A new experimental technique of dynamic compression-shear combined loading based on metamaterials
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摘要: 材料或结构在动态压剪复合加载条件下的力学性能对于其工程应用具有重要影响。然而，现有的动态复合加载实验技术存在压缩波和剪切波难以同步施加到试件、实验设备昂贵等问题。本文中利用压扭超材料进行应力波转化，在一维分离式霍普金森压杆（split Hopkinson pressure bar，SHPB）上实现动态压剪同步复合加载。该实验技术具有荷载精准同步、剪压比可控、简单便捷、低成本等优点。针对当压扭超材料转化出来的扭转波幅值较大，透射杆惯性约束不足情况下出现的扭转信号三角波的问题进行详细讨论，并提出相应的解决方案。选用屈服应力各不相同的金属钛、304不锈钢和316L不锈钢等3种材料进行了实验测试，证实了动态压剪同步复合加载技术的有效性。借助有限元模型，深入分析压扭超材料的几何参数对其压扭系数及承载能力的影响，并结合实验结果讨论了该实验技术的适用范围，预测动态压剪同步复合加载技术能测试的材料强度可达约1 GPa，施加给试件的剪压比可达1.18。Abstract: The mechanical properties of materials or structures under dynamic compression-shear combined loading conditions significantly influence their engineering applications. However, existing experimental methodologies for dynamic combined loading confront challenges, such as the difficulty in synchronously applying compression and shear waves to test specimens, in addition to the high cost of experimental equipment. This study introduces a novel experimental technique that utilizes compression-torsion coupling metamaterials for the conversion of stress waves, enabling synchronous dynamic compression-shear combined loading on a one-dimensional Hopkinson pressure bar. This technique offers several advantages, including precise load synchronization, a controllable shear-compression ratio, simplicity, convenience, and low cost. A detailed discussion is presented on the issue of triangular torsion signals that arise when the amplitude of torsional waves converted from compression-torsion metamaterials reaches considerable levels, coupled with insufficient inertial confinement in the transmission bar of the split Hopkinson pressure bar system. Additionally, corresponding solutions to this issue are proposed. Experimental tests were conducted on three materials with distinct yield stresses: titanium, 304 stainless steel, and 316L stainless steel, validating the effectiveness of this experimental technique. Furthermore, leveraging finite element models, an in-depth analysis was conducted on the influence of the geometric parameters of the compression-torsion coupling metamaterials on their compression-torsion coefficients