High-order Sandwich Plate Theory Research Articles

Global buckling and wrinkling of variable angle tow composite sandwich plates by a modified extended high-order sandwich plate theory

The advanced Automated Fibre Placement (AFP) manufacturing technologies make the synthesis of Variable Angle Tow (VAT) composites, which enable the design of lightweight sandwich structures to possess variable stiffness facesheets. However, both global and local instability phenomena of VAT sandwich plates under in-plane compressive loads are rarely explored till now. The objective of this article is to fill this gap by developing a Rayleigh–Ritz procedure based on a modified extended high-order sandwich plate theory (EHSAPT). The original two-dimensional EHSAPT proposed by Phan et al. (2012) is extended to a three-dimensional case with a minor modification that the first-order shear deformation theory instead of the classical Kirchhoff–Love hypothesis is adopted for each facesheet, which brings several merits such as the conciseness in derivation process and the easiness in program implementation. The Rayleigh–Ritz approach combined with the principle of minimum potential energy is employed to derive the eigenvalue equation that governs the instability problem of VAT sandwich plates under in-plane compressive loads. Both global buckling and wrinkling patterns of VAT sandwich plates can be captured under this proposed analytical model framework. Before instability analysis, the nonuniform prebuckling stresses over the entire sandwich plate are determined under the assumption of membrane prebuckling state. The usage of Lagrange multiplier method in the prebuckling regime removes the restrictions inherent in conventional Rayleigh–Ritz formulation and thus provides a general way to model in-plane boundary conditions. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing with previously published results and FE solutions given by ABAQUS. Effects of core thickness, core orthotropy, and fibre orientation angle of the facesheets on the instability behaviour of VAT sandwich plates are investigated. Finally, the mechanism of steering the fibre trajectory over the facesheets to improve the buckling resistance for the sandwich plate is studied and discussed.

Fluid–structure–soil interaction effects on the free vibrations of functionally graded sandwich plates

This study investigates the effects of fluid–structure and soil–structure interaction on the free vibration response of functionally graded sandwich plates. To this aim, an exemplary problem is analyzed, whereas a metal/ceramic sandwich plate is placed at the bottom of a tank filled in with fluid. Two cases are considered: (i) soft core, i.e., a sandwich plate with metal core and ceramic skins, and (ii) hard core, i.e., a sandwich plate with ceramic core and metal skins. In both cases, the skins are modelled as suitable functionally graded materials (FGMs). The soil is modelled as a Pasternak foundation. The free vibration analysis is carried out according to the extended higher order sandwich plate theory (EHSAPT). The fluid is assumed to be inviscid, incompressible, and irrotational. Hamilton’s principle is exploited to deduce the governing equations and the corresponding boundary conditions. The Rayleigh–Ritz method with two-variable orthogonal polynomials is used to compute the natural frequencies of the sandwich plate. The adopted approach is first validated through comparison with results published in the literature. Then, the effects are studied of several parameters on the dynamic response of the system.

Free vibration analysis of a laminated composite sandwich plate with compressible core placed at the bottom of a tank filled with fluid

The paper addresses the free vibration problem of a laminated composite sandwich plate with compressible core, placed at the bottom of a tank filled in with fluid. The analysis is performed based on the extended higher order sandwich plate theory (EHSAPT). First-order shear deformation theory (FSDT) is used for the face sheets. Cubic and quadratic polynomials are used to describe the in-plane and transverse displacements of the core, respectively. The fluid is assumed inviscid, incompressible, and irrotational. To obtain the kinetic energy of the fluid, its velocity potential is expressed by using the compatibility and boundary conditions. The governing differential equations and corresponding boundary conditions are derived from Hamilton’s principle. A single series expansion is considered with two-variable orthogonal polynomials as a set of admissible functions satisfying the boundary conditions. The natural frequencies of the coupled sandwich plate-fluid system are calculated by using the Rayleigh-Ritz method. Convergence of the adopted strategy is first investigated. Then, comparisons are conducted with previous results reported in the literature. Also, the effects are investigated of several parameters, such as the plate side-to-thickness ratio, the core-to-face sheets thickness ratio, the face sheet-to-core flexural modulus ratio, the height and aspect ratio of the tank.

