Splitting and curling collapse of square sandwich metal tube with aluminum foam core under axial low-velocity impact

Abstract

Experiments on splitting and curling collapse of square sandwich metal tubes with aluminum foam core under axial low-velocity impact are conducted in this paper. Experimental details are described including geometric and test parameters of specimens. Deformation modes and load–displacement curves are obtained through experiment. Analytical solution of quasi-static splitting and curling steady load can be used to predict steady load of square sandwich tube under axial low-velocity impact based on the similarity of quasi-static and low-velocity impact deformation modes.

Appendix 1: Analytical model of steady load of quasi-static splitting and curling collapse of square sandwich metal tubes with aluminum foam core [3]

As shown in Fig. 

Fig. 10

Sketch of steady stage of quasi-static splitting and curling collapse

10, in the steady deformation stage, the plastic curling deformation of inner and outer tubes happens and the metal foam core is compressed.

For the outer tube, from the conservation of energy,

$$ \dot{W}_{out} = \frac{{4M_{20} }}{{R_{2} }}\dot{w} + 4\sigma_{t} H_{2} H\dot{w} $$

(3)

where $M_{20} = \frac{{\sigma_{t} b_{2} H^{2} }}{4}$, $R_{2} = \frac{{H\left( {b_{2} + H} \right)}}{{4H_{1} }}\left[ {\frac{\sin \alpha + \mu \cos \alpha - \mu }{{\sin \left( {\alpha - \beta_{2} } \right) + \mu \cos \left( {\alpha - \beta_{2} - \mu } \right)}} - 1} \right].$

Terms on the right of the equal sign are the rates of bending energy and tearing energy of the outer tube, respectively.

For the inner tube, from the energy conservation,

$$ \dot{W}_{in} = \frac{{4M_{10} }}{{R_{1}^{^{\prime}} }}\dot{w} + 4\sigma_{t} H_{1} H\dot{w} + \frac{\mu F}{{\sin \alpha + \mu \cos \alpha }}\dot{w} $$

(4)

where $M_{10} = \frac{{\sigma_{t} b_{1} H^{2} }}{4}$, $R_{1}^{\prime } = R_{2} + c + H.$

Terms on the right of the equal sign are the rates of bending energy, tearing energy and friction energy of the inner tube, respectively.

The metal foam is compressed until densification strain, and the rate of energy dissipated in the foam compression is

$$ F_{f} \dot{w} = \left[ {\left( {b_{2} - H} \right)^{2} - \left( {b_{1} + H} \right)^{2} } \right]\varepsilon_{f} \sigma_{f} \dot{w} $$

(5)

From the conservation of energy,

$$ F\dot{w} = \dot{W}_{in} + \dot{W}_{out} + F_{f} \dot{w} $$

(6)

Thus,

$$ \begin{gathered} F = \frac{\sin \alpha + \mu \cos \alpha }{{\sin \alpha + \mu \cos \alpha - \mu }}\left\{ {\frac{{\sigma_{t} b_{1} H^{2} }}{{R_{1}^{\prime } }} + \frac{{\sigma_{t} b_{2} H^{2} }}{{R_{2} }} + 4\sigma_{t} \left( {H_{1} + H_{2} } \right)H} \right. \hfill \\ \quad \left. { + \left[ {\left( {b_{2} - H} \right)^{2} - \left( {b_{1} + H} \right)^{2} } \right]\varepsilon_{f} \sigma_{f} } \right\} \hfill \\ \end{gathered} $$

(7)