Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates

A novel asymptotic approach to the theory of non-homogeneous anisotropic plates is suggested. For the problem of linear static deformations we consider solutions, which are slowly varying in the plane of the plate in comparison to the thickness direction. A small parameter is introduced in the general equations of the theory of elasticity. According to the procedure of asymptotic splitting, the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. Three-dimensional conditions of compatibility make the analysis more efficient and straightforward. We obtain the system of equations of classical Kirchhoff's plate theory, including the balance equations, compatibility conditions, elastic relations and kinematic relations between the displacements and strain measures. Subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three-dimensional problem. Matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions. Additional effects, like electromechanical coupling for piezoelectric plates, can easily be incorporated into the model due to the modular structure of the analysis. The results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects, and provide a trustworthy algorithm for computation of the stressed state in the three-dimensional problem. Numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory.     

Dynamic linear viscoelasticity problems for piecewise homogeneous bodies

We consider problems on transient wave processes in linearly viscoelastic piecewise homogeneous bodies in the case of small strains, a bounded perturbation propagation domain, and bounded creep of the materials forming the homogeneous components of the bodies. We study problems related to the construction of solutions of such problems by the method of Laplace integral transform with respect to time and the subsequent inversion. We state assertions about the properties of Laplace transforms of the solutions, which simplify the process of determining the original functions. We also consider relations of correspondence between relaxation kernels that belong to different function classes but still affect transient wave processes in a similar way.     

Viscoelastic vibrations of a triangular plates

The problem of stationary vibrations of a viscoelastic equilateral triangular plate is treated. The vibrations are caused by the action of a uniformly distributed load which varies harmonically and by the motion of the simply supported boundary of the plate as a rigid body which vibrates at the same frequency. The level lines of the vibration amplitude are studied and the graphs of the amplitude distribution along the height of the triangle are given.     

Saint-Venant's problem for linear piezoelectric bodies

We seek a solution for a piezoelectric cylinder acted on the end faces by applied tractions and charges, under the hypothesis that both the stress and electric displacement fields depend linearly on the axial coordinate. The analysis, restricted to monoclinic materials of crystallographic class 2, leads to an explicit solution in terms of the strain and electric fields, which depend on the stress and charge resultants and on two scalar functions determined by the solution of a plane piezoelectric problem.     

On the boundary conditions for transversely isotropic piezoelectric plates

By invoking the theorem of work reciprocity for piezoelectric media, necessary conditions, which the prescribed edge data of the plate must fulfill in order that it should generate a decaying state within the plate, are established through generalizing the method proposed by Gregory and Wan. These decaying state conditions for the case of axisymmetric deformation of a transversely isotropic piezoelectric circular plate when stress and electric displacement conditions are imposed on the plate edge are derived explicitly, which are then used for the formulation of boundary conditions for the plate theory solution (or the interior solution). Also an analytical solution of the axisymmetric decaying state of transversely isotropic piezoelectric circular plates is derived. Furthermore, the corresponding necessary conditions for the axisymmetric deformation of elastic circular plates are indeed reproduced directly.     

Coupled piezoelectric vibrations of quartz plates

The contributions of coupling with the electric field and mass of electrode coatings are taken into account in solutions of equations governing coupled thickness-shear, flexure and face-shear vibrational modes in rotated-Y-cut quartz plates.     

具损伤压电智能层合中厚板的非线性动力稳定性分析

基于各向异性损伤理论和压电理论，采用能量法和Lagrange方程，建立了具损伤压电智能层合中厚板在参数激励下的非线性运动控制方程，应用增量谐波平衡法，求解了具损伤压电智能层合中厚板的非线性动力稳定性问题。算例中，分析了损伤参数和压电效应对压电智能层合板非线性动力稳定性的影响，揭示了力电耦合效应的内在特征，并与有关文献的结果进行了比较。     

The reachability problem for thermoelastic plates

Communicated by C. Dafermos     

An inverse elastoplastic problem for plates

We study an inverse elastoplastic problem of determining the residual stresses, the plasticity zone, and the external loads for a plate for known residual deflections which occur after removal of these loads and elastic unloading. Assuming that the deformation theory of plasticity is valid at the active stage of deformation, we prove the theorem of unique solution. An iterative method of solution is proposed and a variational formulation of the problem is given. Some simple examples are considered. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 186–194, July–August, 1999.     

