Equilibrium Equations Research Articles

Numerical modelling on coupled behavior of calcium leaching and chloride transport in cement-based materials by considering ion desorption

Concrete structures serving in water environment with rich chloride ions are subjected to the coupling action of calcium leaching and chloride corrosion. In this paper, a transport model of calcium and chloride ions coupled with the calcium solid-liquid equilibrium equation is proposed to simulate the behavior of calcium leaching and chloride penetration in concrete. In the model, the effect of free chloride ions bound by solid calcium and the release of bound chloride ions caused by the dissolution of solid calcium is considered. Secondly, the model is numerically solved by using the finite difference method of Crank-Nicolson scheme. Then, the experimental data of porosity and calcium silicon ratio, chloride concentration are compared to the calculated results to validate the model. Finally, a numerical simulation is conducted to analyze the evolution of ion concentration, solid calcium content and porosity of cement paste slice exposed to deionized water and accelerated dissolution solution. The results show that, the concentration of calcium ion decreases with corrosion time, while that of chloride ion increases. There are concentration gradient distributions of calcium and chloride ions in the slice. The higher the concentration of accelerated dissolution solution, the larger the porosity of cement paste and the higher the concentration of chloride ions in the paste. It indicates that, the calcium leaching accelerates the chloride penetration into concrete. In addition, the dissolution process of solid calcium, i.e. calcium hydroxide first dissolves followed by C-S-H gel, are presented.

Three-node Mindlin plate finite element with shear stress resultants derived from equilibrium equations

Three-node Mindlin plate finite element with shear stress resultants derived from equilibrium equations

GRILLAGE ANALYSIS OF CELLULAR PLATE STRUCTURESOF VARYING DEPTH WITH INCL

A static analysis of cellular plate structures of linearly varying depth is presented by using thesimplified grillage method. The cellular plate structure in this study consists of top and bottomflange plates (non-parallel) separated by longitudinal webs having varying depth. A variationalformulation of thin-walled beam element with linearly varying depth has been carried out for thederivation of the stiffness matrix of the grillage members for various warping restraint effects. Theeffect of warping restraint in stiffness is included by two methods, firstly; the implicit method(through the torsional constant) and secondly; by the explicit method (as independent warpingdegree of freedom). Two differer,t approaches are used to derive the torsional stiffness matrix,namely:by the assumed displacement field and by the differential equation of equilibrium. Theeffect of secondary shear stresses from the warping restraint is also considercd.The varying cross section properties of the eguivalent grillage members are derived from theproperties of the actual cellular plate structure.Several numerical examples and applications are presented . The results are compared with thoseobtained from the flat plate / shell elements of MSCA.{ASTRAN program. Acceptable agreement isobtained.

A stable numerical framework for long-time dynamic crack analysis

This paper introduces a stable numerical framework designed to address dynamic crack problems over long-time intervals. The initial step involves the temporal discretization of the governing dynamic equilibrium equations using the arbitrary order Krylov deferred correction method. To ensure precise boundary condition matching, a novel numerical implementation is incorporated into the Krylov deferred correction technique. Subsequently, the resulting system of spatial partial differential equations at each time node is solved using the meshless generalized finite difference method with 4th-order expansions in regions close to the crack-tips and 2nd-order expansions in areas far from the crack-tips. This combined approach capitalizes on the strengths of both the Krylov deferred correction technique and the generalized finite difference method, enabling stable simulations of dynamic cracks with the large time step and without the need for mesh generation. Notably, we refine the collocation nodes near crack tips to attain accurate numerical results for displacement and stress field. Several numerical experiments involving diverse impact loadings are conducted to validate the developed framework. Furthermore, a comparison is made between the dynamic stress intensity factors obtained using our approach and those from existing methods.

Simulation of spring-friction set of freight car truck, taking into account track profile

The purpose of this article is the mathematical modeling of a spring-friction set of a freight car trolley depending on the profile of the railway track. The article uses the methodology of the results of the generalized model of power interaction of friction pairs "pressure beam-friction wedge" and "friction wedge-friction bar" of the spring-friction set of the cart 18-578 in the mechanical system "track-car-load," in a particular case cart from which such models of the e of the freight car 18-100 can be obtained. The manuscript contains a software study using a computer simulation of a freight wagon trolley, which complements the experimental studies. The mathematical (symbolic) method in the computer algebra system MathCAD (a type of computer-aided design system) is used to solve the equations of equilibrium of the supercharger beam as a real object. In the course of the experimental study, a computational mathematical model of the force action of friction pairs of spring-absorbing trolley apparatuses was developed, and the results of calculations of the reaction of the friction bar to the friction wedge were mathematically obtained. The optimal parameters were obtained analytically, and the indicators of the angles of inclination of the contact surfaces of the trolley of the freight car were justified.

