Equilibrium Equations Research Articles

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.

A modified solution for evaluating the sliding direction and stability of 3D slopes

In this paper, we present an improved and efficient analytical solution for evaluating the stability of 3D slopes based on the minimum potential energy method. The equilibrium equation of the landslide is employed to determine the shear stress on the failure surface, and a new formulation for shear potential energy is introduced. The vector-combined direction of the force acting on the landslide is considered equivalent to the sliding direction (SD). Two classic examples are reanalyzed to validate the accuracy and performance of the proposed method. The analysis demonstrates that the obtained results align well with reference solutions and tend to be conservative in engineering applications. Parameter investigations are conducted to explore the influences of SD, shear strength parameters, slope angle, and soil unit weight on the safety factor (SF) of 3D slopes. The findings indicate that a larger SD corresponds to a less stable slope. Finally, the effects of optimization parameters, landslide volume, and surcharge on the location of the failure surface, SF, and SD are also examined. The developed analytical approach offers a novel and straightforward solution with sufficient accuracy and eliminates the need for iteration in determining the SF, making it a valuable complementary method for assessing the stability of 3D slopes.

Optimal strategies design of active target defense differential game

In this paper, the optimal strategies are investigated for the differential game of active target defense. Specifically, we describe a pursuitevasion game with three agents (namely, an attacker, a target and a defender). For the three agents engagement scenario, two different differential game guidance laws–pure pursuit (PP) and proportional navigation (PN)–are presented for defender. Firstly, the relative motion kinematic models of the three agents are established. We subsequently derive the coupled Hamilton-Jacobi (HJ) equations to design the optimal control strategies for the three agents. Through the construction of actor-critic neural networks, optimal solutions are obtained. Then, the stability of the three-agent system is ensured, and the convergence to the Nash equilibrium equations is demonstrated. Finally, we simulate the optimal strategies employed by the attacker and the defender-target team, thereby highlighting the contrasting outcomes and superiority of different defensive approaches.

Discontinuous Galerkin finite element method for the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes

Context. The study of the continuum radiative transfer problem inside circumstellar envelopes is both a theoretical and numerical challenge, especially in the frequency-dependent and multi-dimensional case. While approximate methods are easier to handle numerically, they often fail to accurately describe the radiation field inside complex geometries. For these cases, it is necessary to directly solve the radiative transfer equation numerically. Aims. We investigate the accuracy of the discontinuous Galerkin finite element method (DGFEM hereafter) applied to the frequency-dependent two-dimensional radiative transfer problem, and coupled with the radiative equilibrium equation. We next used this method in the context of axis-symmetric circumstellar envelopes. Methods. The DGFEM is a variant of finite element methods. It employs discontinuous elements and flux integrals along their boundaries, ensuring local flux conservation. However, as opposed to the classical finite element methods, the solution is discontinuous across element edges. We implemented this approach in a code and tested its accuracy by comparing our results with the benchmarks from the literature. Results. For all the tested cases, the temperatures profiles agree within one percent. Additionally, the emerging spectral energy distributions (SEDs) and images, obtained by ray-tracing techniques from the DGFEM emissivity, agree on average within 5% and 10%, respectively. Conclusions. We show that the DGFEM can accurately describe the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes. Consecutively the emerging SEDs and images are also well reproduced. The DGFEM provides an alternative method (other than Monte-Carlo methods for instance) for solving the radiative transfer equation, and it could be used in cases that are more difficult to handle with the other methods.

Analytical, semi-analytical and numerical solutions for global stability analysis in buildings: Double-Beam Systems Timoshenko

In this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings.

Analytical model for flexoelectric sensing of structural response considering bonding compliance

Flexoelectricity has generated huge interest as an alternative to piezoelectricity for developing electromechanical systems such as actuators, sensors, and energy harvesters. This article presents a generic theoretical framework for the sensing mechanism of a flexoelectric sensor bonded to a host beam through an adhesive layer. The model incorporates piezoelectric and flexoelectric effects and considers both shear-lag and peel stresses at the sensor-beam interface. The formulation also includes the electric field gradient terms that are often overlooked. Consistent one-dimensional constitutive relations and governing equations of equilibrium are derived from the electric Gibb’s energy density function and extended Hamilton’s principle. The sensor is assumed to follow the Euler–Bernoulli beam-type membrane and bending deformation behaviour. Closed-form solutions are obtained for the interfacial stresses by analytically solving a seventh-order non-homogeneous ordinary differential equation, satisfying the stress-free boundary conditions at the sensor edges. The induced electric potential at the sensor top is derived by solving a fourth-order differential equation obtained from the charge balance equation, satisfying the electric boundary conditions. For validation, the sensor output is compared with the results of the existing non-rigid bonding piezoelectric sensor model. Numerical results show a significant impact of non-rigid bonding and the electric field gradient terms on the induced electric potential. Further, the importance of bonding compliance on the interfacial stress distributions is illustrated. Finally, the effects of adhesive and transducer thicknesses on the peak amplitudes of interfacial stresses and sensory potential are presented.

