Zigzag Theory Research Articles

Structural analysis of sandwich plates with variable stiffness composite skins and functionally graded cores using refined zig-zag theory

This work investigates the static flexural behavior of new hybrid Sandwich Plates (SP) with Functionally Graded (FG) cores and curvilinear fiber-reinforced Variable Stiffness (VS) skins. For simplicity and computational efficiency, the existing refined zig-zag theory formulation for curvilinear fiber-reinforced composites with three-node finite element (RZT3C) is enhanced to incorporate the thickness-wise material variation of the FG core. Benchmark problems of FG-SP under various boundary conditions are solved, with results compared against 3D Ansys and reference solutions. Contrary to costlier methods, the presented model accurately captures the complex flexural mechanics, paving the way for designing advanced structures.

A local–global optimization approach for maximizing the multiphysics frequency response of laminated functionally graded CNTs reinforced magneto-electro-elastic plates

Smart materials have garnered significant attention due to their distinct properties and potential applications in various engineering fields. In line with this, this study presents a local–global optimization approach aimed at maximizing the multiphysics frequency of laminated functionally graded carbon nanotube reinforced magneto-electro-elastic (FG-CNTMEE) plates. The laminated FG-CNTMEE plate is composed of layers of functionally graded carbon nanotubes embedded in an electro-elastic matrix, rendering it capable of exhibiting exceptional properties in response to magnetic and electric fields. To optimize its performance, a Balance Enhanced Symbiotic Organisms Search (BESOS) algorithm with new modifications of phases is proposed for global optimization, and integrated with a multiphysical isogeometric analysis using refined zigzag theory (MIARZ) for precise local–global analysis of laminated FG-CNTMEE plates within a Computer-Aided Design (CAD) environment. The effectiveness, robustness, and reliability of the proposed approach are demonstrated through numerical examples that highlight its applicability in achieving optimal designs for smart laminated FG-CNTMEE plates.

A new C0 continuous refined zigzag {1,2} finite element formulation for flexural and free vibration analyses of laminated composite beams

This study presents a novel C0 continuous refined zigzag finite element formulation {1,2}, namely RZE{1,2}, for the bending and free vibration analyses of laminated composite beams. The Refined Zigzag Theory (RZT) effectively combines accuracy and computational efficiency, making it a robust approach for thin and thick laminated composite structures. The RZT eliminates the need for shear correction factors, thereby enhancing the overall streamline of the analysis process. The present RZE{1,2} formulation takes into account the transverse stretching by introducing quadratic through-thickness variations of deflection components. The governing equations of the RZT are derived by means of Hamilton’s principle. The through-the-thickness variations of the transverse shear and normal stresses are calculated by integrating the stress equilibrium equations in a post-processing step. Therefore, the Peridynamic Least Squares Minimization (PDLSM) approach is utilized to obtain precise derivatives of axial stresses in the stress equilibrium equations. A new finite element formulation is built up with 3-nodes with a total of 15 DOF. In order to study the influence of transverse stretching, different types of boundary conditions and material variations are applied for laminated composite beams for the bending and free vibration analyses. The distribution of displacements as well as the axial and transverse stresses of the thick beams are extensively examined. The outcomes of RZT are in good agreement with the reference solutions in the literature. The findings of the present study reveal that the presence of the transverse stretching produces more realistic predictions for thick beams where the shear deformations are influential.

Analytical theory of composite beams with CSWs using the state space method

Based on the zig-zag theoretical model of corrugated steel web beam (CSW), the governing equations in the form of state equations are derived and their analytical solutions are obtained for different loads, variable cross-sections and boundary conditions. Through numerical examples, it is verified that the bending moment, deflection and stresses predicted by the present analytical method have a good agreement with the results of the three-dimensional finite element model. The applicability of the present method for non-prismatic continuous beams with CSWs under different loads is also demonstrated. Meanwhile, it is found that the quasi-plane assumption is still valid in non-prismatic continuous beams with CSWs. Through quantitative analysis of the flexural stresses in the top and bottom slabs, it is confirmed that the zig-zag theory in this study can well reflect the accordion effect of CSWs. Because of the inclined bottom slabs in the non-prismatic beams with CSWs, the Resal effect on the shear stresses in the CSWs and bottom slabs, which is caused by the vertical component of the axial force in the bottom slab, is discussed. A refined shear formulation is developed for more accurate shear stresses in the CSWs and bottom slabs, which provides fundamental theoretical support for the strength prediction of the non-prismatic beam with CSWs.

