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Analysis of antisymmetric cross-ply laminates using high-order shear deformation theories: a meshless approach

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Abstract

For many years finite element method (FEM) was the chosen numerical method for the analysis of composite structures. However, in the last 20 years, the scientific community has witnessed the birth and development of several meshless methods, which are more flexible and equally accurate numerical methods. The meshless method used in this work is the natural neighbour radial point interpolation method (NNRPIM). In order to discretize the problem domain, the NNRPIM only requires an unstructured nodal distribution. Then, using the Voronoï mathematical concept, it enforces the nodal connectivity and constructs the background integration mesh. The NNRPIM shape functions are constructed using the radial point interpolation technique. In this work, the displacement field of composite laminated plates is defined by high-order shear deformation theories. In the end, several antisymmetric cross-ply laminates were analysed and the NNRPIM solutions were compared with the literature. The obtained results show the efficiency and accuracy of the NNRPIM formulation.

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Acknowledgements

The authors truly acknowledge the funding provided by Ministério da Ciência, Tecnologia e Ensino Superior—Fundação para a Ciência e a Tecnologia (Portugal), (Grants SFRH/BD/121019/2016 and MIT-EXPL/ISF/0084/2017). Additionally, the authors gratefully acknowledge the funding provided by LAETA, under project UIDB/50022/2020.

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Authors and Affiliations

  1. Institute of Science and Innovation in Mechanical and Industrial Engineering, 4200-465, Porto, Portugal

    D. E. S. Rodrigues, J. Belinha, L. M. J. S. Dinis & R. M. Natal Jorge

  2. Department of Mechanical Engineering, School of Engineering, Polytechnic of Porto (ISEP), 4200-072, Porto, Portugal

    J. Belinha

  3. Department of Mechanical Engineering, Faculty of Engineering, University of Porto (FEUP), 4200-465, Porto, Portugal

    L. M. J. S. Dinis & R. M. Natal Jorge

Authors
  1. D. E. S. Rodrigues
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  2. J. Belinha
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  3. L. M. J. S. Dinis
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  4. R. M. Natal Jorge
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Corresponding author

Correspondence to J. Belinha.

Appendix

Appendix

1.1 Interpolation functions

The NNRPIM uses multiquadric RBFs combined with polynomial basis functions to obtain interpolation functions that possess the delta Kronecker property.

Thus, considering a function \(u(\varvec{x})\) defined in the domain \(\varOmega\), which is discretized by a set of \(N\) nodes, and assuming that only the nodes within the “influence-cell” of the interest point \(\varvec{x}_{I}\) affect the value of the function \(u(\varvec{x}_{I} )\), Eq. (A1) is obtained. The function \(u(\varvec{x})\) passes through all nodes using a radial basis function (RBF).

$$u\left( {\varvec{x}_{I} } \right) = \sum\limits_{i = 1}^{n} {R_{i} \left( {\varvec{x}_{I} } \right)} \,a_{i} \left( {\varvec{x}_{I} } \right) + \sum\limits_{j = 1}^{m} {p_{j} (\varvec{x}_{I} )\,b_{j} (\varvec{x}_{I} )} ,$$
(A1)

where \(n\) is the number of nodes within the “influence-cell” of \(\varvec{x }_{I}\), \(m\) is the basis monomial number, \(R_{i} \left( {\varvec{x}_{I} } \right)\) is the RBF, \(a_{i} \left( {\varvec{x}_{I} } \right)\) and \(b_{j} \left( {\varvec{x}_{I} } \right)\) are non-constant coefficients of \(R_{i} \left( {\varvec{x}_{I} } \right)\) and \(p_{j} \left( {\varvec{x}_{I} } \right)\), the polynomial basis, respectively. In 2D problems, the RBFs are dependent of the vector \(r_{I\,i}\), the Euclidian distance between the interest point \(\varvec{x}_{I}\) and each node \(\varvec{x}_{i}\) within the influence-cell of \(\varvec{x}_{I}\)

$$R(r_{Ii} ) = \left( {r_{Ii}^{2} + c^{2} } \right)^{p} ,$$
(A2)
$$r_{Ii} = \sqrt {(x_{I} - x_{i} )^{2} + (y_{I} - y_{i} )^{2} } .$$
(A3)

Equation (A2) represents the multiquadric (MQ) RBFs proposed initially by Hardy [66], being \(c\) and \(p\) the shape parameters that, according to Ref. [1], should be \(c = 0.0001\) and \(p = 0.9999\).

