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This paper presents a novel analytical framework for investigating the nonlinear dynamic response of functionally graded graphene platelets reinforced composite (FG-GPLRC) cylindrical shells filled with fluids and supported by non-uniform Kerr elastic substrates. The significance of this research lies in its comprehensive exploration of various substrate configurations achieved through discretizing the elastic substrates along the shell’s length. The fluid within the shell is characterized as non-viscous and incompressible, while material properties are rigorously determined using the rule of mixtures and the Halpin–Tsai micromechanical model. At the core of our method lies the establishment of nonlinear kinematic relationships rooted in Donnell shell theory and von Kármán’s nonlinear geometric postulates. We rigorously solve the equations of motion using Galerkin’s technique and the fourth-order Runge–Kutta method. Notably, our enhanced model efficiently accounts for the effect of discontinuities in the elastic foundation’s stiffness solely through integration operations, eliminating the need for intricate algorithms. This approach optimizes computational time and costs. The study conducts a comparative analysis of its results with existing literature, contributing to a thorough understanding of the validity of the proposed approach. We investigate various factors — including material properties, fluid characteristics, and geometric parameters — and their influence on the nonlinear response of nanocomposite shells. Our comprehensive analysis highlights the significant impact of substrate distribution, emphasizing its importance in structural design. Expanding the elastic base area and gradually shifting the elastic foundation toward the central region of the shell positively will affect various factors such as elevating natural frequency and reducing vibrational amplitudes. [ABSTRACT FROM AUTHOR]

The van der Waals (vdW) force, along with the electrostatic force and the first-order fringing field effect, act on the electrostatically actuated nano-cantilever (EANC) when the gap between the deformable electrode and the stationary electrode is less than 20 nanometres. Because of the vdW force, the EANC can undergo a pull-in phenomenon even without the electrostatic force when the nano-cantilever length exceeds its detachment length. The vdW force also results in a significant reduction in static pull-in instability parameters of the slender EANC compared to corresponding parameters obtained when this force is absent. This paper aims to augment the stable static travel range (i.e., the pull-in displacement) of the aforementioned EANC having a length close to its detachment length by varying the beam width. The beam width is assumed to vary in linear and parabolic manners and is controlled using a width variation parameter in each case. The governing equation of the Bernoulli-Euler beam theory and the Galerkin's technique are utilised to obtain the weighted residual statement (GWRS). The GWRS is utilised to obtain static pull-in instability parameters of referential prismatic and variable-width EANCs. Pull-in instability parameters of variable-width EANCs, for various values of width variation parameters and the initial gap between electrodes, have been obtained. The aforementioned results have been validated with corresponding results obtained by three-dimensional finite element simulations performed using COMSOL Multiphysics®. Compared to the referential prismatic EANC, a significant augmentation in the pull-in displacement of the variable-width EANC has been obtained. [ABSTRACT FROM AUTHOR]

Along with the electrostatic force and fringing field effect, the van der Waals (vdW) force acts on the electrostatically actuated nano-cantilever (EANC) when the separation between deformable and stationary electrodes is less than 20 nanometers. The vdW force can cause pull-in of the EANC even without the applied voltage when the nano-cantilever length exceeds its detachment length. The vdW force also causes a substantial curtailment of static pull-in instability parameters of the slender EANC compared to corresponding parameters obtained when this force is absent. This paper aims to utilise particle swarm optimisation (PSO) to arrive at the optimised beam width profile to augment the stable static travel range of the EANC having the length close to its detachment length. Continuous shape function having four parameters is utilised as the variable beam width. The weighted residual statement (GWRS) is obtained using the governing equation of the Bernoulli–Euler beam theory and Galerkin's technique. Static pull-in instability parameters of referential prismatic and variable-width EANCs are evaluated by utilising the GWRS. PSO is employed to arrive at optimised parameters for the variable beam width. The penalty approach is used to impose constraints on design variables. Optimised width profiles of the EANC have been obtained for various values of design constraints and initial separation between electrodes. Static pull-in instability parameters of aforementioned EANCs are validated with corresponding results obtained by three-dimensional finite element simulations performed using COMSOL Multiphysics®. The optimised width profile of the EANC brings a significant augmentation in its stable static travel range. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of cracked Rayleigh beams constructed from bidirectional functionally graded (BDFG) materials under simple boundary conditions. A torsional massless spring is employed to model the beam's open crack type. The vibration equations are obtained using Hamilton's principle. The graded beam material properties are varied throughout the thickness based on the power-law distribution and in the longitudinal direction using the exponential material distribution. To solve the dynamic equations, Galerkin's approach is employed. The paper evaluates the impacts of the axial index, gradient property index, beam modulus ratio, and crack parameters on the natural frequencies of the FG beam. The results indicate that the dimensionless natural frequencies of intact graded beams decrease with increased gradient index k. In contrast, they increase with a rise in the modulus ratio. Additionally, the results demonstrate that an increase in the crack depth ratio decreases dimensionless natural frequencies.

