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%
\documentclass{article}
\gdef\Templatepath{E:/testing/Normalization}
\usepackage{\Templatepath/elsnorm}
\usepackage{bm}
\pageframe{false}
\referencestyle{numbertrue}
\begin{document}
\input testing-macros.tex
\title{A new robust quadrilateral four-node variable kinematics plate element for composite structures}
\author{T.~H.~C.~Le\\
M.~D'Ottavio*\\
P.~Vidal\\
O.~Polit\\
UPL, Univ Paris Nanterre, LEME\\
50 Rue de S{\`e}vres, 92410 Ville d'Avray, France\\
Corresponding author. Email: michele.d\_{o}[email protected]}
\maketitle
\textbf{Abstract}
This paper presents a new four-node quadrilateral finite element for composite plates. A large number of displacement-based, variable kinematics plate models are formulated in the framework of Carrera's Unified Formulation, which encompass Equivalent Single Layer as well as Layer-Wise models with a displacement field that is defined by polynomials up to 4th order along the thickness direction $z$. The main novelty consists in the formulation of a field compatible approximation for the transverse shear strain field, referred to as QC4 interpolation, which eliminates the shear locking pathology by constraining only the $z - $constant transverse shear strain terms. Extensive numerical studies are proposed that demonstrate the absence of spurious modes and of locking problems as well as the enhanced robustness with respect to distorted element shapes in comparison to classical isoparametric approaches. The new QC4 variable kinematics plate element displays excellent convergence rates under different boundary and loading conditions, and it yields accurate displacement and stresses for both, thick and thin composite plates.
\textbf{keywords}
Plate finite element,
Variable kinematics model,
Shear locking,
Field compatibility approach,
Mesh distortion,
Composite structure
\section{Introduction}
The increasing use of composite laminates and sandwich structures in engineering applications drives the need for appropriate analysis and design tools with dedicated computational models. Based on geometric considerations, composite panels are conveniently modeled as two-dimensional plate/shell structures. However, complicating effects $ - $ such as anisotropy, heterogeneity and transverse shear compliance $ - $ call for plate/shell models that go beyond the so-called \emph{classical models}, i.e., those relying on $(i)$ Kirchhoff-Love assumptions and neglecting transverse deformation (Classical Laminate Plate Theory, CLPT), $(ii)$ or on Reissner-Mindlin assumptions and retaining a merely constant transverse shear deformation through the thickness (First-order Shear Deformation Theory, FSDT). Numerous review papers devoted to high-order plate/shell models witness the scientific progress in this specific topic \cite{Noor:Burton:1989,Reddy:Robbins:review:1994,Noor:Burton:sandwichReview:1996,Carrera:ARCME:2002,Sayyad-Ghugal-sandwichReview-2015}.
\section{Variable kinematics plate model\label{sec:vkmodel}}
Let us consider a multilayered plate occupying the domain $V = \Omega \times \left\{ -\frac{e}{2} \leq x_{3} \leq \frac{e}{2} \right\}$ in a Cartesian coordinate system $(x, y, z) = (x_{i})$, see \Fig{plate-coordsys}. Unless otherwise stated, Latin indices range in $\{1, 2,3\}$, Greek indices range in $\{1, 2\}$ and tensorial repeated index convention is employed. $\Omega$ is the reference surface of arbitrary shape lying in the $(x_{1}, x_{2}) - $plane located for convenience at $z = 0$. The plate has constant thickness $e$, which is composed of $k = 1, 2,\ldots N_{L}$ orthotropic, elastic and perfectly bonded layers, each with a thickness $e^{(k)}$ and with an orientation of the material orthotropy axes defined by the rotation angle $\theta^{(k)}$ about the thickness direction $x_{3} \equiv z$.
