Thin Sandwich Plates Research Articles

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

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

An FFT-based homogenization scheme for cohesive zones with an application to adhesives and the core material of thin metal sandwich plates

Cohesive Zone Models with finite thickness are widely used for the fracture mechanical modeling of material layers, e.g., adhesive layers. Within this approach, the whole layer is modeled as a cohesive zone. Moreover, computational homogenization techniques are crucial for the development of advanced engineering materials, which are often heterogeneous. Compared to the commonly used Finite Element Method (FEM), solvers based on the Fast Fourier Transform (FFT) are expected to reduce the computational effort needed for the homogenization. Originated from an existing method for the computational homogenization of cohesive zones using FEM, a novel FFT-based homogenization scheme for cohesive zone models is presented. Our implementation of the FFT solver uses a displacement-based Barzilai–Borwein scheme and a non-local ductile damage model for the fracture behavior. Finally, the practical application of the method is discussed using an adhesive layer and the core material of HybrixTM metal sandwich plates as examples.

Layer-wise solutions for variable stiffness composite laminated sandwich plate using curvilinear fibers

This paper deals with the vibration analysis of variable stiffness composite laminated (VSCL) sandwich plates. The VSCL sandwich plate is composed by soft core and two laminated face sheets, in which the face sheets are considered with curvilinear fiber. The layer-wise plate theory and p-version of finite element method are used to solve the problem formulation. In layer-wise theory, each layer modeled separately as a one p-element and the continuity of displacement fields is based on a linear zig-zag variation of displacements through the thickness of the layers. The obtained results of present model are compared and validated with available solutions, such as equivalent single layer theories, layer-wise theory and three-dimensional theory for sandwich plate with constant stiffness. The frequencies and mode shapes are examined for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers, stacking sequence and boundary conditions. New layer-wise solutions for thin and thick VSCL sandwich plate are offered, in which can be used to establish benchmarks for future comparisons.

Numerical investigating of the effect material and geometrical parameters on the static behavior of sandwich plate

Studying the behavior of sandwich panels is very important due to their widespread use in different industries. Therefore, over the past decades, various theories have been proposed to study the behavior of these panels. In this paper a higher order global-local theory with 3D equilibrium-based corrections is presented to study behavior of thick and thin sandwich plate with flexible and auxetic core. In addition to correcting the results with 3D elasticity equations, another important advantage of the presented theory is the ability to consider the transverse core deformation of the sandwich panels. It should be mentioned that to study the behavior of thick sandwich panels, especially with soft core, the existence of this feature is very necessary and has a great effect on the accuracy of the obtained results. Comparison of the obtained results with those existing in valid references showed that the formulation of the provided finite element had a very good accuracy even for thick and thin sandwich plates. Finally, the effect of different material and geometrical parameters on the behavior of sandwich plates are carefully investigated using the presented theory.

On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates

This paper presents a novel C0 higher-order layerwise finite element model for static and free vibration analysis of functionally graded materials (FGM) sandwich plates. The proposed layerwise model, which is developed for multilayer composite plates, supposes higher-order displacement field for the core and first-order displacement field for the face sheets maintaining a continuity of displacement at layer. Unlike the conventional layerwise models, the present one has an important feature that the number of variables is fixed and does not increase when increasing the number of layers. Thus, based on the suggested model, a computationally efficient C0 eight-node quadrilateral element is developed. Indeed, the new element is free of shear locking phenomenon without requiring any shear correction factors. Three common types of FGM plates, namely, (i) isotropic FGM plates; (ii) sandwich plates with FGM face sheets and homogeneous core and (iii) sandwich plates with homogeneous face sheets and FGM core, are considered in the present work. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume power laws of the constituents. The equations of motion of the FGM sandwich plate are obtained via the classical Hamilton’s principle. Numerical results of present model are compared with 2D, quasi-3D, and 3D analytical solutions and other predicted by advanced finite element models reported in the literature. The results indicate that the developed finite element model is promising in terms of accuracy and fast rate of convergence for both thin and thick FGM sandwich plates. Finally, it can be concluded that the proposed model is accurate and efficient in predicting the bending and free vibration responses of FGM sandwich plates.

