Orthotropic Elastic Plate Research Articles

A contour integral computation of stress intensity factors in the cracked orthotropic elastic plates

Stress intensity factors in the cracked orthotropic elastic plates are obtained numerically using a reciprocal work contour integral technique. The integral representation for calculating the intensities of elastic singular stresses at cracks, notches and corners can be extended to treat the rectilinearly anisotropic crack problems. An extension of the method can be done immediately with identification of the characteristic singular solutions and construction of the corresponding complementary elastic states. A centrally cracked tension plate with various degree of orthotropy is analyzed to assess accuracy of the computational procedures newly adopted. Two cracked orthotropic plates modelling bidirectional fiber-reinforced composites are treated: symmetric Mode I type of double edge crack tension plate and mixed mode type of cantilever plate with a single edge crack.

Cochlear mechanics: analysis for a pure tone.

A three-dimensional hydroelastic model of the cochlea is analyzed, in which the fluid is viscous and the basilar membrane is an inhomogeneous orthotropic elastic plate. After the solution is obtained using a multiple-scale approximation, comparison is made with experiment for the human cochlea.

A boundary-integral equation method for bending of orthotropic plates

A boundary-integral equation method for bending of thin orthotropic linear elastic plates with polygonal planform under quasistatic pressure loading is presented. A change from Huber's differential equation for the deflection to the simpler biharmonic equation of Kirchoff's isotropic plate theory by means of a linear transformation of the plate domain renders a non-classical boundary value problem; the transformed plate is embedded in a basic rectangular domain; coincidence of boundaries is applied as far as possible. Using Green's matrix of the basic isotropic plate, a vectorial boundary integral equation for a fictitious density function vector is developed, which has to be defined on part of the actual boundary only. Solution generally is achieved by a numerical procedure, and Green's matrix is split into a regular and a singular part. Analytic integrals of the latter are independent of the special choice of the basic domain. Having solved the integral equation, all interesting state variables within the isotropic domain are evaluated by means of integration and may be transformed back to the original orthotropic plate in a simple manner. Method is tested in example problems.

Thermal effect on frequencies of an orthotropic rectangular plate of linearly varying thickness

An analysis is presented here of the effect of a constant thermal gradient on free axisymmetric vibrations of an orthotropic elastic rectangular plate of linearly varying thickness. The governing differential equation of motion is solved by Frobenius' method. The frequencies corresponding to the first two modes of vibrations are obtained for an orthotropic rectangular plate with different combinations of boundary conditions for various values of taper constant and the temperature gradient.

On the Liapounov-Movchan stability of equilibrium of elastic orthotropic plates

On the basis of the Liapounov definition of stability and the necessary and sufficient Liapounov-Movchan conditions for a continuous body to remain stable, an elastic orthotropic rectangular plate subject to an uniformly distributed inplane compression, is analyzed. Two types of loading are considered: a conservative, dead, load and a nonconservative, follower, load. Two metrics discussed involve the maximum and the mean-square values of the delfection. The secondary critical load is found to depend heavily on the kind of anisotropy.

Buckle Pattern of Biaxially Compressed Simply Supported Orthotropic Rectangular Plates

Simply supported orthotropic elastic rectangular plates in biaxial compres sion are considered, and it is proved that such plates will buckle with m or n (or both) equal to 1, where m and n are the number of half waves in the two prin cipal directions. While this theorem may be known to many as an empirical computational fact, it does not appear to have been explicitly stated or proved heretofore in the literature on orthotropic plates

A mathematical model of the dynamics of the inner ear

A three-dimensional hydroelastic model of the dynamical motion in the cochlea is analysed. The fluid is Newtonian and incompressible, and the basilar membrane is modelled as an orthotropic elastic plate. Asymptotic expansions are introduced, based on slender-body theory and the relative high frequencies in the hearing range, which reduce the problem to an eigenvalue problem in the transverse cross-section. After this, an example is worked out and a comparison is made with experiment and the earlier low-frequency theory.

On Torsion and Transverse Flexure of Orthotropic Elastic Plates

The equations of transverse bending of shear-deformable plates are used for the derivation of a system of one-dimensional equations for beams with unsymmetrical cross section, with account for warping stiffness, in addition to bending, shearing, and twisting stiffness. Significant results of the analysis include the observation that the rate of change of differential bending moment is given by the difference between torque contribution due to plate twisting moments and torque contribution due to plate shear stress resultants; a formula for shear center location which generalizes a result by Griffith and Taylor so as to account for transverse shear deformability and end-section warping restraint; a second-order compatibility equation for the differential bending moment; a contracted boundary condition of support for unsymmetrical cross-section beam theory in place of an explicit consideration of the warping deformation boundary layer; and construction of a problem where the effect of the conditions of support of the beam is such as to give noncoincident shear center and twist center locations.

