Composite Plate Theory Research Articles

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.

A new mixed model based on the enhanced-Refined Zigzag Theory for the analysis of thick multilayered composite plates

In this paper, a new mixed model based on the enhanced Refined Zigzag Theory for thick multilayered composite plates is formulated. The kinematics is assumed cubic for the in-plane displacements and parabolic for the transverse one. The set of kinematic variables are reduced by using a partial constraint on the transverse shear stress, and a new set of enhanced cubic zigzag functions are obtained. The transverse normal stress is assumed as a smeared cubic function along the laminate thickness. The assumed transverse shear stresses profile is derived from the integration of local three-dimensional equilibrium equations involving a new set of strain variables. The entire formulation is developed by involving the Hellinger-Reissner functional and a penalty term for the strains compatibility; the governing equations and the consistent boundary conditions are derived. Finally, to assess the proposed model's predictive capabilities for general multilayered laminated and sandwich plates, results are compared with exact three-dimensional solutions available in the literature.

Structural optimization to improve the dynamic performance of novel co‐curing damping sandwich composites

AbstractCo‐cured damping composite structures have broad application prospects in high‐tech fields such as aerospace and space vehicles because of their low weight, large damping, and sufficient stiffness. In order to improve dynamic properties of conventional composite structures, a new type of damping composites embedded with multilayer viscoelastic film is proposed in this paper. Based on the complex modulus theory and vibration mechanics of composite plates and shells, vibration differential equation of composite plates with multilayer viscoelastic film is derived. The constitutive relation of the complex structure is established. The consistency of theoretical derivation and numerical simulation results provides a reference for the optimization of dynamic performance of damping sandwich composites. The effects of the distribution of viscoelastic film and the geometric structure parameters on dynamic performance of co‐cured damping composites are also discussed. The results show that the embedding of multi‐layer viscoelastic film can greatly enhance the loss factors of co‐cured composite structures under same damping layer thickness. This work lays a foundation for optimization of dynamic performance of lightweight and large damping composite structures.

Modeling the stress-strain state of variable-thickness composite shells and plates

A model for calculating the stress-strain state of variablethickness composite shells has been developed, based on assumptions such us the classical theory of Timoshenko-Mindlin shells. In the proposed model, the plate thickness is given by a function of curvilinear coordinates and is directly considered in the derivation of the equilibrium equations of the plate. The general equations of the theory of variable-thickness composite plates are derived. The article analyses the solution of the problem of plates bending under uniform pressure considering the variable thickness. For the numerical solution, the finite difference method (FDM) has been applied to the system of differential equations with matrix coefficients. For the resultant algebraic system, the FDM uses the tridiagonal matrix algorithm in computing the solution. The calculation results are compared with a plate of constant thickness. It is shown that the effect of thickness variability is quite significant.

Derivation and Verification of Laminated Composite T-Beam Theory

This study analyzes the composite laminated T-beams using the composite beam and laminated composite plate theories. The theoretical formula was derived assuming that the composite T-beam has one- and two-dimensional (1D and 2D) structures. The 1D analysis was performed according to the Kirchhoff-Love hypothesis, thereby considering only the axial strain to derive a relationship between the strain and displacement. The 2D analysis was performed considering the T-beam as a combination of two composite sheets. The effective stiffness of the beam was derived from the stress-strain and moment-curvature relationships. Furthermore, the deflection of the beam and the stress of each laminate were calculated. A simple support beam, made of AS4/3501-6 carbon/epoxy, was used as a composite laminated T-beam. MSC/NASTRAN finite element software was used for analysis. The accuracy of the theoretical formula and limitations of its use was verified using the finite element analysis. Higher accuracy of the theoretical formula was obtained at a composite beam aspect ratio greater than 15. The formula derived in this study is suitable for thin and long beams.

Evaluation of Laminated Composite Beam Theory Accuracy.

