Equilateral Triangular Plate Research Articles

Understanding Stacking Fault Tetrahedron Formation in FCC Stainless Steel: A Fusion of Transmission Electron Microscopy, Molecular Dynamics, and Machine Learning

The stacking fault tetrahedron (SFT) formation displays a pronounced size effect, progressing from vacancy equilateral triangular plate to perfect SFT, and eventually to truncated SFT, as demonstrated in numerous irradiated face-centered cubic metals. However, the presence of distinct SFT structures in F321 stainless steel has not been reported. This study explored the SFT formation mechanism in irradiated F321 stainless steel using transmission electron microscopy (TEM), molecular dynamics (MD) simulations, and machine learning. SFTs, Frank loops, and Lomer-Cottrell locks were found to be widely generated in the irradiated F321 steel. The critical size for truncated and perfect SFTs was determined using MD simulations; the results were consistent with the theoretical predictions. Additionally, the twin boundaries observed through TEM, which were attributed to the elevated tensile stress near the boundaries, facilitated the formation of perfect SFTs. Moreover, interstitial Frank loops also facilitated the formation of perfect SFTs. This study also explored the influence of variations in Ni and Cr concentrations on the critical size n1 for the transition from vacancy plates to perfect SFTs and n2 for the transition from perfect SFTs to truncated SFTs, using a combination of MD and machine learning methods. As the Ni concentration increased and the Cr concentration decreased, n1 and n2 increased; conversely, the critical sizes decreased when the Ni concentration decreased and the Cr concentration increased. These insights reveal the systematic mechanism of SFT formation under varied conditions, offering new perspectives for understanding the nano-defects in F321 stainless steel.

Auxetics and Other Systems of Anomalous Characteristics

Auxetics and Other Systems of Anomalous Characteristics

Analogies for simply supported nonlocal Kirchhoff plates of polygonal planform

In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green’s function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local “background” Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments.

Shear Deformation in a Class of Thick Hexagonal Plates

The shear deformation characteristics of a simply supported and uniformly loaded regular hexagonal plate is investigated herein, including cases when the plate is made from auxetic materials. Specifically, the shear correction factor for such a plate is obtained by matching its maximum deflections between the Mindlin and Reddy plate theories after a convenient Kirchhoff theory model has been proposed. Results show that the shear deformation is suppressed if the plate material is auxetic; for example, the extent of shear deformation is conserved even if the plate thickness is doubled, provided that the Poisson's ratio reduces from 0.5 to −1. When benchmarked against other plate shapes, the shear deformation for the hexagonal plate is comparable with those of square and equilateral triangular plates. Therefore, the design principles that incorporate shear deformation in thick square plates and in thick equilateral triangular plates can be extended to thick hexagonal plates, and this validity applies for both conventional and auxetic plate materials.

Caustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination

The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.

Bending, Buckling, and Vibration of Equilateral Simply Supported or Clamped Triangular Plates with Rounded Corners

The pb2-Ritz method and the homotopy shape function are used for bending, buckling, and vibration analyses of equilateral triangular plates with rounded corners. On the basis of the comprehensive results obtained, it is found that the deflection shape, buckling load, and fundamental frequency vary nonlinearly with respect to the corner roundedness parameter α and Poisson's ratio for simply supported plates, whereas these aforementioned results vary nonlinearly with respect to α only for clamped plates. The use of the pb2-Ritz method facilitates the development of approximate formulas for the accurate prediction of deflection shapes, buckling loads, and fundamental frequencies for simply supported and clamped rounded-corner equilateral triangular plates. These formulas presented in this paper should be useful for designers working on such plates.

Scattering of electromagnetic waves from surfaces with conformal mapping: An example of a triangular plate.

We discuss a way to exploit conformal mapping to study the response of a finite metallic film of arbitrary shape to an external electromagnetic field at finite frequencies. This provides a simple way to understand different physics issues and provides insights that include the issue of vorticity and eddy current and the nature of the divergent electric field at the boundaries and at corners. We study an example of an equilateral triangular plate and find good agreement with results obtained with traditional numerical techniques.

