the displacement, stress, electric potential and electric displacement fields very accurately including the sharp variations of stresses near the edge, under the potential loading with just two/three terms in the trial function.

... Further, 3D solutions serve as the benchmark for assessing the two-dimensional theories and other numerical methods for analyzing composite and piezoelectric plates accurately. In literature researchers used different approaches, namely Pagano's classical approach by Heyliger and Brooks (1995); Heyliger and Saravanos (1995); Heyliger and Brooks (1996); Pan and Heyliger (2003), the state space approach by Yang et al. (1994Yang et al. ( , 1995 ; Batra and Liang (1997); Xu et al. (1997); Chen and Ding (2002); Sheng et al. (2007), the series expansion approach by Dube et al. (1996a,b); Hussein and Heyliger (1998) and the asymptotic approach by Cheng et al. (1999Cheng et al. ( , 2000; Cheng and Batra (2000a,b), extended Kantorovich method by Kapuria and Kumari (2012); Kumari and Behera (2017a); Behera and Kumari (2019); Kapuria and Dhanesh (2017), to develop analytical 3D solution for plate structures. The limitation of exact and analytical solution is that these solutions are limited only to a certain type of boundary condition, geometry, and configurations due to the mathematical complexity of the problems. ...

For the first time, an accurate analytical solution, based on coupled three-dimensional (3D) piezoelasticity equations, is presented for free vibration analysis of the angle-ply elastic and piezoelectric flat laminated panels under arbitrary boundary conditions. The present analytical solution is applicable to composite, sandwich and hybrid panels having arbitrary angle-ply lay-up, material properties, and boundary conditions. The modified Hamiltons principle approach has been applied to derive the weak form of governing equations where stresses, displacements, electric potential, and electric displacement field variables are considered as primary variables. Thereafter, multi-term multi-field extended Kantorovich approach (MMEKM) is employed to transform the governing equation into two sets of algebraic-ordinary differential equations (ODEs), one along in-plane (x) and other along the thickness (z) direction, respectively. These ODEs are solved in closed-form manner, which ensures the same order of accuracy for all the variables (stresses, displacements, and electric variables) by satisfying the boundary and continuity equations in exact manners. A robust algorithm is developed for extracting the natural frequencies and mode shapes. The numerical results are reported for various configurations such as elastic panels, sandwich panels and piezoelectric panels under different sets of boundary conditions. The effect of ply-angle and thickness to span ratio (s) on the dynamic behavior of the panels are also investigated. The presented 3D analytical solution will be helpful in the assessment of various 1D theories and numerical methods.

... Sheng et al. [23] established the state equation for laminated piezoelectric plate based on the theories of 3D piezoelectroelasticity by assuming appropriate boundary functions. ...

This paper presents a finite element model for a
Sandwich Plate containing a piezoelectric core. A sandwich plate
with a piezoelectric core is constructed using the shear mode of
piezoelectric materials. The orientation of poling vector has a
significant effect on deflection and stress induced in the piezoactuated
adaptive sandwich plate. In the present study, the influence
of this factor for a clamped-clamped-free-free and simple-simplefree-
free square sandwich plate is investigated using Finite Element
Method. The study uses ABAQUS (v.6.7) software to derive the finite
element model of the sandwich plate. By using this model, the study
gives the influences of the poling vector angle on the response of the
smart structure and determines the maximum transverse displacement
and maximum stress induced.

... [13] presented the analytical solution for functionally graded magneto-electro-elastic plane beams. Sheng et al. [14] obtained the state space solution for thick laminated piezoelectric plates with clamped and electric open-circuited boundary conditions. Leung et al. [15] presented a new symplectic approach for piezoelectric cantilever composite plates. ...

A two-dimensional analysis is presented for piezoelectric beam with variable thickness which is simply supported and grounded along its two ends. According to the governing equations of plane stress problems, the displacement solutions, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at two ends of the beam, are derived. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams. The present solutions show a good convergence and the numerical results are presented and compared with those available in the literature. The method could be applied to control engineering and other projects with highly accurate demand on stress and displacement analysis such as the design of micro-mechanical apparatuses.

