Equilibrium Form Research Articles

Large Deflections, Loss of Stability and Over-Critical Behavior of Sloping Panels and Arches of Variable Thickness on an Elastic Base

The problem of large deflections of a flat arch in its plane (or an infinitely long panel) loaded with transverse loads is considered. A variational approach has been applied to solve it. The resolving nonlinear equations are reduced to finding the deflection and longitudinal force, which is considered constant along the length of the arch due to its flatness. An approximate solution method is proposed by decomposing the displacements into a Fourier series. The peculiarity of the approach used is that in order to pass the limit points, it is not necessary to use special algorithms such as the continuation method by parameter. It also allows you to trace the process of supercritical deformation of the arch. The proposed approach allows us to consider problems for arches of variable thicknesses, located on elastic supports, on an elastic base with a variable bed coefficient and various loads. Therefore, it is also convenient in problems of finding, for example, the optimal thickness distribution under restrictions on critical loads, on rigidity stiffness and on maximum compression or tension stresses. The results of numerical calculations are presented. The convergence is studied of the solution depending on the number of terms of the series, into which the desired deflection decomposes. A good agreement with the known analytical results is obtained earlier by solving the equilibrium equations of the arch element and panel in the case of simple types of loading. At the same time, even in more complex cases of loading and supercritical bending of an arch without an elastic base, the results are obtained when holding three, four and five members of the Fourier series, the maximum deflections and critical loads differed by no more than 2.5%. On the basis of numerical experiments, the features of the arch behavior caused by the rearrangement of geometry during loading are revealed. The processes of stability loss of the arch and its supercritical behavior are investigated. The effect of symmetric deformation in the case of kinematic loading is found, namely, it is revealed that at a certain value of the concentrated force, an infinite number of equilibrium forms are possible. Arches on an elastic base, as well as with variable thickness are considered (the results are presented in the form of load–displacement diagrams and in the form of pictures of deformed arch shapes). An interesting effect has been revealed based on numerical experiments. It turned out that the increase in thickness when moving away from the supports does not change the nature of the load–displacement diagram, i.e., with some load, a slap occurs. On the contrary, a decrease in thickness when moving away from the supports leads to a flattening of the diagram, and after a certain thickness value in the center, its further decrease leads to the fact that the arch does not occur.

Factors contributing to phosphorus sorption reversibility from iron-desalinization treatment residue (Fe-DTR)

Phosphorus (P), a non-renewable nutrient, requires effective, innovative recycling practices, especially because of predicted future scarcity. This study focused on the adsorption-desorption of P from iron desalinization treatment residue (Fe-DTR) mixed with dairy wastewater to create an alternative fertilizer (P-Fe/O-DTR). Sequential extraction indicated that P adsorption onto Fe-DTR, at pH 3, predominantly favored adsorption onto iron oxides collected in summer (72 %) versus winter (59 %), due to the elevated use of FeCl3 coagulant in summer in the desalinization pre-treatment. Desorption experiments indicated that the irreversibility of sorption (apparent hysteresis index) was highest under aerobic conditions, using a synthetic P solution, ambient temperature and pH range of 7–3 (98.9–98.3 %). Irreversibility was lowest under anaerobic conditions, using dairy wastewater as a P source, at pH 7 and 40 °C (76.6 %). This suggests that loading settings of organic solution as a P source and high temperature at the sorption stage are favored for extracting P from P-Fe/O-DTR with the greatest potential for release and reuse. Desorption experiments of 3 d, 1 week and 9 weeks indicated that the initial equilibrium forms within 72 h, but resorption and precipitation processes establish a new equilibrium over time (week). Phosphorus desorption was strongly affected by the aeration conditions of the experiment followed by the size of the sorbed reservoir and the nature of the solution (wastewaters versus synthetic solution). Under anaerobic conditions, temperature exerted a high influence on the sorbed P available for desorption.

Backbone assignment of CcdB_G100T toxin from E.coli in complex with the toxin binding C-terminal domain of its cognate antitoxin CcdA.

