Multiobjective Best Theory Diagrams for cross-ply composite plates employing polynomial, zig-zag, trigonometric and exponential thickness expansions

doi:10.1016/j.compstruct.2017.05.055

Composite Structures

Volume 176, 15 September 2017, Pages 860-876

https://doi.org/10.1016/j.compstruct.2017.05.055 Get rights and content

Abstract

This paper presents Best Theory Diagrams (BTDs) for plates considering all the displacement and stress components as objectives. The BTD is a diagram in which the minimum number of terms that have to be used to achieve the desired accuracy can be read. Maclaurin, zig-zag, trigonometric and exponential expansions are employed for the static analysis of cross-ply composite plates. The Equivalent Single Layer (ESL) approach is considered, and the Unified Formulation developed by Carrera is used. The governing equations are derived from the Principle of Virtual Displacements (PVD), and Navier-type closed-form solutions are adopted. BTDs are obtained using the Axiomatic/Asymptotic Method (AAM) and genetic algorithms (GA). The results show that the BTD can be used as a tool to assess the accuracy and computational efficiency of any structural models and to draw guidelines to develop structural models. The inclusion of the multiobjective capability extends the BTD validity to the recognition of the role played by each output parameter in the refinement of a structural model.

Introduction

Composite laminated plates are increasingly common in many engineering applications, such as aerospace, mechanical, marine and civil structures. In fact, composite plates have many favorable mechanical properties, e.g. high stiffness, and low density. The high demand for the use of composite material structures calls for research of efficient and accurate numerical techniques to predict the structural and dynamical behavior of laminated composites.

Classical plate theories (CPT) neglect transverse shear and normal stresses [1], [2]. An extension of this model to multi-layered structures is referred to as the Classical Lamination Theory (CLT) [3], [4]. Due to the increasing use of thick laminated plates in structures, Reissner and Mindlin [5], [6] included transverse shear effects in their well-known First-Order Shear Deformation Theory (FSDT). Although the FSDT is simple to implement and apply for both thick and thin laminated plates, the accuracy strongly depends on shear correction factors and the nonexistence of complicated stress gradients [7]. The limitations of the FSDT weaken or disappear with Higher-order Shear Deformation Theories (HSDT). The HSDTs assume quadratic, cubic, higher-order variations or non-polynomial terms to improve the displacement field along the thickness direction [8], [9], [10], [11], [12], [13], [14]. Further enhancements are useful 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] can deal with these phenomena.

Plate modeling has two main approaches, the Equivalent Single Layer (ESL) and the Layer-Wise (LW) models [18], [19], [20], [21], [22]. Theories based on the ESL assumption offer reduced computational complexity; however, they struggle to model the zig-zag effects typical of laminates. LW theories have quasi-three-dimensional predictive capabilities; however, the computational effort can increase significantly.

The present paper makes use of ESL models and includes non-polynomial terms to Maclaurin expansions. Different non-polynomial kinematics models have been proposed in the literature. 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 symmetrically laminated beams. Mantari and co-workers have recently proposed various extensions to non-polynomial plate models, including ESL and LW trigonometric models [23], [24], HSDTs based on Trigonometric-Exponential terms [25], [26], hybrid Maclaurin-trigonometric models [27], [28], and a generalized hybrid formulation for the study of functionally graded sandwich beams [29], [30]. Thai et al. [31] presented a new first-order shear deformation theory for the bending, free vibration and buckling analysis of functionally graded sandwich plates, which was extended to a generalized formulation by Thai et al. in Ref. [32]. A new simple four-unknown shear and normal deformations theory for static, dynamic and buckling analyses of functionally graded material of isotropic and sandwich plates was proposed recently by Thai et al. [33]. Nguyen et al. [34] developed a unified framework on HSDTs for laminated composite plates. The Carrera Unified Formulation (CUF) has been recently employed to develop non-polynomial structural models [35], [36], [37], [38], [39], [40].

The refined models employed in this paper are based on the CUF. According to CUF, the governing equations are given via the so-called fundamental nuclei whose form does not depend on either the expansion order nor on the choices made for the base functions to generate any structural model [41], [42], [43]. In the CUF framework, Carrera and Petrolo [44], [45] introduced the Axiomatic/Asymptotic Method (AAM) to develop 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 [44], [46], metallic and composite plates [45], [47], shells [48], [49], LW models [50], [51], advanced models based on the Reissner Mixed Variational Theorem [52], piezoelectric plates [53], and thermomechanical problems [54].