Finite element implementation of a seepage-stress coupling method for solid-liquid-gas three phases in porous media considering compressible gas

Some existing coupling models of porous media ignore the gas phase or gas compressibility, making them inadequate in describing the gas transport process and their effects on water infiltration and solid deformation. However, consideration of gas compressibility becomes crucial when involving the confined gas. Furthermore, the unsteady seepage free surface problem may make the numerical solution of seepage-stress coupling more complicated and difficult. To overcome the above problems, a new coupled seepage-stress method considering compressible gas is proposed. Based on the momentum balance equation of solid–liquid–gas, the three-phase model is constructed by making the pore liquid and compressible gas both satisfy the fluid equilibrium equation and continuity equation. Integration of the interface-correction level set method in the solid–liquid–gas three phases model provides a good solution for representing liquid–gas two-phase flow and solid deformation while describing the unsteady free surface between saturated and unsaturated zones. The primary variables in this model are solid displacement and liquid/gas pressure and its governing equations are discretized by finite element strategy. After comprehensively explaining the mathematical framework and numerical implementation, the reliability and effectiveness of the proposed method are validated through several benchmarking examples.

Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations

This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents.

A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models

This study is centered on examining the static bending behavior of sandwich plates featuring functionally graded materials, specifically addressing distinct representations of porosity distribution across their thickness. The composition of the sandwich plate involves a ceramic core and two face sheets with functionally graded properties. Mechanical loads with a sinusoidal distribution are applied to the sandwich plate, and a four-variable shear deformation theory is employed to establish the displacement field. Notably, this theory involves only four unknowns, distinguishing it from alternative shear deformation theories. Equilibrium equations are derived using the virtual work concept, and Navier’s method is applied to obtain the solution. The study addresses the impact of varying porosities, inhomogeneity parameters, aspect ratios, and side-to-thickness ratios on the static bending behavior of the sandwich plates. The influence of various porosities, inhomogeneity parameter, aspect ratio, and side-to-thickness ratio of the sandwich plates are explored and compared in the context of static bending behavior. The three porosity distributions are compared in terms of their influence on the bending behavior of the sandwich plate. The findings indicate that a higher porosity causes larger deflections and Model A has the highest central deflection. Adopting the four-variable shear deformation theory demonstrated its validity since the results were similar to those obtained in the literature. Several important findings have been found, which could be useful in the construction and application of FG sandwich structures. Examples of comparison will be discussed to support the existing theory’s accuracy. Further findings are presented to serve as benchmarks for comparison.

Mechanics of inner core debonding of composite sandwich beam with CFRP hexagonal honeycomb

The inner core debonding of composite sandwich beams with carbon fiber reinforced polymer (CFRP) hexagonal honeycomb core is investigated theoretically and experimentally. Based on the principle of minimum potential energy, a theoretical model is established to reveal the core debonding mechanism with Timoshenko’s beam theory and Extended High-Order Sandwich Panel Theory (EHSAPT) for composite face sheets and CFRP hexagonal honeycomb core, respectively. The theoretical approach of core debonding simulation is based on the framework of the cohesive zone model (CZM). By constructing the constitutive relation of the core interface as the discontinuous stress and displacement boundary conditions, the vertical core debonding can be simulated overcoming the singularity of equilibrium equations. The crack propagation process is decomposed into three stages with the consideration of the double crack propagation mechanism. In experiments, the regular and reinforced hexagonal honeycomb sandwich beams are designed and fabricated, and the quantification of the defect degree of the core bonding interface is achieved by equivalent energy release rate obtained from the homogenized hexagonal honeycomb. The double cantilever beam (DCB) experiments are performed to determine the equivalent energy release rate and reveal the core debonding mechanism of CFRP hexagonal honeycomb sandwich beams. Comparing the load–displacement curves and debonding deformation patterns indicates that the established theoretical model can effectively predict the core debonding mechanical response. Finally, parametric analysis is carried out to discuss the impact of structural dimension parameters, defect location and defect degree on the debonding modes of CFRP hexagonal honeycomb sandwich beams.