A NURBS-based IGA using zig-zag plate theory for nonlinear passive/semi-active damping analysis of laminated FG-CNTRC plates

This study introduces an efficient and versatile numerical approach for simulating passive/semi-active damping treatments on laminated structures composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) plates. This novel approach integrates an isogeometric finite element formulation that leverages basic functions generated from Non-Uniform Rational Basis Spline (NURBS) with the Murakami zig-zag theory, Von Kármán’s nonlinear strain–displacement model, and the electro-mechanical coupling properties of the constraining layers. The proposed isogeometric formulation can be transformed into the Laplace domain using the Golla–Hughes–McTavish model to successfully simulate the time-dependent passive/semi-active damping mechanism of viscoelastic materials. Subsequently, results obtained in the Laplace domain are reconverted into the time domain using inverse techniques. The validation of the proposed model through two numerical examples showcases its high reliability and demonstrates the feasibility of simulating and controlling FG-CNTRC plates with various geometries within a Computer-Aided Design-Computer-Aided Engineering (CAD-CAE) analysis framework.

On the vibrations of sandwich structure with magneto-electro-rheological core and composite face sheet via mesh-free method

Multidirectional functionally graded materials due to high-mechanical resistance with other extraordinary properties in each direction have received much attention in recent years. Thus, in the current research, a numerical solution is presented to study the fundamental frequency characteristics of an MD-FG sector annular plate in a supersonic airflow. For obtaining exact displacement fields, the refined zig-zag theory is employed to obtain the transverse shear strain through layers of the composite lamina. A quasi-3D new refined theory is presented to model the displacement fields of the current sandwich structure. The Legendre-radial point interpolation method shape function is used to estimate the displacements while the mesh-free strong form approach discretizes the equations of motion and boundary conditions. Finally, the results show that Mach number, free stream speed, air yaw angle, temperature, and aerodynamic pressure have important roles in the dynamic stability performance of the sandwich sector annular plate in supersonic airflow. The results of the current report can be used in various high-temperature multidirectional compositionally graded materials for future works.

A multi-physical coupling isogeometric formulation for nonlinear analysis and smart control of laminated CNT-MEE plates

This paper presents a novel and comprehensive numerical approach for nonlinear analysis and smart damping control in laminated functionally graded carbon nanotube reinforced magneto-electro-elastic (FG-CNTMEE) plate structures, taking into account multiple physical fields. The approach employs a multi-physical coupling isogeometric formulation based on Non-Uniform Rational Basis Spline (NURBS) to analyze the behavior of laminated FG-CNTMEE plate structures. It incorporates a refined zig-zag theory to accurately capture the Von Kármán nonlinear strain–displacement relationship, as well as the magneto-electro-elastic coupling properties of laminate layers. To achieve nonlinear damped responses, the smart constrained layer damping (SCLD) treatment is applied. Furthermore, to enable smart control of structures using the time-dependent passive/semi-active damping mechanism of viscoelastic materials, the formulation is transformed into the Laplace domain using the Golla–Hughes–McTavish model. The results are then converted back to the time domain through inverse techniques. The proposed approach has been rigorously validated through verification work and demonstrated its effectiveness in simulating and controlling FG-CNTMEE plates in two numerical examples.

Modified couple stress theory (MCST) and refined zigzag theory (RZT) for wave propagation of embedded smart sandwich plates with magnetorheological fluid core and viscoelastic FG-CNTRC magnetostrictive face sheets