The effect of the non-homogeneity of material properties has been considered the important variation mechanism in the static responses of quasicrystal structures, but the existing theoretical model for it is unable to simulate the material change format beyond the exponential function. In this paper, we create a new model of functionally graded multilayered 1D piezoelectric quasicrystal plates using the state vector approach, in which varying functionally graded electro-elastic properties can be extended from exponential to linear and higher order in the thickness direction. Based on the state equations, an analytical solution for a single plate has been derived, and the result for the corresponding multilayered case is obtained utilizing the propagator matrix method. The present study shows, in particular, that coefficient orders of two varying functions (the power function and the exponential function) of the material gradient provide the ability to tailor the mechanical behaviors in the system’s phonon, phason, and electric fields. Moreover, the insensitive points of phonon stress and electric potential under functionally graded effects in the quasicrystal layer are observed. In addition, the influences of stacking sequences and discontinuity of horizontal stress are explored in the simulation by the new model. The results are very useful for the design and understanding of the characterization of functionally graded piezoelectric quasicrystal materials in their applications to multilayered systems.

Based on state space method, a 2-D model is established for metallic plates with composite patches as reinforcement, the governing equations for the model are derived, and the stress and strain distributions are analyzed. By means of MATLAB, the structure deflection, the adhesive peel stress, and the shear stress under tensile load are figured out. In order to validate the analytical model, the finite element analysis (FEA) is also applied and a FEA model is developed by ABAQUS. Both models give pretty identical results. In the end, using the analytical model, the interlaminar stress and in-plane stress distributions through the thickness of the patch are calculated, which are known to be key contributions to composite failure, and the damage mode is briefly analyzed as well.

A hybrid analysis of the interfacial stresses near the free edges of piezoelectric unsymmetrical laminated plates under uniform extension is presented based on three-dimensional piezoelasticity. A state space equation for cross-ply asymmetric piezoelectric laminates is obtained by considering all the independent elastic and piezoelectric constants of the laminates. The equation satisfies the open-circuit electric, free boundary conditions at two opposite free edges. Using a transfer matrix and a recursive solution approach, a three-dimensional exact solution has been derived and this has been validated by comparing analytical results with those from finite element analyses. Discussion and comments on solutions obtained using the equations proposed are presented. Since all the continuity conditions across the interfaces between material layers are satisfied, the concentrations of interfacial stresses near free edge boundaries are determined precisely. The significant electromechanical influence of the boundary layer effect is found quantitatively.

Free edge effect of laminated plates has been extensively investigated in the past two decades. Due to the boundary condition limitation, very limited work on piezoelectric laminated plates with free edges was carried out. In this paper, coupled and uncoupled analytical analyses on the interlaminar stresses in the vicinity of the free edges of piezoelectric laminated plates are presented on the basis of three-dimensional elasticity and piezoelectricity. The state space equations for cross-ply piezoelectric laminates subjected to uniaxial extension are obtained by considering all independent elastic and piezoelectric constants. The equations satisfy the boundary conditions at free edges and the continuity conditions across the interfaces between plies of the laminates. Three dimensional exact solution is sought and validated by comparing present numerical results with those of existing approximate analytical and finite element models. The singularity of the interlaminar stresses near the free edges is confirmed and the electromechanical coupling effects give much higher interlaminar stresses at the free edges in comparison with those of the corresponding uncoupled cases.

The article is to present an overview of various three-dimensional (3D) analytical approaches for the analysis of multilayered and functionally graded (FG) piezoelectric plates and shells. The reported 3D approaches in the literature are classified as four different approaches, namely, Pagano's classical approach, the state space approach, the series expansion approach and the asymptotic approach. Both the mixed formulation and displacement-based formulation for the 3D analysis of multilayered piezoelectric plates are derived. The analytical process, based on the 3D formulations, for the aforementioned approaches is briefly interpreted. The present formulations of multilayered piezoelectric plates can also be used for the analysis of FG piezoelectric plates, of which material properties are heterogeneous through the thickness coordinate, by artificially dividing the plate as NL-layered plates with constant coefficients in an average sense for each layer. The present formulations can also be extended to the ones of piezoelectric shells using the associated shell coordinates. A comprehensive comparison among the 3D results available in the literature using various approaches is made. For illustration, the through-thickness distributions of various field variables for the simply-supported, multilayered and FG piezoelectric plates are presented using the asymptotic approach and doubly checked with a newly-proposed meshless method. The literature dealing with the 3D analysis of multilayered and FG piezoelectric plates is surveyed and included. This review article contains 191 references.

