Generalized layerwise HSDT and finite element formulation for symmetric laminated and sandwich composite plates

Abstract

A generalized layerwise displacement base higher order shear deformation theory and its finite element formulation for the bending analysis of symmetric laminated and sandwich composite plates are presented. New layerwise HSDTs are presented for the first time. The proposed generalized layerwise HSDT have limited DOFs, because they are independent of the number of layers, and it accounts for non-linear and constant variation of in-plane and transverse displacement respectively through the plate thickness. Results shows that some of the new non-polynomial layerwise HSDTs, having the same or even less DOFs, are more accurate than well-known non-polynomial layerwise HSDTs available in the literature.

Introduction

The use of composite structures is becoming predominant in many engineering fields such as marine, aerospace, aviation, civil, sport, and other applications. Composite materials are made for example by combining materials such as carbon fibers with epoxy, and they have been used in airplane components since the beginning of the uses of this kind of materials [1]. Moreover, recently, large commercial transport airplanes are certified to use composite structures in almost the entire structure, and therefore, it even open more the branch of possible uses of composites structures, such as in larger civil and naval ships [2]. This is because composite structures have high performance and reliability due to high strength-to-weight and high stiffness-to-weight ratios, excellent fatigue strength, resistance to corrosion (e.g. glass fiber composites), and most importantly the design flexibility also known as tailoring the materials for desired applications.

With the increased use of sandwich structures, there is a tremendous need to develop efficient manufacturing techniques, economical and effective repair techniques, and methods to predict the short and long-term behavior of the multilayer composite structures under a variety of loading and environmental conditions. Therefore, it is important to introduce effective macro, micro or perhaps nano mechanical models, and develop new theoretical methodologies for simple and accurate structural response predictions. In what follows a short description of macro theoretical methodologies for classical composites is presented followed by the main contribution of this paper.

In the last forty years, a significant number of higher-order shear deformation theories (HSDTs) for composite laminated plates and shells have been presented. These theories can be classified in different models, such as equivalent single layer (ESL), quasi-layerwise and layerwise models [3], [4]. Another important point in the analysis of composite structures is the variational statement used in the analysis to derive the necessary governing equations and the boundary conditions. Approximate solutions can either be developed using displacement-based theories (when the principle of virtual displacement is used), stress based theories or displacement-stress-based theories (when Reissner mixed variational theorem is used, see [3], [4], [5], [6], [7], [8], [9], [10], [11]).

Following the Carrera’s unified formulation (CUF) [12] or the generalized unified formulation (GUF) proposed by Demasi [3], [4], [9], [10], [11], [13], one can realize that, among equivalent single layer models, there are many existing polynomial class of theories (including the classical plate theory, first shear deformation theory, and the HSDTs) that CUF or GUF can be able to reproduce.

Using the ordinary classification of plate or shell deformations theories (for other classifications of theories, readers may consult [3], [12]), there are mainly three major theories; namely the classical plate theory (CPT), the first order shear deformation theory (FSDT) and the higher shear deformation theory (HSDT). The classical plate theory (CPT) is based on the assumptions of Kirchhoff’s plate theory [14], [15], [16], [17], [18], [19], which neglects the interlaminar shear deformation. First order shear deformation theory (FSDT) is based on the kinematic field assumption postulated by Mindlin [20], and assumes constant transverse shear deformation through the entire thickness of the laminate and violates stress free boundary conditions at the top and bottom surfaces of the plate [21], [22], [23], [24], [25], [26]. Higher order shear deformation theories (HSDT) were developed to improve the analysis of plate and shell responses and extensively used by many researchers [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65].

The abovementioned equivalent single layers theories may be not sufficient if local effects are important or accuracy in the calculation of the transverse stresses is required. For that reason, more advanced plate theories were developed to include zig-zag effects [10], [12], [36]. Layerwise models have also an important role when the abovementioned approaches fail to predict local effects [4], [6], [66]. This advanced method is an accurate refined method but with the disadvantage of expensive computing time. More detailed literature review may be found for various shear deformation theories and finite element approaches for composite and sandwich plates and shells using equivalent single layer and layerwise theories in excellent research papers presented by Reddy [67], [68], Carrera [5], [7], Kreja [40], Altenbach [69], Carrera [12] and Demasi [3], [4], [9], [10], [11], [13].

Carrera [70] concluded that for the accurate analyses of the vibrational response of highly anisotropic, thick and very thick shells a layerwise description and interlaminar continuous transverse shear and normal stresses were crucial for the modified classical models. Carrera and Brischetto [71] presented an extensive survey for a variety of plate theories and evaluated the bending and vibration of sandwich structures. The main drawbacks of the equivalent single layer analysis for the soft-core sandwich structures were specified. Carrera and Petrolo [72] by using CUF and an asymptotic expansion method discussed the effectiveness of higher-order terms in refined beam theories. In Carrera and Petrolo [73], the effect of each displacement variable in the solution was investigated by comparing the error obtained accounting and removing such variable in the plate equations. One important conclusion that can be inferred for the orthotropic plates was the additional term(s) used in the transverse displacement to consider the stretching of the panel highly affects the normal stress (σ_zz) results in a positive way, but they have no influence on the other considered results. In similar fashion, Carrera et al. [74] by using the powerful CUF have found the so-called best plate theory, but this time considering an axiomatic hypothesis method. More detailed information and applications of CUF can be found in the very recent books authored by Carrera et al. [75], [76]. Carrera and Miglioretti [77] analysed various displacement models using the finite elements for static analysis of laminated plates. The authors presented the best plate theories, i.e. very accurate plate theories with few computational efforts (the accuracy of given theories for the selected problem is established in terms of displacement and stress fields).

