Higher-Order Zig-Zag Theory for Laminated Composites With Multiple Delaminations

2000 ◽  
Vol 68 (6) ◽  
pp. 869-877 ◽  
Author(s):  
M. Cho ◽  
J.-S. Kim
Keyword(s):  
Displacement Field ◽  
Transverse Shear ◽  
Dynamic Equilibrium ◽  
Composite Plates ◽  
Present Theory ◽  
Higher Order ◽  
Bottom Surface ◽  
Shear Stresses ◽  
Equilibrium Equations ◽  
Natural Frequency Analysis

A higher-order zig-zag theory has been developed for laminated composite plates with multiple delaminations. By imposing top and bottom surface transverse shear stress-free conditions and interface continuity conditions of transverse shear stresses including delaminated interfaces, the displacement field with minimal degree-of-freedoms are obtained. This displacement field can systematically handle the number, shape, size, and locations of delaminations. Through the dynamic version of variational approach, the dynamic equilibrium equations and variationally consistent boundary conditions are obtained. The delaminated beam finite element is implemented to evaluate the performance of the newly developed theory. Linear buckling and natural frequency analysis demonstrate the accuracy and efficiency of the present theory. The present higher-order zig-zag theory should work as an efficient tool to analyze the static and dynamic behavior of the composite plates with multiple delaminations.

Stress analysis of smart composite plate structures

2020 ◽  
pp. 095440622097544
Author(s):  
Aniket Chanda ◽  
Utkarsh Chandel ◽  
Rosalin Sahoo ◽  
Neeraj Grover
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Laminated Composite ◽  
Composite Plates ◽  
Higher Order ◽  
Hyperbolic Function ◽  
Solution Technique ◽  
Shear Stresses ◽  
Laminated Composite Plates ◽  
Equilibrium Equations

In the present study, the electro-mechanical responses of smart laminated composite plates with piezoelectric materials are derived using a two-dimensional (2 D) displacement-based non-polynomial higher-order shear deformation theory. The kinematics of the mathematical model incorporates the deformation of laminates which account for the effects of transverse shear deformation and a non-linear variation of the in-plane displacements using inverse sine hyperbolic function of the thickness coordinate. The equilibrium equations are obtained using the minimization of energy principle known as the principle of minimum potential energy (PMPE) which is also based on a variational approach and the solutions are obtained using Navier’s solution technique for diaphragm supported smart laminated composite plates. The responses obtained in the form of deflection and stresses are compared with three dimensional (3 D) solutions and also with different polynomial and non-polynomial based higher-order theories in the literature. The transverse shear stresses are obtained using 3 D equilibrium equations of elasticity to enhance the accuracy of the present results. Various examples are numerically solved to establish the efficiency of the present model.

Higher Order Global-Local Theory Coupled Bending and Extension for Laminated Composite Plates and Refined Element Methods

2006 ◽  
Author(s):  
Wanji Chen ◽  
Zhen Wu
Keyword(s):  
Transverse Shear ◽  
Composite Plates ◽  
Higher Order ◽  
Laminated Plates ◽  
Triangular Element ◽  
Local Theory ◽  
Weak Continuity ◽  
Shear Stresses ◽  
Thickness Direction ◽  
Transverse Shear Stresses

In this paper an augmented higher order global-local theories are presented to analyze the laminated plate problems coupled bending and extension. The in-plane displacement field is composed of a mth-order (9 > m > 3) polynomial of global coordinate z in the thickness direction and 1,2-3 order power series of local coordinate ζk in the thickness direction of each layer and a nth-order (5 > n >= 0) polynomial of global coordinate z in the thickness direction of transverse deflection. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns is independent of the layer numbers of the laminate. Based on this theory, a refined three-node triangular element satisfying the requirement of C1 weak-continuity is presented. Numerical results show that present theory can be used to predict accurately in-plane stresses and transverse shear stresses from direct use of the relations of stresses and strains without any postprocessing method. However, to accurately obtain transverse normal stresses, the local equilibrium equation approach in one element is employed herein. It is effective when the number of layers of laminated plates is more than five and up to fourteen, and it can solve the problems for coupling bending and extension. It is also shown that the present refined triangular element possesses higher accuracy.

