Clamped Boundary Conditions Research Articles

Torsional vibration behavior of a restrained non-circular nanowire in an elastic matrix

This study introduces an approach to analyze torsional vibration in non-circular nanowires within magnetic fields, considering various boundary conditions on an elastic foundation. Analytical formulas for natural angular frequencies are obtained using nonlocal strain gradient theory. The analysis covers three non-circular cross-sections, incorporating the warping effect. Elastic springs at the wire ends simulate support conditions, restricting rotation around the wire’s axis. The torsion function around the axis is represented by a Fourier series, discretized at the spring points and linked using the Stokes’ transform alongside the boundary values. This leads to an eigenvalue problem that includes higher-order material size parameters (strain gradient, nonlocal), spring parameters, and the warping function. The study’s novelty lies in effectively solving torsional vibration for non-circular sections, addressing warping function, elastic medium, and size effects under both deformable and non-deformable boundary conditions. The presented solution is capable of determining vibration frequencies for both rigid and deformable boundary conditions. This is accomplished by specifying the torsional spring stiffness values, thereby obviating the necessity for additional recalculations. In order to verify the obtained results and compare them with the existing literature, nanowires with free and clamped boundary conditions are solved numerically by changing the spring parameters in the eigenvalue problem. In the formulation of natural angular frequency; length scale parameters, warping, magnetic field, elastic medium effects are included. In addition, since the support conditions are modeled with elastic springs, the resulting formulas are quite general and can be used to solve different types of torsional vibration problems.

Static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix

A higher-order shear deformable model is extended in this paper for static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix. Principle of virtual work is used to derive governing equations of static bending through the computation of strain energy and external work. A higher-order shear deformable model is used to develop the kinematic relations. The sandwich cylindrical panel is subjected to mechanical and thermal loads. The governing equations are solved using Eigenvalue–Eigenvector method for clamped–clamped boundary conditions. The eigenvalues of the characteristic equation are obtained, and the solution is obtained for real and complex roots. The boundary conditions are applied to obtain integration constants. The axial and radial variation of the displacement, strain, and stress components are presented in terms of significant input parameters such as volume fraction, and folding angle of the graphene origami, various distribution patterns, porosity coefficient, thermal and mechanical loads, and geometric parameters. The results indicate that the stress, strain, and deformation components are increased with an increase in porosity coefficient and folding angle of the graphene. Furthermore, an enhancement in the volume fraction of the graphene leads to a significant decrease in various stress, strain, and deformation components. The results of this work can be used in aerospace science and technology specially in airplanes.

Observations of how boundary conditions affect estimates of Poisson's ratio from simulated mode shapes

This research extends a method for direct estimation of Poisson's ratio from mode shapes of plates with hinged-hinged boundary conditions to include combinations of clamped and free boundary conditions. Using finite element simulations, we observe the effects of boundary conditions on the estimate of Poisson's ratio using the combination of Time-Average Scanning Digital Holography and Cornu's method. First, we introduce the abstraction of the span-to-width ratio to an effective antinode region. This abstraction enables us to extend the direct estimation technique to plates with various boundary conditions. Second, for long and thin enough plates, the effects of boundary conditions become negligible and Cornu's method is sufficient for direct estimation of the true value of Poisson's ratio from an out-of-plane bending mode shape. We note that for many situations, the necessary plate dimensions could be a limitation. Therefore, as a third concept, we discuss the behavior of the direct estimate of Poisson's ratio with respect to each type of boundary condition. Ultimately, this research initiates a path forward for a common processing technique for making a direct estimate of Poisson's ratio across different types of opposing boundary conditions for thin plates.

REFORMULATED STRAIN GRADIENT (RSG) ELASTICITY THEORY FOR FREE VIBRATION OF THERMAL BI-DIRECTIONAL FG MICROBEAM

This research examines the vibrational response of a micro-scale Euler beam made from two-directional functionally graded (2D-FG) materials and subjected to thermal effects. By employing a reformulated strain gradient elasticity (RSGE) approach, the equations of motion using Hamilton’s principle for clamped -clamped and clamped-simply boundary conditions are derived and solved them using Galerkin's approach. The investigation explores the impact of temperature, gradient index, and parameters length scale materials on the bidirectional graded microbeam's dynamic characteristics. Furthermore, the normalized frequency, as based on the current reformulated strain gradient elasticity microbeam model, consistently emerges as higher than that derived from the classical model.

