Element-free IMLS-Ritz Method Research Articles

Surface crack effect on frequency and vibration mode switching of solar-powered aerospace structures

In the engineering design of modern wing panels and their optimal control system, the first step is to find and predict their dynamic characteristics. These characteristics are dependent on material properties, panel geometry, and boundary conditions. In addition, possible fracture existence, imperfections and environmental conditions may unpredictably change the frequency and modes shape of the panel and then the parameters of a control system. In the present paper, the effect of surface fracture depth and orientation on frequency and vibration mode switching of perovskite solar cells-based aircraft wing panels is investigated for the first time. The PSC (perovskite solar cell) panel is made of a polymer substrate reinforced with even-distributed graphene platelets (GPLs) and thin solar layers to enhance the overall electromechanical performance of the whole structure. Equations of motion of the panel are obtained by the Hamilton approach based on the properties of Reddy's third-order shear deformation theory and linear components of the Green-Lagrange strain tensor. The novel aspect is a line-spring model of surface crack by Cartmell et al. adapted to the model of the PSC panel under consideration. The discretization of the established equations of motion for the cracked panel is carried out by the element-free IMLS-Ritz method with mesh-free features. Validation of the obtained solutions is carried out by comparisons with results from previous studies for cracked laminated plates.First, material and geometric properties effect on vibration characteristics of the PSC structure was investigated. Next, partial surface crack depth and orientation angle are examined to show changes in frequency and vibration modes shape of the panel. It is shown, that there is a mode swapping between the second and third modes when transitioning from an intact panel to a cracked one. As the crack length increases, higher-order mode variations become more noticeable, potentially leading to their swapping or the excitation of new ones. Additionally, the findings demonstrate that the surface crack orientation angle significantly impacts the higher-order modes of the panel. When the orientation angle is 90°, the first six modes align with those of the intact plate. Mode shapes and frequency shifts are significant for crack orientation angles between 0° and 90°.

On the flutter of matrix cracked laminated composite plates reinforced with graphene nanoplatelets

This paper investigates graphene nanoplatelets (GPLs) reinforced composite (GPLRC) with matrix cracks by using element-free IMLS-Ritz method. The effective Young's modulus of each GPLRC layer is determined by the modified Halpin-Tsai micromechanics model while its Poisson's ratio and mass density are predicted according to the rule of mixtures. The degraded stiffness of cracked layers is modeled via the self-consistent micromechanical model. The first-order shear deformation theory and first-order piston theory are employed to formulate aeroelastic model. Element-free IMLS-Ritz method is applied to discretize the equation of motion. The accuracy of the IMLS-Ritz results is examined by comparing the natural frequency and critical aerodynamic pressure with those obtained from published values. A comprehensive parametric study is carried out, with a particular focus on the effects of matrix crack density, distribution pattern, weight fraction, total number of layers, geometry of GPLs, and aspect ratio of plates on the flutter bound of matrix cracked GPLRC plates.

Flutter analysis of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method

This paper investigates the flutter characteristics of graphene nanoplatelet (GPL) reinforced laminated composite quadrilateral plates using the element-free IMLS-Ritz method. The modified Halpin-Tsai model and rule of mixture are employed to predict the effective material properties including Young's modulus, mass density and Poisson's ratio. The energy functions of GPL reinforced composite (GPLRC) quadrilateral plates are obtained by the first-order shear deformation theory (FSDT) and first order piston theory. Based on the IMLS-Ritz approximation, the discrete dynamic equation of GPLRC quadrilateral plates is derived. The accuracy of the IMLS-Ritz results is examined by comparing the natural frequencies with those obtained from published values. A comprehensive parametric study is carried out, with a particular focus on the effects of weight fraction, distribution pattern, total number of layers, geometry and size of GPL reinforcements and geometric parameters of quadrilateral plates on the flutter boundary of GPLRC quadrilateral plates.

Geometrically nonlinear analysis of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method

This paper investigates the nonlinear bending of graphene nanoplatelet (GPL) reinforced laminated composite quadrilateral plates using the element-free IMLS-Ritz method. The effective material properties including Young's modulus, mass density and Poisson's ratio are determined by the modified Halpin-Tsai model and rule of mixture. The first-order shear deformation theory (FSDT) and the IMLS-Ritz approximation are employed to obtain the discrete nonlinear governing equation of quadrilateral plates with large deformation. The Newton-Raphson method is used to solve the nonlinear equation. The accuracy of the IMLS-Ritz results is examined by comparing with the published values. A comprehensive parametric study is carried out, with a particular focus on the effects of geometric parameters of quadrilateral plates and GPLs distribution pattern, weight fraction, total number of layers, and geometry and size of GPLs on the nondimensional deflection of GPLs reinforced laminated composite quadrilateral plate.

