Inhomogeneous Plate Research Articles

Low-frequency propagating and evanescent waves in strongly inhomogeneous sandwich plates

The paper aims at studying dispersion of elastic waves in a sandwich plate with the parameters, characteristic of aerogel core and hard skin layers, typical for aerospace applications including optimal design of fuselage structural components. The proposed approach relies on multiparametric analysis, taking into account the effect of strong transverse inhomogeneity. It is demonstrated that both an additional low-frequency propagating wave and a slowly decaying evanescent one appear due to a high contrast in geometric and mechanical parameters of the layers. The key findings include the derivation of two-mode asymptotic expansions of the full dispersion relation at the low-frequency limit, as well as elucidation of the non-trivial link between long-wave evanescent and propagating modes. A sophisticated composite nature of the obtained expansions involving various shortened forms is investigated. The range of validity for each of these forms over frequency and wave-number domains is evaluated. Comparison of asymptotic results with the numerical solution of the full dispersion relation is presented.

Optimization of thermal shock response spectrum as infrared thermography post-processing methodology using Latin hypercube sampling and analytical thermal N-layer model

In this work, we continue to develop and investigate the Thermal Shock Response Spectrum (TSRS) method as an alternative data processing method for infrared thermography (IRT). We focus on improving the current TSRS algorithm and present an optimization methodology for finding the optimal thermal Q-factor and characteristic frequency pair, which is based on the widely applied random sampling method. We show the qualitative relationship between the determined optimal characteristic frequency and the corresponding maximum difference in diffusion length between reference and defective models, as calculated by selecting a specific one-dimensional thermal N-layer model The investigations were performed on an inhomogeneous plate made of carbon fiber reinforced polymer (CFRP) with artificial square defects at different depths. Furthermore, two different heat sources were used: a xenon flash lamp and a laser. These sources are not only distinct by their underlying physics but also generate inherently different pulse shapes. To quantitatively estimate the contrast between defect and non-defect areas, and to compare these results with commonly used infrared thermography (IRT) data post-processing methods such as Pulse Phase Thermography (PPT) and Thermographic Signal Reconstruction (TSR), the Tanimoto criterion (TC) and signal-to-noise ratio (SNR) were used.

Alternative data evaluation methodology for infrared thermography analogous to the Shock Response Spectrum analysis method

We propose an alternative method for processing infrared thermography (IRT) data based on a thermal 1-layer model. The method is based on the analogy of the Shock Response Spectrum (SRS) analysis for mechanical systems (ISO 18431). A thermal Q-factor, similar to the resonance sharpness (Q-factor) of a mechanical oscillator, is introduced as a parameter for selecting a specific thermal 1-layer model. Two main aspects of this method and corresponding properties have been considered: the strictly model-based treatment of this methodology and an empirical introduction of additional bandpass filters. The method is validated with two practical examples: for a homogeneous polymer sheet with blind holes and an inhomogeneous plate made of carbon fiber reinforced polymer (CFRP) with artificial defects, the capability of defect detection is compared with established methods. The comparison of the results with the commonly used data processing methods of IRT as Pulsed Phase Thermography (PPT) and Modified Differential Absolute Contrast (MDAC) shows equivalent or sometimes improved performance.

Impact of the thermal afterglow effect on infrared thermography data evaluation methods

In this work, the influence of the afterglow effects from a source impulse on commonly used data processing methods of infrared thermography (IRT), such as Pulsed Phase Thermography (PPT) and Thermographic Signal Reconstruction (TSR) was investigated. To resolve this issue, we propose the adoption of a modified mathematical model or an empirical function to approximate the flash lamp pulse, which we recorded during the test with a photodiode. Two models were evaluated: one consisting of a combination of exponential and power functions describing the entire pulse and a second, focusing only on the decreasing phase. The models were compared with well-established empirical functions, such as those of Degiovanni and Vageswar, using the widely known statistical scores R2adjusted and Akaike information criterion. The quantitative comparison revealed better fitting results for our proposed models. Additionally, we introduce an algorithm based on Hough transform-clustering for automated determination of the exact pulse length. Statistical analyses were performed to evaluate the reproducibility of the measured pulse shape. The results from applying the first model as pulse correction is compared to another widespread IR data treatment methodology, such as Modified Differential Absolute Contrast (MDAC). We also consider the impact of the afterglow effects on the Source Distribution Image (SDI). Two different specimens were used in the experiments: a homogeneous polymer sheet with blind holes as defects and an in-homogeneous plate made of carbon fiber reinforced polymer (CFRP) with artificial defects made from Tetrafluorethylen-Hexafluorpropylen-Copolymer (FEP) foil inserts. In both experiments a quantitative comparison was made using the Tanimoto criterion (TC), indicating an improved sensitivity. This work forms the basis for a faster and more reliable quantitative evaluation of experimental data, in particular algorithms for the automated detection and assessment of manufacturing defects or in-service damage will benefit from these results.

