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A boundary method using equilibrated basis functions for bending analysis of in-plane heterogeneous thick plates

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Abstract

A simple boundary method is developed for the solution of isotropic thick plates with in-plane arbitrarily variable material properties or thickness. Equilibrated basis functions which have proved to be effective in a variety of problems, are adopted for the bending problem of thick plates. The bases are created through a weighted residual approach over a fictitious rectangular domain so as to approximately satisfy the partial differential equation of the problem. This omits the necessity of the bases to analytically satisfy the equilibrium, thus simplifying the application of the method. Boundary conditions are applied to the approximate solution through a collocation technique, which considerably reduces its computational expenses. Mindlin’s first order and Levinson’s third-order shear deformation theories are adopted for the formulation. To accommodate more complicated geometries, a simple domain decomposition approach is also developed.

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Correspondence to Nima Noormohammadi.

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Appendix

Appendix

The components of (26) are defined as follows. The diagonal matrices \(\mathbf{m }_{x} ,\mathbf{m }_{y} \) import the coefficients resulted from differentiating the variable coefficients of the PDE as,

$$\begin{aligned} \mathbf{m }_{x} =\mathrm{Diag}\left[ {0,1,\ldots ,m_{x} } \right] , \quad \mathbf{m }_{y} =\mathrm{Diag}\left[ {0,1,\ldots ,m_{y} } \right] . \end{aligned}$$
(A.1)