The CcdAB system expressed in the E.coli cells is a prototypical example of the bacterial toxin-antitoxin (TA) systems that ensure the survival of the bacterial population under adverse environmental conditions. The solution and crystal structures of CcdA, CcdB and of CcdB in complex with the toxin-binding C-terminal domain of CcdA have been reported. Our interest lies in the dynamics of CcdB-CcdA complex formation. Solution NMR studies have shown that CcdB_G100T, in presence of saturating concentrations of CcdA-c, a truncated C-terminal fragment of CcdA exists in equilibrium between two major populations. Sequence specific backbone resonance assignments of both equilibrium forms of the ~ 27kDa complex, have been obtained from a suite of triple resonance NMR experiments acquired on 2H, 13C, 15N enriched samples of CcdB_G100T. Analysis of 1H, 13Cα, 13Cβ secondary chemical shifts, shows that both equilibrium forms of CcdB_G100T have five beta-strands and one alpha-helix as the major secondary structural elements in the tertiary structure. The results of these studies are presented below.

Об устойчивости круговых подкрепленных арок в случае пространственной деформации

The work considers a circular arch loaded with uniformly distributed normal pressure directed towards the centre. The ends of the arch are attached with cables, one end of which is attached to the arc of the arch at an appropriate angle, and the distance between the points of attachment of the cables cannot be increased. We estimated pressure values, at which curved equilibrium forms of the arch are possible, and the smallest of these values is found, which is the critical force.

PH-Dependent interactions between β-casein and blueberry anthocyanins based on typical thermal processing conditions: Stability, structural characterization, and binding mechanisms

The color and structural stability of blueberry anthocyanins (BAs) are significantly affected by pH changes during processing. In this study, we present the potential effects of the thermal stability, structural characterization, and binding mechanism of BAs with β-casein (β-CN) at pH 2, 4, and 6 for the first time. The pH was found significantly affect the ability of β-CN to protect BAs from thermal, vitamin C, and sucrose induced degradation. Scanning electron microscopy (SEM) resolved the microstructural differences of β-CN-BAs at varying pH values. Multispectral analysis showed that as pH increases, the electrostatic interaction between β-CN and the major monomer of anthocyanins Cyanidin-3-O-glucoside (C3G) are replaced by hydrogen bonding and van der Waals forces, and then electrostatic interactions are again the dominant force. In vitro gastrointestinal digestion confirmed that the protein stabilized C3G under varying pH conditions. Nuclear magnetic resonance (NMR) spectroscopy reveals the connection between the multiple equilibrium forms of C3G (hemiketal, cis-chalcone, and trans-chalcone) and β-CN on the formation of complexes between multiple equilibrium forms to specifically elucidate the mechanism of its stabilizing effect. Molecular docking simulation revealed the significance of electrostatic interaction and hydrogen bonding, where the highest binding match was achieved at pH 2, with a minimum binding energy. These interactions can effectively promote stable complexation of β-CN with C3G. These findings have practical implications for protein regulation of the relationship between the equilibrium forms of different anthocyanins and can guide the stabilization of anthocyanin-rich products.

A Study of Customer Preference Transmission for New Energy Vehicles Based on a Signaling Game and Separating Equilibrium

Although there are many methods that can be used to obtain customer preferences for new energy vehicles, most studies generally overlook the fact that customer preferences are private information. The purpose of this study is to investigate the transmission mechanism of customer preferences by taking into account situations in which customers lie. Through a signaling game model, this study analyzed the transmission mechanism of customer preference information for the center control touch screen of new energy vehicles based on separation equilibrium. The results show that when inequality (1) remains, such an equilibrium forms: the customers send the real preference signal, the manufacturer then adopts a new sample consistent with the received signal and prices the product accordingly, and, finally, the customers pay for the new NEV. When inequality (2) remains, the following equilibrium forms: customers signal the opposite of their private preference, the manufacturer then adopts a new sample opposite to the received signal, and, finally, customers pay for the new NEV.

