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An enriched 1D finite element for the buckling analysis of sandwich beam-columns

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Abstract

Sandwich constructions have been widely used during the last few decades in various practical applications, especially thanks to the attractive compromise between a lightweight and high mechanical properties. Nevertheless, despite the advances achieved to date, buckling still remains a major failure mode for sandwich materials which often fatally leads to collapse. Recently, one of the authors derived closed-form analytical solutions for the buckling analysis of sandwich beam-columns under compression or pure bending. These solutions are based on a specific hybrid formulation where the faces are represented by Euler–Bernoulli beams and the core layer is described as a 2D continuous medium. When considering more complex loadings or non-trivial boundary conditions, closed-form solutions are no more available and one must resort to numerical models. Instead of using a 2D computationally expensive model, the present paper aims at developing an original enriched beam finite element. It is based on the previous analytical formulation, insofar as the skin layers are modeled by Timoshenko beams whereas the displacement fields in the core layer are described by means of hyperbolic functions, in accordance with the modal displacement fields obtained analytically. By using this 1D finite element, linearized buckling analyses are performed for various loading cases, whose results are confronted to either analytical or numerical reference solutions, for validation purposes.

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Acknowledgments

The authors are grateful to the Nord-Pas-de-Calais Regional Council (France) for its financial support.

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Authors and Affiliations

  1. Mines Douai, Polymers and Composites Technology & Mechanical Engineering Department, 941 rue Charles Bourseul, CS 10838, 59508, Douai Cedex, France

    Kahina Sad Saoud & Philippe Le Grognec

Authors
  1. Kahina Sad Saoud
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  2. Philippe Le Grognec
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Corresponding author

Correspondence to Philippe Le Grognec.

Appendices

Appendix A. Closed-form expressions for the critical displacements of a sandwich column

The critical displacements are given by the following expressions in the antisymmetric and symmetric cases, respectively:

$$\begin{aligned} \lambda _{cr}^A= & {} \left( 4E_s E_c n\pi L^2 h_s\left[ 4n^2\pi ^2 h_s^2+3L^2\right] \cosh ^2\frac{n\pi h_c}{L}\right. \nonumber \\&+ \left[ 3E_c^2 L^5+12E_s E_c n^2\pi ^2 L^3 h_s^2\left( 1-\nu _c\right) \right. \nonumber \\&\left. +\,4E_s^2 n^4\pi ^4 Lh_s^4\left( 3+2\nu _c-\nu _c^2\right) \right] \nonumber \\&\quad \cosh \frac{n\pi h_c}{L}\sinh \frac{n\pi h_c}{L}\nonumber \\&-\,3E_c n\pi L^4\left[ 4E_s h_s+E_c h_c\right] \nonumber \\&+\,12E_s E_c n^3\pi ^3 L^2 h_s^2 h_c\left[ 1+\nu _c\right] \nonumber \\&\left. +\,4E_s^2 n^5\pi ^5 h_s^4 h_c\left[ 1+\nu _c\right] ^2\right) /\nonumber \\&\quad \left( 12E_s n\pi Lh_s\left[ E_c L^2\cosh ^2\frac{n\pi h_c}{L}\right. \right. \nonumber \\&+\,E_s n\pi h_s L\left( 3+2\nu _c-\nu _c^2\right) \cosh \frac{n\pi h_c}{L}\sinh \frac{n\pi h_c}{L}\nonumber \\&+\left. \left. E_s n^2\pi ^2 h_s h_c\left( 1+\nu _c\right) ^2\right] \right) \end{aligned}$$
(33)
$$\begin{aligned} \lambda _{cr}^S= & {} \left( 4E_s E_c n\pi L^2 h_s\left[ [4n^2\pi ^2 h_s^2+3L^2\right] \cosh ^2\frac{n\pi h_c}{L}\right. \nonumber \\&+ \left[ 3E_c^2 L^5+12E_s E_c n^2\pi ^2 L^3 h_s^2\left( 1-\nu _c\right) \right. \nonumber \\&\left. +\,4E_s^2 n^4\pi ^4 Lh_s^4\left( 3+2\nu _c-\nu _c^2\right) \right] \nonumber \\&\times \cosh \frac{n\pi h_c}{L}\sinh \frac{n\pi h_c}{L}\nonumber \\&\left. +\,3E_c^2 n\pi L^4 h_c\right. \nonumber \\&\left. -\,16E_s E_c n^3\pi ^3 L^2 h_s^3\right. \nonumber \\&\left. -\,12E_s E_c n^3\pi ^3 L^2 h_s^2 h_c\left[ 1+\nu _c\right] \right. \nonumber \\&\left. -\,4E_s^2 n^5\pi ^5 h_s^4 h_c\left[ 1+\nu _c\right] ^2\right) /\nonumber \\&\left( 12E_s n\pi Lh_s\left[ E_c L^2\cosh ^2\frac{n\pi h_c}{L}\right. \right. \nonumber \\&+\,E_s n\pi h_s L\left( 3+2\nu _c-\nu _c^2\right) \cosh \frac{n\pi h_c}{L}\sinh \frac{n\pi h_c}{L}\nonumber \\&\left. \left. -\,E_c L^2-E_s n^2\pi ^2 h_s h_c\left( 1+\nu _c\right) ^2\right] \right) \end{aligned}$$
(34)