АНАЛІЗ РІВНОВАГИ ТОНКИХ ОРТОТРОПНИХ ПЛИТА НА ТРИПАРАМЕТРИЧНИЙ ПРУЖНІЙ ОСНОВІ

The paper deals with the problem of studying the static equilibrium of thin orthotropic rectangular plates resting on a three-parameter elastic base. To solve this problem, a mathematical model of a thin reinforced concrete slab as a homogeneous orthotropic plate with an averaged Huber's modulus is constructed. A mathematical model of a three-parameter elastic foundation is proposed, taking into account the friction between the lower surface of the plate and the foundation. The developed method for analyzing the equilibrium of such plates allows obtaining an exact solution of the equilibrium equation, taking into account the boundary and surface conditions at individual nodes. During the numerical implementation of the developed approach, a procedure for generating such nodes is proposed. The solution of the equilibrium equation is presented as the sum of the deflection force functions and its shape functions multiplied by unknown parameters, which are interpreted as the degree of freedom of the plate. This approach made it possible to satisfy the boundary and surface conditions with high accuracy. On the basis of the obtained solutions, the stress-strain state of a thin homogeneous orthotropic square plate completely clamped along the contour is analyzed for the case when the plate is subjected to a distributed load on its upper part and rests on an elastic base. On the basis of the solutions obtained in this work and formulas obtained by other authors, a comparative analysis of the results for the case of three types of elastic bases is carried out: a three-parameter base, a Winkler base, and a plate with a free bottom surface. Based on numerical calculations, it was found that the elastic base significantly reduces the deflection, tilt and moment in the plate. The results obtained for the Winkler model and the three-parameter model differ by 3% and 1,5% for deflections and moments, respectively. It is established that the results obtained within the proposed model practically do not depend on the coefficient of friction between the lower surface of the plate and the foundation.

Zero-sum games subject to uncertain stochastic noncausal systems considering polynomial control functions

An uncertain stochastic noncausal system (USNS) is an uncertain stochastic singular system that is expected to be regular but not impulse-free. This study investigates zero-sum games (ZSGs) under the constraint of nonlinear uncertain stochastic noncausal systems (USNSs) considering polynomial control functions. Firstly, a method is introduced to convert USNSs into subsystems, including forward uncertain stochastic difference equations as well as backward uncertain stochastic difference equations. Equilibrium equations are then derived to determine the saddle-point equilibrium solution of the zero-sum game (ZSG) for USNSs. Subsequently, a unified framework is developed to find the saddle-point equilibrium and equilibrium value of the ZSG. Additionally, a numerical example and an analysis of the treatment of industrial wastewater are provided to demonstrate the effectiveness of the proposed framework.

Research on the Stability of Salt Cavern Hydrogen Storage and Natural Gas Storage under Long-Term Storage Conditions

The stability of salt cavern storage during prolonged operation is a crucial indicator of its safety. This study focuses on an operational underground gas storage facility, conducting comparative numerical simulations for the storage of natural gas and hydrogen. We investigated the evolution of stability for natural gas and hydrogen storage under long-term storage conditions. The main conclusions are as follows: (1) A new equation for stress equilibrium and constitutive relations are derived. (2) At the same storage pressure, the effective stress at the same position in the interlayer is greater for hydrogen storage compared to natural gas storage, signifying a higher level of danger. (3) At the same storage pressure, the displacement at the cavity top for hydrogen storage is greater than that for natural gas storage. The displacement difference between the two is greatest at 9 MPa, amounting to 0.026 m. (4) Due to hydrogen’s lower dynamic viscosity and higher permeability, the depth and extent of the plastic zones within the interlayers are greater compared to natural gas. When the storage pressure is 15 MPa, the depth of the plastic zone within the interlayer can be up to 2.1 m greater than when storing natural gas, occurring in the third interlayer from the top. These research findings may serve as a valuable reference for determining the operational parameters of on-site salt cavern hydrogen storage facilities.