Non-linear oscillations of elastic orthotropic annular plates of variable thickness

A computational analysis of the non-linear oscillations of elastic orthotropic annular plates of variable thickness is presented. The non-linear boundary value problem is converted into a corresponding eigenvalue problem by using a Kantorovich time-averaging method. Then, by a Newton-Raphson iteration scheme in conjunction with the concept of analytical continuation, the solution to the non-linear oscillations of elastic orthotropic annular plates of variable thickness are obtained.

A semianalytic solution for rectangular plate bending

By substituting the basic function satisfying boundary conditions along two opposite edges in one direction of the plate and then using suitable transformation, the plate bending equation is reduced to an ordinary differential equation. The resulting equation is solved by finite difference technique by using a small number of discrete variables. Examples have been presented for a variety of isotropic and orthotropic elastic rectangular plate of different boundary conditions under various loadings. Excellent accuracy has been obtained.

Finite Twisting and Bending of Thin Rectangular Orthotropic Elastic Plates

Finite Twisting and Bending of Thin Rectangular Orthotropic Elastic Plates

Nonlinear bending of rectangular orthotropic plates

Displacement formulation of the governing field equations for nonlinear behaviour of rectilinearly orthotropic elastic plates are presented. These equations are then used to obtain an approximate solution of a simply supported rectangular plate subjected to a uniformly distributed load. The Galerkin-Dubnov technique is used to obtain the approximate solution. Nonlinear behaviour is studied for some specific degrees of orthotropy.

Analysis of a box made of elastic orthotropic plates

Elastic stresses, deflections and critical pressures for various modes of failure of a rectangular box placed in a uniform external pressure field are analyzed by a classical linear plate theory. The field equations governing the deflection of an orthotropic plate are solved by means of a superposition scheme and a method of single series expansion. The over-all solution for the entire box is arrived at by imposing continuity conditions at the edges where the individual plates meet. The method of analysis presented here is also adaptable to boxes on which external compressive loads are not uniformly distributed, but are self-equilibrating.

Asymptotic Theory for the Vibration of Orthotropic Plates

A two-dimensional theory for the vibration of orthotropic elastic plates is derived in a systematic and consistent manner without making any of the usual a priori assumptions regarding the dependence of field variables on the thickness coordinate. The analysis consists of the application of the boundary-layer technique to the equations of three-dimensional orthotropic elasticity. This is done by first introducing plate dimensions and an as yet unspecified length scale via changes in the independent variables. An asymptotic expansion of the field variables as well as the length scale in terms of a small plate parameter λ (the ratio of thickness to a characteristic length) is then performed near λ = 0. By equating corresponding powers of the parameter, successive systems of differential equations are obtained. The lowest-order system, when integrated across the plate thickness and combined with the boundary conditions, yields the simplest thin-plate theory. The higher-order systems yield the higher-order terms in the expansions and constitute thickness corrections.

A Note on a Non-Linear Theory on Bending of Orthotropic Sandwich Plates

SummaryThe derivation of a non-linear theory on bending of orthotropic sandwich plates is carried out by the principle of complementary energy from elasticity. The governing differential equations and natural boundary conditions are obtained. The assumptions used are, namely, the facings are orthotropic thin elastic plates with negligible bending rigidities and are made of the same material; the orthotropic core can take the transverse shear only; and the transverse shortening of the core may be ignored. The geometrical non-linearities are equivalent to von Kármán’s theory for single-layered plates. Through the introduction of a “shear function”, the number of differential equations is reduced to three and the equations are in rather simple form. It appears that the equations obtained here would require, comparatively, the least amount of work for analysing the finite deflection sandwich plate problems having a wide range of properties of practical interest.

Vibration of Corrugated Diaphragm

Formerly, one of us (T.A.) studied the elastostatic properties of the corrugated diaphragm, assuming it as an orthotropic elastic plate by smearing out the effect of the corrugations over a wave length, and found that the calculated results agreed well with those of the experiments. In this paper, analyzing the vibration properties of the corrugated diaphragm under the same assumptions as the static case, we obtained the approximate formulas for the natural frequencies of symmetric and asymmetric modes including the effect of concentrated mass. Furthermore, by the energy method, we studied the effects of initial tension and concentrated mass on the natural frequencies. In order to check this analysis, some experiments were performed. The experimental results showed good coincidence with the correspondig formulas, when the anisotropy parameter α is not so large.