General Solutions for the Statics of Anisotropic, Transversely Inhomogeneous Elastic Plates in Terms of Complex Functions

2006 ◽  
Vol 11 (6) ◽  
pp. 596-628 ◽  
Author(s):  
Kostas P. Soldatos
Keyword(s):  
General Solution ◽  
Elastic Plate ◽  
Plate Theory ◽  
High Order ◽  
Transverse Strain ◽  
Elastic Plates ◽  
General Solutions ◽  
Material Inhomogeneity ◽  
Complex Functions ◽  
Anisotropic Elastic

This paper develops the general solution of high-order partial differential equations (PDEs) that govern the static behavior of transversely inhomogeneous, anisotropic, elastic plates, in terms of complex functions. The basic development deals with the derivation of such a form of general solution for the PDEs associated with the most general, two-dimensional (“equivalent single-layered”), elastic plate theory available in the literature. The theory takes into consideration the effects of bending–stretching coupling due to possible un-symmetric forms of through-thickness material inhomogeneity. Most importantly, it also takes into consideration the effects of both transverse shear and transverse normal deformation in a manner that allows for a posteriori, multiple choices of transverse strain distributions. As a result of this basic and most general development, some interesting specializations yield, as particular cases, relevant general solutions of high-order PDEs associated with all of the conventional, elastic plate theories available in the literature.

The Effect of Transverse Shear Deformation on the Bending of Elastic Plates

1945 ◽  
Vol 12 (2) ◽  
pp. A69-A77 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
General Solution ◽  
Rectangular Plate ◽  
Transverse Shear ◽  
Plate Theory ◽  
Infinite Plate ◽  
Present Theory ◽  
System Of Equations ◽  
Elastic Plates ◽  
Classical Plate Theory

Abstract A system of equations is developed for the theory of bending of thin elastic plates which takes into account the transverse shear deformability of the plate. This system of equations is of such nature that three boundary conditions can and must be prescribed along the edge of the plate. The general solution of the system of equations is obtained in terms of two plane harmonic functions and one function which is the general solution of the equation Δψ − (10/h2)ψ = 0. The general results of the paper are applied (a) to the problem of torsion of a rectangular plate, (b) to the problems of plain bending and pure twisting of an infinite plate with a circular hole. In these two problems important differences are noted between the results of the present theory and the results obtained by means of the classical plate theory. It is indicated that the present theory may be applied to other problems where the deviations from the results of classical plate theory are of interest. Among these other problems is the determination of the reactions along the edges of a simply supported rectangular plate, where the classical theory leads to concentrated reactions at the corners of the plate. These concentrated reactions will not occur in the solution of the foregoing problem by means of the theory given in the present paper.

Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity

2007 ◽  
Vol 463 (2088) ◽  
pp. 3073-3087 ◽  
Author(s):  
Y.B Fu
Keyword(s):  
Plate Theory ◽  
Hamiltonian Formulation ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Elastic Plates ◽  
Spatial Variables ◽  
The Matrix ◽  
Anisotropic Elastic ◽  
Incompressible Elasticity ◽  
Steady Motions

Stroh's sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium/motion is written as a system of first-order differential equations for the displacement and traction in terms of one of the spatial variables. The so-called fundamental elasticity matrix N appearing in this formulation has the property that, when partitioned as a 2×2 block matrix, its 12- and 21-blocks are symmetric matrices and its 11-block is the transpose of its 22-block. This property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of the Stroh formalism in solving a large variety of problems in general anisotropic elasticity. First, we show that the matrix N is guaranteed to have the above property by the fact that the Stroh formulation is in fact a Hamiltonian formulation with one of the spatial variables acting as the time-like variable. This interpretation provides a much desired guide in dealing with other problems for which the governing equations are different, such as incompressible elasticity and problems associated with anisotropic elastic plates as described by the Kirchhoff plate theory. We show that for the last two problems the Hamiltonian interpretation simplifies the derivations significantly, leading to a Stroh formulation in each case which is equivalent to, but much simpler than, what is available in the existing literature.

Forced Axisymmetric Motions of Circular Elastic Plates

1965 ◽  
Vol 32 (4) ◽  
pp. 893-898 ◽  
Author(s):  
R. S. Weiner
Keyword(s):  
Elastic Plate ◽  
Integral Transform ◽  
Plate Theory ◽  
Uniform Pressure ◽  
Elastic Plates ◽  
Pressure Loading ◽  
Concentric Ring ◽  
Simply Supported ◽  
Arbitrary Time Dependence ◽  
Central Load

Axisymmetric motions of a circular elastic plate are considered here according to the Poisson-Kirchhoff plate theory. A concentric ring loading of arbitrary time dependence is examined and used to construct solutions for a concentrated central load and for a uniform pressure loading. The boundary of the plate is considered to be elastically built-in in a manner that prevents transverse edge motion and provides a restoring edge moment linearly related to edge rotation. Thus, limiting cases are a clamped plate and a simply supported plate. Finally, a discussion relating this work to the integral-transform approach of Sneddon is presented to enable physical interpretation and generalization of his approach.