3D hybrid analyses on the interlaminar stresses near the free edges of piezoelectric laminated plates are presented on the basis of three-dimensional piezoelasticity. The state space equations for cross-ply piezoelectric laminates subjected to a uniform axial extension are obtained by considering all the independent elastic and piezoelectric constants. With the application of the transfer matrix and recursive solution approach, the equations satisfy the traction-free and open-circuit boundary conditions at free edges and the continuity conditions across the interfaces between material layers. Three-dimensional exact solution is sought and validated by comparing the present analytical results with those from existing approximate and finite element models. The singularities of interlaminar stresses near free edges are observed and the significant electromechanical influence on the free-edge effect is found.

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green’s functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii’s formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.

In this survey, developments in the theory of both thermoelasticity and fluid mechanics have been reviewed, and appropriate references are supplied throughout. Review of the method of the matrix exponential, which constitutes the basis of the state-space approach, is given.PACS Nos.: 46.35+z, 47.65+aNous passons ici en revue les développements de la mécanique des fluides et de la thermoélasticité et nous donnons les références appropriées. Nous révisons aussi la méthode de l'exposant matriciel qui constitue la base de l'approche par espace d'états.[Traduit par la Rédaction]

A state space formulation is established for the nonaxisymmetric space problem of transversely isotropic piezoelectric media
in a system of cylindrical coordinate by introducing the state vector. Using the Hankel transform and the Fourier series,
the state vector equations are transformed into ordinary differential equations. By the use of the matrix methods, the analytical
solutions of a single piezoelectric layer are presented in the form of the product of initial state variables and transfer
matrix. The applications of state vector solutions are discussed. An analytical solution for a semiinfinite piezoelectric
medium subjected to the vertical point forceP
z, horizontal point forceP
x along x-direction and point electric charge Q at the origin of the surface is presented. According to the continuity conditions
at the interfaces, the general solution formulation forN-layered transversely isotropic piezoelectric media is given.

Discarding any assumptions about displacement or stress models, the state equation for an orthotropic thick walled cylinder is established in the axisymmetric condition. The exact solution is presented for the statics of laminated thick walled cylinder carrying an abitrary distribution of axisymmetric load. Every equation of elasticity can be satisfied and all the elastic constants are taken into account. Arbitrary precision of a desired order can be obtained. Numerical results are calculated and compared with those of SAP5.

Electromechanical transducers, first used in sonar systems, convert electrical energy into mechanical energy, thanks to the piezoelectric effect, then to acoustic energy, with the generation and the radiation of an acoustic wave in a fluid. To design new transducers and to understand their physical behaviour, several physical mechanisms have to be described (piezoelectric- elastic-fluid-structure coupling-acoustic radiation). To solve such problems, analytical and semi-analytical approaches often rely upon simplifying hypotheses, in terms of geometry of the transducers, behaviour of the piezoelectric part of the device, radiation condition or frequency range of interest. Among these approaches, the equivalent electrical scheme (Beranek 1954) is based on a classical lumped constant representation with masses and springs, and the transfer matrix (Neppiras 1973) uses plane wave approximations.With the help of analytical expressions, these approaches become operational tools, leading to the realization of numerous transducers.

In an attempt to develop an efficient analytical approach for the electromechanical analysis of laminated piezoelectric structures, an exact transfer-matrix-based methodology is presented. The state space equations for a three-dimensional piezoelectric lamina are first derived by eliminating the in-plane stresses and electric displacements from the governing equations. The transfer matrix is rendered in block form by judiciously arranging the slate variables and its property is then exploited to minimize the computational effort. Using the approach, an exact analysis of coupled electroelastic behavior of a rectangular piezoelectric plate of 6mm crystal symmetry subjected to a mechanical or electrical load is presented. A simply supported square plate made of barium titanate is analyzed as a numerical example and results are compared to those from the corresponding uncoupled analysis.

A general solution for the equilibrium equations of piezoelectric media under body forces is obtained. With regard to the transversely isotropic piezoelectric material, closed forms for the displacements and electric potential function for an infinite solid loaded with point forces and point charge are then obtained by using the general solution together with potential theory and constructing a kind of harmonic functions. Thus, the fundamental solutions which are utilizable in the boundary element method are obtained.

Closed-form expressions are obtained for the infinite-body Green's functions for a transversely isotropic piezoelectric medium. The four Green's functions represent the coupled elastic and electric response to an applied point force or point charge. The Green's functions are obtained using a formulation where the three displacements and the electric potential are derivable from two potential functions. When piezoelectric coupling is absent, the results reduce to those for uncoupled elasticity and electrostatics.