Elastic analysis of functionally graded porous cylinders under uniform internal pressure by using shear deformation theory

This research aims to analyze cylindrical shells made of functionally graded porous materials, focusing on three types of monotonous, symmetric, and nonsymmetric porosity distributions. Material properties are defined for different porosity distributions, individually. Due to the slight variation of Poisson’s ratio, it is assumed to be constant throughout the thickness of the cylinder. The governing equations are derived using the first-order shear deformation theory and the virtual work principle. The governing equations are solved using the eigenvalue–eigenvector method for clamped–clamped boundary conditions. The solution’s accuracy and convergence are validated by conducting a comparative study. Finally, the numerical results include the components of displacement, axial displacement, radial displacement, circumferential stress, radial stress, and shear stress are reported according to porosity coefficient and different porosity distributions.

Determination of notch stress intensity factors under mode I loading using the 3D-DIC and finite element over-deterministic methods

In this paper, the fracture behaviour of Single Edge Notched Tension Specimens (SENT) is investigated experimentally and numerically using the three-dimensional digital image correlation (3D-DIC) experimental method and the Finite Element Analysis (FEA) technique, respectively. The mode I notch stress intensity factors (NSIFs) for the SENT specimens are derived from the measured and simulated displacement fields using the Finite Element Over-Deterministic (FEOD) method. The step-by-step process of a 3D-DIC experimental system for the SENT specimens is described. The SENT specimens were tested with pinned and clamped boundary conditions (BCs) and different notch initial opening angles. There are three main contributions of the paper: (1) the derivation of the NSIFs under Mode I Loading for thin plates subjected not only to the common pinned BCs, but also to the rare, clamped BCs; (2) the application of the FEOD method to derive the NSIFs from the displacement fields measured using the 3D DIC technique and predicted using FEA; (3) the validation of the NSIFs derived numerically via experimentally-determined NSIFs for notched thin plates subjected to both pinned and clamped BCs. Thus, four sets of NSIF data are provided in the paper and enable confident application of the NSIF values for the conservative (pinned BCs) and light-weight (clamped BCs) design of notched thin plates.

Assessment of machine learning methods predicting the axial vibration frequencies of microbars

AbstractMicrobars are one of the important components of microelectromechanical systems. With the recent increase in their applications, the importance of understanding their mechanical response has become an important topic. In this study, for the first time, the mechanical behavior of microbars based on the strain gradient theory is investigated using a machine learning (ML) approach. Four distinct ML models, namely artificial neural network (ANN), support vector regression (SVR), decision tree regression (DTR), and random forest regression (RFR), are developed for microbars with clamped boundary conditions. The performance of these models is individually assessed using five different metrics: coefficient of determination (R2), Mean Absolute Error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE) and Nash‐Sutcliffe efficiency coefficient (NSE). The best‐performing model is selected based on these comparisons. Additionally, a semi‐analytical approach is employed to determine the natural frequencies of microbars under general elastic boundary conditions using the Fourier sine series and Stokes' transform. While the R2 value for all four models indicated a good fit of 0.999, the percentage difference in MAE and RMSE values between the training and testing data for DTR and RFR models was relatively higher as compared to ANN and SVR. The results showed that ANN and SVR models exhibit the best performance in predicting the natural frequencies on both training and testing data across all three metrics. Finally, a study on the free axial vibration frequencies of microbar under various effects was conducted.