and load-bearing capacities. By integrating these findings with experimental results, the applicability of this experimental technique was discussed, predicting its capability to test materials with strengths up to approximately 1 GPa and to apply shear-compression ratios up to 1.18 to specimens, providing a reference for its application in a broader range of fields. This innovative integration of metamaterials with traditional experimental equipment opens up new avenues for realizing more complex dynamic loading experiments.
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预应力结构是一种常见结构, 一个构件或结构的工作状态总可以看作初始应力在位形上产生的附加变形运动。研究初应力对构件和结构的动态行为有普遍的科学意义[1]。预应力对结构破坏的影响研究较多, 比如预应力对构件振动频率的影响[2], 预应力对层裂的影响[3]。近年来, 结合武器的侵彻和防护工程的需要, 开展了大量混凝土、岩石或者粉末冶金材料在有无预应力或者外界限制下的抗冲击特性研究[4-9], 这些研究对于提高工程设计水平具有较高的价值, 同时也有助于理解材料的损伤演化进程。此类研究一般对样品施加被动或主动围压, 采用霍普金森杆撞击带套筒约束的样品, 结合配套的实验数据处理方法, 获得相应条件下的应力应变曲线, 从这些应力应变曲线中获得相应状态的动态屈服应力等参数, 并希望将这些曲线或者参数引入计算程序进行实际工程损伤破坏过程计算[10-11]。但是, 这些研究没有将应力状态本身引起的动态屈服应力增加项很好地分析出来, 有可能高估围压下的动态本构。
在张世文等[12]的预应力层裂实验中发现, 随着径向压缩预应力的增加, 弹性前驱波幅值相应增加, 该实验可以比较直观地给出预应力对屈服应力的“增加”现象, 这有些类似于围压对材料动态屈服影响实验。这一奇特现象用应变率效应难以获得圆满解释, 因为这些实验均是在近似同一个碰撞速度下开展的, 我们施加的径向应力不及冲击压力的5%, 用静水压力效应同样难以解释。本文中首先简要介绍实验结果, 结合实验特点, 探讨对样品施加静态预应力计算方法, 并采用该方法来复现这一实验现象。
1. 预应力碰撞实验
采用如图 1~2所示热装配方法完成对平面样品施加预应变, 箍紧环内径小于样品半径, 过盈配合量在0.1~0.5 mm之间, 详细介绍见文献[3]。预应变可以通过样品特定线段在装配前后变化计算出来。样品内的径向应力可以通过应变由下式计算:
σ0,r=E1−με0,r (1) 当预应变不再变化时, 按照传统的轻气炮实验将VISAR探头安装好。本文中飞片和预应力样品均为LY12铝合金, 飞片厚2 mm、样品厚8 mm。
2. 实验结果
图 3为不同预应变下采用VISAR技术测得的实验结果。具体实验结果见表 1, 表中v为飞片速度,
0, r为径向压缩应变, σ0, r为径向压缩应力,
为弹性前驱波速度。当飞片均以500 m/s的速度撞击样品时, 预应变为0(无初始预应变)、964.5×10-6和1 886.0×10-6时, 弹性前驱波幅值分别为87.56、95.24、121.03 m/s。
图 3 不同预应变下样品后自由面速度曲线Figure 3. Free surface velocity histories for different pre-strain samples表 1 不同径向预应力下实验结果Table 1. Experimental results of samples under different pre-strainsε0, r/10-6 σ0, r/MPa v/(m·s-1) ue*/(m·s-1) 0 0 497.82 87.56 964.5 99.2 497.06 95.24 1 886.0 194.0 499.60 121.03 3. 数值模拟
3.1 预应力加载技术
借鉴霍普金森压杆(SHPB)测试原理, 可以首先对样品施加径向的预应力, 让应力波在样品中来回反射多次, 达到应力均匀化, 然后约束样品外边界, 此时样品内就能获得稳定的静态预应力, 然后再用飞片撞击样品, 进行下一步计算工作。
