A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates

In this paper, a nth-order shear deformation theory is proposed to analyze the free vibration of laminated composite plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third-order theory can be considered as a special case of present nth-order theory (n = 3). Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties are computed by present nth-order theory and a meshless radial point collocation method based on the thin plate spline radial basis function. The results are compared with available published results which demonstrate the accuracy and efficiency of present nth-order theory.     

A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates

In the present paper, a n-order model for functionally graded and composite sandwich plate is developed. This model uses the n-order polynomial term to represent the displacement field. Zero transverse shear stress boundary conditions at the top and bottom of the plate are satisfied. The third-order theory of Reddy [8] can be considered as a special case of present n-order theory. Natural frequencies of the functionally graded and composite plates with various side-to-thickness ratios, material properties are computed by present n-order theory with different n values. The results are compared with available published results which demonstrate the accuracy and efficiency of present n-order theory.     

Application of the Darboux transform to spectral problems in ocean acoustics

We propose a generalization of the n-order Darboux transform, which can be applied to solving spectral problems in ocean acoustics in cases when the medium can be considered stratified with respect to density. For a second-order Darboux transform with the boundary conditions characteristic of a single-layer shallow sea model, conditions are determined under which the spectra of the initial and transformed spectral problems coincide. It is demonstrated that, within the framework of this model, it is possible to use the n-order Darboux transform for constructing sound velocity profiles admitting exact solutions.     

An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

In this study, a locking-free n-sided C¹ polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third-order shear deformation theory (TSDT). The in-plane displacements are approximated using barycentric form of Lagrange shape functions. The weak-form Galerkin formulation based on the kinematics of TSDT requires the C¹ approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C⁰ Lagrange interpolants over a cubic Bernstein-Bezier patch defined over the n-sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter-element edges. In addition, Eringen's stress-gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.     

A model of composite laminated Reddy plate of the global-local theory based on new modified couple-stress theory

In this article, based on the global-local theory, a model for a composite laminated Reddy plate of new modified couple-stress theory is developed for the first time. This model satisfies free surface conditions and the geometric and stresses continuity conditions at interfaces. Differing from existing modified couple-stress theory, an anisotropic constitutive relation is suggested. There is only one micro material characteristic length constant in each layer of the composite laminated plate. Principle of virtual work is employed to derive the equilibrium equations and the corresponding boundary conditions. With the example of a cross-ply simple-supported Reddy plate subjected to the bending load q₀ = sin?(πx/L)sin?(πy/L), the transverse shear stresses at interfaces are addressed. Additionally, the numerical results show that the present plate model can capture the scale effect, especially the scale effect of the transverse shear stresses at interfaces of the composite laminated plate.     

Linearized buckling analysis of isotropic and composite beam-columns by Carrera Unified Formulation and dynamic stiffness method

Abstract

This article introduces a one-dimensional (1D) higher-order exact formulation for linearized buckling analysis of beam-columns. The Carrera Unified Formulation (CUF) is utilized and the displacement field is expressed as a generic N-order expansion of the generalized unknown displacement field. The principle of virtual displacements is invoked along with CUF to derive the governing equations and the associated natural boundary conditions in terms of fundamental nuclei, which can be systematically expanded according to N by exploiting an extensive index notation. After the closed form solution of the N-order beam-column element is sought, an exact dynamic stiffness (DS) matrix is derived by relating the amplitudes of the loads to those of the responses. The global DS matrix is finally processed through the application of the Wittrick-Williams algorithm to extract the buckling loads of the structure. Isotropic solid and thin-walled cross-section beams as well as laminated composite structures are analyzed in this article. The validity of the formulation and its broad range of applicability are demonstrated through comparisons of results from the literature and by using commercial finite element codes.     

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

A penalty plate-bending element for the analysis of laminated anisotropic composite plates

A C⁰ (penalty) finite element is developed for the equations governing the heterogeneous laminated plate theory of Yang, Norris and Stavsky. The YNS theory is a generalization of Mindlin's theory for homogeneous, isotropic plates to arbitrarily laminated anisotropic plates and includes shear deformation and rotary inertia effects. The present element can also be used in the analysis of thin plates by appropriately specifying the penalty parameter. A variety of problems are solved, including those for which solutions are not available in the literature, to show the material effects and the parametric effects of plate aspect ratio, length-to-thickness ratio, lamination scheme, number of layers and lamination angle on the deflections, stresses, and vibration frequencies. Despite its simplicity, the present element gives very accurate results.     

Triangular finite element for analysis of thick laminated shells

The development of a general curved triangular element based on an assumed displacement potential energy approach is presented for the analysis of arbitrarily laminated thick shells. The associated laminated shell theory assumes transverse inextensibility and layerwise constant shear angle. The present element is a quadratic triangle of C⁰-type in the curvilinear co-ordinate plane, which is then mapped onto a curved surface. Convergence of transverse displacement, moments, stresses and the effect of two Gauss quadrature schemes also form a part of the investigation. Examples of two laminated shell problems demonstrate the accuracy and efficiency of the present element. Comparison of the present LCST (layerwise constant shear-angle theory) based solutions, with those based on the CST (constant shear-angle theory) clearly demonstrates the superiority of the former over the latter, especially in the prediction of the distribution of the surface-parallel displacements and stresses through the laminate thickness.     

