Physical Plane Research Articles

Vibration Study of Functionally Graded Microcantilever Beams in Fluids Based on Modified Couple Stress Theory by Considering the Physical Neutral Plane

Theoretical studies on the vibration of microcantilever beams in fluids, which are commonly used in micro- and nanoelectromechanical systems (MEMS/NEMS). When microcantilever beams are subjected to photo-thermal excitation, they show that the material properties such as the dynamic response and the one-dimensional temperature field will show significant differences from the macroscopic properties when the size appearance of the microbeam decreases to the scale below a dozen micrometers. In this paper, by correcting the scale constants of the beams, the photothermal vibration model of the microbeam is established using the physical neutral plane theory. The one-dimensional heat transfer equations and scale-corrected temperature field of the microcantilever beam under laser excitation are derived, which are solved by Galerkin’s method, based on the theory of thermoelasticity, the hydrodynamic model of the beams vibrating in incompressible liquids proposed by Sader et al. and the theory of Euler–Bernoulli beams. The equations governing the vibration of micro-cantilever beams corrected for scale effects in different fluids when subjected to photothermal excitation are obtained. The results show that the temperature field, resonant frequency, and quality factor of the microbeam will have a significant upward drift when the size of the microbeam is close to the scale parameter, and the scale effect has a non-negligible influence on the macroscopic performance parameters; the upward drift is gradually weakening when the thickness to scale ratio gradually increases. Finally, the property of the beam is almost the same as that of the theory when the thickness of the beam is 10 times the scale constant. The correction of the theory by the scale effect is insignificant.

Vibration Analysis of Porous Cu-Si Microcantilever Beams in Fluids Based on Modified Couple Stress Theory.

The vibrations in functionally graded porous Cu-Si microcantilever beams are investigated based on physical neutral plane theory, modified coupled stress theory, and scale distribution theory (MCST&SDT). Porous microcantilever beams define four pore distributions. Considering the physical neutral plane theory, the material properties of the beams are computed through four different power-law distributions. The material properties of microcantilever beams are corrected by scale effects based on modified coupled stress theory. Considering the fluid driving force, the amplitude-frequency response spectra and resonant frequencies of the porous microcantilever beam in three different fluids are obtained based on the Euler-Bernoulli beam theory. The quality factors of porous microcantilever beams in three different fluids are derived by estimating the equation. The computational analysis shows that the presence of pores in microcantilever beams leads to a decrease in Young's modulus. Different pore distributions affect the material properties to different degrees. The gain effect of the scale effect is weakened, but the one-dimensional temperature field and amplitude-frequency response spectra show an increasing trend. The quality factor is decreased by porosity, and the degree of influence of porosity increases as the beam thickness increases. The gradient factor n has a greater effect on the resonant frequency. The effect of porosity on the resonant frequency is negatively correlated when the gradient factor is small (n<1) but positively correlated when the gradient factor is large (n>1).

A Numerical Expedition through the Mathematical Representation of Complex Braided Morphometry—A Case Study of Brahmaputra River in India

The present work explores the process of mathematical representation for the complex geometry of a wide alluvial river with high braiding intensities. It primarily focuses on an approach to developing a numerical solution algorithm for representing the complex channel geometry of the braided Brahmaputra River. Traditional elliptic PDEs with boundary-fitted coordinate transformation were deployed, converting the non-uniform physical plane into a transformed uniform orthogonal computational plane. This study was conducted for the river channel reach with upstream and downstream nodes at Pandu and Jogighopa (reach length ~100 km), respectively, within the Assam flood plain in India, with fourteen measured river cross-sections for the year of 1997. The geo-referenced image covering the river stretch in 1997 was delineated using a ArcGIS software 9.0 tool by digitizing the bank lines. Stream bed interpolation was conducted by interpolating bed elevation from a bathymetrical database onto code-generated mesh nodes. Discretization of the domain was performed through the developed computer code, and the bed-level matrix was generated by the IDW method as well as the MATLAB tool using the nearest neighborhood technique. A mathematical representation of a digital terrain model was thus developed. This generated model was employed as a geometrical data input to simulate secondary flow utilizing 2D depth-averaged equations with the flow dispersion stress tensor as an extra source component, coming from curvilinear flow patterns caused by severe river braiding. The developed model may further be useful in mathematically representing the geometrical complexities of braided rivers with a relatively realistic assessment of the various parameters involved if deployed with improved river modeling with morphometric evolution.