This paper presents the dynamic and chaotic responses of functionally graded graphene origami (GOri)-enabled auxetic metamaterials (GOEAMs) doubly curved shells featuring stepped thickness profiles in thermal environment. A novel analytical framework is introduced to investigate geometric variations and material distributions, highlighting their crucial roles. Specifically, the investigation delves into structural variations in shell thickness, characterized by abrupt changes in uni- or bi-directional orientations, encompassing both single and double stepped thickness profiles. The five geometric configurations of the stepped structures encompass plates, cylindrical shells, spherical shells, hyperbolic paraboloid shells, and elliptical paraboloid shells. The FG-GOEAM structures, comprising multiple layers with varied GOri distributions across their thickness, are scrutinized using genetic programming-assisted micromechanical models. Central to the present approach is the formulation of nonlinear kinematic relationships using Reddy's third-order shear deformation theory alongside von Kármán's nonlinear geometric assumptions, with equations of motion being solved utilizing Galerkin's technique. Notably, the enhanced model is developed to address non-continuous thickness variation through integral calculus operations, enhancing computational efficiency and obviating the need for complex algorithms. To verify the accuracy of the proposed method, the obtained results are compared with those from published literature. The study thoroughly investigates the influence of material properties, thermal conditions, and geometric parameters on the free vibration and nonlinear behaviors of the structures. Key findings include: The shell stiffness of the functionally graded (FG) shell (X − W G r ) surpasses that of the homogeneous shell (U − W G r ). The transitions from periodic to chaotic states of the stepped structures are discerned through the analysis of the time history response, the phase plane illustrating the deflection-velocity relationship, and the Poincaré map. Increasing the thickness ratios and GOri content while reducing the folding degree of GOri results in a significant increase in the fundamental frequency and critical load, along with a simultaneous decrease in vibrational amplitudes. • Reddy's third-order shear deformation shell theory and Galerkin's method are applied. • Flat plates and four configurations of curved shells are investigated. • Structures with uni- or bi-directional stepped profiles are analyzed. • Material distributions in uniform and functionally graded patterns are investigated. • Elucidate periodic to chaotic states using time history, phase plane, and Poincaré maps. [ABSTRACT FROM AUTHOR]

The mathematical formulation of higher-order, displacement-based and variationally inconsistent refined beam theory variants is presented. These variants are then utilized for the static pull-in instability analysis of electrostatically actuated narrow shear deformable microbeams. In these variants, the axial and transverse displacements of the beam consist of flexural contributors and shearing contributors. Assumed beam displacement functions are such that flexural contributors do not influence the cross-sectional shearing force and shearing contributors do not influence the cross-sectional flexural moment. These variants do not need the shear correction factor. The governing equation of these variants is derived using beam gross equilibrium equations. These variants involve only one governing equation of fourth order with one unknown variable and have the coherence associated with the Bernoulli–Euler beam theory. Beam end conditions including two distinct and physically meaningful clamped end conditions are described for these variants. These variants and the Galerkin's weighted residual technique are then used to study the interaction between the linear restoring force of the shear deformable microbeam and the nonlinear electrostatic force acting between the microbeam and the stationary electrode. Effects of the beam thickness-to-length ratio on pull-in instability parameters of electrostatically actuated microbeams with various fixity conditions are presented. In order to validate obtained numerical results, they are compared with corresponding results obtained by the authors using the Timoshenko beam theory-based spectral finite element analysis technique and with three-dimensional finite element simulations carried out using COMSOL Multiphysics. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive analysis of the nonlinear dynamic response of functionally graded cylindrical shells conveying fluids, surrounded by non-uniform Winkler–Pasternak elastic substrates. The research underscores its significance through the development of a novel analytical framework that explores various substrate configurations. The elastic substrates are discretized into distinct regions along the length of the shell. Within the shell, fluid flow is characterized as non-viscous, incompressible, and irrotational. Material properties are rigorously determined employing the rule of mixtures. Central to our approach is the establishment of nonlinear kinematic relationships utilizing Donnell shell theory in conjunction with von Kármán's nonlinear geometric assumptions. The equations of motion are rigorously solved employing Galerkin's technique and fourth-order Runge–Kutta method. Remarkably, the enhanced model efficiently addresses the effect of discontinuities in the stiffness of the elastic foundation on the cylindrical shell, solely through integration operations, without necessitating the intervention of intricate algorithms. The current method effectively optimizes both computational time and costs. The study's results are compared with those obtained from the existing literature. Various factors, such as the power-law index, fluid flow characteristics, and geometric parameters, are investigated for their effects on the nonlinear response of the composite shells. Our comprehensive analysis highlights the significant influence of substrate distribution, a crucial consideration in structural design. Expanding the area of the elastic base and gradually shifting the elastic foundation towards the central region of the shell positively influence various factors: elevating the natural frequency, reducing vibrational amplitudes, and reinforcing structural stiffness against increased critical flow velocities. • Exploring behavior of fluid-conveying FGM shells using Donnell shell theory. • Analyzing Winkler-Pasternak elastic substrates with varied support areas. • Assessing the impacts of foundation position, dimensions, and stiffness on shells. • Investigating effects of material properties, geometry variations, and fluid medium. [ABSTRACT FROM AUTHOR]