Dynamic characterization of the thickness-tapered laminated plant fiber-reinforced polymer composite plates

Evolution of the laminated woven natural fiber fabric-reinforced polymer composite structures makes a way to the development of the non-uniform laminated composite structures in order to achieve the stiffness variation throughout the structure. An attempt is made in this work to carry out the experimental and numerical investigations on the dynamic characteristics of the thickness-tapered laminated woven jute/epoxy and woven aloe/epoxy composite plates. The governing differential equations of motion for the thickness-tapered laminated composite plate are developed using the h-p version FEM based on higher order shear deformation theory. The validation of the present finite element formulation is carried out by comparing the natural frequencies obtained using the finite element formulation with those natural frequencies determined experimentally. The developed model is further validated with the available literature works on tapered composite plate to confirm the efficiency of h-p version FEM. This work also explores the study of the vibrational characteristics of composite plates under the influence of plant fiber’s transverse isotropic material characteristics and porosity associated with plant fiber composites through the elastic constants evaluated in the author’s previous work. Also the influences of aspect ratios, ply orientations, and taper angles under various end conditions on the natural frequencies of the woven jute/epoxy composite plate are studied using the present finite element formulation. The forced vibration response of the thickness-tapered laminated woven jute/epoxy composite plate under the harmonic force excitation is carried out considering CFCF and CFFF end conditions.     

Boundary tension between amphiphilic layers

The structure and tension of the region of contact where an oil phase, a water phase, and a microemulsion phase meet, are theoretically investigated. The analysis has as a starting point a Landau theory-like expression for the free energy in which, besides the usual gradient of the density squared term, a term proportional to the second derivative of the density squared is present, while the coefficient of the squared gradient tern is taken to be negative in the microemulsion phase. It is shown that a first-order wetting transition exists at which point infinitely many (n=0,1,2,...) surface phases can coexist, each described by the presence of a diferent thickness of the microemulsion phase between the oil and the water phase. This situation physically describes the coexistence of layers of amphiphilic molecules with different thicknesses between a water and an oil bulk phase, the thickness of one amphiphilic layer being equal tol ₀. We present the calculation of the density profile and the boundary tension of the contact region of a coexistingn = 0 andn = 1 surface phase, e.g., between the oil-water interface with a single amphiphilic layer present and the oil-water interface where three amphiphilic layers are present.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994 Boulder Colorado U.S.A.     

A global-local higher order theory including interlaminar stress continuity and C0 plate bending element for cross-ply laminated composite plates

A C⁰-type global-local higher order theory including interlaminar stress continuity is proposed for the cross-ply laminated composite and sandwich plates in this paper, which is able to a priori satisfy the continuity conditions of transverse shear stresses at interfaces. Moreover, total number of unknowns involved in the model is independent of number of layers. Compared to other higher-order theories satisfying the continuity conditions of transverse shear stresses at interfaces, merit of the proposed model is that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To verify the present model, a C⁰ three-node triangular element is used for bending analysis of laminated composite and sandwich plates. It ought to be shown that all variables involved in present model are discretized by only using linear interpolation functions within an element. Numerical results show that the C⁰ plate element based on the present theory may accurately calculate transverse shear stresses without any postprocessing, and the present results agree well with those obtained from the C¹-type higher order theory. Compared with the C¹ plate bending element, the present finite element is simple, convenient to use and accurate enough.     

Exact 3D Thermoelectroelastic Analysis of Piezoelectric Plates through a Sampling Surfaces Method

The article focuses on the use of the method of sampling surfaces (SaS) to exact three-dimensional (3D) solutions of the steady-state problem of thermoelectroelasticity for piezoelectric laminated plates subjected to thermal loading. The SaS method is based on selecting inside the nth layer I_n not equally spaced SaS parallel to the middle surface of the plate in order to choose temperatures, electric potentials, and displacements of these surfaces as basic plate variables. This permits the representation of the proposed thermopiezoelectric plate formulation in a very compact form. The SaS are located inside each layer at Chebyshev polynomial nodes that improves the convergence of the SaS method significantly. As a result, the SaS method can be applied to 3D exact solutions of thermoelectroelasticity for piezoelectric laminated plates with a specified accuracy using the sufficient number of SaS.     

Postbuckling analysis of stiffened laminated composite panels,using a higher-order shear deformation theory

The investigation aims at: (i) constructing a modified higher-order shear deformation theory in which Kirchhoff's hypotheses are relaxed, to allow for shear deformations; (ii) validating the present 5-parameter-smeared-laminate theory by comparing the results with exact solutions; and (iii) applying the theory to a specific problem of the postbuckling behavior of a flat stiffened fiber-reinforced laminated composite plate under compression.The first part of this paper is devoted mainly to the derivation of the pertinent displacement field which obviates the need for shear correction factors. The present displacement field compares satisfactorily with the exact solutions for three layered cross-ply laminates. The distinctive feature of the present smeared laminate theory is that the through-the-thickness transverse shear stresses are calculated directly from the constitutive equations without involving any integration of the equilibrium equations.The second part of this paper demonstrates the applicability of the present modified higher-order shear deformation theory to the post-buckling analysis of stiffened laminated panels under compression. to accomplish this, the finite strip method is employed. A C ²-continuity requirement in the displacement field necessitates a modification of the conventional finite strip element technique by introducing higher-order polynomials in the direction normal to that of the stiffener axes. The finite strip formulation is validated by comparing the numerical solutions for buckling problems of the stiffened panels with some typical experimental results.     