Free vibration of cracked FGM Mindlin plate in fluid

The free vibration of cracked functionally graded material (FGM) plate in fluid is investigated in this paper. The material components are supposed following the power law and formulated by the Voigt model. The Mindlin plate theory (MPT) including the shear deformation effect is established and the physical neutral plane is adopted to simplify the derivation due to the in-plane displacements of this plane assumed as zero. A massless linear rotational spring model (LRSM) is established to simulate the through-width surface crack. The hydrodynamic pressure at the fluid-plate interface is derived by the potential flow theory, and is regarded as the added mass of cracked FGM plate during the analyses of fluid-plate coupled vibration. The derivation and discretization of vibrational equations for cracked FGM plates in fluid are respectively achieved though the Hamilton’s principle and differential quadrature (DQ) method. The frequencies and mode shapes are iteratively calculated with these values of the vacuum case. The influences of immersed depth, fluid density, gradient index, crack depth, crack location, length-to-width ratio, length-to-thickness ratio, and boundaries on vibration characteristics are conducted in the section of numerical results. The present vibration analyses of cracked structures can be applicable to avoid structural resonance and failure.

Complex phase diagram and supercritical matter.

The supercritical region is often described as uniform with no definite transitions. The distinct behaviors of the matter therein, e.g., as liquidlike and gaslike, however, suggest "supercritical boundaries." Here we provide a mathematical description of these phenomena by revisiting the Yang-Lee theory and introducing a complex phase diagram, specifically a four-dimensional (4D) one with complex T and p. While the traditional 2D phase diagram with real temperature T and pressure p values (the physical plane) lacks Lee-Yang (LY) zeros beyond the critical point, preventing the occurrence of criticality, the off-plane zeros in this 4D scenario still induce critical anomalies in various physical properties. This relationship is evidenced by the correlation between the Widom line and LY edges in van der Waals, 2D Ising model, and water. The diverged supercritical boundaries manifest the high-dimensional feature of the phase diagram: e.g., when LY zeros of complex T or p are projected onto the physical plane, boundaries defined by isobaric heat capacity C_{p} or isothermal compression coefficient K_{T} emanates. These results demonstrate the incipient phase transition nature of the supercritical matter.

Bending and free vibration analysis of FG circular plates using a five unknown high order shear deformation theory

In this study, the bending and free vibrations of functionally graded material (FGM) circular plates are investigated using a new quadrilateral finite element based on a five-unknown high-order shear deformation theory. The proposed model accounts for a quadratic distribution of the transverse shear stress through the plate thickness and zero transverse shear stress at the top and bottom surfaces of the plate. The properties of the plate are changed through its thickness according to a power law in which the index varies from 0 value for a fully ceramic plate to infinity for a fully metallic plate. The concept of the physical neutral plane is introduced to avoid membrane-bending coupling. The total potential energy principle and the kinetic energy are used to derive the elementary rigidity and mass matrices. The performance and the convergence of the developed element are demonstrated through examples from the literature. Furthermore, the effect of various parameters, such as the power law index and the radius-to-thickness ratio, on the plate behavior is also considered.

Accuracy Examination of the Fourier Series Approximation for Almost Limiting Gravity Waves on Deep Water

A permanent gravity wave propagating on deep water is a classic mathematical problem. However, the Fourier series approximation (FSA) based on the physical plane was examined to be valid for almost waves at all depths. The accuracy of the FSA for almost-limiting gravity waves remains unevaluated, which is the purpose of this study. We calculate some physical properties of almost-limiting waves on deep water using the FSA and compare them with other studies on the complex plane. The comparison results show that the closer the wave is, the greater the difference. We find that the main reason for this difference is that the wave profile in the FSA retains an original implicit form and is not represented by Fourier series. Therefore, the kinematic and dynamic conditions of the free surface around the wave crest cannot be satisfied at the same time.

