Isotropic Square Research Articles

2D lattice animal formation in the irreversible aggregation of patchy particles

With the self-assembly of patchy colloids, we can engineer many complex structures. In the present work, we are modeling spherical particles with 2 diagonally opposite patches using the Kern-Frenkel potential coupled with isotropic square well potential having different interaction ranges. The isotropic square well potential formed reversible bonds while the patches were irreversibly bonded. A state diagram is calculated where we observed a range of structures from a 1D chain, 2D planar cluster, and 3D random clusters depending on the width of the patch. These structures are controlled by the number of bonds each patch can form for a given patch angle. The range of the patch width can be analytically calculated using geometric arguments to identify different types of clusters, which agree with our simulations. The aggregation process was diffusion-limited, forming clusters with fractal dimension 2 in the low volume fraction limit, otherwise known as flocculation region, and 2.5 in the high volume fraction limit or percolation region, irrespective of the type of clusters formed. In the flocculation limit of the DLCA aggregation model, we expect a fractal dimension of 1.8, contrary to that, we observed the formation of lattice animals having a fractal dimension of 2. The presence of lattice animals is confirmed using the scaling laws of size distribution and chemical length of the clusters formed in the flocculation limit.

Fast analysis and prediction approach for geometrically nonlinear bending analysis of plates and shells using artificial neural networks

For the first time, this paper introduces a Fast Analysis and Prediction (FAP) approach for geometrically nonlinear bending analysis of plates and shells. The proposed approach consists of two phases: fast analysis and prediction using artificial neural networks (ANNs) combined with the Bayesian regularization algorithm. In the first phase (fast analysis), geometrically nonlinear bending analysis of a structure is performed using only several load steps. This enables the efficient computation within a short period of time. In the second phase (prediction using ANNs), the deflections obtained in the first phase are utilized as the data for training the ANNs. The trained networks are then used to quickly predict deflections at any load step. The calculation process in the second phase is also very fast. For the analysis package in the first phase, the formulation of this paper is originated from the literature for geometrically nonlinear analysis of composite shells. High accuracy and efficiency of the FAP approach were demonstrated via solving nonlinear bending problems of plates and shells including an isotropic square plate, a composite cylinder subjected to internal pressure, a FG-CNTRC panel and a pinched cylinder with end diaphragms. The proposed approach offers significantly computational savings (up to 70%) compared to the conventional one. ANNs combined with the Bayesian regularization algorithm, which are proposed for the FAP approach in this work, possess the outstanding advantages compared to many existing neural networks. The FAP approach can be applied to simulations, observations, or experiments where nonlinear responses are present or data is limited. Applying the FAP approach to such areas can significantly save labor, time, and money.

Quantum scars in spin- isotropic Heisenberg clusters

We investigate the influence of the external fields on the statistics of energy levels and towers of eigenstates in spin- isotropic Heisenberg clusters, including chain, ladder, square and triangular lattices. In the presence of uniform field in one direction, the SU(2) symmetry of the system allows that almost whole spectrum consists of a large number of towers with identical level spacing. Exact diagonalization on finite clusters shows that random transverse fields in other two directions drive the level statistics from Poisson to Wigner–Dyson distributions with different values of mean level spacing ratio, indicating the transition from integrability to non-integrability. However, for the three types of clusters, it is found that the largest tower still hold approximately even the symmetry is broken, resulting to a quantum scar. Remarkably, the non-thermalized states cover the Greenberger–Horn–Zeilinger and W states, which maintain the feature of revival while a Neel state decays fast in the dynamic processes. In addition, some dynamic schemes for experimental detection are proposed. Our finding reveals the possibility of quantum information processing that is immune to the thermalization in finite size quantum spin clusters.

