Analysis of projectile penetrating into mortar target with elliptical cross-section
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摘要: 为研究椭圆截面弹体侵彻混凝土靶规律,基于动态球形空腔膨胀理论,建立椭圆截面弹体侵彻受力模型,计算典型椭圆截面弹体阻力规律和侵彻砂浆靶深度。在此基础上,采用弹道炮发射平台,开展相同质量和长度的2种典型椭圆截面弹体及圆截面弹体垂直侵彻半无限砂浆靶实验。结果表明:理论模型能够反映椭圆截面弹体受力情况,并与实验研究结果吻合较好;椭圆截面弹体长短轴参数的改变对侵彻性能影响较为显著。Abstract: In this work, based on the theory of dynamic spherical cavity expansion, we presented a force model of penetrating projectiles with an elliptical cross-section to study their penetration performance and, using this model, calculated the resistance of the elliptical cross section and the penetration depths. Further, we performed a series of experiments of two typical elliptical cross-sectioned and circular cross-sectioned projectiles with the same mass and length penetrating perpendicularly semi-infinite grout targets at a velocity of 700 m/s to 800 m/s. The results show that the established theoretical model reflected the force condition of the elliptical cross-section and the theoretical results agreed well with the experiment, and that the size of the cross-section had a significant influence on its penetration performance.1) “第十一届全国爆炸力学学术会议”推荐论文
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刻槽弹体在侵彻混凝土的过程中会面临结构稳定性问题。国内外一些学者以自由梁动态结构响应为基础开展了一系列弹体结构稳定性的研究工作。陈小伟[1]以空腔膨胀理论为基础,分析了圆柱壳弹体斜侵彻混凝土的受力情况,针对不同撞击速度的细长中空弹体,分析得到不出现弯曲破坏的弹体最大临界倾角和壳体壁厚下限。皮爱国等[2]基于刚塑形模型和理论载荷分析,给出了弹体在横向和轴向载荷作用下的响应行为,并得到弹体任一截面剪力、弯矩以及轴力的分布规律。王一楠等[3]基于自由梁理论和侵彻阻力分析,给出了小攻角情况下弹体弯曲变形分析。
刻槽弹体作为一种新型结构弹体,在高速侵彻情况下,其锥形弹身可产生恢复力矩,提高弹道稳定性,与此同时,弹身壁厚增加,可提高结构稳定性。本文基于刚塑性理论和侵彻载荷理论分析,将刻槽弹体简化为空间自由变截面梁,给出了弹体在侵彻混凝土早期的刚体响应行为,得到了弹体任一截面弯矩、剪力以及屈服函数的分布规律。基于此理论分析,讨论了刻槽弹体壁厚、材料屈服强度、初速及倾角对弹体弯曲的影响规律。
1. 弹体弯曲理论分析
1.1 弹体结构元素计算
刻槽弹体弹身结构如图1所示,在图中坐标系下将分别计算弹体弹身质量、质心及转动惯量等物理量。
弹体弹身质量为
mtotal=∫Lbdm=∫LbρS(x)dx (1) 式中:m为弹体微元质量,ρ为弹体密度,S(x)为弹体横截面积,表达式为
S(x)={πR2x<x0πR2−n(αr2−r2sin2α2+βR2−R2sin2β2)−π(d2)2x≥x0 (2) 式中:
R=R(x)=(x−b)tanθ+a ;d为弹体装药直径;x0为刻槽段起始位置,x0=(H−r−a)/tanθ+b ;n为刻槽个数;r为刻槽半径;H为刻槽刀具圆心距弹体轴线距离;=arccos[(r2+H2−R2)/(2rH)]; = arccos[(R2+H2−r2)/(2RH)]。 弹体弹身质心为
