Inhomogeneous Plate Research Articles

Investigation of natural oscillations of inhomogeneous orthotropic circular plate lying on an inhomogeneous viscoelastic foundation

In the building of large engineering complexes, bridges and overpasses for various purposes and in many other areas the plates of widely different configurations are used. These plates are made of natural and artificial orthotropic materials. Among them, rectangular and circular plates are the most common. According to the above mentioned natural oscillations, engineer-designer and calculator need to properly assess real property of construction element and the influence of the environment, which is in contact during the operation. Therefore, the object of this study is inhomogeneous circular plate lying on inhomogeneous viscoelastic foundations. It is assumed that the moduli of elasticity and the plate density are continuous functions of the current radius. In this case, unlike homogeneous plates, the motion equation is complex differential equation with variable coefficients. In this regard, there is need to build an approximate analytical solution method. In the course of the study we used methods of separation of variables and Bubnov-Galerkin orthogonalization method, which gives effective results with homogeneous boundary conditions. An axisymmetric form of natural oscillations of orthotropic circular plate with inhomogeneous radius lying on an inhomogeneous viscoelastic foundation is considered. The case, when the contour around the plate is rigidly clamped, is studied in detail. Numerical analysis for concrete values of the characteristic parameters is carried out. The motion equation is obtained taking into account inhomogeneity of the plate and the foundation, as well as partial variable coefficients of the fourth order.

A single-element interferometer for measuring parallelism and uniformity of transparent plate

The transparent plates (such as organic glass, plastic plate) are widely used in the construction industry, high-tech products and scientific research applications, and its parallelism and uniformity measurement in the manufacture and quality control become more and more inevitable. Interferometer is a label-free, high-precision, and high-efficient device that can be used in many fields. According to a single-element interferometer, we demonstrate a measurement for the parallelism and uniformity of transparent medium. Beam-splitter cube is a key component. Half of plane wave laser source passes through the measured medium and the remaining half directly passes through the air, then these two halves with different optical paths meet in the beam-splitter cube. The parallelism or uniformity is determined by calculating interference fringe shift number during rotating the measured sample. The coherent beam is divided into two parts by a beam-splitter, one passes through the lens and then arrives at a photoelectric counter, and the other arrives at the observation plane of the charge-coupled device. The photoelectric counter is used to count the integer part of fringe shift number during rotating the sample; and the decimal part can be detected by calculating the phase difference of the two interferograms captured before and after rotation. The measurement principle of the proposed device is analyzed in detail, and the numerical simulations of the fringe shift number and the gray level changing with the sample rotation angle, the thickness and the refractive index of the sample are carried out. The simulation results show that the bigger the rotation angle, thickness and refractive index of the sample, the greater the fringe shift number will be. Therefore, the measurement accuracy can be improved by increasing the rotation angle and the thickness of the sample. In addition, we also simulate the measurement processes of two kinds of samples, which are unparallel and inhomogeneous transparent plates. The simulation results prove the feasibility and high accuracy of the proposed method. Finally, the optical experiment is conducted to demonstrate the practicability of the present device. The parallelism of a cuvette used for more than one year, is tested by our device. The results show that the difference in thickness between the cuvettes is on a micron scale, the peak-valley (PV) value is 9.92 m, and the root mean square (RMS) value is 2.2 m. And the difference between the contrast test results and the results from the proposed method is very small, the PV value is 0.569 m, and the RMS value is 0.131 m. The stability and repeatability of the proposed setup are tested in the experimental condition. The mean value and standard deviation of the fringe shift number during 30 min are 0.0012 and 0.0008, respectively. These results further testify the high accuracy and stability of our method. In conclusion, the performance of our measurement method is demonstrated with numerical simulation and optical experiment.

Reflectance and transmittance of light for weakly inhomogeneous plates in the lower orders of perturbation theory

Reflectance and transmittance of light for weakly inhomogeneous plates in the lower orders of perturbation theory

Reconstruction of the stiffness of an inhomogeneous elastic plate

The paper discusses the problem of reconstructing the inhomogeneous cylindrical, symmetric stiffness distribution of a round plate using information on the bias function for established oscillations, which is measured at a certain point. A solution is constructed to the direct problem using the Galerkin method and to the inverse problem of reconstructing the stiffness using an iterative approach based on the regularized linearization method. We present the results of calculation experiments on reconstructing different types of functions that show the efficiency of the proposed approach and make it possible to estimate changes in stiffness.

