空腔膨胀理论靶体阻力模型及其应用研究进展

刘均伟; 张先锋; 刘闯; 陈海华; 熊玮; 谈梦婷

doi:10.11883/bzycj-2021-0010

空腔膨胀理论靶体阻力模型及其应用研究进展

doi: 10.11883/bzycj-2021-0010

南京理工大学机械工程学院，江苏南京 210094

基金项目: 国家自然科学基金面上项目(11772159)；国家自然科学基金(11790292)

详细信息

作者简介:
刘均伟（1996－　），男，博士研究生，liujunwei@njust.edu.cn

通讯作者:
张先锋（1978－　），男，博士，教授，博士生导师，lynx@njust.edu.cn

中图分类号: O382；TJ410
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出版历程
- 收稿日期: 2021-01-06
- 修回日期: 2021-04-12
- 网络出版日期: 2021-09-06
- 刊出日期: 2021-10-13

Research progress of target resistance model of cavity expansion theory and its application

Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

摘要

摘要: 从静/动态空腔膨胀模型的理论体系出发，介绍了空腔膨胀模型在不同方向上取得的成果，主要涉及理想侵彻条件的空腔膨胀压力计算模型及数值模拟方法和空腔膨胀模型在典型侵彻问题及复杂弹靶条件下的应用。在理想侵彻条件下的空腔膨胀压力计算模型中，主要讨论了靶体材料、屈服准则和状态方程对空腔边界应力的影响规律及空腔膨胀模型的适用性问题；根据数值模拟中初始条件的不同，介绍了空腔表面恒定速度/恒定压力两种数值模拟方法，证明了数值模拟方法的可靠性；整理了空腔膨胀模型的基本假设、适用范围、工程应用特点，列举了其在典型侵彻问题及多层复合靶板、约束靶体、弹体刻槽和异形截面形状弹体等复杂弹靶条件下的应用。针对空腔膨胀模型的研究现状，总结了目前空腔膨胀模型在冲击动力学领域的应用方向，归纳了空腔膨胀模型应用中尚存在的问题，展望了空腔膨胀模型下一步的重点发展方向。
- 侵彻 /
- 空腔膨胀 /
- 阻力模型
Abstract: The cavity expansion theory is one of the main basic theories for the theoretical analysis of penetration problems. It is mainly used to analyze the failure response characteristics of typical target materials under impact load, and then to determine the penetration resistance of the target. It is widely used in the analysis of high-speed impact penetration and failure problems. Domestic and foreign scholars have made abundant research achievements on plastic and (quasi) brittle materials based on the theory of cylindrical and spherical cavity expansion. Starting from the theoretical system of the static/dynamic cavity expansion model, the results of the cavity expansion model in different directions are introduced, mainly involving the cavity expansion pressure theoretical calculation model and numerical simulation method under ideal penetration conditions, and the application of cavity expansion model to typical penetration problems and complex missile target conditions. The theoretical calculation model under ideal penetration conditions based on cavity expansion theory mainly discusses the influence aspects of target material, yield criterion and equation of state on target resistance and the applicability of the cavity expansion model. According to the different initial conditions in the numerical simulation, two numerical simulation methods of cavity surface constant velocity/constant pressure are introduced, and the reliability of the numerical simulation method is proved. The basic assumptions, application scope and engineering application characteristics of the cavity expansion model are summarized, and its applications in typical penetration problems and complex missile targets such as multilayer composite target plate, constrained target, projectile grooves and projectile body with special cross-section are listed. Based on the current status of the cavity expansion model, we summarized the current cavity expansion model application direction in the field of impact dynamics, and the problems existing in the application of the cavity expansion model, as well as the key development direction in the cavity expansion model.
- penetration /
- cavity expansion theory /
- resistance model

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图 1 空腔膨胀模型发展路线图^{[1-3, 9-26]}

Figure 1. The development circuit diagram of cavity expansion model^{[1-3, 9-26]}

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图 2 塑性材料破坏情况^[29]

Figure 2. Failure of plastic material^[29]

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图 3 脆性材料破坏情况^[27]

Figure 3. Failure of brittle material^[27]

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图 4 响应区示意图

Figure 4. Schematic diagram of response area

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图 5 不同速度下的空腔径向应力分布^[11-12]

Figure 5. Radial stress distribution of cavity atdifferentvelocity^[11-12]

