Processing math: 100%
REN Shanliang, WEN Heming, ZHOU Lin. Theoretical Study of the Perforation of Double-Layered Metal Targets without Spacing Struck by Flat-Ended Projectiles[J]. Chinese Journal of High Pressure Physics, 2018, 32(3): 035103. doi: 10.11858/gywlxb.20170695
Citation: YU Pengshan, LIU Zhifang, LI Shiqiang. Design and Energy Absorption Characteristic Analysis of a New Bio-Bamboo Thin-Walled Circular Tube[J]. Chinese Journal of High Pressure Physics, 2021, 35(5): 054205. doi: 10.11858/gywlxb.20210710

Design and Energy Absorption Characteristic Analysis of a New Bio-Bamboo Thin-Walled Circular Tube

doi: 10.11858/gywlxb.20210710
  • Received Date: 19 Jan 2021
  • Rev Recd Date: 04 Feb 2021
  • Inspired by the microstructure of natural bamboo, a new bio-bamboo thin-walled tube was designed by introducing double-rhombic ribs between inner and outer tubes on the basis of the traditional double-circular tube structure. Based on the theory of simplified super folding element, theoretical models of bio-bamboo circular tubes under axial compression were established. Finite element software ABAQUS was used to simulate the axial compression of these models. For crashworthiness and deformation mode of bio-bamboo thin-walled tube, the effects of those number of double-rhombic ribs, diameter of inner tube, wall thickness were analyzed, and it was compared with the traditional structure. The results show that the theoretical prediction is consistent with numerical simulation, and the errors of average compression force and specific energy absorption are less than 10%. Compared with traditional double-circular tube, the specific energy absorption of the bio-bamboo thin-walled tube is increased by 83.61% and the compression force efficiency is increased by 198.65%. The number of ribs has a significant effect on the crashworthiness of the structure. With the number increase of double-rhombic ribs, both specific structural energy absorption and peak crushing force increases. When the number of ribs is small, the structure appears local buckling deformation, which affects the energy absorption crashworthiness. The smaller the diameter of inner tube, the higher the initial peak force, and the larger the diameter of inner tube, the smaller the energy absorption.

     

  • 研究金属靶板在弹丸冲击作用下的响应和破坏对军用武器和防护结构的设计和评估有重要的意义。单层金属靶板在刚性平头弹丸正撞击下的破坏模式可分为:带有整体变形的简单剪切破坏和局部化的绝热剪切冲塞破坏,从能量吸收的角度而言前者优于后者。在工程实践中,常用双层板结构代替单层板。对于单层金属板,很多学者做了相关的理论、实验和数值模拟研究。Wen和Jones[1-2]对刚性平头弹丸低速正撞击下固支的软钢圆板和铝合金圆板的响应和破坏进行了系统的实验研究,并根据实验结果和理论分析提出了刚性平头弹丸正撞下金属靶板低速穿透的Wen-Jones模型。BØrvik等[3]通过实验、理论分析和数值模拟研究了不同厚度固支Weldox460E圆板在平头弹撞击下的变形和穿透,通过初始速度和残余速度求出不同厚度靶板的弹道极限值。Chen等[4]利用刚塑性分析方法研究了平头弹撞击金属圆板的问题,考虑了结构的整体响应和局部剪切破坏。这些研究能够较好地描述和预测平头弹撞击下单层金属靶板的破坏模式和抗弹性能。

    相比于单层板冲击失效响应方面的大量实验、数值和理论的研究文献,在公开发表的文章中仅有少量文章研究了多层板的抗弹性能。理论方面只提出了简单模型,大部分都是实验和数值模拟研究,且得出的结论也大相径庭。Radin等[5]做了很多平头弹穿透单层和多层2024-0铝板的实验,发现单层板的弹道极限均高于等厚度的多层板,同时用理论分析模型计算了弹丸的弹道极限,分析结果和实验一致。张伟等[6]通过实验研究发现厚度较小时,同样厚度的单层钢板的弹道极限要高于等厚接触式双层板的弹道极限。Teng等[7]通过ABAQUS/Explicit研究了金属单层板和等厚双层板的抗弹性能,数值模拟结果表明:双层板弹道极限高于单层板7%~25%。对平头弹而言,Dey等[8]实验得到的等厚接触式双层板的弹道极限约高于单层板弹道极限47.2%,与实验结果相比,数值模拟结果(数值模拟结果约高22.9%)大大低估了双层板的抗弹性能。

