Dynamic response of a sandwich panel cored by butterfly-shaped honeycombs with negative Poisson’s ratio to low-velocity impact
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摘要: 为了研究负泊松比蝴蝶形蜂窝夹芯板在低速冲击下的动力学响应，采用质量-弹簧模型获得了冲击器与蜂窝夹芯板之间的接触力，同时基于哈密顿原理和一阶剪切变形理论推导了负泊松比蝴蝶形蜂窝夹芯板的运动方程，采用Navier法和Duhamel积分对蜂窝板的振动位移进行了理论解析求解。在理论验证方面，蜂窝夹芯板前5阶固有频率的数值模拟结果与理论模型计算结果的最大相对误差为6.52%，蜂窝夹芯板中心最大横向位移的数值模拟结果与理论模型计算结果的最大相对误差为6.84%，理论模型求解的接触力与文献得到的接触力的最大相对误差为8%，验证了理论模型的有效性。结果表明，随着球形冲击器冲击速度的递增，蜂窝夹芯板的最大横向位移呈现递增的规律。而在相同冲击载荷下，蜂窝夹芯板的抗冲击特性随着胞元壁厚的增大而增强，随着胞元角度的增大而减弱；随着负泊松比蝴蝶形蜂窝夹芯板长宽比以及夹芯层与顶部蒙皮层的高度比的增大，蜂窝夹芯板的横向位移减小，冲击器与蜂窝夹芯板之间的接触力增大。当蜂窝夹芯板的宽长比从1∶1变化到1∶2时，蜂窝夹芯板最大横向位移减小6.1%；当顶部蒙皮层与蜂窝芯层的高度比从1∶6变化到1∶14时，蜂窝夹芯板的最大横向位移减小5.4%，这表明蜂窝夹芯板的抗冲击性能增强，吸能效果明显。Abstract: In order to study the dynamic response of a sandwich panel cored by butterfly-shaped honeycomb with negative Poisson’s ratio to low-velocity impact, a mass-spring (MS) model is applied to obtain the contact force between the spherical impactor and the honeycomb sandwich panel. Meanwhile, based on the Hamilton’s principle and the first-order shear deformation theory, the equation of motion for the butterfly-shaped honeycomb sandwich panel with negative Poisson’s ratio is derived. Besides, the Navier method and Duhamel’s integral are used to solve the vibration displacement of the honeycomb sandwich panel. To validate the theoretical model, the results are compared with the results of ABAQUS’ numerical simulation or published literature. It is shown that the maximum relative error between the numerical modeling results of the first five order natural frequencies and the results of theoretical model calculated in this paper is 6.52%, the maximum relative error between the numerical modeling results of the honeycomb sandwich panel under low-velocity impact and the calculated results of the theoretical model in this paper is 6.84%, and the maximum relative error of the contact force between the theoretical model in this paper and the published studies is 8%, thus verifying the validity of the theoretical model. The results show that the maximum lateral displacement of the honeycomb sandwich panel increases with the increasing velocity of the spherical impactor. Under the same impact load, the impact resistance of the honeycomb sandwich panel increases with the increase of the wall thickness of the unit cell, and decreases with the increase of the unit cell angle. The impact resistance of the honeycomb sandwich panel increases by 3.7% when the thickness of the unit cell wall changes from 1 mm to 3 mm. The lateral displacement of the butterfly-shaped honeycomb sandwich panel decreases while the contact force between the impactor and the honeycomb sandwich panel increases with the increase of the length-width ratio and the height ratio. When the width-length ratio of the honeycomb sandwich panel changes from 1∶1 to 1∶2, the maximum lateral displacement of the honeycomb sandwich panel decreases by 6.1%, and when the height ratio of the top skin layer to the honeycomb core layer changes from 1∶6 to 1∶14, the maximum lateral displacement of the honeycomb sandwich panel decreases by 5.4%, which indicates that the impact resistance of the honeycomb sandwich panel is enhanced and the energy absorption effect is obvious.
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图 1 蜂窝夹芯板和胞元结构示意图
Figure 1. Structure schematic diagram of the honeycomb sandwich panel and the unit cell
图 2 球形冲击器与蜂窝夹芯板接触的等效理论模型
Figure 2. An equivalent theoretical model of contact between the spherical impactor and the sandwich panel
图 4 蝴蝶形蜂窝夹芯板低速冲击有限元模型
Figure 4. The finite element model for the butterfly-shaped honeycomb sandwich panel subjected to low-velocity impact
图 6 胞元几何尺寸对蝴蝶形蜂窝夹芯板动力响应的影响
Figure 6. Influences of the unit cell geometry parameters on dynamic responses of the butterfly-shaped honeycomb sandwich panel
图 7 顶部蒙皮层与夹芯层的高度比对蝴蝶形蜂窝夹芯板动力响应的影响
Figure 7. Influences of the height ratio of the top skin layer to the sandwich layer on dynamic responses of the butterfly-shaped honeycomb sandwich panel
图 8 宽长比对蝴蝶形蜂窝夹芯板动力响应的影响
Figure 8. Influences of the width-length ratios on dynamic responses of the butterfly-shaped honeycomb sandwich panel
表 1 网格尺寸对一阶固有频率计算结果的影响
Table 1. Influences of mesh size on the calculated results of the first-order natural frequency
网格尺寸/mm 频率/Hz 15.0 719.26 12.5 699.18 10.0 696.14 7.5 692.37 5.0 691.25 
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表 2 蝴蝶形蜂窝夹芯板的前5阶固有频率
Table 2. The first five order natural frequencies of the butterfly-shaped honeycomb sandwich panel
阶数 频率/Hz 理论模型解 有限元解 1 646.17 691.25 2 1597.40 163.75 3 1648.40 1706.70 4 2570.50 2629.30 5 3181.90 3200.80
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表 3 低速冲击下蜂窝夹芯板中心网格尺寸对中心最大横向位移计算结果的影响
Table 3. Influence of center grid size on the calculated maximum lateral displacement of the sandwich panel center under low-velocity impact
网格尺寸/mm 中心最大横向位移/mm 10.0 0.1928 5.0 0.2129 2.5 0.2218 1.0 0.2516 0.5 0.2556
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表 4 不同冲击速度下蝴蝶形蜂窝夹芯板的横向位移
Table 4. Lateral displacement of the butterfly-shaped honeycomb sandwich panel at different impact velocities
冲击速度/(m·s−1) 中心最大横向位移/mm 本文模型 有限元模拟 6 0.177 0.190 8 0.247 0.252 10 0.320 0.311 12 0.395 0.374
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表 5 矩形板和冲击器的材料参数和几何尺寸[26-27]
Table 5. Material parameters and geometrical sizes of the homogeneous panel and impactor[26-27]
器件 密度/(kg·m−3) 弹性模量/GPa 泊松比 半径/mm 长度/mm 宽度/mm 厚度/mm 冲击器 7971.8 200 0.3 10 矩形板 7971.8 200 0.3 200 200 8
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