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图 6 不同分区下界面移动速度^[30]

Figure 6. Interface moving speed under different partitions^[30]

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图 7 剪切强度-压力数据和屈服准则^[44]

Figure 7. Shear strength-pressure data and yield criteria^[44]

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图 8 压力体积应变测试数据和状态方程^[44]

Figure 8. Pressure-volumetric strain tests data and EOS^[44]

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图 9 粉碎和破碎区域的速度^[10]

Figure 9. Speeds of the comminuted and cracked zones^[10]

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图 10 不同屈服准则下径向应力与速度的关系^[10]

Figure 10. The relationship between radial stress and velocity under different yield criteria^[10]

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图 11 空腔径向应力与空腔速度的关系^[38]

Figure 11. The relation between cavity radial pressure and cavity velocity^[38]

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图 12 空腔速度与弹塑性界面移动速度的关系^[38]

Figure 12. The relation between the velocity of the cavity and elastoplastic interface^[38]

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图 13 刚体侵彻不可压缩理想弹塑性目标的头部区域示意图^[52]

Figure 13. Sketch of the nose region of a rigid projectile penetrating an incompressible elastic-perfectly plastic target^[52]

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图 14 Rankine形弹体的稳态流场^[53]

Figure 14. Steady-state flow field for an ovoid of Rankine shaped projectile^[53]

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图 15 OR、CCE和SCE模型预测的侵彻速度平均轴向阻力应力^[52]

Figure 15. Average axial resistance stress as a function of the penetration velocity predicted by the OR, CCE and SCE models^[52]

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图 16 不同压力时数值模拟与理论值对比^[60]

Figure 16. Comparison between numerical simulation and theoretical values at different pressures^[60]

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图 17 球形空腔膨胀计算的有限元模型

Figure 17. Finite element model for spherical cavity expansion calculation

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图 18 不同膨胀速度下靶体应变分布^[63]

Figure 18. Target strain distribution at different expansion velocities^[63]

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图 19 球形空腔分别以400 m/s和600 m/s的速度膨胀时得到的空腔表面径向应力随时间变化曲线^[5]

Figure 19. Radial stress at the spherical cavity surface versus time for cavity expansion velocities of 400 m/s and 600 m/s^[5]

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图 20 空腔表面径向应力随空腔膨胀速度的变化曲线^[5]

Figure 20. Radius stress at cavity surface versus cavity expansion velocity^[5]

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图 21 长杆弹侵彻坑附近和1/2面积受内压的膨胀空腔附近的速度场^[59]

Figure 21. Velocity field near long rod projectile penetrating crater and 1/2 area expansion cavity under internal pressure^[59]

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图 22 混凝土响应分区形成过程^[65]

Figure 22. Formation process of concrete target response regions^[65]

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图 23 不同侵彻速度下的混凝土等效应变云图^[65]

Figure 23. The equivalent strain diagrams of concrete under different penetration velocities^[65]

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图 24 3种强度混凝土在不同膨胀压力下的空腔边界速度时间历程数值模拟结果^[66]

Figure 24. Simulation results of cavity wall velocities for three strengths concrete with different expansion pressures^[66]

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图 25 抗压强度分别为20、30、40、48、60、80 MPa混凝土的膨胀压力阈值^[66]

Figure 25. Threshold values of expansion pressures for concrete strength of 20, 30, 40, 48, 60, and 80 MPa^[66]

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图 26 4种可能的情况下的腔轮廓原理图^[71]

Figure 26. Schematic of cavity profiles for four-different possible scenarios, in ceramic targets backed by semi-infinite metal^[71]

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图 27 椭圆长短轴比对侵彻阻力、深度的影响规律^[72]

Figure 27. Influence of ellipse axial ratio on penetration resistance and depth^[72]

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图 28 椭圆孔受力状态示意图^[73]

Figure 28. Diagram of stress state of elliptical hole^[73]

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图 29 椭圆空腔边界受力状态^[73]

Figure 29. The state of force at the boundary of an elliptic cavity^[73]

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图 30 压剪联合作用下的空腔分区^[4]

Figure 30. Cavity partition under combined action of compression and shear^[4]

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图 31 弹体侵彻深度理论与实验对比^[4]

Figure 31. Comparison of predicted and experimental DOP data^[4]

