Murakami's Zig-Zag Function Research Articles

Legendre-based node-dependent kinematics shell models for the global–local analysis of homogeneous and layered structures

The present work demonstrates the use of the node-dependent kinematics method to derive and compare several two-dimensional shell theories. The three dimensional displacement field is expressed in terms of generalized coordinates, which are subsequently expanded along the shell thickness using arbitrary functions. The in-plane unknowns, are then discretized through classical finite element approximation. Based on the Carrera Unified Formulation, the proposed method combines in a unique manner the theory of structures and the finite element method; thickness interpolation functions are defined node-wise. As a consequence, the resulting finite element model represents diverse approximation theories at each single node. In this work Taylor-based kinematics (including the Murakami Zig-Zag function) and Legendre-type nodal kinematics are incorporated at the element level without adopting mathematical artifices leading to the global–local strategy, where refined theories are selectively employed in specific areas, while maintaining acceptable computational costs. Numerical cases from the existing literature are employed to establish the effectiveness of node-dependent models in bridging a locally refined theories to global kinematics when local effects need to be considered. The analyses focus on localized loads for both homogeneous and multi-layered structures.

Smart control of functionally graded sandwich plates by incorporating Murakami zig-zag function

This study is aimed at incorporating the zig-zag effect by Murakami zig-zag function in the development of a finite element model for active constraining layer damping treatment of functionally graded sandwich plates. The present sandwich construction consists of functionally graded facings distanced by a ceramic core. The substrate functionally graded plate is subjected to active constraining layer damping treatment, which in itself is a two-layered material system comprised of a viscoelastic layer and a 1–3 piezoelectric composite layer. The deformation kinematics of the functionally graded sandwich plate active constraining layer damping system is shaped using Murakami zig-zag function, and the finite element model is obtained by the virtual work principle. A standard feedback control system has been implemented, and a MATLAB subroutine has been developed to present the open- and closed-loop responses. Substrate plates with functionally graded configurations 1-1-1, 1-2-1, and 2-1-2 are considered to evaluate the effect of active constraining layer damping on damping the frequency responses of these plates. Investigation on damping performance has been carried out, bearing in mind the change in power-law index with top and bottom ceramic-/metal-rich surfaces. Also, the effect of variation in fiber orientation angle (obliquely reinforced) of the piezoelectric composite material on the active constraining layer damping performance has been examined thoroughly.

Geometrically nonlinear bending analysis of variable stiffness composite laminated shell panels with a higher-order theory

A finite element (FE) formulation is developed by extending the third-order shear deformation theory, including the zig-zag effects using the Murakami zig-zag function, for the linear and nonlinear static analyses of variable stiffness composite laminated (VSCL) shell panels. In this work, a nine-noded isoparametric element with thirteen degrees of freedom is employed to model the VSCL shell panels. Geometrical nonlinearity according to von Kármán nonlinear strain-displacement relations is considered for the evaluation of the deflections and stresses. The assumed displacement field considers up to third-order terms for in-plane displacements and second-order terms for the transverse displacement through-the-thickness. The accuracy of the formulation is established by comparing the obtained results with the three-dimensional elasticity and two-dimensional numerical/analytical results available in the literature for static analysis of constant stiffness composite laminates (CSCL) and VSCL. New results are presented for the nonlinear deflection and stresses of VSCL shell panels considering the effect of the fibre path angles, defined by the curvilinear fibre orientation for the VSCL spherical, cylindrical and hyperboloid shells. Further, the effect of curvature ratios, number of layers, lamination configurations and boundary conditions on the nonlinear static behaviour of VSCL shells is presented.

Smart control of laminated plates using Murakami zig-zag functions

The study aims at vibration control of laminated plates by the active constraining layer damping (ACLD) treatment incorporating the Murakami zig-zag function (MZZF) while deriving the displacement fields. The control is achieved by activating the ACLD patches by supplying the control voltage. The ACLD patch has two component layers, namely 1–3 piezoelectric composite (PZC) constraining material layer and the viscoelastic constrained material layer. The complete ACLD plate system is modelled as a three layered structure using a MZZF in the displacement fields of the individual layers, and the FE model is obtained by the virtual work principle. A MATLAB FE code has been developed considering the closed loop feedback system for the control of the input parameters. The present method of modelling the ACLD system has proven to be accurate and cost effective for damping the vibrations of the composite plates. Also, the change in piezo fiber orientation angle on the control and performance of the ACLD patches has been effectively studied in detail.