This paper discusses the wave propagation of a magnetorheological fluid (MRF) micro sandwich plate with magnetostrictive face sheets embedded in the Kerr foundation. The face sheets are reinforced with functionally graded carbon nanotubes (FG-CNT). It was hypothesized that the distribution of carbon nanotubes would be functionally graded along the thickness of face sheets in various patterns such as uniform distribution (UD), and three type of functionally graded (FG), that is, FG-O, FG-X, and FG-AV. The properties of the magnetostrictive layer are considered viscoelastic based on the Kelvin-Voigt model. The refined zigzag theory (RZT) is utilized to simulate displacements of the plate due to the sandwich nature of the system. The modified couple stress theory (MCST) is utilized to predict the influence of small-scale parameter. The equations of motion for the micro sandwich plate are derived by Hamilton's principle, and analytical solutions are utilized to acquire the phase speed, escape, and cut-off frequencies. After model verification, extensive analytical results are presented in detail. They demonstrated that the wave characteristics of the system are influenced by the micro sandwich plate parameters. Consequently, the dimensionless phase speed rises 66% via increasing the magnetic field magnitude from 12.5 to 17.5 G.

Towards composite suspension control arm: Conceptual design, structural analysis, laminate optimization, manufacturing, and experimental testing

The automotive industry needs composite materials to decrease the weight of new-generation vehicles whilst increasing their strength. In this study, one of the critical (load bearing) components of automobiles, i.e., the suspension control arm made of steel, is fully redesigned for its suitable manufacturing using composite materials. To this end, innovative mechanical simulation methods are developed and coupled to perform the design, analysis, and optimization of the automotive suspension control arm. The main design/optimization criteria are set to reduce no less than 75% weight of the metal control arm and increase its safety by at least 60% by using composite materials and a new geometry suitable for mass production. To predict the deformation-stress state of the control arm, a four-node quadrilateral shell element is implemented based on the kinematics of refined zigzag theory (RZT). Once verified numerically, the computer implementation of the RZT is combined with the optimization algorithm to achieve the optimum laminate stacking sequence of the control arm. Accordingly, prototypes of the composite control arms with optimum lamination plans are manufactured and then experimentally tested under the loading and constraint conditions defined at the conceptual design stage. The numerical and full-scale experimental results are compared, and the RZT models are comprehensively validated. Hence, the advantages of the overarching design-analysis-optimization strategy presented herein are revealed for redesigning and manufacturing automobile parts from composite materials.

Nonlinear aeroelastic fluid-structure-acoustic interaction analysis of a coupled composite panel with an acoustic cavity in supersonic flow

This paper is concerned with numerical modeling and nonlinear dynamics analyses of fluid-structure-acoustic interactions of an elastic composite laminated panel with a backed cavity in supersonic flow. To achieve strongly coupled solutions to the aeroelastic fluid-structure-acoustic interaction problem, an implicit partitioned arbitrary Lagrangian-Eulerian solution method is developed based on a finite volume model for the supersonic flow and a monolithic nonlinear finite element model for the composite panel-cavity system. A nonlinear higher-order shear deformation zig-zag theory is adopted to accommodate the large-deformable vibration of the composite panel, and the acoustic wave equation is employed for modeling the compressible fluid in the cavity. The method is verified by comparisons with benchmark results. Physical insights into the effects of the dynamic pressure, cavity depth, and the ply-angle of composite material on the aeroelastic behaviors of the panel-cavity system are provided. Some new phenomena are revealed. A special flutter behavior dominated by acoustic resonance is found for the composite panel with a backed cavity in supersonic flow, which is due to local mode coalescence as the mode frequencies of the acoustic cavity and the composite panel are close to each other. The transition between the aeroelastic flutters of the composite panel induced by the lower-order structural mode coalescence and the acoustic resonance of the cavity can be progressively adjusted by alerting the cavity depth and the ply-angle of the composite panel.

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.

Stability and mode shape analysis of doubly- and cross-helicoidal laminated sandwich plates inspired from dactyl’s club

The present work intends to use higher-order zigzag theory to analyze double- and cross-helicoidal laminated sandwich plates under buckling conditions. Geometry and boundary effects on buckling and mode shapes are examined. The behavior of helicoidal sandwich plates is compared with the cross-ply and quasi-isotropic lamination schemes. In the case of plates with free edges, the bio-inspired laminated sandwich plates outperform the more traditional cross-ply and quasi-isotropic laminated schemes. Numerous more findings are provided as well, all of which may be used as a standard by which to measure the success of comparable future research efforts.