Calculation and Analysis of Equilibrium Position of Aerostatic Bearings Based on Bivariate Interpolation Method

The solution of equilibrium positions is a critical component in the calculation of the dynamic characteristic coefficients of aerostatic bearings. The movement of the rotor in one direction leads to bidirectional variations in the air film force, resulting in low efficiency when using conventional calculation methods. It can even lead to iterative divergence if the initial value is improperly selected. This study concentrates on the orifice throttling aerostatic bearings and proposes a novel method called the bivariate interpolation method (BIM) to calculate the equilibrium position. The equilibrium equation for the rotor under the combined influence of air film forces, gravity, and external loads is established. A calculation program based on the finite difference method is developed to determine the equilibrium position. The process of solving the equilibrium position and the convergence is compared with the secant method and the search method. Furthermore, the variation trend of the equilibrium position and stiffness when the external loads changes are studied based on the BIM. Finally, the correctness of the BIM to solve the equilibrium position is proved by comparing it with the experiment results. The calculation results indicate that the BIM successfully resolves the problem of initial value selection and exhibits superior computational efficiency and accuracy. The equilibrium position initially moves away from the direction of the external load as the load increases, and then this gradually approaches the load direction. The main stiffness increases with increases in the external load, while the variation in cross stiffness depends on the direction of the external load.

Research on numerical solution of statically indeterminate problem of steering tool combined bearing

The steering tool combined bearing is a mechanical component that can support forces, guide movement, and bear vibrations at the same time. In the process of analyzing the contact and load distribution inside the combined bearing, there are several contact points, and the static equilibrium equation is statically indeterminate. Based on dynamics theory and Hertz elastic contact theory, the concept of a load integral and the displacement estimation formula for roller bearings are introduced, and a numerical solution to the statically indeterminate problem of the steering tool combined bearing is given. According to the statically indeterminate equation matrix of the combined bearing, the relationship between the load and displacement of the combined bearing is studied to improve the statically indeterminate equation matrix. Moreover, the simulation results of the maximum stress and displacement distribution results of the inner and outer ring eccentricity from 1 to 6mm are compared with the algorithm prediction results. The error between the numerical solution of the statically indeterminate problem of the steering tool combined bearing and the simulation result is small, improving the calculation efficiency. This provides theoretical support for the design and reasonable application of the steering tool combined bearing.

Improved well-balanced AWENO schemes with hydrostatic reconstruction for the Euler equations under gravitational fields

The Euler equations under gravitational fields allow the hydrostatic equilibrium states, which requires that the numerical scheme of the system should also have this characteristic. In our previous work, a well-balanced finite difference conservative AWENO scheme has been constructed to preserve the isothermal equilibrium state accurately (Fu et al., Appl. Numer. Math., 180:1-15, 2022). However, the scheme fails to maintain more complicated hydrostatic equilibrium states, such as the isentropic equilibrium state. In this study, the improved well-balanced AWENO schemes with hydrostatic reconstruction are designed to maintain the general equilibrium states numerically. The key technologies to achieve the well-balanced property are rewriting the source term into an appropriate form based on the equilibrium equation, acting on the conserved variables with an affine-invariant WENO or linearized WENO interpolation operator, employing hydrostatic reconstruction methods to create auxiliary conserved variables, and modifying the numerical flux with auxiliary conserved variables and higher-order conserved terms based on pressure. The proposed well-balanced schemes maintain the equilibrium solution near the machine precision and capture small perturbations of the equilibrium state without spurious oscillations even at the coarse mesh. Furthermore, they can be applicable to the general hydrostatic equilibrium more than the isothermal equilibrium state. Several numerical experiments are tested to show that the developed well-balanced schemes perform well in maintaining equilibrium states and capturing small perturbations.

A new model for spatial rods incorporating surface energy effects

A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.

Study on influence coefficients of hydrodynamic force on dynamic response of deep-water pier under earthquake

The hydrodynamic force generated by the interaction between deep-water pier and water under earthquake exerts a significant influence on the dynamic response of the pier. At present, there is a lack of characterization methods to measure the influence degree of pier-water interaction on the dynamic response of the deep-water pier. Based on the structural dynamics theory and considering the coupling effect of pier-water interaction, the dynamic equilibrium equations for the deep-water pier under earthquake are established, and the solutions are deduced and validated by the finite element simulation results. Then the comprehensive analysis of influence mechanism is conducted. Further, the displacement influence coefficients and internal force influence coefficients are proposed to evaluate the influence degree of hydrodynamic force on dynamic response of deep-water piers. Validation shows that the proposed coefficients have high accuracy, indicating the important application of the proposed coefficients in seismic design of deep-water piers in real practice.

