Infinite Plate Research Articles

The Behavior of Long Thin Rectangular Plates under Normal Pressure-A Thorough Investigation.

Thin rectangular plates are considered basic structures in various sectors like aerospace, civil, and mechanical engineering. Moreover, isotropic and laminated composite plates subjected to transverse normal loading and undergoing small and large deflections have been extensively studied and published in the literature. Yet, it seems that the particular case of long thin plates having a high aspect ratio appears to be almost ignored by various scholars despite its engineering importance. The present study tries to fill this gap, yielding novel findings regarding the structural behavior of long thin plates in the small- and large-deflection regimes. In contrast to what is normally assumed in the literature, namely that a long plate with a high aspect ratio can be considered an infinitely long plate, the present results clearly show that the structural effects of the ends continue to exist near the remote ends of the long plate. An innovative finding is that long plates would (only on movable boundary conditions for the large-deflection regime) exhibit a larger mid-width displacement in comparison with deflections of infinitely long plates. This innovative higher deflection appears for both small and large-deflection regimes for both all-around simply supported and all-around clamped boundary conditions. This new finding was shown to be valid for both isotropic and orthotropic materials and presents a novel engineering approach for the old assumption well quoted in the literature that a relatively long plate on any boundary condition can be considered an infinite plate. Based on the present research, it is recommended that this assumption should be used carefully as the largest plate mid-deflection might occur at finite aspect ratios.

Soret and Dofour effects on unsteady MHD convection flow over an infinite vertical porous plate

In this paper, the computational examination is carried out on the heat generation, Soret and Dufour’s influence on the unsteady MHD convection flow of an incompressible viscous fluid with chemical reaction. It is due to the exponentially accelerated vertical porous plate embedded in a permeable medium with ramped wall temperature together with surface concentration and also with thermal radiation impacts. The basic governing set of the equations of the fluid dynamics to the flow is converted into nondimensional form by inserting suitable nondimensional parameters and variables. In addition, the resultant equations are solved computationally with the efficient Crank–Nicolsons implicit finite difference methodology. The influences for several imperative substantial parameters for the model on the velocity, temperature and concentration for the fluids, the skin friction coefficient, Nusselt and Sherwood number for together thermal situations have been explored with the help of graphical profiles and tabular forms. It is found that, the increasing quantities of the Dufour, temperature generation and thermal radiation parameters, the fluid temperature as well as velocity enhances. Similarly, it is noted that an escalating Soret parameter causes the fluid’s velocity and concentration whereas the chemical reaction parameter notifies reversal outputs.

A nonlinear analysis of electrically forced vibrations of piezoelectric plates with viscous damping near the thickness-shear mode

Based on the theory of nonlinear piezoelectricity, an approximate solution for electrically forced nonlinear thickness-shear vibrations of piezoelectric plates is introduced. The model considers an infinite piezoelectric plate subjected to an alternative voltage, incorporating 3rd-order and 4th-order elastic constants and viscous damping under finite deformation. The system's differential equations for steady-state forced vibrations are derived from the nonlinear pure thickness-shear vibration problem, and transformed so that the new electrical boundary conditions are free. After the application of Galerkin's method, the transformed differential equations lead to a cubic equation, capturing the nonlinear current-frequency curves of AT-cut quartz plates without the need of the quality factor Q, due to the introduced viscous damping. When the damping effect is ignored, the frequency response curves of the AT-cut quartz plate under a specific voltage are confirmed with the existing literature. The study emphasizes the presence of the critical voltage and the importance of the fourth-order elastic constant (c6666) in the stability, revealing lower c6666 values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability.

Three-Dimensional Singular Stress Fields and Interfacial Crack Path Instability in Bicrystalline Superlattices of Orthorhombic/Tetragonal Symmetries

First, a recently developed eigenfunction expansion technique, based in part on the separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion technique in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of the front of an interfacial crack weakening an infinite bicrystalline superlattice plate, made of orthorhombic (cubic, hexagonal, and tetragonal serving as special cases) phases of finite thickness and subjected to the far-field extension/bending, in-plane shear/twisting, and anti-plane shear loadings, distributed through the thickness. Crack-face boundary and interface contact conditions as well as those that are prescribed on the top and bottom surfaces of the bicrystalline superlattice plate are exactly satisfied. It also extends a recently developed concept of the lattice crack deflection (LCD) barrier to a superlattice, christened superlattice crack deflection (SCD) energy barrier for studying interfacial crack path instability, which can explain crack deflection from a difficult interface to an easier neighboring cleavage system. Additionally, the relationships of the nature (easy/easy, easy/difficult, or difficult/difficult) interfacial cleavage systems based on the present solutions with the structural chemistry aspects of the component phases (such as orthorhombic, tetragonal, hexagonal, as well as FCC (face-centered cubic) transition metals and perovskites) of the superlattice are also investigated. Finally, results pertaining to the through-thickness variations in mode I/II/III stress intensity factors and energy release rates for symmetric hyperbolic sine-distributed loads and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the bicrystalline superlattice plate under investigation also form an important part of the present investigation.