Large amplitude free vibration study of non-uniform plates with in-plane material inhomogeneity

Free vibration study of non-uniform plates with in-plane material inhomogeneity is carried out in the present work considering geometric nonlinearity. Inhomogeneous plates where the material properties vary along only x-axis (unidirectional) and along both x- and y-axis (bidirectional) are considered. The analysis is performed for two boundary conditions namely clamped and simply supported at all edges, under the action of a transverse uniformly distributed load. The large amplitude problem is formulated using nonlinear strain–displacement relations along with a variational form of energy method. A two-step solution procedure is utilised where, in the first part the static problem is solved and undetermined coefficients are found, subsequently the dynamic problem is taken up on the basis of previously determined coefficients. Validity of the results is successfully confirmed by comparison with the works of other researchers. The analysis reveals that the amplitude and taper parameter affect the loaded natural frequencies significantly. Three-dimensional mode shapes for linear and nonlinear cases are presented along with their respective contour plots.

Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle

The equilibrium problems for homogeneous and inhomogeneous plates with a crack is considered. We impose the nonpenetration condition, which has the form of inequality (such as Signorini type condition), on the crack faces. In this paper, we deal with the cases that the crack intersects the external boundary at zero angle (on the mid-plane). Using the fictitious domain method, we establish the unique solvability of four equilibrium problems for different cases of non-Lipschitz domains. In these cases a precise connection between equilibrium problems for a plate contacting with a rigid obstacle and for a plate with a crack is identified.

Existence of an optimal size of a delaminated rigid inclusion embedded in the Kirchhoff-Love plate

We consider equilibrium problems for an inhomogeneous plate with a crack situated at the inclusion-matrix interface. The matrix of the plate is assumed to be elastic. The boundary condition on the crack curve are given in the form of inequalities and describes the mutual nonpenetration of the crack faces. We analyze the dependence of solutions on the size of rigid inclusion. The existence of the solution to the optimal control problem is proved. For that problem the cost functional characterizes the deviation of the displacement vector from a given function, while the size parameter of rigid inclusion is chosen as the control function.

On flat equilibrium shapes of inhomogeneous anisotropic elastic plates and rods

An asymptotic analysis of deformations of anisotropic inhomogeneous thin elastic plates and rods is performed. In the linear formulation, the flat equilibrium shapes of the limiting surface and the limiting line of such thin bodies are shown to be unstable.

Deformations near an elliptical hole with surface effect in a laminated anisotropic thin plate

This work is concerned with the coupled stretching and bending deformation around an elliptical hole with surface energy in a laminated and inhomogeneous anisotropic elastic thin plate within the context of the Kirchhoff theory. A closed‐form full‐field solution is derived by using the octet formalism recently developed by Cheng and Reddy (2002, 2003, 2004, 2005) and by incorporating a simplified version of the surface elasticity model. In particular, explicit real‐form expressions of the hoop membrane stress resultant, hoop bending moment, in‐plane displacements and slopes on the mid‐plane along the edge of the elliptical hole are obtained.

Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes

Theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous material on flexural wave in the plates of continuously varying thickness is developed. For the wedges with thickness increasing as a power law of distance from its edge strong modifications of the wave dynamics with propagation distance are predicted. It is found that nonlinear absorption progressively disappearing with diminishing wave amplitude leads to complete attenuation of acoustic waves in most of the wedges exhibiting black hole phenomenon. It is also demonstrated that black holes exist beyond the geometrical acoustic approximation. Applications include nondestructive evaluation of micro-inhomogeneous materials and vibrations damping.