Exact 3D stress analysis of laminated composite plates by sampling surfaces method

A paper focuses on the use of the efficient approach to exact 3D elasticity solutions of cross-ply and angle-ply laminated composite plates. This approach is based on the new method of sampling surfaces (SaS) developed recently by the authors. We introduce inside the nth layer I_n not equally spaced SaS parallel to the midsurface of the plate and choose displacements of these surfaces as fundamental plate unknowns. Such an idea permits the representation of the proposed higher order layer-wise plate theory in a very compact form. This fact gives in turn the opportunity to derive the exact 3D solutions of elasticity for thick and thin laminated composite plates with a prescribed accuracy by utilizing a sufficiently large number of SaS, which are located at interfaces and Chebyshev polynomial nodes.     

Refined global–local higher‐order theory and finite element for laminated plates

Based on completely three‐dimensional elasticity theory, a refined global–local higher‐order theory is presented as enhanced version of the classical global–local theory proposed by Li and Liu (Int. J. Numer. Meth. Engng. 1997; 40 :1197–1212), in which the effect of transverse normal deformation is enhanced. Compared with the previous higher‐order theory, the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature. However, the previous higher‐order theory will encounter difficulty for this problem. A refined three‐noded triangular element satisfied the requirement of C¹ weak‐continuity conditions in the inter‐element is also presented. The results of numerical examples of moderately thick laminated plates and even thick plates with span/thickness ratios L/h = 2 are given to show that in‐plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique. In order to accurately obtain transverse normal stresses, the local equilibrium equation approach in one element is employed herein. Copyright © 2006 John Wiley & Sons, Ltd.     

A C-type higher-order theory for bending analysis of laminated composite and sandwich plates

In this paper, a C⁰-type higher-order theory is developed for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of number of layers. The continuity conditions of transverse shear stresses at interfaces are a priori enforced. Moreover, the conditions of zero transverse shear stresses on the upper and lower surfaces are also considered. Based on the developed higher order theory, the typical solutions are presented for comparison. It is very important that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields of the proposed model, so that its finite element counterparts may avoid using the C¹ interpolation functions. To assess the developed theory, the C¹-type higher-order theory is chosen for comparison. Numerical results show that the present model can accurately predict the thermal/mechanical response of laminated composite and sandwich plates. Moreover, the present model is able to accurately calculated transverse shear stresses directly from constitutive equations without any postprocessing methods.     

A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation

Based on a modified couple stress theory, a model for composite laminated beam with first order shear deformation is developed. The characteristics of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads of f_w = q₀ sin (πx/L) is adopted and explicit expression of analysis solution is obtained. Numerical results show that the present beam model can capture the scale effects of microstructure, and the deflections and stresses of the present model of couple stress beam are smaller than that by the classical beam mode. Additionally, the present model can be reduced to the classical composite laminated Timoshenko beam model, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam model of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory.     

Lamb wave evaluation and localization of transverse cracks in cross-ply laminates

This paper investigates the effect of transverse cracks on the S ₀ mode velocity in GFRP and CFRP cross-ply laminates, and proposes a new AE source location method that considers the change in the S ₀ mode velocity due to the transverse cracks. We found experimentally that the stiffness and the velocity decreased as the transverse crack density increased. Analytical predictions deduced from the combination of the complete parabolic shear-lag analysis, the classical plate theory and the laminated plate theory are in good agreement with the experimental results. Utilizing this relationship between the velocity and the mechanical damage, we located AE sources of transverse cracks in cross-ply laminates with the calculated in situ velocity. We were able to show that highly accurate source location requires the reduction of the in situ value of the velocity. The present method is simple but quantitative and useful in health-monitoring for detecting and localizing the damage in composite structures.     

A Computationally viable higher-order theory for laminated composite plates

A variational higher-order theory involving all transverse strain and stress components is proposed for the analysis of laminated composite plates. Derived from three-dimensional elasticity with emphasis on developing a viable computational methodology, the theory is well suited for finite element approximations as it incorporates both C⁰ and C^?1 continuous kinematic fields and Poisson boundary conditions. From the theory, a simple three-node stretching-bending finite element is developed and applied to the problem of cylindrical bending of a symmetric carbon/epoxy laminate for which an exact solution is available. Both the analytic and finite element results were found to be in excellent agreement with the exact solution for a wide range of the length-to-thickness ratio. The proposed higher-order theory has the same computational advantages as first-order shear-deformable theories. The present methodology, however, provides greater predictive capabilities, especially, for thick-section composites.