High-order Sandwich Panel Theory Research Articles

Mechanics of inner core debonding of composite sandwich beam with CFRP hexagonal honeycomb

The inner core debonding of composite sandwich beams with carbon fiber reinforced polymer (CFRP) hexagonal honeycomb core is investigated theoretically and experimentally. Based on the principle of minimum potential energy, a theoretical model is established to reveal the core debonding mechanism with Timoshenko’s beam theory and Extended High-Order Sandwich Panel Theory (EHSAPT) for composite face sheets and CFRP hexagonal honeycomb core, respectively. The theoretical approach of core debonding simulation is based on the framework of the cohesive zone model (CZM). By constructing the constitutive relation of the core interface as the discontinuous stress and displacement boundary conditions, the vertical core debonding can be simulated overcoming the singularity of equilibrium equations. The crack propagation process is decomposed into three stages with the consideration of the double crack propagation mechanism. In experiments, the regular and reinforced hexagonal honeycomb sandwich beams are designed and fabricated, and the quantification of the defect degree of the core bonding interface is achieved by equivalent energy release rate obtained from the homogenized hexagonal honeycomb. The double cantilever beam (DCB) experiments are performed to determine the equivalent energy release rate and reveal the core debonding mechanism of CFRP hexagonal honeycomb sandwich beams. Comparing the load–displacement curves and debonding deformation patterns indicates that the established theoretical model can effectively predict the core debonding mechanical response. Finally, parametric analysis is carried out to discuss the impact of structural dimension parameters, defect location and defect degree on the debonding modes of CFRP hexagonal honeycomb sandwich beams.

Uncertainty in the Flexural Behavior of Soft-Core Sandwich Beams

AbstractThe impact of uncertainty on the flexural response of sandwich beams is studied in this paper. The study focuses on the unique features of typical soft-core sandwich beams, including local effects and stresses concentrations. Those are critical for the integrity and performance of the structural system and, to the best knowledge of the authors, were never considered from a stochastic viewpoint. Several structural features of the beam are considered, separately, as uncertain, and the stochastic characteristics of the distributions of the displacements and stresses are investigated. To capture the unique local effects, the extended high-order sandwich panel theory is adopted. Numerical solutions of the stochastic ordinary differential equations use the finite element (FE) method. The parametric uncertainty and its spatial distribution are modeled by random fields. The stochastic analysis uses both the Karhunen–Loeve polynomial-chaos and the perturbation-based stochastic finite element methods. The two approaches are compared and comprehensively validated against Monte Carlo simulations. The numerical results quantify the impact of uncertainty on the response, but show significant disagreements between the two stochastic approaches, and the perturbation-based stochastic finite element method is chosen as the most suitable one. The effects of the coefficient of variation of the uncertain input parameter and its correlation length are also studied. Highly amplified levels of uncertainty are revealed by the stochastic analysis near points of local stress concentration. These localized yet significant uncertainties, reported here for the first time, shed new light on the design, analysis, and safety of such sandwich beams.

Dynamic stability of sandwich panels subjected to periodic axial loads

This paper presents an analysis for the dynamic stability of sandwich beams/wide plates subjected to periodic axial loads. The formulation of the problem is done by use of the Extended High-order Sandwich Panel Theory (EHSAPT). The equations of motion are derived from Hamilton’s principle and are expressed in terms of seven generalized displacements. A sandwich panel with simply supported edges is studied as an example, and the equations of motion for a given harmonic n are further derived. Two time-variations for the axial forces, namely, harmonic axial forces and step-wise periodic axial forces, are considered in this work. By considering the features of these two periodic load profiles, Floquet theory and Bolotin’s method are adopted to perform the dynamic stability analysis. Sandwich panels with different face-to-core thickness ratios are studied. Stability maps for varying frequency and amplitude of the forces are presented. Numerical examples show that when a sandwich panel is subjected to periodic loads, it is possible that it can experience dynamic instability even when the dynamic loads are much lower than the static critical loads.

