Equilibrium Equations Research Articles

Analysis of functionally graded piezoelectric structures by Hermite interpolation element-free Galerkin method

Functionally graded piezoelectric structures (FGPSs) have excellent electromechanical properties and are promising for engineering applications. However, the inhomogeneous material properties and piezoelectric effects create great difficulties in the mechanical analysis of the FGPSs. In this paper, a Hermite interpolation element-free Galerkin method (HIEFGM) is proposed for the numerical analysis of the FGPSs. The HIEFGM utilizes a set of nodes to represent the problem domain, which can ideally reflect the inhomogeneous material properties of the FGPSs. Firstly, the governing equations of the FGPSs are derived through the constitutive equation, equilibrium equation, and boundary conditions. Secondly, the approximating function of the field quantities is obtained by the improved moving least-square method and Hermite interpolation. The HIEFGM formulation of the FGPSs is obtained using the variational principle. Furtherly, the influence of weight function, scaling parameter, and node densities on the HIEFGM of the FGPSs is discussed in detail through a parameter study. Finally, the availability of present method is estimated by several examples with different configurations and boundary conditions. The influence of gradation exponent on the electromechanical responses of the FGPSs is analyzed. The results show that the present method has excellent accuracy and stability in analyzing the FGPSs. As the gradation exponent increases, the distribution of the field quantities of the FGPSs exhibits nonlinear characteristics. This work may provide an effective methodology for the analysis of the FGPSs, and contribute to the theoretical research and engineering application of the FGPSs.

Features of development of mathematical model electric motorcycle thermostating system

BACKGROUND: To maintain the required temperature range of electrical components of electric vehicles, it is necessary to use thermostating systems. The developing mathematical model of the temperature control system allows, even at the design stage of the vehicle, to determine the necessary parameters of the temperature control system, select algorithms for controlling the system, and calibrate the developed algorithm to ensure the necessary heat removal from electrical components. AIMS: Creation of a mathematical model of the electric motorcycle thermostating system, aimed at determining the potential possibilities of optimizing the system by organizing the air flow through the radiator. METHODS: The mathematical model is implemented in the AVL Cruise software environment. The characteristics of the components included in the model were obtained as a result of laboratory studies and supplemented with power equilibrium equations at the inputs and outputs, taking into account their efficiency. RESULTS: Using the developed model of the radiator of the thermostating system, the temperature graphs of the electric motor and inverter of the electric motorcycle during the driving cycle were obtained. It is noted that the applied temperature control system allows to maintain the temperature of electric components in the specified range. CONCLUSIONS: The mathematical model of the radiator of the thermostating system proposed in this paper allows to carry out calculations of the system at the stage of design of electric vehicles. The model also allows for further development in order to more fully determine the parameters of the thermostating system, including the thermal efficiency.

The rotation stability of the tapered rollers in a novel ring-type lapping process

This work introduces a novel ring-type lapping method for precision machining of tapered rollers. In contrast to the conventional centerless grinding, this method allows for the simultaneous machining of a group of tapered rollers. The stable rotation of the tapered rollers during the lapping process is a prerequisite for achieving a high manufacturing accuracy. Therefore, an investigation is conducted to examine the criteria for achieving a stable rotation of the tapered rollers. First, the equilibrium equation for the tapered rollers in the lapping process was established. The stable rotation region for the tapered rollers was determined according to the stable rotation criteria. The impacts of the included angle of straight groove and the contact force of lapping sleeve on the stable rotation region were examined. The results demonstrated that the tapered roller's stable rotation can only be realized when a specific parameter combination was satisfied. Subsequently, to expand the stable rotation region, the electromagnetic driving force was introduced to enhance the lapping process. An electromagnetic lapping sleeve structure was designed according to the critical electromagnetic force required for the tapered roller's lapping process. Furthermore, the structure of the electromagnetic lapping sleeve was optimized by electromagnetic simulation in order to improve the uniformity of the electromagnetic driving force exerted on each tapered roller. Finally, the rotation movement of the tapered rollers and the critical electromagnetic forces exerted on the tapered rollers were validated on a ring-type tapered roller lapping platform. The results demonstrated that the stable rotation of the bearing rollers in the lapping process can be achieved throughout a wide range of processing parameters. The roundness of the tapered rollers was significantly improved from 0.83 μm to 0.47 μm on the stable rotation condition. This study established a solid foundation for the precision machining of tapered rollers.