An incautious use of the well-known Galërkin's technique to find approximate solutions of a differential problem may lead to apparently wrong results. Examples are based on an inverse approach to investigate buckling of compressed axisymmetric circular plates, a subject that is common in courses on mechanics of structures and stability of structural elements. We discuss how the mistake may originate and show how it is possible to recover the expected results, thus providing a means for the students to cross-check their outputs. [ABSTRACT FROM AUTHOR]

Abstract This paper aims to investigate analytically the vibrational characteristics of functionally graded (FG) single-walled carbon nanotube (SWCNTs) based on modified Love shell theory. By using Galerkin's technique, the governing carbon nanotube (CNT) dynamics equation of the motion of the considered system is developed. For theoretical point of view, a nanotube problem is framed by amending some physical terms to accommodate the nature of the problem. For the present problem, vibrations of rotating carbon nanotubes with ring supports are examined. The effects of angular velocity, length- and height-to-radius ratios of rotating armchairs and zigzag SWCNTs with ring supports are fully investigated. Rings supports are attached to CNT in the radial direction. Our investigations show that increasing of length- and thickness-to-radius ratios yields decreasing and increasing frequency behaviors for rotating FG-CNT with clamped-clamped (C-C) and clamped-free (C-F) boundary conditions, respectively. It is found that with increasing the angular speed, frequencies of forward waves decrease and backward waves increase. Moreover, the effect of frequency is also performed to place the ring support at different locations of the CNT. Computational software MATLAB is engaged to characterize the frequencies based on CNT model to estimate the accuracy and effectiveness of this model. A few comparisons of analytical results for non-rotating and rotating CNTs are performed to confirm the validity, efficiency and accuracy of the numerical methodology. [ABSTRACT FROM AUTHOR]

Seismic response of water tanks resting on ground has been a topic of considerable research in academics as well as energy and infrastructure industry. In many cases, it remains essential that they remain functional even after a major ground shaking. State of the art adapted in different national and international code usually recommends a simplified mathematical model that in many cases digresses from field reality. A three dimensional analysis of the tank plus fluid and foundation soil is surely possible based on finite element analysis. However, in most cases, such exhaustive analysis is done away with, often due to lack of supportive software to tackle such problem, but more out of economic and schedule compulsion, where in many cases engineers do not have the luxury to carry out such expensive analysis, to remain competitive in the market. Thus, a search for a better and more realistic model than what is in vogue is still in quest for this particular problem. Present paper attempts to present a mathematical model based on Lagrange's formulation and adapting Galerkin's technique that circumvents many of the problems as cited above. [ABSTRACT FROM AUTHOR]