A novel finite element formulation based on extended higher-order theory for sandwich plates of arbitrary aspect ratio

The extended higher-order sandwich plate theory for plates with arbitrary aspect ratio was formulated for two-dimensional orthotropic sandwich plates. The novelty of the theory is that it considers five generalized co-ordinates in the core (two axial and one transverse displacements at centroid of the core, one rotation at the centroid of the core about x-axis and one rotation at the centroid of the core about y-axis). Theory is very accurate when compared with the exact elasticity solution in terms of stresses and displacement both. In the current paper, a novel two dimensional rectangular element is developed based on the extended higher-order sandwich plate theory. Elemental equations along with the procedure to derive these is given in the paper. Developed finite element model is validated by comparing the results with elasticity solution and the theory itself for two sandwich plate configurations. The comparison shows that results obtained from the proposed finite element are in very good agreement with elasticity in terms of displacements and stresses both. Thus the proposed element is a powerful analysis tool which can be used for accurately analyzing the real world structures involving sandwich plates at a low computational cost.

Free vibration analysis of rectangular sandwich plates with compressible core and various boundary conditions

Extended higher-order sandwich plate theory is used to analyze the free vibrations of rectangular sandwich plates with compressible core. Accordingly, first-order shear deformation theory is used to model the laminated face sheets. Besides, the in-plane and transverse displacements of the core are assumed to be cubic and quadratic functions of the thickness coordinate, respectively. To deduce the governing equations, Hamilton’s principle is used. Then, based on the Rayleigh-Ritz method, single series expansions with two-variable orthogonal polynomials – namely, the orthogonal plate functions – are considered to approximate the displacement components. Lastly, a generalized eigenvalue problem is solved to obtain the free vibrational characteristics of sandwich plates with both symmetric and anti-symmetric lay-ups subjected to various boundary conditions. The method is validated against the results obtained by different methods in the literature. Finally, the effects of the plate side-to-thickness ratio, in-plane aspect ratio, and core-to-face sheets thickness ratio on the natural frequencies are discussed.

Hygrothermal modeling of the buckling behavior of sandwich plates with nanocomposite face sheets resting on a Pasternak foundation

In this work we investigate the buckling response of sandwich plates with a polymeric core and two face sheets reinforced by carbon nanotubes (CNTs). The problem is tackled analytically by means of a higher-order sandwich plate theory, where the face sheets are modeled according to a classical plate theory and modified strain gradient theory with temperature-dependent and moisture-dependent material properties. A Mori–Tanaka method is applied to determine the mechanical properties associated with the face sheets, while considering the agglomeration effect of CNTs. The governing equations of the problem are derived from the Hamilton’s principle, whose solutions are recovered by means of a Navier–Stokes method. A thorough sensitivity study of the structural response to different parameters includes the agglomeration and volume fraction of CNTs, the material length scale parameter, the side and aspect ratios, together with the temperature variation and humidity conditions. The sandwich plates are assumed to be immersed within an orthotropic Pasternak foundation, whose normal and shear moduli can affect the overall buckling response of the structure. Numerical experiments show that sandwich plates with nanocomposite face sheets, resting on orthotropic elastic foundations, feature an increased stiffness, where the proposed formulation yields accurate results, due to the possibility of considering the variation in temperature, humidity and agglomeration of the reinforcing CNTs within the solution.

Free vibration analysis of sandwich plates with compressible core in contact with fluid

In this paper, the extended higher-order sandwich plate's theory (EHSAPT) is used to analyze the free vibration of the sandwich plate with compressible core and different boundary conditions in contact with fluid. First-order shear deformation theory is adopted for the top and bottom face sheets, while the in-plane and transverse displacements of the core are considered to be cubic and quadratic functions of the transverse coordinate, respectively. A single series is considered with two-variable orthogonal polynomials as a set of admissible functions satisfying the boundary conditions. Besides, the fluid is considered to be irrotational, inviscid and incompressible. By taking into account the boundary conditions and compatibility conditions, the fluid velocity potential is acquired. The natural frequencies of the system are calculated by the Rayleigh-Ritz method. An excellent accuracy is obtained between the results in the available literature and the present method. Finally, the effects of various parameters including boundary conditions, side-to-thickness ratio, thickness of the core to thickness of the face sheets ratio, face sheet to core flexural modulus ratio, dimensions of the container, and aspect ratios on the natural frequencies of the sandwich plate are presented and discussed in detail.