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. The penetrating solution determines the internal stress-strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

An analytical model for dynamic analysis of inhomogeneous sandwich plate

Sandwich plates are widely used in the engineering domain and are extensively studied by researchers for their special performances. Among them, the sandwich plate with coordinate-dependent material parameters has excellent vibration isolation performance. However, there are few effective and convenient analytical methods for solving the dynamic response of this type of sandwich plate. To solve this problem, an equivalent spring-based model for the dynamic analysis of the sandwich plate is proposed in this paper. Assuming that the sandwich plate is a linear elastic structure, and considering the core plate as equivalent springs, whose stiffness coefficients are derived based on the classical elasticity theory and Kirchhoff hypothesis. Besides, displacement solutions of base and constrained plates of the sandwich plate in the x, y and z directions are expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series, respectively. Then, according to the orthogonality of trigonometric functions and selectivity of Dirac functions, the displacements of the sandwich plate are solved accurately. By comparing with available analytical and numerical results, the proposed method is proven to have excellent performance in dealing with the sandwich plate with coordinate-dependent material parameters. Finally, the dynamic characteristics and vibration isolation performance of the inhomogeneous sandwich plate are discussed. The results show that the proposed method can be used to design the inhomogeneous sandwich plate and improve their vibration isolation performance significantly.

ON AN OPTIMIZATION PROBLEM FOR A PRESTRESSED PLATE WITH VARIABLE STIFFNESS

Most studies in the field of optimization problems in the mechanics of the deformable solid body are aimed at finding the optimal shape of structural elements, which most often leads to the search for the optimal distributions of geometric characteristics for the considered objects in order to ensure reducing the objects? weight and improve their strength characteristics. At the same time, with the advent of modern materials with variable physical and mechanical properties, new relevant branches arose in the optimization problems, as well as the inverse problems on the reconstruction of non-uniform variable characteristics. In this study, the problem of steady-state oscillations of a prestressed inhomogeneous round elastic plate in an axisymmetric formulation is considered. Variational and weak problem statements are presented. The case of free oscillations of a plate simply supported along the contour is investigated; it is shown that the eigenvalue problem is self-adjoint and quite definite. The problem of finding the optimal distribution of the plates? elastic modulus and the corresponding oscillation shape in order to maximize the first natural frequency is considered. An isoperimetric condition for the variable stiffness function is proposed. Based on the Rayleigh energy ratio, the minimum principle for the first eigenvalue and the corresponding eigenfunctionis formulated. On the basis of the variational principle, using the constructed Rayleigh ratio, the optimality condition is formulated, and an explicit representation is obtained for the functions of the optimal oscillation shape and optimal stiffness, taking into account the initial radial stretching or compression of the plate. The range of possible values for the initial stress is determined. The results of solving the problem via constructing optimal inhomogeneity laws for the stiffness function in the class of smooth functions in the considered range of changes in initial stresses are obtained and analyzed.

Dispersion of an inhomogeneous sandwich plate having imperfect interfaces and supported by the Pasternak foundation

The purpose of this investigation is to see the dispersion of an inhomogeneous sandwich plate with imperfect interfaces between the layers and supported by the two parameters Pasternak foundation under long-wave low-frequency conditions. The governing equation of motion has been considered from the perspective of an anti-plane shear propagation to achieve simplicity. The overall cut-off frequency and the exact dispersion relation (EDR) are computed. In the context of the structure under investigation, one material contrast setup has been considered. The shortened polynomial dispersion relation, which corresponds to the EDR under material contrast setup, has been reported and investigated further. Additionally, the variational effects of the Pasternak foundation parameters as well as the interface imperfect parameter on the lowest dispersion curve subject to the long-wave low-frequency domain have been investigated using numerical simulations and graphical representations. This study is noteworthy because it sheds light on the behavior of elastic waves in multilayered structures and may be utilized to enhance the layout of three-layered structures used in a variety of industrial fields. Furthermore, we have provided the optimum values of the involved parameters to control and confine the sandwich plate’s vibration within the long-wave low-frequency regime.