Time Period Analysis of Orthotropic Skew Plate with 2-D Circular Thickness and 1-D Circular Density

In the present study, the time period of vibration of an orthotropic parallelogram plate with 2 − D (two-dimensional) circular thickness under the effect of 2 − D parabolic temperature is investigated for the first time. The different edge conditions are C C C C , C C C F , C F C F , C S C F , and S F S F boundary conditions, where C, S, and F stands for clamped, simply supported, and free edges of the plate, respectively. The variation in density of plate material is considered to be 1 − D (one-dimensional) circular. The Rayleigh–Ritz technique is used to solve the differential equation and evaluate the time period for the first two modes of vibration. A convergence study of an orthotropic parallelogram, rectangle plate, and square plate for modes of frequency at various edge conditions is also carried out. The authors performed a comparative analysis of the time period and modes of frequency of orthotropic parallelogram, rectangle plate, and isotropic square plate with the available published results at various edge conditions. The main conclusion which we made from this study is that by choosing the above-mentioned plate parameters, we obtained fewer modes of frequency in comparison to other variations mentioned in the literature. Also, the study suggests that the variation in modes of frequency is less in comparison to other variations.

Correlated triple hybrid amplitude and phase holographic encryption based on a metasurface

Metasurface holography is becoming a universal platform that has made a considerable impact on nanophotonics and information optics, due to its advantage of large capacity and multiple functionalities. Here, we propose a correlated triple amplitude and phase holographic encryption based on an all-dielectric metasurface. We develop an optimized holographic algorithm to obtain quantitatively correlated triple holograms, which can encrypt information in multiple wavelength and polarization channels. We apply the “static” and “dynamic” pixels in our design, respectively. Two kinds of isotropic square nanofins are selected, one functioning as a transmitter and the other functioning as a blocker counterintuitively at both working wavelengths, while another anisotropic rectangle nanofin can transmit or block light in co-polarization selectively, mimicking “dynamic” amplitude switches. Meanwhile, such “dynamic” nanofins can simultaneously function as a phase modulator in cross-polarization only at the transmission wavelength. That is, through smart design, different dielectric meta-atoms functioning as spectral filters as well as phase contributors can compositely achieve triple hybrid amplitude and phase holograms. Such strategy promises to be applied in compact large-capacity information storage, colorful holographic displays, optical encryption, multifunctional imaging devices, and so on.

Three Dimensional Photoelastic Investigation for Analyzing Stress Concentration Factor in Isotropic Square Simply Supported Plate with Hole Subjected to Transverse Loading

In this paper, an experimental and numerical study was carried out to examine stress concentration fields near singularities, originating and propagating in accordance with different hole diameter to width ratios (D/A) in a square plate. Experimental three-dimensional photoelastic analyses were performed to predict the reaction of geometric conditions, loading and boundary condition on stress concentration factor (SCF) near the hole. During, experimental analysis, stress was locked inside the plate by the use of stress freezing technique. Further evaluation of locked stress was performed with a polariscope, with the help of slices procured using the slicing method employed on stress frozen plate. The different fringes obtained from monochromatic and white light showed the variation of maximum intensity stresses which were further compared with the stress contours from the finite element model. Also, assessment was conducted with the finite element method to validate the results. Based on the results, percentage variation between experimental and analytical was evaluated and it was found that the variation was between 2 and 5 %.

Series representations and simulations of isotropic random fields in the Euclidean space

This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.

Effect of Membrane Components of Transverse Forces on Magnitudes of Total Transverse Forces in the Nonlinear Stability of Plate Structures.

For an isotropic square plate subject to unidirectional compression in the postbuckling state, components of transverse forces in bending, membrane transverse components and total components of transverse forces were determined within the first-order shear deformation theory (FSDT), the simple first-order shear deformation theory (S-FSDT), the classical plate theory (CPT) and the finite element method (FEM). Special attention was drawn to membrane components of transverse forces, which are expressed with the same formulas for the first three theories and do not depend on membrane deformations. These components are nonlinearly dependent on the plate deflection. The magnitudes of components of transverse forces for the four theories under consideration were compared.