Carbon fiber-reinforced polymer (CFRP) has been widely implemented in electric vehicle bodies and aircraft fuselage structures. The purpose of CFRP is to reduce the weight and impart rigidity in the final product. A beam structure is typically used to bear the structural load, and the rigidity of the beam can be changed by arranging the laminated fibers at different angles. In this study, a composite I-beam is used as an example in engineering components. Because the theoretical model of the superimposed composite I-beam is established, the theoretical formula is based on the theoretical assumptions of the two-dimensional composite beam, and is combined with the traditional composite plate theory to analyze the maximum bending stress, strain, and deflection. During the theoretical derivation, it is assumed that the flanges of the I-beams are divided into narrow and wide flanges. The beams are considered as structures of beams and flatbeds. When a narrow flange is loaded in the side, the wide flange has no lateral deformation, and the lateral moments are neglected. Therefore, the accuracy of this formula needs to be verified. The purpose of this study is to verify the accuracy of theoretical solutions for the deflection and stress analysis of composite beams. A finite element analysis model is used as the basis for comparing the theoretical solutions. The results indicate that when the aspect ratio of the beam is >15, the theoretical solution will have better accuracy. Without the addition of the material, when 0° ply is placed on the outermost layer of the flange of the nonsymmetric beam, the effective rigidity of the beam is increased by 4–5% compared with the symmetrical beam. The accuracy range of the theoretical solution for the composite beams can be accurately defined based on the results of this study.

Vibration Excitation and Suppression of a Composite Laminate Plate Using Piezoelectric Actuators.

Piezoelectric (PZT) actuators bonded on a structure can be used to generate deformation and excite vibration for the shape control and vibration suppression, respectively. This article proposes a theoretical model for predicting vibrational response of a composite laminate plate with PZT actuators. The bending moment induced by the PZT actuator was obtained and applied on the composite laminate plate. Utilizing composite mechanics and plate theory, an analytical solution of the vibrational response of a composite laminate plate excited by the PZT actuator with oscillating voltage was derived. Furthermore, the finite element analysis using ANSYS software (2019 version) was carried out to compare with the proposed model with a good agreement. A parametric study was performed to investigate the influences of PZT location and frequency on the vibration. Numerical results illustrate that mode can be selectively excited provided the PZT actuator is placed in an appropriate location. Moreover, the proposed model was employed to predict the effectiveness of vibration suppression by distributed PZT actuators. The novelty of this work is that a complicated coupling problem between the composite plate and bonded PZT actuator is resolved into two simple problems, leading to a simple analytical solution for the vibrational response of a composite plate induced by PZT actuators. The proposed model has been successfully demonstrated its applications on the vibration excitation and suppression of a composite laminate plate.

Vibration reduction of FG-CNTR piezoelectric laminated composite cantilever plate under aerodynamic load using full-dimensional state observer

A robust control method is developed to suppress the vibrations of the functionally graded carbon nanotube-reinforced (FG-CNTR) piezoelectric laminated composite cantilever plate subjected to the aerodynamic force and thermal environment. The distributions of the carbon nanotubes (CNTs) in the entire plate thickness are classified as the functionally graded (FG) or uniform distributions (UD). The effective material properties are obtained using the rule of mixtures. The classic laminated composite plate theory and Hamilton principle are used to establish the governing equations of motion for the FG-CNTR laminated composite cantilever plate under combined the aerodynamic force and thermal environment. Galerkin method is used to obtain a two-degree-of-freedom ordinary differential control equation. To actively suppress the vibration, the piezoelectric patches are used as the actuators and sensors which are attached to the upper and lower surfaces of the FG-CNTR laminated composite cantilever plate. A full-dimensional state observer is introduced to design the robust controller. To verify the efficiency of the control strategy, a comparison between the robust controller and velocity feedback controller (VFC) indicates that the robust controller has better control efficiency than the VFC. The effects of the CNT distribution, CNT volume fraction, temperature and aspect ratio on the dynamic behaviors of the laminated composite plates are studied. The effectiveness and accuracy of the proposed robust controller are verified through numerical simulations under different cases.

