Nonlinear Plate Bending Research Articles

Numerical and experimental modeling of the behavior of flexible shell elements of structures

This article is devoted to computer and experimental modeling of the behavior of flexible elements of shell structures, and the development of effective algorithms for solving emerging nonlinear boundary value problems of their calculation and optimization of parameters. At the same time, the probability of the results obtained using different approaches to the construction of a nonlinear theory is established. Their comparative analysis, error estimation, which in this case is given by calculation according to linear and corresponding nonlinear theories, is carried out. The results of calculated data and experimental studies of the behavior of real structural elements are compared.
 The results of a comparative analysis of the application of two mathematical models of deformation of flexible shell elements, obtained by direct integration of boundary value problems of shell mechanics, by the finite element method and experimental research, are presented.
 The problems of axisymmetric nonlinear bending of thin ring plates, corrugated membranes of a sinusoidal profile and bellows as a shell of rotation with a complex meridian shape are considered.
 Using the necessary optimality conditions of the principle of maximum L. S. Pontryagin obtained the results of the optimal design of a flexible corrugated membrane with a sinusoidal profile of the highest sensitivity. The results are presented in the form of tables, photos and graphs.

Analysis of Berger nonlinear elastic static plate bending of rectangular plates

This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example.

Modeling and Re‐Examination of Nonlinear Vibration and Nonlinear Bending of Sandwich Plates with Porous FG‐GPLRC Core

In this article, an inhomogeneous model is proposed to predict the material properties of porous GPL‐reinforced composite (GPLRC) and to predict the mechanical responses of a sandwich plate which consists of top and bottom metal face sheets and a porous GPLRC core. The GPLRC core is assumed to be multilayers, and each layer may have different values of porosity coefficient to achieve a piece‐wise functionally graded pattern. Young's moduli along with shear modulus of porous GPLRC core are predicted by a generic Halpin‐Tsai model in which the porosity is included. Thermo‐mechanical properties of both metal face sheets and porous GPLRC core are assumed to be temperature‐dependent. The governing equations of motion for porous sandwich plates are solved by applying a two‐step perturbation approach to obtain the analytical solutions for the two cases of nonlinear vibration and nonlinear bending of porous sandwich plates. Numerical studies are performed to compare the results obtained from the present model and the equivalent isotropic model (EIM). The results reveal that, for most cases, the difference of natural frequencies between two models is over 30%, and the vibration frequency–amplitude curves and the bending load–deflection curves are underestimated by using the EIM.

Integrated effects of auxeticity and pyro-coupling on the nonlinear static behaviour of magneto-electro-elastic sandwich plates subjected to multi-field interactive loads

This work presents a detailed investigation of the effect of auxeticity synergised with the pyro-coupling behaviour of multiphase magneto-electro-elastic (M-MEE) composites using a finite element (FE) framework. The nonlinear deflection and bending problems of sandwich plates with auxetic core and M-MEE skins subjected to multi-physics load (electric, magnetic and thermal) are probed. The plate kinematics is governed by Reddy’s higher-order shear deformation theory (HSDT). The nonlinear relation between the strains and displacements is established through von-Karman’s nonlinear relations. The temperature profiles are considered linearly and nonlinearly varying across the thickness of the plate. Various parametric studies presented in this article highlight the influence of the auxetic cell inclination angle, rib-length ratio, rib thickness, the plate-to-core ratio etc., associated with the pyro-coupling on the nonlinear deflection and bending of sandwich plates. The results reveal that the plate deflection, bending behaviour and degree of pyro-coupling significantly depend on the auxetic unit cell dimensions and magnitude of electro-magnetic loads. The significant influence of temperature profiles on the pyro-coupling are witnessed at lower auxetic cell angle. The nonlinear deflections of the sandwich plate and hence the potentials developed tend to improve with the lower values of plate-to-core ratio and rib thickness. The prominent outcomes of this work related to integrated effects of auxeticity and pyro-coupling are not yet reported in the open literature and are deemed to be utilised as a future reference.