Stochastic model for migration and breakage of detrital and authigenic fines

Mobilisation of attached particles during flow in rocks occurs in geo-energy processes. Particle mobilisation, their migration through rocks and pore plugging yield significant decline in permeability and well injectivity and productivity. While much is currently known about the underlying mechanisms governing the detachment of detrital particles against attracting electrostatic forces, a critical gap exists in the theoretical understanding of detachment by breakage of widely spread authigenic particles, which naturally grow on rock grains during geological times. Previous works derived micro-scale mechanical equilibrium equations for both detrital and authigenic particles, and the upscaling procedure from particle to pore and core scales for detrital fines. In this paper, for the first time we derive a stochastic model for migration and breakage of authigenic fines and authigenic–detrital mixtures. This allows for core-scale transport modelling based on the particle-scale torque balance. We introduce a novel framework for predictive stochastic detachment modelling by particle–rock bond breakage that integrates the beam theory of elastic particle deformation, strength failure criteria and viscous flow around the attached particle. The analytical expressions for stress maxima and stress diagrams for a single particle allow determining the critical failure stresses, breakage points of the beam and breakage flow velocity. The mathematical model describing lab coreflood includes the maximum retention function for both authigenic and detrital fines. The matching laboratory coreflood data under increasing velocity at micro- and core-scales achieved high matching of the experimental data by the model. High matching validates the upscaling and downscaling procedures derived.

Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

– This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method’s effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.

Effect of steel fibers and the fiber-concrete adhesion stress on the nonlinear shear behavior of beams

In this article, a theoretical model in nonlinear elasticity is presented to analyze the influence of the percentage of steel fibers and the effect of the fiber-concrete bond stress on the shear force at the rupture of beams subjected to the combined effect of bending moment, normal force, and shear force. For a given beam section, it is defined by a succession of layers of concrete and longitudinal steel elements. Each layer is defined by its height hi, width bi, and position relative to one end of the section YGi. Each longitudinal steel element is also defined by its cross-sectional area and position relative to one end of the section. The steel fibers are defined by volume percentages of 0.5%, 1%, 1.5%, and 2%, taking into account the mechanical nonlinearity of the materials. This model is based on the multilayer analysis of sections and an iterative solution procedure for each layer and each section, considering a given longitudinal deformation state and shear stress. The global equilibrium of the sections is analyzed under the assumption of flat longitudinal deformations but with, in principle, an interdependence of longitudinal normal stresses and shear stresses. In this study, using the principle of virtual work, equilibrium equations for deformations and stresses, as well as partial compatibility equations between concrete deformations and mean deformations, are derived. Comparative examples between ordinary reinforced concrete beams with variable shapes and reinforcement details, and those reinforced with steel fibers, are presented to demonstrate the accuracy of the proposed model for simulating the nonlinear shear response of beams.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

Free vibration analysis of fiber-reinforced composite multilayer cylindrical shells under hydrostatic pressure