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图 32 球形空腔压力-空腔膨胀速度^[76]

Figure 32. Spherical cavity pressure-cavity expansion velocity diagram^[76]

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图 33 有限球形空腔压力-空腔膨胀速度^[75-76]

Figure 33. Pressure-cavity expansion velocity diagram of a finite spherical cavity^[75-76]

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图 34 柱形/球形空腔膨胀模型下约束强度对空腔边界压力的影响规律^[80]

Figure 34. Effect of confinement strength on boundary pressure of cylindrical/spherical cavity expansion model^[80]

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图 35 不同约束程度下径向应力与空腔速度的关系^[68]

Figure 35. The transmutation discipline of radial stress at cavity wall^[68]

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图 36 流体侵彻示意图

Figure 36. Schematic diagram of fluid penetration

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图 37 弹体结构示意图

Figure 37. Schematic diagram of missile body structure

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图 38 弹体表面单元受力定义

Figure 38. Definition of force on surface element of projectile body

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图 39 尖卵形弹体以459 m/s的速度侵彻石灰石的实验结果与数值模拟结果对比^[91]

Figure 39. Comparison between the experimental and numerical simulation results of projectile penetrating limestone at the speed of 459 m/s^[91]

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图 40 弹体以45°碰撞角斜侵彻金属铝靶的仿真与试验结果对比图^[102]

Figure 40. Comparison of simulation and test results of oblique penetration of projectile into metal aluminum target at 45° impact angle^[102]

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图 41 自由面模型、弹靶分离模型对侵彻弹道的影响^[104]

Figure 41. Influence of free surface model and projectile separation model on penetration trajectory^[104]

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表 1 常见的几种屈服准则与状态方程

Table 1. Several common yield criteria and equations of state

屈服准则/状态方程	函数表达式	备注
Mohr-Coulomb	$\left\| {{\sigma _r} - {\sigma _\theta }} \right\| = Y$	$ {\sigma }_{r} $和$ {\sigma }_{\theta } $分别为径向应力与环向应力，Y为屈服强度
Tresca	$\left\| {{\sigma _r} - {\sigma _\theta }} \right\| = \lambda p + {\tau _0}$	$ {\tau }_{0}=\dfrac{3-\lambda }{3}Y $
Griffith	${\left( {{\sigma _r} - {\sigma _\theta }} \right)^2} = Y\left( {{\sigma _r}{\text{ + }}{\sigma _\theta }} \right)$	$ {\sigma }_{r} $和$ {\sigma }_{\theta } $分别为径向应力与环向应力，Y为屈服强度
Drucker-Prager	$ \left\{ \begin{array}{*{20}{l}} {\sigma _r} - {\sigma _\theta } = \lambda p + {\tau _0}& p \text{＜} {p_{\text{m}}} \\ {\sigma _r} - {\sigma _\theta }{\text{ = }}\left( {{\tau _0}{\text{ + }}\lambda {p_{\text{m}}}} \right)\dfrac{{{p_1} - p}}{{{p_1} - {p_{\text{m}}}}}& {p_{\text{m}}} \text{≤} p \text{≤} {p_1} \\ {\sigma _r} - {\sigma _\theta }{\text{ = }}0 & p \text{＞}{p_1} \end{array} \right. $	$ {\sigma }_{r} $和$ {\sigma }_{\theta } $分别为径向应力与环向应力， Y为屈服强度，p₁、p_m为临界压力
Hoek-Brown	${\left( {{\sigma _r} - {\sigma _\theta }} \right)^2} = Y\left( {{m_0}{\sigma _r}{\text{ + }}{\sigma _\theta }} \right)$	无量纲数m₀与材料强度及脆性程度有关
统一强度理论	$\dfrac{1}{{1{\text{ + }}b}}\left( {{\sigma _1} + b{\sigma _2}} \right) - \gamma {\sigma _3} = {\sigma _{\text{t}}}$	$ \gamma ={\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{c}} $为靶体材料拉压比，$ {\sigma }_{\mathrm{t}} $和$ {\sigma }_{\mathrm{c}} $分别为靶体材料的抗拉和抗压强度，b为中间主应力的效应参数
Voce应变硬化	$\sigma {\text{ = }}\left\{ \begin{array}{*{20}{l}} E\varepsilon & \sigma \text{≤} Y \\ Y + \displaystyle\sum\limits_{i = 1}^2 {{Q_i}} \left( {1 - \exp \left( {{C_i}\varepsilon } \right)} \right) & \sigma \text{＞}Y \end{array} \right.$	Q_i和C_i为硬化参数
线性压力-体积应变	${p_{\text{m}}} = K\varepsilon $	$ \varepsilon $为应变
三段式线性状态方程	${p_{\text{m}}} = \left\{ \begin{array}{*{20}{l}} K\varepsilon & p \text{≤} {p_{\text{c}}} \\ {p_{\text{c}}} + {K_{\text{c}}}\left( {\mu - {\mu _{\text{c}}}} \right)& {p_{\text{c}}} \text{＜}p \text{≤} {p_1} \\ {p_1} + {K_1}\left( {\mu - {\mu _{\text{p}}}} \right)& p \text{＞}{p_1} \end{array} \right.$	K、K_c、K₁为弹性区、孔隙压实区和密实区的体积模量，p_c、p₁为临界压力