A new and efficient constitutive model based on fractional time derivatives for transient analyses of viscoelastic systems

In the open literature, many authors have used the fractional calculus in conjunction with the finite element method to model certain viscoelastic systems. The so-named fractional derivative model may be a better option for transient analyses of systems containing viscoelastic materials due to its causal behavior and its capability to fit accurately the viscoelastic damping properties and to represent properly their fading memory. However, depending on the situation, it leads to costly computations due to the integration of the non-local viscoelastic displacement and stress fields, especially for long time intervals. In this contribution, it is proposed a new and efficient general three-dimensional fractional constitutive formulation based on the use of a recurrence term to give a simplest and low-cost constitutive law to describe the frequency- and temperature-dependent behavior of viscoelastic materials, especially for complex systems. To demonstrate the efficiency and accuracy of the proposed formulation compared with those available in the literature, an academic example formed by a thin three-layer sandwich plate is performed and the main features and capabilities of the proposed methodology are highlighted.

Uncertainty propagation and numerical evaluation of viscoelastic sandwich plates having nonlinear behavior

The dynamic analysis of nonlinear viscoelastic systems in the frequency domain is not an easy task. In most cases, it is due to the frequency- and temperature-dependent properties of the viscoelastic part. Additionally, due to the inherent uncertainties affecting the viscoelastic efficiency in practical situations, their handling in the nonlinear modeling methodology becomes essential nowadays. However, it is still an issue. Thus, this paper presents a numerical modeling methodology intended to perform dynamic analyses in the frequency domain of thin sandwich plates under large displacements. The uncertainties characterizing the nonlinear dynamics of the viscoelastic system are introduced on the random linear and nonlinear finite element matrices by performing the Karhunen–Loève expansion technique. The Latin hypercube sampling method is used herein as the stochastic solver, and the nonlinear frequency responses are computed using the harmonic balance method combined with the Galerkin bases. To overcome the difficulty in solving the resulting complex nonlinear eigenproblem with a frequency-dependent viscoelastic stiffness, making the stochastic nonlinear analyses in the frequency domain very costly, sometimes unfeasible, an efficient and accurate iterative reduction method is proposed to approximate the complex eigenmodes. The envelopes of nonlinear frequency responses demonstrate clearly the relevance of considering the uncertainties in design variables of viscoelastic systems having nonlinear behavior to deal with more realistic situations.

Plastic forming of the doubly curved surfaces of sandwich plates with bi-directionally trapezoidal cores of different sizes

Abstract Plastic forming technology is a low-cost and high-efficiency way to produce doubly curved sandwich plate from a flat plate. However, there are a various potential forming defect modes which may affect the formability of sandwich plate. This paper is focused on the defect-free plastic forming of all-metal sandwich plates with bi-directionally trapezoidal cores, an approximate theoretical model for predicting the global buckling of sandwich plate was developed and the formability of sandwich plate with different geometric parameters were investigated analytically, experimentally and by numerical simulations. The results show that the forming defect mode which dominates the formability of a specified sandwich plate depends on the geometric parameter of the sandwich plate, the formability of the thin sandwich plate with small size core cells is limited by global plastic buckling, while that of the relative thick sandwich plate with relative large size core cells is limited by skin wrinkling and skin dimpling. Significant trends are found that the sandwich plates with thick skins have superior forming performance compared to those with thin ones, and that the limit forming radius is decreased greatly with the decrease of the characteristic sizes of core cell unit. Multi-point forming experiments on bi-directionally trapezoidal sandwich plates were performed, and the numerically simulated results and theoretically analyzed results on the formability were demonstrated finally.