Multi-objective genetic algorithm optimization of composite sandwich plates using a higher-order theory

The main purpose of this paper is multi-objective optimization of soft-core composite sandwich plates using a known accurate high-order sandwich plate theory. In this three-layer theory, high-order kinematic assumptions are dedicated to each face sheet and the core layer. This theory considers the transverse flexibility of the soft core and satisfies transverse shear stresses continuity conditions and zero transverse shear stresses conditions at the upper and lower surfaces of the plate. The governing equations for bending and buckling analyses are derived using Hamilton’s principle, and their analytical solutions are presented for cross-ply plates. Two multi-objective optimization problems consist of weight/deflection optimization and weight/buckling load optimization of the unidirectional and cross-ply sandwich plates being studied. The thicknesses of the core and the face sheet layers are set as problems variables. The genetic algorithm is employed to find the optimal solutions for the problems, and the results are presented in form of the Pareto front. The accuracy of the present modeling and optimization method is evaluated for special case of buckling load maximization of laminated composite plates. For each problem, the optimization process is continued for more than 200 generations, but the results don’t change after 100 generations and the optimized results are obtained. The results confirm that there is no significant difference between the optimal solutions of unidirectional and cross-ply sandwich plates in both optimization problems. The process statistics show that cross-ply layup optimization takes about 25% more time than the unidirectional layup.

Temperature-Dependent Buckling Analysis of Functionally Graded Sandwich Cylinders

This study is limited to study of buckling analysis of a sandwich cylindrical shell with functionally graded face sheets and homogenous core. High-order sandwich plate theory is improved by considering the in-plane stresses of the core that usually are ignored in the analysis of sandwich structures. Assume that all properties of the face sheets and the core are temperature dependent. Strain components are obtained by using the nonlinear Von-Karman type relations. The equilibrium equations are derived via principle of minimum potential energy. Analytical solution for static analysis of simply supported sandwich conical shells with functionally graded face sheets under axial in-plane compressive loads and in the temperature environments is performed by using Navier’s solution. The results show the critical dimensionless static axial loads are affected by the configurations of the constituent materials, compositional profile variations, temperature and the variation of the sandwich geometry. The comparisons show that the present results are in the good agreement with the numerical results.

Bending analysis of sandwich plates with composite face sheets and compliance functionally graded syntactic foam core

In this paper, the problem of a rectangular plate with functionally graded soft core and composite face sheets is considered using high order sandwich plate theory. This theory applies no assumptions on the displacement and stress fields in the core. Face sheets were treated using classical theory and core was exposed to the theory of elasticity. Governing equations and boundary conditions are derived using principle of virtual displacement and the governing equations are based on eight primary variables including six displacements and two shear stresses. This solution is able to present localized displacements and stresses in places where concentrated loads are exerted to the structure since the displacements in the core can take a nonlinear form which could not be seen in the previous theories such as classical and higher order shear theories. This theory is suitable for rectangular plates under all types of loadings distributed or concentrated which can be different on upper and lower face sheets at the same point. The results were compared with the published literature using theory of elasticity and showed good agreement confirming the accuracy of the present theory. Subsequently, the solution for the core with functionally graded material is presented and effectively indicates positive role of functionally graded core.

Free vibration analysis of sandwich plate with a transversely flexible core and FG-CNTs reinforced nanocomposite face sheets subjected to magnetic field and temperature-dependent material properties using SGT

In this research, the free vibration analysis of a sandwich plate with a transversely flexible core and functionally graded – carbon nanotubes (FG-CNTs) reinforced nanocomposite face sheets subjected to magnetic field and temperature-dependent material properties is presented based on high-order sandwich plate theory (HSAPT). The governing equations of motion are derived using Hamilton's principle. Classical plate theory (CPT) is used for modeling the face sheets and the effective properties of them are defined based on the extended mixture rule. Mechanical properties of the core such as Young's and shear modulli are assumed to be function of temperature. The influences of aspect and side ratios, temperature changes, core-to-face sheet thickness ratio, distribution types and volume fraction of carbon nanotubes are presented. The size-dependent mathematical formulation of the face sheets is developed based on the strain gradient theory (SGT). The results show that with increasing the aspect ratio, the non-dimensional natural frequency increases, however the side ratio has a reverse effect. As the temperature of the system increasing, the non-dimensional natural frequency decreases while by applying the magnetic field, the frequency parameter increases. Also it is concluded that the sandwich plate with X distribution type of carbon nanotubes in the face sheets, has higher frequency compared with the other types. Finally it is observed that by considering the size effect and introducing the material length scale parameters, the frequency of the sandwich plate increases. Employing FG-CNTs in face sheets, magnetic field, and size dependent effect leads to increase stiffness of nanostructures, thus the use of this sandwich plate has a prominent role in modern engineering applications.