In this study, the state equation for axisymmetric bending of laminated transversely isotropic circular plates on elastic foundation is established on the basis of three-dimensional elasticity. By using the expansions of Bessel functions, an analytical solution of the problem is presented. As a result, all the fundamental equations of three-dimensional elasticity can be satisfied exactly and all the independent elastic constants can be fully taken into account. Furthermore, the continuity conditions at the interfaces of plies can also be satisfied.

On the basis of three-dimensional elasticity, this paper presents a free vibration analysis of cross-ply laminated rectangular plates with clamped boundaries. The analysis is based on a recursive solution suitable for three-dimensional vibration analysis of simply supported plates. Clamped boundary conditions are imposed by suppressing the edge displacements of a number of planes which are parallel to the mid-plane. This is achieved by coupling a number of different vibration modes of the corresponding simply supported plate using a Lagrange multiplier method. A satisfactory solution can be obtained by choosing suitably larger numbers of the coupled vibration modes and the fixed planes across the thickness of the plate. Numerical results are presented to show the convergence of the solution. Results are also obtained for either isotropic or cross-ply laminated plates having different combinations of simply supported and clamped boundaries.

Classical laminated plate theory (CLPT) is applied to a laminate plate with induced strain actuators, such as piezoceramic patch, bonded to its surface or embedded within the laminate to develop an induced strain actuation theory that allows for the actuator patch to be spatially distributed. When piezoceramic patches are subjected to voltage fields, the equivalent external forces induced by piezoceramic patches can be determined upon the assumption of free constraint for the expansion or contraction of piezoceramic patches. This assumption is generally done in thermal expansion problem. Several examples, including pure bending and pure extension, are illustrated. For the case of pure bending, a comparison between the current work and that of Dimitriadis et al. (1989) is given. In addition, an orthotropic angle-ply laminate with an embedded piezoceramic patch is presented to show the coupling of bending and extension.

The development and experimental verification of the induced strain actuation of plate components of an intelligent structure is presented. Equations relating the actuation strains, created by induced strain actuators, to the strains induced in the actuator/substrate system are derived for isotropic and anisotropic plates. Plate strain energy relations are also developed. Several exact solutions are found for simple actuator/substrate systems, and a general procedure for solving the strain energy equations with a Rayleigh-Ritz technique is formulated. Approximate Ritz solutions lead to both an understanding of the system design parameters and to detailed models of cantilever plate systems. Simple test articles were used to verify the accuracy of the basic induced strain actuator/substrate system models, and cantilever plate test articles were built and tested to verify the ability of the models to predict the strains induced in systems with extensive stiffness coupling and complicated boundary conditions.

A finite element formulation is presented for modeling the behavior of laminated composites with integrated piezoelectric sensors and actuators. This model is valid for both con tinuous and segmented piezoelectric elements that can be either surface bonded or embedded in the laminated plate. The present model takes into account the mass and the stiffness of the piezoelectric patches. The formulation is based on the first-order shear deformation theory, which is applicable for both thin and moderately thick plates. An additional feature of the present model is that it does not introduce the voltage as an additional degree of freedom. The charge/current generated by the sensor and the response of the plate to an actuator voltage can be computed independently. These features are then coupled with a constant-gain negative-velocity/positive-position feedback control algorithm to actively control the transient response of the plate in a closed loop. Numerical results are presented which indicate the increase in damping as the feedback gain is increased. The in fluence of stacking sequence and boundary conditions on the controlled transient response of the plate is examined.

A finite element formulation is presented for modeling the dynamic as well as static response of laminated composites containing distributed piezoelectric ceramics subjected to both mechanical and electrical loadings. The formulation was derived from the variational principle with consideration for both the total potential energy of the structures and the electrical potential energy of the piezoceramics. An eight-node three-dimensional composite brick element was implemented for the analysis, and three-dimensional incompatible modes were introduced to take into account the global bending behavior resulting from the local deformations of the piezoceramics. Experiments were also conducted to verify the analysis and the computer simulations. Overall, the comparisons between the predictions and the data agreed fairly well. Numerical examples were also generated by coupling the analysis with simple control algorithms to control actively the response of the integrated structures in a closed loop.