Novel higher-order zigzag theory for analysis of laminated sandwich beams

2020 ◽  
pp. 146442072095704
Author(s):  
Aman Garg ◽  
HD Chalak
Keyword(s):  
Degrees Of Freedom ◽  
Sandwich Beam ◽  
Present Theory ◽  
Higher Order ◽  
Bottom Surface ◽  
Shear Stresses ◽  
Transverse Displacement ◽  
Sandwich Beams ◽  
Heaviside Step Function ◽  
Zigzag Theory

In the present work, a new higher-order zigzag theory is proposed for the analysis of laminated sandwich beams under static and free vibration conditions. Fourth-order in-plane and transverse displacement fields are chosen along with linear unit Heaviside step function. The present theory satisfies interlaminar transverse stress continuity conditions along with zero value at the top and bottom surface for transverse shear stresses. The proposed approach is also free from any kind of C-1 or penalty requirements. A three-noded one-dimensional finite element having eight degrees of freedom per node is used during analysis. The efficiency of the proposed model is carried out by comparing the present results with those available based on elasticity solutions and zigzag theories in the literature. New results are also reported in the present work, which will serve as a benchmark for future studies. The influence of boundary condition on the nature of stress distribution across the length of beam and frequencies of the beam with different end conditions is also carried out. A comparative study has also been carried out between symmetric and unsymmetric laminated sandwich beam.

scholarly journals Bending of functionally graded plates via a refined quasi-3D shear and normal deformation theory

2018 ◽  
Vol 5 (1) ◽  
pp. 190-200 ◽  
Author(s):  
Asharf M. Zenkour ◽  
Rabab A. Alghanmi
Keyword(s):  
Transverse Shear ◽  
Deformation Theory ◽  
Virtual Work ◽  
Three Dimensional ◽  
Present Theory ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Normal Strain ◽  
Equilibrium Equations ◽  
Normal Deformation

Abstract Bending of functionally graded plate with two reverse simply supported edges is studied based upon a refined quasi three-dimensional (quasi-3D) shear and normal deformation theory using a third-order shape function. The present theory accounts for the distribution of transvers shear stresses that satisfies the free transverse shear stresses condition on the upper and lower surfaces of the plate. Therefore, the strain distribution does not include the unwanted influences of transverse shear correction factor. The effect of transverse normal strain is included. Unlike the traditional normal and shear deformation theories, the present theory have four unknowns only. The equilibrium equations are derived by using the principle of virtual work. The influence of material properties, aspect and side-to-thickness ratios, mechanical loads and inhomogeneity parameter are discussed. The efficiency and correctness of the present theory results are established by comparisons with available theories results.

scholarly journals A Predictor-Corrector Layerwise Model for Thick Laminated Composite Plates and Shells: Bending and Vibration

1997 ◽  
Author(s):  
K. H. Lee ◽  
L. Cao
Keyword(s):  
Transverse Shear ◽  
Laminated Composite ◽  
Composite Plates ◽  
Shear Stresses ◽  
Laminated Composite Plates ◽  
Equilibrium Equations ◽  
Model Account ◽  
Transverse Shear Stresses ◽  
Plates And Shells ◽  
Predictor Corrector

This paper describes a predictor-corrector theory based on a general higher-order layerwise model for the accurate prediction of the linear static and dynamic response of thick laminated composite plates and shells. The general polynomials introduced in the model account for the arbitrary variation of the transverse shear stresses across the thickness of each layer. The main purpose of the approach is to reduce the differences between the assumed variation of the transverse shear stresses provided by the constitutive equations and the computed variation of the same stresses from the equilibrium equations of elasticity. The present predictor-corrector layerwise model satisfies the continuity of the in-plane displacements and the transverse shear stresses at the interfaces. The numerical results for the bending and vibration of thick laminated composite plates and shells show that a high level of accuracy can be achieved with the same number of variables as that in Mindlin’s theory.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

Thermo-mechanical analysis of multilayered composite beams based on a new mixed global-local model

2019 ◽  
Vol 53 (28-30) ◽  
pp. 3963-3978 ◽  
Author(s):  
Qilin Jin ◽  
Ziming Mao ◽  
Xiaofei Hu ◽  
Weian Yao
Keyword(s):  
Transverse Shear ◽  
Order Theory ◽  
Mechanical Analysis ◽  
Composite Beams ◽  
Higher Order ◽  
Shear Stresses ◽  
Mixed Form ◽  
Multilayered Composite ◽  
Proposed Model ◽  
Higher Order Theory