Free vibration analysis of a nanoscale FG-CNTRCs sandwich beam with flexible core: Implementing an extended high order approach

The present work uses an extended high-order technique to examine free vibration analysis of Nanoscale CNTRCs sandwich beams. In this study, face sheets are made of PMMA reinforced with single-wall carbon nanotubes, and the honeycomb structure is selected for the flexible core. First-order shear deformation theory is utilized for the skins. The so-called high-order sandwich panel theory (HSAPT) is used for the core to consider the transverse shear strains in theoretical formulations. Then, skins and cores are subjected to the modified couple stress theory (MCST). For obtaining equations and related boundary conditions, the Hamilton principle is used. Additionally, three forms of beam support are examined: simply supported on both sides (SS), simply supported on one side and clamped supported on the other (SC), and clamped supported on both sides (CC). The material properties in skins are simulated to be graded in the thickness direction. The shooting method, as the numerical solution, is adopted to solve the governing differential equations to determine the natural frequencies of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams in which boundary conditions are converted into initial conditions. Compared to other techniques, this method is a reliable way to solve complicated equations and boundary conditions; therefore, its applicability, convergence, and accuracy are verified by comparing obtained results with those existing ones in the literature. Furthermore, the influences of distribution pattern and volume fraction of carbon nanotubes, boundary conditions, the size dependency parameter, core thickness, and slenderness ratio on vibration responses are discussed. It is concluded that natural frequencies rise with an increase in skin thicknesses and length scale parameter values while decreasing when beams’ length and core thicknesses increase.

On the nonlinear post-critical responses of VAT sandwich beams with variable stiffness composite skins under axial compression

In this article, the nonlinear post-critical problem of Variable Angle Tow (VAT) sandwich beams with variable stiffness composite skins under axial compression is studied for the first time. The present sandwich beam model is constructed from a modified version of extended high-order sandwich panel theory (EHSAPT), in which the first-order shear deformation theory with von Kármán’s strain–displacement relation is utilized for the skin layer, whereas the high-order theory with linear strain–displacement is applied to characterize the deformation of the core layer. Based on the principal of virtual work, a system of differential equations with variable coefficients is firstly formulated for the postbuckling problem of VAT sandwich beams with variable stiffness composite skins under plane-strain state. Meanwhile, a more efficient Rayleigh–Ritz approach combined with variational formulas is employed to derive the nonlinear algebraic equations in weak forms, and thereafter a sophisticated path-tracing technique is applied to determine the postbuckling equilibrium path. Both super-critical and sub-critical postbuckling states of VAT sandwich beams can be readily captured under the present analytical model framework. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing against the results available in literature. Effects of variation in skin stiffness on both the critical instability and post-critical behaviours of VAT sandwich beams are discussed in numerical examples. It is found that the change in skin stiffness plays a determinative role in the formation of nonperiodic wrinkling patterns of VAT sandwich beams. In addition, the change in skin stiffness may cause global and local instability loads to be close to each other, and accordingly, the possibility of encountering interactive buckling is increased. The results presented herein may be beneficial for the design of VAT sandwich structures under compression.

High-order torsion analysis of sandwich beams of rectangular cross-section

The structural response of soft-core sandwich beams of rectangular cross-section subjected to torsion is investigated. The paper uses the concepts of the extended high-order sandwich panel theory and generalizes the approach to account for the torsional response and its unique mechanisms. An analytical model that utilizes this generalized approach is formulated. The results of the model are presented and compared with the results of a reference model and a finite element analysis. Emphasis is placed on the torsional resistance mechanisms, their interaction, and the torsional-related localized effects. The results of the present model are in good agreement with the finite element analysis in terms of displacements and stresses, including the localized effects attributed to the soft-core. It is also shown that the present model can identify and quantify the localized effects while reference models available in the literature fail to do so. The results of the model contribute to the understanding of the torsional response of soft-core sandwich beams, including the transition between the warping and Saint-Venant torsional mechanisms and the localized effects associated with the soft-core and the application of boundary conditions to the different layers.

