Interlaminar Shear Stress Continuity Research Articles

An efficient C0 FE model for the analysis of composites and sandwich laminates with general layup

A C0 continuous finite element model is developed to model the refined higher order shear deformation theory. The proposed element is an upgraded version of an element based on higher order shear deformation theory. The C0 continuity of the present element is compensated in the stiffness matrix calculations. The computational efficiency is achieved by the C0 continuous finite element model by satisfying the inter-laminar shear stress continuity at the interfaces and zero transverse shear stress conditions at plate top and bottom. The performance of the upgraded element is illustrated with many numerical examples.

Vibration Control of Smart Plates Based on Mixed Variational Theorem

Vibration control of smart plates with distributed piezoelectric sensor and actuator layers is considered in this article. The mathematical model is based on the Reissner mixed variational theorem (RMVT), which ensures inter-laminar shear stress continuity and the zig-zag nature of displacement fields. The problem of smart laminated composite plates involves inter-dependent mechanical and electric fields. In the present work, displacements, electric potential, and transverse stresses are considered independently and are taken as degrees of freedom. An isoparametric eight-noded plate element is used for the finite element modeling. Numerical examples are solved to study the vibration control of smart structures.

Higher-order layerwise laminate theory for the prediction of interlaminar shear stresses in thick composite and sandwich composite plates

A higher-order layerwise theoretical framework is presented, which enables prediction of the static response of thick composite and sandwich composite plates. The displacement field in each discrete layer through the thickness of the laminate includes quadratic and cubic polynomial distributions of the in-plane displacements, in addition to the linear approximations assumed by linear layerwise theories. In-plane and interlaminar shear stiffness matrices of each discrete layer are formulated and interlaminar shear stress compatibility conditions are subsequently imposed to ensure continuity of interlaminar shear stresses through the thickness. A Ritz-type exact solution is further implemented to yield the structural response of thick composite and sandwich composite plates. The advantage of the present formulation in comparison to linear layerwise theories lies in the small number of discrete layers used to model the thick composite laminate through-thickness and in the prediction of interlaminar shear stresses at the interface.

Vibration characteristics of composite/sandwich laminates with piezoelectric layers using a refined hybrid plate model

The free vibration characteristics of laminated composite and sandwich plates with embedded and/or surface-bonded piezoelectric layers is studied, where a hybrid plate theory is proposed for modelling the structural system. It involves a problem of coupled electro-mechanical field. The variation of mechanical/structural displacements across the thickness are modelled by an efficient plate theory, which ensures inter-laminar shear stress continuity as well as stress free condition at the plate top and bottom surfaces. At the same time, this plate model has the distinct advantage of involving unknowns at the reference plane (plate mid-plane) only. For the electrical potential, the through thickness variation is modelled by the layer-wise theory, where the unknowns are taken at all the interfaces including top and bottom surfaces of the plate. However, the degrees of freedom for the electric potential are finally eliminated from the element matrices through condensation. In order to have generality in analysis, the finite element technique is used to approximate the in-plane variation of displacement parameters at the reference plane and electric potential at the different interfaces. For this purpose, an attempt has been made to develop a simple and efficient rectangular element, which satisfies C 1 continuity of transverse displacement at the inter-element boundaries. To validate the proposed model, some numerical examples are solved and the results obtained are compared to the published results. As the number of such published results is not sufficient, new examples covering various features are solved to study different aspects of the proposed model.

Formula omitted] plate element for global/local analysis of multilayered composites, based on a 3D zig-zag model and strain energy updating