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表 2 不同$R_{\rm t} $的表达式

Table 2. Different values of R_t

来源	R_t表达式	备注
Bishop^[1]	${R_{\text{t}}} = \dfrac{{{\sigma _{\text{y}}}}}{{\sqrt 3 }}\left\{ {1 + \ln \left[ {\dfrac{{\sqrt 3 E}}{{\left( {5 - 4v} \right){\sigma _{\text{y}}}}}} \right]} \right\}$	$ {\sigma }_{\mathrm{y}} $为靶体屈服强度
Rubin^[89]	${R_{\text{t}}} = \ln \left( {4{\varsigma ^2}} \right){Y_{\text{p}}} - {f_{\text{t}}} - \left( {2/3 + \ln 4} \right){\sigma _{\text{y}}}$	Y_p为弹体屈服强度，$ {\sigma }_{\mathrm{y}} $为靶体屈服强度， f_t为与靶体材料相关的常数
Godwin^[85]	${R_{\text{t}}} = \left( {{\text{2 + }}2\sqrt {1{\text{ - }}{Y_{\text{p}}}/{Y_{\text{t}}}} } \right){\sigma _{\text{y}}}$	Y_p为弹体屈服强度，$ {\sigma }_{\mathrm{y}} $为靶体屈服强度
A-W模型^[84]	$ {R_{\text{t}}} = \dfrac{7}{3}\ln \left( {{\alpha _{\text{k}}}} \right){\sigma _{\text{t}}} $	$ {\alpha }_{k} $为与靶体材料相关的常数
S-W-Z-S模型^[90]	$ {R_{\text{t}}} = \dfrac{2}{3}{\sigma _{\text{y}}}\left( {1 + \ln \dfrac{{2E}}{{3{\sigma _{\text{y}}}}}} \right) + \dfrac{2}{{27}}{{\rm{\pi }}^2}E $	$ {\sigma }_{\mathrm{y}} $为靶体屈服强度，E为弹性模量
L-W模型^[88]	$ \begin{aligned} {R_{\text{t}}} =\,& S + C{\rho _{\text{t}}}{\left( {{U_{{\text{F0}}}}\exp \left( { - {{\left( {\dfrac{{u - {U_{{\text{F0}}}}}}{{n{U_{{\text{F0}}}}}}} \right)}^2}} \right)} \right)^2} -\\ & \dfrac{1}{2}{\rho _{\text{t}}}{\left( {u - {U_{{\text{F0}}}}\exp \left( { - {{\left( {\dfrac{{u - {U_{{\text{F0}}}}}}{{n{U_{{\text{F0}}}}}}} \right)}^2}} \right)} \right)^2} \end{aligned} $	$ {U_{{\text{F0}}}} = \sqrt {{Y_{\text{H}}}/{\rho _{\text{t}}}} $，S为靶体静态阻力，Y_H为材料动态屈服强度