An accurate mixed-form global-local higher-order theory including transverse normal thermal deformation is developed for thermo-mechanical analysis of multilayered composite beams. Although transverse normal deformation is considered, the number of displacement parameters is not increased. The proposed mixed-form global-local higher-order theory is derived using the displacement assumptions of global-local higher-order theory in conjunction with the Reissner mixed variational theorem. Moreover, the mixed-form global-local higher-order theory retains a fixed number of displacement variables regardless of the number of layers. In order to obtain the improved transverse shear stresses, the three-dimensional equilibrium equation is used. It is significant that the second-order derivatives of in-plane displacement variables have been eliminated from the transverse shear stress field, such that the finite element implementation is greatly simplified. The benefit of the proposed mixed-form global-local higher-order theory is that no post-processing integration procedure is needed to accurately calculate the transverse shear stresses. The equilibrium equations and analytical solution of the proposed model can be obtained based on the Reissner mixed variational equation. The performance of the proposed model is assessed through different numerical examples, and the results show that the proposed model can better predict the thermo-mechanical responses of multilayered composite beams.

Analytical modeling of laminated composite plates integrated with piezoelectric layer using Trigonometric Zigzag theory

2020 ◽  
Vol 54 (29) ◽  
pp. 4691-4708
Author(s):  
Aniket Chanda ◽  
Rosalin Sahoo
Keyword(s):  
Transverse Shear ◽  
Piezoelectric Layer ◽  
Composite Plate ◽  
Laminated Composite ◽  
Composite Plates ◽  
Solution Technique ◽  
Shear Stresses ◽  
Smart Composite ◽  
Laminated Composite Plate ◽  
Zigzag Theory

The analytical solution for static analysis of laminated composite plate integrated with piezoelectric fiber reinforced composite actuator is obtained using a recently developed Trigonometric Zigzag theory. The kinematic field consists of five independent field variables accommodating non-linear variation of transverse shear strains through the thickness of the laminated composite plate. The principle of minimum potential energy is adopted to derive the governing equations of equilibrium. Navier’s solution technique is employed to convert the system of coupled partial differential equations into a system of algebraic equations. The electric potential is assumed to vary linearly through the thickness of the piezoelectric layer. The analytical formulation also does not include voltage as an additional primary variable. The response in the form of deflection and stresses are obtained for smart composite plates subjected to electro-mechanical loads and compared with the elasticity solutions and available results reported by other researchers in the existing literature. The transverse shear stresses are accurately determined by an efficient post-processing technique of integrating the equilibrium equations of elasticity. Parametric studies on actuation in the response of the smart composite plate are also presented graphically in order to have a clear understanding of the static behavior.

scholarly journals Higher-order semi-layerwise models for doubly curved delaminated composite shells

2020 ◽  
Author(s):  
András Szekrényes
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Higher Order ◽  
Continuity Condition ◽  
Shear Stresses ◽  
J Integral ◽  
Composite Shells ◽  
Equilibrium Equations ◽  
Energy Release Rates ◽  
Doubly Curved

Abstract This work deals w ith the development and extension of higher-order models for delaminated doubly curved composite shells with constant radii of curvatures. The mechanical model is based on the method of four equivalent single layers and the system of exact kinematic conditions. A remarkable addition of this work compared to some previous ones, is a modified and improved continuity condition between the delaminated and undelaminated parts of the shell. Using the principle of virtual work, the equilibrium equations of the shell systems are brought to the stage and solved by using the classical Lévy plate formulation under simply supported conditions. Four different scenarios of elliptic and hyperbolic delaminated shells are investigated providing the solutions for the mechanical fields as well as for the J-integral. The analytical results are compared to 3D finite element calculations, and excellent agreement was obtained for the displacement components and normal stresses. On the contrary, it was found that the transverse shear stresses are captured quite differently by the proposed method and the finite element models. Although the role of shear stresses should not be underrated, they seem to be marginal because the distributions of the J-integral components are in very good agreement with the numerically determined energy release rates.

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