Around-language discourse during the Russian-Ukrainian war

The goal of this study was to find out how the discourse related to the language issue, which has developed, in particular, in the Ukrainian segment of the Internet, changed after Russia’s full-scale attack on Ukraine. For this purpose, the material was collected during the period of 2012—2021, with a comparison of theses and arguments in the discussions that arose around the appearing issues of the language situation with those recorded during the year of the Ukrainian-Russian war. Nowadays, the discourse surrounding the language issue has changed. Even those political forces that used it as a tool of pre-election campaigning, speculating on the topic of the division of society on the basis of language, immediately changed their rhetoric. The language-related discourse has lost its former characteristic of debate; it has become more radical, highly symbolised, and sometimes pejorative; the former polarity of thoughts has disappeared. It displays public reflections on the nation-building function of language. More and more Ukrainians are coming to understand that “language matters” and “language is a weapon”. The symbolic “war of languages” has transformed in the physical plane into a “war with the help of language”. The language has become a weapon of the military. At the same time, Ukrainian journalists shock the society by covering scandals surrounding the issue of language, as the term ‘Movosrach’, which is a manipulative denomination for language-related discussions and which emphasises contempt for those who raise the issue of language functioning, arose long before the start of the full-scale war. Many people are convinced that the language situation in Ukraine will undergo a radical change after the war. This refers, at a minimum, to the levelling of the status, prestige, and ultimately the prevalence of the Russian language. Expert opinions carry significant weight when it comes to the language issue. However, a part of society still hopes that in the language policy, the state will adhere to the position of non-interference in the language practices of society. Based on the current language policies and the prevailing language attitudes, the development of the language situation in the country is likely to be influenced by the ongoing events; however, it is expected that the trends related to language communication guidelines and the vision for the language development will remain inertial. Keywords: metalinguistic discourse, Ukrainian language, language issue, rhetoric, RussianUkrainian war.

SurfShare

Shared Mixed Reality experiences allow two co-located users to collaborate on both physical and digital tasks with familiar social protocols. However, extending the same to remote collaboration is limited by cumbersome setups for aligning distinct physical environments and the lack of access to remote physical artifacts. We present SurfShare, a general-purpose symmetric remote collaboration system with mixed-reality head-mounted displays (HMDs). Our system shares a spatially consistent physical-virtual workspace between two remote users, anchored on a physical plane in each environment (e.g., a desk or wall). The video feed of each user's physical surface is overlaid virtually on the other side, creating a shared view of the physical space. We integrate the physical and virtual workspace through virtual replication. Users can transmute physical objects to the virtual space as virtual replicas. Our system is lightweight, implemented using only the capabilities of the headset, without requiring any modifications to the environment (e.g. cameras or motion tracking hardware). We discuss the design, implementation, and interaction capabilities of our prototype, and demonstrate the utility of SurfShare through four example applications. In a user experiment with a comprehensive prototyping task, we found that SurfShare provides a physical-virtual workspace that supports low-fi prototyping with flexible proxemics and fluid collaboration dynamics.

A layer-wise theory for the dynamic analysis of functionally graded composite nanobeams under thermal environment using nonlocal boundary conditions and Chebyshev collocation technique

In this study, a layer-wise theory satisfying displacement continuity at each layer’s interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen’s nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method.

The growth and decay of nonlinear wave in polytropic reacting gas with small solid dust particles

The aim of this study is to examine the evolutionary process of shock waves along the characteristic path under the effect of dust particles in a polytropic reacting gas. Testing the nature of the governing quasilinear of this system as a coordinate, we reconstruct the governing equations as Bernoulli’s differential equation and obtain the results. The results in the characteristic plane can display a non-linear behavior in the physical plane. It is shown how the reacting gas parameter influences the evolutionary process of the compressive and expansive waves, respectively, in reacting gas flows. Also, it is shown how the existence of dust in reacting gas influences the growth and decay of the compressive and expansive waves. The transport equation leading to the evolution of shock waves is determined and used to derive the condition for the shock formation. The comparative study of the effect of reacting gas parameters and dust particles on the flow patterning and distorting of shock waves for planar, cylindrically and spherically symmetric flows is also performed.

Аналитическое решение задачи определения границы растекания потока

A mathematical model of a two-dimensional high-speed flow in terms of justification and taking into account several physical assumptions is formulated. The problem is solved analytically in the plane of the velocity hodograph and in the physical plane to determine all parameters in terms of flow flow. The coupling of a uniform flow with a general flow in the form of a "simple wave" made it possible to achieve a reduction in the error of the mathematical model. The adequacy of the pre-sented method is shown.&#x0D; The existing models are described that are insufficiently acceptable and adequate in terms of the geometry of the flow boundary, but with a large mismatch in local depths and velocities. The adequacy of the new model as a whole is characterized by the convergence of the model parameters both in geometry (flow spreading boundaries) and kinematics (depth and flow velocity) increased to 18% in both directions.&#x0D; The section of the "simple wave" should be well combined with the real flow, taking into ac-count the forces of resistance to the flow. The boundaries of the use of the proposed model belong to the section of the expansion of the flow in 3-7 b as required in the reference literature, and have been clarified in earlier works. The proposed model, as shown in the article, takes into account real (experimental) flow spreading and is consistent with previously performed theoretical studies. An important conclusion in the article is that the values of the Froude criterion in the new model can be any in the range from 1 to infinity, and at the same time the section "$X_D^I$" can increase with in-creasing Froude number.