Investigating the Influence of the Relative Roughness of the Riverbanks to the Riverbed on Equilibrium Channel Geometry in Alluvial Rivers: A Variational Approach

The roughness of a river’s boundary significantly influences the sediment transport process and the ultimate configuration of the river’s stable cross-section. This interplay between boundary roughness and river morphology is crucial to a river’s overall behavior and form. This study aims to analyze the influence of the relative roughness of riverbanks to a riverbed λ on the equilibrium form of alluvial rivers using a variational method. The results show the following: (1) As the parameter λ transitions from smaller to larger values, noteworthy variations are observed in a river’s characteristics. Specifically, there is a discernible reduction in the calculated maximum sediment discharge, coupled with a corresponding expansion in the optimal width–depth ratio. For instance, when λ changes from 1 to 0.1, the optimal width–depth ratio increases by 45%, while the calculated maximum sediment discharge experiences a decrease of 1.62%. (2) An examination of hydraulic geometric relationships, derived by assigning distinct values to the relative roughness of riverbanks to the riverbed, highlights the significant influence of this relative roughness on the ultimate equilibrium configuration of the river channel. Remarkably, this effect remains consistent and stands independently of other variables such as sediment discharge, flow discharge, channel gradient, and sediment size. (3) The critical and average hydraulic geometric relationships deduced in this study closely align with previous research findings. Notably, this research contributes to addressing the existing gap in understanding the mechanistic underpinnings of how river boundary conditions impact the equilibrium forms of rivers, thereby advancing our knowledge of river morphology. Nevertheless, it is imperative to emphasize that while this study provides valuable theoretical insights, the practical application of these findings in the context of river morphological evolution necessitates further in-depth research. It calls for a more comprehensive exploration of the transition from theoretical constructs to real-world applications, thus promoting a deeper understanding of the dynamics that shape river systems.

Synthesis, characterization, DNA binding interactions, DFT calculations, and Covid-19 molecular docking of novel bioactive copper(I) complexes developed via unexpected reduction of azo-hydrazo ligands

In this work, we focused on the 3rd goal of the sustainable development plan: achieving good health and supporting well-being. Two redox-active hydrazo ligands namely, phenylcarbonohydrazonoyldicyanide (PCHD) and pyridin-4-ylcarbonohydrazonoyl-dicyanide (PyCHD), and their copper(I) complexes have been synthesized and characterized. The analytical data indicates the formation of copper(I) complexes despite starting with copper(II) perchlorate salt. The 1H-NMR and UV–visible spectral studies in DMSO revealed that PyCHD mainly exists in its azo-form, while PCHD exists in azo ↔ hydrazo equilibrium form, and confirmed the copper(I) oxidation state. XPS, spectral and electrochemistry data indicated the existence of copper(I) valence of both complexes. Cyclic voltammetry of PCHD and its copper(I) complex supported the reduction power of the ligand. The antimicrobial activity, cytotoxicity against the mammalian breast carcinoma cell line (MCF7), and DNA interaction of the compounds are investigated. All compounds showed high antimicrobial, and cytotoxic activities, relative to the standard drugs. Upon studying the wheat DNA binding, PCHD and PyCHD were found to bind through external contacts, while both [Cu(PCHD)2]ClO4.H2O and [Cu(PyCHD)2]ClO4.H2O were intercalated binding. In-silico molecular docking simulations against Estrogen Receptor Alpha Ligand Binding Domain (ID: 6CBZ) were performed on all produced compounds and confirmed the invitro experimentally best anticancer activity of [Cu(PyCHD)2]ClO4.H2O. The molecular docking tests against SARS-CoV-2 main protease (ID: 6 WTT) showed promising activity in the order of total binding energy values: [Cu(PCHD)2]ClO4.H2O > [Cu(PyCHD)2]ClO4.H2O > PCHD > PyCHD.