    目前对接触式双层板在平头弹撞击下的抗弹性能和破坏模式尚未有统一的结论,仍需要进行大量的研究工作。我们对等厚接触式双层金属板在平头弹撞击下的穿透破坏进行了理论研究,基于先前单层金属板的穿透理论和实验观察提出一个等厚接触式双层金属板穿透的新模型,并与相关实验数据和其他理论模型进行比较和讨论。

    图 1给出了单层板和等厚接触式双层板在平头弹撞击下的示意图。单层板厚度为H,双层板总厚为H,上、下板等厚度,各为H/2,板的半径为R,平头弹弹径为d(d=2a)。由于靶板间的相互作用,双层板的变形和破坏相较单层板要复杂很多。

    图  1  平头弹加载固支金属圆板示意图
    Figure  1.  Schematic of fully-clamped circular metal plates loaded by flat-ended projectile

    F为平头弹作用在双层板上时所受的总作用力,F1为第一层板所受的作用力,F2为第二层板所受的作用力,则F=F1+F2;第一层板的总体变形Wo1,第二层板的总体变形为Wo2。根据文献[1],有

    对第一层板:

    F1=K1mWo1+F1c
    (1)

    对第二层板:

    F2=K2mWo2+F2c
    (2)

    式中:K1mK2m分别为第一层和第二层板的薄膜刚度,且有K1m=K2m=Km=2πN0/ln(R/a),N0为靶板单位长度的薄膜力,且有N0=σyH/2,σy为靶板材料的屈服应力;F1cF2c为两层板的静态极限载荷,且有F1c=F2c=Fc=(4/3)πM0[1+(1+3/2)/ln(R/a)]M0=σy(H/2)2/4为靶板单位长度的极限塑性弯矩。第一层板穿透破坏之前,第二层板紧贴第一层板,该过程第一层板和第二块板的整体变形基本相同,可以表示为

    Wo1Wo2=Wo
    (3)

    结合F=F1+F2和方程(1)、方程(2),有

    F=2KmWo+2Fc
    (4)

    对于固支金属圆板,等效应变εe[9]可以表达为

    ε2e=43(ε2r+εrεθ+ε2θ)+13γ2rz
    (5)

    式中:εrεθγrz分别为径向应变、周向应变和横向剪应变,对于厚度为H/2的靶板,其值可表示为

    εr=W202a2ln2(a/R)+W0(H/2)2a2ln(a/R)
    (6)
    εθ=W0(H/2)2a2ln(a/R)
    (7)

    第一层板横向剪应变[1]可以表示为

    γrz=Δ/e=(F/Fu)1/nγc
    (8)

    式中: γc为临界剪应变;Δ为弹丸压入深度,Δc为临界压入深度;e为剪切带半宽度;Fu为发生剪切冲塞破坏的临界剪切力,Fu=τuAsτu=σu[0.41H/(2d)+0.42],AsdH/2,σu为极限拉伸应力。令(5)式中的等效应变εe等于拉伸破坏应变εf(εe=εf),就可以从方程(5)~方程(8)求得第一层板穿透破坏时的最大整体变形Wo1f

    ε2f=163(H2d)4[(2Wo1f/H)4ln4(a/R)+(2Wo1f/H)3ln3(a/R)+(2Wo1f/H)2ln2(a/R)]+13{2λ[0.41H/(2d)+0.42]{2Wo1f/dln(R/a)+13[1+3/2ln(R/a)]H2d}}2/nγ2c
    (9)

    将由(9)式求得的不同总厚度H对应的第一层板的最大整体变形Wo1f代入方程(10)、方程(11)即可求得第一层板的整体变形耗能(Ebm1)

    Ebm1=Wo1f0F1dS=Wo1f0(KmW01+Fc)dWo1+Km2W2o1f+FcWo1f
    (10)

    和局部剪切耗能(Es1)

    Es1=Δc0FsdΔ=FuΔcn+1(KmWo1f+FcFu)(n+1)/n
    (11)