Rewriting Eq. (A1) in a matrix form,

$$u\left( {\varvec{x}_{I} } \right) = \varvec{R}^{\text{T}} \left( {\varvec{x}_{I} } \right)\,\,\varvec{a}\left( {\varvec{x}_{I} } \right) + \,\varvec{p}^{\text{T}} \left( {\varvec{x}_{I} } \right)\,\,\varvec{b}\left( {\varvec{x}_{I} } \right) = \left\{ {\begin{array}{*{20}c} {\varvec{R}^{\text{T}} \left( {\varvec{x}_{I} } \right)} & {\varvec{p}^{\text{T}} \left( {\varvec{x}_{I} } \right)} \\ \end{array} } \right\}\left\{ {\begin{array}{*{20}c} {\varvec{a}\left( {\varvec{x}_{I} } \right)} \\ {\varvec{b}\left( {\varvec{x}_{I} } \right)} \\ \end{array} } \right\},$$
(A4)

being the vectors of Eq. (A4),

$$\varvec{R}\left( {\varvec{x}_{I} } \right) = \left\{ {R_{1} \left( {\varvec{x}_{I} } \right),\,\,R_{2} \left( {\varvec{x}_{I} } \right),\,\,\, \ldots ,\,\,\,R_{n} \left( {\varvec{x}_{I} } \right)} \right\}^{\text{T}} ,$$
(A5)
$$\varvec{p}\left( {\varvec{x}_{I} } \right) = \left\{ {p_{1} \left( {\varvec{x}_{I} } \right),\,\,p_{2} \left( {\varvec{x}_{I} } \right),\,\,\, \ldots ,\,\,\,p_{m} \left( {\varvec{x}_{I} } \right)\,} \right\}^{\text{T}} ,$$
(A6)
$$\varvec{a}\left( {\varvec{x}_{I} } \right) = \left\{ {a_{1} \left( {\varvec{x}_{I} } \right),\,\,a_{2} \left( {\varvec{x}_{I} } \right),\,\,\, \ldots ,\,\,\,a_{n} \left( {\varvec{x}_{I} } \right)\,} \right\}^{\text{T}} ,$$
(A7)
$$\varvec{b}\left( {\varvec{x}_{I} } \right) = \left\{ {b_{1} \left( {\varvec{x}_{I} } \right),\,\,b_{2} \left( {\varvec{x}_{I} } \right),\,\, \ldots ,\,\,b_{m} \left( {\varvec{x}_{I} } \right)} \right\}^{\text{T}} .$$
(A8)

The polynomial basis function has sequence of terms presented in Eq. (A9)

$$\varvec{p}^{\text{T}} \left( {\varvec{x}_{I} } \right) = \left\{ {1,\,x,\,y,\,x^{2} ,\,\,xy,\,y^{2} ,\,\, \ldots \,} \right\}.$$
(A9)

The number of terms depends on the chosen monomial number, \(m\), which should be \(m < n\) in order to obtain a more stable function,

$$\begin{aligned} & {\text{Null }}\,{\text{basis}}\text{ (}m = 0\text{)}:\,\,\,\,\,\,\,\,\,\,\,\,\,\varvec{p}^{T} (\varvec{x}) = \left\{ 0 \right\}\,\,,\,\,m = 0, \hfill \\ & {\text{Constant }}\,{\text{basis}}\text{ (}m = 1\text{)}:\,\,\,\,\,\,\varvec{p}^{T} (\varvec{x}) = \left\{ 1 \right\}\,\,,\,\,m = 1, \hfill \\ & {\text{Linear}}\,{\text{ basis}}\text{ (}m = 3\text{)}:\,\,\,\,\,\,\,\,\,\,\varvec{p}^{T} (\varvec{x}) = \left\{ {\begin{array}{*{20}c} 1 & x & y \\ \end{array} } \right\}\,,\,\,m = 3, \hfill \\ & {\text{Quadratic}}\,{\text{ basis}}\text{ (}m = 6\text{)}:\,\,\,\varvec{p}^{T} (\varvec{x}) = \left\{ {\begin{array}{*{20}c} 1 & x & y & {x^{2} } & {xy} & {y^{2} } \\ \end{array} } \right\}\,\,,\,\,m = 6. \hfill \\ \end{aligned}$$
(A10)

A “null basis” is the absence of polynomial basis. If it is chosen a polynomial basis with \(m > 0\), an extra requirement need to be satisfied in order to guarantee a unique approximation,

$$\sum\limits_{i = 1}^{n} {p_{j} (\varvec{x}_{i} )\,a_{i} (\varvec{x}_{i} )} = 0 \Leftrightarrow \varvec{p}^{\text{T}} (\varvec{x}_{i} )\,\,\varvec{a}(\varvec{x}_{i} ) = 0,\quad j = 1,2, \ldots ,m.$$
(A11)

