Thin Anisotropic Plates Research Articles

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

A homotopy-based wavelet method for extreme large bending analysis of heterogeneous anisotropic plate with variable thickness on orthotropic foundation

A novel homotopy-based decoupled wavelet numerical strategy is developed to investigate the large-deflection bending of variable-thickness anisotropic thin plate constituted by heterogeneous material resting on orthotropic foundation. The anisotropic plates are described undergoing sinusoidally initial imperfections with inhomogeneous thickness and exponentially varying elastic modules, while the subgrade reaction involves the nonlinear support effects by Winkler-type foundation along with Pasternak-type orthotropy. Geometric nonlinearity by uneven thickness and large deformation, material nonlinearity by heterogeneity and contact nonlinearity of orthotropic foundation are simultaneously considered with the highly coupled and strongly nonlinear variable-coefficient Von Kármán equations derived. The accuracy and convergence of the present formulation are validated in excellent agreement with published results, while the new results of extreme bending solutions are provided, which are few given by other methods and further verify the effectiveness and superiority of proposed wavelet procedures. Parametric investigations of influences of material orthotropy and inhomogeneity, unevenness of thickness, amplitude of initial imperfection along with orthotropy of nonlinear Winkler–Pasternak foundation have been studied, which are of great significance on the extremely large bending properties.

A three‐phase laminated anisotropic elliptical inhomogeneity with internal uniform stress resultants

AbstractWe use the Stroh octet formalism for the coupled stretching and bending deformations of Kirchhoff laminated anisotropic thin plates together with the complex variable formulations of the decoupled stretching and bending deformations of Kirchhoff homogeneous isotropic thin plates to study the internal stress resultants inside a laminated anisotropic elliptical inhomogeneity bonded to a surrounding infinite laminated anisotropic matrix through a confocal interphase layer composed of a homogeneous isotropic plate when the matrix is subjected to uniform remote membrane stress resultants and bending moments. We prove that for given geometric and material parameters of the composite, the internal stress resultants inside the elliptical inhomogeneity are uniform when four specific conditions on the remote loading are satisfied. Furthermore, the internal uniform elastic field is determined in real‐form in terms of the two 8 × 8 fundamental elastic plate matrices for both the inhomogeneity and the matrix and the three 4 × 4 real matrices S, H, and L for the matrix.

New Straightforward Benchmark Solutions for Bending and Free Vibration of Clamped Anisotropic Rectangular Thin Plates

Abstract This study presents new straightforward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method. Being different from the previous studies that took pure trigonometric functions as the integral kernels, the exponential functions are adopted, and the unknowns to be determined are constituted after the integral transform, which overcomes the difficulty in solving the governing higher-order partial differential equations with odd derivatives with respect to both the in-plane coordinate variables, thus goes beyond the limit of conventional finite integral transforms that are only applicable to isotropic or orthotropic plates. The present study provides an easy-to-implement approach for similar complex problems, extending the scope of finite integral transforms with applications to plate problems. The validity of the method and accuracy of the new solutions that can serve as benchmarks are well confirmed by satisfactory comparison with the numerical solutions.

Analysis of ellipsometric parameters in the reflection of circularly polarized light from a thin anisotropic plate

The article presents the results of the analysis of the angular spectra of the ellipsometric parameters of the reflected wave when a circularly polarized light wave is incident on an anisotropic plate. The given dependences show a very high sensitivity of the ellipsometric parameters of the reflected light on the angle of incidence and the angle between the optical axis and the normal to the plate boundary. The energy reflection spectra themselves show much less variability when these parameters change. It should be especially emphasized the nature of the change in the ellipsometric angle Δ, which is responsible for the type of elliptical polarization - when Δ> 0, the polarization is left-handed, and when Δ <0, it is right-handed. It is shown that a thin anisotropic plate at certain angles can serve as a polarization converter of the incident radiation. The ellipsometry parameter ρ characterizes the degree of compression of the ellipse - when ρ = 1, the ellipse is transformed into a circle, and the light is circularly polarized in this case. Thus, a thin anisotropic plate can not only convert left-handed polarization to right-handed, but it can also control the very shape of the polarization ellipse. Such a plate can be used in conjunction with a layered medium, for example, a one-dimensional photonic crystal, to control the polarization of the incident circularly polarized light.