Higher-order shear deformation of very thick simply supported equilateral triangular plates under uniform load

ABSTRACTWhere deflections of thick plates are concerned, sufficient studies have been performed on circular and rectangular plates using first-order shear deformation theory (FSDT). Less attention, however, has been placed on the use of higher-order theories and plates of other shapes. This article proposes a deflection model for a simply supported and under uniformly loaded equilateral triangular plate that fulfils the third-order shear deformation theory (TSDT). Comparison of maximum plate deflections using this exact TSDT model with the FSDT and the simplified TSDT models reveals that the former exhibits better correlation agreement with the exact TSDT. Using the plate deflection from the exact TSDT, a refined shear correction factor, which is a function of Poisson's ratio, relative thickness, and location on plate, is extracted for the FSDT model. Results suggest that the usual shear correction model of 5/6 in Mindlin plates is highly accurate when dealing with moderately thick triangular plates made from negative Poisson's ratio materials, while the prescription of the refined shear correction factor obtained herein is advised for very thick triangular plates made from large Poisson's ratio materials. The results avail an FSDT deflection model for very thick triangular plates with the accuracy of exact TSDT.

Bending Stresses in Triangular Auxetic Plates

This technical brief considers bending stress minimization as a basis for obtaining the optimal Poisson's ratio of simply supported equilateral triangular plates under (a) bending loads at the plate boundary, (b) uniform load throughout the entire plate, and (c) concentrated load at the plate center. Results suggest that the use of auxetic materials is not appropriate for triangular plates under applied bending at the boundary, while mildly auxetic and highly auxetic materials are appropriate for triangular plates under uniform and central point loads, respectively. In addition, obtained results show that the optimal Poisson's ratios for circular and square auxetic plates are not necessarily applicable for triangular plates. The use of auxetic materials offers an additional choice for decreasing bending stresses under specific boundary conditions and loading patterns.

SHAPE EFFECT ON FREE VIBRATION OF FUNCTIONALLY GRADED PLATES

In this article, shape effect on free vibration behavior of functionally graded plates is investigated. Square, rectangular, skew, circular, elliptical, annular and equilateral triangular plates with the same surface area and thickness are considered. Frequency values of these plates are compared for simply supported and clamped boundary conditions. Finite element method (FEM) is used in calculating frequency values and mode shapes. Since commercial codes do not allow inputting functionally graded material properties directly, MATLAB code was developed for FEM solution. Findings of this study can be useful for designers that have freedom to choose the plate shape in engineering applications

Large Amplitude Free Vibrations of Plates Resting on a Pasternak Type Elastic Foundation

In this paper following Berger's approximate plate theory for large deflections, large amplitude free vibrations of Rectangular Plate, isosceles right-angled triangular plate, Equilateral triangular plate and circular plate resting on a Pasternak-type elastic foundation have been discussed.

PIV measurements of turbulent jets issuing from triangular and circular orifice plates

The present study experimentally investigated the near-field flow mixing characteristics of two turbulent jets issuing from equilateral triangular and circular orifice plates into effectively unbounded surroundings, respectively. Planar particle image velocimetry (PIV) was applied to measure the velocity field at the same Reynolds number of Re=50,000, where Re = U e D e/ν with U e being the exit bulk velocity and ν the kinematic viscosity of fluid, D e the equivalent diameters. The instantaneous velocity, mean velocity, Reynolds stresses were obtained. From the mean velocity field, the centreline velocity decay rate and half-velocity width were derived. Comparing the mixing characteristics of the two jets, it is found that the triangular jet has a faster mixing rate than the circular counterpart. The triangular jet entrainments with the ambient fluid at a higher rate in the near field. This is evidenced by a shorter unmixed core, faster Reynolds stress and centreline turbulence intensity growth. The primary coherent structures in the near field are found to break down more rapidly in the triangular jet as compared to the circular jet. Over the entire measurement region, the triangular jet maintained a higher rate of decay and spread. Moreover, all components of Reynolds stress of the triangular jet appear to reach their peaks earlier, and then decay more rapidly than those of the circular jet. In addition, the axis-switching phenomenon is observed in the triangular jet.

