Displacement Equations Research Articles

Novel analytical perspectives on nonlinear instabilities of viscoelastic Bingham fluids in MHD flow fields

The nonlinear stability of a plane interface separating two Bingham fluids and fully saturated in porous media is inspected in the existing work. The two fluids are compressed by a normal magnetic field. The two fluids have diverse viscoelasticity, densities, magnetic, and porosity medium, with the existence of surface tension at the interface. The motivation of applied physics and engineering relations has encouraged the discussion of the current paper. Because the mathematical behavior is rather complex, the viscoelasticity involvement is reproduced only at the surface of separation, which is well-known as the viscous potential theory. Thereby, the equations of movement are scrutinized in a linear form, whereas a set of nonlinear boundary conditions are supposed. This procedure produces a nonlinear expressive nonlinear partial differential equation of the interface displacement. The non-perturbative approach which is based on the He’s frequency formula is employed to transform the nonlinear distinguishing ordinary differential equation with complex coefficients into a linear one. A novel process relying on the non-perturbative approach is utilized to examine the nonlinear stability and scrutinize the interface presentation. A non-dimensional analysis produces several dimensionless physical numerals. To validate the new approach, a comparison between the non-perturbative approach and its corresponding linear ordinary differential equation via the Mathematica Software is described and interpreted through a set of diagrams. Additionally, the Polar graphs have been elucidated. It is found that the mechanism of the stability does not change in the cases of real and complex coefficients.

Creep stress analysis of functionally graded transversely isotropic piezoelectric disc with variable thickness under rotation

Materials exhibiting different piezoelectric properties throughout their volume are referred to as functionally graded piezoelectric materials (FGPM). Due to their graded composition, the characteristics of these materials can be customized to fit particular needs in a variety of applications. Functionally graded piezoelectric materials are beneficial in many technical applications because of their versatility that enables increased performance, efficiency, and adaptability in a variety of disciplines. The objective of the current study is to analyse creep stresses in an annular disc composed of transversely isotropic, functionally graded piezoelectric material with varying thickness parameters. Creep stresses are evaluated analytically using the approach of Seth's transition theory. By using the stress-strain relations, the equations for mechanical stresses and electrical displacement are determined. Substituting these relations into the equilibrium equation a nonlinear differential equation is obtained. The findings are presented both numerically and graphically, demonstrating how the thickness parameter affects the circumferential stresses in the intermediate surface of the rotating disc. The disc made of transversely isotropic piezoelectric material PZT-4 exhibits higher creep stresses than other materials under consideration according to all of the numerical discussions and calculations. This work may provide an effective methodology for the analysis of functionally graded piezoelectric rotating discs and contribute to theoretical research and engineering applications.

Fourier Transform Approach to Boundary Domain Integral Equations for Elastic Composites With Domain Decomposition and Multi Reference Parameters

ABSTRACTIn this article, displacement and strain periodic boundary domain integral equations for homogenization problems of elastic composites are derived in the context of FFT homogenization methods. The resolution methods based on regular grids and discrete Green's tensors are presented. The displacement based equations can be used to solve problems in arbitrary domains under periodic and non‐periodic boundary conditions. The strain based integral equation is obtained from the combination of the displacement based equations for different domains, each one having its own reference elasticity tensor. In the latter, the strain values inside every phases are connected to material mismatch parameters on the phase boundary. It was shown that by decomposing suitably domains by stiffness and using adapted reference parameters, the iteration schemes converge faster.

A New Method for Displacement Modelling of Serial Robots Using Finite Screw

Kinematics is a hot topic in robotic research, serving as a foundational step in the synthesis and analysis of robots. Forward kinematics and inverse kinematics are the prerequisite and foundation for motion control, trajectory planning, dynamic simulation, and precision guarantee of robotic manipulators. Both of them depend on the displacement models. Compared with the previous work, finite screw is proven to be the simplest and nonredundant mathematical tool for displacement description. Thus, it is used for displacement modelling of serial robots in this paper. Firstly, a finite-screw-based method for formulating displacement model is proposed, which is applicable for any serial robot. Secondly, the procedures for forward and inverse kinematics by solving the formulated displacement equation are discussed. Then, two typical serial robots with three translations and two rotations are taken as examples to illustrate the proposed method. Finally, through Matlab simulation, the obtained analytical expressions of kinematics are verified. The main contribution of the proposed method is that finite-screw-based displacement model is highly related with instantaneous-screw-based kinematic and dynamic models, providing an integrated modelling and analysis methodology for robotic mechanisms.