An Eshelby inclusion of parabolic shape in a Kirchhoff laminated anisotropic thin plate

An Eshelby inclusion of parabolic shape in a Kirchhoff laminated anisotropic thin plate

A MODEL OF A LINEAR VISCOELASTIC ANISOTROPIC PLATE, CONSIDERING AN ARBITRARY INHOMOGENEITY OF THE MATERIAL

The paper derives a model of an arbitrary inhomogeneous, anisotropic thin plate with linear viscoelastic components under a slowly varying in time loading; the components of a plate are also supposed anisotropic. The displacements of an inhomogeneous plate are developed following the integral formula approach of composite mechanics, by establishing an interrelation between Laplace transformants of displacement fields of the inhomogeneous plate and a homogeneous one with the corresponding type of anisotropy, so-called concomitant. The interrelation comes from an integral formula approximation by the structural functions technique; in this paper, the first-order approximation of the integral formula is considered. A system of partial differential equations for the first-order structural functions is adjusted for the case of linear viscoelastic components, both in the Laplace and time domain. For a concomitant body, Timoshenko plate equations are generalized onto the anisotropic case. An explicit form of displacements and strains of an inhomogeneous plate is obtained. An example−an instantly applied bending of a simply supported plate, composed of isotropic, asymmetrically laid layers−is considered.

A hierarchical wavelet method for nonlinear bending of materially and geometrically anisotropic thin plate

The paper aims at the application of a hierarchical wavelet homotopy methodology for solving nonlinear bending of anisotropic thin plates. The newly mechanical models of materially anisotropic plate and geometrically orthotropic combined plate at ship double bottom in large deflection have been formulated. The dimensionless coupled and nonlinear governing equations have been derived and decomposed into linear differential terms, while the solutions are expanded in wavelet series as a requirement for implementing the Homotopy recursive system of algebra equations by Galerkin method. Highly accurate Coiflet-type solutions have been obtained with good computational efficiency in excellent agreement with other numerical results in published literatures. Materially and geometrically orthotropic analysis have been conducted with different ratios of elasticity modulus or inertia moment in orthogonal directions, which indicate both orthotropy play an important role in largely deformed anisotropic plate under extreme loads. Convergent solutions over wider bending range of isotropic or anisotropic plates subjected to different boundary conditions are given which reveals our proposed wavelet methodology performs good superiority and versatility dealing with strongly nonlinear problems.

Koncentracija napona oko kružnih i eliptičnih rupa u drvenim laminatima

This paper examines the effects of material natural anisotropy of wood on stress concentration surrounding circular and elliptic holes. A thin anisotropic plate of wood was loaded with a constant pressure in one direction. The relationships between the stress components on the edge of the elliptic and circular holes were derived. For unloaded holes, normal stress values in small areas perpendicular to the edge of the hole were calculated. When calculating stress intensity, the relationships between the direction of applied pressure and the principal directions of elastic symmetry in spruce, ash, poplar, and oak wood were investigated.

Two-Dimensional Model of Second-Order Accuracy for an Anisotropic Plate

The deformation of a thin elastic anisotropic plate nonuniform in thickness is considered in a linear approximation. A two-dimensional model of second-order accuracy with respect to the small thickness parameter is constructed for a plate with general-type anisotropy (described by 21 elasticity moduli) via the asymptotic integration of the three-dimensional equations of elasticity theory. A system of differential equations describing the displacements of the middle layer with a differential order coinciding with the order of the Timoshenko-Reissner model is derived. The constructed model is suitable for studying the statics, dynamics, and stability of multilayered and functionally graded plates. The models of second-order accuracy for isotropic plates and plates with partial types of anisotropy were constructed earlier. This is the first time the model of second-order accuracy has been considered for general type anisotropy.