Combining Eq. (A11) with Eq. (A4), the following system of equations can be established,

$$\left\{ {\begin{array}{*{20}c} {\varvec{u}_{s} } \\ {\mathbf{0}} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\varvec{R}^{\text{T}} \left( {\varvec{x}_{I} } \right)} & {\varvec{p}\left( {\varvec{x}_{I} } \right)} \\ {\varvec{p}^{T} \left( {\varvec{x}_{I} } \right)} & {\mathbf{0}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varvec{a}\left( {\varvec{x}_{I} } \right)} \\ {\varvec{b}\left( {\varvec{x}_{I} } \right)} \\ \end{array} } \right\} = \varvec{G}\left\{ {\begin{array}{*{20}c} {\varvec{a}\left( {\varvec{x}_{I} } \right)} \\ {\varvec{b}\left( {\varvec{x}_{I} } \right)} \\ \end{array} } \right\},$$
(A12)

where the vector \(\varvec{u}_{s}\) is defined as \(\varvec{u}_{s} = \left\{ {u_{1} ,u_{2} , \ldots ,u_{n} } \right\}^{T}\) and the matrixes \(\varvec{R}\) [\(n \times n\)] and \(\varvec{p}\) [\(n \times m\)] are given

$$\varvec{R} = \left[ {\begin{array}{*{20}c} {R(r_{11} )} & {R(r_{12} )} & \ldots & {R(r_{1n} )} \\ {R(r_{21} )} & {R(r_{22} )} & \ldots & {R(r_{2n} )} \\ \vdots & \vdots & \ddots & \vdots \\ {R(r_{n1} )} & {R(r_{n2} )} & \cdots & {R(r_{nn} )} \\ \end{array} } \right],$$
(A13)
$$\varvec{p} = \left[ {\begin{array}{*{20}c} {p_{\text{1}} (\varvec{x}_{\text{1}} )} & {p_{\text{2}} (\varvec{x}_{\text{1}} )} & \ldots & {p_{m} (\varvec{x}_{\text{1}} )} \\ {p_{\text{1}} (\varvec{x}_{\text{2}} )} & {p_{\text{2}} (\varvec{x}_{\text{2}} )} & \ldots & {p_{m} (\varvec{x}_{\text{2}} )} \\ \vdots & \vdots & \ddots & \vdots \\ {p_{\text{1}} (\varvec{x}_{n} )} & {p_{\text{2}} (\varvec{x}_{n} )} & \cdots & {p_{m} (\varvec{x}_{n} )} \\ \end{array} } \right].$$
(A14)

The distance between the interest points and the nodes that belong to their “influence-cell” is directional independent. Thus, the matrix \(\varvec{G}\) is symmetric. Solving Eq. (A12) in order to the non-constant coefficients,

$$\left\{ {\begin{array}{*{20}c} {\varvec{a}\left( {\varvec{x}_{I} } \right)} \\ {\varvec{b}\left( {\varvec{x}_{I} } \right)} \\ \end{array} } \right\}\varvec{ = G}^{{ - \text{1}}} \left\{ {\begin{array}{*{20}c} {\varvec{u}_{s} } \\ {\mathbf{0}} \\ \end{array} } \right\},$$
(A15)

and substituting Eq. (A15) into Eq. (A4), the interpolation functions, \(\varvec{\varphi }\,(\varvec{x}_{I} )\), are finally determined,

$$u(\varvec{x}_{I} ) = \left\{ {\varvec{R}^{T} (\varvec{x}_{I} )\,,\,\varvec{p}^{T} (\varvec{x}_{I} )} \right\}\,\,\,\varvec{G}^{ - 1} \,\left\{ {\begin{array}{*{20}c} {\varvec{u}_{\text{s}} } \\ \text{0} \\ \end{array} } \right\} = \left\{ {\varphi (\varvec{x}_{I} ),\psi (\varvec{x}_{I} )} \right\}\,\varvec{u}_{s} ,$$
(A16)
$$\varvec{\varphi }(\varvec{x}_{I} ) = \left\{ {\varvec{R}^{T} (\varvec{x}_{I} )\,,\,\varvec{p}^{T} (\varvec{x}_{I} )} \right\}\,\,\,\varvec{G}^{ - 1} = \left\{ {\varphi_{1} (\varvec{x}_{I} ),\varphi_{2} (\varvec{x}_{I} ), \ldots ,\varphi_{n} (\varvec{x}_{I} )} \right\}.$$
(A17)

Vector \(\psi (\varvec{x}_{I} )\) is a residual vector without a significant physical meaning. In Ref. [1], it can be found a detailed explanation about the construction procedure of the shape functions, its partial derivatives and most relevant mathematical properties.

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Rodrigues, D.E.S., Belinha, J., Dinis, L.M.J.S. et al. Analysis of antisymmetric cross-ply laminates using high-order shear deformation theories: a meshless approach. Acta Mech. Sin. 36, 1078–1098 (2020). https://doi.org/10.1007/s10409-020-00990-z

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