Analytical seismic model for a tunnel with segmental liner buried in the half-space soil

In this paper, by using the wave function expansion (WFE) and the expansion of the cylindrical wave into plane wave (ECPW) methods, an analytical method for the dynamic response of a circular tunnel with segmental liner buried in the half-space soil to harmonic elastic waves is developed. The liner of the tunnel is supposed to consist of several segments and joints. Both the segments and joints are treated as open cylindrical shells, and the segment and joint shells together thus form an equivalent continuous shell (ECS) liner, and the thin shell theory is utilized to describe its vibration. The scattered wave field due to the presence of the tunnel and boundary of the half-space soil is divided into two parts, namely, the direct scattered wave field and secondary scattered wave field. The expressions for the direct scattered cylindrical waves in the soil is determined via the WFE method, while the secondary scattered waves form the boundary of the half-space soil are obtained by the ECPW method together with the application of the boundary condition along the soil surface. Applying the cylindrical shell theory on the ECS liner and using the Fourier series expansions for the variables and parameters of the ECS liner along the azimuthal direction together with introducing the Fourier component constitutive relation for the ECS liner, a system of equations for the Fourier component ECS displacements are derived with the differential equations for the ECS liner displacements. By using the system of equations, the expressions for the free and scattered wave fields and the continuity conditions at the interface between the liner and soil, the system of equations for the ECS liner coupled with the soil are derived. By using the developed analytical method for the tunnel, some results of the ECS tunnel under different incident waves are given.

Features of distribution of electric field strength and current density in the reactor during treatment of liquid media with high-voltage pulse discharges

Purpose. Development and use of a mathematical model of the stages of formation of high-voltage pulse discharges in gas bubbles in the discharge gap «rod-plane» to identify the features of the electric field intensity distribution in the reactor and determine the current density in the load during disinfection and purification of liquid media by high-voltage pulse discharges and find the most rational treatment. Methodology. To achieve this goal, we used computer modeling using the finite element method as a method of numerical analysis. An experimental reactor model was created that takes into account the dynamics of discharges in gas bubbles in water. The equations describing the system include the generalized Ampere equation, the Poisson equation and the electric displacement equation, taking into account the corresponding initial and boundary conditions, as well as the properties of materials. The dependence of the potential of a high-voltage electrode on time has the form of a damped sinusoid, and the specific electrical conductivity in a gas bubble is a function of time. Processes occurring at the front of the voltage pulse from 0 to 20 ns are considered. Results. It is shown that with an increase in conductivity and high-voltage potential to amplitude values in a gas bubble, the electric field strength in the water layer in the reactor reaches 70 kV/cm, and it is in the water layer that there is a strong electric field. The calculations show that already by 19th ns the density of conduction currents in water prevails over that of displacement currents. At the same time, additional inclusions in the water significantly affect the distribution of electric field strength and current density, creating a significant difference in their values at the boundaries of the interface between the bubble, conductive element and water. Originality. A simulation of the dynamics of transient discharge processes in a gas bubble and a layer of water with impurities was carried out, including an analysis of the distribution of the electric field strength and current density in a system with rod-plane electrodes in the phase transition section of a gas bubble-water. This approach allows us to reveal the features of processes in reactors and to investigate the influence of phase transitions on the distribution of electrophysical quantities. Practical value. Computer simulations confirm the prospect of using nanosecond discharges generated in gas bubbles within a volume of water for widespread industrial use and are of great interest for further experimental and theoretical research. References 25, figures 9.

Bending of bidirectional functionally graded nonlocal stress-driven beam

This paper provides a comprehensive analysis, using nonlocal stress-driven integral theory, of the static behavior of a nanoscale beam of bidirectionally graded materials. After a brief explanation of the mathematical formulation of BDFGMs, the work done and strain energy expressions derived from the displacement field are discussed. Variational formulations and Hamilton's principle are used to develop the equilibrium equation. An analytical development of the nonlocal kernel for stress-driven integral theory and formulated governing equation which was nondimensionalized later. Explicit equations for displacement and moment are obtained by solving this equation using the Laplace transformation. Three different boundary conditions are examined, and differences in the maximum displacement with respect to the nonlocal parameter and the two material FGM parameters are displayed both visually and in table form. The results exhibit excellent agreement and provide a standard for further research when they are closely compared to the existing numerical data. This work contributes to the knowledge of BDFGMs under nonlocal effects generated by stress-driven integral theory and offers solutions that have been confirmed for further investigation.