Setting the Linear Oscillations of Structural Heterogeneity Viscoelastic Lamellar Systems with Point Relations

The paper solves the problem of the variation formulation of the steady-linear oscillations of structurally inhomogeneous viscoelastic plate system with point connections. Under the influence of surface forces, range of motion and effort varies harmonically. The problem is reduced to solving a system of algebraic equations with complex parameters. The system of inhomogeneous linear equations is solved by the Gauss method with the release of the main elements in columns and rows of the matrix. For some specific problems, the amplitude-frequency characteristics are obtained.

Finite Integral Transform Method in Static Problems for Inhomogeneous Plates

A new modification of the finite integral transform method for solving two-dimensional linear boundary-value problems of general form is proposed. This method involves construction of two integral transformations over different variables of the domain such that the kernel of one of them is the transform (map) of the other and vice versa. The method is tested by solving bending problems for homogeneous and inhomogeneous plates

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method (SSDQM). The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model (ALM) is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.

Singularities of meniscus at the V-shaped edge

An understanding of the capillary rise of the meniscus formed on the V-shaped fibers is crucial for many applications. We classified the cases when the meniscus cannot be smooth by analyzing the local behavior of the solutions to the Laplace equation of capillarity near the sharp edge. The V-angle and two contact angles that the meniscus forms on two chemically different sides of the fiber form a 3D phase space. Smooth menisci constitute a special domain in this 3D space. The constructed diagram allows one to separate the solutions with smooth and non-smooth menisci. The obtained criteria were illustrated using chemically inhomogeneous plates, blades, square corners, and Janus V-shaped edges.

Analogies between Kirchhoff plates and functionally graded Saint-Venant beams under torsion

Exact solutions of elastic Kirchhoff plates are available only for special geometries, loadings and kinematic boundary constraints. An effective solution procedure, based on an analogy between functionally graded orthotropic Saint-Venant beams under torsion and inhomogeneous isotropic Kirchhoff plates, with no kinematic boundary constraints, is proposed. The result extends the one contributed in Barretta (Acta Mech 224(12):2955–2964, 2013) for the special case of homogeneous Saint-Venant beams under torsion. Closed-form solutions for displacement, bending–twisting moment and curvature fields of an elliptic plate, corresponding to a functionally graded orthotropic beam, are evaluated. A new benchmark for computational mechanics is thus provided.

The Inhomogeneous Plate with a Hole: Kirsch's Problem

There is a solution of the problem of stress-strain state determining in the inhomogeneous plate with a circular hole. The inhomogeneity can be induced by the temperature field, the explosion load or the neutron fluency. The problem reduces to a differential equation system with variable coefficients. The allowance for the variable Young's module lets to arrive to a more accurate solution.

A Mechanical Model for the Detachment of Barnacles under Tangential Forces

A model for the detachment of a barnacle under tangential forces has been developed by simulating the barnacle's base plate as an inhomogeneous plate contacting a coating bonded to a rigid substrate. It has been revealed that the stress intensity factors at the plate's ends depend on the elastic inhomogeneity of the plate as well as the ratio between the plate length and its thickness, but almost independent of the thickness of the coating. It may invalidate the applicability of the Kendall model to evaluate the antifouling performance of coatings.

Natural Oscillations of Viscoelastic Lamellar Mechanical Systems with Point Communications

We investigated the natural oscillations of dissipative inhomogeneous plate mechanical systems with point connections. Based on the principle of virtual displacements, we equate to zero the sum of all active work force, including the force of inertia which obtain equations vibrations of mechanical systems. Frequency equation is solved numerically by the method of Muller. According to the result of numerical analysis we established nonmonotonic dependence damping coefficients of the system parameters.

Asymptotic Analysis of Solution of a Nonlinear Problem of Nonstationary Heat Conduction of Lamellar Anisotropic Inhomogeneous Shells with Mixed Boundary Conditions on Faces

An external asymptotic expansion of the solution of a nonlinear problem of nonstationary heat conduction of lamellar anisotropic inhomogeneous shells and plates with mixed boundary conditions on faces has been constructed with account of the thermal response of the layers′ materials. The resulting two-dimensional resolvents have been analyzed, and the asymptotic properties of solutions of the heat-conduction problem have been investigated. A physical explanation has been provided for certain features of the asymptotic expansion of temperature.

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in Kalamkarov (1987) and Kalamkarov (1992), and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.