Vibration of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method

This paper investigates free vibration of graphene nanoplatelet (GPL) reinforced laminated composite quadrilateral plates using the element-free IMLS-Ritz method. The effective material properties including Young's modulus, mass density and Poisson's ratio are determined by the modified Halpin–Tsai model and rule of mixture. The first-order shear deformation theory (FSDT) is employed for formulation of the energy functional. Based on the IMLS-Ritz approximation, the discrete vibration equation of the laminated composite quadrilateral plates is derived. The accuracy of the IMLS-Ritz results is examined by comparing with the published values. A comprehensive parametric study is carried out, with a particular focus on the effects of weight fraction, distribution pattern, geometry and size of GPL reinforcements, total number of layers and geometric parameters of quadrilateral plates on the natural frequencies of GPL reinforced laminated composite quadrilateral plate.

A multiscale framework for large deformation modeling of RBC membranes

In the present contribution, a multiscale framework for nonlinear analysis of finite deformation of red blood cell (RBC) membrane is developed. The first-order Cauchy–Born rule is adopted to establish an atomistic enriched hyperelastic constitutive model and to develop macroscale stress–strain relation of the RBC membrane. In order to circumvent the inherent limitations of utilizing mesh-based methods for large deformation analysis, we systematically coupled the 3D multiscale scheme with the element-free IMLS-Ritz method for numerical modeling of RBC deformability by simulating the optical tweezers experiment. This development was partly motivated by the need for a more precise scheme for modeling membrane structures. The effectiveness of the proposed approach is affirmed by the better prediction of RBC membrane deformability in comparison with experimental and numerical results found in literature and a significant reduction in computational cost. Our approach enables precise characterization of the effect of varying microstructure parameters, physiological, and osmolality conditions on the deformability of RBC membrane.

Mechanical behavior of laminated CNT-reinforced composite skew plates subjected to dynamic loading

Abstract This paper presents the first known mechanical behavior of laminated CNT-reinforced composite skew plates subjected to a transverse sudden dynamic load. The plate is composed of multilayers of nanocomposite reinforced with single-walled carbon nanotubes (SWCNTs). The problem is formulated using the first-order shear deformation theory (FSDT), and solution to the problem is obtained through the element-free IMLS-Ritz method. The elastodynamic behavior is furnished by employing the Newmark-β method. Material properties of CNT-reinforced composites are predicted through the Mori–Tanaka approach. The stability and precision of the IMLS-Ritz method are validated by convergence and comparison studies. The effects of skew angles, width-to-thickness ratio, CNT-volume fraction, CNT-distribution along the layer thickness, CNT-fiber orientation and boundary conditions on the elastodynamic responses of laminated CNT-reinforced composite skew plates are examined.

An element-free based IMLS-Ritz method for buckling analysis of nanocomposite plates of polygonal planform

The buckling behavior of nanocomposite plates of polygonal planform under in-plane loads is examined. The plate under consideration is reinforced by single-walled carbon nanotubes (CNTs). The governing eigenvalue equation to this problem is derived based on the first-order shear deformation plate theory (FSDT) with a set of element-free shape functions in approximating the two-dimensional displacement fields. To solve this eigenvalue equation, the element-free IMLS-Ritz method is employed to furnish the buckling solution. The convergence of the solution for the CNT-reinforced composite plates is examined. Comparison study is further carried out to validate the accuracy of the solution. A parametric study is performed by varying the CNT volume fraction, CNT distribution, plate thickness-to-apothem ratio and boundary conditions. This first known buckling solution may serve as benchmarks for future research.

Elastodynamic analysis of regular polygonal CNT-reinforced composite plates via FSDT element-free method

A comprehensive study on elastodynamic behavior of regular polygonal carbon nanotube (CNT) reinforced functionally graded composite plates subjected to a sudden transverse dynamic load is presented in this paper. This analysis uses as its base the first-order shear deformation theory (FSDT) and the element-free IMLS-Ritz method, to conduct a numerical simulation. A numerical time integration for the dynamic problems is performed using the Newmark-β method. Convergence studies are carried out for an isotropic square plate with different support sizes and different numbers of nodes. Parametric studies are further carried out to investigate the effect of four special regular polygons with different CNT volume fractions, CNT distribution types, plate apothem-to-thickness ratios and boundary conditions on the dynamic behaviors of regular polygonal CNT-reinforced composite plates.

Geometrically nonlinear large deformation of CNT-reinforced composite plates with internal column supports

Abstract The geometrical nonlinear analysis of internally supported nanocomposite plates subjected to a uniformly distributed load is carried out. This study investigates the effects of internal point/column supports on the large deformation bending of nanocomposite plates reinforced by carbon nanotubes (CNTs) with different types of distributions, namely, uniform and two kinds of functionally graded distributions through the thickness of the plates. Two-dimensional displacement field of the plate is approximated by a set of Improved Moving Least Squares (IMLS) functions. The arc-length iterative algorithm with the modified Newton method is employed to obtain the nonlinear response of nanocomposite plates. Convergence studies indicate the validity and effectiveness of the element-free IMLS-Ritz method. The effects of plate thickness-to-width ratio, volume fraction ratio, and plate aspect ratio on the large deformation behavior of nanocomposite plates under various boundary conditions are examined. To the best of the authors’ knowledge, the problem has not been attempted in the open literature.

