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Theoretical model for a porous projectile striking on flat rigid anvil
LIU Hu , LIU Hua , YANG Jialing
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-07-14; Accepted: 2015-08-06; Published online: 2015-09-17 10:43
Foundation item: National Natural Science Foundation of China (11472035, 11472034)
Corresponding author. Tel.: 010-82317507 E-mail: liuhuarui@buaa.edu.cn
Abstract: Taylor impact is often applied to the determination of the dynamic yield stress of materials. For theoretical analysis of the Taylor impact of porous projectiles, the relationship between the density of a compressed porous projectile and the compressive plastic strain is very important. This paper proposes an exact density model for the compressible porous projectile by inducing the plastic Poisson's ratio, and further, an analytical model is established for the compressible porous projectile striking on a flat rigid anvil. As the plastic Poisson's ratio is a constant, the first order Taylor series expansion of the compression density ratio model can be reduced to the existing model. As the plastic Poisson's ratio is a function of the compressive plastic strain and the relative density, the relative density has a major influence on the impact response and the final deformation of the projectile, but the duration of impact-contact process is almost unaffected. The initial velocity of the projectile has considerable effects on both the final deformation of the projectile and the duration of impact-contact process. The present theoretical model is useful in analyzing the dynamic behavior of porous materials.
Key words: Taylor impact     porous material     plastic     plastic Poisson's ratio     rigid

1948年，Taylor[3]率先提出了Taylor测试的基本原理，他利用圆柱形子弹撞击刚性靶板，通过子弹的最终变形获得了材料高应变率状态下的力学特性。同年，Whiffin[4]利用该方法对多种金属材料进行了试验分析，通过子弹的最终变形得到了材料动态屈服强度，此后多名学者对Taylor模型进行了改进。例如，Hawkyard[5]利用能量守恒的办法代替动量定理对塑性波前进行了分析，得到的子弹变形模式比Taylor模型更加接近于试验结果。Jones等[6-7]同时也分析了应变率和应变强化效应对Taylor撞击结果的影响。另外还有不少学者对Taylor测试进行了有限元仿真分析[8-10]

1 问题描述及基本方程

 图 1 圆柱体泡沫子弹撞击刚性靶板前、撞击中间阶段及子弹的最终变形 Fig. 1 A cylindrical porous projectile striking on a rigid anvil before impact, in intermediate stage of deformation, and in final stage of deformation

 (1)
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1.1 子弹未变形区域的速度

 (11)

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1.2 子弹未变形区域的长度

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1.3 子弹塑性变形区域的长度

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1.4 撞击持续时间

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2 压缩密度比的退化 2.1 压缩密度比的一阶泰勒展开

 (25)

Lu等[14]通过多孔材料试验，认为子弹的压缩密度比与应变e满足，其中α为常数。显然这里的α与式(25)中的1-2μp相对应，这也说明在Lu等[13]的分析模型中，子弹的塑性泊松比μp为常数。

 (26)
 (27)
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μp为常数时，，令1-2μp=α，式(26)~式(30)可以退化到Lu等[13]的分析模型。

2.2 压缩密度比的二阶泰勒展开

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3 结果与讨论 3.1 塑性泊松比为常数

 图 2 μp=0.25, e0=0.3时的冲击响应 Fig. 2 Impact responses when μp=0.25 and e0=0.3
3.2 塑性泊松比为压缩应变的函数

 (37)

 (38)

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 图 3 压缩密度比与压缩应变关系 Fig. 3 Relations between compressive density ratio and compressive strain

 图 4 不同压缩密度比模型的冲击响应比较(ζ=0.2, λ0=0.204 35) Fig. 4 Comparison of impact responses predicted by different compressive density ratio models (ζ=0.2, λ0=0.204 35)

 图 5 ζ不同时的冲击响应-时间曲线(λ0=0.204 35) Fig. 5 Impact responses versus time for different ζ (λ0=0.204 35)

 图 6 ζ不同时的最终冲击响应-λ0曲线 Fig. 6 Final impact responses versus λ0 for different ζ
4 结论

1)当塑性泊松比为常数时，压缩密度比的一阶泰勒展开式可以退化到Lu的分析模型。同时，通过分析发现，当压缩应变较小时，撞击过程中的压缩密度比与压缩应变近似呈线性关系，这与Lu的试验分析结果一致，但压缩应变较大时，这种线性关系将不再成立。而后又计算了压缩密度比二阶泰勒展开的结果，发现它们的冲击响应历程具有相似的变化规律，同时随着展开阶次的提高，计算结果将逐渐向着原始表达式的结果靠近。

2)当塑性泊松比为压缩应变的函数时，通过塑性泊松比与压缩应变和相对密度的关系式(37)，分析了子弹相对密度对冲击响应的影响，发现相对密度会对泡沫子弹的冲击响应历程及最终变形产生影响，但对冲击响应持续时间影响较小。

3)子弹的初始速度会对子弹最终变形和冲击响应持续时间产生重要影响。

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#### 文章信息

LIU Hu, LIU Hua, YANG Jialing

Theoretical model for a porous projectile striking on flat rigid anvil

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(7): 1461-1468
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0471