Higher Order Zigzag Theory Research Articles

Generalization of the C0-type zigzag theory for accurate thermomechanical analysis of laminated composites

Abstract A new method based on the C0-type efficient higher-order zigzag theory (EHOZT) is proposed for the efficient and accurate thermomechanical analysis of laminated composite plates. In this method, the transverse shear strain energy is modified according to the mixed variational theorem (MVT). The transverse stress field is defined by the arbitrary m-th order zigzag model including the transverse normal strain effect to improve the accuracy, while the displacement field is obtained from the C0-type EHOZT to amplify the benefits of the numerical efficiency. The transverse displacement field is assumed to be a smooth parabolic distribution to consider the transverse normal strain effect, which has a significant role in predicting the thermal deformation. Derivative components of the transverse displacement field are eliminated from the in-plane displacement field of the C0-type EHOZT by the transverse shear stress condition. The resulting strain energy expressions are collectively referred to as the C0-type EHOZT via the MVT. The proposed theory has the computational advantage of utilizing the C0-continuity condition for finite element implementation while allowing local distributions of the transverse stress to be improved via the recovery procedure. The obtained displacements and stress distributions were compared with data available in the literature for validation.

Viscoelastic behavior of Naghdi shell model based on efficient higher-order zig-zag theory

This paper proposes a method based on efficient higher-order zig-zag theory to analyze the viscoelastic response of doubly-curved laminated shell structures. In the general curvilinear coordinates, displacement fields are obtained by imposing a varying cubic displacement field on a varying linear zig-zag field. Then, the transverse shear stress-free condition at the top and bottom surfaces and the continuity condition at the interfaces are employed to reduce the number of unknown variables. The Laplace transformation is then used to simplify the integral-formed constitutive equation for viscoelastic material in the real time domain into a linear system equation in the Laplace domain so that all computations can be carried out in the Laplace domain. Therefore, the equilibrium equation for general viscoelastic Naghdi shell model can be obtained by converting the virtual work principle into the Laplace domain. Finally, solutions for the long-term viscoelastic properties in the real-time domain are obtained by using numerical inverse Laplace techniques. To simplify the formulation and conveniently evaluate the method proposed in the present study and to compare its outcomes with those of an elastic laminated composite shell, several numerical examples for a cylindrical shallow shell model are investigated.

A triangular finite element using Laplace transform for viscoelastic laminated composite plates based on efficient higher-order zigzag theory

To predict the time-dependent behaviors of viscoelastic laminated composites, a three-node multilayered plate element is developed based on the efficient higher-order plate theory (EHOPT), which was originally proposed by Cho and Parmerter. With the help of the Laplace transform, the integral form of the constitutive equation in the time domain is reduced to an algebraic equation in the Laplace domain. Thus, the structures and advantages of the EHOPT can be preserved in viscoelastic laminated composites in the Laplace domain. Since the time dimension is transformed to Laplace domain, the finite element discretization is only used in the spatial domain. A nonconforming three-node triangular element is employed to implement the viscoelastic EHOPT for finite element analysis. To pass the proper bending and shear patch tests in arbitrary mesh configurations, the modified shape function developed by Specht is applied and converted into Laplace domain. Therefore, the final numerical results, which is obtained by using inverse Laplace techniques, always converge to the corresponding analytical solutions. In order to verify the efficiency and accuracy of the present study, some numerical examples for long-term creep and relaxation processed are performed. The present viscoelastic finite element of composite laminates provides a powerful tool to accurately investigate the responses of the viscoelastic and time-dependent mechanical behaviors of composite laminates.

Improved Viscoelastic Analysis of Laminated Composite and Sandwich Plates With an Enhanced First-Order Shear Deformation Theory

An enhanced first-order shear deformation theory (EFSDT) is developed for linear viscoelastic analysis of laminated composite and sandwich plates. Improved strain energy expression of the conventional Reissner/Mindlin first-order shear deformation theory (FSDT) through strain energy transformation is derived in the Laplace domain by minimizing the strain energy difference between FSDT and an efficient higher-order zigzag theory (EHOPT). The convolution theorem of Laplace transformation is applied to circumvent the complexity of dealing with linear viscoelastic materials. The present EFSDT with the Laplace domain approach has the same computational advantage of the conventional FSDT while improving upon the accuracy of the viscoelastic response by utilizing the postprocess recovery procedure. The accuracy and efficiency of the proposed theory are demonstrated through the numerical results obtained herein by comparing to those available in the open literature.