Jacobian-free Newton-Krylov method for multilevel nonlocal thermal equilibrium radiative transfer problems

Context. The calculation of the emerging radiation from a model atmosphere requires knowledge of the emissivity and absorption coefficients, which are proportional to the atomic level population densities of the levels involved in each transition. Due to the intricate interdependence of the radiation field and the physical state of the atoms, iterative methods are required in order to calculate the atomic level population densities. A variety of different methods have been proposed to solve this problem, which is known as the nonlocal thermodynamical equilibrium (NLTE) problem. Aims. Our goal is to develop an efficient and rapidly converging method to solve the NLTE problem under the assumption of statistical equilibrium. In particular, we explore whether the Jacobian-Free Newton-Krylov (JFNK) method can be used. This method does not require an explicit construction of the Jacobian matrix because it estimates the new correction with the Krylov-subspace method. Methods. We implemented an NLTE radiative transfer code with overlapping bound-bound and bound-free transitions. This solved the statistical equilibrium equations using a JFNK method, assuming a depth-stratified plane-parallel atmosphere. As a reference, we also implemented the Rybicki & Hummer (1992) method based on linearization and operator splitting. Results. Our tests with the Fontenla, Avrett and Loeser C model atmosphere (FAL-C) and two different six-level Ca II and H I atoms show that the JFNK method can converge faster than our reference case by up to a factor 2. This number is evaluated in terms of the total number of evaluations of the formal solution of the radiative transfer equation for all frequencies and directions. This method can also reach a lower residual error compared to the reference case. Conclusions. The JFNK method we developed poses a new alternative to solving the NLTE problem. Because it is not based on operator splitting with a local approximate operator, it can improve the convergence of the NLTE problem in highly scattering cases. One major advantage of this method is that it is expected to allow for a direct implementation of more complex problems, such as overlapping transitions from different active atoms, charge conservation, or a more efficient treatment of partial redistribution, without having to explicitly linearize the equations.

Form-Finding of Tensegrity Basic Unit with Equal Cable Length

Tensegrity is a lightweight, self-stressing, and self-stabilizing structure made up of cables and bars, with each member bearing either tension or compression but not affected by shear stress. This design allows for optimal utilization of the material properties of the members. In a tensegrity basic unit, the bar members are of equal length, while the cable members come in three lengths: lower-end surface horizontal cable, upper-end surface horizontal cable, and stayed cable. The tensegrity basic unit with equal cable length simplifies this further by ensuring that all cables are the same length, resulting in a structure with only two member lengths, i.e., bar length and cable length, enhancing interchangeability. In order to find the form without the action of external forces, the force density coefficient ratio is introduced. By performing a force balance analysis on any node of the unit, the equilibrium equation of the structure is determined, incorporating the additional constraint of equal cable length. Two methods are employed to ascertain the force density coefficient ratio of each member in the unit: the theoretical derivation method based on the stable configuration condition of the tensegrity basic unit with equal cable length, and the method of solving the characteristic equations of the force density matrix. A program is developed to validate the form-finding method using basic units with three, four, five, and six bars as examples. The results show that the model accurately represents the physical structure, confirming the reliability of the form-finding methods.

Thermal buckling of organic nanobeams resting on viscous elastic foundation

This study presents an analytical solution for organic solar nanobeams by including the new high-order shear deformation theory and considering the impact of size effects using non-local elasticity theory. The beam is supported by a viscoelastic base and experiences thermal loads that are distributed in accordance with uniform and nonlinear principles. The calculation formulae are derived from the consideration of significant deformations in the beam, resulting in increased complexity of the calculations. The equilibrium equation is derived from the fundamental principle of maximum work potential. The explicit equation for the critical thermal load is derived, providing a convenient means for the calculation and analysis of the impacts of relevant factors. The critical thermal buckling load encompasses both real and imaginary components, with the real component corresponding to the thermal load responsible for inducing beam instability, and the imaginary component associated with the dissipation of this load. This consideration is particularly pertinent when accounting for the foundation resistance. This finding enhances the level of interest in the research outcomes. This study also examined the impact of certain characteristic factors of the foundation and organic beams on the thermal buckling behavior of the beam, establishing a scientific basis for practical design applications.