xc=∫Lbxdm/mtotal (3) 弹体弹身转动惯量为
J=∫Lb(x−xc)2dm (4) 1.2 载荷分析
弹体斜侵彻半无限混凝土靶体示意图如图2所示。
横向载荷参照陈小伟[1]给出的刚性弹体斜侵彻混凝土靶时的平均侧向作用力
ˉF=12πd024(Sfc+N∗ρ0v02)sinβ0(5) Forrestal等[4]给出的轴向载荷作用力
Fx=πd024(Sfc+N∗ρ0v02cos2β0)(6) 式中:d0为刻槽弹体小端直径;fc为混凝土无约束抗压强度;
S=72.0f−0.5c ;ρ0为靶体密度;v0为弹体初速;β0为弹体倾角;N*为弹头形状因子,对于卵形弹头,N*=1/(3ψ)-1/(24ψ2),ψ为弹体头部卵形系数。1.3 弹体弹身各个位置处的剪力和弯矩
塑性铰出现之前弹身刚体运动示意图如图3所示。
在低载情况下,将刻槽弹体简化为变截面自由梁的刚体运动,运动模式如图3所示,将运动分解为绕质心转动和平动,u为端点位移,θ0为端点转角。由刚体动力学可得其运动方程为:
{ˉF(xc−b)=J¨θ0ˉF=∫Lb[¨u−¨θ0(x−b)]dm (7) 由于式(7)积分得不到解析解,因此需采用数值解法求得
¨θ0 和¨u 。弹身各个位置处的剪力表达式为:Q(x)=ˉF−∫xb[¨u−¨θ0(x−b)]dm (8) 弹身各个位置处的弯矩表达式为:
M(x)=∫xbQ(x)dx (9) 同样采用数值积分法求得弹身各截面位置处的剪力和弯矩分布。
1.4 弹体弹身各个位置处的屈服函数
理想夹层梁弹塑性材料在弹性范围内承受轴力NA和弯矩载荷M共同作用梁截面的屈服条件为[5]:
φe=|NA|/NY+|M|/MY−1<0 (10) 式中:φe为屈服函数,NY及MY分别为截面的分离弹性屈服极限。
针对本文所提刻槽弹体,将各物理参量代入到屈服函数可得:
φe(x)=Fx[1−m(x)/mtotal]NY(x)+M(x)MY(x)−1 (11) 其中
NY(x)=S(x)σcr,MY(x)=σcrI(x)ymax (12) 式中:
S(x) 为各截面的面积,I(x) 为各截面的惯性矩,{y_{\max }}(x) 为截面上各点距中心轴的最大距离,{\sigma _{{\rm{cr}}}} 为弹体材料屈服强度。对于弹身圆锥段截面形状,其惯性矩
I(x) 的表达式为:I(x) = \pi {R^4}(x)/4 (13) 而对于图4所示弹身刻槽段截面形状,其惯性矩
I(x) 的表达式为:I(x) = \frac{{\pi {{(2R)}^4}}}{{64}}\left[ {1 - {{\left( {\frac{d}{{2R}}} \right)}^4}} \right] - 2\left[ {{I_1}(x) + {I_2}(x)} \right] (14) 其中
{I_1} = \int_{{y_{\rm{G}}}}^{{y_{\rm{K}}}} {2{y^2}} \left[ {\sqrt {{R^2} - {y^2}} - (H\cos {\alpha _0} - \sqrt {{r^2} - {{(y - H\sin {\alpha _0})}^2}} )} \right]dy {I_2} = \int_{{y_{\rm{N}}}}^{{y_{\rm{W}}}} {2{y^2}} \sqrt {{r^2} - {{(y - H)}^2}} {\rm d}y + \int_{{y_{\rm{W}}}}^R {2{y^2}\sqrt {{R^2} - {y^2}} } {\rm d}y {y_{\rm{G}}} = \frac{{4H\sin {\alpha _0}({R^2} + {H^2} - {r^2}) - \sqrt { - 16{H^2}({R^2} + {H^2} - {r^2}){{\cos }^2}{\alpha _0} + 64{H^4}{{\cos }^2}{\alpha _0}{R^2}} }}{{8{H^2}}} {y_{\rm{K}}} = \frac{{4H\sin {\alpha _0}({R^2} + {H^2} - {r^2}) + \sqrt { - 16{H^2}({R^2} + {H^2} - {r^2}){{\cos }^2}{\alpha _0} + 64{H^4}{{\cos }^2}{\alpha _0}{R^2}} }}{{8{H^2}}} {y_{\rm{N}}} = H - r {y_{\rm{W}}} = ({R^2} + {H^2} - {r^2})/(2H) 式中:
{\alpha _0} =30°,如图4所示。2. 刻槽弹体斜侵彻混凝土结构响应研究
基于1.2节建立的刚塑性分析模型和经过验证合理的轴力弯矩耦合屈服函数[5],以文献[6]中的刻槽弹体为例,计算斜侵彻工况下弹身各截面剪力和弯矩分布。弹靶结构参数如表1所示。
表 1 弹靶结构参数Table 1. Parameters of the projectile and concretemtotal/kg ψ L/m D0/m fc/MPa ρ/(kg∙m−3) θ/(°) 1.43 3 0.231 0.045 50 2 450 2 注:D0为弹体大端直径。 图5和6分别为刻槽弹体斜侵彻混凝土靶体时(
{v_0} =1 000 m/s,{\beta _0} = {15^ \circ } )弹身截面剪力和弯矩分布。由图可得,最大弯矩截面位于距离弹体头部0.49L处,其值为0.108 6\bar PL ,P为弹身小端面所受横向力。图7~10分别为不同速度、倾角、弹体内径及弹体材料屈服强度条件下,弹体各截面的无量纲屈服函数。可以看出,对于文中所提刻槽弹体斜侵彻混凝土靶体时的危险截面位于刻槽段的起始截面。由图7可得,随着弹体初速增大,危险截面屈服函数值随之增大,当弹体初速大于1 200 m/s时,危险截面的屈服函数值大于0,表明该处应力已超过弹体材料屈服极限,当作用载荷继续增大时,弹体将发生结构弯曲变形。由图8可得,随着弹体倾角增大,危险截面屈服函数值随之增大,当倾角大于20°时,危险截面的屈服函数值大于0,表明该截面应力已大于材料屈服强度。由图9可得,随着内径的增大,即壁厚减小,危险截面屈服函数值随之增大,当内径大于22.225 mm时,危险截面的屈服函数值大于0,表明该截面应力已大于材料屈服强度。由图10可得,随着弹体材料屈服强度增大,危险截面屈服函数值随之减小,当屈服强度小于1 500 MPa时,危险截面的屈服函数值大于0,表明该截面应力已大于材料屈服强度。因此,通过本文理论,可计算文献[6]中刻槽弹体在各种条件下保持结构稳定性的临界条件。
以文献[6]中刻槽弹体为基准,变化刻槽尺寸大小,即
r 的大小,可分别求得各弹体弹身截面屈服函数值分布(假定装药长度和刻槽长度相等),其中弹体初速{v_0} =1 000 m/s,倾角{\beta _0} = {15^ \circ } ,弹体材料屈服强度{\sigma _{\rm {cr}}} =1 500 MPa,靶体无约束抗压强度{f_{\rm{c}}} =50 MPa,计算结果如图11所示。由上述计算结果可得,随着刻槽半径的增大,弹体危险截面的屈服函数值也随之增大,当刻槽半径增大到7.35 mm时,危险截面屈服函数值大于0,即弹体发生弯曲;当刻槽半径小于6.85 mm时,弹体的危险截面位于刻槽段起始位置,而当刻槽半径为7.35 mm时,弹体的危险截面向后偏移,位于距弹体头部0.35L处。
3. 结 语
基于刚塑性理论和侵彻载荷理论分析,将刻槽弹体简化为空间自由变截面梁,给出了弹体在侵彻混凝土早期的刚体响应行为。对于本文所提刻槽弹体,基于上述理论可计算得到不同条件下弹体不发生弯曲的临界壁厚、材料屈服强度、初速及倾角,同时得到了不同刻槽半径所对应的屈服函数分布规律。
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表 1 实验数据与理论模型结果对比
Table 1. Comparison between experimental and theoretical data
编号 m/g γ/(°) v0/(m·s-1) dmax/mm dmin/mm h/mm z/mm ε/% 实验 理论 L1-2 460 1.6 744 430 370 66 596 590 -1.00 L1-3 451 0.6 754 380 320 60 555 593 6.85 T1-1 448 2.5 755 540 480 71 573 591 3.14 T1-2 450 1.4 747 526 416 75 553 582 5.24 T1-3 447 1.1 748 260 230 66 570 580 1.75 T2-1 451 3.3 714 480 450 69 462 443 4.11 T2-2 443 1.7 740 490 440 70 472 468 -0.85 T2-3 451 0.9 741 470 440 68 471 477 1.27 -
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