Complete Solutions in the Dilatation Theory of Elasticity with a Representation for Axisymmetry

In this paper, we present certain complete solutions in the dilatation theory of elasticity. This model can be derived as a special case of Eringen’s linear theory of microstretched elastic solids when microrotations are absent. It is also a version of the theory of materials with voids. The dilatation theory can be considered the simplest theoretical model of microstructured materials and is suitable for investigating various phenomena that occur in engineering, geomechanics, and biomechanics. We establish three complete solutions to the displacement equations of equilibrium that are the counterpart of the Green–Lamé (GL), Boussinesq–Papkovich–Neuber (BPN), and Cauchy–Kowalevski–Somigliana (CKS) solutions of classical elasticity. The links between these BPN and CKS solutions are established. Then, we present a representation of the BPN solution in the case of axisymmetry. The results presented here are useful for solving axisymmetric problems in semi-infinite and infinite domains.

Inspection of the nonlinear instability of electrified Casson fluids: a novel approach

The nonlinear electrohydrodynamic (EHD) instability of a circular cylindrical interface, separating two Casson prototypes is the aim of the present inspection. The motivation for the issue is owing to the increasing significance of several engineering and practicle physics. Because of the complicated mathematical processes, the contribution is performed through the viscous potential theory (VPT). The methodology resulted in a nonlinear characteristic equation of the interface displacement. This equation is transformed to simulate a hybrid Rayleigh-Helmholtz Duffing oscillator (HRDO) with complex coefficients. Guaranteeing the stability of fluid interfaces is crucial in industrial coating operations to provide consistent and even coatings. Gaining comprehension of the nonlinear stability criterion helps to enhance the management of coating thickness and uniformity. A novel methodology via the Non-perturbative approach (NPA) is employed. Through the case of the real coefficients, the later form is validated using a numerical solution (NS). It is concluded that the Ohnesorge contribution has a destabilizing effect on the stability zone in contrast with permeability factor. The electrical coefficients are found to be decaying factors in the stability configuration. It is also established that the viscosity decays the nonlinear steadiness requirement of the structure. The approach is a simple, promising, effective, and interesting.

3D electro-elastic static analysis of advanced plates and shells

The proposed three-dimensional (3D) coupled electro-elastic shell model allows the static analysis of smart structures embedding classical piezoelectric and functionally graded piezoelectric layers. Plates, cylindrical and spherical panels are investigated in both sensor and actuator configurations. The primary variables of the coupled model are displacement components and the electric potential. Therefore, displacements, stresses, strains, electric potential and electric displacements are calculated for smart structures used as sensors (applied mechanical load) and smart structures used as actuators (applied electric potential). In the case of spherical shells, 3D equilibrium equations are coupled with the 3D divergence electric displacement equation. The obtained coupled system is also valid for cylindrical shells and plates thanks to the use of a mixed curvilinear orthogonal reference system. The partial differential governing equations are solved using the Navier harmonic form and the exponential matrix method for simply supported structures. Preliminary assessments are proposed to validate the model and further benchmarks are analyzed to discuss the effects connected with the thickness ratio, the lamination scheme, the load conditions, the geometry of the structures and the employed materials. The main innovation point of the proposed model is the possibility to analyze several geometries including different materials and applied loads by means of a unique and general formulation where partial differential equations are solved in closed form. The use of functionally graded piezoelectric materials allows to eliminate all stress component discontinuities at each layer interface.

Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

In this paper, we consider the time fractional thermoporoelasticity problem in fractured and heterogeneous media. The mathematical model with a time memory formalism is described by a coupled system of equations for pressure, temperature and displacements. We use an implicit finite difference approximation for temporal discretization. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) for two-dimensional model problems. Further, we use the Generalized Multiscale Finite Element Method (GMsFEM) for coarse grid approximation. The primary concept behind the proposed method is to streamline the complexity inherent in the thermoporoelasticity problem. Given that our model equation incorporates multiple fractional powers, leading to multiple unknowns with memory effects, we aim to address this intricacy by optimizing the problem’s dimensionality. As a result, the solution is sought on a coarse grid, a strategic choice that not only simplifies the computational cost but also contributes to significant time savings. We present numerical results for the two-dimensional model problems in heterogeneous fractured porous media. We derive relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results confirm that the proposed method is able to achieve good accuracy with a few degrees of freedoms on the coarse grid.

Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells

In the present paper, a coupled 3D exact electro-elastic shell model for Functionally Graded (FG) and composite piezoelectric structures is proposed. Primary variables of the electro-elastic model are the electric potential and the three displacement components. The model allows to evaluate the piezoelectric effect in terms of frequencies and vibration modes. Both closed and open circuit configurations are analyzed and compared. The 3D equilibrium equations and the 3D divergence electric displacement equation for spherical shells give the set of partial differential equations for the electro-elastic problem. The proposed model for spherical shells automatically degenerates into simpler models for plates and cylindrical shells via properly considerations on radii of curvature along in-plane directions. The orthogonal mixed curvilinear coordinates α, β and z are employed. The partial differential governing equations have constant coefficients considering fictitious layers and they are solved using the Navier harmonic form and the exponential matrix method. These features lead to an exact solution for simply-supported boundary conditions. Free vibration analyses are conducted and circular frequencies for the first three thickness vibration modes are computed. After a global assessment phase to verify the correctness of the developed model, new benchmarks are proposed: different thickness ratios and material configurations are investigated. The present work can be intended as a reference general solution for those scientists interested in the study of piezoelectric structures via 2D analytical and numerical formulations.

Single domain Chebyshev spectral method for analyses of the vibroacoustic characteristics of baffled irregularly shaped plates

In this study a Chebyshev spectral method is introduced to analyse the vibroacoustic characteristics of plates with arbitrary shape. This approach involves a unique expansion of the governing equations for plate displacements using Chebyshev polynomials, coupled with Gauss-Chebyshev-Lobatto sampling. A significant advancement in computational efficiency is the use of a tensor product configuration. To manage irregular shapes, the elements of the inner product matrix corresponding to the open regions were set to zero. For precise shape representation, a substantial number of Chebyshev-Gauss-Lobatto points, surpassing the polynomial truncation count, are carefully selected. the elastic boundary conditions of the plate were simulated by applying artificial spring techniques at different points. The prediction accuracy of the vibroacoustic phenomena, including those of the mean square velocity and sound power, were rigorously confirmed through comprehensive comparisons with results from previous studies and those obtained using the finite element method. Furthermore, the accuracy and versatility of the method are demonstrated through a variety of numerical examples encompassing plates with geometries shaped like rectangules, triangules, clouds, and pigs geometries, and different boundary conditions.

A method for predicting deformation field of deep foundation pit considering spatial effect

Urban underground spaces have experienced significant development and utilization due to the rapid progress of urban construction and continuous advancements in construction technology. The number of foundation-pit projects immediately adjacent to buildings, structures, and pipelines has been increasing. Ensuring the safety of construction and surrounding facilities, it is essential to predict deformation caused by foundation pit excavations. In this study, a simple calculation method was proposed for predicting deformation of an envelope structure and surface settlement caused by deep foundation excavation with spatial effects. Firstly, the principle of minimum potential energy was applied and the lateral displacement equation of the envelope structure was derived, considering spatial deformation effects. Secondly, based on linear elasticity theory, a surface settlement expression was formulated utilizing the plane strain boundary. Finally, the stratigraphic loss method was used to establish a connection between horizontal envelope deformation and surface settlement, allowing for calculations of surface settlement at any position behind the envelope structure. The effectiveness and practicability of the proposed method were demonstrated by comparison of two existing foundation-pit engineering examples.

Analysis of fluid force and flow fields during gliding in swimming using smoothed particle hydrodynamics method

Gliding is a crucial phase in swimming, yet the understanding of fluid force and flow fields during gliding remains incomplete. This study analyzes gliding through Computational Fluid Dynamics simulations. Specifically, a numerical model based on the Smoothed Particle Hydrodynamics (SPH) method for flow-object interactions is established. Fluid motion is governed by continuity, Navier-Stokes, state, and displacement equations. Modified dynamic boundary particles are used to implement solid boundaries, and steady and uniform flows are generated with inflow and outflow conditions. The reliability of the SPH model is validated by replicating a documented laboratory experiment on a circular cylinder advancing steadily beneath a free surface. Reasonable agreement is observed between the numerical and experimental drag force and lift force. After the validation, the SPH model is employed to analyze the passive drag, vertical force, and pitching moment acting on a streamlined gliding 2D swimmer model as well as the surrounding velocity and vorticity fields, spanning gliding velocities from 1 m/s to 2.5 m/s, submergence depths from 0.2 m to 1 m, and attack angles from −10° to 10°. The results indicate that with the increasing gliding velocity, passive drag and pitching moment increase whereas vertical force decreases. The wake flow and free surface demonstrate signs of instability. Conversely, as the submergence depth increases, there is a decrease in passive drag and pitching moment, accompanied by an increase in vertical force. The undulation of the free surface and its interference in flow fields diminish. With the increase in the attack angle, passive drag and vertical force decrease whereas pitching moment increases, along with the alteration in wake direction and the increasing complexity of the free surface. These outcomes offer valuable insights into gliding dynamics, furnishing swimmers with a scientific basis for selecting appropriate submergence depth and attack angle.