Behaviour of sandwich laminates subjected to thermal loading using higher-order zig-zag theory

In the present work, a higher-order zig-zag laminate theory is utilized to investigate the behaviour of sandwich laminates having low-density core materials subjected to thermal loading. The theory satisfies continuity condition of transverse shear stress at the layer interfaces and zero transverse shear stress at the top and bottom surfaces of the laminated plate. The assumed in-plane displacements are having cubic variations through the entire thickness of the plate. On the other hand, the variation of the transverse displacement is taken to be quadratic within the core and constant through the facesheets. In order to reduce the computation effort, an efficient C0 finite element formulation for the above sandwich plate model is adopted. Some examples of laminated composite and sandwich plate with different material properties, core thickness ratios, aspect ratios, boundary conditions, number of layers and ply orientations are considered for the analysis. The efficacy of the present model in predicting various responses is verified by comparing with three-dimensional elasticity solutions along with the available published results.

Vibro-acoustic analysis of multilayered shells of revolution based on a general higher-order shear deformable zig-zag theory

This paper presents a semi-analytical approach for predicting the vibration and acoustic responses of an arbitrarily shaped, multilayered shell of revolution immersed in a light or heavy unbounded fluid. A higher-order shear deformable zig-zag shell theory with general shape functions is proposed to describe the displacement field of a multilayered shell with arbitrary curvatures, which provides a theoretical unification of most thin and shear deformable shell theories in the literature. Based on the higher-order zig-zag theory, the structure model of the multilayered shell is formulated by using a modified variational method combined with a multi-segment technique, whereas a Chebyshev spectral Kirchhoff–Helmholtz integral formulation is employed to model the exterior acoustic fluid. The displacement field of the shell and the sound pressure of the fluid are expanded by Fourier series and Chebyshev orthogonal polynomials. Such a treatment reduces the size of the problem and permits a semi-analytical solution for the displacement and acoustic variables. A set of collocation nodes distributed over the roots of Chebyshev polynomials are used to establish the algebraic system of the acoustic integral equations, and the non-uniqueness solution is eliminated by means of interior CHIEF points. Numerical examples are given for vibration and acoustic radiation analyses of multilayered spherical, cylindrical and conical shells. Comparison studies are performed to evaluate the accuracy of various shell theories. The validity of the present method for acoustic analyses of multilayered shells is demonstrated by comparing the results with exact solutions and those obtained from the coupled finite element/boundary element method. Individual contributions of circumferential modes to the radiated sound of multilayered shells are examined.

Vibration response of laminated composite plate having weakly bonded layers

Free and forced vibration response of laminated composite and sandwich plate having weakly bonded layer is studied by using an efficient C0 continuous two dimensional finite element (FE) model. This FE model is based on higher order zigzag theory (HOZT) in which, in-plane displacements are obtained by superposing a global displacement field having cubical variation across thickness, on a zigzag displacement field having linear variation with a different slope in each layer. The out of plane displacement is assumed constant across the thickness of the plate. In the proposed model, transverse shear stresses are continuous at the layer interfaces and vanish at the top and bottom surfaces of the plate. At the same time, it is computationally as elegant as any single layer plate model since it requires the usual unknowns at the reference plane only. The inter-laminar imperfections are represented by in-plane displacement jumps at the layer interfaces and characterized by a linear spring layer model. In order to circumvent the problem of C1 continuity associated with the above plate theory at the time of FE implementation, first derivatives of the transverse displacement have been treated as independent variables. It is interesting to note that the plate model having all these refined features requires unknowns at the reference plane only. The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on HOZT for forced vibration problems. Numerical problems on free and forced vibration having different geometric and material properties of laminated composite and sandwich plates are solved. Some new problems have also been solved and new results have been presented which would be useful for future research.

