Sandwich Plate Theory Research Articles

Geometrically Nonlinear Deformation of Circular Sandwich Plates under Transversely Non-Uniform Temperature Rise

Geometrically nonlinear bending and buckling of circular sandwich plates subjected to transversely non-uniform temperature rise is investigated in this paper. On the basis of sandwich plate theory, nonlinear equations governing the large thermal axis-symmetric deformations of circular sandwich plate in terms of the middle plane’s displacements are derived. Numerical solutions of the nonlinear boundary value problem are obtained by using the shooting method. Equilibrium paths and configurations for different boundary conditions and different values of materials and geometry parameters are illustrated. Numerical results show that the boundary conditions and the stiffness greatly effect critical buckling loads.

Laser-welded web-core sandwich plates under patch loading

This paper presents an approach to the structural analysis of patch-loaded, laser-welded web-core sandwich plates. This approach is especially suitable for the concept design phase of ship structures. The method consists of two parts; the global bending response is evaluated using sandwich plate theory and the local response of the patch-loaded face plate is determined with the help of basic plate theory. Both analyses utilise analytical and numerical methods and include the effect of the rotation stiffness of the laser-welded T-joint between the face and web plate. Combining the results of these two analyses gives the total response of the web-core sandwich plate. The results of the proposed approach are compared to those given by 3D solid and shell element FE analyses and the agreement of deflections and normal stresses is found to be good. The sensitivity of the response to patch-load size, T-joint rotation stiffness, and plate aspect ratio is studied. The main benefits of the proposed approach are its short analysis time and the good accuracy of deflections and stresses. This considered advantageous when applied to the design of large ship structures such as hoistable car decks.

Dynamic Response of In-plane Pre-stressed Sandwich Panels with a Viscoelastic Flexible Core and Different Boundary Conditions

A new mathematical method and double Fourier series set of functions is developed to find out the local and global transverse and in-plane mode shapes, damped natural frequencies and modal loss factors in composite sandwich panels with multilayered laminated face sheets, viscoelastic flexible core and arbitrary boundary conditions. An improved high order sandwich plate theory (IHSAPT) is applied to investigate the dynamic behavior of fiber reinforced plastic (FRP) sandwich thin and thick panels. The present analysis is very general and valid for any type of core, any type of boundary conditions as well as for the cases where the conditions at the upper face sheet are different from those at the lower one along the same edge. Interaction between static in-plane initial stress and dynamic response of a sandwich panel are studied in this article. For different boundary conditions, solutions are obtained using the simple proposed identical double Fourier series functions and Stokes’s transformation technique. The effect of some parameters such as panel aspect ratio, thickness of core-thickness of panel ratio, and initial in-plane stresses on dynamic response of composite sandwich panel are discussed and studied in detail.

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.

Response of In-plane Linearly Prestressed Composite Sandwich Panels with Transversely Flexible Core to Low-velocity Impact

The problem of transverse low-velocity impact on a linearly in-plane prestressed sandwich panel has been considered in this article. The panel may be subjected to initial in-plane biaxial normal and shear stresses along the edges of the panel. Impact is assumed to occur normally over the top or the bottom face-sheets, at arbitrary location. The boundary conditions of the top and the bottom face-sheets are independent. The interaction between the impactor and the panel is modeled with the help of the system having three-degrees-of-freedom (TDOF) consisting of springs-masses-dashpot (SMD) or springs-masses (SM). The effects of initial biaxial stresses and transverse flexibility of the core are considered analytically. In order to determine all components of the displacements, stresses and strains in the face-sheets and the core, a numerical procedure based on improved higher-order sandwich plate theory (IHSAPT) and Galerkin’s method, are employed. The deflections and the corresponding stresses in the panel with higher initial stresses are smaller than those with lower initial stresses. The effects of in-plane initial shear stresses on impact response are slightly greater than that of the other in-plane normal loading states.

Response of composite sandwich panels with transversely flexible core to low-velocity transverse impact: A new dynamic model

A new computational procedure based on improved higher order sandwich plate theory (IHSAPT) and two models representing contact behavior between the impactor and the panel are adopted to study the low velocity impact phenomenon of sandwich panels comprising of a transversely flexible core and laminated composite face-sheets. The interaction between the impactor and the panel is modeled with the help of a new system having three-degrees-of-freedom, consisting of spring–mass–damper–dashpot (SMDD) or spring–mass–damper (SMD). The effects of transverse flexibility of the core, and structural damping are considered. The present analysis yields analytic functions describing the history of contact force as well as the deflections of the impactor and the panel in the transverse direction. In order to determine all components of the displacements, stresses and strains in the face-sheets and the core, a numerical procedure based on improved higher order sandwich plate theory (IHSAPT) and Galerkin's method is employed for modeling the layered sandwich panel (without the impactor), while the analytic force function developed on the basis of SMDD or SMD model, can be used for the contact force between the impactor and the panel. The contact force is considered to be distributed uniformly over a contact patch whose size depends on the magnitude of the impact load as well as the elastic properties and geometry of the impactor. Various boundary conditions for the sandwich panel have also been considered. Finally, the numerical results of the analysis have been compared either with the available experimental results or with some theoretical results.