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表 3 侵彻深度理论预测公式

Table 3. Theory prediction formula of penetration depth

来源	侵彻深度预测公式
Frew等^[101]	$ \begin{aligned} \dfrac{P}{{\left( {L + {{2a} / 3}} \right)}} =\,& \dfrac{1}{C}\left( {\dfrac{{{\rho _{\text{p}}}}}{{{\rho _0}}}} \right)\left\{ {\ln \left[ {1 + \dfrac{{2B}}{{3A}}\left( {\sqrt {\dfrac{{{\rho _0}}}{Y}v} } \right) + \dfrac{C}{{2A}}{{\left( {\sqrt {\dfrac{{{\rho _0}}}{Y}v} } \right)}^2}} \right]} \right.+ \\ &\left. { \dfrac{{4B}}{{\sqrt {18AC - 4{B^2}} }}\left[ {{\rm{arctan}}\dfrac{{2B}}{{\sqrt {18AC - 4{B^2}} }} - {\rm{arctan}}\left[ {\dfrac{{3C\sqrt {{{{\rho _0}} / Y}} v + 2B}}{{\sqrt {18AC - 4{B^2}} }}} \right]} \right]} \right\} \end{aligned} $
Warren^[91]	$ P = \dfrac{m}{{2{\text{π }}{a^2}{\rho _0}N}}\ln \left( {1 + \dfrac{{N{\rho _0}{v^2}}}{R}} \right) + 4a $
Wen^[92]	$P = \left\{ \begin{array}{*{20}{l}} \left( {\sqrt {4\psi - 1} - 2\psi \cos \psi } \right)a & P \text{≤} {L_{\text{N}}} \\ \dfrac{P}{{L + 8{\psi ^3}\eta a}} = \left( {\dfrac{{{\rho _{\text{p}}}}}{{{\rho _{\text{t}}}}}} \right)\dfrac{{{\rho _{\text{t}}}{v^2}}}{{{\sigma _{\text{e}}}}}\dfrac{1}{{2\left[ {1 + \beta \sqrt {\dfrac{{{\rho _{\text{t}}}}}{{{\sigma _{\text{e}}}}}} v} \right]}} + \dfrac{{\left( {\sqrt {4\psi - 1} - 8{\psi ^3}\eta } \right)a}}{{L + 8{\psi ^3}\eta a}}& P\text{＞} {L_{\text{N}}} \end{array} \right. $
Teland等^[96]	$ P = \dfrac{2}{{\text{π }}}\dfrac{M}{N}\ln \left[ {\dfrac{{\left[ {1 - \dfrac{{\text{π }}}{4}\dfrac{{{R^2}}}{M}{X_1}} \right]\dfrac{{v_0^2}}{S} + \dfrac{M}{N} - \dfrac{{\text{π }}}{4}\dfrac{{{R^2}}}{M}{X_1}}}{{\dfrac{M}{N} + \dfrac{{\text{π }}}{4}{X_1}}}} \right]{\text{ + }}{X_1} $
Kong等^[17]	$ \begin{aligned} \dfrac{P}{{{l_{{\text{eff}}}}}} =\,& \dfrac{{{\rho _{\text{p}}}}}{{2{N_2}C{\rho _0}}}\ln \left( {\dfrac{{A{f_{\text{c}}}{N_0} + {N_1}B\sqrt {{\rho _0}{f_{\text{c}}}} v + {N_2}C{\rho _0}{v^2}}}{{A{f_c}{N_0}}}} \right) + \dfrac{{{\rho _{\text{p}}}{N_1}B\sqrt {{\rho _0}{f_{\text{c}}}} }}{{{N_2}C{\rho _0}\sqrt {{\rho _0}{f_{\text{c}}}\left( {4AC{N_0}{N_2} - N_1^2{B^2}} \right)} }}\times \\ &\left[ {\arctan \left( {\dfrac{{{N_1}B\sqrt {{\rho _0}{f_{\text{c}}}} }}{{\sqrt {{\rho _0}{f_{\text{c}}}\left( {4AC{N_0}{N_2} - N_1^2{B^2}} \right)} }}} \right) - \arctan \left( {\dfrac{{{N_1}B\sqrt {{\rho _0}{f_{\text{c}}}} + 2{N_2}C{\rho _0}v}}{{\sqrt {{\rho _0}{f_{\text{c}}}\left( {4AC{N_0}{N_2} - N_1^2{B^2}} \right)} }}} \right)} \right] + \dfrac{{kd}}{{{l_{{\text{eff}}}}}} \end{aligned} $
注：表中参数与上文一致，A、B、C、N₁、N₂、k、d、M、l_eff等为与材料或弹体结构相关的常数。

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