Efficient higher-order zig-zag theory for viscoelastic laminated composite plates

An efficient higher-order plate theory for viscoelastic materials is developed to predict the time-dependent mechanical behaviors of composite laminates. In-plane displacement fields are constructed by superimposing a cubic varying displacement field on a linear zig-zag varying field. Time-dependent relaxation moduli have the form of Prony series, which can be determined by the master curve based on experimental data. The constitutive equation of linear viscoelastic materials in the form of the Boltzmann superposition integral is simplified by the convolution theorem of the Laplace transform to avoid direct integration as well as to improve both computational accuracy and efficiency. By using the equivalent linear elastic stress–strain relationship in the corresponding Laplace domain, the transverse shear stress-free conditions at the top and bottom surfaces and the transverse shear stress continuity conditions at the interfaces between layers are satisfied conveniently. Finally, the viscoelastic responses in the time domain are obtained through various numerical inverse Laplace transforms. To validate the present theory, the numerical results for graphite/epoxy GY70/339 material are obtained and compared with the solutions of elastic composite laminated plates.

Efficient Failure Analysis of Laminated Composites and Sandwich Cylindrical Shells Based on Higher-Order Zigzag Theory

AbstractAn efficient failure analysis of laminated composite and sandwich cylindrical shell is done for the first time using an efficient C0 finite-element (FE) model based on higher-order zigzag theory (HOZT). The FE implementation based on HOZT to study the failure of composite and sandwich shells incorporating all three radii of curvature is presented. The proposed two-dimensional (2D) FE model satisfies the interlaminar shear stress continuity at the layer interfaces and also ensures zero transverse shear stress conditions at the shell top and bottom. The problem of C1 continuity associated with the HOZT is circumvented by using an appropriate C0 FE formulation. The piecewise parabolic shear stress variation across thickness of each layer is considered and hence, shear deformation is accurately modeled for composites and sandwich shells. The failure load results for laminated composite and sandwich shells obtained by using the present 2D FE model are quite close to the 3D elasticity results. Many new ...

Accurate dynamic response of laminated composites and sandwich shells using higher order zigzag theory

Forced vibration response of laminated composite and sandwich shell is studied by using a 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the forced vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at each layer interface in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C0 finite element formulation has been done to overcome the problem of C1 continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells besides normal curvature shells like cylindrical, spherical shells etc. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.

VIBRATION AND BUCKLING ANALYSIS OF LAMINATED SANDWICH PLATE HAVING SOFT CORE

Free vibration and buckling of laminated sandwich plate having soft core is studied by using an efficient C0 continuous finite element (FE) model based on higher-order zigzag theory (HOZT). In this theory, the in-plane displacement field for both the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zigzag linearly varying displacement field with a different slope in each layer. The transverse displacement is assumed to be quadratic within the core while it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the plate. The nodal field variables are chosen in an efficient manner to overcome the problem of C1 continuity requirement of the transverse displacement. Numerical examples on free vibration and buckling covering different geometric and material features of laminated composite and sandwich plates are presented. Many new results are also presented which should be useful for future research.

C0 FE model based on HOZT for the analysis of laminated soft core skew sandwich plates: Bending and vibration

Bending and free vibration behaviour of laminated soft core skew sandwich plate with stiff laminate face sheets is investigated using a recently developed C0 finite element (FE) model based on higher order zigzag theory (HOZT) in this paper. The in-plane displacement fields are assumed as a combination of a linear zigzag function with different slopes at each layer and a cubically varying function over the entire thickness. The out of plane displacement is considered to be quadratic within the core and constant in the face sheets. The plate theory ensures a shear stress-free condition at the top and bottom surfaces of the plate. Thus, the plate theory has all of the features required for accurate modelling of laminated skew sandwich plates. As very few element model based on this plate theory (HOZT) exist and they possess certain disadvantages, an attempt has been made to check the applicability of the refined element model. The nodal field variables are chosen in such a manner that there is no need to impose any penalty stiffness in the formulation. Refined C0 finite element model has been utilized to study some interesting problems on static and free vibration analysis of laminated skew sandwich plates.

Vibration of laminated composites and sandwich shells based on higher order zigzag theory

Free vibration response of laminated composite and sandwich shell is studied by using an efficient 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders’ approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at the interfaces in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C0 finite element formulation has been done efficiently to overcome the problem of C1 continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells, etc., besides normal curvature shells like cylindrical, and spherical shells. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.