Natural element hierarchic models for laminated composite plates

This paper presents the hierarchic models for laminated composite plates in the framework of 2-D natural element method (NEM), aiming at the accurate and effective numerical analysis of stress distributions and buckling loads. The current study was inspired by the fact that the classical composite plate theories are restricted to thin structures and most of hierarchic models have been implemented by FEM. In the hierarchic models, the displacement field is expressed in the product of the 2-D in-plane unknown functions and the 1-D out-of-plane assumed monomials. The in-plane unknown functions, which are supported on the mid-surface of composite plate, are approximated by the 2-D locking-free NEM. The approximated strains and stresses are smoothened by the CPU time-effective strain recovery technique, and furthermore the transverse shear stresses are enhanced by the equilibrated post-processing technique. The proposed hierarchic models are applied to two types of laminated composites, the conventional orthotropic laminated plate and the laminated plate with functionally graded CNTRC layers. Through the numerical experiments, the fundamental features of hierarchic models are investigated in terms of the model level, the structure thickness and the grid density. And the deflections, stresses and buckling load parameters obtained by the present models were compared with 3-D exact solutions and the advanced plate theories. From the comparison, the reliability and excellence of the proposed hierarchic models have been justified.

Elastothermodynamic damping modeling of three-layer Kirchhoff–Love microplate considering three-dimensional heat conduction

Elastothermodynamic damping is one of the most important energy loss mechanisms in micro/nano mechanical resonators, which is related to the high sensitivity and high resonance frequency characteristics of resonators. With the wide application of resonators, multilayer microplate resonators have been applied. In this paper, the elastothermodynamic damping models of three-layer of Kirchhoff–Love microplate under three typical boundary support conditions of Clamped-Clamped-Clamped-Clamped (C-C-C-C), Clamped-Free-Clamped-Free (C-F-C-F) and Clamped-Free-Free-Free (C-F-F-F) are established. Firstly, the thermoelastic mechanical equations of the microplate are given, and then the three-dimensional temperature distribution functions of the microplate are obtained based on the generalized orthogonal function method. Finally, the elastothermodynamic damping models are established based on the composite multilayer plate theory. The model in this paper is compared with the previous model and the finite element results.

Modeling of stresses in inorganic composite plates under non uniform high temperature heating

An asymptotic theory of thin composite plates with multi-stage high-temperature phase transformations is proposed. The theory is based on an asymptotic analysis of 3-dimensional equations of the mechanics of composite materials, taking into account phase transformations. Phase transformations are described by a system of kinetic equations, which is solved in conjunction with the equations of internal heat and mass transfer. The so-called local problems for plate theory are obtained and the averaged equations of plate theory with phase transformations are derived. An example of a composite based on an aluminum-chromo-phosphate matrix under uneven heating is considered. A numerical-analytical solution showed that although the heating is uneven due to the peculiarities of phase transformations, the stress-strain state changes significantly in time and is variable across the plate thickness. The developed theory allows us to calculate the stress distribution in the plate with high accuracy.

Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Variational Principles of Elasticity Theory 2. Particular Low-Order Theories

In a geometrically linear formulation based on the use of variation principles of elasticity theory, the twodimensional equations of statics and the boundary conditions at the edge of an orthotropic bending plate are obtained within the framework of the classical theory and two low-order nonclassical theories: Reddy–Nemirovskii and Margueere–Timoshenko–Naghdi ones. It is shown that using the hypotheses of the classical theory, the two-dimensional static equations and the corresponding boundary conditions ensure the equilibrium of an arbitrary subarea of the structure and of the entire plate as a rigid whole; within the framework of the Reddy–Nemirovskii theory, this equilibrium is disturbed in moments and forces in the transverse direction; within the framework of the Margueere–Timoshenko–Naghdi theory, the equilibrium is disturbed in the membrane forces and moments. An example of cylindrical bending of rectangular elongate orthotropic and isotropic plates clamped along one longitudinal edge and loaded in the transverse direction on the other longitudinal edge is considered. It is shown that, within the framework of the classical, Reissner, and Ambartsumyan theories, this problem is statically determinate and the shearing force, according to these theories, is determined elementary by the method of sections. According to the Reddy–Nemirovskii theory, this problem is statically indeterminate and has a rather awkward solution. It is demonstrated that, within the framework of the Reddy–Nemirovskii theory, the shearing force is zero on the clamped edge, i.e., the plate as a whole cannot be in equilibrium. In the vicinity of such an edge, there arises a pronounced false boundary effect, which does not take place in reality, because it is located inside the zone of the boundary layer and cannot be calculated by this theory.

