Equivalent Single Layer Theories Research Articles

New Accomplishments on the Equivalence of the First-Order Displacement-Based Zigzag Theories through a Unified Formulation

The paper presents a critical review and new accomplishments on the equivalence of the first-order displacement-based zigzag theories for laminated composite and sandwich structures. Zigzag theories (ZZTs) have widely spread among researchers over the last few decades thanks to their accuracy in predicting the response of multilayered composite and sandwich structures while retaining the simplicity of their underlying equivalent single-layer (ESL) theory. The displacement field consists of two main contributions: the global one, able to describe the overall structural behaviour, and the local layer-wise one that considers the transverse shear continuity at the layer interfaces that describe the “zigzag” displacement pattern typical of multilayered structures. In the framework of displacement-based linear ZZTs, various assumptions have been made on the local contribution, and different theories have been deduced. This paper aims to provide a unified formulation for first-order ZZTs, highlighting some common aspects and underlying equivalencies with existing formulations. The mathematical demonstrations and the numerical examples prove the equivalence of the approaches to characterising local zigzag enrichment. Finally, it is demonstrated that the kinematic assumptions are the discriminants of the ZZTs’ accuracy.

Aeroelasticity analysis of anisogrid lattice sandwich cylindrical shells: Extended potential and piston theories

Based on the extended potential and piston aerodynamics theories, buckling and vibration analysis of sandwich cylindrical shells subjected to external airflow in both subsonic and supersonic regimes is presented. Sandwich cylindrical shells are considered to be made of same homogenous face sheets and an anisogrid composite lattice core with hexagonal cells. An equivalent single layer (ESL) theory in the framework of classical Love’s shell theory is used to formulate the global mechanical behavior of such shells. According to the extended potential and piston aerodynamic theories, aerodynamic pressure distributions are determined for steady, inviscid, irrotational compressible, and external airflow. The closed–form solutions of critical upstream pressure and natural frequency are developed for different boundary conditions using the Galerkin method. Finally, case studies are presented to compare between results of extended potential and piston theories and investigate the effects of upstream speed, geometric ratios, boundary conditions, and lattice specifications on various response quantities. Findings indicate that the differences in results of extended potential and piston theories are less than 3% in the supersonic regime with 2.3 ≤ M ∞ ≤ 2.7 for intermediate-length shells with a wide range of R . The piston theory is inadequate for analyzing aeroelasticity behavior of very long thin sandwich cylindrical shells.

A unified framework for equivalent single layer theories of composite beams, plates, and shells: a review

This article presents a comprehensive review of the unified framework for the equivalent single layer theories of composite beams, plates and shells. First, the approach to unified displacement fields has two main categories: one is to start from the assumption of linear displacement field variables. And the another is to start from the assumption of transverse strains and transverse displacement variable. In terms of unified shape functions, the shape function is defined as the function that exist in the transverse strain rather than the in-plane displacement variables. Moreover, the basic requirements for the choose of shape functions are also given and analysed in this paper. The unified shape functions are introduced in terms of the several functions and the unified polynomial model. In addition, a construction method for new shear strain shape functions is described in this review. Finally, the current research progress and future prospects are also given.

Mathematical modeling and dynamic analysis of carbon fiber reinforced plastic shaft-disk-bearing system considering internal damping effect

The internal damping has significant influence on the dynamic characteristics of carbon fiber reinforced plastic drive shaft. In this paper, a dynamic model of the single-span multi-disk carbon fiber reinforced plastic shaft with elastic support is established based on layer-wise theory, damping specific capacity model, and Euler–Bernoulli beam theory. The model takes into account physical factors such as rigid body displacement caused by elastic support deformation, gyroscopic effect of concentrated mass disk, viscous internal damping of the carbon fiber reinforced plastic shaft, and lamination parameters. The model is used to analyze the natural angular frequency and critical instability speed of carbon fiber reinforced plastic shaft-disk-bearing system. The analysis results are compared with those obtained from the equivalent modulus beam theory, equivalent single layer theory, simplified homogenized beam theory, and quadrature element method. All the results demonstrate the accuracy of the proposed model, thereby validating its effectiveness. Furthermore, the model is utilized to investigate the influence of lamination parameters and mass ratio on the natural angular frequency and critical instability speed. This study provides valuable insights for the dynamics and stability design of the carbon fiber reinforced plastic shaft-disk-bearing system.