An exact three-dimensional solution for the problem of a simply supported rectangular homogeneous piezoelectric plate is obtained, in the framework of the linear theory of piezoelectricity. The plate is made of a transversely isotropic material, is earthed on the lateral boundary, and is subjected to prescribed surface charge and tractions on the end faces. The limit of this solution as the plate thickness aspect ratio approaches zero is explicitly carried out. The analytical results obtained may constitute a reference case when developing or applying two-dimensional plate theories for the analysis of more complex piezoelectric problems. A numerical investigation in the case of a square uniformly loaded plate is also performed, in order to evaluate the influence of the thickness-to-side ratio on the three-dimensional solution of the plate problem.

In an attempt to develop an efficient analytical approach for the electromechanical analysis of laminated piezoelectric structures, an exact transfer-matrix-based methodology is presented. The state space equations for a three-dimensional piezoelectric lamina are first derived by eliminating the in-plane stresses and electric displacements from the governing equations. The transfer matrix is rendered in block form by judiciously arranging the state variables and its property is then exploited to minimize the computational effort. Using the approach, an exact analysis of coupled electroelastic behavior of a rectangular piezoelectric plate of 6mm crystal symmetry subjected to a mechanical or electrical load is presented. A simply supported square plate made of barium titanate is analyzed as a numerical example and results are compared to those from the corresponding uncoupled analysis.

An exact elasticity solution for an orthotropic cylindrical shell with piezoelectric layers is obtained in this paper. The stress and displacement distributions are presented. The influence of the piezoelectric layers on the mechanical behavior of structures is studied. Both the direct piezoelectric effect and the converse piezoelectrical effect of the piezoelectric material are investigated. Results presented in this paper can be used to study various approximate shell theories used in the numerical simulations of piezoelectric structures.

This paper presents a unified exact analysis for the statics and dynamics of a class of thick laminates. A three-dimensional, linear, small deformation theory of elasticity solution is developed for the bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. All the nine elastic constants of orthotropy are taken into account. The solution is formally exact and leads to simple infinite series for stresses and displacements in flexure, forced vibration and “beam-column” type problems and to closed form characteristic equations for free vibration and buckling problems. For free vibration of plates, the present analysis yields a triply infinite spectrum of frequencies instead of only one doubly infinite spectrum by thin plate theory or three doubly infinite spectra by Reissner-Mindlin type analyses. Some numerical results are presented for plates and laminates. Comparison of results from thin plate, Reissner and Mindlin analyses with these yield some important conclusions regarding the validity and effects of the assumptions made in the approximate theories.

Exact solutions are developed for predicting the coupled electromechanical vibration characteristics of simply supported laminated piezoelectric plates, composed of orthorhombic layers. The three-dimensional equations of motion and the charge equation are solved using the assumptions of the linear theory of piezoelectricity. The through-thickness distributions for the displacements and electrostatic potential are functions of eight constants for each layer of the laminate. Enforcing the continuity and surface conditions results in a linear system of equations representing the behavior of the complete laminate. The determinant of this system must be zero at a resonant frequency. The natural frequencies are found numerically by first incrementally stepping through the frequency spectrum and refining the final frequencies using bisection. Representative frequencies and mode shapes are presented for a variety of lamination schemes and aspect ratios. (C) 1995 Acoustical Society of America.

A sandwich plate with a piezoelectric core is constructed using the shear mode of piezoelectric materials. The Raleigh-Ritz formulation for the proposed plate is developed based on the principle of stationary potential energy. An approximate solution for a clamped-clamped-free-free square sandwich plate is obtained based on the Raleigh-Ritz formulation. Computationally efficient finite element models for a clamped-clamped-free-free and a cantilever sandwich plate are developed. The solutions of the Raleigh-Ritz formulation are compared with those of finite element analysis. It is shown that the Raleigh-Ritz formulation is applicable to the proposed adaptive sandwich plate.

An exact three-dimensional state space solution is obtained for the static cylindrical bending of simply supported laminated plates with embedded shear mode piezoelectric actuators, and subjected to mechanical and electric loading on the upper and lower surfaces. Each layer of the laminate is made of either an orthotropic elastic material or a piezoelectric material whose poling direction lies in the plane of the plate, with perfect bonding between the adjoining layers. The displacements and stresses for a homogeneous piezoelectric plate for various length-to-thickness ratios are compared with those obtained by the first-order shear deformation theory. Results are also presented for a hybrid laminate with a shear mode piezoelectric core sandwiched between two elastic layers. A comparison of stresses with those in the corresponding surface-mounted extension actuation configuration shows that the longitudinal and shear stresses within the actuator are significantly smaller for the shear actuation mechanism. The analytical results can be used to assess the accuracy of different plate theories and/or validating finite element codes.