Smart damping of skew composite plates using Murakami zig-zag function

The present work is aimed at deriving a finite element model for active constraining layer damping treatment (ACLD) of layered skew plates by incorporating zig-zag behaviour using a Murakami zig-zag function (MZZF). The ACLD in skew patch form comprises of 1–3 PZC material and viscoelastic material in the layer form placed on substrate skew plate. The overall skew substrate ACLD system deformation kinematics are derived using MZZF and the equations of motion for the same are derived by virtual work method. A MATLAB subroutine for the overall skew plate ACLD system has been developed to present the closed loop frequency responses by successful implementation of closed-loop feedback system. The substrate skew plates with different lamination schemes namely symmetric/antisymmetric cross-ply and antisymmetric angle-ply are considered to assess the damping behavior of the skew plates undergoing ACLD. Also, the piezo-fiber angle (obliquely reinforced) variation of the PZC layer on the damping responses of the skew plates have been thoroughly examined.

Smart control of cylindrical shells incorporating Murakami Zig-Zag function

The work is devoted towards incorporating the Murakami zig-zag function (MZZF) for modelling the active constraining layer damping (ACLD) treatment of ‘N’ layered laminate cylindrical shells. The ACLD treatment in the patch form comprises of two material layers with the viscoelastic material layer constrained between the host shell and the smart 1–3 piezoelectric composite (PZC) material layer. The kinematics of deformation are derived successfully implementing the MZZF in the displacement fields. Virtual work principle has been employed for the finite element (FE) model to deduce governing equations of shell ACLD system in complete form. A feed-back control system has been considered to arrive at closed loop governing equations of motion. Considering the above, a MATLAB code for the same is generated to emphasise on dynamics of the cylindrical shell ACLD system. Cylindrical shells with various lamination schemes have been considered to assess the validity and implementation of MZZF for ACLD treatment of shells. Also, the oblique orientation of the 1–3 PZC material layer and its effect on the damping behavior of the cylindrical shell has been thoroughly explored.

Large displacement models for composites based on Murakami's Zig-Zag Function, Green-Lagrange Strain Tensor, and Generalized Unified Formulation

Abstract Murakami's Zig-Zag Function in conjunction of Green-Lagrange Strain- and Second-Piola Kirchhoff Stress-tensors and Generalized Unified Formulation are introduced for the first time to present a class of Zig-Zag theories for composite structures. Transverse strain effects are retained and modeled, making the present approach also suitable for thick structures. Each displacement variable is independently represented with user-selected order of expansion. The stress recovery is achieved with a special procedure in which combination of deformation gradient and Second-Piola Kichhoff transverse stresses are integrated along the thickness of the undeformed geometry. Comparison with available data and commercial codes indicate the effectiveness of the proposed framework for variable kinematics Zig-Zag theories in the presence of large displacements.