Flexural–torsional buckling with enforced axis of twist and related problems

The classical lateral–torsional instability of a cantilever beam with continuous elastic lateral restraint and a transverse point-load applied at the free end is discussed here through the lens of asymptotic simplifications. One of our main goals is to provide analytical approximations for the critical buckling load, as well as its dependence on various key non-dimensional groups. The first part of this study is concerned with a scenario in which a doubly symmetric beam is constrained to rotate about a fixed axis situated at an arbitrary height above the shear centroidal axis. The second part examines a beam that features a continuous elastic lateral restraint spanning its entire length. Assuming that the stiffness of the constraint, κ (say), is finite, the buckling equations in the second case are described by a system of two coupled fourth-order differential equations in the lateral displacement and the angle of twist. We show that as κ → ∞ , the problem discussed in the first part provides an outer solution for the aforementioned system; the relevant boundary conditions for the next-to-leading-order outer approximation are also derived using matched asymptotics. Our theoretical findings are confirmed by comparisons with direct numerical simulations of the full buckling problem.

Photothermoelastic interaction in a two-dimensional semiconductor with nonlocal stress theory

The present contribution enlightens a viable simulation which describes the photothermoelastic interactions for a two-dimensional thermoelastic transversely isotropic thick semiconducting plate due to the presence of a varying heat source. The upper surface of the semiconductor is traction-free, subjected to prescribed surface temperature and prescribed carrier density flux while the lower surface of the plate rests on a rigid foundation and is thermally insulated. In this new model, the conventional Fourier law has been modified assimilating the memory-dependent derivative within a slipping interval in association with the nonlocal stress theory. On utilizing Laplace and Fourier transformation mechanism, the governing equations for displacement, carrier density and temperature have been obtained in the Laplace–Fourier transform domain. In order to have the solutions in the space-time domain, the inversion of the double transform have been carried out numerically based on the method of Fourier series expansion. From the graphical representations corresponding to the numerical results, the significance of effective parameters such as nonlocality parameter, delay-time parameter is discussed. Noteworthy differences in the results have been demonstrated for different kernel functions. Also, the superiority of a nonlinear kernel function and its effectiveness is also reported in the study.

The effect of imperfect bonding on stress distribution in fibrous composites

An attempt was made to map the distribution of stress in fibrous composites with imperfect bonding. Two analytical micro-mechanics models were developed. In the first model, the composite was subjected to axial tensile loading, parallel to the fiber direction, and the assumption of iso-strain was employed to derive the control equations. In the second model, the composite was loaded in the direction transverse to the fiber. An iso-stress condition was employed, and Airy stress function was utilized to articulate the stress and displacement equations. An assumption of how the stress is transferred between the matrix and the fiber was introduced in both models. To investigate and validate the models, specimens were fabricated using a carbon plain weave fabric and a geopolymer matrix. Single fiber pullout and three-point bending tests were carried out. The maximum average tensile stress obtained from the three-point bending tests, as well as the mechanical properties of the fiber and geopolymer, served as input for the models. Results indicate that the effect of the level of bonding is very high in the transverse direction while almost negligible in the axial direction. The difference in the maximum value of the axial tensile stress at the fiber-matrix interface was used to calculate the numerical value of the interfacial shear strength, and the numerical result matched the data obtained from the single fiber pullout test.

Optimal stability for laminated beams with Kelvin–Voigt damping and Fourier’s law

This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin–Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart–Prüss–Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.

Dynamic model for an independent suspension

The paper presents the dynamic model with 4 degrees of freedom of an independent suspension with MacPherson type coil springs, taking into account the anti-roll bar. Starting from the constructive scheme, a kinematic analysis necessary to obtain the dynamic model is carried out. The differential displacement equations are obtained starting from the expressions of kinetic and potential energy, by applying Lagrange’s equations. Next, the expressions of the inertia, stiffness and damping matrices and, finally, the matrix expression of pitch-roll vibrations are presented. At the end of the paper, in the numerical application carried out with a calculation program in MATLAB, the influence of the values of the damping constants on the dynamic forces at the wheels is studied. The numerical results are presented in the form of diagrams.