Paul R. Heyliger

Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates

Laminate and structural mechanics for the analysis of laminated composite plate structures with piezoelectric actuators and sensors are presented. The theories implement layerwise representations of displacements and electric potential, and can model both the global and local electromechanical response of smart composite laminates. Finite-element formulations are developed for the quasi-static and dynamic analysis of smart composite structures containing piezoelectric layers. Comparisons with an exact solution illustrate the accuracy, robustness and capability of the developed mechanics to capture the global and local response of thin and/or thick laminated piezoelectric plates. Additional correlations and numerical applications demonstrate the unique capabilities of the mechanics in analyzing the static and free-vibration response of composite plates with distributed piezoelectric actuators and sensors.

Mechanics and Computational Models for Laminated Piezoelectric Beams, Plates, and Shells

A considerable number of laminate theories, analytical approaches, numerical solutions and computational models have been reported for the analysis of laminates and structures with piezoelectric actuators or sensors. This article provides a review of published work in this area of mechanics. The reported laminate theories and structural mechanics are classified based on fundamental assumptions, the approximation of the through-the-thickness variation of the electromechanical state variables, the method of representation of piezoelectric layers, and their capability to model curvilinear geometries and thermal effects. The performance, advantages and limitations of the various categories of laminate theories are subsequently assessed by correlating results obtained by representative average models. The capability of each theory to model global structural response, local through-the-thickness variations of electromechanical variables, stresses, and piezoelectric laminates of high thickness is also quantified. This review article includes 103 references.

A higher order beam finite element for bending and vibration problems

Abstract The finite element equations for a variationally consistent higher order beam theory are presented for the static and dynamic behavior of rectangular beams. The higher order theory correctly accounts for the stress-free conditions on the upper and lower surfaces of the beam while retaining the parabolic shear strain distribution. The need for a shear correction coefficient is therefore eliminated. Full integration of the shear stiffness terms is shown to result in the recovery of the Kirchhoff constraint for thin beams without introducing spurious locking constraints. The accuracy of this formulation is demonstrated by using several numerical examples for the cases of small and large displacements. For a hinged-hinged beam, the linear thickness-shear mode frequency can be matched with the Timoshenko frequency to yield a shear coefficient of 0·824. Matching the bending frequencies between the two theories indicates a shear coefficient for the Timoshenko theory that changes with mode number and slenderness ratio. The influence of in-plane inertia and slenderness ratio on the non-linear frequency is examined for beams with a number of different support conditions.

Exact Solutions for Simply Supported Laminated Piezoelectric Plates

Exact solutions are presented for the static behavior of laminated piezoelectric plates with simple support. The upper and lower surfaces of the laminate can be subjected to a number of applied loadings, confined in this study to an applied transverse load or a specified surface potential. Each layer of the laminate can be piezoelectric, elastic/dielectric, or conducting, with perfect bonding assumed between each interface. Expressions are obtained for the components of displacement, stress, electric displacement, and potential through the thickness of the laminate. Representative examples are shown to demonstrate the fundamental behavior, and the influence of the piezoelectric coefficients and internal electrodes are discussed.

EXACT FREE-VIBRATION ANALYSIS OF LAMINATED PLATES WITH EMBEDDED PIEZOELECTRIC LAYERS

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.

Coupled Layerwise Analysis of Composite Beams with Embedded Piezoelectric Sensors and Actuators

Unified mechanics are developed with the capability to model both sensory and active composite laminates with embedded piezoelectric layers. Two discretelayer (or layerwise) formulations enable analysis of both global and local electromechanical response. The first assumes constant through-the-thickness displacement, while the second permits piecewise continuous variation. The mechanics include the contributions from elastic, piezoelectric and dielectric components. The incorporation of electric potential into the state variables permits representation of general electromechanical boundary conditions. Approximate finite element solutions for the static and freevibration analysis of beams are presented. Applications on composite beams demonstrate the capability to represent either sensory or active structures, and to model the complicated stressstrain fields, the interactions between passive/active layers and interfacial phenomena between sensors and composite plies. The capability to predict the dynamic characteristics under various electrical boundary conditions is demonstrated. Some advantages of the variable transverse displacement formulation on the freevibration response of sensory structures are also shown.

Exact Solutions for Laminated Piezoelectric Plates in Cylindrical Bending

Exact solutions are presented for the problem of piezoelectric laminates in cylindrical bending under an applied surface traction or potential. An arbitrary number of elastic or piezoelectric layers can be considered in this analysis. Example problems are considered for several representative cases, with resulting displacement, potential, stress, and electric displacement distributions shown to demonstrate the effects of the electroelastic coupling.

Free vibration of piezoelectric laminates in cylindrical bending

Exact solutions are presented for the free vibration behavior of piezoelectric laminates in cylindrical bending. The laminates can be composed of an arbitrary number of elastic and piezoelectric layers. The natural frequencies and through-thickness modal distributions are computed for the case where the upper and lower surfaces of the laminate are traction free. The electrostatic potential or the normal electric displacement is specified to be zero at these surfaces. All appropriate interface conditions are also satisfied. The resulting determinant equation is iteratively solved for the resonant frequencies, with the mode distributions of the elastic and electric field variables also computed. Representative examples are studied for thick and thin laminate geometries.

Layerwise mechanics and finite element model for laminated piezoelectric shells

A discrete-layer shell theory and associated finite element model is constructed for general laminated piezoelectric composite shells. The discrete-layer shell theory is based on linear piezoelectricity and accounts for general through-thickness variations of displacement and electrostatic potential by implementing one-dimensional piece-wise continuous Lagrange interpolation approximations over a specified number of sublayers. The formulation applies to shells of general shape and lamination. Initially, the static and dynamic behavior of a simply supported flat plate is studied to compare with available exact solutions, with excellent agreement being obtained. Static loading and free vibration of a cylindrical ring are then considered to evaluate the element and to study the fundamental behavior of active/sensory piezoelectric shells.

Discrete Layer Solution to Free Vibrations of Functionally Graded Magneto-Electro-Elastic Plates

Natural frequencies of orthotropic magneto-electro-elastic graded composite plates are determined using a discrete layer model with two different approaches. In the first, the functions describing the gradation of the materials properties through the thickness of the plate are incorporated into the governing equations. In the second approach, the plate is divided into a finite number of homogeneous layers. The model is validated by comparing the natural frequencies of a simply supported Al/ZrO2 graded square plate with the exact solution. Excellent agreement is obtained. Rectangular plates with different boundary conditions, aspect ratios, and made of different types of composite materials are also considered: Al/ZrO2 and BaTiO3/CoFe2O4 plates for which the volume fraction of the phases change as a function of the z coordinate, graphite/epoxy plates with the orientation of the fibers changing through the thickness of the plate, and plates having an exponential variation of the material properties. Applicability of the proposed model is not limited to specific boundary conditions and gradation functions.