Structure and Spectroscopic Studies of Bis Azo Compounds with S/SO2 Bridges and Containing Cyclic-1,3-Dicarbonyl Groups

A series of new bis azo dyes were synthesized starting from aromatic diamine with S/SO2 bridges and cyclic-1,3-dicarbonyl groups; contain active methylene group, namely; 3,3-dimethyl-1,5-cyclohexadione (dimedone), 1,3-cyclohexadione and 2,2-dimethyl-1,3-dioxane-4,6-dione to give 1-6. Whether the reaction is complete and the purity of the compounds were checked with TLC. The structure of the newly synthezed compounds was investigated by using, FT-IR, 1HNMR, LC–MS/MS and UV–Visible spectrophotometric methods, CH3OH was used in the LC-MS/MS experiments, and the molecular masses of the compounds were seen as M and M+1 protonated in the spectrum. In the UV-Vis spectra of these compounds, the solvent effect by using different solvents and also the acid-base effect by adding HCl or KOH to the medium were investigated. The tautomeric equilibrium forms (enol-imine O-H….N, keto-amine O….H-N forms) of the compounds were investigated according to the electronic spectrum recorded in chloroform, dimethylformamide (DMF), acetic acid, and methanol solvents. The results of the UV-visible and 1H-NMR spectra indicated that the compounds in chloroform, dimethylformamide (DMF), acetic acid and methanol were found in the hydrazo tautomer (keto-amine O….H-N forms) of the proposed tautomeric forms.

Average Turbulence Dynamics from a One-Parameter Kinetic Theory

We show theoretically that the mean turbulent dynamics can be described by a kinetic theory representation with a single free relaxation time that depends on space and time. A proper kinetic equation is constructed from the Klimontovich-type kinetic equation for fluid elements, which satisfies the Navier–Stokes hydrodynamics exactly. In a suitably averaged form, the turbulent kinetic energy plays the role of temperature in standard molecular thermodynamics. We show that the dynamics of turbulent fluctuations resembles a collision process that asymptotically drives the mean distribution towards a Gaussian (Maxwell–Boltzmann) equilibrium form. Non-Gaussianity arises directly from non-equilibrium shear effects. The present framework overcomes the bane of most conventional turbulence models and theoretical frameworks arising from the lack of scale separation between the mean and fluctuating scales of the Navier-Stokes equation with an eddy viscous term. An averaged turbulent flow in the present framework behaves more like a flow of finite Knudsen number with finite relaxation time, and is thus more suitably described in a kinetic theory representation.

A note on simple bifurcation of equilibrium forms of an elastic rod on a deformable foundation

We study bifurcation of equilibrium states of an elastic rod on a two-parameter Winkler foundation. In [5] the existence of simple bifurcation points was proved by the use of the Crandall-Rabinowitz theorem. In this paper we want to present an alternative proof of this fact based on the Krasnosielski theorem, which seems to be simpler than that of [5].

Time-reversal symmetries and equilibriumlike Langevin equations.

Graham has shown [Z. Phys. B 26, 397 (1977)0340-224X10.1007/BF01570750] that a fluctuation-dissipation relation can be imposed on a class of nonequilibrium Markovian Langevin equationsthat admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equationis associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may lose its time-reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time-reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. We find instances where fluctuations due to noise are solely responsible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.

Еволюція форми лінії дислокації в багатокомпонентному сплаві під навантаженням

The evolution of the dislocation line shape in a multicomponent alloy CrCoNiFeMn under loading was investigated by the method of discrete dislocation dynamics. It was found in a numerical experiment that the best approximation for the shape of the average bulge of the dislocation line would be a sinusoidal shape rather than a parabolic or arc shape. The equilibrium form of dislocation at zero load fits well into a band with a width of three correlation lengths of the short-wave component of the shear stress field created by dissolved atoms in the glide plane. In this case the dislocation line waviness on the scale of the correlation length of the long-wave component is not observed. It has been found that dislocation segments can overcome internal stress barriers with external applied stress assistance. This is an irreversible process of new equilibrium bulges formation. One of these bulges becomes nonequilibrium, increases and releases the dislocation from the initial fixation at a critical stress, which can be conditionally considered to be the yield strength. The external stress, which assists to the dislocation segments to overcome the internal stress barriers, can to some extent compensate for the short-wave component of the shear stress field. Then, as the numerical experiment shows, the dislocation line waviness on the scale of the correlation length of the long-wave component will be activated. Thus, the two components of the shear stress field affect the shape of the dislocation line separately and sequentially with increasing external load. Keywords: shear stresses, solid solution, glide plane, dislocation.