\begin{figure}
\centering
\caption{Coordinates and notation used for the description of the composite plate.}
\label{fig:plate-coordsys}
\end{figure}
\subsection{The weak form of the boundary value problem}
The Principle of Virtual Displacement (PVD) is employed to obtain the weak form for a displacement-based plate approximation. The volume boundary is split as $\partial V = \Gamma_{u} \bigcup \Gamma_{\sigma}$, where $\Gamma_{u}$ is the portion with an imposed displacement field and $\Gamma_{\sigma}$ is the portion with imposed tractions; without loss of generality we shall neglect body forces and pose $\Gamma_{u} = \partial \Omega \times \left\{ -\frac{e}{2} \leq x_{3} \leq \frac{e}{2} \right\}$ and $\Gamma_{\sigma} = \Omega \times \{ - \frac{e}{2}, \frac{e}{2}\}$. The weak formulation of the problem thus reads:
For all admissible virtual displacement $\delta u_{i} \in \delta \mathbb{U}$, find the displacement field $u_{i} \in \mathbb{U}$ (space of admissible displacements) such that:
\begin{equation}\label{weakform}
- \int_{V} \epsilon_{ij}(\delta u_{l}) \, \sigma_{ij}(u_{m}) \; \ud x_{i}
+ \int_{\Gamma_{\sigma}} \delta u_{i} \,\bar{t}_{i} \; \ud x_{\alpha}= 0
\end{equation}
where $\bar{t}_{i}$ are the surface loads at $\Gamma_{\sigma}$, a bar denoting prescribed values; $\epsilon_{ij} (\delta u_{l})$ is the compatible virtual strain tensor and $\sigma_{ij}(u_{m})$ the stress tensor defined by means of the linear elastic constitutive law in terms of the actual strains $\epsilon_{rs}(u_{m})$.
\subsection{Variable kinematics assumptions}
Carrera's Unified Formulation (CUF) is a technique that permits to implement a large number of bi-dimensional models in a unified manner by means of an extensive use of a compact index notation \cite{Carrera:Demasi:2002, Carrera:etAl:CUFbook:2014}. Within the displacement-based approach, all plate theories are defined in CUF by specifying $(i)$ whether the displacement field is described in Equivalent Single Layer (ESL) or Layer-Wise (LW) manner, and $(ii)$ the order $N$ of the polynomial expansion used for approximating the behavior along the thickness direction. The two-dimensional variable kinematics plate model is generally formulated upon separating the in-plane variables $x_{\alpha}$ from the thickness direction $z$, along which the displacement field is \emph{a priori} postulated by known functions $F(z)$:
\begin{equation}\label{cufExp}
u_{i} (x_{\alpha}, z) = F_{\tau}(z) \, \hat{u}_{i_{\tau}} (x_{\alpha}),
\end{equation}
where $\tau = 0, 1,\ldots, N$ is the summation index and the order of expansion $N$ is a free parameter of the formulation. In this work $N$ can range from 1 to 4, in agreement with the classical CUF implementation \cite{Carrera:Demasi:2002}.
In order to deal with both ESL and LW descriptions within a unique notation, it is convenient to refer to a layer-specific thickness coordinate $z_{k} \in \{ z^{(k)}_{b} , z^{(k)}_{t} \}$ that ranges between the $z$-coordinates of the bottom (subscript $b$) and top (subscript $t$) planes delimiting the $k^{\mathrm{th}}$ layer, see \Fig{plate-coordsys}. \Eq{cufExp} can thus be formally re-written for each layer as
\begin{equation}\label{cufExp2}
u_{i}^{(k)}(x_{\alpha}, z_{k}) = F_{t}(z_{k}) \hat{u}_{i_{t}}^{(k)}(x_{\alpha}) + F_{b}(z_{k}) \hat{u}_{i_{b}}^{(k)}(x_{\alpha}) + F_{r}(z_{k}) \hat{u}_{i_{r}}^{(k)}(x_{\alpha})
\end{equation}
with $\tau =t, b, r$ and $r = 2, \ldots N$. The displacement field $u_{i}$ for the whole multilayered stack is then defined through an opportune assembly procedure of the layer-specific contributions $u_{i}^{(k)}$, which depends on the ESL or LW description.