With the development and application of composite materials in deepwater submersibles, the influence of high external pressure from deepwater environments on the vibration characteristics of composite pressure-resistant cylindrical shells cannot be ignored. In this paper, the free vibration characteristics of prestressed composite multilayer shells are methodically examined by establishing partial differential equations based on the three-dimensional elasticity theory. To solve these equations, the state-space technique and double Fourier series expansion are employed to transform them into state-space ordinary differential equations. These equations are utilized to analyze vibration problems of simply supported composite multilayer cylindrical shells with arbitrary layered angles. The initial stress equilibrium equation is solved to determine the prestress induced by the hydrostatic pressure. This overcomes the limitation of membrane stress and considers non-uniform distributions of radial and interlaminar prestresses. The proposed theoretical model and solution strategies are validated by comparing with theoretical and experimental results from the existing literature and the finite element method. Additionally, the present investigation reveals that the natural frequencies of thick shells can be calculated more accurately using the prestress derived by the proposed approach, especially subjected to noticeable hydrostatic pressure. Finally, the performed investigations indicate that the circumferential prestressing has a more pronounced effect on the shell's natural frequencies than axial prestressing.

Variable angle tow-steered curvilinear fibres-based rotating arbitrarily layered composite beams- a coupled vibration of chordwise-flapwise-torsional motions by higher-order beam theory

In the present work, variable angle tow-steered curvilinear fibres-based straight composite rotating beam is studied for its coupled vibrational behaviours from different beam planes by a higher-order beam formulation satisfying plane stress situation and combining with finite element methodology. The structural theory is extended to include all the vibrational motions including chord- and flap-wise motions, axial and torsional vibrations of beam. The centrifugal force stiffening and gyroscopic effects stemming from rotational motions are taken into the formulation. The generic differential equations of equilibrium for the proposed structural theory are evolved adopting virtual dynamic work, Hamilton’s principle and a suitable C1 continuity based elemental equations are obtained. The newly constructed finite element formulation is tested for solved problems that are known in the literature. Based on in-depth analysis, vibration features of curvilinear fibre-based rotating generally layered composite beam including coupled dynamic motions in terms of vibrational modes and their frequencies pertaining to chord- and flap-wise, axial, torsional motions are detailed presuming variable angles along the curvilinear fibre path at the centre and layer edge, short and long beams, hub-radius, beam cross-section, and high rotational speed effects.

Deasphalting of crude oils: A molecule-based model to explore an optimization approach and alternative solvents

Actual motivation to reduce environmental impact leads to an increase in the study of green technologies, even those used in processes with fossil fuels. However, far from the proposed reduction of such energies, heavier crude oils are being processed to fulfill the energy demand. Heavy crude oils require an upgrading process to mitigate the effect of noxious components or contaminants such as asphaltenes and heavy metals. In this work, a deasphalting process was studied by simulation as a pretreatment to eliminate the heavy compounds of crude oil. The simulation of such a deasphalting process was carried out by Aspen Plus® software by combining a model for crude oil, formed by 33 molecules and able to describe its complex composition, mainly in the asphaltene fraction, and a model for the phase equilibria that use liquid-liquid equilibrium equations and the modified UNIFAC (Dortmund) model.A new objective function, including extraction yield, asphaltene reduction, and the ratio solvent/crude oil, was proposed and used to optimize extraction conditions and compare the performance of several solvents. Up to 12 pure solvents of different nature (tert-butanol, n-butanol, iso-propanol, methanol, ethylene glycol, acetone, methyl-isobutyl ketone, per-fluoro heptane, anisole, 1-metil-2-pyrrolidone, sulfolane, and n-heptane) were checked as well as some binary mixtures of them to explore more sustainable alternatives against conventional paraffin as precipitating agents. Satisfactory results were obtained when alcohols or ketones were used. A deeper study was developed to understand the effects of the alcohols' molecular structure (branching, chain length, and molecular weight) as solvents. Higher branching reduces the solvent/oil ratio, while higher molecular weight leads to higher extraction yield but poorer asphaltene reduction.