Low velocity impact analysis of sandwich plates with functionally graded face sheets

ABSTRACTThis article deals with the study of low velocity impact response on sandwich plates with functionally graded face sheets. High-order sandwich plate theory is improved by considering the in-plane stresses of the core that usually are ignored in the analysis of sandwich structures. A new approach is used to reduce the equations of motion from 27 equations to 15 equations and then solving them for both unsymmetric and symmetric sandwich plates. The model is also checked by finite element simulation and by comparing with other references for validation. A parametric study is done for various geometrical and mechanical properties.

The influence of correlating material parameters of gradient foam core on free vibration of sandwich panel

For the sandwich panel with mass density gradient (DG) foam core, the Young's modulus of the core varies with the mass density along the thickness direction. To characterize the correlative effect of Young's modulus and mass density of the DG closed-cell foam material, a simplified formula is presented. Subsequently, based on a high-order sandwich plate theory for sandwich panel with homogeneous core, a new gradient sandwich model is developed by introducing a gradient expression of material properties. Finite element (FE) simulation is carried out in order to verify this model. The results show that the proposed model can predict well the free vibration of composite sandwich panel with the gradient core. Finally, the correlating effects of material parameters of the DG foam core on the natural frequencies of sandwich panel are investigated. It is found that the natural frequencies of sandwich panels decrease as the gradient changes of the DG foam cores increase under the condition of that the core masses keep constant.

A spectral element for wave propagation in honeycomb sandwich construction considering core flexibility

Spectral elements are found to be extremely resourceful to study the wave propagation characteristics of structures at high frequencies. Most of the aerospace structures use honeycomb sandwich constructions. The existing spectral elements use single layer theories for a sandwich construction wherein the two face sheets vibrate together and this model is sufficient for low frequency excitations. At high frequencies, the two face sheets vibrate independently. The Extended Higher order SAndwich Plate theory (EHSaPT) is suitable for representing the independent motion of the face sheets. A 1D spectral element based on EHSaPT is developed in this work. The wave number and the wave speed characteristics are obtained using the developed spectral element. It is shown that the developed spectral element is capable of representing independent wave motions of the face sheets. The propagation speeds of a high frequency modulated pulse in the face sheets and the core of a honeycomb sandwich are demonstrated. Responses of a typical honeycomb sandwich beam to high frequency shock loads are obtained using the developed spectral element and the response match very well with the finite element results. It is shown that the developed spectral element is able to represent the flexibility of the core resulting into independent wave motions in the face sheets, for which a finite element method needs huge degrees of freedom.

Twist in soft-core sandwich plates

This paper studies the twist behavior of soft-core sandwich plates. The paper adopts a comparative analytical approach, derives a geometrically nonlinear extended high-order sandwich plate theory, and compares this theory with the classical high-order one. The extended theory takes all stiffness components of the core and its Poisson effect into account and considers the direct contribution of the in-plane shear stresses in the core to the twist resistance mechanism. The classical high-order sandwich plate theory, which is presented for comparison, neglects the in-plane shear and normal stiffness of the core and attributes the twist mechanism to the composite action of the face sheets and to the ability of the core to resist out-of-plane shear and out-of-plane normal stresses. Both theories account for large displacements, moderate rotations, and small strains in the face sheets and serve as platforms for the development of specially tailored finite elements. Along with the development of the extended sandwich plate theory, the paper studies the twist behavior, compares the two theories, compares the results with experimental ones taken from the literature, and looks into the impact of the geometrical nonlinearity on the twist response.

Geometrically Nonlinear Behavior of Sandwich Plates

The geometrically nonlinear behavior of modern soft-core sandwich plates is studied. For this purpose, a specially tailored geometrically nonlinear finite element that is based on a high-order sandwich plate theory is developed. The theory uses a von Kármán type of geometrical nonlinearity in the face sheets and account for shear and through the thickness deformability of the core. The conversion of the theory to a specially tailored finite element extends its applicability to a wide range of structural layouts, allows the use of standard numerical techniques, and simplifies the coupling with other elements. Yet it avoids the need for meshing through the thickness of the plate. The application of the specially tailored finite element to the geometrically nonlinear analysis of in-plane and out-of-plane loaded sandwich plates explores many interesting physical phenomena. Among them, the evolution of localized instabilities during a globally stable load-deflection behavior, the development of localized diagonal wrinkling patterns, and their impact on the interfacial stresses are listed. The development of unique load resisting mechanisms and, particularly, the evolution of these mechanisms along the geometrically nonlinear response path are also detected. The paper discusses these effects and their role in the geometrically nonlinear behavior of the soft-core sandwich plate.