    第一层板穿透破坏吸收的总能量Ep1为整体变形耗能Ebm1和局部剪切耗能Es1之和,即Ep1=Ebm1+Es1

    在贯穿第一层板后,平头弹前端附着第一层板的塞块撞击第二层板。图 2给出了第二层板组合弹丸(平头弹+塞块)作用下破坏示意图。第一层板的塞块在平头弹和第二层板作用下边缘厚度变薄,剖面近似于四分之一圆,中间部分近似成平面。在组合弹丸(平头弹+塞块)作用下,第二层板的整体变形增大,因薄膜拉伸造成局部厚度变薄。

    图  2  第二层板破坏示意图
    Figure  2.  Schematic of second plate failure

    图 2所示,根据实验和数值模拟结果第一层板塞块的厚度约为初始厚度的0.9倍,即H1≈0.45H。第二层板的初始厚度为H0=H/2,发生破坏时破坏处的厚度为H2,由塑性变形体积不变可以有πr20H0=πr22H2,即r2/r0=H0/H2,根据工程应变和真实应变的关系可得第二层圆板在破坏处的径向真实应变,即εr1=lnH0/H2=(1/2)ln(2H2/H)

    根据文献[9],将第二层板破坏处的径向应变近似表示为εr2=W2o2f2r21ln2(r1/R)+Wo2fH4r21ln(r1/R),其中r1为破坏处距离弹丸中心的距离,根据本模型可得r1=aH1+H1sin(π/4),由εr1=εr2可求得第二层板破坏时的最大整体变形Wo2f与总厚度H及第二层板破坏时最终厚度H2之间的函数关系

    12ln(2H2H)=W2o2f2r21ln2(r1/R)+Wo2fH4r21ln(r1/R)
    (12)

    而厚度为H/2的单层板在平头弹撞击下的最大整体变形Wof[8]可以表示为

    ε2f=163(H2d)4[2(W0f/H)4ln4(a/R)+(2W0f/H)3ln3(a/R)+(2W0f/H)2ln2(a/R)]+13{1λ[0.41H/(2d)+0.42]{2W0f/dln(R/a)+13[1+1+3/2ln(R/a)]H2d}}2/nγ2c
    (13)

    对于不同厚度的双层板,靶板的厚度越小,第一层板的塞块厚度H1越小,r1=aH1+H1sin(π/4)的值与平头弹半径a越接近。当靶板厚度趋于零时,有r1|H→0=a,此时第二层板与平头弹穿透厚为H/2单层板的速度场和整体变形场相同[1],有εr2=εr,即(Wo2f=Wof)|H→0,文字表述为当靶板厚度趋于零时第二层板的整体变形和平头弹穿透厚度为H/2的单层靶板的整体变形相同。由于公式较为复杂且包含隐式形式,求解较为复杂,用Matlab软件求解隐式方程组(12)式、(13)式,得到第二层板的最终厚度H2以及(12)式、(13)式对应的函数关系,(12)式、(13)式的函数图像在H=0处相交。将求得的H2代入(12)式即可求得双层靶第二层靶板的最大整体变形Wo2f与总厚度H间的关系,进而可求得第二层板的总体变形耗能Ebm2

    Ebm2=Km2W2o2f+FcWo2f
    (14)

    和局部拉伸耗能Et2的近似值

    Et2=EVσyεr1πr21H2
    (15)

    第二层板穿透的能量消耗Ep2=Ebm2+Et2,则可以得到平头弹穿透双层板所消耗的总能量Ep

    Ep=Ep1+Ep2=Ebm1+Es1+Ebm2+Et2
    (16)

    以上得到的是准静态条件下的穿透能量,动态冲击下需要考虑材料的应变率效应。材料的应变率效应可以用Cowper-Symonds经验公式来描述, 即

    σd=σy[1+(˙εm/D)1/q]
    (17)

    式中: σd为材料的动态屈服应力,Dq为描述材料应变率的敏感性常数,˙εm为靶板的平均应变率。对双层靶而言,第一、二层板的平均应变率[1]可分别写为

    ˙εm1=2Wo1fvbl32Raln2(a/R)
    (18)
    ˙εm2=2Wo2fvbl32Rr1ln2(r1/R)
    (19)

    式中: vbl为弹道极限。将能量公式中的静态屈服应力σy用动态屈服应力σd代替,可得到动态情况下平头弹穿透靶板所消耗的能量Edp。令Edp=Mv2bl/2,可以得到平头弹撞击下双层靶的弹道极限, 即

    vbl=2EdpM
    (20)