Economic Pricing in Peer-to-Peer Electrical Trading for a Sustainable Electricity Supply Chain Industry in Thailand

The state-owned power Electricity Generating Authority of Thailand (EGAT), a monopoly market in charge of producing, distributing, and wholesaling power, is the focal point of Thailand’s electricity market. Although the government has encouraged people to install on-grid solar panels to sell electricity as producers and retail consumers, the price mechanism, i.e., purchasing price and selling prices, is still unilaterally determined by the government. Therefore, we are interested in studying the case where blockchain can be used as a free trading platform. Without involving buying or selling from the government, this research presents a model of fully traded price mechanisms. Based on the study results of the double auction system, data on buying and selling prices of electrical energy in Thailand were used as the initial data for the electricity peer-to-peer free-trading model. Then, information was obtained to analyze the trading price trends by using the law of demand and supply in addition to the principle of the bipartite graph. The price trend results agree well with those of price equilibrium equations. Therefore, we firmly believe that the model we offer can be traded in a closed system of free-trade platforms. In addition, the players in the system can help to determine the price trend that will occur according to various parameters and will cause true fairness in the sustainable electricity supply chain industry in Thailand.

Equilibrium solubility, solvent effect, and equation correlations of cyclosporine in twenty mono solvents and four binary mixtures

In this study, the solid–liquid equilibrium solubility of cyclosporine in 20 pure solvents and four binary mixtures at 298.15 K were determined using shake-flask technique. Solid-state characterization exhibited that no solvate formation or polymorphic transformation occurred during the dissolution of cyclosporine in any of the solvent systems investigated. The experimental solubilities of cyclosporine in 20 pure mono solvents were ranked as follows: DEGME (2.21 × 10−1) > ethanol (1.22 × 10−1) > NMP (1.15 × 10−1) = 1-butanol (1.15 × 10−1) > dichloromethane (1.07 × 10−1) > acetonitrile (1.02 × 10−1) > PEG 200 (9.78 × 10−2) > PEG 300 (9.12 × 10−2) > chloroform (8.98 × 10−2) > PEG 400 (8.11 × 10−2) > DMF (7.65 × 10−2) > 2-propanol (7.09 × 10−2) > 1-propanol (6.57 × 10−2) > methanol (6.14 × 10−2) > propylene glycol (6.08 × 10−2) > tetrahydrofuran (4.69 × 10−2) > DMSO (2.64 × 10−2) > acetone (1.28 × 10−2) > glycerol (1.35 × 10−5) > water (4.29 × 10−7). In the four binary solvents (DEGME + water, PEG 200 + water, PEG 300 + water, and PEG 400 + water), the mole fraction solubility of cyclosporine increased with increasing mass fractions of the four different organic solvents. Based on the Kamlet-Taft linear solvation energy relationship (KAT-LSER) model, the solubility of cyclosporine was mainly affected by the self-cohesiveness of the solvent, which is related to the interaction between the solvent and the solvent. Additionally, the solubility parameter was used to explain and understand the solubility of cyclosporine in four different binary solvents. To correlate the solubility data of cyclosporine, two models (the Yalkowsky Log-Linear and simplified CNIBS/R–K models) were applied. The calculated data based on two mathematical models and experimental mole fraction solubility data for cyclosporine could provide information for the separation, extraction, purification, crystallization, and selection of suitable a solvent system.

Improved softened truss model of prestressed concrete box girders subjected to combined bending and torsion

Prestressed concrete (PC) box girder is an innovative structure that is widely used in large-span bridges. Under eccentric loads such as vehicles, PC girders are inevitably under combined bending and torsion. To study its bending-torsional behavior, a refined full-range analysis model based on the combined action softening truss model (CA-STM) was proposed, where a simple and effective solution procedure and failure criteria were provided. The equilibrium equation, initial estimation equation for torque, and material constitutive model were modified by considering the prestressing effect to improve the CA-STM. An optimized algorithm was also employed to simplify the solution procedure instead of the traditional trial-and-error method, thus increasing the solution rate and stability. The theoretical curves predicted from the improved CA-STM exhibited great agreement with the available experimental results, and the predicted cracking and ultimate loads were also close to the experimental values. Hence, the improved CA-STM can reasonably predict the full-range mechanical response and failure modes of PC box girders subjected to combined bending and torsion, which provides great support to the design evaluation of such bridges.