Finite element analysis of laminated composite and sandwich shells using higher order zigzag theory

Static analysis of laminated composite and sandwich shell is presented by developing a C0 finite element (FE) formulation based on higher order zigzag theory (HOZT) using Sander’s approximations. The proposed model satisfies the inter-laminar shear stress continuity at the interfaces; and zero transverse shear stress conditions at shell top and bottom. This is the first finite element implementation of the HOZT to solve the problem of shells incorporating cross curvature effects in shells. The present HOZT predicts transverse shear stresses more accurately than FE results previously published in literature. Numerical results show that the present 2D model is very efficient in predicting the static response of laminated composite and sandwich shallow shell very close to 3D elasticity solutions.

Exponential basis functions in the solution of laminated plates using a higher-order Zig–Zag theory

The solution method presented in this paper is of Trefftz type which uses exponential basis functions (EBFs) to solve composite plate problems. Following its success in the solution of plates using well-known theories (Composite Struct 2011;93:3112–9, 94:84–91 and 2012;94:2263–68), here we aim to apply the method to higher order shear deformation theories. In this paper we demonstrate the way that one can generally evaluate the EBFs for a laminated plate using a Zig–Zag theory. We present explicit relations for three-layer sandwich plates which have a wide range of applications in structural/mechanical engineering fields. The results of our numerical experiments on the bending analysis of composites with different boundary conditions and different configurations are provided and compared with those available in the literature. For further sudies we present some results for sandwich composite plates, including those with soft cores, as new benchmarks using the Zig–Zag theory.

Free Vibration Analysis of Laminated Soft Core Sandwich Plates

Free vibration behavior of laminated soft core sandwich plates with stiff laminated face sheets is investigated using a new C0 finite element (FE) model based on higher order zigzag theory (HOZT) in this paper. The in-plane displacement variations are considered to be cubic for both the face sheets and the core, while the transverse displacement is assumed to vary quadratically within the core and remains constant in the faces beyond the core. The plate theory ensures a shear stress-free condition at the top and bottom surfaces of the plate. Thus, the plate theory has all of the features required for an accurate modeling of laminated sandwich plates. As very few elements based on this plate theory (HOZT) exist and they possess certain disadvantages, an attempt has been made to develop this new element. The nodal field variables are chosen in such a manner to overcome the problem of continuity requirement of the derivatives of transverse displacements, i.e., no need to impose any penalty stiffness in the formulation. A nine node C0 quadratic plate finite element is implemented to model the HOZT for the present analysis. A new C0 element has been utilized to study some interesting problems on free vibration analysis of laminated sandwich plates. Many new results are also presented which should be useful for future research.

Buckling analysis of laminated composite plates using an efficient C0 FE model

Buckling analysis of laminated composite plates is carried out by using an efficient C0 FE model developed based on higher order zigzag theory. In this model the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C1 continuity associated with the FE implementation of the plate theory. The C0 continuity of the present FE model is compensated in the stiffness matrix calculations by using penalty parameter approach. Numerical results and comparison with other existing solutions show that the present model is very efficient in predicting the buckling responses of laminated composites.

Buckling analysis of laminated sandwich beam with soft core

Stability analysis of laminated soft core sandwich beam has been studied by a C0 FE model developed by the authors based on higher order zigzag theory (HOZT). The in-plane displacement variation is considered to be cubic for the face sheets and the core, while transverse displacement is quadratic within the core and constant in the faces beyond the core. The proposed model satisfies the condition of stress continuity at the layer interfaces and the zero stress condition at the top and bottom of the beam for transverse shear. Numerical examples are presented to illustrate the accuracy of the present model.

Nonlinear damping and forced vibration analysis of laminated composite beams

The purpose of the present work it to study the damping and forced vibrations of three-layered, symmetric laminated composite beams. In the analytical formulation, both normal and shear deformations are considered in the core by using the higher-order zig-zag theories. The harmonic balance method is coupled with a one mode Galerkin procedure for a simply supported beam. The geometrically nonlinear coupling leads to a nonlinear frequency amplitude equation governed by several complex coefficients. In the first part of the paper, linear and nonlinear damping parameters of laminated composite beams are obtained. In the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. The frequency response curves are presented and discussed for various geometric and material properties.

Vibration of laminated sandwich beams having soft core

Free vibration response of laminated sandwich beams having a soft core is studied by using a recently developed C0 finite element beam model. The model has been developed based on higher order zigzag theory where the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to be quadratic within the core while it remains constant in the faces beyond the core. The model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. Numerical examples on free vibration covering different features of laminated composite and sandwich beams are presented. Many new results are also presented which should be useful for future research.