Torsional vibration of tapered pile including circumferential support of surrounding soil

The tapered pile is usually divided into finite segments along the vertical direction due to its variable cross-section when investigating its torsional dynamic characteristics. However, an annular projection would be formed at the contact surface of the adjacent pile segments due to the difference in radius, through which the surrounding soil will apply the circumferential support on the pile segment. In this study, the Voigt model is adopted to model the circumferential support of the surrounding soil acting on the annular projection. On this basis, an amended impedance function transfer method is established combined with the circumferential stress equilibrium equations and circumferential displacement continuity conditions at the contact surface of the adjacent pile segments. The shear complex stiffness transfer method is employed to consider the construction disturbance during pile installation. Then, an improved analytical solution for the torsional vibration of the tapered pile considering the circumferential support of the surrounding soil is established by solving the equations for the soil and pile. Based on the solution, the torsional vibration of the tapered pile is investigated and the influence of the circumferential support of the surrounding soil is revealed under different pile parameters and construction disturbance conditions.

Preparation of modified atmosphere packaging based on the respiratory characteristics of cherry tomato and its freshness preservation application

Cherry tomato (Lycopersicon esculentum Mill.) belongs to the respiratory leap type of fruit, post-harvest respiratory metabolism exuberant, easy to lead to nutrient loss, tissue destruction, and then rotting and deterioration, seriously affecting its commercial value. Therefore, research on post-harvest preservation technology of cherry tomatoes is crucial for related industries. Spontaneous modified atmosphere film packaging technology can extend the storage period of fruits and vegetables by adjusting the gas concentration of the storage environment, so that the fruits and vegetables can be maintained at a level suitable for their storage. In this study, the respiration model of cherry tomatoes at different temperatures was established by measuring the respiration rate of cherry tomatoes at different temperatures. It was found that the respiration rate of cherry tomatoes increased with increasing temperature and gradually decreased with increasing storage time. T-test results showed that there was no significant difference between the experimental values and the predicted values. The chemical kinetic model and its equilibrium equation were used to determine the parameters of modified atmosphere packaging of cherry tomatoes. PEG/PVDF membranes and SPEEK/PEG/PVDF blend membranes were prepared with sulfonated polyether ether ketone (SPEEK), polyvinylidene fluoride (PVDF) and polyethylene glycol (PEG) according to the parameters of modified atmosphere membranes to keep cherry tomatoes fresh. The results showed that there was a significant effect of package method on the preservation parameters of cherry tomatoes, the SPEEK/PEG/PVDF membrane treatment group significantly reduced the weight loss rate of cherry tomatoes and maintained the lower film permeability and higher sound fruit rate, hardness, titratable acid (TA) content, VC content and total soluble solids (TSS) content of cherry tomatoes. It can effectively delay the ripening and senescence of cherry tomatoes and extend the fruit storage period.

Mass–radius relation of neutron stars and massive pulsars with realistic equation of state

We use up-to-date constraints on the mass and radius of 15 neutron star objects and pulsars, from electromagnetic and gravitational wave observables and different theoretical schemes, to extend the nuclear equation of state (EOS) based on realistic Michigan-three-Yukawa (M3Y) nucleon–nucleon interaction, which truly accounts for the low-density EOS of nuclear matter (NM), to describe dense NM. The considered EOSs are employed to map the mass–radius profiles using the Tolman–Oppenheimer–Volkoff equations of hydrostatic equilibrium. We found that the EOSs from CDM3Y-230 to CDM3Y-270, with saturation incompressibility K 0 = 230–270 MeV, successfully reproduce most of the recent constraints on the NS masses and radii. Based on both M3Y-Paris and M3Y-Reid NN interactions, these EOSs indicate radius of 11.67 ± 0.34 km for the NS of 1.4 M ʘ mass, and the expected maximum NS mass (M max) to be 1.93 ± 0.21 M ʘ. The upper limits of constraints indicated stiffer EOSs of K 0 = 300–330 MeV, which have estimated 1arger radii of 12.29 ± 0.14 km for NS (1.4 M ʘ) and heavier M max of 2.31 ± 0.14 M ʘ. Increasing the stiffness of the employed EOS is found to increase the indicated maximum mass of NS, its radius and maximum compactness, the core–crust transition density, the speed of sound in its interior, and slightly the transition proton-fraction, but to decrease the abundance of the proton, muon, and electron over npeμ core matter of NS, as well as the estimated central density.