Bending Behavior of Clamped Skew Plates

The deflection analyses performed using MSC/NASTRAN on skew plates under different loads such as (concentrated and uniformly distributed loads) for clamped boundary conditions are presented in the present study. The CQUAD4 and CQUAD8 components of MSC/NASTRAN are verified using values from the literature. The CQUAD8 element was employed in the present experiments because it had been found to produce more effective results than the CQUAD4 element. For isotropic skew plates, the variation of deflection with regards to aspect ratio and length to thickness ratio is provided. With an increase in the skew angle, it appears that the deflections decrease.

Design of manufacturable variable stiffness composite laminates using spectral Chebyshev and normalized cut segmentation methods

The in-plane fiber orientations of variable stiffness (VS) laminates can be tailored to achieve enhanced structural properties compared to conventional constant stiffness (CS) laminates. However, VS laminate manufacturing faces challenges such as wrinkles, gaps, and overlaps. To address these challenges, we present a novel three-step design methodology. First, the laminate is modeled using lamination parameters (LPs) and the spectral Chebyshev method, and the optimal LPs are determined to maximize the fundamental frequency. Then, the discrete fiber angles are retrieved using the optimal LP distribution. Lastly, a normalized-cut segmentation method is applied to divide the domain into clusters and to generate manufacturable curvilinear fiber paths. Case studies focusing on designing clusters containing both straight and curvilinear fiber paths demonstrate that the designed VS composites can significantly enhance the dynamic performance with up to 20% enhancement in the fundamental frequency compared to CS laminates, under fully clamped boundary conditions with manufacturing constraints.

A separation‐of‐variable method for eigenbuckling analysis of closed and open circular cylindrical shells

AbstractThe analytical solutions for buckling of closed and open cylindrical shells under different boundary conditions are challenging. A separation‐of‐variable method with closed‐form eigensolution under Donnell–Mushtari theory is proposed for eigenbuckling analysis in cylindrical shells. The modal function is adopted as the form W = eμαeλβ. It can be applied to the closed and open cylindrical shells with different boundary conditions, and the whole calculation process is explicit. The critical buckling capacities of short cylindrical shells are close under the simply supported or clamped boundary conditions. The predictions are in good agreement with those in literature and the finite element model.

Vibroacoustics of a membrane-finite cavity resonator: Traditional and alternative approaches for analysis

This paper investigates a vibroacoustic model consisting of a flexible rectangular membrane backed by an acoustic air cavity of large lateral extent. A traditional methodology (termed Approach I) for analyzing the interaction between the acoustic airfield in the cavity and the flexible membrane is initially revisited and employed. In this approach, the transversal deformation of the mid-membrane is represented as an infinite series of eigenfunctions (normal modes) that satisfy the clamped boundary conditions for the in-vacuo membrane, regardless of the presence of the cavity. However, numerical challenges and limitations are encountered in obtaining direct insights into system characteristics, as this method heavily relies on matrix inversions and computing the roots of determinants. These operations can be computationally expensive and prone to inaccuracies and numerical instabilities, particularly when dealing with high-dimensional, non-sparse (dense) and nearly singular matrices. To overcome these limitations, an alternative methodology (termed Approach II) is proposed, which enables a precise analysis of the acoustically forced membrane and provides explicit expressions for the steady-state response, mode shapes, governed by a single transcendental characteristic equation for determining the vibroacoustic natural frequencies. Through a comparative analysis, we demonstrate the effectiveness and reliability of the proposed approach, showcasing its potential advantages over the traditional method in addressing numerical challenges. The influence of cavity depth on the vibroacoustic natural frequencies is examined across various levels of membrane stiffness, including low, intermediate, and high values. The analysis reveals the presence of both softening and hardening effects with respect to the in-vacuo case, suggesting that the air pressure plays a dual role in the system, i.e., influencing both the overall stiffness and inertia. The vibroacoustic membrane mode shapes exhibit significant deviations for soft and intermediate-stiff membranes but resemble in-vacuo mode shapes for stiff membranes, with the resemblance dependent on the selected cavity depth. Additionally, two qualitatively distinct mode families for the acoustic pressure, namely, propagating, and localized modes, are identified. The propagating modes are spatially extended and are characterized by the deformation of the volume of the medium, rendering them volume modes. In contrast, the localized modes are distinguished by localized vibrations occurring in the near field close the membrane surface, without significant energy transport or sound wave propagation in the cavity; as a result, they can be regarded as surface modes. The proposed methodology reveals features common to more complex vibroacoustic systems and offers a comprehensive and efficient approach for predictively designing the dynamics of such systems.