Nonlinear transient response of elastoplastic sandwich beam in underwater blast and the fluid-structure interaction

This work focuses on a new theory on the transient response of elastoplastic sandwich beam in underwater blast, and two fully coupled fluid-structure interaction (FSI) models are proposed, to gain insight into the propagation and dissipation of underwater blast wave, and the optimal design for sandwich structure with excellent underwater blast resistance. Based on the nonlinear dynamic high-order sandwich panel theory (EHSAPT) according to the authors’ preceding work [1], the governing equation of motion is constructed and modified by integrating the rarefaction-wave-related term into the structural damping matrix. So, the underwater blast analysis is transformed into the dynamic response of sandwich structure with structural damping under the superposition of incident and reflected waves in vacuum. Furthermore, this governing equation is simplified to get the analytical solution of impulse transferred to the sandwich structure, by implementing displacement and core rigidity assumptions. Finite element simulation and two existing FSI models are used to validate the accuracy. Conclusions are drawn that, the key aspects dominating FSI mechanism and time-domain response include pressure signatures, fluid characteristics, front-face mass, core density, and core elastic modulus, and the quantitative relationships are provided. Moreover, the core elastic modulus dominates the impulse transmitted to the sandwich structure, and the core strength and strain hardening govern the pressure delivered to the supports.

Large deformation of sandwich wide panel

This work addresses the general nonlinear static and dynamic formulations of sandwich panels with compressible cores under large deformation, within the framework of Extended High-order Sandwich Panel Theory. The commonly-used assumption named large displacement with moderate rotation, is assessed. Within a total Lagrange manner, the weak form of the nonlinear equation of equilibrium is derived via the principle of virtual work and further linearized through the Newton-Raphson iteration. Both the static and dynamic governing equations are acquired with respect to the nodal degrees of freedom. Numerical techniques involving Newton-Raphson method for the static analysis, and Newmark-Beta method for the time domain integration are employed to get the solutions. The static and transient mechanical response of simply supported sandwich panels under static pressure and dynamic blast is conducted respectively. The closed linear solutions and finite element models are accomplished to validate. Conclusions are drawn that the deformation gradient, as well as the displacement and rotation, critically determines the structural stiffness and internal force, and their relationship is originally provided in this paper. The structural stiffness at large deformation is stronger than that at small deformation. The assumption of large displacement with moderate rotation is inadequate for sandwich structure in large deformation.

Large amplitude free vibration of sandwich beams with flexible core and FG Graphene Platelet Reinforced Composite (FG-GPLRC) face sheets based on extended higher-order sandwich panel theory

The present article deals with the nonlinear free vibration analysis of sandwich beams with flexible core integrated with face sheets which are reinforced with graphene platelets (GPLs). Each layer of the composite media may have a different amount of graphene platelets which leads to a functionally graded (FG) distribution. The elastic modulus of the reinforced composite (RC) is calculated via the Halpin–Tsai model. Following the extended higher order sandwich panel theory (EHSAPT) which captures the core thickness compressibility effect and the axial stress of the core, the complete set of governing equations is obtained using the Lagrangian formulation. To analyze the effect of the inclusion of nonlinear terms in the core, hitherto not reported in the literature, results are computed considering the von Kármán type of geometrical nonlinearity assumptions for the core and face sheets. With the aid of the Ritz-based method, Chebyshev functions are implemented as admissible functions of the displacement fields. Furthermore, the nonlinear governing equations are solved using a direct displacement control technique. As the EHSAPT can be applied with any combination of the core and face sheets and not only the soft cores that the other theories demand, the results for the sandwich beam with stiff core and consist of functionally graded carbon nanotube-reinforced composite face sheets provided for validation. Since this study is the first in the open literature to develop nonlinear vibration of the sandwich beam based on EHSAPT, Timoshenko beam theory (TBT) is also applied to investigate the problem and to extract nonlinear frequencies with various stiffness of the core, meanwhile, the results are compared with those obtained by the EHSAPT. Parametric studies are conducted in detail to examine the influences of weight fraction and distribution pattern of graphene platelets, core to face sheet thickness ratio, boundary conditions, length to thickness ratio, and amplitude of vibrations. It is found that by increasing the stiffness of the core, the inclusion of nonlinear effects in the core as well as the face sheet strain nonlinearity is more significant. Finally, the nonlinear vibration analysis of sandwich beams can change the variation trend of frequency predicted in the linear analysis, and the different trends are observed depending on the vibration amplitude.