Laminated and sandwich composites, which find applications as primary structures where weight saving is crucial, absorb energy through a variety of local failure modes. Since this damage accumulation could adversely affect the structural performances and the service life, an intensive research activity has been oriented towards suitable computational models. To support the optimization processes within the damage-tolerant design approach, low computational cost, refined zig-zag models and postprocessing procedures enabling the prediction of interlaminar stresses with the desired accuracy have been recently developed. In this paper, a 3D zig-zag plate model based upon the kinematics and nodal degrees of freedom of classical plate models, corresponding finite element and related postprocessing procedures are developed to efficiently and accurately predict ply level stresses in laminated and sandwich composites. To overcome the C 2 continuity requirement for shape functions which results from enforcement of the continuity of interlaminar shear and normal stresses and of the transverse normal stress gradient at the layer interfaces, the higher-order energy contributions brought about by this model are incorporated through strain energy updating into a C o parent eight-node plate element based on the first-order shear deformation plate theory. This energy updating is made possible by the coinciding functional degrees of freedom of the two models. A postprocessing iterative procedure is developed to obtain the suited interdependent interpolation of displacement and stresses required for capturing interlaminar stresses at the ply level with the desired accuracy. The accuracy of the present element is assessed by comparing its predictions with the stress fields of the elasticity solution for a very thick, simply supported sandwich beam with laminated faces, loaded by a sinusoidal heap loading. In addition, a comparison is made with the predictions of a mixed solid element recently developed by the author, in terms of accuracy and costs by a discrete-layer model. To test the present element in a situation of practical use, the damage of stiffened panels undergoing impact loads is evaluated by different criteria and compared with that detected by ultrasonic inspection, the exact elasticity solution not being available in this case. The present element appears cost-effective and able to accurately predict the interlaminar stresses also when composites are thick, and suitable for predicting impact-induced damage.

Coupled High-Order Shear Layerwise Analysis of Adaptive Sandwich Piezoelectric Composite Beams.

A high-order discrete layer theoretical framework and a finite element are presented for predicting the electrostatic response of multifunctional beams with piezoelectric layers. A new layerwise coupled piezoelectric laminate theory is developed, in which quadratic and cubic fields are added to the in-plane displacement and electric potential approximation in each discrete layer. Interlaminar shear stress continuity is imposed through the thickness. Stiffness, piezoelectric, and permittivity matrices are formulated from ply to structural level. A finite element method and a beam element are further developed and used to predict the electrostatic response of piezocomposite beams. Numerical results and comparisons with linear layerwise beam finite element predictions illustrate the accuracy and capability of the developed mechanics to capture the local electric and shear stress response of smart piezoelectric composite beams efficiently and accurately, including beams of high thickness, beams with sandwiched foam cores, and/or beams with compliant shear layers.

High-order layerwise mechanics and finite element for the damped dynamic characteristics of sandwich composite beams

A high-order discrete-layer theory and a finite element are presented for predicting the damping of laminated composite sandwich beams. The new layerwise laminate theory involves quadratic and cubic terms for approximation of the in-plane displacement in each discrete layer, while interlaminar shear stress continuity is imposed through the thickness. Integrated damping mechanics are formulated and both laminate and structural stiffness, mass and damping matrices are formed. A finite element method and a beam element are further developed for predicting the free vibration response, including modal frequencies, modal loss factors and through-thickness mode shapes. Numerical results and evaluations of the present model are shown. Modal frequencies and damping of sandwich composite beams are measured and correlated with predicted values. Finally, parametric studies illustrate the effect of core thickness and face lamination on modal damping and frequency values.

An efficient hybrid plate model for analysis and control of smart sandwich laminates

The problem of smart laminated sandwich plates involves inter-dependent mechanical and electric fields. The mechanical/structural component is modelled by an elegant plate theory, which ensures inter-laminar shear stress continuity and satisfies stress free condition at the plate surfaces. Moreover, this theory contains unknowns at the reference plane only. The electric field is modelled by layer-wise theory to retain generality. The solution of the resulting problem is based on an element developed for this purpose. It satisfies C 1 continuity at element interfaces as demanded by the present plate theory. Numerical examples are solved to study the static response and its control.

Finite Element Analysis of Composite and Sandwich Plates Using a Continuous Inter-laminar Shear Stress Model

A four noded plate element based on a refined higher order shear deformation theory is developed for the analysis of composite and sandwich plates. This plate theory satisfies the conditions of inter-laminar shear stress continuity and stress free top and bottom surfaces of the plate. Moreover, the number of independent unknowns is the same as that in the first order shear deformation theory. The inter-elemental continuity, as desired by the plate theory, is fully satisfied in the present element. Numerical examples of composite and sandwich plates are solved to validate the element. Some new results are also presented.