Influence of coupled fields on free vibration and static behavior of functionally graded magneto-electro-thermo-elastic plate

2017 ◽  
Vol 29 (7) ◽  
pp. 1430-1455 ◽  
Author(s):  
Vinyas Mahesh ◽  
Piyush J Sagar ◽  
Subhaschandra Kattimani
Keyword(s):  
Finite Element ◽  
Electric Fields ◽  
Elastic Plate ◽  
Coupling Effect ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Finite Element Formulation ◽  
Geometrical Parameters ◽  
Element Formulation ◽  
Elastic Plates

In this article, the influence of full coupling between thermal, elastic, magnetic, and electric fields on the natural frequency of functionally graded magneto-electro-thermo-elastic plates has been investigated using finite element methods. The contribution of overall coupling effect as well as individual elastic, piezoelectric, piezomagnetic, and thermal phases toward the stiffness of magneto-electro-thermo-elastic plates is evaluated. A finite element formulation is derived using Hamilton’s principle and coupled constitutive equations of magneto-electro-thermo-elastic material. Based on the first-order shear deformation theory, kinematics relations are established and the corresponding finite element model is developed. Furthermore, the static studies of magneto-electro-elastic plate have been carried out by reducing the fully coupled finite element formulation to partially coupled state. Particular attention has been paid to investigate the influence of thermal fields, electric fields, and magnetic fields on the behavior of magneto-electro-elastic plate. In addition, the effect of pyrocoupling on the magneto-electro-elastic plate has also been studied. Furthermore, the effect of geometrical parameters such as aspect ratio, length-to-thickness ratio, stacking sequence, and boundary conditions is studied in detail. The investigation may contribute significantly in enhancing the performance and applicability of functionally graded magneto-electro-thermo-elastic structures in the field of sensors and actuators.

Large-Amplitude Oscillations of Unsymmetrically Laminated Anisotropic Rectangular Plates Including Shear, Rotatory Inertia, and Transverse Normal Stress

1985 ◽  
Vol 52 (3) ◽  
pp. 536-542 ◽  
Author(s):  
K. S. Sivakumaran ◽  
C. Y. Chia
Keyword(s):  
Boundary Conditions ◽  
Normal Stress ◽  
Transverse Shear ◽  
Plate Theory ◽  
Orientation Angle ◽  
Nonlinear Frequency ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Rotatory Inertia ◽  
Transverse Normal Stress

This paper is concerned with nonlinear free vibrations of generally laminated anisotropic elastic plates. Based on Reissner’s variational principle a nonlinear plate theory is developed. The effects of transverse shear, rotatory inertia, transverse normal stress, and transverse normal contraction or extension are included in this theory. Using the Galerkin procedure and principle of harmonic balance, approximate solutions to governing equations of unsymmetrically laminated rectangular plates including transverse shear, rotatory inertia, and transverse normal stress are formulated for various boundary conditions. Numerical results for the ratio of nonlinear frequency to linear frequency of unsymmetric angle-ply and cross-ply laminates are presented graphically for various values of elastic properties, fiber orientation angle, number of layers, and aspect ratio and for different boundary conditions. Present results are also compared with available data.

Nonlocal and Initial Problems for Quasilinear, Nonstrictly Hyperbolic Equations with General Solutions Represented by Superposition of Arbitrary Functions

2003 ◽  
Vol 10 (4) ◽  
pp. 687-707
Author(s):  
J. Gvazava
Keyword(s):  
Cauchy Problem ◽  
General Solution ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Quasilinear Equations ◽  
General Solutions ◽  
Parabolic Degeneracy

Abstract We have selected a class of hyperbolic quasilinear equations of second order, admitting parabolic degeneracy by the following criterion: they have a general solution represented by superposition of two arbitrary functions. For equations of this class we consider the initial Cauchy problem and nonlocal characteristic problems for which sufficient conditions are established for the solution solvability and uniquness; the domains of solution definition are described.

The Stroh Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
General Solution ◽  
Elastic Body ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Stroh's Formalism ◽  
Elasticity Problems ◽  
Anisotropic Elastic ◽  
Stroh’S Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

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