Kearsley-type instabilities in finite deformations of transversely isotropic and incompressible hyperelastic materials

We assume that the strain energy density, W, for a transversely isotropic and incompressible hyperelastic solid is a complete quadratic function of components of the right Cauchy-Green strain tensor, C. We first discuss restrictions on the seven material parameters appearing in the expression for W. It is shown that for the reference configuration to be stress-free, both terms linear in (I4 − 1) and (I5 − 1) must simultaneously appear in the expression for W where I4 = A • CA, I5 = A • C2A, and A is a unit vector along the axis of transverse isotropy. Subsequently, we show that for a prismatic body comprised of this material deformed either in dead-load or displacement-controlled uniaxial tension/compression along A, depending upon the sign of the material parameter h14 multiplying the term (I1 − 3)(I4 − 1) in the expression for W, the two lateral stretches may bifurcate from being equal to being unequal for a stable solution defined as the one that has lower free energy. Here I1 = tr(C) where tr is the trace operator. This is similar to Kearsley's instability in an isotropic square Mooney-Rivlin membrane deformed with equal biaxial dead loads on the edges for which the stable deformed configuration shifts from having equal to unequal biaxial stretches.

A Novel Refined Shear Deformation Theory for the Buckling Analysis of Thick Isotropic and Orthotropic Plates on Two-Parameter Pasternak’s Foundations

Buckling analysis of isotropic and orthotropic plates resting on two-parameter Pasternak’s foundations using the four-variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate; hence, it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The nonlinear strain–displacement of Von Karman relations is also taken into consideration. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results obtained by the present theory are compared with classical plate theory solutions, first-order shear deformable theory solutions, higher-order shear deformation theory and available exact solutions in the literature. The effects of the foundation parameters, side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are considered in this analysis. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order shear deformable theory.

Effects of the Radius of Curvature on Natural Frequency and Mode Shape

In this study, the effect of radius of curvature on the natural frequency of isotropic square thin plate is investigated. The models examined are obtained by simply changing the radius of curvature by keeping the length constant from the flat plate to the semicircle model. The free vibration analysis of the plates fixed on two straight edges is performed by the finite element method. As a finite element type, the four-node quadrilateral rectangular element type, which has a total of 24 generalized coordinates, is used. Out-of-plane theory and in-plane theory are used together to form the curve model. The accuracy and validity of the theory are controlled with the data obtained from the finite element program. The effect of the curvature radius on the first five natural frequencies and the mode shapes of these natural frequencies is given in tables and graphs.

The SAPBC Method on local, non-Cluster updates algorithms of Monte Carlo simulation: A study on more convergence of spin correlation at critical temperature.

In this method, we propose computational technique of Screw-Antisymmetric Periodic Boundary Condition (SAPBC Method) on local, non-cluster update algorithms of Isotropic square lattice Ising model of Monte Carlo simulations, as well. The SAPBC Method, actually, is an extended mixed method of Screw (helical) and Antisymmetric periodic boundary conditions beyond connection from of nearest neighbor spin of the main lattice to even far away block of the outer (foreign) neighbor spin arrays. Here, meanwhile of description of geometry exact details of method and way of spin interaction, have applied to critical slowing down in order to achieve more convergence of spin correlation at critical temperature. Actually, in general, at critical temperature algorithms performed by using SAPBC Method have faster correlation and much shorter autocorrelation time than algorithms performed by using PBC Method. We will also see that autocorrelation function for the typewriter Metropolis algorithm was found to be zero at high temperatures. For low temperatures it fell to zero and stayed there. The SAPBC Method also confirms and consists with the law of the spatial correlation length with its dynamical critical exponent. Therefore, it can be used as a trenchant method applied to boundary conditions of Monte Carlo simulation problems extending on a variety of other models such as XY-Pots-Heisenberg model and also cluster algorithms such as Wolf, Swendsen-Wangas, Hoshen-Koppelman as well.

Benchmark Vibration Frequencies of Square Thin Plates with all Possible Combinations of Classical Boundary Conditions

In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: C — clamped, S — Simply supported, F — free, and G — guided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution is composed of three parts, the sum of which satisfies exactly both the field equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the first six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the first time.

Series representations of isotropic vector random fields on balls

This paper deals with a class of second-order vector random fields in the unit ball of Rd, whose direct/cross covariances are invariant or isotropic with respect to a distance defined on the ball, and gives a series representation of such an isotropic vector random field. A necessary format of covariance matrix functions is also derived for isotropic and mean square continuous vector random fields on the ball.

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isotropic covariance matrix function on the sphere or the real projective space.

