Best Theory Diagrams for cross-ply composite plates using polynomial, trigonometric and exponential thickness expansions
Introduction
Laminated composite plates are extensively used in many engineering applications due to their high strength-to-weight ratio, high stiffness-to weight ratio, environmental resistance and the ability to tailor properties for desired applications. An accurate analysis of composite structures is fundamental for a reliable structural design. Several researchers have investigated the modelling of the laminated composites over the past few decades and some structural models have been developed for their analysis.
Classical plate theories (CPT), originally developed for thin isotropic plates [1], [2], neglect transverse shear and normal stresses. An extension of this model to multi-layered structures is referred to as the Classical Lamination Theory (CLT) [3], [4]. Reissner and Mindlin [5], [6] included transverse shear effects in their well-known First Order Shear Deformation Theory (FSDT). More accurate theories such as higher order theories (HOT) assume quadratic, cubic, higher variations or non-polynomial terms to improve the displacement field along the thickness direction [7], [8], [9], [10], [11], [12], [13], [14].
However, the abovementioned theories may be not sufficient if local effects are important or accuracy in the calculation of the transverse stresses is required. The Zig-Zag models [15], [16] and mixed variational tools [17] have been proposed to deal with these phenomena. Among the plate models for laminated structures two different approach can be distinguished: the Equivalent Single Layer (ESL) and the Layer-Wise (LW) models. Excellent reviews of existing ESL and LW models can be found in [18], [19], [20], [21], [22].
This paper makes use of trigonometric and exponential expansions to build refined plate models. Shimpi and Ghugal [12], proposed a LW trigonometric shear deformation theory for the analysis of composite beams. Arya et al. [13] developed a Zig-Zag model using a sine term to represent the non-linear displacement field across the thickness in symmetric laminated beams. Ferreira et al. [14] presented a LW plate model using a meshless discretization method for symmetric composite plates. Mantari et al. [23] developed a new ESL plate model in which a parameter m was included on the trigonometric functions to obtain 3D like elasticity solutions. Mantari et al. [24] extended [23] to a LW plate model for finite element analysis of sandwich and composite laminated plate. Thai et al. [25], [26] presented isogeometric finite element formulations for static, free vibration and buckling analysis of laminated composite and sandwich plates. This was extended to a generalized shear deformation theory by Thai et al. [27]. Hybrid Maclaurin-trigonometric models were proposed by Mantari et al. [28], [29] for bending, free vibration and buckling analysis of laminated beams. Mantari et al. [30] presented a generalized hybrid formulation for the study of functionally graded sandwich beams, which was extended to the Finite Element Method (FEM) by Yarasca et al. [31]. A unified framework on higher order shear deformation theories of laminated composite plates was proposed by Nguyen et al. [32]. Ramos et al. [33] developed refined theories based on non-polynomial kinematics via the Carrera Unified Formulation to deal with thermal problems, which was extended by Mantari et al. [34] to investigate the static behavior of FGM.
The refined models employed in this paper are based on the Carrera Unified Formulation (CUF). According to CUF, the governing equations are given regarding the so-called fundamental nuclei whose form does not depend on either the expansion order nor on the choices made for the base functions. This important feature allows to analyze any number of kinematic models in a single formulation and software. ESL and LW models were successfully developed in CUF, as reported in [35]. More details on CUF can be found in [36], [37]. To developed accurate refined theories with lower computational effort, Carrera and Petrolo [38], [39] introduced the Axiomatic/Asymptotic Method (AAM). This method consists of discarding all terms that do not contribute to the plate response analysis once a Ref. solution is defined. This leads to the development of reduced models whose accuracies are equivalent to those of full higher-order models. The AAM has been applied to several problems, including: static and free vibration of beams [38], [40], metallic and composite plates [39], [41], shells [42], [43], LW models [44], [45], advanced models based on the Reissner Mixed Variational Theorem [46], and piezoelectric plates [47].