Vibration analysis of CNT-reinforced thick laminated composite plates based on Reddy’s higher-order shear deformation theory

In this paper, free vibration behavior of carbon nanotube (CNT) reinforced functionally graded thick laminated composite plates utilizing Reddy’s higher-order shear deformation theory (HSDT) is studied. To the best of authors’ knowledge, this paper is the first to incorporate HSDT with one of the element-free approaches to investigate this issue. The element-free IMLS-Ritz method is employed and four types of CNT distributions are considered. The resulting effective material properties of the CNT-reinforced composite are estimated by a detailed and straightforward Mori–Tanaka approach. The numerical results have been compared with the literature showing excellent agreement. Considering various CNT orientation angles, a parametric study showing the effects of CNT volume fraction, plate aspect ratio, plate width-to-thickness ratio and number of plate’s layers on the non-dimensional natural frequencies is investigated. Finally, the influence of boundary conditions on the sequence of the first six mode shapes for various lamination arrangements is presented.

Geometrically nonlinear large deformation analysis of triangular CNT-reinforced composite plates

A first known investigation on the geometrically nonlinear large deformation behavior of triangular carbon nanotube (CNT) reinforced functionally graded composite plates under transversely distributed loads is investigated. The analysis is carried out using the element-free IMLS-Ritz method. In this study, the first-order shear deformation theory (FSDT) and von Kármán assumption are employed to account for transverse shear strains, rotary inertia and moderate rotations. A convergence study is conducted by varying the supporting size and number of nodes. The effects of transverse shear deformation, CNT distribution and CNT volume fraction on the nonlinear bending characteristics under different boundary conditions are examined.

Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory

Literature researching the active vibration control of functionally graded material (FGM) plates with piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) using any of the element-free methods does not exist. To the best of the authors’ knowledge, this paper is the first to use Reddy’s HSDT with the element-free IMLS-Ritz method to investigate this problem. In this study, seven mechanical degrees of freedom (DOF) and one additional electrical DOF are considered for each node of the discretized domain. The natural frequency results of two FGM plates with top and bottom piezoelectric layers are compared with the literature in terms of various electrical and mechanical boundary conditions, volume fraction exponent (n) and dimension ratios, with obvious agreement. Furthermore, parametric studies are performed, for the first time, to study the effects of mechanical boundary conditions, n value, FGM plate thickness-to-width ratio and piezoelectric layer thickness to FGM plate thickness ratio on the natural frequency increment between open and closed circuit conditions. For the purpose of active vibration control, a constant velocity feedback approach is utilized. The effectiveness of two proposed positions, of piezoelectric sensor and actuator layers, to control the vibration of FGM plates is investigated.

Free vibration analysis of triangular CNT-reinforced composite plates subjected to in-plane stresses using FSDT element-free method

This study conducts the first-known vibration analysis of carbon nanotube (CNT) reinforced functionally graded composite triangular plates subjected to in-plane stresses. The first-order shear deformation theory is applied to study the effect of the transverse shear deformation of the plates and the element-free IMLS-Ritz method is used for numerical computation. Numerical experiments are carried out to examine the convergence and accuracy of the element-free IMLS-Ritz method in its application to the vibration of CNT-reinforced functionally graded composite plates subjected to in-plane stresses. In this study, sets of vibration frequency parameters and mode shapes for various CNT-reinforced functionally graded composite triangular plates are illustrated. The influences of CNT volume fraction, plate thickness-to-width ratio, CNT distribution type, isosceles triangular angle, in-plane stress ratio and boundary conditions on the plate’s vibration behaviors are investigated in detail. These results may serve as a benchmark for future studies.

Elastodynamic analysis of quadrilateral CNT-reinforced functionally graded composite plates using FSDT element-free method

An elastodynamic analysis of quadrilateral carbon nanotube (CNT) reinforced functionally graded composite plates subjected to a transverse sudden dynamic load is studied. Based on the first-order shear deformation theory (FSDT), the element-free IMLS-Ritz method is extended to conduct a numerical simulation for this study. A numerical time integration for the dynamic problems is performed using the Newmark-β method. To verify the stability and accuracy of the element-free IMLS-Ritz method, convergence studies are carried out for an isotropic square plate with different support sizes and different numbers of nodes. Detailed numerical simulations are further carried out to investigate the effect of CNT volume fraction, CNT distribution type, side angles, area of quadrilateral plate, plate width-to-thickness ratio and boundary condition on the dynamic behaviors of quadrilateral CNT reinforced composite plates.