Limit analysis of plates with piecewise linear yield surface

Summary Rigid-plastic plates having a piecewise linear yield surface are studied by limit analysis. Extremum principles for the evaluation of specific dissipation power are defined by means of linear programming concepts. The law governing the plastic collapse of plates is formulated on the basis of the well-known kinematic theorem of limit analysis. A general procedure for the approximate determination of the collapse load is proposed. The paper ends with a brief numeric investigation of the uniformly loaded square plate.

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Sommario Si studiano attraverso l'analisi limite le piastre costituite di materiale rigido-plastico aventi superficie di snervamento di tipo poliedrico. I principi di estremo riguardanti la valutazione della potenza specifica di dissipazione vengono definiti attraverso i concetti della programmazione lineare. Attraverso il noto teorema cinematico dell'analisi limite, si formula la legge che regola il collasso plastico delle piastre. Viene proposto un procedimento per la determinazione approssimata del carico di collasso del tutto generale. Si conclude infine con una breve indagine numerica relativa al caso di piastra quadrata con carico uniforme.

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The results presented in the paper form part of a Research supported by the National Research Council (C.N.R.).     

A quasi-three-dimensional approach for homogeneous anisotropic plates undergoing extensional deformations

Summary In present paper, the Kane-Mindlin assumptions are applied to homogeneous anisotropic plates undergoing extensional deformations. A complex potential approach is formulated;to that end, a third complex potential is added to the two Lekhniskii complex potentials, [2]. The components of displacement and stress as well as boundary conditions are represented in terms of these three complex potentials. The advantage of the proposed approach is that problems of anisotropic plates under extensional deformations are treated as a quasi-three-dimensional problem by considering the effect of the transverse normal strain.     

Modelling of thin piezoelectric layers on plates

The derivation of plate equations for a plate consisting of two layers, one anisotropic elastic and one piezoelectric, is considered. Power series expansions in the thickness coordinate for the displacement components and the electric potential lead to recursion relations among the expansion functions. Using these in the boundary and interface conditions, a set of equations are obtained for some of the lowest-order expansion functions. This set is reduced to six equations corresponding to the symmetric (in-plane) and antisymmetric (bending) motions of the elastic layer. These equations are given to linear (for the symmetric equations) or quadratic (for the antisymmetric equations) order in the thickness. It is noted that it is, in principle, possible to go to any order, and that it is believed that the corresponding equations are asymptotically correct. A few numerical results for guided waves along the plate and a 1D actuator case illustrate the accuracy.     

Plane infinitesimal waves in homogeneous elastic plates

Using the theory of elastic Cosserat plates, we obtain conditions for reality of the frequencies of vibration.

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Résumé En utilisant la théorie des plaques élastiques de Cosserat, on obtient les conditions pour réalité des fréquences de vibration.

     

Bending of functionally graded piezoelectric rectangular plates

By introducing two displacement functions as well as two stress functions, two independent state equations with variable coefficients are derived from the three-dimensional theory equations of piezoelasticity for transverse isotropy. A laminated approximation is used to transform the state equations to those with constant coefficients in each sub-layer. The bending problem of a functionally graded rectangular plate is then analyzed based on the state equations. Numerical results are presented and the effect of material gradient index is discussed. Supported by the National Natural Sciences Foundation of China (No. 10002016).     

Viscoelastic shear horizontal wave in graded and layered plates

In this paper, viscoelastic shear horizontal (SH) wave propagation in functionally graded material (FGM) plates and laminated plates are investigated. The controlling differential equation in terms of displacements is deduced based on the Kelvin–Voigt viscoelastic theory. The SH wave characteristics is controlled by two elastic constants and their corresponding viscous coefficients. By the Legendre polynomial series method, the asymptotic solutions are obtained. In order to verify the validity of the method, a homogeneous plate is calculated to make a comparison with available data. Through three different graded plates, the influences of gradient shapes on dispersion and attenuation are discussed. The viscous effects on the displacement and stress shapes are illustrated. The different boundary conditions are analyzed. The influential factors of the viscous effect are analyzed. Finally, two multilayered (two layer and five layer) viscoelastic plates that are composed of the same material volume fraction are calculated to show their differences from the graded plate.