Elsevier

Composite Structures

Volume 172, 15 July 2017, Pages 1-14
Composite Structures

Global-local analysis of laminated plates by node-dependent kinematic finite elements with variable ESL/LW capabilities

https://doi.org/10.1016/j.compstruct.2017.03.057Get rights and content

Abstract

This work presents a class of plate finite elements (FEs) formulated with node-dependent kinematics, which can be used to construct global-local models with high numerical efficiency. Taking advantage of Carrera Unified Formulation (CUF), plate theory kinematics can be individually defined on each FE node, realizing a variation of refinement levels within the in-plane domain of one element. When used in the bridging zone between a global model and a locally refined one, an efficient global-local model can be constructed. Elements with variable ESL/LW kinematics from node to node are developed and applied in the global-local analysis of laminated structures. This work includes numerical examples in which LW models with refined kinematics are employed in local regions while ESL models are adopted in the less critical area, and modeling domains are connected by transition zone composed of elements with node-dependent kinematics. The obtained results are compared with solutions from literature and 3D FE modeling. For laminated plates with local effects to be considered, the proposed plate models can reduce the computational costs significantly while guaranteeing numerical accuracy without using special global-local coupling methods.

Introduction

Application of composite laminated structures to improve the structural efficiency is drawing increasing attention in many engineering fields especially in aerospace. Sophisticated local effects that cause stress concentration in laminated structures raise the demands for efficient numerical solution approaches.

The simplest 2D model is the well-known Classical Plate Theory (CPT) [1] based on Kirchoff-Love’s hypothesis. To take the transverse strains into consideration, First-order Shear Deformation Theory (FSDT) [2] was proposed, which can also been referred to as Reissner–Mindlin model. In the last two decades, a variety of Higher-order Theories (HOT) have also been suggested for the analysis of thin-walled structures. Carrera [3] proposed Unified Formulation (CUF) as a new framework to build refined 2D models. CUF introduces thickness functions Fτ(z) (employing either series expansion or interpolation polynomials) to formulate the kinematics through the thickness, with which both Equivalent Single Layer (ESL) and Layer-wise (LW) models can be described in a unified manner. Numerical accuracy can be improved by increasing the theory approximation order inherently without the cumbersome derivation of governing equations thanks to the employment of fundamental nuclei (FNs), see [4], [5]. Such an advantage leads to a variety of models with variable kinematics, such as those presented by Cinefra et al. [6] and Cinefra and Valvano [7].

In FE analysis, traditionally, to increase the numerical accuracy and capture local stress concentration, the h-version approach [8] is used to increase the mesh refinement while the p-version refinement [9] uses higher order polynomials as shape functions. In contrast, the h-p-version method combines these two approaches [10]. Since in this type of FE models only one set of mesh employing a globally defined kinematic theory exists, and the focus is on the refinement of mesh or element order rather than kinematic theories, they can be distinguished as mono-model approaches.

In multi-model methods, kinematically inconsistent models (e.g., 2D/3D FEs [11], [12], or classical 2D/HOTs, etc.) are combined in a global-local scenario. In a sequential multi-model approach, a subsequent locally refined model is subjected to boundary conditions determined by a global model with less-refined kinematics in a previous step. Due to their intrinsic characteristics, sequential multi-model methods are difficult to be extended to nonlinear cases. This drawback can be overcome by iterative sequential methods, in which the global and local models are solved simultaneously, and an equilibrium needs to be established on their boundaries. Whitcomb and Woo [13] extended this method to geometrically nonlinear problems.

A variety of simultaneous multi-model approaches have been proposed for global-local analysis, in which different FE models are employed separately in different regions, then the compatibility is enforced at region interfaces or in the overlapping zone. Fish et al. [14] developed an accelerated multi-grid method with an iterative process of information sharing between coarse and fine meshes. Fish [15] presented s-version method to improve the accuracy in the local domain by superimposing additional elements with higher-order kinematics on the global model, in which homogeneous boundary conditions on the superimposed field were used to guarantee the continuity of displacement. Park et al. [16] proposed a similar method which refined the local mesh. The s-version FE method can also be used in combination with h-version and p-version approaches in a simultaneous way, as summarized by Reddy and Robbins [17] and Reddy [18].

Three-field formulations introduce displacements at domain interfaces, then enforce the displacement compatibility with Lagrange multipliers. Prager [19] proposed an interface potential utilizing Lagrange multipliers. Aminpour et al. [20], and Ransom [21] adopted a spline method to couple two domains with different meshes. Adoption of similar approaches was also reported by Brezzi and Marini [22]. Blanco et al. [23], [24] presented an eXtended Variational Formulation (XVF) to couple non-matching kinematic models through two newly introduced Lagrange multiplier fields, which was lately employed by Wenzel et al. [25] in building global-local models. Carrera and Pagani [26], [27], [28] also utilized Lagrange multipliers in refined beam models for global-local analysis.

Two solutions domains can also be connected by an overlapping zone. Dhia [29] and Dhia and Rateau [30] suggested the Arlequin method to enforce compatibility within the overlapping domain with Lagrange multipliers. The Arlequin method has also been implemented in in the framework of CUF by Biscani et al. [31] for beam models and plate models [32], [33]. He et al.[34] adopted Arlequin method to bridge low- and high-order models constructed with CUF, and Constrained Variational Principle (CVP) was used to derive beam elements for multi-layered structures with individual kinematics in each layer.