Optimum design of sandwich composite structures under various dynamic loads using the equivalent static load method

Sandwich composite structures are high-performance materials with greater specific strength and specific stiffness properties, than widely used conventional metals. In this study, sandwich composite structures under dynamic loads were optimized using the equivalent static load method (ESLM). First, a finite element analysis was performed on the sandwich composite structure, using the high-order sandwich panel theory (HSAPT), considering the core material to be compressible. Further, it was optimized by applying the ESLM to three different sandwich composite structures under different types of dynamic loads. The results of the proposed method and conventional dynamic response optimization method were compared. The ESLM was observed to attain a convergence at a rate 1/24 times faster than the conventional method.

Face-core interfacial debonding characterization model of an all-composite sandwich beam with a hexagonal honeycomb core

In this paper, the mode I face-core interfacial debonding of an all-composite sandwich beam with carbon fiber reinforced polymer (CFRP) hexagonal honeycomb core is characterized through a theoretical model and double cantilever beam (DCB) tests. The theoretical model is established by combining Timoshenko’s beam theory for the face sheets and Extended High Order Sandwich Panel Theory (EHSAPT) for the CFRP hexagonal honeycomb core and the nonlinear exponential cohesive zone model (CZM) is used to model the face-core interfacial constitutive relation, which can couple the normal and the tangential tractions. Then three groups of all-composite sandwich beams with CFRP hexagonal honeycomb core are fabricated and DCB tests are conducted to characterize the mode I face-core interfacial debonding behaviors. The theoretical model is validated by comparing the results of fracture toughness obtained from theoretical calibration and the MBT method. The crack tip relative displacement and crack tip stress indicate the DCB test is mode I and mode II coupled debonding process and it is governed by mode I fracture. Moreover, the mechanism of crack nucleation and propagation between the face-core interface is revealed in detail. Investigation from this paper seeks and to explain the physical phenomenon of the debonding mechanism of all-composite sandwich beams. Finally, a parametric study is carried out to evaluate the effect of fracture toughness, characteristic length, face sheet thickness and fracture factors on the load–displacement response as well as the effect of fracture toughness, characteristic length, face sheet thickness and core thickness on the interfacial strength and damage zone length.

Free vibration analysis of deep doubly curved soft-core sandwich panels with different boundary conditions

This paper presents a free vibration analysis of deep doubly curved soft-core sandwich panels where the face sheets are made of laminated composite, and the core is considered to be compressible. The compressibility of the core is formulated using the Extended Higher-Order Sandwich Panel Theory (EHSAPT). Accordingly, the first-order shear deformation theory is used to model the face sheets, and third- and second-order functions are used to model the in-plane and transverse displacements of the core, respectively. The equations of motion and relevant boundary conditions are derived using Hamilton’s principle. The Rayleigh-Ritz method with single series expansions with two-variable orthogonal polynomials as trial functions is employed to obtain the panel's natural frequencies and mode shapes. Convergence and comparison studies are carried out to demonstrate the efficiency and accuracy of the solution, and the method is validated against the results obtained through different approaches in the literature. Finally, a parametric study illustrates the effect of various parameters on vibration characteristics of cylindrical and spherical sandwich panels with different boundary conditions, which may be used as benchmark results for future studies.

Vibration analysis of sandwich beam with honeycomb core and piezoelectric facesheets affected by PD controller

Free vibration analysis of a sandwich beam with honeycomb core and piezoelectric face sheets, which is rested on the viscoelastic foundation is investigated. The thermal environment and the electric field are applied to this structure. Also, it is affected by the proportional-derivative (PD) controller. The amount of gain in this controller can affects the vibration frequency. The displacement components are expressed by improved high-order sandwich panel theory (IHSAPT) that considers continuity conditions for transverse shear stress at the interfaces and the zero transverse shear stresses conditions on the upper and lower surfaces of the beam and core flexibility. The motion equations are derived and solved by Hamilton's principle and Navier's method, respectively. This paper examines the effects of various parameters, such as the internal aspect ratio and the cell angle/thickness of the honeycomb core, temperature variations, viscoelastic environment, electric load, and control gain on its natural frequencies. The results show when the honeycomb core's to face sheet's thickness ratio increases, the beam dimensionless frequency increases, too. Also, by increasing the internal aspect ratio of honeycomb core, the frequency of sandwich beam decreases. The results of this study can be used to vibration control in aerospace engineering and constructions.

Optimization of a higher-order sandwich composite beam under uncertainties

The present paper investigates the reliable and robust optimum design of a higher-order sandwich composite beam under the effect of uncertainty in material properties. The sandwich beam is modeled using the extended higher-order sandwich panel theory. The optimization approaches employed in this work are reliability-based design optimization, robust design optimization, and a hybrid reliability-based robust design optimization. The efficiency of the optimization process is enhanced by using a novel time domain spectral element method-based polynomial chaos surrogate model. The performance of the surrogate model is further improved by using Sobol indices based sensitivity analysis. The optimization procedure is performed using an accelerated particle swarm optimization. The numerical results of the reliable and robust optimal design are presented for both the soft and stiff core sandwich beam. Furthermore, the effect of load density and allowable deflection on the optimal design is also examined.

Dynamic interfacial crack propagation and kinking in sandwich panels

The dynamic interfacial crack propagation and crack kinking phenomena in sandwich panels are studied. The paper derives an analytical approach based on the Extended High-Order Sandwich Panel Theory (EHSAPT) and a cohesive interface modeling. The mathematical modeling of the panel makes a distinction between the face sheets and the core. The core is further divided into two bodies aiming to allow the nucleation of a shear crack in the core or the kinking of a core-face sheet crack into the core. The panel therefore includes four layers that are interconnected through three interfaces. The First-Order Shear Deformation Theory (FSDT) is adopted for the two face sheets and the EHSAPT is applied to the two core layers. Translational, rotational, and high order inertial effects are considered in all components. A geometrical nonlinear behavior with linear elastic laws is adopted for the face sheets while the core layers use linear kinematic relations and linear constitutive laws. The interfaces use nonlinear cohesive laws to model the dynamic nucleation, propagation, and kinking of cracks. The dynamic model is used for a numerical study of sandwich panels subjected to three point bending and end shortening loading conditions. The study explores the coupling of the crack propagation with the dynamic response. A comparison of the dynamic results with their static counterparts further explains the physical response of the sandwich panel and the dynamic characteristics of its failure mechanism.

Nonlinear static analysis of plates with arbitrary aspect ratios using Extended Higher Order Sandwich Panel Theory

A nonlinear static analysis of the sandwich plate using variational techniques and the Ritz method is presented. The kinematic description developed for a sandwich plate undergoing small strains and moderate rotations within the framework of Extended Higher Order Sandwich Panel Theory is considered. Employing the Ritz method, the total potential energy of the system is developed. Four different cases and combinations of boundary conditions are studied and assumed solutions satisfying the geometric boundary conditions are developed. The total potential energy results in nonlinear algebraic equations for the unknown coefficients in the assumed solution. The nonlinear equations are then solved using the Newton–Raphson method. A convergence study is then carried out to study the effect of variation of the number of terms in the assumed solution. In order to analyze the effect of inclusion of nonlinear effects in the core, results are computed considering only the facesheets with nonlinear strains and the core is considered to remain in the linear range. A stress analysis is carried out to compare the stresses predicted by the linear theory against the nonlinear results. After calculating and comparing the results for the simply supported case, three more cases for different sets of boundary conditions are also considered and presented. Since EHSAPT allows for analysis of plates with variable aspect ratios a further case where it is assumed that the depth of the plate is half the width is also considered and the results are presented for various geometric configurations.

Dynamic Stability of Sandwich Beams/Wide Plates Subjected to Axial Impulsive Loads

Abstract This paper presents an analysis for the dynamic stability of sandwich beams/wide plates subjected to axial impulsive loads. The formulation and solution of the problem is done by use of the extended high-order sandwich panel theory (EHSAPT). With the initial geometric imperfection included, the equations of motion in terms of seven generalized displacements are derived. The dynamic response of sandwich panels subjected to three different types of impulsive loads, namely, step, linear decay, and triangular impulse, is studied. Furthermore, the effects of the oscillation mode number, face/core materials, and geometries are investigated. It is observed that all measurements of the dynamic response, such as the maximum displacements, strains, and stresses, change at the same rate as the change of the impulse load magnitude and duration, for a specific impulse load profile. When the impulse load is lower than the static buckling load, the dynamic response is bounded no matter how long the load is applied. A step impulsive axial load with magnitude lower than the static buckling load can lead a sandwich panel to have a dynamic response as high as twice the static response. When the impulse load is higher than the static critical load, the dynamic response is unbounded with increasing load duration. However, it is possible that the dynamic response can be controlled at a low level if the duration of the impulse load is short enough, and thus, in this case, the load can safely exceed the static critical load.

Three-dimensional pulse response of curved composite sandwich panel with compressible core using non-linear higher-order theory

In this paper, the dynamic response of cylindrical sandwich panels with compressible core is obtained using the extended non-linear higher-order sandwich panel theory. It is assumed that the sandwich panel has simply supported boundary at all edges and is consisted of orthotropic face sheets and viscoelastic core layer. To describe the mechanical properties of the viscoelastic foam core, the Kelvin-Voigt linear viscoelastic model was applied. Three-dimensional linear equations of motions were used to describe the sandwich panel deformations. The effects of various parameters including the panel span, core and facing thickness, the viscous damping factor, pulse duration, and maximum pressure on the dynamic response of the sandwich cylindrical panel are studied. The results obtained from present method are compared with finite element solutions and those reported in the literature, and consequently, agreement among the results could be observed. The results shown that applied viscoelastic model has a signification effect on the panel response and reduces the magnitude of vibrations. The presented programming code (DQ) needs less computational time and computer hardware capacity and is faster than finite element solution.

Modeling of interfacial crack propagation and kinking in sandwich panels

The paper presents an analytical model for the nucleation, propagation, and kinking of cracks in sandwich panels. The analytical approach adopts the variational principle of minimum of the total potential energy and it is based on cohesive interface modeling. The mathematical modeling considers two face sheets, a core layer, and an interfacial core-face sheet crack that can kink into the core and form a diagonal shear crack. This phenomenon, which has been observed in experiments, divides the core into two bodies. The core is therefore modeled using such two body approach. Three interfaces are introduced between the four components and appropriate interfacial laws are used to describe the interfacial phenomenon. The First-Order Shear Deformation Theory with geometrically nonlinear kinematic relations is adopted for the face sheets. The two core layers use linear geometrical relations and assume the high order polynomial displacement fields of the Extended High-Order Sandwich Panel Theory. Linear elastic laws are considered for the two face sheets and the core. The model is validated through a comparison with experimental observations of pre-cracked Double Cantilever Beam sandwich specimens that show a diversity of crack paths. The comparison and the quantitative investigation explore the capabilities and the limitations of the approach and shed light on the nucleation, debonding, and kinking phenomena in soft-core sandwich panels.

A new sandwich element for modeling of partially delaminated sandwich beam structures

In this paper, a new sandwich beam element is introduced for analyzing sandwich beam structures with a flexible core and partially delaminated regions. In this element, interfaces between the core and face sheets are modeled by two independent layers. The model uses a high-order sandwich panel theory to consider flexibility of the core with nonlinearities associated with geometry and real contact characteristics of the delaminated regions. The proposed motion field takes advantage of both displacement and displacement gradients in the core boundaries. Therefore, the kinematics allow continuity conditions for displacements and rotations at the interfaces to be exactly satisfied in fully bounded and delamination regions. By using finite element (FE) formulation with Hermite shape functions, elemental vectors and matrices are derived in the framework of Hamilton’s principle. FE governing equations are solved by Newton-Raphson iterative scheme. A 2D FE method is also developed to verify predictions of the sandwich model for various delamination cases. Comparison studies show that results from both sandwich and 2D FE models are in a good agreement. They can predict large-deformation results for the sandwich behaviors much better than the simplified model available in the literature. A set of parametric study is devoted to provide an insight into the influence of boundary condition, number and position of delaminated regions on deformations, stresses and instability of fully clamped and cantilever sandwich beams. The developed formulation is not only more computationally efficient than 2D models usually used for such analysis, but also at the same time is accurate, simple and robust. It is also found that modeling of the delaminated zone core and stress distribution at each interface independently is crucial to accurately analyze instability behaviors of sandwich structures.