Reduced form of statistical equilibrium equations and radiative transfer equations

Abstract An extension of the vector algebra for irreducible spherical tensor operators (ITOs) is proposed, which involves coupling two ITOs into a third one (V (kv)= T (kt)× U (ku)*), with the additional condition that one of the operators is a complex conjugate and therefore is not an ITO. The correctness condition is considered for this extension. The new coupling rule treats the density matrices ρ(Kl) and all tensors related to light polarization (J (Kr) or Τ (K)) equally.
This approach has been applied to rewrite the fundamental statistical equilibrium equations and radiation transfer equations relevant to astrophysical polarimetry. The revised equations are more concise and eliminate the need for the nj symbols. Instead, they utilize vector and scalar products of ITOs. All rate coefficients in the equations consist of two independent factors, the reduced matrix element from dipole moments and the product (vector or scalar) of the tensors ρ(Kl) and J (Kr) (or Τ (K)).

New analytical model for multi-layered composite plates with imperfect interfaces under thermomechanical loading

AbstractA new analytical model for thermoelastic responses of a multi-layered composite plate with imperfect interfaces is developed. The composite plate contains an arbitrary number of layers of dissimilar materials and is subjected to general mechanical loads (both distributed internally and applied on edges for each layer) and temperature changes, which can vary from layer to layer and along two in-plane directions. Each layer is regarded as a Kirchhoff plate, and each imperfect interface is described using a spring-layer interface model, which can capture discontinuities in the displacement and stress fields across the interface. Unlike existing models, the governing equations and boundary conditions are simultaneously derived for each layer by using a variational procedure based on the first and second laws of thermodynamics, which are then combined to obtain the global equilibrium equations and boundary conditions for the multi-layered composite plate. A general analytical solution is developed for a symmetrically loaded composite square plate with an arbitrary number of layers and imperfect interfaces by using a new approach that first determines the interfacial normal and shear stress components on one interface. Closed-form solutions for two- and three-layer composite square plates are obtained as examples by directly applying the general analytical solution. Numerical results for two-, three- and five-layer composite plates under different loading and boundary conditions predicted by the current model are provided, which compare well with those obtained from finite element simulations using COMSOL, thereby validating the newly developed analytical model.

Solvation free energy in governing equations for DNA hybridization, protein-ligand binding, and protein folding.

This work examines the thermodynamics of model biomolecular interactions using a governing equation that accounts for the participation of bulk water in the equilibria. In the first example, the binding affinities of two DNA duplexes, one of nine and one of 10 base pairs in length, are measured and characterized by isothermal titration calorimetry (ITC) as a function of concentration. The results indicate that the change in solvation free energy that accompanies duplex formation (ΔGS) is large and unfavorable. The duplex with the larger number of G:C pairings yields the largest change in solvation free energy, ΔGS = +460 kcal·mol-1per base pair at 25 °C. A van't Hoff analysis of the data is complicated by the varying degree of intramolecular base stacking within each DNA strand as a function of temperature. A modeling study reveals how the solvation free energy alters the output of a typical ITC experiment and leads to a good, though misleading, fit to the classical equilibrium equation. The same thermodynamic framework is applied to a model protein-ligand interaction, the binding of ribonuclease A with the nucleotide inhibitor 3'-UMP, and to a conformational equilibrium, the change in tertiary structure of α-lactalbumin in molar guanidinium chloride solutions. The ribonuclease study yields a value of ΔGS = +160 kcal·mol-1, whereas the folding equilibrium yields ΔGS ≈ 0, an apparent characteristic of hydrophobic interactions. These examples provide insight on the role of solvation energy in binding equilibria and suggest a pivot in the fundamental application of thermodynamics to solution chemistry.

Optimistic value models of saddle point equilibrium control problems for uncertain jump systems

The saddle point equilibrium control problem is of great importance in differential game theory. In this paper, we investigated the optimistic value model for the saddle point equilibrium control problem based on multi-dimensional uncertain differential equations with jumps. The equilibrium equation for the proposed model is presented. Then a linear quadratic game optimistic value model and a bang-bang game optimistic value model are discussed, respectively. Finally, some applications are analysed by the results obtained in this paper.

Design and research of face gear central limited slip differential and analysis of its influence on vehicle performance

In this paper, a new design scheme of face gear central limited-slip differential (F-LSD) is proposed, which aims to unveil the technical veil of central limited-slip differential and fill the research blank of face gear application in the field of central limited-slip differential. Through the analysis of structural principle and motion characteristics, the dynamic equilibrium equation and locking coefficient model of F-LSD are established. The Simulink simulation model based on F-LSD dynamics and the torque distribution model of the whole transmission system are established. A 7-DOF vehicle dynamics model was established, and the performance of F-LSD vehicle in four conditions of acceleration, climbing, chassis twist sine wave and double line change was studied by using simulation software. At the same time, four groups of four-wheel drive vehicles with different central differential speeds were set as the control test group. The simulation results show that F-LSD system can effectively improve the vehicle's dynamic performance and extrication performance, and will not reduce the vehicle's handling stability, which provides a reference for the practical application of the face gear central limited slip differential.