In an ESL approach, the thickness functions are defined as Taylor-type expansion and only one variable $\hat{u}_{i_{\tau}}$ is used for the whole multilayer, i.e., the layer index $(k)$ in \Eq{cufExp2} may be dropped off and the following thickness functions are used:
\begin{equation}\label{eslF}
F_{b} = 1, \quad F_{t} = z^{N}, \quad F_{r} = z^{r} \quad (r=2,\ldots N-1)
\end{equation}
The ESL description can be enhanced by including the Zig-Zag function $F_{ZZ}(z)$ proposed by Murakami in order to allow slope discontinuities at layers' interfaces \cite{Murakami:1986}. In this case, the Zig-Zag function replaces the highest expansion order and the following functions are used:
\begin{equation}\label{eslFZZ}
F_{b} = 1, \quad F_{t} = F_{ZZ}(z) , \quad F_{r} = z^{r} \quad (r=2,\ldots N-1)
\end{equation}
where Murakami's ZigZag Function (MZZF) is defined as
\begin{equation}\label{MZZF}
F_{ZZ} (z) = (-1)^{k} \, \zeta_{k}(z)
\quad \text{with} \quad
\zeta_{k}(z) = \frac{2}{z^{(k)}_{t} - z^{(k)}_{b}} \left( z - \frac{z^{(k)}_{t} + z^{(k)}_{b}}{2}\right)
\end{equation}
Note that $F_{ZZ}(z)$ is expressed in terms of the non-dimensional layer-specific coordinate $ - 1 \leq \zeta_{k} \leq +1$ and it provides a linear piecewise function of bi-unit amplitude across the thickness of each layer $k$. More details about the use of MZZF in variable kinematics PVD-based models can be found in the paper by Demasi \cite{Demasi:partialZZ:2012}.
The assumptions for a LW description are formulated in each layer $k$ as in \Eq{cufExp2}, where the thickness functions are defined by linear combinations of Legendre polynomials $P_{r}(\zeta_{k})$ as follows:
\begin{equation}\label{polLeg}
\begin{array}{rl}
F_{t}(\zeta_{k}) = \dfrac{P_{0}(\zeta_{k}) + P_{1}(\zeta_{k})}{2}; \quad F_{b}(\zeta_{k}) = \dfrac{P_{0}(\zeta_{k}) - P_{1}(\zeta_{k})}{2};\\
F_{r}(\zeta_{k}) = P_{r}(\zeta_{k}) - P_{r-2}(\zeta_{k}) \qquad (r=2, \ldots N)
\end{array}
\end{equation}
where $\zeta_{k}$ is the non-dimensional coordinate introduced in \Eq{MZZF}. The Legendre polynomials of degree 0 and 1 are $P_{0}(\zeta_{k}) = 1$ and $P_{1}(\zeta_{k}) = \zeta_{k}$, respectively; higher-order polynomials are defined according to the following recursive formula:
\begin{equation}\label{polLegDef}
P_{n+1}(\zeta_{k}) = \frac{(2n+1)\zeta_{k}\,P_{n}(\zeta_{k}) - nP_{n-1}(\zeta_{k})}{n+1}
\end{equation}
which leads to the following expressions for the polynomials employed if $N = 4$:
\begin{equation}\label{polLegDef-4}
P_{2}(\zeta_{k}) = \dfrac{3\zeta_{k}^{2} - 1}{2}; \quad
P_{3}(\zeta_{k}) = \dfrac{5\zeta_{k}^{3} - 3 \zeta_{k}}{2}; \quad
P_{4}(\zeta_{k}) = \dfrac{35\zeta_{k}^{4}}{8} - \dfrac{15 \zeta_{k}^{2}}{4} + \dfrac{3}{8}
\end{equation}
It is finally emphasized that the chosen thickness functions for a LW model satisfy the following properties
\begin{align}\label{eq10}
\zeta_{k} = 1 &: \quad F_{t} = 1,\quad F_{b} = 0, \quad F_{r} = 0
\end{align}
\begin{align}
\zeta_{k} = -1 &: \quad F_{t} = 0,\quad F_{b} = 1, \quad F_{r} = 0
\end{align}
Therefore, $\hat{u}_{i_{t}}^{(k)}$ and $\hat{u}_{i_{b}}^{(k)}$ are the physical displacement components at the top and bottom of the $k^{\mathrm{th}}$ layer, respectively, and $F_{t}(\zeta_{k})$ and $F_{b}(\zeta_{k})$ are the corresponding linear Lagrange interpolation functions.