Bound states in the continuum for antisymmetric lamb modes in composite plates made of isotropic materials

In this study, we report a numerical design and observation of bound states in the continuum (BICs) for Lamb waves. BICs for elastic-wave systems, especially in non-periodic configurations, are difficult to obtain due to their intricate polarization states. However, the study in this matter has become very important, especially in the field of opto-mechanics or other multi-field couplings at micro or nanoscales. To illustrate the design concept, we simulate the introduction of a piece of silica (SiO2) into a thin infinite Si plate and show that, for specific aspect ratios, BICs for elastic waves can be predicted. We present numerical results for both two-dimensional (2D) rectangular plates and three-dimensional (3D) disk structures. Moreover, we also investigate the modal contributions of both the background and inclusion media during the occurrence of BICs, further verifying the physical background of our design strategy. Although we have focused our work on asymmetric Lamb modes, the current method can also be applied to construct other types of elastic-wave BICs, providing a powerful tool for metamaterial device prototyping based on the control or guiding of elastic waves.

Evaluation of T-stress in an infinite plate with double-crack under compression and shear

The T-stress at the crack tip under compression and shear has received more and more attention, but most of the theoretical researches have focused on single-crack, with little reported on the closed form solutions of T-stress at the double-crack or multi-crack tips. Therefore, the single-crack problem under compression and shear is first analyzed in this paper, the formulas for calculating the three components of T-stress (Tx, Ty and Txy) are obtained. According to the principle of superposition, the origins of the three components of T-stress are examined. Then the basic equations for solving the double-crack problems under compression and shear are established, leading to the formulas for calculating the three components of T-stress at the double-crack tips. Finally, the effects of the spacing between two cracks, the ratio of double-crack half-length, the dip angle of rock-bridge and the absolute angle between two cracks on the three components of T-stress are discussed. The results indicate that the Tx and Ty are unaffected by the friction on the crack surface, but the Txy is significantly correlated with it. Each component of T-stress at the inner and outer tips of double-crack is influenced by different geometrical factors and shows a different variation curve, which may be a potential factor contributing to the complexity of the double-crack or multi-crack problems under compression shear.

Dissipative heat transfer on MHD micropolar hybrid nanofluid flow through infinite parallel squeezing plates

The hybrid nanoparticles suspended in an ethylene glycol water mixture are modeled as micropolar nanofluids. Studying the properties of such a mixture through a squeezing channel is vital in heat exchangers and as a cooling agent in producing sheets out of molten solids. The study aims to regulate the heat transfer rate through a hybrid nanofluid of carbon nanotubes and ferrous oxide nanoparticles immersed in ethylene glycol water. The motion of the fluid and the heat transfer with the plates are studied through the Navier-Stokes equation and the first law of thermodynamics by assuming that the base fluid has micropolar fluid properties. Apart from the usual viscous dissipation, heat dissipation due to the fluid’s micropolar nature is vigorously studied by solving the governing equations through a shooting technique with Runge-Kutta 4th-order algorithm. The results of the current study are compared against the ones with earlier studies as exceptional cases, and are very close. It is noticed that an increase in the micropolar viscosity naturally lowers the momentum in the radial direction and, at the same time, it enhances the axial momentum due to squeezing phenomena. Further, an escalating behavior is noticed in both temperature and microrotation, with an increase in micropolar parameters due to the squeezing force.

A novel investigation of Casson fluid flow over an infinite plate with constant temperature by constructing the absorbing boundary conditions

Casson fluid has been widely used in blood, honey, chocolate, and various polymer fluids, for which the investigation of its underlying flow and heat transfer mechanisms in complex environments is of vital significance. In this paper, the Casson fluid flow over an infinite moving plate with constant temperature is studied, which leads to a coupled system of the velocity and temperature fields. To treat the infinite coupled system more accurately, the artificial boundary conditions in terms of fractional derivatives are developed using the Laplace transform, which transfers the problem in the infinite domain to a model in the bounded domain. The finite difference method is applied to solve the new bounded coupled system. Furthermore, three numerical experiments are presented to demonstrate the validity and accuracy of the transferred model and the numerical scheme. The influence of some important physical parameters on the velocity and temperature fields is also explored. The main findings are that the viscosity of the Casson fluid decreases with an increasing Grashof number leading to an accelerated flow, the thermal conductivity diminishes as the Prandtl number increases resulting in a weakened thermal effect within the flow, and the velocity at the fixed position decreases as the Casson parameter increases.