Free vibration analysis of sandwich plates with functionally graded face sheets and temperature-dependent material properties: A new approach

Abstract Improved high-order sandwich plate theory is used to analyze the free vibration of sandwich plates with functionally graded (FG) face sheets in various thermal environments. The material properties of FG face sheets are assumed to be temperature-dependent by a third-order function of temperature and vary continuously through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. Also, the material properties of the core are assumed to be temperature-dependent by a third-order function of temperature. The governing equations of motion in free natural vibration are derived using Hamilton's principle. A new approach is used to reduce the equations of motion and then solved them for both un-symmetric and symmetric sandwich plates. In-plane stresses of the core that usually are ignored in the vibration characteristics of the sandwich structures are considered in this formulation. The results show that the fundamental frequency parameter increases and decreases with increasing the volume fraction index for soft core and hard core sandwich plates, respectively. The results indicate that as the side-to-thickness ratio, the core-to-face sheet thickness ratio and the temperature are changed, a significant effect on the fundamental frequency parameter is observed. Good agreement is found between the theoretical predictions of the fundamental frequency parameters and the results obtained from other references for simply supported sandwich plates with functionally graded face sheets in the literature.

Improvement of thermo-mechanical properties of transversely flexible sandwich panels by functionally graded skins

In this article bending response of the sandwich panel with functionally graded (FG) skins under thermal and/or mechanical loading is studied. Higher order sandwich plate theory has been exploited. In addition, by using Fourier conduction equation and obtaining temperature distribution, thermo-elastic behavior of such structure is investigated. Comparison between the results of analytical and FEM solutions is indicative of robustness of the analytical solution for prediction of in-plane and out-of-plane stresses. It is shown a wise selection for FG skin materials is helpful to have better conditions under thermo-mechanical loading and to decrease the delamination failure possibility.

Predicting the Sound Transmission Loss of Sandwich Panels by Statistical Energy Analysis Approach

Abstract A statistical energy analysis (SEA) approach is used to predict the sound transmission loss (STL) of sandwich panels numerically. Unlike conventional SEA studies of the STL of sandwich panels, which consider only the antisymmetric (bending) motion of the sandwich panel, the present approach accounts for both antisymmetric and symmetric (dilatational) motions. Using the consistent higher-order sandwich plate theory, the wave numbers of the waves propagating in the sandwich panel were calculated. Using these wave numbers, the wave speed of the propagating waves, the modal density, and the radiation efficiency of the sandwich panels were determined. Finally, the sound transmission losses of two sandwich panels were calculated and compared with the experimentally measured values, as well as with conventional SEA predictions. The comparisons with the experimental data showed good agreement, and the superiority of the present approach relative to other approaches is discussed and analyzed.

Higher-order dynamic response of composite sandwich panels with flexible core under simultaneous low-velocity impacts of multiple small masses

Small mass impactors, such as runway debris and hailstones may result in a wave controlled local response, which is essentially independent of boundary conditions. The higher-order impact model of sandwich beams presented by Mijia and Pizhong [Mijia, Y., Pizhong, Q., 2005. Higher-order impact modeling of sandwich structures with flexible core. International Journal of Solids and Structures 42 (10), 5460–5490] is developed and enhanced to impact analysis of sandwich panels with transversely flexible cores. Therefore, an improved fully dynamic higher-order impact theory is developed to analyze the low-velocity impact dynamic of a system which consists of a composite sandwich panel with transversely flexible core and multiple small impactors with small masses. Impacts are assumed to occur normally and simultaneously over the top face-sheet with arbitrary different masses and initial velocities of impactors. The contact forces between the panel and the impactors are treated as the internal forces of the system. First shear deformation theory (FSDT) is used for the face-sheets while three-dimensional elasticity is used for the soft core. The fully dynamic effects of the core layer and the face-sheets are considered in this study. Contact area can be varied with contact duration. The results in multiple mass impacts over sandwich panels that are hitherto not reported in the literature are presented based on proposed improved higher-order sandwich plate theory (IHSAPT). Finally, for the case study of the single mass impact, the numerical results of the analysis have been compared either with the available experimental results or with some theoretical results. As no literature could be found on the impact of multiple impactors over sandwich panels, the present formulation is validated indirectly by comparing the response of two cases of double small masses and single small mass impacts. Also, in order to demonstrate the applicability of the validation, the analytical relation of minimum distance between two impactors is derived based on Olsson’s wave control principle in this paper.