Numerical and theoretical studies about in-plane impact properties of Semi-Reentrant structures

Abstract

The in-plane impact and crushing properties have been studied for the Semi Reentrant structure (SRS) by using numerical simulations and theoretical studies. Different deformation modes during impact process could be observed under the various impact velocities, and there were three and two deformation modes in x₁ and x₂ directions, respectively, no matter for single or double sidewall conditions. The influence of arm length l and cell wall thickness t on the critical impact velocities of different deformation modes was investigated. With the increase of cell wall thickness, the critical velocity in both directions increased and power-law relations could be established. With the increase of inclined arms length l, the critical velocity in x₁ direction was almost the constant while that in x₂ direction generally decreased and was described by the power-law relationship. The influence of impact velocity on the dynamic elastic moduli was studied and the fitting curves between the normalized velocity and moduli were characterized by using piecewise functions. Moreover, the effect of impact velocity on the plateau stress, densification strain and densification strain energy were also analyzed, and related fitting expressions were obtained. Besides, theoretical models for predicting static and dynamic plateau stress and densification strain energy for SRS were established based on the previous study. The analytical model of dynamic plateau stress was then revised according to the fitting equations with a large enhancement of predicting accuracy.

Appendix

For the Semi Reentrant structures (see Fig. 2c), the plateau stress could be considered as the combination of Honeycomb and Reentrarnt. The basic force diagram of Semi Reentrant structure in x₁ direction is shown in Fig. 16.

When the plateau stress ${\left({\sigma }_{0}\right)}_{1}$ is reached, the force applied on the top inclined arm (AB) could be expressed as:

$${P}_{U}={\left({\sigma }_{0}\right)}_{1}(h+lsin\alpha )d$$

(27)

The moment of force around hinge points A and B could be expressed as:

$${{M}_{U}=P}_{U}lsin\alpha ={\left({\sigma }_{0}\right)}_{1}(h+lsin\alpha )dlsin\alpha$$

(28)

When the plastic hinge points A and B plastically rotate the angle ф, the plastic work done at the hinges are:

$${W}_{U}={M}_{U}{\Phi}={\left({\sigma }_{0}\right)}_{1}(h+lsin\alpha ){\Phi}dlsin\alpha$$

(29)

And this plastic work should be equal to the two times of work done by the pure plastic moment ${M}_{p}$ of the cell wall AB, and the pure plastic moment ${M}_{p}$ is generally described by,

$${M}_{p}=\frac{1}{4}{\sigma }_{y}d{t}^{2}$$

(30)

where ${\sigma }_{y}$ is the yielding stress of the cell wall materials.

Hence, the work done by pure plastic moment ${M}_{p}$ of the cell wall AB is

$${W}_{p}={M}_{p}{\Phi}=\frac{1}{4}{\Phi}{\sigma }_{y}d{t}^{2}$$

(31)

Hence, there exists following equation:

$${W}_{U}=2{W}_{p}$$

(32)

Then for the bottom inclined arm (CD) could be expressed as:

$${P}_{B}={\left({\sigma }_{0}\right)}_{1}(h-lsin\beta )d$$

(33)

The moment of force around hinge points C and D could be expressed as:

$${{M}_{B}=P}_{B}lsin\beta ={\left({\sigma }_{0}\right)}_{1}(h-lsin\beta )dlsin\beta$$

(34)

When the plastic hinge points C and D plastically rotate the angle ф, the plastic work done at the hinges are:

$${W}_{B}={M}_{B}{\Phi}={\left({\sigma }_{0}\right)}_{1}(h-lsin\beta ){\Phi}dlsin\beta$$

(35)

And this plastic work is also equal to the two times of work done by the pure plastic moment ${M}_{p}$ of the cell wall CD:

$${W}_{B}=2{W}_{p}$$

(36)

By combining Eqs. (32) and (36), the total plastic work could be obtained for the Semi Reentrant cell,

$${W}_{T}={W}_{B}+{W}_{U}=4{W}_{p}$$

(37)

The by substituting Eqs. (29), (30) and (35), the following equation is obtained,

$${\left({\sigma }_{0}\right)}_{1}\left(h+lsin\alpha \right){\Phi}dlsin\alpha +{\left({\sigma }_{0}\right)}_{1}\left(h-lsin\beta \right){\Phi}dlsin\beta ={\Phi}{\sigma }_{y}d{t}^{2}$$

(38)

Thus the plateau stress in x₁ direction could be arranged as follows,

$${\left({\sigma }_{0}\right)}_{1}={\sigma }_{y}{\left(\frac{t}{l}\right)}^{2}\frac{1}{2\left(\frac{h}{l}\left(sin\alpha +sin\beta \right)+{sin}^{2}\alpha -{sin}^{2}\beta \right)}\quad \left( {{\text{in}}\;x_{1} \;{\text{direction}}} \right)$$

(39)

Fig. 16

The basic force diagram of Semi Reentrant structure in x1 direction