Effect of physical and geometrical parameters on transverse low-velocity impact response of sandwich panels with a transversely flexible core

The effects of important physical and geometrical parameters on transverse low-velocity impact response of composite sandwich panels have been studied in this paper. Impacts are assumed to occur normally over the top and/or the bottom face sheets, at arbitrary locations and with different impactor masses and initial velocities. For deriving closed-form solutions for the contact force, displacements of the impactor and the panel in the transverse direction, the sandwich panel has been modeled as a discrete three-degrees-of-freedom dynamic system with equivalent masses and springs (SM). The dynamic response of the panel is based on the improved higher-order sandwich plate theory (IHSAPT) and both thick and thin panels have been analyzed. The effects of transverse flexibility of the core, and boundary conditions are considered. Also, the area of the contact patch between the impactor and the panel can be varied as it changes with contact duration. The numerical results of the analysis have been compared either with the available experimental results or with some theoretical results. It is established that the dynamic behavior of the sandwich panel depends on various parameters, such as the aspect ratio and the length-to-thickness ratio of the panel, core thickness, boundary conditions of the panel and impactor parameters like its potential energy, velocity and the location of contact point, etc.

Analysis of sandwich TPS panel with functionally graded foam core by Galerkin method

The method of Fourier analysis is combined with the Galerkin method for solving the two-dimensional elasticity equations for a sandwich thermal protection system (TPS) insulation panel with foam core subjected to transverse loads. The variation of the Young’s modulus through the thickness is given by a polynomial in the thickness coordinate and the Poisson’s ratio is assumed to be constant. The Fourier series method is used to reduce the partial differential equations to a pair of ordinary differential equations, which are solved by using the Galerkin method. The method will be useful in analyzing functionally graded TPS structures with arbitrary variation of thermomechanical properties in the thickness direction. The analysis was also performed using sandwich plate theory. Significant differences were found in the results suggesting that the sandwich theory may not be suitable for the analysis of thick sandwich TPS panel.

Bending behavior of low-cost sandwich plates

A simple sandwich plate theory, which is an extension of an existing sandwich beam theory, is presented. The solution for a sandwich plate simply supported along two opposite edges, free along the two other opposite edges, and loaded by a line load is developed with a superposition approach that has been used for classic isotropic plates. The construction of a low-cost sandwich plate using glass/PET facesheets and a nylon/PET core bonded together with a urethane adhesive is discussed. The experimental set-up to test the plate under the conditions assumed is described. Out-of-plane displacements using dial gage and shadow moire measurements are compared with predictions of the theory. The lack of ideal simple supports and accurate facesheet mechanical properties results in a less than ideal correlation between predictions and measurements. However, these issues are accounted for and a satisfactory correlation is obtained.

On buckling loads for edge-loaded orthotropic plates including transverse shear

Corrugated board usually exhibits low transverse shear stiffness, especially across the corrugations. In the present study the transverse shear is included in an analysis to predict the critical buckling load of an edge-loaded orthotropic linear elastic sandwich plate with all edges simply supported. In the analysis, effective (homogenised) properties of the corrugated core are used. Classical elastic buckling theory of orthotropic sandwich plates predicts that such plates have a finite buckling coefficient when the aspect ratio, i.e. the ratio between the height and width of the plate, becomes small. However, inclusion in the governing equilibrium equations of the additional moments, produced by the membrane stresses in the plate at large transverse shear deformations, gives a buckling coefficient which approaches infinity when the aspect ratio goes to zero. This improvement was first included in the buckling theory of helical springs by Harinx [Proc. Konjlike Nederland Akademie Wettenschappen, vol. 45, 1942, Amsterdam, Holland, pp. 533–539, 650–654] and later applied to orthotropic plates by Bert and Chang [J. Eng. Mech. Division, Proc. Am. Soc. Civil Engrs. 98 (EM6) (1972) 1499–1509]. Some inconsistencies in the latter analysis have been considered. The critical buckling load calculated with corrected analysis is compared with a predicted load obtained using finite element analysis of a corrugated board panel, and also with the critical buckling load obtained from panel compression tests.