An Accurate Solution for Buckling and Nonlinear Analysis of a Sandwich Beam with Functionally Graded Material by Considering Zigzag Displacements

In this paper, critical buckling analysis and nonlinear load-deflection curve for sandwich beam with functionally graded material are presented based on refined zigzag theory (RZT). By using the variational principle, the equilibrium equations in buckling analysis are given based on the RZT formulations as well as the nonlinear strain-displacement relations. The solutions are also derived for eigenvalue problems in critical buckling load calculations and for load-deflection relations with initial geometric imperfection. The solutions are presented analytically, and the mathematical properties during the derivation process have been proven in order to keep the mathematical rigor. The present analytical RZT critical buckling loads are validated by the RZT FEM, which is the finite element solutions of the sandwich beam meshed by the beam elements based on RZT. These solutions are also compared by commercial software ANSYS, resulting that this approach can obtain an accurate critical buckling load. Various parameters such as aspect ratio, thickness ratio and modulus ratio are considered to investigate their effects on the critical buckling loads. The present results are compared to the beams with higher-order shear deformation theory (HSDT). From the comparisons of the RZT and the HSDT results, it is seen that both theories approach to CBT for slender beam. The results show that the HSDT overestimates the stiffness in the load-deflection curve. It is shown that the RZT exhibits the zigzag displacements at high accuracy, resulting in accurate calculation in critical buckling loads, mode shapes and nonlinear load-deflection curves than HSDT. The superiority of the RZT solutions is presented especially for the case of FGM sandwich beam with soft middle layer.

Delamination Detection and Localization in Vibrating Composite Plates and Shells Using the Inverse Finite Element Method.

Delamination damage is one of the most critical damage modes of composite materials. It takes place through the thickness of the laminated composites and does not show subtle surface effects. In the present study, a delamination detection approach based on equivalent von Mises strains is demonstrated for vibrating laminated (i.e., unidirectional fabric) composite plates. In this context, the governing relations of the inverse finite element method were recast according to the refined zigzag theory. Using the in situ strain measurements obtained from the surface and through the thickness of the composite shell, the inverse analysis was performed, and the strain field of the composite shell was reconstructed. The implementation of the proposed methodology is demonstrated for two numerical case studies associated with the harmonic and random vibrations of composite shells. The findings of this study show that the present damage detection method is capable of real-time monitoring of damage and providing information about the exact location, shape, and extent of the delamination damage in the vibrating composite plate. Finally, the robustness of the proposed method in response to resonance and extreme load variations is shown.

A comprehensive analysis for real-time shape and strain sensing of composite thin-walled structure

This study presents an effective and reliable shape sensing strategy to model the flat cross section deformation of laminated composite thin-walled structure. The proposed strategy is derived by using the customized isogeometric I-beam elements and Murakami zigzag theory (MZT), named IG-MZT(I). Compared to the traditional reconstruction model, the IG-MZT(I) offers numerous advantages for displacement and strain analysis of composite thin-walled structure such as high-order continuity representation, simple mesh refinement, and fewer strain sensors. Initially, the theoretical framework of composite I-beam element based on the MZT deformation theory is presented. Then, the isogeometric analysis is introduced to approximate the displacement field, and the order of B-spline basis functions can be determined by the constitutive equations.Furthermore, an intuitive strain conversion model is established for solving the geometric complexities in strain measurements. Finally, utilizing a least-square variational function within IG-MZT(I) enables the establishment of the shape sensing formula. Several numerical analyses and experiment results demonstrate the outstanding accuracy and reliability of the proposed approach for reconstructing composite I-beams.Overall,the present model is simple and independent of the traditional composite reconstruction models that require embedded sensors,making the IG-MZT(I)sufficiently general to analyse the composite structure in service.