Robust layerwise C0 finite element approach based on a variable separation method for the modeling of composite and sandwich plates

This paper deals with a new approach using both the variable separation and a robust C0 eight-node finite element for the modeling of composite plates. The displacement field is approximated as a sum of separated functions of the in-plane coordinates x, y and the transverse coordinate z. This choice yields to an iterative process that consists of solving a 2D and 1D problem successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, the main novelty consists in the formulation of a field compatible approximation for the transverse shear strain field, referred to as the CL8 interpolation. This latter has to be adapted to the particular framework of the separated representation. It allows us to eliminate the shear locking pathology by constraining only the z−constant transverse shear strain terms.Numerical assessments show the absence of locking problems as well as the enhanced robustness with respect to distorted element shapes in comparison to classical isoparametric approaches. This new CL8 plate element provides excellent convergence rates under different boundary and loading conditions, and it yields accurate displacements and stresses for both thick and thin composite and sandwich plates.

Failure characteristics of a thin metallic sandwich plate with metallic sheared dimple cores under low-velocity impact loading

The functionality and the performance of a lightweight sandwich plate with periodic cellular metal (PCM) cores are influenced by failure characteristics of the lightweight sandwich plate. The goal of this paper is to investigate the failure characteristics of a thin metallic sandwich plate with metallic sheared dimple cores under low-velocity impact loading through nonlinear finite element analysis. The influence of the imperfection mode of the cores on the impact response of a thin sandwich plate is examined to obtain a proper simulation model. Repeated finite element analyses are performed to examine the failure modes of the sandwich plate for different impact conditions. Normalized impact parameters are defined in order to generalize a dynamic failure map (DFM). DFMs of the thin metallic sandwich plate have been estimated via the regression analysis of critical impact conditions for each failure mode. In addition, the influence of the normalized impact parameters on the failure mode has been investigated using the DFM.

A New Rectangular Finite Element Formulation Based on Higher Order Displacement Theory for Thick and Thin Composite and Sandwich Plates

A new displacement based higher order element has been formulated that is ideally suitable for shear deformable composite and sandwich plates. Suitable functions for displacements and rotations for each node have been selected so that the element shows rapid convergence, an excellent response against transverse shear loading and requires no shear correction factors. It is completely lock-free and behaves extremely well for thin to thick plates. To make the element rapidly convergent and to capture warping effects for composites, higher order displacement terms in the displacement kinematics have been considered for each node. The element has eleven degrees of freedom per node. Shear deformation has also been considered in the formulation by taking into account shear strains ( rxz and ryz) as nodal unknowns. The element is very simple to formulate and could be coded up in research software. A small Fortran code has been developed to implement the element and various examples of isotropic and composite plates have been analyzed to show the effectiveness of the element.

Natural frequencies of sandwich plates with FGM core via variable-kinematic 2-D Ritz models

This paper deals with the formulation of advanced two-dimensional Ritz-based models for accurate prediction of natural frequencies of thin and thick sandwich plates with core made of functionally graded material (FGM). The formulation is rather general due to its invariant properties with respect to the underlying plate kinematic theory and the admissible set of Ritz functions. Convergence and accuracy of the method are investigated in this work using an entire family of higher-order layerwise and equivalent single-layer theories, whose corresponding displacement variables are approximated by series of Chebyshev polynomials multiplied by appropriate boundary functions. Results are presented for rectangular sandwich FGM plates with various thickness-to-length ratios and combinations of clamped, free and simply-supported edges.