\subsection{The stress and strain fields}
The contributions to the strain and stress fields in each layer $k$ are identified with respect to the bending ($b$), transverse normal ($n$) and transverse shear ($s$) deformation of the plate:
\begin{equation}\label{voigt}
\begin{aligned}
\bepsilon_{b}^{(k)} &= \left[ \epsilon_{1}^{(k)} \, \epsilon_{2}^{(k)} \, \epsilon_{6}^{(k)} \right]; \quad
&\bepsilon_{n}^{(k)} &= \epsilon_{3}^{(k)}; \quad
&\bepsilon_{s}^{(k)} &= \left[ \epsilon_{5}^{(k)} \, \epsilon_{4}^{(k)} \right] \\
\bsigma_{b}^{(k)} &= \left[ \sigma_{1}^{(k)} \, \sigma_{2}^{(k)} \, \sigma_{6}^{(k)} \right]; \quad
&\bsigma_{n}^{(k)} &= \sigma_{3}^{(k)}; \quad
&\bsigma_{s}^{(k)} &= \left[ \sigma_{5}^{(k)} \, \sigma_{4}^{(k)} \right]
\end{aligned}\end{equation}
where Voigt's contracted notation for the symmetric strain and stress tensors has been invoked. The generic layer $k$ is assumed to have a monoclinic material symmetry in the plate's reference frame $(x_{\alpha}, z)$; thus the linear elastic constitutive law reads in matrix form as follows
\begin{equation}\label{hooke}
\begin{bmatrix} \bsigma_{b}^{(k)} \\ \bsigma_{n}^{(k)} \\ \bsigma_{s}^{(k)} \end{bmatrix}
=
\begin{bmatrix}
\tilde{\mathbf{C}}_{bb}^{(k)} & \tilde{\mathbf{C}}_{bn}^{(k)} & \mathbf{0} \\
{\tilde{\mathbf{C}}_{bn}^{(k)^{T}}} & \tilde{\mathbf{C}}_{nn}^{(k)} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \tilde{\mathbf{C}}_{ss}^{(k)}
\end{bmatrix}
\begin{bmatrix} \bepsilon_{b}^{(k)} \\ \bepsilon_{n}^{(k)} \\ \bepsilon_{s}^{(k)} \end{bmatrix}
\end{equation}
where superscript $T$ is the transposition operator. The constitutive law is obviously defined for each layer $k$ for it depends on the layer's orthotropic elastic properties $C_{PQ}^{(k)}$ ($P, Q \in \{1, 6\}$) and on the orientation angle $\theta^{(k)}$.
Within the small strain and small displacement approximation, the linearized strain components are defined as
\begin{equation}\label{strainDef}\begin{aligned}
\epsilon_{i} &= u_{i,_{i}} \quad &\text{for}& \;\;i=1,2,3;\\
\epsilon_{(9-i-j)} &= u_{i,_{j}}+u_{j,_{i}} \quad &\text{for}& \;\;i,j=1,2,3\;\; \text{and} \;\;i \neq j
\end{aligned}\end{equation}
\begin{thebibliography}{55}
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\bibitem{Noor:Burton:1989}
A.~K. Noor, W.~S. Burton, Assessment of shear deformation theories for
multilayered composite plates, Appl.~Mech.~Rev. 42 (1989) 1--13.
\bibitem{Reddy:Robbins:review:1994}
J.~N. Reddy, D.~H. {Robbins Jr}, Theories and computational models for
composite laminates, Appl.~Mech.~Rev. 47 (1994) 147--169.
\bibitem{Noor:Burton:sandwichReview:1996}
A.~K. Noor, W.~S. Burton, Computational models for sandwich panels and shells,
Appl.~Mech.~Rev. 49 (1996) 155--199.
\bibitem{Carrera:ARCME:2002}
E.~Carrera, Theories and finite elements for multilayered, anisotropic,
composite plates and shells, Arch.~Comput.~Meth.~Eng. 9 (2002) 87--140.
\end{thebibliography}
\end{document}