This paper provides the methodology on the use of piezoelectric patches for the repair of delaminated beams subjected to static loading, considering moment equilibrium. The singularity of the shear stresses induced at the crack tip of the delamination in the beam may result in sliding mode fracture, and a method to erase this singularity via piezoelectric materials is developed. An index defined to measure the effectiveness of repair is proposed. To facilitate design of the repair, a numerical solution obtained using the commercial finite element software ABAQUS is used, and simplified expressions based on simple beam theory are derived and validated. Parametric studies based on the geometry and material properties of the delaminated beam and piezoelectric materials are performed numerically to demonstrate the effectiveness of the proposed repair methodology.

Analytical formulations are presented which account for the coupled mechanical, electrical, and thermal response of piezoelectric composite laminates and plate structures. A robust layerwise theory is formulated with the inherent capability to explicitly model the active and sensory response of piezoelectric composite plates having general laminations in thermal environments. Finite element equations are derived and implemented for a bilinear 4-noded plate element. Applications demonstrate the capability to manage thermally induced bending and twisting deformations in symmetric and antisymmetric composite plates with piezoelectric actuators and attain thermal stability. The resultant stresses in the thermal piezoelectric composite laminates are also investigated.

This paper is concerned with the problem of point force and point charge applied in the interior of an infinite two-phase transversely isotropic piezoelectric solid. Based on the general solutions, by using the method of the image source, a series of displacement functions are constructed. The Green's functions are obtained when arbitrary constants are determined by the boundary conditions on the interface. Furthermore, we reduce the present solutions to the extension of Mindlin results and of Lorentz results for semi-infinite transversely isotropic piezoelectric materials by suitable substitutions of boundary conditions on the interface.

In this paper, a refined theory of laminated composite plates with piezoelectric laminae is developed. The equations of motion of the theory are developed using an energy principle. This formulation is based on linear piezoelectricity, and includes the coupling between mechanical deformations and the charge equations of electrostatics. The theory developed herein is hybrid in the sense that an equivalent single-layer theory is used for the mechanical displacement field, whereas the potential function for piezoelectric laminae is modeled using a layerwise discretization in the thickness direction. For the equivalent single layer, the third-order shear deformation theory of Reddy is used. This hybrid feature is good in that it demonstrates a way in which multilayered smart skin piezoelectric structures may be analysed to accommodate multiple voltage inputs and/or sensor outputs.

Analytical solutions are derived for the cylindrical bending of multilayered, linear, and anisotropic magneto-electro-elastic plates under simple-supported edge conditions. We construct the general solution in terms of a simple formalism for any homogeneous layer, from which any physical quantities can be solved for the given boundary conditions. For multilayered plates, we derive the solution in terms of the propagator matrices. A special feature of cylindrical bending, which distinguishes itself from the three-dimensional plate problem, is that the associated eigenvalues for any homogeneous layer are independent of the sinusoidal mode, and thus need to be solved only once. Typical numerical examples are also presented for a piezomagnetic plate, a two-layered piezoelectric/piezomagnetic plate, and a four layered piezoelectric/piezomagnetic plate, with different span-to-thickness ratios. In particular, the piezoelectric and piezomagnetic fields show certain interesting features, which give guidance on the development of piezoelectric/piezomagnetic thin-plate theories. Furthermore, it is shown that the variations of the elastic, electric, and magnetic quantities with thickness depend strongly upon the material property and layering, which could be useful in the analysis and design of smart composite structures with sensors/actuators.

Analytical solutions are derived for free vibrations of three-dimensional, linear anisotropic, magneto-electro-elastic, and multilayered rectangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism, from which any physical quantities can be solved for given boundary conditions. In particular, the dispersion equation that characterizes the relationship between the natural frequency and wavenumber can be obtained in a simple form. For multilayered plates, we derive the dispersion relation in terms of the propagator matrices. The present solution includes all previous solutions, such as piezoelectric, piezomagnetic, and purely elastic solutions as special cases, and can serve as benchmarks to various thick plate theories and numerical methods used for the modelling of layered composite structures. Typical natural frequencies and mode shapes are presented for sandwich piezoelectric/piezomagnetic plates. It is shown clearly that some of the modes are purely elastic while others are fully coupled with piezoelectric/piezomagnetic quantities, with the latter depending strongly upon the material property and stacking sequence. These frequency and mode shape features could be of particular interest to the analysis and design of various “smart” sensors/actuators constructed from magneto-electro-elastic composite laminates.