Calculation of three-layer composite beams using difference equations of successive approximation method

Beams are considered the most popular bending elements used in building structures. With the load-bearing capacity and production characteristics, it is possible to use many different materials to combine them into beams to receive high economic-technical efficiency. Besides, because of the development of structural material industry, the manufacturing and construction technology, the form of beams is as diverse as composite wood beams, foundation beams, force-sensitive composite beams prior to installation grafting, steel-concrete combination beams. So there are many researches about solving the problems of multi-layer structures in general and multi-layer beams in particular by different calculation methods. In this paper, based on the theory of multi-layer composite rods and plates of A.R Rzhanitsyn the solutions of calculation analysis of three-layer beams that is subjected to discontinuous loads using difference equations of successive approximation method (MSA) are presented. The obtained results with good convergence show high accuracy of the numerical method with the use of difference equations of successive approximation method.

Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Variational Principles of Elasticity Theory 1. General Theories of High Order

In a geometrically linear formulation based on the use of variational principles of elasticity theory and the generalized Galerkin method, three approaches to constructing equations of static bending of orthotropic plates with a constant thickness are considered. The displacements of plate points are approximated by polynomials in the transverse coordinate with unknown coefficients of expansion. The application of variational principles leads to the first two approaches for constructing these equations. In the first approach, the force boundary conditions on faces of the plates are not taken into account, but the resulting two-dimensional equations and the corresponding boundary conditions on their edges are consistent, since, according to these equations, any subarea of the plates and the entire structures as a whole are in equilibrium. The two-dimensional equations of statics and boundary conditions on edges of the plates obtained within this approach can also be obtained by the generalized Galerkin method. The simplest theory using this approach is the Reissner theory. The second approach takes into account the force boundary conditions on faces of the plates. The latter circumstance leads to the necessity for using dependent variations of kinematic variables or introduction of undetermined Lagrange multipliers. It is shown that the resulting two-dimensional static equations and boundary conditions on plate edges are inconsistent, since any arbitrary subarea of the plate and the entire structure as a whole are in the nonequilibrium state. The two-dimensional statics equations and boundary conditions constructed in this approach cannot be obtained by the generalized Galerkin method. The simplest theories using this approach are the Reddy–Nemirovskii and the Margueere–Timoshenko–Naghdi theories. The third approach takes into account the force boundary conditions on faces of the plates, but the two-dimensional equations and the corresponding boundary conditions on the edges are obtained by the generalized Galerkin method. The homogeneous polynomials in the transverse coordinate are used as weight functions. The two-dimensional equations and boundary conditions obtained in this approach are consistent. The simplest theory that uses this approach is the Ambartsumyan theory. The relevance of the study is determined by the fact that the choice of a simple, but adequate, theory of bending can be of fundamental importance in solving optimization problems for composite plates.

Development and applications of shear deformation theories for laminated composite plates: An overview

This review, overviews the development and applications of shear deformation (SD) laminated composite plate theories. The present overview mainly focuses on theoretical models based on SD theories of laminated composite plates. This comprehensive review has been categorized with an equivalent single-layer laminate and layerwise laminate theories. An equivalent single-layer laminate theory includes classical laminated plate theory, first-order shear deformation laminated theory, second-order shear deformation laminated theory, third-order shear deformation laminated theory, parabolic shear deformation laminated theory, trigonometric shear deformation laminated theory, hyperbolic shear deformation laminated theory, and higher order shear deformation laminated theory. The layerwise laminate theories include discrete-layer theories and zig-zag theories. In addition, the 3D elasticity solutions theory, the mixed (hybrid) plate theories, unified theories, advanced shear deformation theories, and the recently developed smart composite are also reviewed.

