Low-velocity impact of laminated composites using a layerwise theory

A layer-wise theory is used to study the low velocity impact response of laminated plates. The forced-vibration analysis is developed by the modal superposition technique. Six different models are introduced for representation of the impact pressure distribution. The first five models, in which the contact area is assumed to be known, result in a nonlinear integral equation similar to the one obtained by Timoshenko in 1913. The resulting nonlinear integral equation is discretised using a time-finite-element scheme. Two different interpolation functions, namely: (i) Lagrangian and (ii) Hermite are used to express the impact force. The Hermitian-polynomials based representation, obviously, more sophisticated, is introduced to verify the Lagrangian based representation. Due to its modular nature the present numerical technique is preferable to the existing numerical methods in the literature. The final loading model, in which the time dependence of the contact area is taken into account according to the Hertzian contact law, resulted in a relatively more complicated but more relalistic, nonlinear integral equation. The analytical developments concerning this model are all new and reported for the first time in this paper. Also a simple, but accurate, numerical technique is developed for solving our new nonlinear integral equation which results in the time-history of the impact force. Our numerical results are first tested with a series of existing example problems. Then a detailed study concerning all the response quantities, including the in-plane and interlaminar stresses, is carried out for cross-ply laminates and important conclusions are reached concerning the usefulness and accuracy of the various plate theories.     

Modelling of laminates using a layerwise element with enhanced strains

A layerwise finite element with enhanced strains is developed for the analysis of laminates with special emphasis on determination of interlaminar forces and study of delaminations. An interface model using the penalty function method is developed to calculate strain energy release rates. Since the interface model provides the facility for the closure of delamination by a small amount, strain energy release rates were evaluated by actual crack closure and by virtual crack closure methods for a comparative study. Numerical examples of delamination are presented to illustrate the accuracy of the computational approaches developed herein. © 1998 John Wiley & Sons, Ltd.     

Dynamic analysis of isotropic composite plates using a layerwise theory

The paper is concerned with small amplitude vibrations of moderately thick composite plates. Polygonal and simply supported plates of three asymmetrically arranged layers in perfect bond are considered. The homogeneous layers are assumed to consist of isotropic linear elastic materials. The Mindlin–Reissner kinematic assumptions are implemented separately to each layer, and as such model both the global and local elastic response of composite plates. The continuity of the transverse shear stress across the interfaces is prescribed according to Hooke's law. For plates composed of layers with a uniform ratio of mass density to shear modulus and a common Poisson's ratio the lateral deflection can be calculated independently from the in-plane displacements. It is shown that the lateral deflection is governed by a boundary value problem of the fourth order. Alternatively, by neglecting the longitudinal as well as rotary inertia the complex problem reduces to the simpler case of a homogenized shear-deformable plate with effective stiffness and corresponding set of boundary conditions. The theory is applied to statically and dynamically loaded rectangular simply supported composite plates with various dimensions and material properties.     

The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise models

A continually growing interest in the response of unsymmetric multilayered plates is apparent. Analyses were recently completed addressing the load-deflection behaviour of these plate geometries. The characteristic feature of the analyses is the use of nonlinear strain-displacement relations, even at low loading levels, in reaction to the large-deflection effect enhanced by the bending-extension and twisting-shearing coupling. Approaches where use is made of Higher Order Shear Deformation Theories (HSDT) for predicting global quantities, such as deflections and critical loads, are not found in the open literature. Such modelling approaches, in particular those of the layerwise type, are reserved to predict distributions across the thickness. Thus, a further assessment of the influence of the transverse shear effect on global quantities should be required. To give some preliminary contributions on this subject, the load-deflection behaviour of a [90₄/0₄] cross-ply plate with pinned edges, subjected to cylindrical bending under uniform transverse pressure, is investigated. Use is made of the Layerwise Higher Order Shear Deformation Layerwise Theory (RHSD) to serve this purpose. From the numerical results presented, the influence of modelling is enhanced or reduced, depending on the sign of loading. It is concluded that, depending on the loading, boundary conditions and lay-up, higher-order approaches can be used for predicting global quantities in unsymmetric multilayered plates. In order to investigate stability, nonlinear equations are developed where critical points are located under boundary and combined loading conditions which vary during perturbation.     

