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Plastic Buckling and Postbuckling Analysis of Plates Using 3D Incompatible and Standard Elements

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Abstract

In the current study, plastic buckling and postbuckling behavior of aluminum alloy plates under uniaxial, biaxial and combined compressive/shear loadings using 3D standard and incompatible elements are investigated. In this study, the finite element code considering geometrically and material nonlinearities is developed based on the incremental theory of plasticity. Obtained results show that the bifurcation point and postbuckling behavior in the models with linear standard elements have significant differences in models with incompatible and quadratic elements at the same mesh size. Furthermore, the buckling and postbuckling analysis using incompatible elements have faster convergence rate compared to linear and quadratic elements.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sahand University of Technology, Sahand New Town, Tabriz, Iran

    Hamed Mahmoud Soltani & Mahsa Kharazi

Authors
  1. Hamed Mahmoud Soltani
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  2. Mahsa Kharazi
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Correspondence to Mahsa Kharazi.

Appendix

Appendix

Matrices used in 3D standard element based on Total Lagrangian formulation:

According to Eq. (7a), linear strain–displacement matrix can be written as follows (Bathe 1996):

$${}_{0}^{t} B^{\text{L}} = {}_{0}^{t} B^{{\text{L0}}} + {}_{0}^{t} B^{{\text{L1}}}$$
(32)
$${}_{0}^{t} B^{{\text{L0}}} = \left[ {\begin{array}{*{20}l} {{}_{0}H_{k,1} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {{}_{0}H_{k,2} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{}_{0}H_{k,3} } \hfill \\ {{}_{0}H_{k,2} } \hfill & {{}_{0}H_{k,1} } \hfill & 0 \hfill \\ 0 \hfill & {{}_{0}H_{k,3} } \hfill & {{}_{0}H_{k,2} } \hfill \\ {{}_{0}H_{k,3} } \hfill & 0 \hfill & {{}_{0}H_{k,1} } \hfill \\ \end{array} } \right]\quad k = 1,2, \ldots ,8$$
(33)
$${}_{0}^{t} B^{{\text{L1}}} = \left[ {\begin{array}{*{20}l} {l_{11} {}_{0}H_{k,1} } \hfill & {l_{21} {}_{0}H_{k,1} } \hfill & {l_{31} {}_{0}H_{k,1} } \hfill \\ {l_{12} {}_{0}H_{k,2} } \hfill & {l_{22} {}_{0}H_{k,2} } \hfill & {l_{32} {}_{0}H_{k,2} } \hfill \\ {l_{13} {}_{0}H_{k,3} } \hfill & {l_{23} {}_{0}H_{k,3} } \hfill & {l_{33} {}_{0}H_{k,3} } \hfill \\ {(l_{11} {}_{0}H_{k,2} + l_{12} {}_{0}H_{k,1} )} \hfill & {(l_{21} {}_{0}H_{k,2} + l_{22} {}_{0}H_{k,1} )} \hfill & {(l_{31} {}_{0}H_{k,2} + l_{32} {}_{0}H_{k,1} )} \hfill \\ {(l_{12} {}_{0}H_{k,3} + l_{13} {}_{0}H_{k,2} )} \hfill & {(l_{22} {}_{0}H_{k,3} + l_{23} {}_{0}H_{k,2} )} \hfill & {(l_{32} {}_{0}H_{k,3} + l_{33} {}_{0}H_{k,2} )} \hfill \\ {(l_{11} {}_{0}H_{k,3} + l_{13} {}_{0}H_{k,1} )} \hfill & {(l_{21} {}_{0}H_{k,3} + l_{23} {}_{0}H_{k,1} )} \hfill & {(l_{31} {}_{0}H_{k,3} + l_{33} {}_{0}H_{k,1} )} \hfill \\ \end{array} } \right]\quad k = 1,2, \ldots ,8$$
(34)
$$l_{ij} = \sum\limits_{k = 1}^{8} {{}_{0}H_{k,j} } {}^{t}u_{i}^{k} \quad i,j = 1,2,3$$
(35)

in addition, according to Eq. (7b) nonlinear strain–displacement matrix can be written as follows:

$${}_{0}^{t} B^{{\,\text{NL}}} = \left[ {\begin{array}{*{20}l} {{}_{0}^{t} \tilde{B}^{{\,\text{NL}}} } \hfill & {\tilde{0}} \hfill & {\tilde{0}} \hfill \\ {\tilde{0}} \hfill & {{}_{0}^{t} \tilde{B}^{{\,\text{NL}}} } \hfill & {\tilde{0}} \hfill \\ {\tilde{0}} \hfill & {\tilde{0}} \hfill & {{}_{0}^{t} \tilde{B}^{{\,\text{NL}}} } \hfill \\ \end{array} } \right]$$
(36)
$${}_{0}^{t} \tilde{B}^{{\,\text{NL}}} = \left[ {\begin{array}{*{20}l} {{}_{0}H_{k,1} } \hfill & 0 \hfill & 0 \hfill \\ {{}_{0}H_{k,1} } \hfill & 0 \hfill & 0 \hfill \\ {{}_{0}H_{k,1} } \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]\quad k = 1,2, \ldots ,8$$
(37)
$$\tilde{0}\, = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \right]$$
(38)

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Soltani, H.M., Kharazi, M. Plastic Buckling and Postbuckling Analysis of Plates Using 3D Incompatible and Standard Elements. Iran J Sci Technol Trans Mech Eng 44, 881–903 (2020). https://doi.org/10.1007/s40997-019-00316-w

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