Nonlinear Bending of Sandwich Plates with Graphene Nanoplatelets Reinforced Porous Composite Core under Various Loads and Boundary Conditions

The nonlinear bending of the sandwich plates with graphene nanoplatelets (GPLs) reinforced porous composite (GNRPC) core and two metal skins subjected to different boundary conditions and various loads, such as the concentrated load at the center, linear loads with different slopes passing through the center, linear eccentric loads, uniform loads, and trapezoidal loads, has been presented. The popular four-unknown refined theory accounting for the thickness stretching effects has been employed to model the mechanics of the sandwich plates. The governing equations have been derived from the nonlinear Von Karman strain–displacement relationship and principle of virtual work with subsequent solution by employing the classical finite element method in combination with the Newton downhill method. The convergence of the numerical results has been checked. The accuracy and efficiency of the theory have been confirmed by comparing the obtained results with those available in the literature. Furthermore, a parametric study has been carried out to analyze the effects of load type, boundary conditions, porosity coefficient, GPLs weight fraction, GPLs geometry, and concentrated load radius on the nonlinear central bending deflections of the sandwich plates. In addition, the numerical results reveal that the adopted higher order theory can significantly improve the simulation of the transverse deflection in the thickness direction.

Modeling and simulation of thin sheet folding

The article addresses the mathematical modeling of the folding of a thin elastic sheet along a prescribed curved arc. A rigorous model reduction from a general hyperelastic material description is carried out under appropriate scaling conditions on the energy and the geometric properties of the folding arc in dependence on the small sheet thickness. The resulting two-dimensional model is a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the folding arc. A discontinuous Galerkin method and an iterative scheme are devised for the accurate numerical approximation of large deformations.

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.

Nonlinear Static Bending and Free Vibration Analysis of Bidirectional Functionally Graded Material Plates

Bidirectional functionally graded material (2D-FGM) plates have mechanical properties that vary continuously in both the thickness and one-edge directions; these plates are more and more widely used in design and engineering applications. When these structures are subjected to strong loads, they can be largely deformed; therefore, nonlinear calculations, in this case, are necessary. In this paper, nonlinear static bending and nonlinear free vibration behaviors of 2D-FGM plates are studied by using the finite element method based on the third-order shear deformation theory; the Newton-Raphson method is used to solve this problem. The accuracy of this approach is confirmed by comparing the results with respect to other papers. The effects of some numerical aspect ratios such as volume fraction index and thickness-to-length ratio on nonlinear static bending and free vibration of the plates are explored. This study shows that there is a big difference between the numerical results obtained from the nonlinear problem and those from the linear one.

Wavelet-based homotopy method for analysis of nonlinear bending of variable-thickness plate on elastic foundations

A unified bran-new wavelet algorithm has been proposed for solving large bending of clamped or simply supported variable-thickness plates resting on nonlinear triparametric elastic foundations. The strongly nonlinear variable-coefficient governing equations for plates with arbitrary distributed thickness have been successfully transformed and solved under the formulation of thickness distributed function meeting circled homogeneous Neumann boundary. The highly accurate wavelet solutions of dimensionless deflection and bending stresses are finally obtained by wavelet homotopy approach, which are difficult and rarely obtained by other numerical methods. Nonlinear bending of characteristic variable-thickness plates of prismatic, tapered, pitted and raised types are further investigated with different parameters of thickness profiles and elastic foundations. Nonlinear effect of inhomogeneous thickness turns to be more evident on changing deflection and stresses distributions of rectangular tapered plates, but only affect extreme values for square tapered one. Resisting bending capability of prismatic plate will be enhanced by improvement of tapered slope but weakened adding polynomial orders of thickness profile. Difference of deflection and bending stresses between raised and pitted plates are enlarged by small deflection theory but gradually narrowed by large deformation theory, which reveals thickness discrepancy has little effect at large bending stage.

A hierarchical wavelet method for nonlinear bending of materially and geometrically anisotropic thin plate

The paper aims at the application of a hierarchical wavelet homotopy methodology for solving nonlinear bending of anisotropic thin plates. The newly mechanical models of materially anisotropic plate and geometrically orthotropic combined plate at ship double bottom in large deflection have been formulated. The dimensionless coupled and nonlinear governing equations have been derived and decomposed into linear differential terms, while the solutions are expanded in wavelet series as a requirement for implementing the Homotopy recursive system of algebra equations by Galerkin method. Highly accurate Coiflet-type solutions have been obtained with good computational efficiency in excellent agreement with other numerical results in published literatures. Materially and geometrically orthotropic analysis have been conducted with different ratios of elasticity modulus or inertia moment in orthogonal directions, which indicate both orthotropy play an important role in largely deformed anisotropic plate under extreme loads. Convergent solutions over wider bending range of isotropic or anisotropic plates subjected to different boundary conditions are given which reveals our proposed wavelet methodology performs good superiority and versatility dealing with strongly nonlinear problems.

Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach

In this paper, an efficient numerical strategy is used to study the geometrically nonlinear static bending of functionally graded graphene platelet-reinforced composite (FG-GPLRC) porous plates with arbitrary shape. Porous nanocomposite plates including cutout with various shapes can be modeled by the present approach. Four types of porous distribution scheme and four GPL dispersion patterns are selected, and the material properties are calculated based on the closed-cell Gaussian random field scheme, the Halpin–Tsai micromechanical model together with the rule of mixture. First, the variational statement of governing equations based on the virtual work principle and higher-order shear deformation theory (HSDT) is derived and presented in vector–matrix form for computational aims. Then, using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a numerical approach called as VDQ-FEM is used to address the considered problem. In VDQ-FEM, the domain of problem is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to obtain the set of Studying effects of porosity and GPL distributions and porosity coefficient matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of the present work in implementing VDQ-FEM is proposing an efficient way based on mixed-formulation to guarantee the continuity condition of first-order derivatives in entire domain for the used HSDT model. A detailed parametric study is conducted to investigate the nonlinear bending of FG-GPLRC porous plates with different shapes. In the numerical results, the effects of porosity coefficient, porosity distribution pattern, GPL distribution pattern and boundary conditions on the nonlinear bending response of plates are analyzed.

Geometrically nonlinear plate bending under the action of moving load

Considerable scientific interest is the development of mathematical models that describe the behavior of materials that are sensitive to deformation rate and can improve the accuracy of analytical calculations of their deformation in the region of noticeable changes of loading rates. Nonetheless, in most works, the problems were solved under the assumption of small displacements (geometrically linear statement of the problem). Meanwhile, in practice, this is not always true and bending of cover can be commensurable with its thickness, this article approximately solves the problem of geometrically nonlinear deformation of a thin elastic plate in aquasistatic setting under the action of an infinite normal uniformly distributed load moving along its surface at a constant speed. In the article, the methods of mathematical modeling, the analytical method, as well as the methods of spatial characteristics and bicharacteristics are used. The problem is solved in the quasistatic formulation and is reduced to a system of two nonlinear differential equations for deflections of the plate and the stress function, which include the speed of the load as a parameter. The results of methodological calculations are presented; based on these solutions of linear and nonlinear problems, they were compared, and the influence of finiteness of displacements on the critical speeds of the forces was determined. Materials of the article can be useful in the study of wave dynamics, aircraft, mechanics, and engineering.

A Novel Method for Numerical Analysis of 3D Nonlinear Thermo-Mechanical Bending of Annular and Circular Plates with Asymmetric Boundary Conditions Using SAPM

This study is the first report of numerical solution of nonlinear bending analysis for annular and circular plates based on 3D elasticity theory with asymmetric boundary conditions using semi-analytical polynomial method (SAPM). Orthotropic annular and circular plates are subjected to transverse loading and 3D bending analysis in the presence of symmetric and asymmetric boundary conditions is studied. For asymmetry cases, the plate boundaries are divided to two or three parts and various boundary conditions such as clamped, simply-supported and free edges are defined for each part. The asymmetry in one and two directions is studied. The influence of elastic foundations, mechanical and thermo-mechanical loadings are examined. Regarding this fact that no study has been done in the case of asymmetric boundary conditions, the obtained results are compared with FEM results by ABAQUS. The results show good agreement with the literatures and FEM results, which it shows that the presented method can use to analyze the 3D bending of plates under asymmetric conditions. Also, it is observed that 3D elasticity estimates some higher deflections than other theories. But, the obtained results by 3D elasticity theory and those obtained by FEM analysis in the case of asymmetric conditions are so close.

Geometrically non-linear bending of plates: Implications in curved building façades

Curved glazed façades of buildings are often realized by cold bending of flat plates. This paper is devoted to experimental tests under static loading on the geometrically non-linear anticlastic bending of aluminium plates, whose constituent material is used as a phantom material to mimic the elastic mechanical behaviour of glass. The 3D displacement field of the plates is reconstructed through the use of the DIC (Digital Image Correlation) technique and several numerical FE analyses are performed to simulate the experimental results. A critical value, corresponding to a configuration of the plate where a change in its curvature sign occurs, of the applied displacement at the plate corner is determined. The results are discussed in terms of critical applied displacement against the plate thickness for different plate sizes. It is shown that the critical displacements are almost independent on the plate size, but linearly depend on the plate thickness.