Multi-Conceptual Mechanical Design Optimization of Capacitive Pressure Sensors via Finite Element Analysis with use of Anisotropic Behavior of Silicon &lt;111&gt; Crystal: Summary of Design Optimization Approaches

In the world of micro-mechanical design of micro-sensors, up to date, there has not been substantial considerations given to the actual mechanical or structural aspect of the designs. Hence, most of the currently available designs are challenged to linearize the “non-linear” sensor’s output by utilization of electronic circuitry. In this research work, a micro-pressure diaphragm which possess linear pressure-deflection behaviour is designed via FEM optimization techniques. The diaphragm is modelled as a Silicon (111) plane, which possess plane isotropic properties. A circular centre boss section is added to the diaphragm and optimization is carried out, to achieve an optimum diaphragm geometry that would allow for flat or rigid deflection of this boss section under the applied surface pressure loading. The approximate closed−form deflection solutions are developed using the anisotropic thin plate theory and the diaphragm deflection behaviour of the FEM optimized design is compared with this thin plate theory model. This diaphragm design is proposed to be used as the top electrode plate of a capacitive pressure sensor, where linear pressure-capacitance change behaviour would become present. This pressure diaphragm has a pressure range of 0 to 206843 Pa (30 psi) with a pressure resolution of 689.5 Pa (0.1 psi).

EDGE STATE METHOD IN MECHANICS PROBLEMS CONCERNING ANISOTROPIC THIN PLATES

ObjectivesThe aims of the study are to expand the edge state method for solving bending and torsion problems concerning anisotropic thin plates, develop the theory of construction of the bases of spaces of interior and edge states based on a general approximate solution to the plate bending problem, formulate the relationships determining the desired elastic state and implement the developed theory in solving specific problems.MethodsThe fulfilment of the tasks is assumed to be based on the edge state method. The state spaces comprising the methodological basis are formed according to the fundamental system of Weierstrass polynomials.ResultsAn isomorphism of interior and edge state spaces is demonstrated, allowing a correspondence between the elements of these spaces to be unambiguously established. The isomorphism of spaces allows the process of finding the internal state to be reduced to the study of the edge state isomorphic to it. The mechanical characteristics are represented in the form of a Fourier series. In the case of the first and second fundamental mechanics problems, the Fourier coefficients are represented by scalar products having an energy implication: in the space of edge states, this consists in the work of external forces; in the space of internal states, it is the internal energy of elastic deformation. In the case of mixed mechanical problems, the search for an elastic state in the terms of the edge state method is reduced to solving an infinite system of algebraic equations. ConclusionThe solution of the first-tested basic problem of bending with torsion for a rectangular fibreglass plate with corresponding conclusions is given, as well as the problems of torsion for a plate of nontrivial form. Commentaryis provided concerning the unreasonableness of the solution of the second fundamental problem, as well as the problem with mixed boundary conditions for a rectangular plate where twisting and bending forces are defined simultaneously on the one face, while the opposite face is squeezed. Both explicit and indirect signs of the convergence of the solution to the problem are presented along with a graphical visualisation of the results.

A decomposition method for nonlinear bending of anisotropic thin plates

A modified version of the Adomian decomposition method is applied to solve the differential equations of nonlinear bending of rectangular anisotropic thin plates. The von Kármám hypotheses are considered in the plate's kinematics. The nonlinear differential equations are decomposed into linear and nonlinear parts and the solution is expanded into series. Essentially, the approach is to embed the nonlinear effects into the anisotropic linear solution. Such incorporations are performed in a recursive fashion along with the Adomian's polynomials, which are responsible for the approximation of the nonlinear effects. In spite of the problem being three-dimensional, no partial solutions are required due to a modification of the standard method, which considers the inversion of the entire linear operator. As a result, the obtained solution respects the boundary conditions set and is independent of its symmetries. The pb-2 Rayleigh-Ritz method is used to obtain the plate's equilibrium path. A multistep procedure is proposed in order to increase the decomposition method convergence radius aiming at global convergence. Numeric results are obtained and compared to those found in the literature.

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. A notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. A closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.

Solving the Problem of Linear Viscoelasticity for Piecewise-Homogeneous Anisotropic Plates

An approximate method for solving the problem of linear viscoelasticity for thin anisotropic plates subject to transverse bending is proposed. The method of small parameter is used to reduce the problem to a sequence of boundary problems of applied theory of bending of plates solved using complex potentials. The general form of complex potentials in approximations and the boundary conditions for determining them are obtained. Problems for a plate with elliptic elastic inclusions are solved as an example. The numerical results for a plate with one, two elliptical (circular), and linear inclusions are analyzed.