Efficient zigzag theory-based spectral element model for guided waves in composite structures containing delaminations

Lamb wave-based structural health monitoring offers excellent potential for real-time delamination detection in laminated structures. It necessitates fast and accurate numerical simulation of guided wave interaction with delaminations. Towards this objective, we present an efficient layerwise zigzag theory (ZIGT) based time-domain spectral element for wave propagation analysis of composite beam- and panel-type structures (strips) containing delamination at arbitrary locations. The ZIGT delivers accuracy like layerwise theories by allowing slope discontinuities in the in-plane displacement field at layer interfaces while maintaining computational efficiency like equivalent single-layer (ESL) theories, making it ideal for such analyses. The high-order elemental nodes at Lobatto points have only four displacement variables, irrespective of the number of layers in the laminate. Delamination is modelled using the region approach, adapting the recently proposed hybrid point-least squares continuity method to satisfy the nonlinear displacement field continuity at the delamination fronts. The model’s efficacy is examined vis-à-vis elasticity-based elements, ESL theory-based elements, and the ZIGT-based standard finite element for free vibration and guided wave propagation responses of composite strips featuring multiple delaminations. The present model shows superior overall efficiency, accuracy, and convergence. An application of the model is also explored for identifying delamination locations from Lamb wave velocity fields.

Numerical prediction of thermoacoustic responses of CNT reinforced natural (luffa) fibre/epoxy hybrid composite and experimental verification

This research reported the thermoacoustic radiation characteristics of luffa fibre/epoxy hybridised with MW-CNT (multi-walled carbon nanotube) composite flat panel structures. The acoustic responses are evaluated under harmonic excitation and uniform thermal environments. The panel model is derived from a coupled scheme by merging two different computational techniques, i.e., finite element method, FEM (structural modelling) and boundary element method, BEM (fluid part), via higher-order deformation kinematics. Additionally, the higher-order panel model utilises an equivalent single-layer theory for computational purposes. The responses are obtained through MATLAB computer code from the derived mathematical formulation by customising the parameters (geometry and material). A few frequency responses with and without environmental effects are compared with experimental data from home-grown experimentation facilities. The impact of individual parameters is thoroughly analysed, and highlighted their significance in controlling the panel’s acoustic radiation behaviour. The findings of this study contribute to the understanding of the acoustic radiation properties of luffa fibre/epoxy hybridised with MW-CNT composite flat panels. The higher-order coupled FEM-BEM scheme-based tailored MATLAB code also provides a reliable computational tool for analysing various composite structural components for futuristic design. This research would be a valuable reference for optimising composite flat panel structural design and performance in engineering applications requiring enhanced vibroacoustic control.

Determination of Strength Characteristics of Basalt Composite in COMSOL Multiphysics

Some strength characteristics of basalt melt, and basalt fiber are considered. The analysis of micromechanics and stress analysis is carried out on the example of composite cylindrical reinforcement based on multilayer and equivalent single-layer theory. The distributions of the stress wave field of three different values of natural frequencies and mode shapes in three samples of objects from the melt of basalt and basalt fiber are studied using the COMSOL Multiphysics software package. It has been established that three parameters of basalt fiber and basalt melt mainly influence the natural frequency and shape of the modes: these are the volume fraction, the density of the basalt fiber and the density of the basalt melt. It has also been found that the concentration of basalt melt, and basalt fiber affect the strength of the composite material.

A seven-parameter high-order finite element model for multi-stable analysis of variable stiffness laminated shells

Variable angle fiber composites provide large design space for multi-stable structures. However, they require reliable numerical models to accurately predict their mechanical behavior characterized by a complex material response and large deformation. To this end, we propose an accurate yet efficient model for stability analysis of variable angle fiber laminated shells. In this model, first a seven-parameter shell formulation with Equivalent Single Layer theory, which is suitable for three-dimensional constitutive laws, is utilized to accurately characterize the large deformation. The fiber orientation angle is described by a linear variation from the center to the end of the shell. Then, to improve the accuracy and to reduce the computational cost, the spectral/hp element with high-order polynomial basis functions is adopted for discretization. Finally, complex nonlinear equilibrium paths are effectively tracked by the Riks method. By comparison with ABAQUS solutions, the proposed model shows to be efficient and accurate for variable angle fiber laminates under various conditions, in which both the buckling, the snap-through and tristability behaviors are studied. Moreover, variable angle fiber shells provide more design possibilities than straight fiber shells. By opportunely varying the fibers angle of each lamina, it is possible to control the stable states as well as to obtain multiple stable configurations (bi- and tri-stability).

