Clamped Boundary Conditions Research Articles

Free and forced vibrations of circular plates made of functionally graded graphene origami-enabled auxetic metamaterials

Presented herein is a numerical investigation into the free and forced vibrational characteristics of circular plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs). The material properties of FG-GOEAM are estimated using genetic programing-assisted micromechanical models considering two graphene content distribution patterns. The effects of graphene content, size and folding degree on the material properties are taken into account by the models. The formulation of the paper is in the context of first-order shear deformation plate theory, and the governing equations of motion are achieved utilizing Hamilton’s principle. In the solution approach, the variational differential quadrature (VDQ) method is employed by which the variational statement of problem is directly derived and discretized using matrix differential and integral operators. Selected numerical results are presented to study the effects of graphene origami (GOri) content, folding degree, and distribution pattern on the free and forced vibration responses of FG-GOEAM circular plates with clamped and hinged boundary conditions.

Dynamic behavior and stability analysis of fluid-conveying nano-spiral shells: Modeling, simulation, and parametric study

This paper presents a study on a fluid-conveying nano-spiral shell’s free vibration and stability analysis. The nano-spiral shell with Archimedean cross-section shape is modeled according to the first-order shear deformation (FSDT) and modified coupled stress (MCST) theories. Van der Waals force is modeled based on the Leonard Jones potential for atomic interactions. The fluid-structure interaction equation is derived based on the assumption of slip boundary conditions for the fluid. The Rayleigh-Ritz method is used to obtain the natural frequencies of a macro-size spiral shell to be validated with the finite element method (FEM). The parametric vibration analysis is performed for a nano-spiral shell conveying fluid. The results show that the natural frequencies decrease as the distance between the layers of the nano-spiral shell increases. Additionally, an increase in length results in a decrease in natural frequencies. Stability analysis is conducted, and flutter and divergence bifurcations are accurately studied. The results show that the shell with clamped boundary conditions is more stable than the supported boundary condition due to its higher bending stiffness. As the distance between the layers of the fluid-conveying nano-spiral shell increases, instability occurs at a lower fluid velocity. Conversely, as the thickness of the shell increases, the nanostructure becomes unstable at higher fluid velocities.

Dynamic optimization design of thin plate structure based on surrogate model

Abstract To enhance dynamic optimization efficiency in traditional thin plate structures, surrogate models are integrated for structural dynamic size optimization, reducing dependence on high-precision finite element simulations. This paper focuses on optimizing the aluminum alloy sheet as the subject of research. The parameters of the specimen are optimized through sensitivity analysis, based on the comparison between the free mode Finite Element Analysis (FEA) and Experimental Modal Analysis (EMA) results. The modified specimen undergoes validation under a four-edge clamped boundary condition. Subsequently, random vibration analysis is conducted to determine the root mean square (RMS) value of the acceleration response. Employing surrogate model technology, four models are constructed using parametric modeling and experimental design methods. These models are then compared for their fitting effectiveness. The results indicate that the RBF model exhibits the highest accuracy in fitting each response, effectively replacing dynamic simulation. Finally, the RBF combined with the NSGA-II method, optimizes the specimen’s structural size, resulting in a 21.6% reduction in mass and an 18.60% decrease in acceleration response. These outcomes demonstrate satisfactory optimization results, ensuring accuracy while enhancing the dynamic characteristics of the specimen structure. This research offers valuable insights for related engineering applications.

Anisotropic error analysis of weak Galerkin finite element method for singularly perturbed biharmonic problems

We consider the weak Galerkin finite element approximation of the singularly perturbed biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent H2− norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.

A Gauss kernel non-local stress-driven plate theory

This study presents a novel stress-driven formulation, incorporating Kirchhoff’s plate assumptions and accounting for the non-local size-dependent feature in both plate geometric axes, which is practical, especially near the origin of the axis in the polar coordinates. The presented formulation does not impose any additional constraints on the elastic radial curvatures, which are known as constitutive boundary conditions. Therefore, taking into account the non-local behavior in both radial and circumferential directions does not lead to an overconstrained problem. The study focuses on the analysis of a circular plate with an outer radius (R), thickness (h), and the length scale parameter (L). The governing differential equations are derived in Cartesian coordinates and then transformed to the polar coordinates to analyze circular plates. When R tends to infinity, the governing equation is solved analytically. Using the Galerkin technique, a finite element solution is presented for the analysis of a bounded circular plate with a clamped boundary condition at the outer radius R. The study discusses the size effects and the gradient properties of functionally graded plates in stress-driven non-local theory. It reveals that while increasing the non-local parameter results in a reduction of transverse displacement, the moments remain unaffected.