Hygrothermal buckling analysis of thin rectangular FGM plate with variable thickness

PurposeThis paper aims to contribute novel insights into the analysis of thin functionally graded material (FGM) plates with variable thickness, considering both temperature-dependent and independent material properties, focusing on critical linear buckling temperature rise and the effect of critical linear moisture for various moisture concentrations.Design/methodology/approachThe study derives stability and equilibrium equations for thin rectangular FGM plates under hygrothermal loading, employing classical plate theory (CPT). Buckling behavior is examined using Galerkin’s method to obtain pre-buckling force resultants.FindingsThe findings highlight significant increases in critical buckling temperature with aspect ratio, distinct temperature sensitivity between materials and increasing moisture susceptibility with larger aspect ratios. These insights inform material selection and design optimization for FGM plates under hygrothermal loading, enhancing engineering applications.Research limitations/implicationsThis research primarily focuses on hypothetical scenarios and mathematical model development and analysis.Originality/valueThis paper presents original contributions in the field by addressing the hygrothermal buckling analysis of thin FGM rectangular plates with variable thickness, utilizing CPT, thereby enriching the understanding of structural behavior in varying environmental conditions.

Coupled thermomechanical analysis using isoparametric curved shell elements

This study focuses on the coupled thermo-mechanical formulation in isoparametric shell elements and its implementation in curved laminar structures. The formulation is based on the direct isoparametric formulation, specifically designed for flat and curved shell elements, and relies on the theory of degenerated solids. The research provides a comprehensive description of the coupled thermo-mechanical analysis, including the mechanical equilibrium equations, the interrelations between stress, temperature, displacement, deformation, and the conservation of thermal energy. It adopts a simultaneous approach to addressing the impacts of temperature and mechanical loads. The paper elaborates on various numerical examples that substantiate the formulation. These examples include simulations that encompass linear scenarios, which are then compared to analytical solutions and results derived from other numerical software. The examples highlighted include one-dimensional temperature diffusion, radial diffusion, and the thermal behavior of a cylindrical cover and a pipe under a linear temperature gradient. These simulations demonstrate the formulation's capacity to accurately capture the interactions between temperature variations and structural displacements. They also confirm the alignment of the proposed model with existing analytical and numerical solutions. In conclusion, the research provides a good framework for coupled thermo-mechanical analysis. The flexibility of the formulation is evidenced across different configurations and scenarios, accommodating various types of boundary conditions such as constant and linear temperature fields, distributed loads, and constraints on displacements and rotations. It effectively manages temperature flows across one, two, and three dimensions, highlighting its extensive applicability in the realm of structural analysis. This versatility underscores its broad applicability in the field of structural analysis.

Heat and Mass Transfer of Water During Activated Carbon Adsorption

AbstractIn order to investigate the dynamic process of the heat and mass transfer phenomenon of water vapor on activated carbon, a fixed‐column adsorption unit and two kinds of activated carbon were used as adsorbent for water vapor adsorption under the adsorption temperature 293.15 K, 303.15 K and 313.15 K, respectively. A coupled model was developed by mass and energy balance, mass transfer and simplified DO adsorption equilibrium equation. The effects of physical parameters on heat and mass transfer were theoretically investigated. Results show that the numerical model match well with the experimental data (R2>0.993). The axial diffusion coefficient (DL) increases from 1.226×10−6 m2 s−1 to 4.062×10−6 m2 s−1 for RAC and 1.425×10−6 m2 s−1 to 4.030×10−6 m2 s−1 for MAC, mass transfer coefficient (k) increases from 8.171 s−1 to 27.083 s−1 for RAC and 9.499 s−1 to 26.864 s−1 for MAC. Concurrently, the breakthrough time of adsorption water vapor gradually shorten from 1000 s to 780 s with temperature increasing from 293.15 K to 313.15 K. The effect of axial diffusion coefficient (DL) on mass transfer is not obvious. Furthermore, the influence of the mass transfer coefficient (k) on temperature variation rate surpasses that of the internal heat transfer coefficient (h).