Interlaminar Stress Analysis of Shell Structures with Piezoelectric Patch Including Thermal Loading

Interlaminar stress distribution in smart composite shells using a coupled thermal-piezoelectric-mechanical model is investigated. To maintain local accuracy of stress distributions, the trial displacement field is assumed layerwise higher order and C 0 continuous through the entire laminate thickness, accommodating zigzag in-plane warping and interlaminar shear stress continuity. The temperature and electrical fields are modeled using higher-order descriptions that can satisfy surface flux boundary conditions at structural surfaces and equipotential conditions at electrode surfaces. These assumptions ensure computational efficiency. A variational principle, addressing the interaction between thermal, piezoelectric, and mechanical fields, is used to derive the governing equations of equilibrium. The proposed theory is used to investigate the cylindrical bending problem of simply supported composite host structures with attached piezoelectric actuators, subject to a combination of mechanical, piezo-electric, and thermal loading. The interlaminar stress distributions under comprehensive loading are presented for different geometries and stacking sequences. The effects of two-way piezoelectric and thermal coupling on the stress distributions are investigated. The significance of the thermal mismatch effect on interlaminar stress distribution is also discussed. The results from present theory are validated with available exact elasticity solutions.

Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner-Mindlin formulation

A finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented. An underlying shell model is a direct extension of the first-order shear-deformation theory of Reissner–Mindlin type. A refined theory with seven unknown kinematic fields is developed: (i) by introducing an assumption of a zig-zag (i.e. layer-wise linear) variation of displacement field through the thickness, and (ii) by assuming an independent transverse shear stress fields in each layer in the framework of Reissner's mixed variational principle. The introduced transverse shear stress unknowns are eliminated on the cross-section level. At this process, the interlaminar equilibrium conditions (i.e. the interlaminar shear stress continuity conditions) are imposed. As a result, the weak form of constitutive equations (the so-called weak form of Hooke's law) is obtained for the transverse strains–transverse stress resultants relation. A finite element approximation is based on the four-noded isoparametric element. To eliminate the shear locking effect, the assumed strain variational concept is used. Performance of the derived finite element is illustrated with some numerical examples. The results are compared with the exact three-dimensional solutions, as well as with the analytical and numerical solutions obtained by the classical, the first-order and some representative refined models. Copyright © 2000 John Wiley & Sons, Ltd.

A family of shear-deformable shell finite elements for composite structures

The paper presents some aspects of refined analysis of multilayered composite plates and shells. Refined shell theory is based on a zig-zag (piece-wise linear) variation of displacement field through the thickness, and on a Reissner’s mixed variational formulation which allows an introduction of independent transverse shear fields in each layer and an enforcement of the interlaminar shear stress continuity. By relaxing basic assumptions, three additional models of interest are obtained. For the finite element approximation we use a four-noded mixed finite element on the basis of assumed strain variational approach.

C 0 Zig-Zag Finite Element for Analysis of Laminated Composite Beams

A new C0 finite element for accurate analysis of laminated composite beam structures is developed. The element formulation is based on a quadratic zig-zag layerwise theory developed previously by the writers. The theory assumes a zig-zag distribution of the in-plane displacement field through the thickness and satisfies the interlaminar shear stress continuity across the layer interfaces. In developing the finite-element formulation, the shear strain fields are made field consistent, and thus the shear locking phenomenon is eliminated. A new transverse normal strain is derived by assuming the transverse normal stress to be constant through the thickness of the laminate. This assumption is shown to remove Poisson's ratio stiffening. The results obtained from the present finite element are found to be in good agreement with exact elasticity solutions available for simply supported beams. A multisublaminate approach that is simple to implement with the present element is shown to improve the predictions of t...

Laminated composite structures by continuum-based shell elements with transverse deformation

In this paper a first-order shear/fourth-order transverse deformation theory of laminated composite shells is presented. A nonlinear continuum-based (degenerated 3D) finite element model with a strain/stress enhancement technique is developed in such a way that the nonzero surface traction boundary conditions and the interlaminar shear stress continuity conditions are all satisfied identically. Analytical integration through the shell thickness is performed. The resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the shell laminate characteristics in the normal direction can be evaluated precisely and the computational cost of the overall analysis is reduced. The numerical results are compared with analytical solutions and other finite element solutions to demonstrate the effectiveness of the theory and the computational procedure developed herein.