Solving the Problem of Bending of Multiply Connected Plates with Elastic Inclusions

This paper describes a method for determining the strain state of a thin anisotropic plate with elastic arbitrarily arranged elliptical inclusions. Complex potentials are used to reduce the problem to determining functions of generalized complex variables, which, in turn, comes down to an overdetermined system of linear algebraic equations, solved by singular expansions. This paper presents the results of numerical calculations that helped establish the influence of rigidity of elastic inclusions, distances between inclusions, and their geometric characteristics on the bending moments occurring in the plate. It is found that the specific properties of distribution of moments near the apexes of linear elastic inclusions, characterized by moment intensity coefficients, occur only in the case of sufficiently rigid and elastic inclusions.

Electrical Conductivity Tensor Modeling of Stratified Woven-Fabric Carbon Fiber Reinforced Polymer Composite Materials

In this paper, a new multi-scale approach to predict the effective electrical conductivity tensor of a woven-fabric composite layer for frequencies lower than 10 MHz is presented. The result is then used to model the behavior of a stratified composite material composed by a pileup of woven-fabric layers with different relative orientations. To take into account the orientation and the anisotropy of the material, a 3-D multilayer hexahedral degenerated elements method is used to model the anisotropic thin plate. This approach enables to take into account random distribution of carbon fibers in the yarns, 3-D effects, and electric percolation threshold with an acceptable numerical cost. The developed model is then validated by results for one anisotropic thin layer.

A recursive methodology for the solution of semi-analytical rectangular anisotropic thin plates in linear bending

• Novel application of Adomian Decomposition Method for elasticity problems. • New semi-analytical solutions of anisotropic thin plate in bending. • A new recursive methodology driven by a constitutive decomposition. • A starting isotropic solution progressively enhanced by anisotropic influence. • Plate results for several aspect and anisotropic ratios and boundary conditions. A new recursive methodology is introduced to solve anisotropic thin plates bending problems, which is based on three concepts: (a) the plate differential operator additively decomposed obeying a material constitutive hierarchy; (b) the plate displacement field expanded into an infinite series and (c) an homotopy-like scheme applied to determine each term of the series. The pb-2 Rayleigh–Ritz Method is adopted to construct the solution space. Convergence conditions are presented and related to the differential operator decomposition and material’s anisotropy degree. Different cases of geometry, loading and boundary conditions were studied using the methodology and excellent agreement with available solutions was found.

Natural vibrations of anisotropic plates with an internal curve with hinges

The objective of this paper is to propose a general algorithm to obtain approximate analytical solutions for the study of the free vibrations of a rectangular anisotropic thin plate with an internal curved line hinge and general restraints. In this system, there exists an intermediate condition that requires the continuity of the transverse displacements. It is well known that the difficulty in choosing admissible functions has been the most significant drawback of the Ritz method. To relax the admissibility requirement the Ritz method, with polynomials as coordinate functions, in conjunction with the Penalty Function method is proposed. This study is focus on different problems related the curved hinge and a natural parametrization is used to treat the mentioned curves. The accuracy of the formulation is ensured by comparing some numerical examples with those available in the literature. Cases not previously treated are particularly analyzed. Frequencies parameters and several sets of vibration mode shapes are included, to provide a better understanding of the dynamical behavior of these plates.

Application of Anisotropic Thin Plate Theory for Development of Displacements for Silicon &lt;111&gt; Crystals Plates in Cylindrical Systems

Silicon &lt;111&gt; crystal is one of the crystal orientations, which shows potential for application in micro device developments of pressure diaphragms for measurement purposes. To date, no theoretical basis has been examined to develop the deflections for computational analysis purposes [1]. This paper presents the development of the diaphragm deflections for Silicon &lt;111&gt; Crystal in Cylindrical coordinates system. The Silicon &lt;111&gt; crystal possesses transverse isotropic properties. Thus, an anisotropic thin plate theory is used here to develop the plate deflection. A numerical example is given to compare the theoretical results with Finite Element Analysis (FEA) results.