Variable angle tow-steered fibres based rotating composite beam with rotary oscillations – Parametric excitation/instability analysis

This study is concerned with the stability analysis for various vibrational motions of tow-steered variable angle fibres based composite rotating beam with time varying speed. The formulation combines a higher order theory introducing sine function along with finite element methodology. It satisfies the plane stress condition in width direction of beam. The displacement kinematics are generic in the sense it includes various possible vibrational motions like chord and flap wise motions, torsional and axial vibration behaviours. The equilibrium equations for the proposed problem evolved adopting virtual dynamic work are then transformed into the equations of Mathieu-Hill type considering time varying harmonic rotational speed. Using Bolotin’s procedure coupled with C1 continuity-based beam element, the parametric resonance zones along with boundary limits are numerically evaluated. The eigenvalue type of solution methodology applied for the detailed study is validated for accuracy and computational efficiency against the time domain dynamic response analysis. Based on in-depth analysis, dynamic instability characteristics in terms of resonance range and origin of instability of composite beam subjected to the chord and flap wise, torsional and axial motions are studied considering curvilinear fibre path angles, beam cross section, hub radius, thickness ratio and mean rotary speed.

Mechanism design and mechanical analysis of pipeline inspection robot

Purpose This study aims to address the issues of limited pipe diameter adaptability and low inspection efficiency of current pipeline inspection robots, a new type of pipeline inspection robot capable of adapting to various pipe diameters was designed. Design/methodology/approach The diameter-changing mechanism uses a multilink elastic telescopic structure consisting of telescopic rods, connecting rods and wheel frames, driven by a single motor with a helical drive scheme. A geometric model of the position relationships of the hinge points was established based on the two extreme positions of the diameter-changing mechanism. Findings A pipeline inspection robot was designed using a simple linkage agency, which significantly reduced the weight of the robot and enhanced its adaptive pipe diameter ability. The analysis determined that the robot could accommodate pipe diameters ranging from 332 mm to 438 mm. A static equilibrium equation was established for the robot in the hovering state, and the minimum pressing force of the wheels against the pipe wall was determined to be 36.68 N. After experimental testing, the robots could successfully pass a height of 15 mm, demonstrating the good obstacle capacity of the robot. Practical implications This paper explores and proposes a new type of multilink elastic telescopic variable diameter pipeline inspection robot, which has the characteristics of strong adaptability and flexible operation, which makes it more competitive in the field of pipeline inspection robots and has great potential market value. Originality/value The robot is characterized by the innovative design of a multilink elastic telescopic structure and the use of a single motor to drive the wheel for spiral motion. On the basis of reducing the weight of the robot, it has good pipeline adaptability, climbing ability and obstacle-crossing ability.

Dissolution Mechanisms and Surface Charge of Clay Mineral Nanoparticles: Insights from Kinetic Monte Carlo Simulations

The widespread use of clay minerals and clays in environmental engineering, industry, medicine, and cosmetics largely stems from their adsorption properties and surface charge, as well as their ability to react with water. The dissolution and growth of minerals as a function of pH are closely related to acid–base reactions at their surface sites and their surface charge. The vivid tapestry of different types of surface sites across different types of clay minerals generates difficulties in experimental studies of structure–property relationships. The aim of this paper is to demonstrate how a mesoscale stochastic kinetic Monte Carlo (kMC) approach altogether with atomistic acid-base models and empirical data can be used for understanding the mechanisms of dissolution and surface charge behavior of clay minerals. The surface charge is modeled based on equilibrium equations for de/protonated site populations, which are defined by the pH and site-specific acidity constants (pKas). Lowered activation energy barriers for these sites in de/protonated states introduce pH-dependent effects into the dissolution kinetics. The V-shaped curve observed in laboratory experiments is reproduced with the new kMC model. A generic rate law for clay mineral dissolution as a function of pH is derived from this study. Thus, the kMC approach can be used as a hypothesis-testing tool for the verification of acid–base models for clay and other minerals and their influence on the kinetics of mineral dissolution and growth.