Then we have,

$$\begin{aligned} \mathbf{A }_{h}^{1}= & {} \left[ {\mathbf{A }_{3} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{A }_{2} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{2} } \right] _{k}^{T} +\left[ {\mathbf{A }_{7} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{C} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} \nonumber \\&\quad +\left[ {\mathbf{A }_{6} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{C} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \mathbf{m }_{y} \left[ {\mathbf{B }_{777} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \mathbf{m }_{y} \left[ {\mathbf{B }_{6} } \right] _{k}^{T} \nonumber \\&\quad -\left[ {\mathbf{A }_{44} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} -\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} -\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{444} } \right] _{k}^{T} -\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{3} } \right] _{k}^{T} \nonumber \\&\quad +\left[ {\mathbf{A }_{22} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} +\left[ {\mathbf{A }_{66} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} +\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{222} } \right] _{k}^{T} +\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathrm{\mathbf{B}}_{666} } \right] _{k}^{T} \nonumber \\&\quad +\nu \left( {\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{A }_{77} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{A }_{333} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{33} } \right] _{k}^{T} +\left[ {\mathbf{A }_{333} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathbf{B }_{77} } \right] _{k}^{T} } \right) \nonumber \\&\quad +\frac{\left( {1-\nu } \right) }{2}\left( {\left[ {\mathbf{A }_{44} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{22} } \right] _{k}^{T} +\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{A }_{44} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathbf{B }_{66} } \right] _{k}^{T} +\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathbf{B }_{777} } \right] _{k}^{T} } \right. \nonumber \\&\quad \left. {+\left[ {\mathbf{A }_{333} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{33} } \right] _{k}^{T} +\left[ {\mathbf{A }_{222} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{444} } \right] _{k}^{T} \text{+ }\left[ {\mathbf{A }_{777} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathbf{B }_{33} } \right] _{k}^{T} +\left[ {\mathbf{A }_{666} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathrm{\mathbf{B}}_{444} } \right] _{k}^{T} } \right) \end{aligned}$$
(A.2)
$$\begin{aligned} \mathbf{A }_{h}^{2}= & {} -\frac{1}{3}\left\{ {\left[ {\mathbf{A }_{3} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{A }_{2} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{2} } \right] _{k}^{T} +\left[ {\mathbf{A }_{7} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{C} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} } \right. \nonumber \\&\quad +\left[ {\mathbf{A }_{6} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{C} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \mathbf{m }_{y} \left[ {\mathbf{B }_{777} } \right] _{k}^{T} +\left[ {\mathbf{A }_{4} } \right] _{\ell } \mathbf{C }_{C} \mathbf{m }_{y} \left[ {\mathbf{B }_{6} } \right] _{k}^{T} -\left[ {\mathbf{A }_{44} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} \nonumber \\&\quad -\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} \nonumber \\&\quad \left. {-\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{444} } \right] _{k}^{T} -\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{C} \left[ {\mathbf{B }_{3} } \right] _{k}^{T} } \right\} \nonumber \\&\quad -\frac{1}{5}\left\{ {\left[ {\mathbf{A }_{22} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} +\left[ {\mathbf{A }_{11} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} +\left[ {\mathbf{A }_{66} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathbf{B }_{44} } \right] _{k}^{T} +\left[ {\mathbf{A }_{55} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathbf{B }_{4} } \right] _{k}^{T} } \right. \nonumber \\&\quad +\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{222} } \right] _{k}^{T} +\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{1} } \right] _{k}^{T} +\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathbf{B }_{666} } \right] _{k}^{T} +\left[ {\mathbf{A }_{444} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathbf{B }_{5} } \right] _{k}^{T} \nonumber \\&\quad +\nu \left( {\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{D} (\left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{B }_{2} } \right] _{k}^{T} )+\left[ {\mathbf{A }_{77} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} (\left[ {\mathbf{B }_{333} } \right] _{k}^{T} +\left[ {\mathbf{B }_{2} } \right] _{k}^{T} )+\left[ {\mathbf{A }_{333} } \right] _{\ell } \mathbf{C }_{D} (\left[ {\mathbf{B }_{33} } \right] _{k}^{T} +\mathbf{m }_{y} } \right. \nonumber \\&\quad \left. {\left[ {\mathbf{B }_{77} } \right] _{k}^{T} )+\left[ {\mathbf{A }_{222} } \right] _{\ell } \mathbf{C }_{D} (\left[ {\mathbf{B }_{3} } \right] _{k}^{T} +\mathbf{m }_{y} \left[ {\mathbf{B }_{7} } \right] _{k}^{T} )} \right) \nonumber \\&\quad +\frac{\left( {1-\nu } \right) }{2}\left( {\left[ {\mathbf{A }_{44} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{22} } \right] _{k}^{T} +\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{D} (\left[ {\mathbf{B }_{333} } \right] _{k}^{T} +2\left[ {\mathbf{B }_{2} } \right] _{k}^{T} )+\left[ {\mathbf{A }_{44} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} \left[ {\mathbf{B }_{66} } \right] _{k}^{T} } \right. \nonumber \\&\quad +\left[ {\mathbf{A }_{33} } \right] _{\ell } \mathbf{C }_{D} \mathbf{m }_{y} (\left[ {\mathbf{B }_{777} } \right] _{k}^{T} \text{+2 }\left[ {\mathbf{B }_{6} } \right] _{k}^{T} )+\left[ {\mathbf{A }_{333} } \right] _{\ell } \mathbf{C }_{D} \left[ {\mathbf{B }_{33} } \right] _{k}^{T} +\left[ {\mathbf{A }_{222} } \right] _{\ell } \mathbf{C }_{D} (\left[ {\mathbf{B }_{444} } \right] _{k}^{T} \text{+2 }\left[ {\mathbf{B }_{3} } \right] _{k}^{T} ) \nonumber \\&\quad \left. {\left. {\text{+ }\left[ {\mathbf{A }_{777} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} \left[ {\mathbf{B }_{33} } \right] _{k}^{T} +\left[ {\mathbf{A }_{666} } \right] _{\ell } \mathbf{m }_{x} \mathbf{C }_{D} (\left[ {\mathbf{B }_{444} } \right] _{k}^{T} +2\left[ {\mathbf{B }_{3} } \right] _{k}^{T} )} \right) } \right\} \end{aligned}$$
(A.3)

\(\left[ {\mathbf{X }} \right] _{i} \) means all of the entries of the three-index array \(\mathbf{X }\) for its first component equal to i (in terms of a 2D matrix).

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Noormohammadi, N., Boroomand, B. A boundary method using equilibrated basis functions for bending analysis of in-plane heterogeneous thick plates. Arch Appl Mech 91, 487–507 (2021). https://doi.org/10.1007/s00419-020-01784-2

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