Low-velocity impact response of a thin sandwich plate with a metallic sheared dimple core subjected to a free boundary condition

► Excessive crushing and shear of core commence when core compressive strain is larger than 0.39. ► Energy absorption characteristics of the sandwich plate are influenced by the core compressive strain. ► Specific dissipated energy of the sandwich plate is 13.2–24.8% larger than that of DP780. ► Benefits of the sandwich plate can be maintained as bending dominant deformation is induced. Various studies of the impact and crushing characteristics of sandwich plates with metallic structured cores have been undertaken to apply sandwich plates to structural materials of transport vehicles. The goal of this paper is to investigate into low-velocity impact responses of a thin sandwich plate with metallic sheared dimple cores subjected to a free boundary condition through drop impact experiments. In order to examine the influence of the impact parameter, including the impact energy and the nose diameter of the impact head, on the deformation, failure and energy absorption characteristics of the thin sandwich plate in a free boundary condition, drawing type drop impact experiments have been performed. The results of the experiments have been shown that excessive extension of the sheared section and significant crushing of the core as well as excessive deformation of the front face-sheet in the vicinity of the impacted region commence when the nose diameter is 20 mm and the impact energy is larger than 65.3 J. In addition, it has been noted that the energy absorption rate of the thin sandwich plate is influenced by the core compressive strain and that the core crushing should be reduced to improve the energy absorption characteristics of the thin sandwich plate. The performance and benefits of the thin sandwich plate have been discussed through the comparison of the thin sandwich plate with a DP780 monocoque plate. From the results of the comparison, it has been shown that the thin sandwich plate can improve energy absorption characteristics in spite of the reduced weight and weaker properties.

Experimental determination of elastic properties of the core in a thin sandwich plate with a metallic truss core

The deformation behavior and stress-strain relationships for a lightweight sandwich plate in the elastic regions are mainly dependent on the elastic properties of the core in the sandwich plate. The aim of this paper is to determine experimentally the elastic properties of a core in a thin sandwich plate with a metallic truss core. Three-points bending experiments were performed to obtain a linear relationship between compliance per span length and the square of the span length for the designed sandwich plate. The specimen was manufactured from the continuous multi-points resistance welding of SUS 3O4H sheets and crimped metallic truss cores. The elastic and shear moduli of the core were estimated by the slopes and the intercepts of the linear relationship, respectively. Through the comparison of the shear moduli and force-deflection considering the core stiffness with those disregarding the core stiffness, it was shown that the core stiffness should be considered to estimate accurately the elastic and shear moduli of a core as well as the force-deflection curves of a thin sandwich plate with a metallic truss core. The contribution ratios of the bending and shear deflections of the thin sandwich plate to the total deflection were investigated. In addition, the influence of the crimping angle on the elastic properties, the force-deflection curves, and the contribution ratios of both bending and shear deflections to the total deflection was examined.

Nonlinear dynamic behaviour of a preloaded thin sandwich plate incorporating visco-hyperelastic layers

The nonlinear dynamic behaviour of preloaded traditional multilayer plates incorporating visco-hyperelastic materials is investigated. The visco-hyperelastic material was confined between stiff layers and worked as a damping layer. The full structure is submitted to shear vibrations under a compression preload. We present herein a nonlinear model describing the tangent dynamic shear modulus as function of both preload and frequency. A Navier approach is used to calculate the stress and the deformation fields which are written by a transfer matrix formulation. The dynamic shear modulus for elastomers is expressed by a nonlinear visco-hyperelastic model based on a deformation energy density. The effects of the elastomer's nonlinear behaviour, the frequency and the preload on the structure's overall dynamic shear modulus is the main object of this study. Verification of the model is performed on visco-hyperelastic multilayer through a series of dynamic shear tests under various compression preload levels. A comparison with both finite element calculations and discreet models is discussed.