A new sandwich beam model with layer-to-layer boundary modified displacements based on higher-order absolute nodal coordinate formulation

The absolute nodal coordinate formulation (ANCF) is one of the efficient modeling methods to solve the dynamics of flexible multibody systems. Zig-zag theory can accurately calculate the stress variation in the sandwich structure with only a few unknown variables, thus providing a good balance between computational efficiency and computational accuracy. Combining ANCF and zig-zag theory, this paper proposes a new sandwich beam model, called HANCF-ZZ, by introducing layer-to-layer boundary modified displacements in a high-order ANCF beam model. The capability of HANCF-ZZ for static deformation analysis and free vibration analysis is verified by comparing experimental, analytical, and numerical results in other literature. The results show that the HANCF-ZZ corrects the global stiffness overestimated by the equivalent single-layer theory, eliminates the locking problem of the low-order ANCF, and captures the nonlinear bifurcation phenomenon in large deformation problems under large loads. The HANCF-ZZ is not only an efficient numerical method for zig-zag theory but also enriches the ANCF element types to enable its application to the dynamics analysis of soft core sandwich beams.

A Novel Higher-Order Zigzag Function Applied to Refined Unified Beam Theory for the Analysis of Composite Laminated Materials

Highly efficient materials and structures are becoming increasingly common in military, aeronautical, aerospace, mechanical, and civil engineering applications. Composite materials have been developed to address the need to combine two or more materials to achieve superior properties. Many structural elements, such as laminated beams, use composite materials, but an accurate mathematical model of the bending behavior is required due to the abrupt changes in material properties in the interlaminar zones. This accurate model can be achieved using zigzag theory. This theory is one of the most commonly used formulations for modeling laminated beams. This theory is an improvement of the equivalent single-layer theory as an additional term called the “zigzag function” is used to represent the variation in the axial displacement along the cross section. This paper proposes a novel high-order zigzag function in a sinusoidal format. Several higher-order beam theories are combined with the proposed functions, and their performances are compared with those of other functions in the literature. The results reveal excellent agreement between the proposed formulation and the reference solution as well as a more effective combination of zigzag functions and beam theory.

A homogenization method for natural frequencies and damping analysis of composite pyramidal lattice truss sandwich structures based on representative volume elements

A homogenization method for the vibration analysis of periodic sandwich panels is presented. Periodic boundary conditions are applied to a representative volume element (RVE) of the structures. The complex responses of the RVE under eight sinusoidal excitations are calculated and the complex stiffness matrix of the sandwich panels is obtained by the direct-solution steady-state dynamic analysis (DSDA) using the finite element method (FEM) in the frequency domain. Based on equivalent single-layer (ESL) theory and finite element-modal strain energy (FE-MSE) method, the natural frequencies, the mode shapes and the loss factors of the structures are obtained by shell models. The effectiveness of the homogenization method is validated by using pyramidal lattice truss sandwich panels made of carbon fiber reinforced plastics (CFRP). The results obtained by the homogenization method are compared with those obtained by experiments and solid models. The effects of the number of cells and damping layers are discussed.

Free vibration analysis of functionally graded composite rectangular plates reinforced with graphene nanoplatelets (GPLs) using full layerwise finite element method

The current investigation is the first endeavor to apply the full layerwise finite element method (FEM) in free vibration analysis of functionally graded (FG) composite plates reinforced with graphene nanoplatelets (GPLs). Unlike the equivalent single-layer (ESL) theories, the layerwise FEM focuses on all three-dimensional (3D) effects. Therefore, it can compute interlaminar stresses and other local effects with the same accuracy as 3D FEM, requiring a lower computational cost. The GPLs weight fraction is presumed invariable in each layer but varies through the plate thickness in a layerwise model. The modified Halpin-Tsai model is employed to acquire the effective Young’s modulus. The rule of mixtures is applied to specify the effective Poisson’s ratio and mass density. First, the current method is validated by comparing the numerical results with those stated in the available works. Next, a thorough numerical study is performed to examine how various factors involving the pattern of distribution, weight fraction, geometry, and size of GPLs, together with the thickness-to-span ratio and boundary conditions of the plate, affect its free vibration behaviors. Numerical results demonstrate that employing a small percentage of GPL as reinforcement considerably grows the natural frequencies of the pure epoxy. Also, distributing more square-shaped GPLs, involving a smaller amount of graphene layers, and vicinity to the upper and lower surfaces make it the most efficient method to enhance the free vibration behaviors of the plate.