Modelling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions

In this paper we use a trigonometric layerwise deformation theory for modelling symmetric composite plates. We use a meshless discretization method based on global multiquadric radial basis functions. The results obtained are compared with solutions derived from other models and numerical techniques. The results show that the use of trigonometric layerwise deformation theory discretized with multiquadrics provides very good solutions for composite plates and excellent solutions for sandwich plates.     

Classical laminate theory model for twill weave fabric composites

The present paper is concerned with the development of an analytical model, based on Classical Lamination Theory, to predict the stiffness of twill in composites. A very simple yet quite general model is developed to obtain the elastic properties, i.e. extensional stiffness (A), extension–bending coupling stiffness (B) and bending stiffness (D). The model takes into account effects of the fabric structure by considering tow undulations and continuity along both the fill and warp directions. Various tow cross sections are possible and can be easily incorporated for a particular application. The model results in simple formulae for calculating in-plane and bending elastic constants, which can be used further in structural analysis.     

Static analysis of delaminated composite shell panels using layerwise theory

In this paper, a layerwise theory and associated finite element model are presented to study the bending deformations of delaminated composite shell panels. The layerwise theory is based on the assumption of first-order shear deformation theory in each layer, and it satisfies the displacement continuity at layer interfaces. The presence of multiple discrete delaminations is modeled through the use of Heaviside step functions. Delamination movements (relative slip and opening) are included in the theory as additional degrees of freedom. The finite element model developed is validated by comparing the present results with those available in the literature. This model based on the layerwise theory is subsequently used for the first time in the literature to study the static response and delamination movements such as interfacial slipping and opening of delaminated composite shell panels.     

Composite laminate failure analysis using multicontinuum theory

Damage in a composite material typically begins at the constituent level and may, in fact, be limited to only one constituent in some situations. An accurate prediction of constituent damage at sampling points throughout a laminate provides a genesis for progressively analyzing failure of a composite structure from start to finish. Multicontinuum Theory is a micromechanics based theory and associated numerical algorithm for extracting, virtually without a time penalty, the stress and strain fields for a composites’ constituents during a routine finite element analysis. A constituent stress-based failure criterion is used to construct a nonlinear progressive failure algorithm for investigating the material failure strengths of composite laminates. The proposed failure analysis methodology was used to simulate the nonlinear laminate behavior and progressive damage of selected laminates under both uniaxial and biaxial load conditions up to their ultimate strength. This effort was part of a broader project to compare the predictive capability of current composite failure criteria.     

Fast and reliable analysis of free‐edge stress fields in a thermally loaded composite strip by a layerwise laminate theory

A layerwise theory for the analysis of free‐edge effects in thermally loaded symmetric laminates with arbitrary layups is developed. The laminate under investigation is decomposed into an arbitrary number of mathematical layers through the thickness. The theory approach employs displacement terms according to Classical Laminate Plate Theory which are upgraded by layerwise displacement functions. The additional layerwise displacement functions consist of unknown inplane functions and linear thickness terms. The principle of minimum potential energy yields the governing Euler–Lagrange equations which allow for a closed‐form analytical solution for the inplane functions, thus characterizing this method as a semi‐analytical one. The underlying boundary conditions of traction free edges are fulfilled in an integral sense. The present results are in excellent agreement with accompanying finite element computations. Copyright © 2006 John Wiley & Sons, Ltd.     

A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory

A study of bending deformations of sandwich plates using a layerwise theory of laminated or sandwich plates is presented. The analysis is based on a wavelet collocation technique to produce highly accurate results. Numerical results for symmetric laminated composite and sandwich plates are presented and discussed.     

Vibration characteristics of laminated composite beams with magnetorheological layer using layerwise theory

Vibration characteristics of laminated composite beams with magnetorheological (MR) layer are investigated using layerwise theory. In most studies, shear strain across the thickness of MR layer has been considered as a constant value, which does not precisely describe the shear strain. In this study, layerwise theory is employed to develop a finite element formulation to investigate MR-laminated beams. Experimental tests under different magnetic fields are carried out to verify the numerical results. Layerwise numerical results are compared with the experimental results and other theories. An empirical expression for complex shear modulus is presented. The effects of MR layer thickness on vibration of MR-laminated beams are examined.     