Critical Loads of Uniformly Compressed Orthotropic Rectangular Plate on an Elastic Base

Introduction. The problem of critical loads of a compressed orthotropic rectangular plate on an elastic base was considered. The following orthotropy parameters were set for the plate: Poisson coefficients, Young's modules for the main directions, and the shear modulus of the plate material. The components of the compressive load were uniformly distributed along two opposite edges of the plate and acted parallel to the coordinate axes. The edges of the plate were loosely pinched or pivotally supported. Cases were also considered when two parallel edges of the plate were free from loads, and the other two were freely pinched or pivotally supported.Materials and Methods. The problem was studied on the basis of a system of nonlinear Kármán-type equilibrium equations. The critical values of the load parameter were determined from a linearized problem based on a trivial solution. At the same time, the variational method in combination with the finite difference method was used to solve the boundary eigenvalue problem.Results. The problem was reduced to solving a parametric linear boundary eigenvalue problem. In case of boundary conditions of a movable hinge support, exact formulas of eigenvalues and eigenfunctions were given. While in case of free edge pinching, a variational method was used in combination with a finite-difference method, and a computer program for solving the problem was built. It was established that one or two eigenfunctions expressing the deflection of the plate could correspond to the critical value of the compressive load parameter at which the stability of the compressed plate was lost. The results of numerical calculations of the critical values of the compressive load at different values of the orthotropy parameters were presented, and graphs of the corresponding equilibrium forms were constructed. For the case of a long orthotropic plate on an elastic base, it was established that the main term of the asymptotic expansion of the solution to the linear eigenvalue problem was determined from the problem of critical loads of a compressed beam on an elastic base with an elastic modulus that coincides with the elastic modulus of the plate in the longitudinal direction.Discussion and Conclusions. The problem of critical loads of an orthotropic plate compressed in two directions lying on an elastic base was investigated. As the compressive load component increased along one direction, the critical value of the load compressing the plate along the other direction decreased. If an orthotropic plate was compressed by a load along a direction that corresponded to a greater bending stiffness, then the critical value of the loss of stability was greater than the critical value of the compressive load acting along the direction of a lesser bending stiffness. The presence of an elastic foundation increased the bearing capacity of the compressed plate.

Influence of substrate properties on a fluid drop’s free translational oscillations

The liquid drop’s natural translational oscillations are considered. The equilibrium form of this drop is a circular cylinder. Its axis of symmetry is perpendicular to two parallel solid substrates. The properties (wetting, roughness etc.) of these surfaces differ from each other. The drop is in another liquid. The contact angles’s changes are linearly proportional to the velocities of both contact lines. The Fourier series form by Laplace’s operator eigenfunctions are used for the problem solution. A system of complex equations of eigenvalue problem is solved numerically. The main frequency of the translational mode becomes zero after a critical Hocking’s parameter in situation of identical plates. The branching point of a decrement curve agrees with the zero point of a fundamental frequency. This frequency may not be vanishing on nonidentical surfaces of plates.

The stress function basis of the upper bound theorem of plasticity

The discontinuous slip-line form of upper bound plasticity analysis is considered using an equilibrium of forces approach. It is demonstrated that the underlying basis of the approach can be written in terms of stress functions that provide a continuum stress state interpretation of the upper bound solution. An alternative proof of the upper bound theorem, applicable to both associative and non-associative materials, using stress functions is presented. The broader nature of the equilibrium form and the strict conditions under which it is valid are discussed, including examination of the apparent omission of moment equilibrium and associativity in many equilibrium form solutions. Finally, the relationship of the stress function formulation to the output of the computational limit analysis method discontinuity layout optimisation (DLO) and the potential to use the stress function formulation to derive a form of lower bound solution from an upper bound analysis are demonstrated.