Visibility of Albert Einstein’s Four-Dimensional Plane of Existence in Sri Aurobindo’s Yogic Vision of ‘Space’ in His Epic “Savitri”

Albert Einstein’s mathematical discovery of the four-dimensional plane of existence, the ‘Spacetime’ or ‘Space-Time Continuum’, as it is called, is an evolutionary breakthrough in the realm of Science. Since the plane of ‘Spacetime’ or ‘Space-Time Continuum’ is a mathematical structure, the most celebrated scientist finds it to be a physical plane of existence. Mysteriously enough, it is not physically visible. Physics is answerless on its invisibility. Transcending Einstein, Sri Aurobindo, a ‘yogi of the unknown eternal heights’ reveals ‘Space’, in his epic Savitri, to be ‘a vast experiment of Soul’, the Divine Consciousness in the being of Man. Furthermore, it is the Soul of the Universe, ‘Vishwatman’, as the Rishis call it. In such a vision of space the mystery of the invisibility of Spacetime disappears. The plane of Spacetime is the plane of integral existence to which mankind is yet to evolve. Man is subject to the divisive principle of Consciousness as a result of which the plane of Spacetime is not visible to him. On account of the subjection to the divisive principle of consciousness Newton could find Space and Time to be two incompatible realities. Man is required to evolve his latent spiritual consciousness to attain the integrality of Consciousness for the vision of the plane of Spacetime. He has to undertake the yoga to attain, first, the silent state of mind to ascend the planes of the spiritual and Supramental Consciousnesses, says Sri Aurobindo. In the silent state of mind, the divisiveness of consciousness ceases to have its hold as a result of which man is able to visualise the plane of Spacetime, the Spacetime which is all the more clearly visible when he continues to ascend the higher planes of the spiritual consciousness. On the plane of the spiritual consciousness there is a movement towards the integrality of consciousness, and consequently ‘Space’ is found to be ‘a vast experiment of Soul’ in this movement, the movement which is, in fact, the ascent of the yogic consciousness.

Nonlinear Differential Equations of Flow Motion Considering Resistance Forces

For a stationary potential 2D planar open high-velocity water flow of the ideal liquid, we propose a closed system of nonlinear equations considering the resistance forces to the flow from the channel bottom. Tangential stresses on jet interfaces are ignored. The resistance force components are expressed in terms of velocity components. In this case, the flow equations can be solved through the method of characteristics, and the surface forces are reduced to equivalent volumetric forces. The system of non-linear equations is solved in the velocity hodograph plane; further, the transition to the physical plane takes place. Since the value of the hydrodynamic pressure decreases downstream of the flow, the friction forces to the flow in the first approximation can be considered by using the integral laws of resistance. At that, the form of the equations of motion in the plane of the velocity hodograph does not change. This fact is proved in the article. An example of calculating the water flow is provided. The kinecity, ordinates, and velocities of the flow along its extreme line are calculated without considering resistance forces. Validation of the model in the real flow is performed. Acceptable accuracy relative to experimental data is obtained.

Performance Analysis of Zhao and Durbin Numerical Inversion Methods of Laplace Transform

The Laplace transform is essential to satisfy the independence of time for the analysis of the transient response of the composite or functionally-graded materials. The time independent boundary value problem may be solved then either by numerically or analytically. The solutions should be inverted to the physical plane using inverse Laplace transform. Therefore, the selected numerical inversion method may be crucial to obtain the high accuracy throughout the whole analysis steps. In the present study, Zhao’s Method I, Zhao’s Method II, Durbin’s Method and Modified Durbin’s Method are applied to dynamic loading conditions. The analysis results show that the accurate and stable solutions even for long time inversion have been obtained by Modified Durbin’s Method and Zhao’s Methods. However, compared with the methods of Zhao’s, the computational and programing load of Durbin’s Methods are minimum.