Development of an accurate finite element model for N-layer MR-laminated beams using a layerwise theory

Laminated composite beams incorporated with magneto-rheological fluid are being used in variety of critical applications. An N-layer magneto-rheological-laminated beam based on layerwise theory has been developed to study the dynamic characteristics. For simulation purpose, an MR-laminated beam with five layers is considered in which two layers filled with magneto-rheological and three layers are made of composite materials. The results of simulations are compared with existing layerwise, first-order shear-deformation theory and experimental tests where it shows the accuracy and functionality of the present model. The complex shear modulus of magneto-rheological fluid has been determined using ASTM E756-98 standard test.     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations

In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies.     

Modelling the creep of a pipe weld using Cosserat theory

A mathematical model is developed which describes the steady state creep in a welded pipe which is subjected to a constant uniaxial end load and/or uniform internal and external pressure. The model is based on the Cosserat theory of plates and shells and a generalisation of Norton's law. Both asymptotic and analytical solutions are found and the results reveal that bending and thinning of the pipe take place on different length scales.     

Simulated stiffness determination from simple compression tests on a thick laminate

A set of experimental compression tests were simulated with a 3D finite element (FE) model to evaluate a technique for determination of engineering constants (Young’s moduli and Poisson’s ratios). During the experiments prism specimens of E-glass UD weave/PET were compressed between parallel platens. The prisms were equipped with four 0°/90° strain gauges. The simulations include friction between prism and platens and measured specimen misalignment. The simulated stress/strain and strain/strain responses are in good agreement with the linear part of the experimental response. The simulations confirm that good estimates of engineering constants can be obtained by proper strain averaging and extrapolation of the non-oscillating last part of the differentiated stress/strain and strain/strain curves.     

Giant magneto-electric effect in laminate composites

It has been discovered that laminate composites of longitudinally magnetized magnetostrictive and transversely poled piezoelectric layers (a L-T laminate composite) have a giant magneto-electric (ME) effect under a low magnetic bias. The ME voltage coefficient is over 110 mV/Oe at a magnetic bias H=500 Oe. This value is 5-10 times higher than that previously reported for transverse magnetized/transverse polarized (T-T) laminates of the same layer compositions at the same bias. In this paper, we also report the magneto-elasto-electric bieffect equivalent circuit of the L-T laminate composite and the corresponding theoretical formula of the magneto-electric voltage coefficient.     

An accurate determination of stresses in thick laminates using a generalized plate theory

Analytical solutions for displacements and stresses in composite laminates are developed using the laminate plate theory of Reddy. The theory accounts for a desired degree of approximation of the displacements through the laminate thickness, allowing for piecewise approximation of the inplane deformation through individual laminae. The solutions are compared with the 3-D elasticity solutions for the simply supported case and excellent agreement is found. Analytical solutions are also presented for other boundary conditions. The results indicate that the generalized shear deformation plate theory predicts accurate stress distributions in thick composite laminates.     

A simple laminate theory approach to the prediction of the tensile impact strength of woven hybrid composites

Simple laminate theory is used to predict the stress distribution in plain weave hybrid carbon/glass-reinforced epoxy composites under tensile loading in a direction parallel to the direction of the weave. The tensile load to cause initial failure in the carbon-reinforced plies is predicted in terms of the two-dimensional Tsai-Wu failure criterion and the measured strengths of the constituent carbon- and glass-reinforced plies. The load at final failure is predicted using the same criterion for the failure of the glass plies and assuming a reduced tensile stiffness in the carbon plies following initial failure. The theory is tested against experimental results for three woven reinforced hybrid carbon/glass composites at a quasi-static and an impact rate of strain. Reasonable agreement is obtained for the overall strength at failure, but the strain at failure is significantly overestimated.     

Cold roll bonding of multi-layered bi-metal laminate composites

Multi-layered laminate composites of dissimilar metals have assumed importance industrially. Cold roll bonding can produce multi-layered sheet composites. Study of the effect of rolling and material variables on the bonding characteristics needs to be studied in order to predict the optimum bonding conditions and the final composition of the laminate sheets. In this work, cold roll bonding of multi-layered bimetals has been modeled using the slab method. The effect of anisotropy has been included. Effects of different process and material variables are analyzed. Novel experiments were performed on multilayered Ti–Al system and the numerical results from the model were compared with the experimental results. A good agreement was observed between the model and experimental results.