The AAM method was adopted to build the BTD by Carrera et al. [48]. The BTD is a curve in which the minimum number of expansion terms - i.e. unknown variables - required to meet a given accuracy can be read; or, conversely, the best accuracy provided by a given amount of variables can be read. To construct BTDs with a lower computational cost, a genetic algorithm was employed by Carrera and Miglioretti [49]. Petrolo et al. [50] presented BTDs for ESL and LW composite plate models based on Maclaurin and Legendre polynomial expansions of the unknown variables along the thickness.
The research works in Refs. [48], [49], [50] focused on the construction of Best Theory Diagrams (BTDs) for composite plates based only on Polynomial expansions. The AAM was employed via genetic algorithm (GA) to reduce the computational cost. In the present paper, the effect of non-polynomial terms on plate models is investigated employing the full capacity of CUF, AAM and GA. This paper main hypothesis is that, if robust plate theories are analyzed, the contribution of a vast number of functions can be tested via the AAM-GA method in a single run. The computational complications with robust theories can be ignored since only focused on plate theories with low number of unknown variables (thanks to the AAM). The two main contributions in this paper are: (a) a method to establish the relevance of any number of test functions (in this case Maclaurin, trigonometric and exponential) in a single analysis via the AAM and GA. This is an improvement to previous studies in HSDT where the influence of a small number of non-polynomial functions are studied; (b) the expansion with only polynomial terms presented in Refs. [48], [49], [50], is a particular case of this robust expansion. The results show that refined theories which employed polynomial and non-polynomial terms obtain better BTDs.
Overall, the present work presents BTDs using Maclaurin, trigonometric and exponential thickness expansions for the analysis of laminated composite plates. The functions employed in this paper were selected according to Filippi et al. [51], [52]. Genetic algorithms are employed to reduce the computational cost related to the definition of the BTD.
The present paper is organized as follows: a description of the adopted formulation is provided in Section 2; the governing equations and closed-form solution is presented in Section 3; the AAM is presented in Section 4; the BTD is introduced in Section 5; the results are presented in Section 6, and the conclusions are drawn in Section 7.
Section snippets
Carrera unified formulation for plates
The geometry and the coordinate system of the multilayered plate of L layers are shown in Fig. 1. The integer k denotes the layer number that starts from the plate-bottom, x and y are the in-plane coordinates while z is the thickness coordinate.
In the framework of CUF, the displacement of a plate model can be described as:where is the displacement vector () whose components are the displacements along the x, y and z reference axes. are the expansion
Governing equations and Closed-form solution
Geometrical relations enable to express the in-plane and the out-planes strains in terms of the displacement .where , and are differential operators whose components are:
Stress components for a generic k layer can be obtained using the Hooke law,
where matrices , and are:
Axiomatic/Asymptotic method
The introduction of high order terms in a plate model offers significant advantages in terms of improved structural response analysis at the expense of higher computational cost. The Axiomatic/Asymptotic Method (AAM) allows us to decrease the computational cost of a model and at the same time preserve the accuracy of a high order model. The AAM procedure can be summarized as follows:
- (1)
Parameters such as geometry, boundary conditions, loadings, materials and layer layouts are fixed.
- (2)
A set of output
Best theory diagram
The construction of reduced models through the AAM allows one to obtain a diagram, which for a given problem, each reduced model is associated with the number of active terms and its error computed on a Ref. solution. This diagram allows editing an arbitrary given theory to get a lower number of terms for a given error, or to increase the accuracy while keeping the computational cost constant. Considering all the reduced models, it is possible to recognize that some of them provide the lowest
Results and discussion
A bisinusoidal load is applied to the top surface of the simply supported laminated plate:where . is the applied load amplitude, and and are equal to 1. The reduced models are developed for and . The axial and shear stress are computed at and , with . is the total thickness of the plate. The stresses are normalized according to:
The material properties are:
Conclusion
Best Theory Diagrams (BTDs) for cross-ply laminated plates have been presented in this paper. The BTD is a curve in which, for a given probelem, the most accurate plate models for a given number of unknown variables can be read. The Axiomatic/Asymptotic Method and genetic algorithms have been employed together with the Carrera Unified Formulation to develop refined ESL models (See Table 19). In particular, a combination of Maclaurin, trigonometric and exponential polynomials has been used to
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