Size effects in a power law creeping layer under compression or shear, and implications for deformation mechanisms of lithium films

The axisymmetric compression of a power law creeping metallic sandwich layer of micron-scale thickness is analysed. Account is taken of the elevation in flow strength due to the presence of a spatial gradient in plastic strain rate. Numerical and analytical solutions reveal that the average compressive traction is enhanced by a combination of strain rate gradients and plastic constraint. A similar size effect is predicted for simple shear of the creeping sandwich layer. The difference in responses for compression and shear is traced to the different profiles of shear strain rate through the thickness of the layer. The sensitivity of compressive and shear strengths to the choice of higher-order boundary condition is explored, and good agreement with recent experiments on compression and shear of a thin sandwich layer of lithium is achieved by assuming fully clamped higher-order boundary conditions and a material length scale on the order of 3−5μm in the strain gradient-based creep theory.

MODAL BEHAVIOUR OF LONGITUDINALLY PERFORATED NANOBEAMS

Nano-electro-mechanical systems (NEMS) require perforated beams for structural integrity. Hole sizes, hole numbers, and scale effects need to be modelled appropriately in their design. This paper presents a new finite element model to investigate the modal behaviour of longitudinally perforated nanobeams (LPNBs) using the classical Euler–Bernoulli beam theory. A symmetric array of holes arranged parallel to the length direction of the beam with equal spacing was assumed for the perforation. The non-local Eringen’s differential form was used to incorporate the nanoscale sizes. The accuracy of the proposed model was verified by comparing the obtained results with the available analytical solutions for fully filled nanobeams. The effects of aspect ratios, non-local parameters, boundary conditions, and perforation characteristics on the modal behaviour of LPNBs were investigated. The non-local parameter reduced the natural frequency owing to a decrease in the stiffness of the structures. However, the perforation filling ratio led to higher values of the fundamental frequency. Furthermore, compared with other boundary conditions, clamped–clamped boundary conditions demonstrated the best performance in terms of the maximum frequency.

Nonlinear dynamics response of porous functionally graded annular plates using modified higher order shear deformation theory

In this investigation, the nonlinear dynamics of porous annular plates that have been functionally graded are examined under varying time-dependent loads. Both simple supported and fully clamped boundary conditions are taken into account. The mechanical properties of the functionally graded plate in its thickness are considered according to the distribution function of the modified law. In addition, porosity, as a stress-relieving property, is also applied throughout the plate thickness by different functions and pore volume fractions. By adopting Hamilton’s principle, the equations of motion are obtained based on the modified higher-order shear deformation plate theory. Then these partial differential equations are solved using the viscous dynamic relaxation method in conjunction with Newmark’s implicit integration method. The present findings are compared and successfully confirmed with those available in the literature. Finally, the effects of some key factors such as porosity distribution type, pore volume fraction, power-law index, loading conditions, and thickness-to-radius ratio on the dynamic behavior of both simple and fully clamped plates have been studied in detail. The study’s findings indicate that porosity pattern X, characterized by a greater prevalence of interconnected pores, exhibited enhanced resistance to deflection. Conversely, pattern O, featuring a more uniformly distributed pore size, effectively mitigated stress in FG porous materials. These observations offer valuable insights for optimizing plate design and elevating its performance.

Vibrational frequency of triangular plate having circular thickness

In the current research, modes of frequency of isotropic tapered triangular plate having 1-D (one dimensional) circular thickness and 1-D (one dimensional) linear temperature profile for clamped boundary conditions are discussed. Authors implemented Rayleigh Ritz technique to solve the frequency equation of isotropic triangular plate and computed the first four modes with a distinct combination of plate parameters. Authors have performed the convergence study of modes of frequency of the isotropic triangular plate. Also, conducted comparative analysis of modes of frequency of the current study with available published papers and the results presented in tabular form. The aim of the present study is to show the impact of a one dimensional circular thickness and one dimensional linear temperature on modes of frequency of vibration of an isotropic tapered triangular plate.