This paper attempts to develop a mathematical model for estimating the seismic response of a cylindrical shaped nuclear reactor building resting in an elastic halfspace considering foundation compliance. Most of the research carried out on this topic has either been carried out by resorting to finite element method (FEM) which makes the computational cost expensive or based on the simplifying assumption of assuming the cylindrical structure as a multi degree lumped mass stick model with soil coupled as boundary springs. In the present paper an analytical model has been developed in which the deformation of the cylindrical body (including its shear deformation characteristics) has been taken into cognizance and then coupling with foundation stiffness a comprehensive solution has been sought based on Galerkin's weighted residual technique. The results are finally compared with FEM to check the reliability of the same. The results are found to be in good agreement with the detailed finite element analysis. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a new theoretical model to characterize the superconducting microstrip line and carefully studies the effects of the nonlinearity of superconductors, the strip thickness and losses on circuit performances. The microstrip line has been considered as a multilayered structure. The integral equation for the electrical field has been formulated, in the spectral domain, using the exact dyadic Green’s function of bianisotropic planar media. The Galerkin’s technique has been used for solving this integral equation. Obtained results concern the effective permittivity constant and the attenuation constant versus frequency and temperature rate. [Copyright &y& Elsevier]

The problem of Rayleigh-Bénard convection with temperature dependent viscosity is investigated for all combinations of rigid and dynamically free boundaries. The principle of exchange of stabilities is shown to be valid for the problem which yields that the onset of convection is through stationary mode. Further, Rayleigh numbers for each possible combination of boundary conditions derived using Galerkin's technique. The effect of variable viscosity on the onset stationary convection for each case of boundary combination is computed numerically and depicted graphically. The present analysis yields that the positive values of temperature-dependent viscosity parameter has stabilizing effect on the onset of stationary convection while negative values have destabilizing effect. Further, various conclusions and results for the problem of Rayleigh-Bénard convection with constant viscosity for different cases of boundary conditions have been worked out and are discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper we present a new theoretical model for the modelling of the microstrip line as well as two types of discontinuities: regular (open end, step, bend and T- and cross-junctions) and irregular (stub and bent-stub). The two-dimensional exact dyadic Green function of a grounded dielectric slab has been used with the Galerkin's technique. The subdivision of the discontinuity in a network of juxtaposed unit cells, characterized by their own longitudinal and transversal current distributions, allowed the treatment of a large class of irregular discontinuities in addition to the regular discontinuities. The obtained results have been commented and compared with those of different approaches and with experimental results where a good concordance has been observed. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Postbuckling and postbuckled vibration analysis of antisymmetric angle-ply composite plates subjected to a thermo-mechanical loading is investigated in this article. The nonlinear partial differential equations of equilibrium are derived based on higher-order shear deformation theory, incorporating von Kármán nonlinear strain displacement relations and initial geometric imperfections. The solutions to the governing nonlinear partial differential equations are sought using the multi-term Galerkin technique. The nonlinear equilibrium paths are traced using the Newton-Raphson method in conjunction with the Riks approach. The free vibration frequencies of postbuckled plate about the static equilibrium state are reported by solving the associated linear eigenvalue problem. Results are presented for a simply supported antisymmetric angle-ply laminated plate, clearly bringing out the effects of uniform through-thickness temperature distribution and mechanical edge load on postbuckling and vibration behavior of the plate. [ABSTRACT FROM AUTHOR]

A finite element solution for the Navier-Stokes equations for steady flow under the porosity effects through a double branched two-dimensional section of a three-dimensional model of a canine aorta was obtained. The numerical solution involves transforming a physical coordinates to a curvilinear boundary fitted coordinate system. The steady flow, branch flow and shear stress under the porous effects were discussed in detail. The shear stress at the wall was calculated for Reynolds number of 1000 with branch to main aortic flow rate ratio as a parameter. The results are compared with earlier works involving experimental data and it has been observed that our results are very close to the exact solutions. This work is in fact an improvement of the work of Sharma et al. (2001) in the sense that computational technique is economic and Reynolds number is large. [ABSTRACT FROM AUTHOR]

The total Lagrangian formulation and implementation of the element-free Galerkin method (EFG) is presented for the analysis of contact-impact problems with large deformations and for the simulation of metal forming processes. An integration scheme based on stress points is used, so no mesh is needed. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. Numerical results for Taylor bar impact are compared to finite element solutions and agree well. Solutions to upsetting, extrusion of metals with large material distortions are given to show the effectiveness of the algorithm. In particular, EFG is shown to be more capable of treating motions of the workpiece around corners of the punch than finite element methods. [ABSTRACT FROM AUTHOR]