An analysis of a simply supported rectangular elastic plate forced into bending vibrations by the application of time harmonic voltages to piezoelectric actuators attached to its bottom and top surfaces is performed by using the equations of linear elasticity. The actuators have been modeled as thin surface films and mixed edge conditions are employed to simulate simple supports.

We use the three-dimensional linear theory of elasticity to analyse the steady-state vibrations of a simply-supported rectangular linear elastic laminated plate with embedded PZT layers. Some of these PZT layers act as actuators while the remaining act as sensors. It is assumed that there is perfect bonding between different layers. Numerical results for a thin and a thick plate containing one embedded actuator layer and one embedded sensor layer are presented. For the former case, the optimum location of the centroid of the excited rectangular region that will result in the maximum out-of-plane displacement for a given distribution of the applied voltage is also determined. Equivalently, an equal and opposite voltage applied to this region of a vibrating plate will be most effective in diminishing these vibrations. The maximum shear stress at the interface between the sensor and the lamina is lower than that between the actuator and the lamina. The point of maximum output voltage from the sensor coincides with that of peak out-of-plane displacement. The variations of displacement and stress components through the thickness for the thin and thick plates are similar.

In this study, the three-dimensional state equation for the jth ply of a laminated thick orthotropic plate is established in the local coordinate system according to the knowledge which has been introduced in the paper by Sundara Raja Iyengar and Pandya (1983, Fiber Sci. Technol.18, 19–36). Because all the physical quantities appearing in the state equation are just the compatible quantities of the interfaces, it is extremly convenient to develop the state equation of the whole plate. Furthermore, the number of unknowns included in the final equations has no relationship with that of the plies of the plate. Exact solutions are presented for the statics and dynamics of a three-ply orthotropic thick plates with simply supported edges. Numerical results are obtained and compared with those of Srinivas and Rao (1970, Int. J. Solids Structures6, 1463–1481) and thin plate theory.

The topic of vibration control with distributed actuators has been the subject of much recent research. A technique by which the random vibration of a rectangular panel is controlled by means of a pair of thin-sheet piezoceramic actuators is presented. Two accelerometers provide the feedback signals, which are processed digitally, according to a multivariable direct feedthrough feedback control law, to determine the control voltages. The control law design process decouples the feedback system into two actuator/sensor pairs. The procedure used to determine actuator and sensor locations, based on the modes to be controlled, is discussed. The method employed to select model orders and optimise feedback gains is also described. Finally, experimental results which demonstrate that significant reductions in vibration levels are achieved globally are presented.

Analytical formulations are presented which account for the coupled mechanical, electrical, and thermal response of piezoelectric composite shell structures. A new mixed multi-field laminate theory is developed which combines "single layer" assumptions for the displacements along with layerwise fields for the electric potential and temperature. This laminate theory is formulated using curvilinear coordinates and is based on the principles of linear thermopiezoelectricity. The mechanics have the inherent capability to explicitly model both the active and sensory responses of piezoelectric composite shells in thermal environment. Finite element equations are derived and implemented for an eight-noded shell element. Numerical studies are conducted to investigate both the sensory and active responses of piezoelectric composite shell structures subjected to thermal loads. Results for a cantilevered plate with an attached piezoelectric layer are com- pared with corresponding results from a commercial finite element code and a previously developed program. Additional studies are conducted on a cylindrical shell with an attached piezoelectric layer to demonstrate capabilities to achieve thermal shape control on curved piezoelectric structures.

Plane problem of electroelastic theory for a piezoelectric plates Laminated theory for spatially distributed induced strain actuators

- Ia Vekovishcheva
- Wang Bt
- Rogers

Vekovishcheva IA. Plane problem of electroelastic theory for a piezoelectric plates. Soviet Applied Mechanics 1976;11(2): 180–3. [6] Wang BT, Rogers CA. Laminated theory for spatially distributed induced strain actuators. Journal of Composite Materials 1991;25: 433–52.

The laminated thick walled cylinder in a state of non-plain strain A new approach to the thick laminated plates with clamped edges

- Hy Sheng
- Jr Fan
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