Boundary Discontinuous Fourier analysis of clamped isotropic and cross-ply laminated plates via Unified Formulation

This paper presents an analytical solution for the static analysis of plates with clamped boundary conditions prescribed at the edges. The displacement field is expressed via the Carrera Unified Formulation (CUF) where an Equivalent-Single-Layer (ESL) approach is adopted. The governing equations are obtained by employing the principle of virtual displacements (PVD) statement. The main novelty is the use of the boundary-discontinuous Fourier-based approach to provide accurate numerical solutions. From thick to thin isotropic, cross-ply laminated and sandwich plates with different side-to-thickness ratios and stacking sequences are studied. Furthermore, the out-of-plane stresses are calculated via both the constitutive relation and the stress recovery technique. The accuracy of the proposed solution is verified by comparing the numerical results obtained with those from the literature and 3D FEM solutions. The present approach seems capable of handling not just fully clamped conditions but also mixed external conditions, which may include clamped and simply-supported edges. The solution approach provided in this article is unique, hence the proposed results might be useful as a benchmark for validating new plate theories and finite elements.

Elastic graphs with clamped boundary and length constraints

We study two minimization problems concerning the elastic energy on curves given by graphs subject to symmetric clamped boundary conditions. In the first, the inextensible problem, we fix the length of the curves while in the second, the extensible problem, we add a term penalizing the length. This can be considered as a one-dimensional version of the Helfrich energy. In both cases, we prove existence, uniqueness and qualitative properties of the minimizers. A key ingredient in our analysis is the use of Noether identities valid for critical points of the energy and derived from the invariance of the energy functional with respect to translations. These identities allow us also to prove curvature bounds and ordering of the minimizers even though the problem is of fourth order and hence in general does not allow for comparison principles.

A mixed finite element method using a biorthogonal system for optimal control problems governed by a biharmonic equation

In this article, we consider an optimal control problem governed by a biharmonic equation with clamped boundary conditions. We use the Ciarlet--Raviart formulation combined with a biorthogonal system to obtain an efficient numerical scheme. We discuss the a priori error analysis and present results of the numerical experiments that validate the theoretical estimates.

A unified weighted inequality for fourth-order partial differential operators and applications

In this paper, we establish a fundamental inequality for fourth order partial differential operator P=△α∂s+β∂ss+Δ2 (▪) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator P with some suitable boundary conditions in the case of β<0 or α≠0,β=0. As application, we obtain a resolvent estimate for P, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.

Unraveling thermoelastic performance of the multi-directional functionally graded composite structure by data-driven deep-learning approach

Multi-directional functionally graded materials (MD-FGMs) are a brand-new family of sophisticated composites with continually changing structural and mechanical properties but no internal boundaries and interfacial stress concentrations. This study is the first to examine the effects of an instantaneous thermal shock on the thermo-elastodynamic response of a multi-directional functionally graded (MD-FG) doubly curved panel resting on an elastic substrate due to the remarkable properties of these advanced materials. The peripheral discontinuity at the boundaries between the loaded and unloaded surfaces as well as the shear coupling between the various Winkler-spring components are both taken into account in this proposed mode of the Kerr-type elastic substrate. Through the use of 3D-DQA, the system’s equations may be solved for both the simply supported and the completely clamped boundary conditions. For the first time, it is examined how several thermal shock models, such as cosinusoidal, sinusoidal, and Heaviside functions, impact how the system’s stress and deflection parameters fluctuate. The authors of this paper provide a unique strategy for enhancing the computing efficiency of the approach by constructing a deep-learning-based method on the data produced by 3D-DQT and Laplace transform at specified locations due to the dynamic nature of the issue. In order to decrease the mechanical excitation of the existing curved structure brought on by thermal shock, some advice for relevant industries is offered at the conclusion.