Hygrothermoelastic Analysis of Orthotropic Cylindrical Shells

IN THIS WORK, THE COMBINED EFFECT OF MOISTURE AND TEMPERATURE ON THE BENDING BEHAVIOR OF SIMPLY SUPPORTED ORTHOTROPIC CYLINDRICAL SHELLS HAS BEEN INVESTIGATED. INITIALLY THREE DIMENSIONAL EQUILIBRIUM EQUA-TIONS OF THERMOELASTICITY, SIMPLIFIED TO THE CASE OF GENERALIZED PLANE STRAIN DEFORMATIONS IN THE AXIAL DIRECTION ARE SOLVED ANALYTICALLY FOR AN ORTHOTROPIC CYLINDRICAL SHELL STRIP UNDER THERMAL LOADING. BASED ON THE REALISTIC VARIATION OF DISPLACEMENTS FROM THE ELASTICITY APPROACH, A NEW HIGHER ORDER SHEAR DEFORMATION THEORY WAS PROPOSED FOR THE ANALYSIS OF AN ORTHOTROPIC CYLINDRICAL SHELL STRIP UNDER HYGROTHERMAL AND MECHANICAL LOADING. THE ZIGZAG FORM OF THE DISPLACEMENT IS INCOR-PORATED VIA THE MURAKAMI ZIGZAG FUNCTION. RESULTS ARE PRESENTED FOR MECHANICAL AND THERMAL LOADING FOR VARIOUS LAYUPS AND THEY ARE VALI-DATED AGAINST THE DERIVED ELASTICITY SOLUTION. THE SIGNIFICANCE OF RE-TAINING VARIOUS HIGHER-ORDER TERMS IN THE PRESENT MODEL, IN EVALUATING THE STRESSES AND DEFLECTION FOR COMPOSITE LAMINATES IS BROUGHT OUT CLEARLY THROUGH PARAMETRIC STUDY. USEFUL RESULTS FOR COMBINED HYGROTHERMAL LOADING ARE PRESENTED IN TABULAR AND GRAPHICAL FORM. IT IS EXPECTED THAT THE NUMERICAL RESULTS PRESENTED HEREIN WILL SERVE AS BENCH MARK IN FUTURE.

Hygrothermoelastic analysis of anisotropic cylindrical shells

In this work, the combined effect of moisture and temperature on the bending behavior of simply supported anisotropic cylindrical shells has been investigated. Initially three dimensional equilibrium equations of thermoelasticity, simplified to the case of a generalized plane strain deformations in the axial direction are solved analytically for an anisotropic cylindrical shell strip under thermal loading. Based on the realistic variation of displacements from the elasticity approach, a new 13 term higher order shear deformation theory HSDT13 was proposed for the analysis of an anisotropic cylindrical shell strip under hygrothermal and mechanical loading. The zigzag form of the displacement is incorporated via the Murakami zigzag function. Results are presented for mechanical and thermal loading for various layups and they are validated against the derived elasticity solution. The significance of retaining various higher-order terms in the present model HSDT13, in evaluating the stresses and deflection for composite laminates is brought out clearly through parametric study. Useful results for combined hygrothermal loading are presented in tabular and graphical form. It is expected that the numerical results presented herein will serve as benchmark in future.

On the Use of Zigzag Functions in Equivalent Single Layer Theories for Laminated Composite and Sandwich Beams: A Comparative Study and Some Observations on External Weak Layers

The paper presents a comparison between two existing zigzag functions that are used to improve equivalent single layer (ESL) theories for the analysis of multilayered composite and sandwich beams. ESL theories are easy to implement and computationally affordable but, in order to correctly describe the mechanical behavior of laminated structures (especially those exhibiting high transverse anisotropy or high thickness-to-side length ratios), the displacement field needs to be enriched by a through-the-thickness piecewise linear contribution denoted as “zigzag.” The zigzag term of the displacement field is used to model the local distortion of the cross section in each lamina of multilayered structures and is related to the continuity of transverse stresses. The paper considers two zigzag functions that have been proposed in the open literature (namely Murakami's zigzag function and the refined zigzag function) and compares their performances when they are used to improve the classical Timoshenko beam theory; both displacement-based and mixed formulations are considered. To the best of the author's knowledge, such a comparative study has never been published. The problem of a simply supported beam subjected to a transverse distributed load is considered as a test case. Several stacking sequences, ranging from monolithic to sandwich-like and from symmetric to arbitrary, are considered. The special case of laminates with external weak layers is also investigated and the effects of these lay-ups on the derivation of the refined zigzag function are analyzed for the first time. The capability of the tested zigzag functions to help evaluate the overall deflection and model the through-the-thickness distribution of the axial displacement and stress is investigated. It has been recognized that the refined zigzag function is more accurate, especially for unsymmetric and arbitrary lay-ups and can be adopted to efficiently introduce zigzag kinematics into any ESL theory.