For multi-layered structures, a simple approach to reduce the computational consumption is grouping the plies into sub-laminates [35], [36] each can be further characterized by independent kinematic theory [37]. Robbins and Reddy [38] proposed variable kinematic finite elements (VKFE) by superposing different types of assumed displacement fields (LW and ESL) within the same element domain, in which homogeneous essential boundary conditions are imposed on the optional and incremental LW variables to the basic ESL ones to maintain the displacement continuity between different types of elements.

CUF provides the convenience to implement node-dependent kinematics by writing theory approximation order as a function of the approximation domain and by making extensive use of index notation, which allows the governing equations in the form of FE arrays to be written in a compact form, see [39]. CUF-based FE models increase the number of degrees of freedom at each node to better approximate the structural responses. For refined plate models, different nodal thickness functions Fτi can be integrated with the nodal Lagrangian shape functions Ni to construct advanced elements. Such a methodology permits the connection of domains with different kinematic models by commonly used Lagrangian shape functions, keeping the continuity of displacement field without any ad hoc coupling method, and reduces the complexity of the numerical methods greatly. A natural and important application of node-dependent kinematics is building global-local FE models. Carrera and Zappino [40] firstly proposed such an approach, then extended it to the analysis of composite structures with advanced beam models [41]. Plate elements adopting node-dependent kinematics with variable ESL models for multi-layered structures were also proposed by Carrera et al. [42].

In the present article, node-dependent kinematics are applied to construct plate FE models for the global-local analysis of multi-layered structures, and particular attention is paid to the application of elements with variable ESL/LE capabilities used to bridge a locally refined model to a global less-refined one. The FE governing equations for plate models with node-dependent kinematics are derived by applying the Principle of Virtual Displacement (PVD). The assembly of FE stiffness matrix and load vector is elaborated. The proposed approach is assessed with three numerical cases including both laminated composite plates and sandwich structure. The obtained results are compared against solutions provided by literature or 3D FE modeling.

Section snippets

Preliminaries

The reference system and the geometry of the multi-layered plate are given in Fig. 1. For displacement-based plate theories, the strain and stress components can be arranged as follows:pT={xx,yy,xy},nT={xz,yz,zz}.σpT={σxx,σyy,σxy},σnT={σxz,σyz,σzz}.where the subscript p and n indicate the in-plane and out-of-plane components, respectively. The strain vector p and n can be obtained via the geometrical equations:p=Dpu,n=(Dnp+Dnz)u.the explicit expressions of the differential operator

Carrera Unified Formulation

According to CUF, the displacement field u={u,v,w}T can be expressed by means of approximation functions Fτ(z) as follows:u(x,y,z)=F0(z)u0(x,y)+F1(z)u1(x,y)++FN(z)uN(x,y)v(x,y,z)=F0(z)v0(x,y)+F1(z)v1(x,y)++FN(z)vN(x,y)w(x,y,z)=F0(z)w0(x,y)+F1(z)w1(x,y)++FN(z)wN(x,y)

In a more compact form, CUF can be written as shown in Eq. (8) for ESL models and Eq. (9) for LW models, respectively:u(x,y,z)=Fτ(z)uτ(x,y)τ=0,1,,Nuk(x,y,ζk)=Fτk(ζk)uτ(x,y);τ=0,1,,Nwhere k is the layer index in laminated plates.

Plate elements with node-dependent kinematics

As discussed in the previous section, CUF introduces thickness functions Fτ(k) to build refined plate models. At the same time, commonly used plate finite elements employ Lagrangian shape functions to approximate the in-plane displacements. For a plate element with M nodes, the 3D displacement field can be discretized as follows:u(k)(x,y,z)=Ni(x,y)Fτ(k)(z)uiττ=1,,N;i=1,,M.in which i is the node index, and M represents the number of nodes in a plate element. uiτ is the vector of nodal primary

Numerical results

In this section, results on laminated structures obtained with node-dependent kinematic plate models are presented. Corresponding global-local models with variable ESL/LW kinematics are denoted by TEnZ-LEm, in which TEnZ refers to the less-refined kinematic models based on Taylor series used in the region of less interest, and LEm stands for the refined models with Lagrange polynomials employed in the local zone. The transition zone is as wide as the size of one element. Since FSDT give a

Conclusions

Based on Carrera Unified Formulation (CUF), node-dependent kinematics is presented as an innovative approach to constructing advanced elements, which can be applied to build FE models for global-local analysis. Plate elements with node-dependent variable ESL/LW kinematics are developed and implemented in the global-local analysis of laminated structures. Through the numerical investigation, the following conclusions can be drawn:

  • Node-dependent kinematics provides a convenient approach to

Acknowledgment

This research work has been carried out within the project FULLCOMP (FULLy analysis, design, manufacturing, and health monitoring of COMPosite structures), funded by the European Union Horizon 2020 Research and Innovation program under the Marie Skłodowska Curie grant agreement No. 642121.

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