The AAM has led to the BTD [55]. The BTD allows one to determine the minimum number of expansion terms – i.e. unknown variables – required to meet a given accuracy; or, conversely, the best accuracy provided by a given amount of variables. To construct BTDs with a lower computational cost, a genetic algorithm was employed by Carrera and Miglioretti [56]. In particular, BTDs were built by minimizing the number of the expansion terms and the error on an output parameter, such as a displacement or stress component. Petrolo et al. [57] presented BTDs for ESL and LW composite plate models based on Maclaurin and Legendre polynomial expansions of the unknown variables along the thickness. Recently, Carrera et al. have extended the BTD to multifield problems [58].

The present work presents a method to develop BTDs considering multiple objectives simultaneously; in particular, the three displacement components and the six stress ones. The BTDs are therefore the Pareto fronts of the optimization of the expansions to minimize the error on each displacement and stress component. A Maclaurin expansion with zig-zag terms and a hybrid Maclaurin, zig-zag, trigonometric and exponential expansion are considered. The non-polynomial terms in the latter are selected according to Filippi et al. [39].

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 for multiple output parameters 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 are shown in Fig. 1, where x and y are the in-plane coordinates while z is the thickness coordinate. The integer k denotes the layer number. In the framework of the CUF, the displacement components of a plate model is $u (x, y, z) = F_{τ} (z) \cdot u_{τ} (x, y) τ = 1, 2, \dots, N + 1$ where u is the displacement vector $(u_{x}, u_{y}, u_{z})$ . $F_{τ}$ are the expansion functions. $u_{τ}$ $(u_{x_{τ}}, u_{y_{τ}}, u_{z_{τ}})$ is the vector of the displacements variables. In the ESL case, $F_{τ}$ functions can be

Governing equations and Closed-form solution

Geometrical relations enable to express the in-plane $ε_{p}^{k}$ and the out-of-plane $ε_{n}^{k}$ strains in terms of the displacement u, $\begin{matrix} ε_{p}^{k} = {[ε_{xx}^{k}, ε_{yy}^{k}, ε_{xy}^{k}]}^{T} = (D_{p}^{k}) u^{k}, \\ ε_{n}^{k} = {[ε_{xz}^{k}, ε_{yz}^{k}, ε_{zz}^{k}]}^{T} = (D_{np}^{k} + D_{nz}^{k}) u^{k} \end{matrix}$ where $D_{p}^{k}$ , $D_{np}^{k}$ and $D_{nz}^{k}$ are differential operators whose components are: $\begin{matrix} D_{p}^{k} = [\begin{matrix} \frac{\partial}{\partial x} & 0 & 0 \\ 0 & \frac{\partial}{\partial y} & 0 \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \end{matrix}], \\ D_{np}^{k} = [\begin{matrix} 0 & 0 & \frac{\partial}{\partial x} \\ 0 & 0 & \frac{\partial}{\partial y} \\ 0 & 0 & 0 \end{matrix}], \\ D_{nz}^{k} = [\begin{matrix} \frac{\partial}{\partial z} & 0 & 0 \\ 0 & \frac{\partial}{\partial z} & 0 \\ 0 & 0 & \frac{\partial}{\partial z} \end{matrix}] \end{matrix}$

Stress components for a generic k layer can be obtained using the constitutive law, $\begin{matrix} σ_{p}^{k} = C_{pp}^{k} ε_{p}^{k} + C_{pn}^{k} ε_{n}^{k} \\ σ_{n}^{k} = C_{np}^{k} ε_{p}^{k} + C_{nn}^{k} ε_{n}^{k} \end{matrix}$ where $C_{pp}^{k}$ , $C_{pn}^{k}$ , $C_{np}^{k}$ and $C_{nn}^{k}$ are: $\begin{matrix} C_{pp}^{k} = [\begin{matrix} C_{11}^{k} & C_{12}^{k} & C_{16}^{k} \\ C_{12}^{k} & C_{22}^{k} & C_{} \end{matrix}] \end{matrix}$

