Over the last few decades, man-made honeycombs have been widely used in many industries due to their properties such as high strength to weight ratio and good energy absorption capabilities. Honeycombs are manufactured from materials such as aluminum, nomex, polymer, and ceramic. Aluminum honeycombs can be used as industrial products as well as core materials in sandwich panels in various fields of engineering such as aerospace, aircraft, automotive, and naval engineering [1
A number of studies have been conducted on the out-of-plane compression of aluminum honeycombs at low and intermediate strain rates [3
]. Zhou and Mayer [3
], Wu and Jiang [4
] and Baker et al.
] conducted compression tests on aluminum honeycombs at different strain rates in the out-of-plane direction and found that the plateau stress, σpl
, increased with strain rate,
. Both Xu et al.
] and Ashab et al.
] found that with the increase of t/l
ratio (cell wall thickness to edge length ratio) and strain rate, plateau stress, σpl
, increased. Vijayasimha Reddy et al.
] concluded that energy absorption capacity of aluminum honeycombs increased with the impact velocity under out-of-plane compression load. Alavi and Sadeghi [9
] conducted experiments on foam-filled aluminum hexagonal honeycombs under the out-of-plane compression loads. They observed that the crushing strength of bare honeycombs and foam-filled honeycombs increased with strain rate and bare honeycombs were more sensitive to strain rate than foam-filled honeycombs. Mozafari et al.
] employed ABAQUS software and observed that the mean crushing strength and energy absorption of foam-filled honeycomb were greater than the sum of those of bare honeycomb and foam.
Along with the experimental investigation, finite element analysis (FEA) has also been conducted by various researchers [11
] to study the mechanical behavior of aluminum honeycombs. Guo and Gibson [11
] conducted numerical analysis of intact and damaged honeycomb properties in the in-plane direction and reported that modulus and strength decreased due to the effect of single and isolated defects of various sizes. They also investigated the separation distance between two defects and its effect on the plastic collapse strength and Young’s modulus. Ruan et al.
] employed ABAQUS to investigate the effects of t/l
ratio and impact velocity on the in-plane deformation mode and plateau stress. They derived an empirical formula to describe the relationship between the plateau stress, t/l
ratio and velocity. Hu et al.
] conducted experiments as well as finite element analysis to study in-plane crushing of aluminum honeycombs. They proposed a dynamic sensitivity index to describe crushing strength and energy absorption.
Deqiang et al.
] used ANSYS/LS-DYNA [16
] to study the out-of-plane dynamic properties of aluminum hexagonal honeycomb cores in compression. They found that the out-of-plane dynamic plateau stresses of honeycombs were related to the impact velocity, t/l
ratio, and expanding angle θ
of honeycombs by power laws. Yamashita and Gotoh [17
] conducted both experimental and numerical analyses on the compression of aluminum honeycombs. The crushing strength was related to the t/l
ratio of honeycombs by a power law with the exponent of 5/3, which was the same as the theoretical equation derived by Wierzbicki [18
]. The computer simulation carried out by Xu et al.
] also found a power law relationship between the out-of-plane compressive strength of aluminum honeycombs and the strain rate and t/l
A limited number of experiments were conducted on aluminum honeycombs subjected to indentation [3
] at very low and intermediate strain rates. Zhou and Mayer [3
] conducted quasi-static indentation tests on aluminum honeycombs to study the influence of specimen size on the force versus
displacement curve. They found flatter and lower indentation force for the larger specimen. This was because larger specimens had a larger amount of surrounding cells, which provided stiffer support and resulted in fewer cells to be involved in tearing. The four primary deformation mechanisms were shear, tearing initiation, tearing, and compression. Zhou and Mayer also used different indenters, such as square, rectangular, and circular, to study the effect of indenter shape. Ashab et al.
] conducted indentation tests on three types of aluminum honeycombs at strain rates from 10 to 102
and found that the tearing energy increased with the t/l
ratio of honeycomb and strain rate. However, due to the limited honeycombs and testing machines available, previous studies were not able to draw quantitative conclusions on the effects of t/l
ratio and strain rate on the tearing energy of honeycombs.
In the present paper, numerical simulation is performed using ANSYS/LS-DYNA [16
] to study the dynamic out-of-plane properties of aluminum hexagonal honeycombs with various t/l
ratios subjected to indentation. Compression of honeycombs is also simulated in order to calculate the tearing energy in indentation. Full-scale FE models of honeycombs are verified by the previous experimental results. The verified FE models are then used to investigate the effects of t/l
ratio and strain rate on the plateau stress and tearing energy of honeycombs subjected to indentation. Empirical equations are proposed.