Real-Time Shape Sensing of Composite and Sandwich Structures by Coupling Inverse Finite-Element Method with Refined Zigzag Theory

Real-Time Shape Sensing of Composite and Sandwich Structures by Coupling Inverse Finite-Element Method with Refined Zigzag Theory

First-Ply Failure Analysis of Helicoidal/Bouligand Bio-Inspired Laminated Composite Plates

Among the first, the present paper focuses on studying the first-ply failure load for bio-inspired helicoidal laminated composite plates. Five different layup schemes, namely, helicoidal recursive, helicoidal semicircular, helicoidal exponential, Fibonacci helicoidal, and linear helicoidal, were studied. The performance of the five helicoidal layups was compared with the commonly used cross-ply and quasi-isotropic lamination schemes. Maximum stress, maximum strain, Tsai–Hill, Tsai-Wu, and Hoffman failure theories were employed in the framework of the recently proposed finite-element-based higher-order zigzag theory to determine the first-ply failure load. It was observed that the maximum strain theory predicted the lowest value of the first-ply failure load, whereas Tsai–Wu predicted the highest one. For some cases, Tsai–Hill and Tsai–Wu theories were not able to predict the first-ply failure load. For plates having free edges, the bio-inspired plates exhibited the first-ply failure load compared to the conventional layup sequences.

First-Ply Failure Analysis of Bioinspired Double and Cross-Helicoidal Laminated Sandwich Plates

Inspired from the helicoidal schemes available from biological creatures, the present paper aims to predict the first-ply load for laminated composite and sandwich plates inspired from biological creatures. The laminated composite and sandwich plates are assumed to be made up of double- and cross-helicoidal lamination schemes. The first-ply failure load is determined using the recently proposed higher-order zigzag theory. Five different failure criteria (namely, maximum stress, maximum strain, Tsai–Hill, Tsai–Wu, and Hoffman’s theory) are used to predict the first-ply failure load. The performance of the helicoidal plate is compared with the cross-ply and quasi-isotropic lamination schemes. The influences of the end conditions, the ratio of the thickness of the core to the face layers, and the thickness-to-side ratio on the first-ply failure load are carried out. It has been observed that for helicoidal plates, a higher value of the first-ply failure load is observed as compared to quasi-isotropic and cross-ply laminates, especially for plates having free edges. The maximum strain theory should not be used for predicting the failure load for helicoidal laminated plates because it undermines the strength of the plate.

A new and efficient zigzag theory for laminated composite plates

This paper develops a new and efficient zigzag theory named as C-311 for laminated composite plates. The new local zigzag displacement model of C-311 nonlinearly distributes through the thickness coordinate and satisfies the piece-wise continuity at the layer surface, and combines three-order displacement model to form the initial displacement field of the present theory. By enforcing the inter-laminar transverse stress continuity and the free surface conditions, the local displacement unknown variables can be eliminated, and the final displacement field of C-311 is expressed by a fixed number of unknown variables. In addition, the equilibrium equations and boundary conditions of the present theory can be obtained from the virtual work principle. To test the prediction accuracy of C-311, a simple-supported cross-ply square plate is investigated under mechanical loading. Numerical results show that compared to several classical laminated plates theories, the present theory can predict accurately mechanical behaviors of thick multilayered composite plates and heterogeneous laminated plates. Especially the C-311 can give an acceptable accuracy of the transverse shear stress derived from the constitutive equations directly, and that is benefit for practical engineering applications in the finite element method.

Nonlinear damping and forced vibration analysis of sandwich functionally graded material beams with composite viscoelastic core layer

In the present article, an analytical model is proposed to assess the nonlinear damping and frequency curves of the three layered sandwich beams, constituted with FGM faces and viscoelastic core layers. The proposed analytical model is formulated in the context of forced vibration analysis. The kinematical model developed for the purpose is based on the higher order zig-zag theories coupled with the higher order shear deformation theory (HSDT), employing shear functions, formulated by Timoshenko, Reddy, and Touratier. The frequency response curves and the nonlinear loss factors are provided by resourcing to analytical solution, derived from the harmonic balance technique coupled with the one mode of Galerkin’s method. The elastic moduli of the viscoelastic material are described with complex numbers in the frequency domain. The material proprieties of the FGM beams vary gradually along the thickness direction, transiting from metal to ceramic materials. The effect of the geometrical and material parameters of the FGM and viscoelastic composite layers on the damping and frequency response curves of the sandwich beams are studied according to the first and higher-order shear deformation theories. Basing on the proposed analytical model, numerical results are presented to study the effect of the FGM parameters and the fiber orientation angle of the composite core on the nonlinear damping evolution with the vibration amplitude and the frequency curves.