Axiomatic/Asymptotic method

The introduction of higher-order terms in a plate model offers significant advantages in terms of improved structural response prediction at the expense of higher computational costs. The axiomatic/asymptotic method (AAM) allows us to lower the computational cost of a model without affecting its accuracy. A typical AAM analysis has the following steps:

(1)
Parameters such as geometry, boundary conditions, loadings, materials, and stacking sequences are fixed.
(2)
A set of output parameters is chosen,

Best Theory Diagram

The construction of refined models through the AAM allows one to obtain a diagram in which each refined model is associated with the number of active terms and the error on a given displacement or stress output variable on a reference solution. Best models are those that, for a given error, require the minimum number of variables; or, for a given number of variables, provide the best accuracy. Best models represent a Pareto front of an optimization problem in which the objectives are the

Results and discussion

A bisinusoidal load was applied to the top surface of the simply supported laminated plate, $p = \overline{p_{z}} \cdot \sin (\frac{m π x}{a}) \sin (\frac{n π y}{b})$ where $a = b = 0.1 m$ , $\overline{p_{z}} = 1 kPa$ , and m,n = 1. The reduced models were developed considering the displacements $u_{x}$ , $u_{y}$ , $u_{z}$ , and the stresses $σ_{xx}$ , $σ_{yy}$ , $σ_{zz}$ , $τ_{xy}$ , $τ_{xz}$ , and $τ_{yz}$ . The following dimensionless quantities were defined for the displacements and stresses: $\begin{matrix} {\overline{u}}_{x} = \frac{u_{x} E_{2}^{k = 1} \cdot h^{2}}{{\overline{p}}_{z} \cdot a^{3}}, {\overline{u}}_{y} = \frac{u_{y} E_{2}^{k = 1} \cdot h^{2}}{{\overline{p}}_{z} \cdot a^{3}}, {\overline{u}}_{z} = \frac{u_{z} 100 \cdot E_{2}^{k = 1} \cdot h^{3}}{{\overline{p}}_{z} \cdot a^{4}}, \\ {\overline{σ}}_{xx, yy} = \frac{σ_{xx, yy}}{{\overline{p}}_{z} \cdot {(a / h)}^{2}}, {\overline{σ}}_{zz} = \frac{σ_{zz}}{{\overline{p}}_{z}}, {\overline{τ}}_{xy} = \frac{τ_{xy}}{{\overline{p}}_{z} \cdot {(a / h)}^{2}}, {\overline{τ}}_{xz, yz} = \frac{τ_{xz, yz}}{{\overline{p}}_{z}} \end{matrix}$

Conclusion

Best Theory Diagrams (BTDs) for cross-ply laminated plates considering multiple output parameters have been presented in this paper. The BTD is a curve in which, for a given problem, the most accurate plate models for a given number of unknown variables can be read. In this work, BTDs consider multiple objectives simultaneously; in particular, the three displacement components and the six stress ones. The BTDs are therefore the Pareto fronts of the optimization of the expansions to minimize the

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Multiobjective Best Theory Diagrams for cross-ply composite plates employing polynomial, zig-zag, trigonometric and exponential thickness expansions

Abstract

Introduction

Section snippets

Carrera Unified Formulation for plates

Governing equations and Closed-form solution

Axiomatic/Asymptotic method

Best Theory Diagram

Results and discussion

Conclusion

Comput Struct

Int J Eng Sci

Compos Sci Technol

Compos Struct

Comp Struct

Int J Sol Struct

Comp Struct

Compos Struct

Compos B

Aerosp Sci Technol

Compos Struct

Compos Struct

Compos Struct

Eur J Mech A Solids

Comput Struct

Compos Struct

Int J Mech Sci

Compos B

Eur J Mech A Solids

Compos Struct

Compos Struct

J Sound Vib

Compos Struct

Compos Struct

Compos B

Compos Struct

Reliab Eng Syst Saf

Sure l’equilibre et le movement d’une plaque solide

Excercies Matematique

Memoire sur l’equilibre et le mouvement des corps elastique

Mem l’Acad Sci

Über das Gleichgewicht und die Bewegung einer elastishen Scheibe

J Angew Math

4th ed.