Adaptive deep homogenization theory for periodic heterogeneous materials

We present an adaptive physics-informed deep homogenization neural network (DHN) approach to formulate a full-field micromechanics model for elastic and thermoelastic periodic arrays with different microstructures. The unit cell solution is approximated by fully connected multilayers via minimizing a loss function formulated in terms of the sum of residuals from the stress equilibrium and heat conduction partial differential equations (PDEs), together with interfacial traction-free or adiabatic boundary conditions. In comparison, periodicity boundary conditions are directly satisfied by introducing a network layer with sinusoidal functions. Fully trainable weights are applied on all collocation points, which are simultaneously trained alongside the network weights. Hence, the network automatically assigns higher weights to the collocation points in the vicinity of the interface (particularly challenging regions of the unit cell solution) in the loss function. This compels the neural networks to enhance their performance at these specific points. The accuracy of adaptive DHN is verified against the finite element and the elasticity solution respectively for elliptical and circular cylindrical pores/fibers. The advantage of the adaptive DHN over the original DHN technique is justified by considering locally irregular porous architecture where pore–pore interaction makes training the network particularly slow and hard to optimize.

Axisymmetric hybrid Vlasov equilibria with applications to tokamak plasmas

We derive axisymmetric equilibrium equations in the context of the hybrid Vlasov model with kinetic ions and massless fluid electrons, assuming isothermal electrons and deformed Maxwellian distribution functions for the kinetic ions. The equilibrium system comprises a Grad–Shafranov partial differential equation and an integral equation. These equations can be utilized to calculate the equilibrium magnetic field and ion distribution function, respectively, for given particle density or given ion and electron toroidal current density profiles. The resulting solutions describe states characterized by toroidal plasma rotation and toroidal electric current density. Additionally, due to the presence of fluid electrons, these equilibria also exhibit a poloidal current density component. This is in contrast to the fully kinetic Vlasov model, where axisymmetric Jeans equilibria can only accommodate toroidal currents and flows, given the absence of a third integral of the microscopic motion.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

A wave model of high frequency vibrational energy transfer in a wedge-shaped combined plate structure

Wedge shaped combined plate structure (WCPS) is broadly applied in large-sized transport equipment such as aerospace and shipping. However, with the increase of equipment operating running, it is easy to cause high-frequency vibration of structure components. Therefore, studying the energy transfer and vibration response of high-frequency vibration in WCPS is significant for the structural design of aerospace and shipping. The promising method is energy finite element analysis (EFEA). However, the energy transfer coefficient at joint of WCPS is needed in EFEA. The current studies about the energy transfer coefficients are focus on the joint between two uniform thickness plate. But it is some different that getting the energy transfer coefficient for WCPS, because the near-field solve representation for wedge-shaped plate is complicated due to the non-uniform thickness of wedge-shaped plate member in WCPS. In this research, geometric acoustic approximation and an equilibrium equation on both ends of joints were used to derive the equation of the energy transfer coefficient for the WCPS. Then by setting coupling nodes, construction of a coupling matrix to characterize discontinuity of energy density at the coupling position of the WCPS and energy density governing equation of wedge-shaped plates were integrated. In doing so, the EFEA modeling of the WCPS was developed and the energy density distribution on each node of the WCPS could be obtained. Same-sized uniformly coupled structures is analyzed, the vibration energy distribution characteristics of two types of coupling structures were compared. The results show that the energy density value of the wedge plate member attenuate more slowly than that of the uniform plate member due to the acoustic black hole aggregation effect in the wedge plate part, and there is an abrupt change in the energy density value at the joint of WCPS. As a result, the factors influencing high-frequency vibration energy transfer were investigated.