INTERLAMINAR SHEAR STRESS ANALYSIS OF COMPOSITE LAMINATE WITH LAYER REDUCTION TECHNIQUE

Interlaminar stress plays an important role in the delamination failure of laminated composites. A recently presented theory, the Interlaminar Shear Stress Continuity Theory (ISSCT), can directly and accurately predict interlaminar shear stresses in laminated composites by the constitutive equations. The present study further generalizes the derivation to a complete form from which many displacement-based laminate theories can be derived. Most of all, both the single-layer and multiple-layer approaches are incorporated into the analysis in the thickness direction. The laminate is discretized into several sublaminates and, then, a layerwise theory is applied in the analysis of this reduced laminate. This reduction in the number of layers used in the analysis makes the calculation of interlaminar shear stresses on the interested interface in a thick laminate more efficient. In addition, numerical solutions in closed-form and finite element form for laminates under cylindrical bending and bidirectional bending are examined. It is found that in the thickness direction the cubic order of the interpolation function and the discretization with four to six sublaminates can reduce the computational efforts dramatically and retain the accuracy of the predicted stresses within ±8 per cent.

Nonlinear displacement-based finite-element analyses of composite shells—A new total Lagrangian formulation

Presented here is a new total Lagrangian finite-element formulation for general laminated composite shells undergoing large-displacement, large-rotation, and large-strain motion. The theory fully accounts for geometric nonlinearities (large rotations), large in-plane strains, general initial curvatures, transverse shear deformations, and interlaminar normal stresses by using Jaumann stress and strain measures, an exact coordinate transformation, and a new concept of orthogonal virtual rotations. In addition, with the inclusion of transverse normal and shear stresses, the theory accounts for three-dimensional stress effects and gives accurate effective structural stiffnesses. Because of the inclusion of all possible initial curvatures, the developed strain-displacement expressions can be used for any surface structures and are fully nonlinear, which contain von Karman strains as a special case. Moreover, the theory accounts for the continuity of interlaminar shear and peeling stresses and the normal and shear stress conditions on the bonding surfaces by using a new layer-wise local displacement field, which results in a higher-order shear theory that contains most shear deformation theories as special cases. In this paper, we develop a fully nonlinear displacement-based finite-element formulation based on the derived shell theory. The verification and accuracy of the theory will be presented in a subsequent paper by comparing obtained numerical results with some exact solutions.

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.

Stress analysis of imperfect composite laminates with an interlaminar bonding theory

Delamination is a major damage mode in laminated composites since it can cause severe structural degradation. Based on an interlaminar shear stress continuity theory and a linear shear slip theory, a so-called Interlayer Shear Slip Theory was presented in a previous study. This theory was verified to be feasible for shearing-mode delamination analysis. However, in order to account for opening-mode delamination in laminated composites, the continuity of interlaminar normal stress and the modelling of normal separation on the composite interface should also be considered. The present study gives a complete discussion on the Interlaminar Bonding Theory. The effects of interlaminar bonding condition on the laminate deformation and stress distribution are also presented. It is concluded from numerical results that the present theory is suitable for analysis of composite laminates with imperfect interfaces.

An Interlaminar Bonding Theory for Delamination and Nonrigid Interface Analysis

This study presents a theory for investigating composite laminates with delamination or nonrigid interfaces. It is extended from a previous study, the so-called Interlayer Shear Slip Theory (ISST), in examining composite laminates with shear slip in terfaces. The ISST consists of the Interlaminar Shear Stress Continuity Theory (ISSCT) and a shear slip theory. It has been proved to be feasible for studying composite laminates with shearing-mode (mode II and mode III) fractures. Based on a previous study on the Interlaminar Stresses Continuity Theory (ISCT), the present study adds both shear slip and normal separation to the composite interfaces. It then is useful for studying open- mode (mode I) and mixed-mode fracture problems.

A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors

A fully nonlinear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large-rotation and small-strain vibrations is presented. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation. Moreover, the model accounts for continuity of interlaminar shear stresses, extensionality, orthotropic properties of piezoelectric actuators, dependence of piezoelectric strain constants on induced strains, and arbitrary orientations of the integrated actuators and sensors. Extension and shearing forces and bending and twisting moments are introduced onto the plate along the boundaries of the piezoelectric actuators. Five nonlinear partial differential equations describing the extension-extension-bending-shear-shear vibrations of laminated plates are obtained, which display linear elastic and nonlinear geometric couplings among all motions. Piezoelectric actuator-induced warping is also addressed, and comparisons with other simplified models and nonlinear theories are made.