Vibration suppression of cantilevered piezoelectric laminated composite rectangular plate subjected to aerodynamic force in hygrothermal environment

In this paper, a robust control method is proposed for the vibration suppression of the piezoelectric laminated composite cantilever rectangular plate subjected to the aerodynamic force in the hygrothermal environment. The laminated composite cantilever rectangular plate is placed on the piezoelectric actuator and sensor for the upper and lower surfaces. The classical laminated composite plate theory and Hamilton's principle are applied to derive the dynamic equation of motion for the piezoelectric laminated composite cantilever rectangular plate under the aerodynamic force and hygrothermal loads. The structural damping is considered for the piezoelectric laminated composite cantilever rectangular plate. Galerkin method is used to obtain a two-degree-of-freedom discrete ordinary differential control equation of motion. For the active vibration suppression, a robust controller for the uncertain systems is designed through the obtained ordinary differential equation of motion. Moreover, the full-dimensional state observer is constructed to calculate the close-loop system. The influences of the moisture, temperature and geometric parameters on the dynamics behaviors of the piezoelectric laminated composite cantilever rectangular plate are investigated. The accuracy and effectiveness of the robust controller are verified in terms of the moisture concentration, temperature, aspect ratio, damping and parameter uncertainty by numerical simulations.

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.

Free vibration analysis of the hard-coating splitter blisk using the energy-based finite element method

This paper presents the free vibration analysis of the hard-coating splitter blisk (known as the splitter blisk deposited hard coating) and the damping-improved performance of the hard coating for vibration suppression of the splitter blisk. Based on the proposed energy-based finite element method by the energy considerations, complex modulus theory and composite Mindlin plate theory, the equations of motion of hard-coating splitter blisk are formulated. And, the complex solutions for free vibration problem are then solved, whose real and imaginary parts are closely related to natural frequencies and loss factors. Considering the academic splitter blisk deposited NiCoCrAlY+YSZ hard coating as the benchmark, the comparisons for vibration characteristics of hard-coating splitter blisk originating from different sources are performed for model validation, and the influence of hard coating on the dynamics of splitter blisk are also investigated. The numerical results reveal that the vibration amplitudes of splitter blisk are suppressed remarkably by the hard coating without altering natural frequencies significantly.

Specific damping capacity calculation of composite plates with delamination based on higher-order Zig-Zag theory

Damping is focused in a mass of engineering applications at present. In this paper a new laminate element rests on the higher-order zig-zag theory for composite plates were presented. And then, viscoelasticity damping and frictional damping models in delaminated composites were established. The damping changes of delaminated composites with different boundary conditions were investigated via the laminate element, the effects of area and location on damping were also researched. The results revealed that viscoelasticity damping and frictional damping are in the same order of magnitude even delamination area is small, and frictional damping increase significantly when delamination area enlarged, frictional damping is needed to be considered in the damping research of delaminated composites.

Nonlinear finite element modeling and vibration analysis of the blisk deposited strain-dependent hard coating

Abstract This study focuses on the nonlinear vibration analysis of the integrally bladed disk (the blisk for short) deposited strain-dependent hard coating (also known as the hard-coating blisk) and the influence of hard coating on dynamics of the blisk. The high-order polynomials are employed to characterize the experimental mechanical parameters of the strain-dependent hard coating. The strain-dependent model of hard-coating blisk is constructed by the energy-based FE (finite element) method consisting of the complex modulus and composite Mindlin plate theory. And, a specified iterative solution procedure based on the Newton-Raphson method is proposed to solve strain-dependent dynamics. A specific benchmark of an academic blisk deposited NiCoCrAlY + YSZ hard coating on blades is chosen to perform numerical calculation and experimental test, and the specific comparisons between both numerical and experimental results are then conducted to validate the developed FE model. Moreover, the specific influence of hard coating with or without strain-dependent manner on the blade and blisk are investigate here.