Three-dimensional isogeometric model for nonlinear vibration analysis of graded inhomogeneous nanocomposite plates with inconstant thickness

Three-dimensional isogeometric model for nonlinear vibration analysis of graded inhomogeneous nanocomposite plates with inconstant thickness

Electro-thermo-mechanical multiple fields coupled guided wave in a piezoelectric semiconductor plate

Based on a fully coupled model of electro-thermo-mechanical multiple physical fields, the guided wave in a piezoelectric semiconductor (PSC) plate is studied. The non-Fourier heat conduction (G-N III-type) and non-Fick diffusion are included in the model. By the appropriate definition of the extended displacements and stresses, the system of second-order differential coupled equations degrades to a first-order matrix differential equation. The transfer matrix of the PSC plate is derived from the first-order matrix differential equation. Some combinations of mechanical, electric and carrier boundary conditions are applied to obtain the dispersion relations. The influences of various boundary conditions and typical parameters on the dispersion and attenuation are investigated numerically. The gradient profiles of the inhomogeneous plate are also discussed. Based on the numerical results, it is found that the dispersion characteristics are mainly affected by the mechanical boundary conditions, while the attenuation characteristics are mainly affected by the transport of carrier and heat. In addition, the gradient profiles of inhomogeneous plates affect both dispersion and attenuation characteristics.

Physical nonlinearity in porous functionally graded kirchhoff nano-plates: Modeling and numerical experiment

In this paper, mathematical models of porous functionally graded inhomogeneous plates were constructed. The following hypotheses were used as the foundation: the kinematic Kirchhoff model was used; nano effects were considered by the modified couple stress theory; material properties depend on coordinates (x,y,z), and for metals, they also hinge on the stress-strain state at a point in the plate volume, i.e. there is a dependence σi(ei). This dependence can be any analytical function derived from experimentation, which means that it is possible to take into account elastic-plastic deformations according to the deformation theory of plasticity. The ceramic material is resilient. The required variational and differential equations and the boundary conditions were derived from the Lagrange functional. The plate material's porosity was characterised by six types. For numerical solutions, the methods of reduction of the partial differential equation to the ordinary differential equation were applied, as well as iterative methods nested one into another, based on the following techniques: variational iterations method (VIM), Agranovskii-Baglai-Smirnov method (ABSM) and Birger's method of variable parameters of elasticity. For each of these methods, there are theorems proving their convergence. Each problem that describes the stress-strain state of Kirchhoff nano-plates was solved using the following analytical methods: Exact Navier's solution (ENS), Bubnov-Galerkin method (BGM), Levy solution (LS), numerically analytical Kantorovich-Vlasov (KVM) methods, variational iterations method (VIM), Vaindiner method (VaM) and numerical finite difference method (FDM). The convergence of these methods was investigated and the optimal method for calculating such structures, both in terms of accuracy and speed, was determined for the elliptic 2D PDEs of arbitrary area Ω. Numerical results for both elastic nano-plates and elastoplastic nano-plates have been presented.

Instability and its prediction of an inverted inhomogeneous plate loaded by a subsonic airflow: Theory and experiment

This paper presents a study on the instability of an inverted cantilevered plate with an inhomogeneous cross-section in axial flow from both theory and experiment. The plate is free at the leading edge and clamped at the trailing one. It has a cross-section with discontinuity, bringing difficulty in solving the fluid–structure interaction equation by the traditional methods. We study this problem in state space and transfer its solution into a function approximation problem. We bring no approximation at the first equation level, and the derived instability equation is on the continuum. A numerical scheme is proposed to solve the instability equation. Series expansion and least square method apply to the numerical solutions. The convergence and reliability of the present method are entirely tested. This present modeling method preserves the original information of the problem and facilitates the exploration of the effects of inhomogeneity. Based on the present continuum instability equation, prediction formulas for critical dynamic pressure and instability slope of an inhomogeneous plate are first given and verified. They can apply to the stability prediction of arbitrary cross-sectional plates and have application potential for the plate design. In addition, we report an optimized stiffness distribution of the plate for the maximum critical dynamic pressure. Finally, a wind tunnel test is designed and implemented. A comparative study shows that the present method and prediction formula agree well with the experiment. The present modeling method for a fluid–structure interaction problem in a continuous sense, avoiding approximation at the beginning level, can serve as another new way to address the static instability problem of plates.