Element-free geometrically nonlinear analysis of quadrilateral functionally graded material plates with internal column supports

A nonlinear deflection analysis is carried out for internal column supported functionally graded material (FGM) arbitrarily straight-sided quadrilateral plates under a uniformly distributed loading. In order to achieve this aim, the formulation of a discrete nonlinear governing equation for large deformation is undertaken based on the element-free IMLS-Ritz method. The first-order shear deformation theory and von Kármán assumption are adopted in the formulation. In this study, the computation involves a geometric transformation from the physical quadrilateral domain to a square computational domain. Several example problems are selected to illustrate the effect of internal column support distributions on the maximum deflection and deformed shape of the FGM plates. In addition, parametric studies on the effect of volume fraction ratio, geometry and length-to-thickness ratio on the large deflection behavior of FGM plates are examined.

Postbuckling of carbon nanotube reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression

Abstract This paper presents, to the authors’ knowledge, a first known postbuckling analysis of carbon nanotube (CNT) reinforced functionally graded plates with edges elastically restrained against translation and rotation. The plate considered is of moderate thickness and, hence, the first-order shear deformation theory (FSDT) and von Karman assumption are adopted to incorporate the effects of transverse shear strains, rotary inertia and moderate rotations. The element-free IMLS-Ritz method is employed. The cubic spline weight function and linear basis are utilized in the approximation. In this study, the bending stiffness is evaluated through the nodal integration scheme. The postbuckling path is traced using the arc-length method combined with the modified Newton–Raphson technique. Parametric studies on the postbuckling behavior of CNT reinforced functionally graded plates are conducted to examine the effects of CNT content by volume, plate width-to-thickness ratio and plate aspect ratio by varying the elastically restrained parameters of translation and rotation on the boundaries. The results of the present study are obtained for simplified cases so that comparison studies can be undertaken using the values reported in the literature.

An element-free IMLS-Ritz method for numerical solution of three-dimensional wave equations

Abstract This paper presents an element-free based numerical approach for solving three-dimensional wave equations based on the Ritz minimization procedure. The method involves the use of a set of orthogonal shape functions to approximate its field variables. In this study, an improved moving least-squares (IMLS) is used to generate sets of orthogonal shape functions that reduced the number of unknown coefficients in the trial functions. The entire approximation procedure can be easily implemented numerically. The accuracy of the approximation can be enhanced by increasing the number of nodes used in the computation. As the result of the above procedures, a final algebraic equation system is derived through discretizing the constructed functional. The functional is established by enforcing the Dirichlet boundary conditions via the penalty approach. The simplicity and applicability of the element-free method are demonstrated by solving several selected linear and nonlinear three-dimensional wave equations. Besides the convergence study, the present results are compared with the available published solutions from the literature, where possible, verifying the accuracy, efficiency and reliability of the IMLS-Ritz method.

Large deflection analysis of FG-CNT reinforced composite skew plates resting on Pasternak foundations using an element-free approach

The first known geometrically nonlinear large deformation analysis of functionally graded carbon nanotube (FG-CNT) reinforced composite skew plates resting on Pasternak foundations is presented. The skew plate studied is of moderate thickness and, hence, the first-order shear deformation theory (FSDT) and von Kármán assumption are adopted to take care of the transverse shear strains, rotary inertia and moderate rotations. The element-free IMLS-Ritz method is employed in the present analysis. Parametric studies are conducted to examine the effects of CNT content by volume, elastic foundation, skew angle, plate width-to-thickness ratio, plate aspect ratio and boundary conditions on the nonlinear responses of the FG-CNT reinforced composite skew plates. The results of the present study are obtained for simplified cases so that comparison studies can be made with the values reported in the literature.

Buckling of FG-CNT reinforced composite thick skew plates resting on Pasternak foundations based on an element-free approach

Buckling behavior of functionally graded carbon nanotube (FG-CNT) reinforced composite thick skew plates is studied. The element-free IMLS-Ritz method is used to obtain the buckling solutions to this problem. The first-order shear deformation theory (FSDT) is employed for formulation of the energy functional to incorporate the effects of transverse shear deformation and rotary inertia. Using the IMLS approximation in the field variables and minimizing the energy functional via the Ritz procedure, a discretized eigenvalue equation of the problem is derived. The buckling solution can be obtained through solving this eigenvalue problem. The numerical stability of the IMLS-Ritz method is validated through convergence studies. The accuracy of the IMLS-Ritz results is examined by comparing with the known solutions. Close agreement is found from the comparison study. Besides, parametric studies are conducted for various types of CNTs distributions, CNT ratios, aspect ratios, plate geometries and thickness-to-height ratios under different boundary conditions.