Appendix B. Useful expressions in the core layer

The displacement gradient tensor components in the foam core write:

$$\begin{aligned} {\mathcal {H}}_{XX}^c= & {} U^c_{0,_X}+U^c_{1,_X}\sinh \left( \frac{\pi }{L}Y\right) +\phi _{1,_X}\cosh (\alpha Y)\nonumber \\&+\, \phi _{2,_X}\sinh (\alpha Y)\nonumber \\&+\, \phi _{3,_X} Y\cosh (\alpha Y)+\phi _{4,_X} Y\sinh (\alpha Y)\nonumber \\ {\mathcal {H}}_{XY}^c= & {} U^c_1 \frac{\pi }{L}\cosh \left( \frac{\pi }{L}Y\right) +\phi _1 \alpha \sinh (\alpha Y)+\phi _2 \alpha \cosh (\alpha Y)\nonumber \\&+\,\phi _3[\cosh (\alpha Y)+\, Y\alpha \sinh (\alpha Y)]\nonumber \\&+\,\phi _4[\sinh (\alpha Y)+Y\alpha \cosh (\alpha Y)]\nonumber \\ {\mathcal {H}}_{YX}^c= & {} V^c_{0,_X} \cosh \left( \frac{\pi }{L}Y\right) +V^c_{1,_X} Y+\phi _{5,_X} \cosh (\alpha Y)\nonumber \\&+\,\phi _{6,_X} \sinh (\alpha Y)+\phi _{7,_X} Y \cosh (\alpha Y)\nonumber \\&+\,\phi _{8,_X} Y \sinh (\alpha Y)\nonumber \\ {\mathcal {H}}_{YY}^c= & {} V^c_0 \frac{\pi }{L} \sinh \left( \frac{\pi }{L} Y\right) +V^c_1+\phi _5 \alpha \sinh (\alpha Y)\nonumber \\&+\,\phi _6 \alpha \cosh (\alpha Y)+\phi _7[\cosh (\alpha Y)+Y\alpha \sinh (\alpha Y)]\nonumber \\&+\,\phi _8[\sinh (\alpha Y)+Y\alpha \cosh (\alpha Y)] \end{aligned}$$
(35)