Polarization Scattering Properties of Inhomogeneous Plasma-Covered Metal Plate

The plasma sheath generated during the reentry of the hypersonic vehicle will significantly influence the amplitude and polarization property of the scattered wave and lead to abnormal scattering characteristics of the hypersonic vehicle. Therefore, an electromagnetic (EM) simulation model of 3-D inhomogeneous plasma-covered metal plate is deployed to study the scattered properties of linear polarization plane wave obliquely incident on the inhomogeneous plasma. Based on the dispersive media’s convolutional perfect matched layer (CPML) and taking the Gaussian pulse linear polarization wave as an example, this article studies the effects of the plasma parameters, including the electron density and the collision frequency, on the scattered properties of plasma covered metal plate. Then, the influences of incident angle and EM wave polarization mode on the polarization properties of the scattered wave are emphatically analyzed. Numerical results indicate that the inhomogeneous plasma presents the cross-polarization isolation effect when the electric field of the plane wave is parallel to the surface of the plasma. On the contrary, the inhomogeneous plasma presents the cross-polarization depolarization effect when the electric field vector intersects with the surface of the plasma. The results provide a theoretical basis for the research on radar cross section (RCS) measurement, polarization matching, and EM stealth of plasma-sheath-covered hypersonic vehicles.

Spectral Optimization of Inhomogeneous Plates

This article is devoted to the study of spectral optimisation for inhomogeneous plates. In particular, we optimise the first eigenvalue of a vibrating plate with respect to its thickness and/or density. Our result is threefold. First, we prove existence of an optimal thickness, using fine tools hinging on topological properties of rearrangement classes. Second, in the case of a circular plate, we provide a characterisation of this optimal thickness by means of Talenti inequalities. Finally, we prove a stability result when assuming that the thickness and the density of the plate are linearly related. This proof relies on H-convergence tools applied to biharmonic operators.

Dispersion of elastic waves in the three-layered inhomogeneous sandwich plate embedded in the Winkler foundations.

This research article investigates the behavior of elastic waves in an inhomogeneous isotropic three-layered sandwich plate with soft-core and stiff-skin layers embedded in Winkler foundations, using anti-plane shear motion. The study establishes the exact antisymmetric dispersion relation, low-frequency spectrum, and overall cut-off frequency of the wave propagation. A shortened polynomial dispersion relation is developed for the long-wave low-frequency regime by considering the contrasting material setup and compared with the exact dispersion relation. This article also provides exact and asymptotic formulas for stresses and displacements in each layer of the plate, as well as approximate one-dimensional equations of motion. The results suggest that the approximate equations of motion for the three-layered sandwich plate are valid throughout the entire low-frequency range, despite the presence of Winkler foundations on both sides of the plate. This research is significant as it provides insights into the behavior of waves in composite materials and can be used to improve the design of sandwich structures used in various engineering applications. Additionally, the findings that the approximate equations of motion for the three-layered sandwich plate are valid throughout the low-frequency range, despite the presence of Winkler foundations, can help simplify calculations and make the design process more efficient.

Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

The Analytical Expressions to Describe Wave Propagation and Heat Release During Microwave Treatment of Porous Inhomogeneous Plate Based on the W.K.B. Solution Model

Abstract In this paper study for wave propagation in non-homogeneous porous plate sample with slowly varying refractive index is presented. It is based on simple symmetric solution of the wave equation for linearly polarized electromagnetic wave aligned into the porous plate perpendicularly to the external surface. Using correct boundary conditions both the transverse electric (T.E) and transverse magnetic (T.M) modes, named shortly by (T.E.M) mode for electromagnetic wave, are considered. The Wentzel-Kramers-Brillouin (W.K.B.) solutions for symmetric incident irradiation of fixed power generated at the plate surfaces was obtained. It is done the analysis of the reflection and transmission coefficients on the surfaces of plate.

Solid mechanics models in application to ophthalmology

The review presents the models on a change in the stress-strain state of the eye shell after vision correction operations and a change in intraocular pressure with the introduction of drugs into the vitreous body. All mathematical models were developed in collaboration with ophthalmologists. Mathematical models that describe the process of determining the true intraocular pressure (IOP) by applanation methods are discussed. New results in solid mechanics were obtained while studying ocular biomechanics problems. For example, the problem of the stability of a spherical shell subjected to a concentrated force and internal pressure was solved and the stability of the axisymmetric equilibrium state of annular inhomogeneous orthotropic plates under normal pressure was studied. The stress-strain states of transversely isotropic spherical and cylindrical layers subjected to internal and external pressures are discussed and the solutions obtained in the framework of the three-dimensional theory are compared with the solutions obtained by non-classical shell theories. This comparison made it possible to assess the accuracy of approximate theories.

Boundary Controllability of a Coupled Inhomogeneous Plates System

Boundary Controllability of a Coupled Inhomogeneous Plates System

Effects of the external magnetic field on propagation of thickness-twist waves in inhomogeneous plates

Effects of the external magnetic field on propagation of thickness-twist waves in inhomogeneous plates