Analysis of sandwich plates by radial basis functions collocation, according to Murakami's Zig-Zag theory

In this article, the static analysis of sandwich plates is performed by radial basis functions collocation, according to the Murakami's Zig-Zag function theory. The Murakami's Zig-Zag function theory accounts for through-the-thickness deformation, by considering a Zig-Zag evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by the Carrera's unified formulation, and further interpolated by collocation with radial basis functions.

Two higher order Zig-Zag theories for the accurate analysis of bending, vibration and buckling response of laminated plates by radial basis functions collocation and a unified formulation

In this article, we combine the Carrera's Unified Formulation, CUF (Carrera E. Theories and Finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Methods Eng., 2003; 10: 215–297.) and a radial basis function collocation technique for predicting the static deformations, free vibrations and buckling behavior of thin and thick cross-ply laminated plates. We develop by the CUF two Zig-Zag theories according to Murakami's Zig-Zag function. Both theories account for through-the-thickness deformations, allowing the analysis of thick plates. The accuracy and efficiency of this collocation technique for static, vibration, and buckling problems are demonstrated through numerical examples.

2D, Quasi 3D and 3D Exact Solutions for Bending of Thick and Thin Sandwich Plates

A previous author's work related to the exact solution of a single layer isotropic plate is extended to the case of multilayered plate structures composed of isotropic layers. The method solves a first order linear system of differential equations in the unknown amplitudes of the displacements and stresses. An eigenvalue problem, in which analytical expressions for the basis of eigenvectors and generalized eigenvectors are available, is then formulated. This makes the coding of exact solution of multilayered structures very simple and effective compared with other exact methods present in the literature. The paper is mainly concerned about the analysis of sandwich structures made of isotropic layers. Challenging cases of thick and moderately thick plates (a/h = 1, 4, 10) and thin plates (a/h = 100) are analyzed in detail and the displacements and stresses are found and plotted through the thickness of the plate. An extensive study of the ratio between the elastic modulus of the skins and the elastic modulus of the core, as well as the change in the displacements and stresses with that parameter, is conducted. The exact three-dimensional results are also validated with quasi-3D results obtained by adopting a mixed assiomatic theory of order 9 presented here for the first time. Twenty three two-dimensional theories are assessed, five of which are presented here for the first time. The use of Murakami's zig—zag function and the consequent improvement of the ESL theories is also analyzed and discussed. All the results of this work are an useful tool to test and compare approximated methods, such as Finite Element Method.

Mixed Finite Elements for Thermoelastic Analysis of Multilayered Anisotropic Plates

This paper presents some finite elements for the thermoelastic analysis of multilayered plates which are based on unified formulation and the Reissner mixed variational theorem (RMVT). The pure mechanical statement of the theorem is herein extended to thermoelastic analysis and the related mixed constitutive equations have been derived. Assumptions are made for the displacement fields in thickness direction and the RMVT is employed to obtain finite element (FE) matrices. The unified formulation allows to easily derive the FE matrices in terms of a few fundamental nuclei whose form is not affected by: number of the nodes of the elements; order of the expansion for the displacements-variable description (Layer-Wise, LW, and Equivalent-Single Layer, ESL, cases are both addressed). The Murakami Zig-Zag function is used to introduce Zig-Zag effects in the framework of ESL descriptions. Comparisons to exact 3D solutions as well as to classical finite elements have proved the accuracy of the proposed elements and thier capability to accurately describe the stress field of layered plates subjected to thermal loadings.