Rotating MHD flow and heat transfer of generalized Maxwell fluid through an infinite plate with Hall effect

Rotating MHD flow and heat transfer of generalized Maxwell fluid through an infinite plate with Hall effect

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

Stress intensity factor and T-stress around hypocycloidal-shaped holes in finite sheets

Hypocycloidal-shaped holes are seen in sheet structures in special designs or because of the crack growth in particular fatigue loadings. The study focused on the determination of stress intensity factor (SIF) and T-stress around these holes’ cusps. Three-cusp and four-cusp hypocycloid holes with various lengths in various plate sizes were simulated in ANSYS. Results showed both SIF and T-stress decrease with the increase in the plate size. The SIF in a finite plate gets close to the infinite plate SIF where the fraction of plate length to hole length is around 10. For the T-stress, a considerable difference was seen between finite and infinite plates. Finally, the numerical results were experimentally validated using the DIC technique.

Analytical solution to stress intensity factors of cracks on both sides of a triangular hole in an infinite plate

This paper presents an analytical solution for determining the stress intensity factors (SIFs) of double-sided cracks emanating from a triangular hole in an infinite plate. A conformal function is introduced, which can be applied to these cracks, and the SIFs are analytically derived using the Muskhelishvili method. By combining the Schwarz-Christoffel integral with the conformal transformation for cracks originating from circular holes, a mapping function for the cracked triangular hole is presented. The accuracy of the SIF at the single right-sided crack of the triangular hole is validated using finite element method (FEM) and compared with existing references. This study investigates the influence of biaxial loading factors and lengths of cracks on SIFs. The SIF of the hole-edge crack is related to the structure from which the crack emanates. This analytical approach provides a reliable solution for calculating SIFs of double-sided cracks emanating from a triangular hole.

Radiation absorption effect on MHD natural convective flow past an impulsively started infinite vertical plate with chemical reaction and thermal radiation

AbstractThe current work is interested on investigating the impacts of thermal radiation, chemical reaction, and absorption radiation of a hydromagnetic convection‐free mass and heat transfer flow in case of an electrically conducting fluid that passes through a vertical plate moving impulsively. The analytical solutions of the governing momentum, energy, and species concentration equations with the initial and boundary conditions are obtained by the Laplace transformation technique. Graphs for fluid characteristics are used to analyze the impact of changing parametric quantities such as M, N, Sc, Kc, Q, Gr, Gm, and t on the temperature, velocity, concentration, and Sherwood number. We derive the engineering curiosity expressions for the Nusselt number and stress, and at the end, we tabulate and discuss the consequences of new parameters. The magnetic field effect and the chemical reaction are seen to diminish the fluid velocity and concentration, respectively, but in contrast, the absorption radiation effect is seen to accelerate both velocity and temperature. It is closely studied that the Nusselt number and skin friction values for hydrogen consistently exceed those for carbon monoxide.

Casimir Forces in CFT with Defects and Boundaries

We investigate the quantum forces occurring between the defects and/or boundaries of a conformal field theory (CFT). We propose to model imperfect defects and boundaries as localized relevant double-trace operators that deform the CFT. Our focus is on pointlike and codimension-one planar defects. In the case of two parallel membranes, we point out that the CFT 2-point function tends to get confined and develops a tower of resonances with a constant decay rate when the operator dimension approaches the free field dimension. Using a functional formalism, we compute the quantum forces induced by the CFT between a variety of configurations of pointlike defects, infinite plates and membranes. Consistency arguments imply that these quantum forces are attractive at any distance. Forces of the Casimir–Polder type appear in the UV (ultraviolet), while forces of the Casimir type appear in the IR (infrared), in which case the CFT gets repelled from the defects. Most of the forces behave as a non-integer power of the separation, controlled by the dimension of the double-trace deformation. In the Casimir regime of the membrane–membrane configuration, the quantum pressure behaves universally as 1/ℓd; however, information about the double-trace nature of the defects still remains encoded in the strength of the pressure.

Effects of thermal boundary conditions on Stokes' second problem

In this investigation, an infinite flat plate is employed to examine the flow behavior of Casson fluid and the associated heat transfer phenomena. The plate undergoes an initial acceleration with a constant velocity until it eventually decelerates to rest. The energy equation incorporates the effects of viscous dissipation. An appropriate similarity transformation transforms the governing equations into coupled nonlinear ordinary differential equations (ODEs). A closed-form solution for the velocity profile is obtained, while the energy equation is solved using the built-in function NDSOLVE in Mathematica. The study investigates the influence of governing parameters on dimensionless velocity, temperature, skin friction, and local heat transfer rate under two thermal boundary conditions: Newtonian heating and convective boundary conditions. The fluid's thermophysical properties remain constant throughout the study, with the surface temperature of the plate assumed to be fixed at a constant value. A graphical analysis examines the flow behavior and temperature distribution, revealing the impact of non-dimensional parameters. This study reveals that thermal boundary conditions significantly influence heat transfer rates, with Newtonian heating leading to an increase and convective heating causing a decrease. This is attributed to the direct application of heat at the boundary in Newtonian heating, which enhances thermal energy transfer. In contrast, convective heating disperses heat through fluid motion, limiting transfer rates.