2. Finite Element (FE) Modeling
In the present paper, numerical analysis of aluminum honeycombs was carried out using ANSYS/LS-DYNA [16
]. Two types of honeycombs, differing in cell size and cell wall thickness, were simulated. The honeycombs are named as H31 and H42 for honeycombs 3.1-3/16-5052-.001N, 4.2-3/8-5052-.003N, respectively. The specifications of the honeycombs, provided by the manufacturer, are listed in Table 1
. The dimensions of each honeycomb model are the same as those of the actual specimen used in the previous experiments [7
]. The height of all honeycombs, h
, was 50 mm. The in-plane dimensions of all honeycomb specimens were 180 mm × 180 mm in indentation simulation (Figure 1
a) and 90 mm × 90 mm in compression simulation (Figure 1
Aluminum honeycomb walls were simulated using a bilinear kinematic hardening material model. The corresponding material properties are listed in Table 2
. Belytschko-Tsay Shell 163 elements with five integration points were employed to simulate the honeycomb cell walls for high computational efficiency [19
]. In each honeycomb cell, single wall thickness was employed for the four oblique walls and double wall thickness was employed for the two vertical walls. To identify the optimum element size, a convergence test was carried out. Five different element sizes—2.1 mm, 1.4 mm, 0.7 mm, 0.3 mm, and 0.15 mm—were used to simulate compression of honeycombs at 5 ms−1
. No significant difference (less than 7%) was observed between the results for element sizes 0.7 mm and 0.15 mm. Therefore, in this FE analysis of aluminum honeycombs an element size of 0.7 mm was employed. Since tearing of cell walls happened in honeycombs under indentation, MAT_ADD_EROSION failure criterion with a maximum effective strain of 0.3 [21
] was used in the indentation models. All degrees of freedom of one node at a corner of the honeycomb were fixed to keep the honeycomb in place (i.e.
, no rigid body movement).
In physical experiments, honeycomb specimens were placed on a fixed lower plate and crushed by an upper plate (in compression) or indenter (in indentation). In FE models, the lower plate was simulated by a rigid plate while the upper plate and indenter were simulated by rigid bodies. The lower plate was 1 mm (thickness) × 200 mm × 200 mm. The upper plate and indenter were cuboids with dimensions of 50 mm (height) × 90 mm × 90 mm, the same as in the previous experimental study [7
]. The material properties used for the plates and indenter are listed in Table 3
For the lower plate, all degrees of freedom were fixed. For the upper plate (in compression) and indenter (in indentation), all three rotational movements and two transitional movements in the X and Z directions were fixed. The upper plate or indenter could move in the negative Y direction at a constant velocity to compress or indent honeycombs.
A tiny gap (0.1 mm) between the fixed lower plate and the honeycomb was employed to avoid the initial penetration at the beginning of the simulation. For the same reason, an initial gap of 5 mm was also introduced between the upper plate or the indenter and the honeycomb. SURFACE_TO_SURFACE contacts were employed between the plates or indenter and honeycomb. Typical finite element models of indentation and compression of honeycombs in the out-of-plane direction are shown in Figure 1
In this finite element analysis, different honeycomb models have been developed by using ANSYS/LS-DYNA to study the mechanical behavior of honeycombs under out-of-plane indentation and compression loads over a wide range of high strain rates from 1 × 102 to 1 × 104 s−1. The FE models have been validated by the previous experimental results (compression and indentation) in terms of deformation, stress-strain curves, plateau stress, and dissipated energy. A reasonable agreement between the FEA and experimental results has been found for both honeycombs H31 and H42.
It is found that the plateau stress, dissipated energy, and tearing energy increase with the t/l ratio. For a constant strain rate of 1 × 103 s−1, the plateau stresses increase with t/l ratio by power laws with exponents of 1.47 and 1.36 for compression and indentation, respectively.
Moreover, the plateau stress, dissipated energy, and tearing energy increase gradually for low and intermediate strain rates. Significant enhancement in the plateau stress, dissipated energy, and tearing energy is observed at high strain rates for honeycombs subjected to either compression or indentation loads. An empirical formula is proposed for the tearing energy per unit fracture area in terms of strain rate and relative density of honeycombs.
The current FEA reveals that at velocities at 5 ms−1, under indentation, plastic buckling of the honeycomb cell walls occurs from the end that is adjacent to the indenter, while under compression the buckling of honeycomb cell walls occurs from both ends of the honeycomb.
It is found that under quasi-static indentation, the empirical formula proposed by Shi et al. for foam can be used for honeycombs as well.