Forced vibration of sandwich pipes with zero Poisson's ratio honeycomb core under moving pressure

Based on Love shell theory and equivalent single layer theory, a new method is proposed in this paper to analyze the dynamic response of honeycomb sandwich pipe under the action of moving pressure. The results of displacement response under two boundary conditions are within 3.5% error range with the FEM results. The theoretical model takes into account the effect of thickness, and the calculation results are more accurate compared with the FSDT method, which can be used for both thin-walled and thick-walled pipes. By analyzing the effects of honeycomb structure parameters and pipe dimensions on the critical velocity and displacement response of honeycomb sandwich pipe, theoretical guidance is provided for the design of new pipeline structures.

Numerical thermal frequency prediction of smart composite structure and experimental validation

Thermal eigenvalue characteristics of layered composite panels are obtained numerically utilizing the higher-order deformation kinematics, and the exactness of the solution has been verified with in-house experimentation. The laminated panel configuration bonded with and without shape memory alloy (SMA) fibres is modelled mathematical using the equivalent single-layer theory and third-order displacement kinematics. The functional composite structural properties under the elevated temperature are introduced in the current model for the necessary activation via the marching technique (to include the material nonlinearity due to SMA). The structural governing equation of motion is obtained by Hamilton’s principle and discretized using the isoperimetric finite element steps. Further, the computational modal responses are predicted using the in-house MATLAB computer code. The numerical solutions’ accuracy and stability have been compared with the earlier research data, including a few experimental verifications of thermal/ambient frequencies for laminated composite without SMA fibre. Further, a few numerical experiments have been carried out to show the significance of SMA fibre on the frequency responses of the parent structure, including temperature-dependent properties. The structural/material inputs, i.e., geometrical shapes, properties, geometrical parameters, and SMA-relevant data (volume fractions, VSMA; pre-strain, and recovery stress), are utilized for the computation of new results. The final insights have shown a significant improvement of frequency values, i.e., 4 %, 6 %, and 8 % of the laminated composite, while the volume fractions of SMA increase VSMA = 10 %, 15 % and 20 % for 3 % prestrain, respectively.

Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

The transverse stretching vibration of thick sandwich plates, which is attributed to largely different stiffness at the adjacent layers, is a challenging issue, and efficient approach for such issue is less reported in the published literature. Thus, natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design, which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies. This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes. The proposed model is classified in the displacement-based equivalent single-layer theory, as the number of displacement parameters in the proposed model is independent of the layer number. The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures. To demonstrate the capability of the proposed model, typical examples are analyzed by utilizing the proposed model, the three-dimensional finite element method and the chosen higher-order models. By comparing with the exact three-dimensional elasticity solutions, it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models. Moreover, the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail, which can provide a reference for the accurate prediction of the higher-order frequencies.

Variable-kinematic finite elements for the aero-thermo-elastic analysis of variable stiffness composite laminates

This paper explores the aero-thermo-elastic properties of variable stiffness composite laminates (VSCLs) with fibers steered along curved trajectories using a refined 1 D model. High-order solutions are generated by the p-version finite element method in conjunction with the Carrera Unified Formulation (CUF), which is applied to structural modeling by employing the improved hierarchical Legendre expansion (IHLE) in particular. The merit of using such a kind of expansion lies in that both the Equivalent Single Layer (ESL) and Layer-Wise (LW) theories can be formulated in a compact and straightforward manner. The linear piston theory is utilized to model the aerodynamic force, whereas the steady-state temperature field is applied during the flutter analysis. The weak-form governing equations are derived by taking a perturbation of the equilibrium configuration, and divergence and flutter instabilities are determined by solving the related eigenvalue problem. The accuracy of the proposed model has been verified through several test cases involving flutter, thermal buckling, and vibration of symmetric and antisymmetric VSCLs. The obtained results were compared to those obtained using the ABAQUS software and the available literature. Finally, the effect of the kinematic order, shear coefficient, and temperature variation on the critical dynamic pressure of VSCLs is thoroughly investigated.