Numerical Analysis of Unsteady Vibrations of a Plate Resting on an Elastic Isotropic Half-Space

The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. Previously, the approximate solution of this problem was obtained analytically without using mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray series was written down. In this work, a two-layer medium with different properties was investigated using an algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and weak discontinuity surfaces.

Role of suction/injection on free convective flow in a vertical porous channel with non‐Fourier effects in the presence of periodic boundary

AbstractThe key attention of this work is on the impact of suction/injection on the free convective flow of a viscous fluid passing between two infinite, parallel, vertical and porous plates with non‐Fourier effects. The analysis focuses on a porous channel with boundaries featuring a steady–periodic temperature regime. The governing equations, representing velocity(momentum) and temperature(energy) fields, are well‐stated in dimensional form. Employing a separation technique, the momentum and energy equations are separated into steady and periodic components. On solving, the resulting second‐order ordinary differential equations derived, revealed the expressions for velocity and temperature in dimensionless form. However, the study investigates the influence of various flow parameters used in this work, including suction/injection , heat source/sink , Strouhal number , Prandtl number , and dimensionless relaxation time , on the velocity and temperature distributions. Also not left out is the rate of heat transfer on the flow performance and the skin‐friction coefficient on the plates. The findings are visualized using MATLAB‐generated graphs. An interesting observation found during the course of investigation in that the introduction of suction/injection enhances the flow velocity and fluid temperature within the channel, and they are both seen as declining functions of the Strouhal number, which measures the frequency of periodic heating on the plates. Furthermore, when the suction/injection parameter(s) is being relaxed, this study demonstrates a strong agreement with the findings of Ajibade and Mukhtar.

Comparison of approximation methods for bending of finite and infinite elastic plates: The traction boundary value problem

A generalized Fourier series method is designed to approximate the solution of the Neumann problem in a finite domain and in an infinite domain with a hole, for the mathematical model of bending of elastic plates with transverse shear deformation. The results in the two cases are compared and contrasted, and the analysis is illustrated by examples with specific boundary conditions.

Aeroelastic tailoring of stiffened cantilever plate using composites and structural layouts: A parametric study

AbstractThis paper studies the aeroelasticity of a stiffened cantilever plate using composite material and novel structural layouts. A comprehensive parametric study is conducted to determine the influence of different design parameters on the aeroelastic boundaries. Design parameters include plate sweep angle, ply orientation, stringer cross-section and stringer sweep angle. Nastran is used to run the aeroelastic analysis, and the process is automated using ${\rm{Matla}}{{\rm{b}}^{}}$ . The structure of the plate is modelled using laminate elements whereas the stringers are modelled using the Euler-Bernoulli beam elements. The unsteady aerodynamic loads are modelled using doublet-lattice method (DLM) and the structural and aerodynamic meshes are connected using an infinite plate surface (IPS) spline. A mesh sensitivity analysis is conducted to ensure fine meshes for the aerodynamics and structure. The study’s findings demonstrate the benefits of employing forward swept (Fw) stringers since it increases flutter speed by almost 38% compared to the unswept stringers case and prevents divergence. Moreover, the static aeroelastic analysis illustrates that the utilisation of Fw swept stringers can reduce the average tip displacement and tip twist effectively. T-shaped stringers are recommended to stiffen the plate due to their lower impact on the total mass of the plate. In some configurations, the structural layout has a much higher effect on the aeroelastic instabilities when compared to the material effect (ply orientation). However, results suggest combining both for some cases to get balanced washin and washout effects.

Thermal and mass stratification effects on unsteady parabolic flow past an infinite vertical plate with exponential decaying temperature and variable mass diffusion in porous medium

AbstractThis study looks at how thermal and mass stratification affect the unsteady parabolic flow past an infinite vertical plate with exponentially decaying temperature and variable mass diffusion in a porous medium. The Laplace transform method is applied to solve the governing equations for flow, energy, and mass diffusion. The effects of the both stratification on the flow field, temperature, and mass diffusion are then investigated. The research suggests that the existence of thermal and mass stratification has a considerable influence on the profiles of velocity, temperature, and mass diffusion. It is also found that a stable state is reached for the velocity when both stratification parameters are increased, while stable states are reached for the temperature and concentration when the mass stratification is increased but the thermal stratification is decreased.