Viscoelastic vibrations of a triangular plates

The problem of stationary vibrations of a viscoelastic equilateral triangular plate is treated. The vibrations are caused by the action of a uniformly distributed load which varies harmonically and by the motion of the simply supported boundary of the plate as a rigid body which vibrates at the same frequency. The level lines of the vibration amplitude are studied and the graphs of the amplitude distribution along the height of the triangle are given.

Benchmarks of simple, generic, shaped plates for validation of low-frequency electromagnetic computational codes

The validation of low-frequency measurements and electromagnetic (EM) scattering computations for several simple, generic shapes, such as an equilateral-triangular plate, an equilateral-triangular plate with a concentric equilateral-triangular hole, and diamond- and hexagonal-shaped plates, is discussed. The plates were constructed from a thin aluminium sheet with a thickness of 0.08 cm. EM scattering by the planar plates was measured in the experimental test range (ETR) facility of NASA Langley Research Center. The dimensions of the plates were selected such that, over the frequency range of interest, the dimensions were in the range of lambda /sub 0/ to 3 lambda /sub 0/. In addition, the triangular plate with a triangular hole was selected to study internal-hole resonances.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The capacitance of the regular tetrahedron and equilateral triangle

We calculate the capacitance of an isolated regular tetrahedral conductor and of an isolated equilateral triangular plate by the charge simulation method. By using careful extrapolation, we obtain the results C/4πϵ 0l = 0.35688 ± 0.0001 and 0.25096 ± 0.00002 respectively, using a small number of unknowns. Here, l is the length of the side. The first result suggests that the inversion method is rather poor.

A boundary element solution of the plate buckling problem

Boundary integral equations are obtained from the variational equation governing the buckling mode of a thin elastic plate through the well established procedure of introducing the two fundamental solutions of the classical plate bending problem to the variational functionals. The existence in these equations of domain integrals containing the unknown lateral post-buckling deflection of the plate requires the introduction of domain in addition to boundary unknowns in the formulation. Through the boundary and domain discretisation, two linear algebraic systems of equations are obtained from which the boundary unknowns are eliminated and the eigenvalue problem for the buckling load and the buckling mode is thus formulated. Two alternative approximations of the post-buckling deflection have been tried for the evaluation of the domain integrals. The first involves finite element type triangular cells, the second trigonometric series. The various examples solved include simply supported or clamped square plates under a variety of edge loadings as well as an equilateral triangular plate under uniform edge compression. The numerical results obtained demonstrate the validity of the formulation through the convergence of the approximate model indicating at the same time the degree of accuracy which can be achieved.

Wide band random vibration of an equilateral triangular plate

The distribution of mean-square velocity response over a lightly damped uniform plate excited at a point by a wide-band random force tends to be more or less uniform except in certain special zones or lanes of intensified or decreased response. The locations of these zones depend on the symmetry of the plate shape and on the position of the driving-point. In the case of an equilateral triangular plate there is intensified response along a median whenever the driving-point lies on that median. When the driving-point is at the centroid of the triangle there is intensified response along all three medians with a localized concentration at the driving-point. It is shown that the driving point intensification ratio increases slowly, but without limit, as the excitation bandwidth increases due to the increasing multiplicity of the higher-frequency plate resonances. Quantitative estimates for the driving-point response are obtained from a computer evaluation of the multiplicities of the first 32 000 resonances and from an asymptotic approximation based on the theory of numbers.

Study on the vibration of a piezoelectric equilateral triangular plate by the finite element method.

Study on the vibration of a piezoelectric equilateral triangular plate by the finite element method.

Large amplitude free vibrations of simply supported equilateral triangular plate using tri-linear co-ordinates

Large amplitude free vibrations of simply supported equilateral triangular plate using tri-linear co-ordinates

Corner supported equilateral triangular plate subjected to a concentrated load at the center

The bending by a concentrated load acting at the center of a corner supported equilateral triangular plate is presented. The problem is solved under the assumptions that the total bending moment is zero, and the supplemented shear vanishes at discrete points along the edges of the plate. The results, presented in graphs, are compared with the measurements taken earlier on a steel plate model.