The instability of a helical vortex filament under a free surface

We perform a theoretical investigation of the instability of a helical vortex filament beneath a free surface in a semi-infinite ideal fluid. The focus is on the leading-order free-surface boundary effect upon the equilibrium form and instability of the vortex. This effect is characterised by the Froude number $F_r = U(gh^*)^{-{1}/{2}}$ where $g$ is gravity, and $U = \varGamma /(2{\rm \pi} b^*)$ with $\varGamma$ being the strength, $2{\rm \pi} b^*$ the pitch and $h^*$ the centre submergence of the helical vortex. In the case of $F_r \rightarrow 0$ corresponding to the presence of a rigid boundary, a new approximate equilibrium form is found if the vortex possesses a non-zero rotational velocity. Compared with the infinite fluid case (Widnall, J. Fluid Mech., vol. 54, no. 4, 1972, pp. 641–663), the vortex is destabilised (or stabilised) to relatively short- (or long-)wavelength sub-harmonic perturbations, but remains stable to super-harmonic perturbations. The wall-boundary effect becomes stronger for smaller helix angle and could dominate over the self-induced flow effect depending on the submergence. In the case of $F_r > 0$ , we obtain the surface wave solution induced by the vortex in the context of linearised potential-flow theory. The wave elevation is unbounded when the $m$ th wave mode becomes resonant as $F_r$ approaches the critical Froude numbers ${\mathcal {F}} (m) = (C_0^*/U)^{-1} (mh^*/b^*)^{-{1}/{2}}$ , $m=1, 2, \ldots,$ where $C_0^*$ is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if $F_r < {\mathcal {F}}(1)$ . Compared with the infinite fluid and $F_r \rightarrow 0$ cases, the wave effect causes the vortex to be destabilised to super-harmonic and long-wavelength sub-harmonic perturbations with generally faster growth rate for greater $F_r$ and smaller helix angle.

Limit state according to the condition of loss of stability of equilibrium forms

Objective. The purpose of the study is to determine the group of the limiting state according to the condition of loss of stability of the equilibrium form of structures. Method. The study is based on the provisions of the theory of stability of equilibrium states of building structures; branching theory of solutions of nonlinear equations; perturbation method; methods of catastrophe theory.Result. The results of the analysis of the post-critical behavior of structures based on the solution of the problem in higher approximations and from the fundamental provisions of the theory of catastrophes are generalized. It is proved that the study of the stability of equilibrium forms of structures using algebraic means and geometric images of the theory of catastrophes makes it possible to unambiguously determine the type of critical bifurcation points, predict the nature of the behavior of the structure, and determine the limit state group to which the state reached by the structure should be attributed. Conclusion. It seems necessary to rename the ordinal numbers of the types of critical points of bifurcations so that they coincide with the numbers of the groups of limit states corresponding to them.

On non-axisymmetrical buckling modes ofcircular plates under normal pressure

The paper presents the results of a study of the bifurcation of axisymmetric equilibrium forms of round plates under various conditions for fixing the outer edge. It is shown that for the case of sliding edge, the analytical, asymptotic and finite element approaches give close results. When the edge of the plate is hinged, the buckling to a non-axisymmetric state occurs at a much higher load and with the formation of a smaller number of waves than for a sliding boundary conditions. Difficulties in obtaining a numerical solution based on the analytical approach are apparently associated with the need for a more accurate description of the stress-strain subcritical state of the plate.

Application of elastic stability of flat and spatial shapes of rods in agriculture

In various elements of building structures, pipelines and tanks, ships and submarines, aircraft and spacecraft, in the oil and gas industry, in agriculture, elastic rods, shells and plates are widely used, for which the stability of the forms of equilibrium determines the conditions for their trouble-free operation. Today, the literature contains a huge number of works devoted to both the solution of individual particular problems and the development of general methods for analyzing the stability of equilibrium forms of elastic systems. Despite this, many problems still need to be studied. In particular, in the well-known works on the theory of stability of elastic systems, the most difficult problems include the problems of bending of heavy elastic rods, experiencing the joint action of inhomogeneous force factors. These tasks remain poorly understood, despite their importance for many agricultural and industrial processes. The article investigates the stability of elastic rods (stems), loaded at the ends by concentrated moments, and investigates the influence of the characteristics of distributed stiffness on the value of critical moments, which is a mathematical model of maintaining the stability of the stem of a cereal plant during lodging. For stems with the same bending stiffness in different planes, within the framework of the static approach, the influence of the variability of the distributions along the rods of bending stiff nesses on the critical values of the torques at which the loss of stability occurs is studied.