Analysis of laminated piezoelectric composite plates using an inverse hyperbolic coupled plate theory

This paper presents a non-polynomial coupled plate theory for smart composite structures employing inverse hyperbolic displacement and electric potential functions. The theory is utilized towards analysis of composite piezoelectric plates operating in sensor and actuator modes. Particularly, the following three cases are studied: (i) passive laminated composite structure, (ii) composite piezoelectric plate actuator and (iii) unimorph and bimorph piezoelectric plate sensors. Analytical solutions are obtained for simply supported plates under static electrical and mechanical loads. These results are validated with existing 3D elasticity solutions and compared with other plate theory solutions. Furthermore, parametric studies are performed to determine the effect of loading, span-to-thickness ratio and lamination sequence on the response of the piezoelectric plate. Finally, the theory is applied to a transverse shear sensing device which utilizes transverse shear-electric field coupling in piezoelectric materials. This effect is often ignored in literature.It is observed that the maximum percentage error of the present theory, when compared with 3D results, is less than 3%, which is lower than other higher order plate theories.     

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping

In this paper we consider a one-dimensional linear thermoelastic system of Timoshenko type, where the heat conduction is given by Green and Naghdi theories. We prove a general decay result, from which the exponential and polynomial decays are only special cases.     

Optimization of vertical axisymmetric thermal displacements in a round plate using heat sources

We consider the problem of using heat sources for optimal control of the distribution of the vertical axisymmetric thermal displacements of a thin round plate with edge clamped at an angle of revolution. On the basis of the method of the inverse thermoelasticity problem we construct a solution of the control problem. For specific cases of heating the plate we carry out a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Poly a, Issue 36, 1992, pp. 105–110.     

Bending,buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory

In the present study, higher order shear and normal deformable plate theory is developed for analysis of incompressible functionally graded rectangular thick plates. Also, The effect of incompressibility is studied on the static, dynamic and stability responses of thick plate. It is assumed that plate is incompressible and the incompressibility condition is considered in addition to the governing equations for determining the unknowns. Since the plate is thick, higher order shear and normal deformable theory is applied so that the Legendre polynomials are used for expansion of displacement field components in the thickness direction. Also, it is supposed that material properties vary through the thickness based on the power law function. Utilizing the variational approach, governing equations for static, stability and dynamic analysis of plate are derived. Resulted equations are solved analytically for simply supported plates. Finally, the effects of material properties and dimensions on the response of incompressible plates are investigated in details.     

Optimization of a cylindrical composite shell with a viscoelastic core in axial compression

The problem of the optimal design of a composite shell in creep is formulated. The progressive buckling of a cross-wound reinforced cylindrical shell supported on a viscoelastic core is considered as a particular case. The reinforcement structure and shell thickness corresponding to minimum weight for a given load and service life are found.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 442–446, May–June, 1975.     

On subcritical development of a shear crack in a composite with viscoelastic components

We present results of an investigation of the development of a transverse shear crack in a composite material with linearly viscoelastic components under external shear load. The solution is divided into the following two main stages: determination of the time dependence of the crack tip opening displacement and determination of the crack-growth kinetics as a result of the solution of integral equations. In the first stage, we use the solution of the corresponding elastic problem of determination of the crack opening displacement and the problem of determination of the effective moduli of the composite reinforced with unidirectional discrete fibers. Using the theoretically proved principle of elasto-viscoelastic analogy and the method of Laplace inverse transformation, we obtain a solution in a time domain. In the second stage, using the criterion of critical crack opening displacement for a transverse shear crack and an equation for the viscoelastic crack opening displacement of this crack, we construct an equation of crack growth. We present results of the numerical solution, which illustrate the influence of relations between the relaxation parameters of the materials of the components on the durability of the body with a crack.     

Optimization of dynamic mechanical response of a composite plate using multi-field coupling with thermal constraints

We consider the problem of optimizing the dynamic response of a mechanically loaded, rectangular, electrically conductive anisotropic composite plate by applying an electromagnetic field, which exploits the electro-magneto-mechanical field coupling phenomenon. An important aspect of the formulated nonlinear partial differential equation (PDE)-constrained optimization model is the presence of a thermal constraint that prevents polymer matrix degradation in the composite material due to Joule heating produced by the electromagnetic field. A black-box optimization approach based on the active set algorithm is employed. A system of governing PDEs is solved using a series of sequential numerical procedures that includes the method of lines, Newmark time-stepping scheme, quasilinearization, integration of two-point boundary-value problems, and a superposition method followed by orthonormalization. Implementation in hyper-dual arithmetics facilitated automatic differentiation and computation of the gradient. Optimization results show that application of an electromagnetic field with optimal characteristics enables one to significantly reduce the amplitude of the plate vibrations while controlling for Joule heating.     

Bifurcation of the self-excited oscillations of a plate with slight damping in a supersonic gas flow

A non-linear boundary-value problem is considered which simulates the oscillations of a plate in a supersonic gas flow. The classical version of the formulation of the problem, proposed by Bolotin, as well as several of its modifications considered by Holmes and Marsden, are taken as a basis. The oscillations of the plate are studied assuming that the damping coefficient is small. This version of the formulation of the problem leads to the need to investigate the bifurcations of the self-excited oscillations in a non-linear boundary-value problem in a case which is close to the critical case of a double pair of pure imaginary values of the stability spectrum. The bifurcation problem is reduced to the investigation of a complex second order non-linear differential equation by the method of normal forms. All the stages in the investigation are carried out without using the Bubnov method.     

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term,nonlocal boundary condition,and localized damping term

The paper deals with the existence of a global solution of a singular one-dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized frictional damping a(x)u_t using the potential well theory. Furthermore, the general decay result is proved. We construct a suitable Lyapunov functional and make use of the perturbed energy method.     

Concentration of electroelastic fields in a composite piezoceramic plate with defects crossing the interface

The characteristics of fracture and strength of a composite piezoceramic plate with defects in the form of cracks and holes situated in both of the plate components are investigated. The corresponding boundary-value problems of electroelasticity are reduced to systems of singular integral equations by constructing integral representations of the complex potentials. The results of numerical realization of the constructed algorithms are reported.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 777–786, November–December, 1998.     

Concentration of electroelastic fields in a composite piezoceramic plate with a hole intersecting the interface of materials

A plane problem of electroelasticity is considered for an infinite compound plate with a hole located in both constituents of the plate. The corresponding boundary value problems is reduced to a system of singular integral equations of second kind, which is solved in numerical quadratures. Calculation results are presented that describe the concentration of electroelastic fields near the hole upon action at infinity of the vectors of mechanical stresses and electric field strength.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 359–366, May–June, 1999.     

On the stress distribution in a rectangular thick plate of a composite material with curved structure under forced vibration

The stress distribution in a rectangular plate of a multilayer composite material with a periodically curved structure under forced vibration is studied. It is assumed that the plate is hinge supported at opposite sides. The investigation is carried out within the exact three-dimensional linear theory of elasticity. The mechanical relationships of the plate material are described by the continuum theory of Akbarov and Guz'. The numerical results obtained by the finite element method show that even in low-frequency dynamic loading of the plate the extreme values of stresses, which appear as a result of the curving in the plate structure, considerably exceed those in the corresponding static loading.Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, Baku, Azerbaijan. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 447–454, July–August, 1999.