Calling attention to non-polynomial shear strain shape functions in displacement based layerwise formulations, it is known that the first non-polynomial (trigonometric) shear strain shape function was introduced by Levy [78], as mentioned in [79]. The TPT (trigonometric plate theory) formed by using the sinus shear shape strain function were corroborated and assessed by Stein [80] (after one century) and later extensively used by Touratier [55] and co-workers. Later, this ESL HSDT was intensively used in several layerwise and zig-zag HSDTs [49], [50], [51], [52]. Moreover, advanced numerical calculations such as finite element [52], [81] and meshless methods [42], [53], [57] were also implemented by using Levy’s HSDT [78]. However, the other even more accurate non-polynomial ESL HSDT such as the ones reported in [58], [62], [63], [64] were not extended to layerwise formulations. Therefore, in order to implement new shear strain shape functions in layerwise HSDTs, a generalized HSDT is presented in this paper.

Recently, for the first time, Neves et al. [82], [83] and Ferreira et al. [84] presented meshless solutions by using CUF with non-polynomial shape strain functions (sinus) to analyze static and dynamic behavior of classical and advanced (e.g. functionally graded) composites. Therefore, it can be said that there are evidences of the demand of non-polynomial shear deformation theories, mainly because they are richer than polynomial functions, simple, accurate, and the free surface boundary conditions can be guaranteed a priori.

Non-polynomial layerwise HSDTs with improved accuracy and accounting for the continuity of the transverse shear stresses having a fixed number DOFs (the DOFs does not depend on the number of layers of the multilayer panel) were recently developed by Mantari et al. [85] with the ideas based on the works presented by Shimpi and Ghugal [50], Arya et al. [51] and Roque et al. [53]. The computation time is not problem any more with this type of layerwise theory. However, not much attention was devoted to this type of advanced theories. According to the authors, intensive research activities are needed in order to obtain a more robust layerwise FEM than the new one presented in this paper. Fortunately, representative research work in accuracy and generalization of FEM formulations have been presented in [3], [4], [9], [10], [11], [12], [13]. Therefore, join research activities to improve the present generalized layerwise HSDT with perhaps the compact CUF (which were recently extended to include non-polynomial shear shape strain functions, and used to solve the bending problem of a multilayer structures by meshless numerical methods [82], [83], [84]), GUF, and others, can be of great importance in the improvement of the present generalized layerwise HSDT and its finite element formulation.

In this paper a novel generalized layerwise displacement-based higher order shear deformation theory and its finite element formulation for the bending analysis of symmetric sandwich and composite plates are presented. New non-polynomial layerwise HSDTs are presented for the first time. The proposed generalized layerwise HSDT have limited DOFs, are independent of the number of layers, and accounts for non-linear and constant variation of in-plane and transverse displacement respectively through the plate thickness. The new generalized layerwise displacement field accounts for approximately parabolic distribution of the transverse shear deformation and the transverse shear traction free conditions on the top and bottom surfaces of the plate can be satisfied, thus a shear correction factor is not required. By enforcing the free conditions on upper and lower surfaces and interlaminar continuity conditions of the transverse shear stresses, the number of total unknowns does not depend on the number of layers of the laminate. Therefore, the discrete element chosen is a four-nodded quadrilateral with five-degrees-of-freedom per node. For the finite element formulation, the five-degrees-of-freedom $\left(w_o\text{,}\partial w/\partial x\text{,}\partial w/\partial y\text{,}{\theta }_{c^{\prime }}^x\text{,}{\theta }_{c^{\prime }}^y\right)$ are assumed and they are independent of each other. Therefore, a C⁰ Lagrangian isoparametric faceted quadrilateral element to analyze the static behavior of the general laminated composite and sandwich plates is introduced. The accuracy of the present code is ascertained by comparing it with Srinivas [86] exact solution and with various numerical methods available in the literature, such as the finite element [85], [87], [88] and the meshless solutions [42], [53], [57], [89]. In addition, the accuracy of the present generalized layerwise FEM code is ascertained by comparing it with Pagano [90] exact solution and with various numerical calculations available in the literature [31], [42], [63], [64], [65], [89], [91], [92], [93], [94] for the problem of rectangular orthotropic laminated plate subjected to a laterally distributed load. Results show than some of the new non-polynomial layerwise HSDTs, having the same or even lesser DOFs than others HSDTs, are more accurate than for example well-known non-polynomial layerwise HSDTs.