Treatment of Unsymmetric Adhesively Bonded Composite Sandwich Panels-To-Flange Joints

ABSTRACT An analytical solution for the determination of stresses in unsymmetric adhesively bonded joints is presented. The classical sandwich plate theory and an adhesive interface constitutive model are employed for this development. In addition, an analytical solution for the consideration of the effect of fracture in the joints, using the modified crack closure also is developed. Joints with various configurations (symmetrical and/or unsymmetrical, single skin and/or sandwich composites), and joints having different thickness, length, and lay up, including the effect of crack tip position, can be treated with these solutions. FEA is used to confirm the integrity of these solutions. The results from the proposed solution agree well with the results obtained from finite element analyses (FEA).

Weight Optimization of Large Scale Sandwich Structures with Acoustic and Mechanical Constraints

A generic model of a railway car body in sandwich design was weight optimized subject to various constraints. These constraints were global natural frequency constraints, buckling and wrinkling constraints, and sound reduction constraints. A finite element model was used for global analysis, while sandwich plate theory was used for local structural analyses. The acoustic model for sound reduction was a sandwich plate model. The optimization procedure started with an infeasible initial design and produced a final design which fulfilled all requirements, yet had a reduced weight. The optimization was performed with and without acoustic constraints and the results imply that the acoustic requirements will produce a heavier design than when the optimization is performed with mechanical constraints only.

High-order behavior of fully bonded and delaminated circular sandwich plates with laminated face sheets and a “soft” core

The bending behavior of circular, fully or partially bonded (delaminated), sandwich plates with a “soft” core and composite laminated face sheets of general layup is investigated. The analytical model derived follows the principles of the High-Order Sandwich Plate Theory and it is based on equilibrium and compatibility requirements. The variational principle of virtual work is employed for the derivation of the field equations along with the boundary and continuity conditions. The delamination considered forms a penny shaped disbonded zone, in which the delaminated faces can slip horizontally with respect one to another, yet they may be in contact accommodating vertical normal compressive stresses. The core is assumed a 3D elastic medium and the composite face sheets are modeled using the classical lamination theory, which yields coordinate dependent constitutive relations. The governing equations take the form of partial differential equations with variable coefficients. The solution procedure adopts the Galerkin method with truncated Fourier series in the circumferential direction and the multiple-points shooting method in the radial direction. Numerical results concerning typical fully and partially bonded sandwich plates are presented and discussed. The results reveal the effect of the anisotropy of the face sheets on the response of the structure and the influence of the delaminated area on the behavior of the sandwich plate.

The contact behavior between a foam core sandwich plate and a rigid indentor

The purpose of this paper is to determine the indentation produced by a rigid indentor falling on a sandwich plate with a height–density closed cellular foam core. Two different indentor shapes are considered: a rigid sphere and a cylindrical rigid punch. By assuming that the time of complete indentation of the indentors is much larger than the time required by the elastic waves to propagate from the point of first contact to the boundary of the plate, a static analysis is performed. A distribution of surface pressures reproducing the contact law of the rigid indentor on an elastic half-space is introduced and the sandwich plate theory proposed by Dundrová, Kovarı́k and Slapák (1970) is adopted; the explicit solutions with pre-assigned surface pressures are obtained and the contact force–indentation relations, before the sandwich plate is damaged, are found. The contact laws are obtained for both a simply supported and a clamped circular sandwich plate and the relevant influence of boundary conditions on the elastic response of the sandwich plate is shown.

A higher-order sandwich plate theory accounting for 3-D stresses

In this paper we extend a layerwise higher-order shear-deformation theory to model a sandwich plate impacting with an elastic foundation at a low velocity. A new concept of sublaminates is introduced, and the new sandwich plate theory satisfies the continuity conditions of interlaminar shear and normal stresses, accommodates the normal and shear stresses on the bonding surfaces, and accounts for non-uniform distributions of transverse shear stresses in each layer. Moreover, the use of sublaminates enables the modeling of shear warpings that change with the spatial location, vibration frequency, and loading and boundary conditions. A finite-element model based on this sandwich plate theory is derived for performing direct transient analyses to predict the initiation and location of critical matrix crack and the threshold of impact damage. Moreover, analytical shear warping functions, shear coupling functions, and normal strain functions due to in-plane stretching, bending, transverse shearing, and surface loading are presented.