STRESSES IN COMPOSITE PLATES WITH RIVETED BARS USED FOR AIRCRAFT CONSTRUCTION

Reinforcement of plates with rod systems is widely used in engineering, especially in aircraft construction. For example, An-178 aircraft. Removable panels on the lower surface of the wing half, located between the rear spar and the flaps. The method of calculating stresses and deformations in composite plates reinforced with rods is developed in the work. It is assumed that the rod is elastic, attached with rivets. Rivets were considered as rigid inclusions to which unknown forces were applied. These forces were determined from the condition of compatibility of plate and rod deformations. The singular integral equations' method was used to determine stresses and strains in the plate. Integral equations were solved numerically and reduced to a system of algebraic equations. To obtain the forces and moments acting on the rivets, the equations of equilibrium of the rivets and the conditions that ensure the same displacements between the rivets in the plate and in the rods are added to these equations. Examples of calculating stresses near circular and elliptical rivets, magnitudes of forces acting on rivets depending on the rigidity of the rods are given. The reduction of the stress concentration near the elliptical hole, which is placed between two rods, was studied.

Intrinsic correlation between the generalized phase equilibrium condition and mechanical behaviors in hydrate-bearing sediments

The phase equilibrium and mechanical behaviors of natural gas hydrate-bearing sediment are essential for gas recovery from hydrate reservoirs. In heating closed systems, the temperature-pressure path of hydrate-bearing sediment deviates from that of pure bulk hydrate, reflecting the porous media effect in phase equilibrium. A generalized phase equilibrium equation was established for hydrate-bearing sediments, which indicates that both capillary and osmotic pressures cause the phase equilibrium curve to shift leftward on the temperature-pressure plane. In contrast to bulk hydrate, hydrate-bearing sediment always contains a certain amount of unhydrated water, which keeps phase equilibrium with the hydrate within the hydrate stability field. With changes in temperature and pressure, a portion of pore hydrate and unhydrated water may transform into each other, affecting the shear strength of hydrate-bearing sediment. A shear strength model is proposed to consider not only hydrate saturation but also the change in temperature and pressure of hydrate-bearing sediment. The model is validated by experimental data with various hydrate saturation, temperature and pressure conditions. The deformation induced by partial dissociation was studied through depressurization tests under constant effective stress. The reduction in gas pressure within the hydrate stability field indeed caused sediment deformation. The dissociation-induced deformation can be reasonably estimated as the difference in volume between hydrate-bearing and hydrate-free sediments from the compression curves.

Collisional effects in modeling solar polarized lines

Context. Rigorous implementation of the effects of collisions in modeling the formation of the polarized solar lines is of utmost importance in order to realistically analyze the available, highly sensitive solar spectropolarimetric observations. Indeed, even when an observation seems to fit well with theory, one can misinterpret results if important effects due to collisions are not correctly implemented in the modeling process. Aims. We point out inconsistencies in the models adopted to implement the Paschen Back effect together with collisional effects on the solar linear polarization formed by scattering of anisotropic radiation. Because the significance of these inconsistencies increases as polarization becomes increasingly responsive to collisions, we investigate the range of hydrogen densities nH to which the polarization is sensitive. Methods. We used the density matrix formalism in the tensorial irreducible basis, which was developed within the theory of atom-radiation interaction and of atomic collisions. We solved the statistical equilibrium equations for multi-level atoms with hyperfine structure (HFS) in order to evaluate the collisional depolarization of levels of the D1-D2 lines of the K I atom. Results. We find that collisions play a prominent role, particularly at hydrogen densities of between 1013 and 1016 cm−3. Conclusions. So far, analyses of polarized lines formed in the presence of solar magnetic field have incorporated, if at all, collisional rates calculated assuming zero magnetic field. This could be a good approximation in the Hanle regime but not in the Paschen Back regime. For typical quiet Sun magnetic fields, the latter regime could be reached, and level-crossing takes place in several atomic systems. Therefore, one must be careful when using collisional rates calculated in the zero-field case to interpret linear polarization formed in magnetized media.