BENDING BEHAVIOR OF SIMPLY SUPPOTED METALLIC SANDWICH PLATES WITH DIMPLED CORES

Metallic sandwich plates are lightweight structural materials with load-bearing and multi-functional characteristics. Previous analytic studies have shown that the bendability of these plates increases as the thickness decreases. Due to difficulty in the manufacture of thin sandwich plates, dimpled cores (structures called egg-box cores) are employed as a sandwich core. High-precision dimpled cores are easily fabricated in a sectional forming process. The cores are then bonded with skin sheets by multi-point resistance welding. The bending characteristics of simply supported plates were observed by the defining measure, including the radius ratio of the small dimple, the thickness of a sandwich plate, and the pattern angle (0°/90°, 45°). Experimental results revealed that sandwich plates with a thickness of 2.2 mm and a pattern angle of 0°/90° showed good bendability as the punch stroke under a collapse load was longer than other cases. In addition, the gap between attachment points was found to be an important parameter for the improvement of the bendability. Finally, sandwich plates with dimpled cores were bent with a radius of curvature of 330 mm for the sheet thickness of 2.2 mm using an incremental bending apparatus.

2D, Quasi 3D and 3D Exact Solutions for Bending of Thick and Thin Sandwich Plates

A previous author's work related to the exact solution of a single layer isotropic plate is extended to the case of multilayered plate structures composed of isotropic layers. The method solves a first order linear system of differential equations in the unknown amplitudes of the displacements and stresses. An eigenvalue problem, in which analytical expressions for the basis of eigenvectors and generalized eigenvectors are available, is then formulated. This makes the coding of exact solution of multilayered structures very simple and effective compared with other exact methods present in the literature. The paper is mainly concerned about the analysis of sandwich structures made of isotropic layers. Challenging cases of thick and moderately thick plates (a/h = 1, 4, 10) and thin plates (a/h = 100) are analyzed in detail and the displacements and stresses are found and plotted through the thickness of the plate. An extensive study of the ratio between the elastic modulus of the skins and the elastic modulus of the core, as well as the change in the displacements and stresses with that parameter, is conducted. The exact three-dimensional results are also validated with quasi-3D results obtained by adopting a mixed assiomatic theory of order 9 presented here for the first time. Twenty three two-dimensional theories are assessed, five of which are presented here for the first time. The use of Murakami's zig—zag function and the consequent improvement of the ESL theories is also analyzed and discussed. All the results of this work are an useful tool to test and compare approximated methods, such as Finite Element Method.

Consistent higher-order free vibration analysis of composite sandwich plates

Abstract The consistent higher-order dynamic formulation for foam-type (soft) core sandwich beams was extended to the case of composite sandwich plates. Eight dynamic governing equations and the corresponding boundary conditions were derived through the application of Hamilton’s principle. The extended formulation was applied to the free vibration analysis of soft-core and honeycomb-core sandwich plates with anti-symmetric and symmetric lay-ups. The vibration results for the thin and thick composite sandwich plates obtained using the extended formulation were consistent with the predictions of the higher order mixed layerwise theory for laminated and sandwich plates. To simplify the formulation for the case of symmetric sandwich plates, the general dynamic formulation was decoupled into two systems of equations representing symmetric and anti-symmetric vibrations. The numerical study demonstrates the importance of the present formulation for the prediction of higher mode vibration response of composite sandwich plates.

Numerical solution to the buckling problem for a sandwich plate under uniaxial compression

A thin rectangular sandwich plate with isotropic linear elastic layers is considered. The plate is in a plane-strain state under uniaxial compression. An exact statement of the buckling problem is given. Its approximate solution is found by the finite-difference method. The concept of base scheme is used to formulate discrete problems in explicit and compact form. As an example, the critical parameters of the plate are calculated using a computation optimization procedure. Its efficiency is demonstrated

Stopping Power of ERA Sandwiches as a Function of Explosive Layer Thickness or Plate Velocities

AbstractThe stopping power rises in symmetric sandwiches with increasing explosive layer thickness, which is equivalent to higher plate velocities. Further the additional weight is a burden for add‐on armor. Therefore a study was made with very thin sandwich plates of 1 mm as more or less the minimum, what can be used in practice. The reduction effect is more reduced with thicker plates at constant explosive layer thickness. For the penetration reduction the product of plate thickness and plate velocity seems important, as a thumb rule.