A decomposition method for nonlinear bending of anisotropic thin plates

A modified version of the Adomian decomposition method is applied to solve the differential equations of nonlinear bending of rectangular anisotropic thin plates. The von Kármám hypotheses are considered in the plate's kinematics. The nonlinear differential equations are decomposed into linear and nonlinear parts and the solution is expanded into series. Essentially, the approach is to embed the nonlinear effects into the anisotropic linear solution. Such incorporations are performed in a recursive fashion along with the Adomian's polynomials, which are responsible for the approximation of the nonlinear effects. In spite of the problem being three-dimensional, no partial solutions are required due to a modification of the standard method, which considers the inversion of the entire linear operator. As a result, the obtained solution respects the boundary conditions set and is independent of its symmetries. The pb-2 Rayleigh-Ritz method is used to obtain the plate's equilibrium path. A multistep procedure is proposed in order to increase the decomposition method convergence radius aiming at global convergence. Numeric results are obtained and compared to those found in the literature.

Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness

Abstract With employing the von Karman plate theory, and considering the linearly thickness variation in one direction, the bending problem of a rectangular magnetoelectroelastic plates with linear variable thickness is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction for the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates with linear variable thickness are established based on the Hamilton’s principle. The Galerkin procedure is taken to translate a set of differential equations into algebraic equations. The numerical examples are presented to discuss the influences of the aspect ratio and span–thickness ratio on the nonlinear load–deflection curves for magnetoelectroelastic plates with linear variable thickness. In addition, the induced electric and magnetic potentials are also presented with the various values of the taper constants.

Influence of the ambient pressure on thin plate and film bending

The distributed lateral load on a plate located in a gas medium has been determined. The difference in the gas pressures of the plate surfaces results in a pressure drop and a lateral force, which depends on the middle surface curvature. It is shown that both these components of the lateral force must be taken into account in the general case. The influence of the second load component is small at a small ratio of the average pressure to the elastic modulus of the plate material and at a large relative thickness of the plate. At a small relative plate thickness and a large ratio of the average pressure of the medium to the elastic modulus of the plate material, the influence of the second component of the lateral load on the bending becomes significant. Cases of both linear and nonlinear plate bending are considered.

Nonlinear bending and vibration analyses of quadrilateral composite plates

Linear and nonlinear analyses of shear deformable thin and thick arbitrary straight-sided quadrilateral plates are reported here using smoothed finite element technique. The Reissner-Mindlin plates are discretized with quadrilateral background cells. Then membrane and bending stiffness matrices of background quadrilateral cells are evaluated using edge-based smoothed finite element method(ES-FEM). The shear stiffness matrix is calculated based on "smoothed shear strain approach" and the performance is compared with "edge-consistent four-node quadrilateral finite element". The convergence, accuracy and sensitivity to mesh distortion of the present quadrilateral element is examined. Thereafter, the nonlinear bending and vibration analyses of trapezoidal and arbitrary straight-sided quadrilateral composite plates are presented for which only limited analytical results are available in the literature.

Nonlinear Bending and Thermal Post-buckling Analysis of FGM Super Elliptical Thin Plates

Super elliptical plates which are defined by shapes between an ellipse and a rectangle have a wide range of use in engineering applications. Investigations on non-linear behaviors of super elliptical isotropic plates are available in the literature, while investigations on nonlinear behaviors of FGM super elliptical plates haven’t been reported at present. In this paper, nonlinear bending and thermal post-buckling analysis are first presented for functionally graded super elliptical plates based on classical plate theory. Material properties are assumed to be temperature-dependent and graded in the thickness direction. The numerical illustrations concern the nonlinear behaviors of functional graded plates with immovable simply supported edge and immovable clamped edge. Influences played by different supported boundaries, thermal environmental conditions, and volume fraction index are discussed in detail using Ritz method.

Nonlinear bending and stability of circular plates under the liquid pressure and temperature

Cases of substantial difference between working and initial temperatures are examined. Peculiarities of the deformation process at the sub- and post-critical temperatures and liquid pressure are revealed.