The equations coupled by Von Karman system with thermoelasticity

The aim of this paper is to study the model of the von Karman evolution coupled to the thermoelastic equation, with rotational inertia and clamped boundary conditions. We establish the existence and uniqueness of a weak solution related to the dynamic model. Towards the conclusion, we employ the finite difference method to approximate the solution to our problem.

Vibrational analysis of polyurethane-sandwiched bridge decks with variable skew angles

Bridge decks are the surface of bridges that carry the weight of the vehicles and pedestrians crossing over them. The design of bridge decks varies depending on the span, traffic volume, and material availability. But nowadays, the need for a sustainable approach is required. So, use of a sustainable material for construction and retrofitting purposes is the need of the hour. In the present study, a novel synthetic material polyurethane has been used in bridges. The study deals with the variation in skew angles to determine the response of the bridge deck. The response of natural frequencies on the bridge deck due to the variation in skewness and thickness of steel are analysed under simply supported and clamped boundary conditions. Further, the bridge deck is sandwiched using steel and polyurethane having different thicknesses, and the responses are recorded. Afterwards, a bridge deck is modelled using polyurethane as a special case, to pursue sustainability and justify the RRR (reduce, reuse, and recycle) concept of waste management. A comparative study is also performed between the isotropic steel deck and sandwiched deck by varying the skewness. The skew angle is varied from 0° to 60° with a difference of 10°, i.e., 0°, 10°, 20°, 30°, 40°, 50°, and 60°. The frequencies of the isotropic steel and sandwiched decks are increasing when the skewness is increased. Also, the decks may be modelled in a way to enhances the vibration behaviour by sandwiching the existing steel decks. The free vibration frequencies of the sandwiched decks are comparable to the steel deck of similar thickness, which shows the use of polyurethane as the core material does not affect the vibrational characteristics of the deck, while at the same time reducing the cost significantly. The research lays the groundwork for the creation of engineering recommendations that practitioners can use.

Modeling and Vibration Analysis of Carbon Nanotubes as Nanomechanical Resonators for Force Sensing.

Carbon nanotubes (CNTs) have attracted considerable attention as nanomechanical resonators because of their exceptional mechanical properties and nanoscale dimensions. In this study, a novel CNT-based probe is proposed as an efficient nanoforce sensing nanomaterial that detects external pressure. The CNT probe was designed to be fixed by clamping tunable outer CNTs. By using the mobile-supported outer CNT, the position of the partially clamped outer CNT can be controllably shifted, effectively tuning its resonant frequency. This study comprehensively investigates the modeling and vibration analysis of gigahertz frequencies with loaded CNTs used in sensing applications. The vibration frequency of a partially clamped CNT probe under axial loading was modeled using continuum mechanics, considering various parameters such as the clamping location, length, and boundary conditions. In addition, the interaction between external forces and CNT resonators was investigated to evaluate their sensitivity for force sensing. Our results provide valuable insights into the design and optimization of CNT-based nanomechanical resonators for high-performance force sensing applications.

Singularities of the hyperbolic elastic flow: convergence, quantization and blow-ups

We study the elastic flow of closed curves and of open curves with clamped boundary conditions in the hyperbolic plane. While global existence and convergence toward critical points for initial data with sufficiently small energy is already known, this study pioneers an investigation into the flow’s singular behavior. We prove a convergence theorem without assuming smallness of the initial energy, coupled with a quantification of potential singularities: Each singularity carries an energy cost of at least 8. Moreover, the blow-ups of the singularities are explicitly classified. A further contribution is an explicit understanding of the singular limit of the elastic flow of λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-figure-eights, a class of curves that previously served in showing sharpness of the energy threshold 16 for the smooth convergence of the elastic flow of closed curves.

Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions

Abstract In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary.