Nonlinear bending of a sandwich beam with metal foam and GPLRC face-sheets using Chebyshev–Ritz method: Effects of agglomeration and internal pore

In this study, a sandwich structure composed of functionally graded graphene platelet (GPL)-reinforced composite (FG-GPLRC) face-sheets and a metal foam core considering GPL agglomeration is proposed. The face-sheets are multilayered, and the GPL volume fraction is constant in each layer. However, this volume varies according to a layer-wise rule in the thickness direction. A two-parameter model supported by the Mori–Tanaka approach is developed to provide solutions that quantifiably consider the negative influence of GPL agglomeration on the elastic properties of nanocomposites. The core is made of a metal foam with closed cells whose effective elastic properties are determined using the Gibson–Ashby model. Two types of porosity distributions, uniform and non-uniform variations in thickness, are considered. The nonlinear bending behaviors of this sandwich beam are investigated. Within the framework of an equivalent single-layer theory, the formulations governing the nonlinear bending problems are derived from a third-order shear deformation theory supported by the von Kármán nonlinearity. The Chebyshev–Ritz method, a numerically stable method, is applied to describe the beam under various classical boundary conditions. An extensive numerical study is implemented to examine the effects of the agglomerations and volume fraction of GPLs in the face-sheets, internal pores, and core-to-face ratio on the nonlinear bending response of sandwich beams.

Transverse stretching vibration analysis of cross-ply composite and sandwich beams by means of an equivalent single-layer theory

Due to largely different stiffness between the face sheets and the core, the vibration modes corresponding to the transverse stretching displacement will occur in the sandwich plates. The third-order models will encounter difficulty to reasonably produce these frequencies corresponding to transverse stretching displacement modes, which will influence engineering design. Therefore, a global-local higher-order theory including transverse stretching effect has been proposed to study transverse stretching vibration, which can meet in advance the continuity conditions and the free surface conditions of transverse shear stresses. In the light of Hamilton’s principle, the equation of motion is presented, and analytical solutions of the simply-supported beams can be acquired by using Navier’s technique. In order to verify the capability of the proposed model, free vibration of composite beams is firstly investigated and the results have been compared to two-dimensional elasticity solutions. After the proposed model is assessed by two-dimensional elasticity solutions, the proposed model is extended to study dynamic behaviors corresponding to the transverse stretching displacement vibration modes, which will be verified by utilizing the three-dimensional finite element (3 D-FEM). Furthermore, variation of natural frequencies of the sandwich beams is also investigated by altering the thickness of the face sheets, which can further illustrate the capability of the proposed model.

FREE VIBRATION OF SKEW LAMINATES – A BRIEF REVIEW AND SOME BENCHMARK RESULTS

This study investigates and reviews prior research works on skew composite laminates. The equivalent single layer theories are explored and discussed. An exhaustive review on static and dynamic analysis of composite skew laminates is also presented. Subsequently, a nine node isoparametric plate bending element is used for free vibration analysis of laminated composite skew plate with central skew cut out. The effect of shear deformation is incorporated in the formulation considering first order shear deformation theory. Two types of mass lumping schemes are analysed to study the effect of rotary inertia. Certain numerical examples of plates having different skew angles, skew cut out sizes, boundary conditions, thickness ratios (h/a), aspect ratios (a/b), fiber orientations and number of layers are solved which will be useful for benchmarking of future studies.

Critical speeds analysis of spinning laminated composite shaft based on isogeometric analysis

This article presents the critical speed analysis of spinning laminated composite shaft under the different coupling mechanism effects based on isogeometric analysis. The analysis of the layered tubular composite shaft is usually performed using shell elements, but the present study concerns with the equivalent single layer theory (ESLT) obtained from the transformation of the cylindrical coordinates to the Cartesian coordinates and therefore, it avoids the use of non-uniform rational Bsplines. The results obtained are in good agreement with the modified EMBT and the modified SHBT. It is found that the proposed approach can yield highly accurate solutions compared with other existing methods available in the literature. In addition, the numerical results of critical speeds are obtained and investigated under the Poisson’s effects and different mechanism coupling effects, length-to-mean diameter ratios, and stacking sequence of layers.