    式中: M为平头弹质量。

    将本研究的理论模型结果与文献中的相关实验结果进行比较和讨论。针对Dey等[8]做的平头弹撞击等厚接触式双层Weldox700E钢板的实验,模型中的相关参数值见表 1,根据本研究模型可以求得不同厚度的第一层板和第二层板的最大整体变形(见图 3),并得到第二层板破坏处的最终厚度为H2≈0.41H。从图 3(a)可以看出,双层靶中的第一层板的总体变形随着总厚度的增加而减少,而第二层板正好相反,其总体变形随着厚度的增加而增加。图 3(b)给出了单层板在平头弹作用下总体变形随厚度的变化情况, 即其总体变形随着厚度的增加而逐步减少。

    表  1  Weldox700E钢板相关参数值[3, 9, 11]
    Table  1.  Parameters for Weldox700E steel plates[3, 9, 11]
    σy/MPa σu/MPa ρt/(kg·m-3) n γc D/s-1
    859 877 7 850 0.12 1.4 4.6×107
    d/mm M/g R/mm a/mm q εf
    20 197 250 10 7.33 1.05
    下载: 导出CSV 
    | 显示表格
    图  3  双层Weldox700E钢板的最大整体变形
    Figure  3.  Maximum global deformation of double-layered plates of Weldox700E

    图 4给出了本研究双层板理论模型求得的等厚接触式双层Weldox700E钢板的弹道极限与Dey等的实验结果及单层板的Wen-Jones模型[1]和绝热剪切模型[10]的对比。从图 4可以看出, 本研究模型能较好地预测等厚接触式双层板的弹道极限值。且对于单层板,当发生局部化的绝热剪切破坏时,等厚接触式双层板的弹道极限要明显大于单层板的弹道极限;当发生带有整体变形的简单剪切破坏时,等厚接触式双层板和单层板的弹道极限几乎相同。

    图  4  本研究理论模型与Weldox700E板实验数据和单层板模型的比较
    Figure  4.  Comparison of present model with experimental data for Weldoc700E plates and theoretical models for monolithic plates

    针对张伟等[6]的平头弹撞击等厚接触式双层Q235钢板的实验,模型中的相关参数值见表 2,用同样的方法可以求得不同厚度的第一层板和第二层板的最大整体变形(见图 5),根据本研究理论可得第二层板破坏处的最终厚度为H2≈0.315H。接触式双层Q235钢板中第一层板和第二层板的总体变形规律与接触式双层Weldox700E钢板类似,见图 5

    表  2  Q235钢板相关参数值[6, 9, 12]
    Table  2.  Parameters for Q235 steel plates[6, 9, 12]
    σy/MPa σu/MPa ρt/(kg·m-3) n γc D/s-1
    229 335 7 800 0.08 0.8 1 400
    d/mm M/g R/mm a/mm q εf
    12.7 34.5 85 6.35 1.5 1.05
    下载: 导出CSV 
    | 显示表格
    图  5  双层Q235钢板的最大整体变形
    Figure  5.  Maximum global deformation of double-layered plates of Q235

    图 6给出了用本研究能量模型求得的等厚接触式双层Q235钢板的弹道极限与张伟等的实验结果及单层板的Wen-Jones模型[1]和绝热剪切模型[10]的对比。从图 4可以发现, 本研究模型与张伟等的等厚接触式双层板的实验弹道极限值吻合得很好。且对于单层板,当发生带有整体变形的简单剪切破坏时,等厚接触式双层板和单层板的弹道极限几乎相同;当发生局部化的绝热剪切破坏时,等厚接触式双层板的弹道极限要明显大于单层板的弹道极限。

    图  6  本研究理论模型与Q235钢板实验数据和单层板模型的比较
    Figure  6.  Comparison of present model with experimental data for Q235 plates and theoretical models for monolithic plates

    通过分析等厚接触式双层板的破坏模式,基于Wen-Jones模型和应变失效准则得到了接触式双层板穿透的理论模型。结果表明:理论预测与有限的实验数据结果吻合得很好;当总厚度大于单层板绝热剪切冲塞临界厚度值时,双层板的弹道极限明显高于单层板的弹道极限;小于该值时,双层板的弹道极限与单层板的弹道极限差别不大。

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