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

A numerical study on tensile strength of low-density Kagome networks made of brittle fibers

In this paper, we present a numerical study on the tensile strength of low-density Kagome networks made of brittle fibers. First, an elastic beam model is employed to analytically predict the effective elastic properties and tensile strength, as well as the critical condition for buckling of the fibers in Kagome networks. A series of finite element analyses are then conducted to simulate the elastic deformation and failure of Kagome networks under tension. The numerical results based on unit-cell models reveal four possible failure modes of the Kagome networks subject to uniaxial tension, summarized in a phase diagram in terms of the relative density and the fiber strength. The pre-buckling failure mode is restricted to cases with relatively high density and low fiber strength. A low-density Kagome network is likely to fail by one of the post-buckling modes, with an effective tensile strength much lower than the prediction by the elastic beam model. For Kagome networks consisting of a large number of unit cells, the effect of boundary conditions on the tensile strength is examined. Under periodic boundary conditions, the effective tensile strength is nearly identical to that predicted by the unit-cell model, independent of the model size. Under a roller boundary condition, with damage initiation near the free edges followed by a diffusive damage progression, the effective tensile strength is lower than that under periodic boundary conditions for the cases of relatively low fiber strengths. Under a clamped boundary condition, the effective tensile strength is higher than that under periodic boundary conditions for the cases of relatively high fiber strengths, where fiber buckling is largely suppressed by the clamped boundaries. Finally, the effect of a crack-like defect on the effective tensile strength is studied for Kagome networks under the clamped boundary condition. With a small defect, the effective strength is nearly independent of the defect size. In contrast, with a relatively long defect, the effective strength decreases almost linearly with the length of the crack-like defect. The effective toughness for damage initiation and steady-state damage progression in the Kagome networks is discussed from an energetic perspective.

Existence and convergence of the length-preserving elastic flow of clamped curves

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long-time existence and proving a constrained Łojasiewicz–Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

Hybrid isogeometric-based analysis and experimental investigation on the dynamic response characteristics of a clamped circular plate partially in contact with fluid

The aim of this study was to analyze the effect of fluid presence on the dynamic response characteristics of the thin circular plate and the distortions associated with the wet mode shapes. To this end, series of numerical calculations and experimental measurements were performed to investigate the free vibration characteristics of clamped circular plate under in-vacuo conditions and when it is partially in contact with fluid. In the experimental studies, the clamped boundary conditions are imposed on the circular end plate of the horizontal rigid cylindrical tank by closely-spaced bolts, and measurements were performed based on roving hammer impact technique. The proposed numerical approach was divided into two parts based on linear hydroelasticity theory. In the first stage, the thin circular plate is considered to be under in-vacuo conditions, and the mathematical model for this problem is developed based on the isogeometric finite element method (IGAFEM). The fluid environment is introduced in the second stage of the study in which the generalized in-vacuo modal displacements constitute the boundary conditions of the potential flow problem. The influence of fluid medium is incorporated in the system of equations in the form of fluid added mass, and the corresponding fluid forces are calculated by the isogeometric boundary element method (IGABEM). It is observed that the fluid presence has significant effects on the dynamic response characteristics of the test structure, and the specifically, the distortions of wet mode shapes were noticeable due to presence of free surface of the water. Overall, it is found that the natural frequencies and corresponding mode shapes obtained by conducted experiments and adapted numerical framework are in favorable agreement.

Exact solutions for clamped spherical and cylindrical panels via a unified formulation and boundary discontinuous method

Numerous works in both recent and historical literature have concentrated on formulating theories to perform static analysis on simply-supported shell structures. However, it is worth noting that obtaining analytical solutions for clamped boundary conditions presents a strong challenge. In this paper, closed-form solutions for clamped cross-ply laminated and sandwich shells are achieved by employing a robust and hybrid methodology not previously reported in the literature. The high versatility of the Carrera Unified Formulation (CUF), based on the Equivalent-Single-Layer (ESL) description, is utilized to implement several refined shell theories. The Principle of Virtual Displacements (PVD) is utilized to derive the strong form of the governing equations in terms of displacement variables. As the main novelty, these equations are solved by the Boundary Discontinuous Fourier-based method (BDM) which provides highly accurate analytical solutions. The validity and robustness of the proposed methodology are assessed through a detailed comparison with references available in the open literature, as well as with FEM 3D results obtained with commercial software. Furthermore, the stress recovery technique is exploited to fulfill zero-stress and interlaminar continuity (IC) conditions. The findings might be useful in training artificial intelligence (AI) models, which, for instance, could facilitate the development of digital twin structures.