Non-uniform Compact Schemes for Parallel Computing

Here, the performance of a sixth order accurate Non-Uniform Compact Scheme (NUCS6) in a parallel framework using the domain decomposition method is analyzed. Traditional compact schemes are used for uniform grids. However, study of flows involving complicated geometries requires transformation of computational domain onto a uniform grid. Compact schemes developed for physical plane computations eliminate the need for such grid transformations. The first successful implementation of uniform compact schemes on a parallelized framework with Schwartz domain decomposition demonstrated the effect of overlap points in eliminating spurious signals at inter-processor boundaries Sengupta et al. [Jour. Comp. Phy. 220 (2007)]. This paper presents the performance of a high order non-uniform compact scheme (NUCS6) in parallel computations with the domain decomposition method. The results of the benchmark problem of flow in a 2D square lid-driven cavity at post-Hopf bifurcation Reynolds number of 10000 is presented to show how the width of overlap region affects the growth and evolution of gyrating polygonal vortex at the centre of the cavity.

Propagators and widths of physical and purely virtual particles in a finite interval of time

We study the free and dressed propagators of physical and purely virtual particles in a finite interval of time τ and on a compact space manifold Ω, using coherent states. In the free-field limit, the propagators are described by the entire function (ez– 1 – z)/z2, whose shape on the real axis is similar to the one of a Breit-Wigner function, with an effective width around 1/τ. The real part is positive, in agreement with unitarity, and remains so after including the radiative corrections, which shift the function into the physical half plane. We investigate the effects of the restriction to finite τ on the problem of unstable particles vs resonances, and show that the muon observation emerges from the right physical process, differently from what happens at τ = ∞. We also study the case of purely virtual particles, and show that, if τ is small enough, there exists a situation where the geometric series of the self-energies is always convergent. The plots of the dressed propagators show testable differences: while physical particles are characterized by the usual, single peak, purely virtual particles are characterized by twin peaks.

TERRESTRIAL SPACES AND CELESTIAL SYMBOLISM IN CEMETERIES: AN EMPIRICAL EXAMINATION OF SPACES OF THE DECEASED

The open terrestrial swathes are by far the most collaborative of spaces shared by human beings as they partake of the spatial and temporal experience. Their experiential reality is shaped by the sensory experience rooted in the physical plane of existence, palpably elusive of the metaphysical. Cemeteries however hold a unique terrestrial position as not simply seeking out open spaces on earth, but also symbolize the metaphysical experience; transcending the earthly in death. This unique ambivalence of the earthly and the celestial entrenched in the cemeteries is going to be a major concern of this research article. The article seeks to deconstruct how cemeteries which command sanctity can be powerful celestial signs in contradiction to ordinary cemeteries which however are bracketed with a relatively less celebrated and overtly mundane experience of death and Thanatos. The article will dwell into the role of select cemeteries as metaphors of power emanating from the treatment meted out to the cemeteries. The centre-periphery, life-death and celestial-terrestrial binaries will be explored through the epithets conferred upon rites of death, cemeteries and phenomena of afterlife in different cultures. Keywords: Celestial, Terrestrial, Ambivalence, Juxtaposition, Binaries, Thanatos.

Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.

Clarification of Faber series and related applications to complex variable methods in two-dimensional elasticity

Faber series are used extensively in the application of complex variable methods to two-dimensional elasticity theory, for example, in the mechanical analysis of composite materials where Faber series representations of complex potentials lead to convenient expressions for the corresponding displacement and stress distributions. In many cases, the use of the Faber series is combined with conformal mapping techniques which “transfer” a boundary value problem defined in the elastic body (physical plane) to a simpler problem posed in an imaginary plane characterized by the conformal mapping. In several instances in the literature, however, little attention has been paid to the domain of definition of the Faber series in the imaginary plane leading often to misunderstandings and erroneous conclusions regarding the concept and feasibility of the use of the Faber series. In this paper, we present a thorough and rigorous examination of the representation of the Faber series in both the physical (occupied by the material) and imaginary (defined by the conformal mapping) plane. In addition, we show that replacing a truncated Faber series by a truncated Taylor series does not induce any additional errors in the numerical analysis of the corresponding boundary value problem. We anticipate that the discussion in this paper will help clarify any existing misinterpretations regarding the application of the Faber series and help further extend their use to a range of problems in composite mechanics.