Positive definite matrix functions on spheres defined by hypergeometric functions

ABSTRACTPositive definite matrix functions on spheres arise naturally in multivariate approximation and spatial statistics. The construction of strictly positive definite models has become one of the most important goals for the analysis and study of vector valued random fields on spheres. This paper derives strictly positive definite models based on hypergeometric functions. In statistical nomenclature, it describes strictly positive definite covariance functions of stationary, isotropic and mean square continuous Gaussian vector random fields on spheres in which the isotropic part of the covariance functions are set through hypergeometric functions.

A resummed thermodynamic perturbation theory for positive and negative hydrogen bond cooperativity in water

In this paper a resummed thermodynamic perturbation theory is developed which accounts for both positive and negative hydrogen bond cooperativity in water. The theory is developed in Wertheim’s multi-density statistical mechanics. We demonstrate that the hydrogen bonded structure of water is controlled by positive hydrogen bond cooperativity in homodromic hydrogen bonded clusters. Inclusion of negative anti-cooperativity in antidromic hydrogen bonded structures has little effect on the underlying hydrogen bond structure of liquid water. The resummed perturbation theory is shown to give hydrogen bond statistics consistent with first principles ab initio molecular dynamics simulations. In addition, the theory is shown to be in good agreement with experiment in the prediction of the average number of hydrogen bonds per water molecule in a liquid at ambient conditions. Finally, we develop a full thermodynamic model for water by including contributions to the free energy for isotropic square well attractions. We demonstrate that the model gives good representation of saturated liquid density, hydrogen bonding and vapor pressure. In addition, we show the full model predicts the anomalous minima in isothermal compressibility and isobaric heat capacity.

Mechanical Stability Investigation of Advanced Composite Plates Resting on Elastic Foundations Using a New Four-Unknown Refined Theory

A refined and simple shear deformation theory for mechanical buckling of composite plate resting on two-parameter Pasternak’s foundations is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces.Therefore, there is no need to use shear correction factor. The number of independent unknowns of present theory is four, as against five in other shear deformation theories.It is assumed that the warping of the cross sections generated by transverse shear is presented by a hyperbolic function. The stability equations are determined using the present theory and based on the existence of material symmetry with respect to the median plane.The nonlinear strain-displacement of Von Karman relations are also taken into consideration. .The boundary conditions for the plate are assumed to be simply supported in all edges. Closed-form solutions are presented to calculate the critical load of mecanical buckling, which are useful for engineers in design. The effects of the foundation parameters,side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are considered in this analysis.are presented comprehensively for the mechanical buckling of rectangular composite plates.

Polarization-Diverse Holographic Metasurfaces

This letter presents the design and fabrication of polarization-diverse metasurface leaky-wave antennas that are composed of simple isotropic square patches. Multiple vertical monopoles are employed to excite the metasurface. Each source, or a combination of sources, is designed to enable the metasurface to form a pencil beam with the desired horizontal, vertical, or circular polarization. A prototype of the antenna was realized, using a 3.175 mm thick substrate of dielectric constant 2.2, to operate at 12 GHz. It is also shown that, by employing only backward leaky waves to form the pencil beam, frequency scanning with a much larger scan-angle range can be achieved.

ПРОГИБЫ И ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ СОСТАВНЫХ МНОГОСЛОЙНЫХ КВАДРАТНЫХ ИЗОТРОПНЫХ ПЛАСТИН С ШАРНИРНЫМ ОПИРАНИЕМ ПО КОНТУРУ ПРИ ИЗМЕНЕНИИ ЖЕСТКОСТИ СВЯЗЕЙ СДВИГА

The relationship between the fundamental frequency of free transverse vibrations ω of multilayer isotropic square plates in an unloaded state and their maximum deflections W 0 under the action of a uniformly distributed load depending on the number of layers and the rigidity of shear bonds is considered. The number of layers in a composite plate varies from two to ten. In the work, numerical studies of composite two-layer plates on compliant bonds were carried out using the finite element method. A finite element diagram of a multilayer plate is developed. The calculation was performed in the SCAD software package. The curves for different numbers of layers of multilayer plates “Deflection - stiffness of shear bonds” and “Frequency of transverse vibrations - stiffness of shear bonds” are plotted. It is shown that for composite multilayer square-shaped plates, the coefficient K coincides with an accuracy of 5.4% with the same coefficient for square plates of whole section.