Analysis of the dynamic response for Kirchhoff plates by the element-free Galerkin method

Dynamic response analysis involves examining structural behavior under dynamic loading conditions. Transient dynamic analysis emerges as a method employed to assess the responses of deformable bodies. Due to the depth analysis of transient responses for thin plates, the associated error analysis work becomes particularly important. In this work, we conduct stability and convergence analysis of the element-free Galerkin (EFG) method for the dynamic response of Kirchhoff plate model. We start by discretizing the thin plate dynamics using the EFG method for the spatial domain and the central difference scheme for the temporal domain. Stability and convergence analysis for transient responses, including deflection and moment responses, are derived similarly to those in two-dimensional transient structural dynamic analysis. The key to the analysis is transforming the governing equations and clamped boundary conditions into a weak formulation with penalty terms using the penalty method. The main results demonstrate a significant correlation between the error estimate and both nodal spacing and time step size. Additionally, the continuity of the approximation functions and the selection of penalty factors significantly affect numerical accuracy. To validate the theoretical results, we conduct numerical experiments involving clamped square and L-shaped plates with uniform and nonuniform node distributions, as well as a perforated plate model. Furthermore, we investigate the impact of node influence domains and penalty factors on numerical accuracy, facilitating parameter selection for practical engineering applications.

The Behavior of Long Thin Rectangular Plates under Normal Pressure-A Thorough Investigation.

Thin rectangular plates are considered basic structures in various sectors like aerospace, civil, and mechanical engineering. Moreover, isotropic and laminated composite plates subjected to transverse normal loading and undergoing small and large deflections have been extensively studied and published in the literature. Yet, it seems that the particular case of long thin plates having a high aspect ratio appears to be almost ignored by various scholars despite its engineering importance. The present study tries to fill this gap, yielding novel findings regarding the structural behavior of long thin plates in the small- and large-deflection regimes. In contrast to what is normally assumed in the literature, namely that a long plate with a high aspect ratio can be considered an infinitely long plate, the present results clearly show that the structural effects of the ends continue to exist near the remote ends of the long plate. An innovative finding is that long plates would (only on movable boundary conditions for the large-deflection regime) exhibit a larger mid-width displacement in comparison with deflections of infinitely long plates. This innovative higher deflection appears for both small and large-deflection regimes for both all-around simply supported and all-around clamped boundary conditions. This new finding was shown to be valid for both isotropic and orthotropic materials and presents a novel engineering approach for the old assumption well quoted in the literature that a relatively long plate on any boundary condition can be considered an infinite plate. Based on the present research, it is recommended that this assumption should be used carefully as the largest plate mid-deflection might occur at finite aspect ratios.

Vibration and buckling analysis of porous metal foam thin-walled beams with closed section

This paper investigates the vibration and buckling analysis of porous metal foam thin-walled box beams. These beams exhibit a unique structural configuration with symmetrical and asymmetrical porosity distributions along their wall thickness, thereby altering the effective mechanical properties. The first-order shear deformable beam theory is employed and the governing equations are derived using the Hamilton’s principle. Numerical results are presented for porous metal foam thin-walled box beams under simply-supported, clamped–clamped and clamped-free boundary conditions. The effects of various porosity parameters, length-to-side and side-to-wall-thickness ratios on the beams’ performance are also examined. A comprehensive comparison between the porous metal foam thin-walled box beams and their counterparts in the form of equivalent homogeneous are presented.

Numerical analysis of optimal control problems governed by fourth‐order linear elliptic equations using the Hessian discretization method

AbstractThis article focuses on the optimal control problems governed by fourth‐order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretization method (HDM). The HDM is an abstract framework that enables the convergence analysis of several numerical methods such as the conforming finite element methods, the Adini and Morley nonconforming finite element methods (ncFEMs), the method based on gradient recovery (GR) operators, and the finite volume methods (FVMs). Basic error estimates and superconvergence results are established for the state, adjoint, and control variables. A companion operator for the GR method with specific property is designed. The article concludes with numerical results that illustrate the theoretical convergence rates for GR method, Adini ncFEM, and FVM.

A unified mixed finite element method for fourth-order time-dependent problems using biorthogonal systems

This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper.