Extended 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.

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.

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.

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.

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.

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.

Geometric and material nonlinearities of sandwich beams under static loads

A new theory developed from extended high-order sandwich panel theory (EHSAPT) is set up to assess the static response of sandwich panels by considering the geometrical and material nonlinearities simultaneously. The geometrical nonlinearity is considered by adopting the Green–Lagrange-type strain for the face sheets and core. The material nonlinearity is included as a piecewise function matched to the experimental stress–strain curve using a polynomial fitting technique. A Ritz technique is applied to solve the governing equations. The results show that the stress stiffening feature is well captured in the geometric nonlinear analysis. The effect of the geometric nonlinearity in the face sheets on the displacement response is more significant when the stiffness ratio of the face sheets to the core is large. The geometric nonlinearity decreases the shear stress and increases the normal stress in the sandwich core. By comparison with open literature and finite element simulations, the present nonlinear EHSAPT is shown to be sufficiently precise for estimating the nonlinear static response of sandwich beams by considering the geometric and material nonlinearities simultaneously.

Nonlinear Stability Analysis of Sandwich Wide Panels—Part II: Postbuckling Response

The nonlinear post-buckling response of sandwich panels based on the extended high-order sandwich panel theory (EHSAPT) is presented. The model includes the transverse compressibility, the axial rigidity, and the shear effect of the core. Both faces and core are considered undergoing large displacements with moderate rotations. Based on the nonlinear weak form governing equations, the post-buckling response is obtained by the arc-length continuation method together with the branch switching technique. Also, the post-buckling response with imperfections is studied. The numerical examples discuss the post-buckling response corresponding to global buckling and wrinkling. It is found that due to the interaction between faces and core, localized effects may be easily initiated by imperfections after the sandwich structure has buckled globally. Furthermore, this could destabilize the post-buckling response. The post-buckling response verifies the critical load and buckling mode given by the buckling analysis in part I. The axial rigidity of the core, although it is very small compared to that of the faces, has a significant effect on the post-buckling response.

Nonlinear Stability Analysis of Sandwich Wide Panels—Part I: Buckling Behavior

This is a series of two papers in which the nonlinear stability behavior of sandwich panels is investigated. This part presents the buckling behavior and focuses on the critical load and the buckling mode. The buckling analysis is based on the extended high-order sandwich panel theory (EHSAPT) which takes transverse compressibility and axial rigidity of the core into account. It allows for the interaction between the faces and the core. The geometric nonlinearity, i.e., large displacement with moderate rotation, is considered in both faces and core. The weak form governing equations are derived based on the EHSAPT-based element. Detailed formulations and analysis procedures are provided. It presents a general approach for arbitrary buckling type without decoupling it into isolated global buckling and wrinkling. There are no additional assumptions made about the prebuckling state and buckling mode shape, which are commonly presumed in the literature. In addition, edge effects which are also commonly neglected are included. The prebuckling state is determined via a nonlinear static analysis. Solving an eigenvalue problem yields the critical load and the corresponding eigenvector gives the buckling mode. Sandwich panels with different lengths are studied as examples. Both global buckling and wrinkling are observed. It shows that the axial rigidity of the core has a pronounced effect on both the critical load and the buckling mode.

Non-linear wrinkling of a sandwich panel with functionally graded core – Extended high-order approach

The non-linear response of a sandwich panel with the core that consists of a functionally graded material (FGM) undergoing in-plane loading is investigated. The panel consists of two face sheets, metallic or composite laminated, and an FGM core that is a medium whose mechanical properties change through the depth facilitating a desirable response of the structure. The effect of the FGM core is introduced through the constitutive relations that affect conventional and high-order stress resultants and stress couples in the panel. The formulation employs the Extended High-Order Sandwich Panel Theory (EHSAPT) to assess the effect of the FGM material of the core. A variational approach is adopted to derive the linear and non-linear governing equations with a prescribed FGM distribution through the depth of the core. The wrinkling study of FGM panels includes two loading scenarios where in-plane loads are applied through a rigid edge beam connected to the core only and where the loads are applied through a rigid edge structure attached to both the face sheets and core causing uniform end shortening. The post-wrinkling behavior is also considered to prove that the initial pattern of wrinkles is not affected.

Geometrically nonlinear analysis of sandwich beams under low velocity impact: analytical and experimental investigation

Nonlinear low velocity impact response of sandwich beam with laminated composite face sheets and soft core is studied based on Extended High Order Sandwich Panel Theory (EHSAPT). The face sheets follow the Third order shear deformation beam theory (TSDT) that has hitherto not reported in conventional EHSAPT. Besides, the two dimensional elasticity is used for the core. The nonlinear Von Karman type relations for strains of face sheets and the core are adopted. Contact force between the impactor and the beam is obtained using the modified Hertz law. The field equations are derived via the Ritz based applied to the total energy of the system. The solution is obtained in the time domain by implementing the well-known Runge-Kutta method. The effects of boundary conditions, core-to-face sheet thickness ratio, initial velocity of the impactor, the impactor mass and position of the impactor are studied in detail. It is found that each of these parameters have significant effect on the impact characteristics which should be considered. Finally, some low velocity impact tests have been carried out by Drop Hammer Testing Machine. The contact force histories predicted by EHSAPT are in good agreement with that obtained by experimental results.

Analytical, numerical and experimental investigation of low velocity impact response of laminated composite sandwich plates using extended high order sandwich panel theory

The Nonlinear dynamic response of a sandwich plate subjected to the low velocity impact is theoretically and experimentally investigated. The Hertz law between the impactor and the plate is taken into account. Using the Extended High Order Sandwich Panel Theory (EHSAPT) and the Ritz energy method, the governing equations are derived. The skins follow the Third order shear deformation theory (TSDT) that has hitherto not reported in conventional EHSAPT. Besides, the three dimensional elasticity is used for the core. The nonlinear Von Karman relations for strains of skins and the core are adopted. Time domain solution of such equations is extracted by means of the well-known fourth-order Runge-Kutta method. The effects of core-to-skin thickness ratio, initial velocity of the impactor, the impactor mass and position of the impactor are studied in detail. It is found that these parameters play significant role in the impact force and dynamic response of the sandwich plate. Finally, some low velocity impact tests have been carried out by Drop Hammer Testing Machine. The results are compared with experimental data acquired by impact testing on sandwich plates as well as the results of finite element simulation.

Nonlinear dynamic response of sandwich wide panels

This work focuses on the general nonlinear dynamic formulation of sandwich panels based on the Extended High-order Sandwich Panel Theory. The faces and the core are both considered undergoing large deformation with moderate rotation. In the literature, when it comes to nonlinear dynamic response, various simplifying assumptions are adopted in the nonlinear kinematic relations. A critical assessment of these simplifications, as well as a comprehensive investigation of the geometric nonlinearity effects on the dynamic response of sandwich beams/wide panels is presented. The Extended High-order Sandwich Panel Theory (EHSAPT)-based finite element is used to formulate the equations of motion, and the time response is obtained by the central difference method. The transient response of a sandwich panel subjected to a distributed blast loading is studied as a numerical example. By neglecting the nonlinear terms, the nonlinear dynamic analysis is reduced to the linear dynamic small deformation analysis, for which a closed-form elasticity solution exists. This, the linear response is first compared with the closed-form linear dynamic elasticity solution. It is shown that the linear EHSAPT yields an identical response to the dynamic elasticity analysis. Subsequently, the dynamic response is evaluated by considering the geometric nonlinearities. Several simplified nonlinear models with partial nonlinear terms included are considered. These models reflect the assumptions commonly adopted in the literature. Numerical results are presented to investigate the effects of geometric nonlinearities on the displacements and stresses. It is shown that the geometric nonlinearities have a significant effect on the dynamic response of sandwich panels and, in particular, the nonlinear terms in the core kinematic equations are especially important.

Low velocity impact response of sandwich beams with soft cores and carbon nanotube reinforced face sheets based on Extended High Order Sandwich Panel Theory

Response of sandwich beam with carbon nanotube reinforced composite (CNTRC) face sheets and soft core subjected to the action of an impacting mass based on the Extended High Order Sandwich Panel Theory (EHSAPT) is first analyzed. Distribution of fibers through the thickness of the face sheets could be uniform or functionally graded (FG). Contact force between the impactor and the beam is obtained using the conventional Hertz law. The field equations are derived via the Ritz based applied to the total energy of the system. The solution is obtained in the time domain by implementing the well-known Runge–Kutta method. After examining the validity of the present solution, the effects of distribution of Carbon Nanotubes (CNTs), nanotube volume fraction, core-to-face sheet thickness ratio, initial velocity of the impactor and the impactor mass are studied in detail. Finally, it is concluded that, the highest peak contact force and the lowest indentation of the top face sheet belong to the sandwich beam with V distribution figure of face sheet, followed by the UD and Λ-ones, respectively. Also, numerical results reveal that employing FG-CNTs in face sheets has a prominent role on impact response of the sandwich panel with soft core.

Dynamic extended high order sandwich panel theory for transient response of sandwich beams with carbon nanotube reinforced face sheets

The dynamic version of the Extended High Order Sandwich Panel Theory (EHSAPT) for a sandwich beam with soft core and carbon nanotube reinforced composite (CNTRC) face sheets is first formulated. Distribution of fibers through the thickness of the face sheets could be uniform or functionally graded (FG). The influences of boundary conditions on dynamic response of the sandwich panel are investigated. In each type of boundary condition the effects of nanotube volume fraction and their distribution pattern, core-to-face sheet thickness ratio, on many essential involved parameters of the sandwich beam with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets are studied in detail. The results for a transient displacement of sandwich beam with isotropic face sheets reveal that the EHSAPT is very accurate during the initial, transient phase of dynamic loading in comparison with conventional high order sandwich panel theory (HSAPT). Finally, it is concluded that the sandwich beam with V distribution figure of face sheets is the strongest and the smallest transverse displacement, followed by the X, UD, O and ∧-ones, respectively.

Static analysis of sandwich panels with non-homogeneous soft-cores by novel weak form quadrature element method

The extended high order sandwich panel theory (EHSAPT) is employed to analyze the static behavior of sandwich beams with non-homogeneous soft-cores. The material properties of the orthotropic core vary along the thickness direction according to a power-law form starting from its middle plane to the upper or bottom faces. Due to complexity of the theory, the solutions are obtained numerically by using the weak form quadrature element method. Based on the expanded EHSAPT and differential quadrature rule, a novel N-node weak form quadrature sandwich beam element with Gauss Lobatto Legendre (GLL) nodes is established. The element stiffness matrix and work equivalent load are obtained by using GLL quadrature. Detailed formulations are worked out. Different materials for the face sheets and core of the sandwich beam structure are considered. Numerical results are presented to investigate effects of the power-law exponents, boundary conditions, different types of material system and loading on the displacements and stresses of the sandwich panel with non-homogeneous soft core.

Non-linear response of curved sandwich panels – extended high-order approach

The geometrical non-linear behavior a curved sandwich panel with a stiff or compliant core when subjected to a pressure load using the Extended High-Order Sandwich Panel theory (EHSAPT), is presented. The formulation follows the EHSAPT procedure where the in-plane. i.e circumferential rigidity of the core is considered and the distribution of the displacements through the depth of the core are presumed. These displacement distributions are the closed-form solutions of the 2D governing equations of the curved core without circumferential rigidity that appear in the HSAPT curved sandwich panel model. The mathematical formulation includes the field equations along with the appropriate boundary and continuity conditions that take into account the high-order stress resultants in the core due to the presumed distributions. Finally a numerical study is conducted for a panel loaded by a distributed pressure at the upper face sheet. It reveals that the post-buckling response of a curved sandwich panels is associated with shallow to deep wrinkling deformations of the upper face sheet in the case of a simply-supported panel or a general non-linear pattern without wrinkles in the case of pinned supports with a short span. In both cases a stable post-buckling response is observed similar to that of a plate one.

Extended high order sandwich panel theory for bending analysis of sandwich beams with carbon nanotube reinforced face sheets

Bending analysis of a sandwich beam with soft core and carbon nanotube reinforced composite (CNTRC) face sheets in the literature is presented based on Extended High order Sandwich Panel Theory (EHSAPT). Distribution of fibers through the thickness of the face sheets could be uniform or functionally graded (FG). In this theory the face sheets follow the first order shear deformation theory (FSDT). Besides, the two dimensional elasticity is used for the core. The field equations are derived via the Ritz based solution which is suitable for any essential boundary condition. The influences of boundary conditions on bending response of the sandwich panel with soft core and CNTRC face sheet are investigated. In each type of boundary condition the effect of distribution pattern of CNTRCs on many essential involved parameters of the sandwich beam with functionally graded carbon nanotube reinforced composite (FG- CNTRC) face sheets are studied in detail. Finally, experimental result have been compared with those obtained based on developed solution method. It is concluded that, the sandwich beam with X distribution figure of face sheets is the strongest with the smallest transverse displacement, and followed by the UD, O and ∧-ones, respectively.

Geometrically nonlinear extended high order analysis for sandwich beams based on elastic–plastic core shear behavior

Geometrically nonlinear response of a sandwich panel based on elastic–linear plastic (bilinear) shear behavior of the core is presented. The extended high order sandwich panel theory (EHSAPT) is applied to obtain governing equations. The nonlinear Von Karman type relations for strains of face sheets and the core are adopted. The face sheets follow first order shear deformation theory along with the linear elastic assumption. A Ritz based solution which is suitable for any essential boundary condition is implemented. Two types of boundary conditions are considered simply supported and immovable clamped at both edges. In each type of boundary conditions, the effects of plastic shear modulus of bilinear material behavior on transverse shear stress distribution along with detection of the elastic and plastic regions within the core, force and moment resultants, transverse displacement distributions in the face sheets, in-plane and vertical displacements through the thickness of the core are studied in detail. Besides, all results based on the bilinear material assumption of the core are compared with the linear elastic ones. Finally, for the case study of the simply supported beam, the results of the constitutive model are compared with finite element simulation.

Static analysis of higher order sandwich beams by weak form quadrature element method

Based on the extended high order sandwich panel theory (EHSAPT) and differential quadrature rule, an N-node novel weak form quadrature sandwich beam element is established. Gauss Lobatto Legendre (GLL) points are utilized as element nodes and GLL quadrature is used to obtain the element stiffness matrix and work equivalent load. Detailed formulations are given. Convergence study is performed. Numerical results are presented for sandwich beams with different boundary conditions subjected to distributed loadings. Different materials for the face sheets and core of the beam structure are considered. It is shown that the proposed beam element can yield very accurate displacements and stresses as compared to theoretical solutions.