Unified Formulation for Finite Element Thermoelastic Analysis of Multilayered Anisotropic Composite Plates

ABSTRACT This article presents various finite plate elements for the thermal stress analysis of multilayered anisotropic structures. Assumptions are made for the displacement fields in thickness direction and the principle of virtual displacements (PVD) is employed to derive finite element (FE) matrices. The unified formulation is employed so that these FE matrices have been derived in terms of a few fundamental nuclei whose form is not affected by number of the nodes of the elements or order of the expansion for the displacements-variable description (layer-wise (LW) and equivalent single layer (ESL) cases are both addressed). The Murakami zig-zag function is used to introduce zig-zag effects in the framework of ESL descriptions. The performances of the derived finite elements, in terms of displacement and stress fields, are shown by solving thermal stress problems related to cross-ply laminated plates for various thickness ratio values. Comparison to closed form solutions as well as to available 3-D exact analyses have shown the effectiveness of the proposed elements and their capability to trace quasi 3-D descriptions of thermal stresses in layered plates. Communicated by Liviu Librescu on November 25, 2004.

A unified formulation to assess multilayered theories for piezoelectric plates

This paper proposes a unified formulation for the electro-mechanical analysis of multilayered plates embedding piezo-layers. Both equivalent single layer (ESL) and layer-wise (LW) models have been considered. Zig-zag effects have been described by employing the Murakami zig-zag function (MZZF). Linear up to fourth-order expansions are used for displacement variables. Electric potential has been restricted to LW descriptions. Governing equations have been derived in terms of a few ‘fundamental nuclei’ with only nine terms each. These equations have been presented in a manner which is ‘formally’ not affected by the assumptions that have been introduced for displacement and electrical variables. Closed-form solutions and related numerical verifications show that the unified formulation could lead to a quasi-three-dimensional description of global and local characteristic of static and dynamic response of pieloelectric plates. The convenience of implementing the MZZF has been underlined. Quite different accuracy has been obtained by different theories with respect to mechanical and electrical loadings respectively. Layer-wise analyses could be required in thick plate geometries. Furthermore, an accurate description of the electro-mechanical coupling demands a layer-wise description of the displacement or at least the separated modeling of the piezoelectric and the structural layers, even though thin plates are considered.

Refined multilayered plate elements based on Murakami zig–zag functions

This paper discusses the FEM use of Murakami zig–zag function (MZZF) in the two-dimensional modeling of multilayered plates. A literature overview is first conducted on the available works in which MZZF has been applied. MZZF is used to introduce the zig–zag effect in classical and higher order theories, which are formulated using only displacements as unknowns. MZZF is also considered to introduce the zig–zag effect in those theories, which are formulated on the basis of both displacement and transverse stress assumptions (mixed formulations). The present FEM formulation is validated by comparing the results with some available papers from the literature. A very thick plate ( a/ h = 4) is studied and the results are compared with the commercial code NASTRAN. Numerical results are presented to show both the effectiveness and limitations of MZZF in the modeling of layered plates. Linear up to forth order expansion for in-plane and out-of-plane displacements, in the thickness plate direction, has been compared. It has been concluded that MZZF consists of a valuable tool to enhance the performances of both classical and advanced theories. In particular, the conducted numerical evaluations have shown that it can be more convenient to enhance a plate theory by introducing the MZZF than refining it by adding two or three higher order terms. However, in order to well approximate local effects or study thick plates, advanced models (layer-wise or three-dimensional) are required.

On the use of the Murakami's zig-zag function in the modeling of layered plates and shells

This paper discusses the use of the Murakami zig-zag function (MZZF) in the two-dimensional modeling of multilayered plates and shells. A literature overview of the available works is first presented. A `simple use' of the MZZF is discussed: the MZZF is used to introduce the zig-zag effect in classical and higher order theories which are formulated with only displacement unknowns. An `advanced use' of the MZZF is then considered to introduce the zig-zag effect in those theories which are formulated on the basis of both displacement and transverse stress assumptions. A number of new plate/shell theories has been considered. Numerical results encompassing, static, dynamic and thermally loaded orthotropic, simply supported plates and shells are presented to show both the effectiveness and limitations of the MZZF in the modeling of layered structures. Linear up to forth-order expansions for the in-plane and the out-of-plane displacements, in the thickness plate/shell direction, have been compared. It has been concluded that the MZZF is a valuable tool to enhance the performances of both classical and advanced theories. The conducted numerical evaluations have shown in particular that multilayered plate and shell theories can be greatly improved by the use of MZZF.