Response of Orthotropic Sandwich Plates to Concentrated Loading

Sandwich plates are often analyzed with either classical or first order shear deformation theory. However experiments have shown that for lower stiffness cores and with concentrated loadings, these theories omit important features of the response. Recently a higher order theory for sandwich beams and plates has appeared in the literature that appears to better predict the response of sandwich structures. This theory treats the faces of the sandwich as independent orthotropic plates, coupled together by means of a simplified elasticity solution for the core that includes shear deformation and through-the-thickness strains. Analytical solutions for transverse static loading of plates can be obtained for simply-supported boundary conditions, analogous to the Navier solution. However the accuracy of this theory has not been sufficiently examined. In this paper the exact elasticity solution available for this problem is used to compare with the higher order sandwich theory. The results show that classical and first order shear deformation theory under-predict stresses and strains in the sandwich faces by more than an order of magnitude, while the higher order shear deformation theory is much more accurate. Although the higher order theory gives reasonable agreement with the elasticity solution, and is believed to be useful in practical application, it still gives errors on the order of 10 to 25% in face strains and interface shear stresses. The elasticity solution shows that the shear stress varies through the core thickness in the vicinity of localized loads, contrary to the assumptions of the higher order sandwich theory.

THEORETICAL CHARACTERISTICS OF THE VIBRATION OF SANDWICH PLATES WITH IN-PLANE NEGATIVE POISSON'S RATIO VALUES

This paper is concerned with re-entrant cell honeycombs which show in-plane negative Poisson's ratio values, in which their anisotropic mechanical properties are described using the cellular material theory. Out-of-plane shear moduli are affected by the unit cell geometric parameters and, for some ranges of the latter, it is possible to obtain higher values of the shear moduli compared to those of a regular hexagonal honeycomb, in particular for cell geometries with a negative Poisson's ratio. A first order sandwich plate theory is applied in order to obtain the fundamental frequencies of sandwich laminates in cylindrical bending and for the simply supported case. Sensitivities of the frequencies per unit mass versus the geometric cell parameters are also calculated. The results suggest that the dynamic performance of a sandwich structure could be significantly improved with a proper design of the unit cell shape of the honeycomb. In particular, re-entrant cell cores offer improvements in bending stiffness capabilities for particular cell parameter ranges.

Performance of Sandwich Structures with Composite Reinforced Core

Sandwich panels with a foam core reinforced with composite laminates were investigated. The effective material properties of the reinforced core were obtained by homogenization. The general equations of equilibrium for sandwich plates were derived and used for parametric studies of the behavior of the sandwich plates. Three Point Bending tests were performed to experimentally measure the bending stiffness along with the strength of sandwich beams. The test results were also used to verify the analytical solution obtained by using the sandwich plate theory. It was found that local indentation must be included in the theoretical model. Static punching tests (SPT) and low velocity impact (LVI) tests were performed to investigate the behavior of sandwich plates under static and dynamic point loads. The force-displacement relations were modeled based on indentation data. The damage modes, effects of the core reinforcement, and effects of the foam were investigated. A second manufacturing method for constructing the reinforced core was presented, and its properties were compared with those obtained using the original manufacturing method.

Performance of Sandwich Structures with Composite Reinforced Core

Sandwich panels with a foam core reinforced with composite laminates were investigated. The effective material properties of the reinforced core were obtained by homogenization. The general equations of equilibrium for sandwich plates were derived and used for parametric studies of the behavior of the sandwich plates. Three Point Bending tests were performed to experimentally measure the bending stiffness along with the strength of sandwich beams. The test results were also used to verify the analytical solution obtained by using the sandwich plate theory. It was found that local indentation must be included in the theoretical model. Static punching tests (SPT) and low velocity impact (LVI) tests were performed to investigate the behavior of sandwich plates under static and dynamic point loads. The force-displacement relations were modeled based on indentation data. The damage modes, effects of the core reinforcement, and effects of the foam were investigated. A second manufacturing method for constructing the reinforced core was presented, and its properties were compared with those obtained using the original manufacturing method.

Buckling and failure analysis of FRP faced sandwich plates

A thin-walled sandwich plate having laminated composite faces and subject to axial compression is studied. The first part of the work is devoted to the discussion of the 2-D geometrically nonlinear formulation of governing equations with the use of the Hamilton and the Lagrange variational principles. It allows us to compare and verify values of buckling loads and natural frequencies obtained with the use of different variants of sandwich plate theory. The analytical results are also confronted with the FE computations conducted with the use of the NISA II package. Then, we consider also the problems of local failure damage of sandwiches basing on the 3-D FE analysis. The aim of those investigations is to point out the effects of the normal stresses and their influence on the sandwich behaviour.