Taking into account the displacement continuity constraints at the interfaces, \( \phi _{1,_X} \), \( \phi _{2,_X} \), \( \phi _{5,_X} \) and \( \phi _{6,_X} \) can be given by the following expressions:

$$\begin{aligned} \phi _{1,_X}= & {} \frac{1}{\cosh (\alpha h_c)}\left( \frac{1}{2}(U_{,_X}^b+U_{,_X}^a)+\frac{h_s}{2}(\theta _{,_X}^b-\theta _{,_X}^a)\right. \nonumber \\&\left. -\,U^c_{0,_X} -\phi _{4,_X} h_c \sinh (\alpha h_c)\right) \nonumber \\ \phi _{2,_X}= & {} \frac{1}{\sinh (\alpha h_c)}\left( \frac{1}{2}(U_{,_X}^b-U_{,_X}^a)+\frac{h_s}{2}(\theta _{,_X}^b+\theta _{,_X}^a)\right. \nonumber \\&\left. -\,U^c_{1,_X}\sinh (\frac{\pi }{L}h_c)-\phi _{3,_X} h_c\cosh (\alpha h_c)\right) \nonumber \\ \phi _{5,_X}= & {} \frac{1}{\cosh (\alpha h_c)}\left( \frac{1}{2}(V_{,_X}^b+V_{,_X}^a)-V^c_{0,_X}\cosh (\frac{\pi }{L}h_c)\right. \nonumber \\&\left. -\,\phi _{8,_X} h_c \sinh (\alpha h_c)\right) \nonumber \\ \phi _{6,_X}= & {} \frac{1}{\sinh (\alpha h_c)}\left( \frac{1}{2}(V_{,_X}^b-V_{,_X}^a)-V^c_{1,_X} h_c\right. \nonumber \\&\left. -\,\phi _{7,_X} h_c\cosh (\alpha h_c)\right) \end{aligned}$$
(36)

Appendix C. Useful matrices

The non-zero components of matrix \( \mathbf H \) are:

$$\begin{aligned}&H(1,4)=1 \qquad \qquad H(1,12)=-Y\\&H(2,8)=1 \qquad \qquad H(2,10)=-1\\&H(3,3)=\frac{\cosh (\alpha Y)}{2\cosh (\alpha h_c)}+\frac{\sinh (\alpha Y)}{2\sinh (\alpha h_c)}\\&H(3,4)=\frac{\cosh (\alpha Y)}{2\cosh (\alpha h_c)}-\frac{\sinh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(3,11)=\frac{h_s \cosh (\alpha Y)}{2\cosh (\alpha h_c)}+\frac{h_s\sinh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(3,12)=\frac{h_s\sinh (\alpha Y)}{2\sinh (\alpha h_c)}-\frac{h_s\cosh (\alpha Y)}{2\cosh (\alpha h_c)} \\&H(3,14)=\sinh \left( \frac{\pi }{L}Y\right) -\frac{\sinh (\alpha Y)\sinh (\frac{\pi }{L}h_c)}{\sinh (\alpha h_c)} \\&H(3,21)=Y\cosh (\alpha Y)-\frac{h_c\sinh (\alpha Y)}{\tanh (\alpha h_c)} \\&H(3,22)=Y\sinh (\alpha Y)-h_c \tanh (\alpha h_c)\cosh (\alpha Y) \\&H(3,26)=1-\frac{\cosh (\alpha Y)}{\cosh (\alpha h_c)} \\&H(4,5)=\frac{\alpha \sinh (\alpha Y)}{2\cosh (\alpha h_c)}+\frac{\alpha \cosh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(4,6)=\frac{\alpha \sinh (\alpha Y)}{2\cosh (\alpha h_c)}-\frac{\alpha \cosh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(4,15)=\frac{\pi }{L}\sinh \left( \frac{\pi }{L} Y\right) -\frac{\alpha \sinh (\alpha Y)\cosh (\frac{\pi }{L}h_c)}{\cosh (\alpha h_c)} \\&H(4,19)=\cosh (\alpha Y)+Y\alpha \sinh (\alpha Y)-\frac{h_c\alpha \cosh (\alpha Y)}{\tanh (\alpha h_c)} \\&H(4,20)=\sinh (\alpha Y)+Y\alpha \cosh (\alpha Y)\\&\quad -h_c\tanh (\alpha h_c)\alpha \sinh (\alpha Y) \\ \end{aligned}$$
$$\begin{aligned}&H(4,27)=1-\frac{h_c\alpha \cosh (\alpha Y)}{\sinh (\alpha h_c)}\\&H(5,1)=\frac{\alpha \sinh (\alpha Y)}{2\cosh (\alpha h_c)}+\frac{\alpha \cosh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(5,2)=\frac{\alpha \sinh (\alpha Y)}{2\cosh (\alpha h_c)}-\frac{\alpha \cosh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(5,7)= \frac{\sinh (\alpha Y)}{2\sinh (\alpha h_c)}+\frac{\cosh (\alpha Y)}{2\cosh (\alpha h_c)} \\&H(5,8)= \frac{\cosh (\alpha Y)}{2\cosh (\alpha h_c)}-\frac{\sinh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(5,9)=\frac{h_s\alpha \sinh (\alpha Y)}{2\cosh (\alpha h_c)}+\frac{h_s\alpha \cosh (\alpha Y)}{2\sinh (\alpha h_c)} \\&H(5,10)=\frac{h_s\alpha \cosh (\alpha Y)}{2\sinh (\alpha h_c)}-\frac{h_s\alpha \sinh (\alpha Y)}{2\cosh (\alpha h_c)} \\&H(5,13)=\frac{\pi }{L}\cosh \left( \frac{\pi }{L}Y\right) -\frac{\alpha \sinh (\frac{\pi }{L}h_c)\cosh (\alpha Y)}{\sinh (\alpha h_c)} \\&H(5,16)= \cosh \left( \frac{\pi }{L}Y\right) -\frac{\cosh (\frac{\pi }{L}h_c)\cosh (\alpha Y)}{\cosh (\alpha h_c)} \\ \end{aligned}$$
$$\begin{aligned}&H(5,17)=\cosh (\alpha Y)+Y\alpha \sinh (\alpha Y)-\frac{h_c\alpha \cosh (\alpha Y)}{\tanh (\alpha h_c)} \nonumber \\&H(5,18)=\sinh (\alpha Y)+Y\alpha \cosh (\alpha Y)\nonumber \\&\quad -h_c\alpha \tanh (\alpha h_c)\sinh (\alpha Y) \nonumber \\&H(5,23)=Y\cosh (\alpha Y)-\frac{h_c\sinh (\alpha Y)}{\tanh (\alpha h_c)}\nonumber \\&H(5,24)=Y\sinh (\alpha Y)-h_c\tanh (\alpha h_c)\cosh (\alpha Y)\nonumber \\&H(5,25)=-\frac{\sinh (\alpha Y)}{\cosh (\alpha h_c)} \nonumber \\&H(5,28)=Y-\frac{h_c\sinh (\alpha Y)}{\sinh (\alpha h_c)} \nonumber \\&H(6,3)=1 \qquad \qquad H(6,11)=-Y\nonumber \\&H(7,7)=1 \qquad \qquad H(7,9)=-1\nonumber \\ \end{aligned}$$
(37)

The constitutive matrix \( \varvec{{\mathcal {L}}} \) is defined by the following non-zero components:

$$\begin{aligned}&{\mathcal {L}}(1,1)=E_s, \qquad \qquad {\mathcal {L}}(2,2)=\mu _s, \quad \quad {\mathcal {L}}(3,3)=\Lambda _c^*+2\mu _c \nonumber \\&{\mathcal {L}}(3,4)={\mathcal {L}}(4,3)=\Lambda _c^*, \qquad \qquad \qquad {\mathcal {L}}(4,4)=\Lambda _c^*+2\mu _c, \nonumber \\&{\mathcal {L}}(5,5)=\mu _c, \qquad {\mathcal {L}}(6,6)=E_s, \qquad \qquad {\mathcal {L}}(7,7)=\mu _s\nonumber \\ \end{aligned}$$
(38)

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Sad Saoud, K., Le Grognec, P. An enriched 1D finite element for the buckling analysis of sandwich beam-columns. Comput Mech 57, 887–900 (2016). https://doi.org/10.1007/s00466-016-1267-1

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