此方法面临2个问题:样品尺寸问题和如何施加预应力。如果按照样品实际直径进行建模计算, 应力波在样品径向方向达到均匀化所需时间较长, 前期计算时间较长。为解决这个问题, 可将模型径向尺寸减小。根据一维应变碰撞时径向应变为0这一结论, 董杰等[13]采用径向方向只有1个单元, 通过固定节点非速度向的位移和转动, 模拟一维应变单元, 成功模拟了平板撞击产生的层裂现象。类似地, 当对样品施加了静态径向预应力, 在飞片撞击过程中, 样品径向同样来不及膨胀, 即样品外边界(此时已有固定的位移, 该位移可以通过弹性力学解求出)将不会有增加的位移出现, 数值模拟也证实了这一结论, 因此可以对样品外边界施加此恒定位移, 飞片仍采用零位移约束。根据对称性, 1/4三维建模, 计算模型中样品和飞片直径为0.4 mm、飞片厚2 mm、样品厚8 mm, 直径方向4等分、样品厚度方向320等分、飞片40等分, 共计4 320个单元, 如图 4所示, 其中单元1为中心点位置, 单元2为外边界位置。
图 4 计算模型(右图为样品截面上网格划分)Figure 4. Calculation model (mesh of sample section on the right)在施加位移约束时, 位移约束时间曲线对计算结果影响明显, 不能直接给样品施加上升沿较窄的脉冲位移。施加位移应满足从0上升到指定位移的上升沿时间较长, 然后固定该位移, 图 5为样品外边界施加上升沿为1×10-11、10-4 s的两种位移历史曲线。图 6为径向应力计算结果, 从图中可以看出, 当上升沿较短时, 沿着样品径向方向应力并不均匀(图 6中曲线a1和a2), 这是因为对样品外边界施加突加载荷时, 该部位存在相应的应力。但是, 在远离外边界的样品中心部位, 并没有相应的应力, 如同泰勒杆撞击一样, 在撞击端具有一定的变形, 但远离撞击的另一端既没有应变也不存在应力。样品中沿径向分布的应力不均匀即受到加载上升沿的影响, 又受到计算模型尺寸的影响。
图 5 样品外边界施加不同上升沿的位移曲线Figure 5. Two kinds of displace-time curves loading on samples' outer boundary
图 6 样品中心和外边界部位的径向预应力Figure 6. Samples' radical pre-stress (center-element 1, and outer boundary-element 2)不过, 当位移加载较为缓慢时, 样品中心点(单元1)和边界点(单元2)的径向应力相同(图 6中曲线b1和b2), 与弹性力学理论解完全一样。施加的位移可下式确定:
dr=rεr=rσr/(E/(1−μ)) (2) 式中:dr为径向位移, r为计算模型样品半径,
r为径向应力,
r为径向应变, E为弹性模量,
为泊松比。
图 6表明, 对样品外边界施加上升沿10-4 s的位移历史曲线能够对样品施加稳定的径向预应力。
3.2 材料本构及断裂判据
计算中飞片和样品均为LY12铝合金, 采用Grüneisen状态方程和含断裂判据的Johnson-Cook本构关系, Grüneisen状态方程分为2个阶段:
当材料处于压缩状态, 有:
p=ρ0c2μ[1+(1−γ02)μ−a2μ2][1−(s1−1)μ−s2μ2μ+1−s3μ3(μ+1)2]2+(γ0+aμ)E (3) 当材料处于拉伸状态, 有:
p=ρ0c2μ+(γ0+aμ)E (4) 式中:c为vs-vp曲线截距(vs为冲击波速度, vp为粒子速度), 近似为体积声速, s1、s2、s3为vs-vp曲线相关系数,
0为Grüneisen系数, a为
0一阶体积修正系数,
= ρ/ρ0-1为比容, E为内能。LY12铝合金计算参数取值:c=5.328 km/s, s1=1.338,
0=2.0。其余均取为0。
考虑应变率效应的JC本构关系及参数定义见文献[14]。与JC本构关系配套的用于拉伸断裂计算的判据有3种模式[14], 一是压力限制模型, 当压力p超过设定值-pmin(该值一般为负值, 相当于拉伸应力)时, 压力p将降低到-pmin而不是0, 明显不适合层裂分析。第2种为最大拉应力失效模式, 当单元拉伸应力达到设定值后, 单元即失效, 拉应力降为0。第3种为应变失效模型, 当单元的应变达到某一特定值后, 单元删除, 该模型在单向拉伸时效果比较明显, 遗憾的是, 该模型没有将拉伸阶段和压缩阶段分别处理, 当材料处于压缩状态时, 单元的应变已经达到失效应变而删除, 无法进行压力的传输, 影响计算结果。本文中选择最大拉应力判据进行计算。
3.3 典型样品后自由面速度曲线计算结果
图 7为不同预应变样品后自由面速度曲线的数值模拟结果, 在计算中均采用同一本构模型和同一最大拉应力判据进行计算。由计算结果可知, 随着径向预压应力的增加, 弹性前驱波速度也相应地增加, 这与实验结果趋势一致, 但也说明了对于LY12铝合金, 通常采用的Johnson-Cook模型描述的屈服应力由于应变率相对较低, 低于轻气炮实验获得的弹性极限。关于预应力对层裂强度的影响已在另文中讨论[3, 15]。
图 7 不同预应变样品后自由面速度曲线计算结果Figure 7. Free surface velocity histories for different pre-strain samples4. 应力状态对弹性前驱波影响分析
4.1 一维应变的弹性前驱波速度
对于一维平面应变, 根据单一曲线假设, Mises条件可以写为:
σx−σr=Y0 (5) 式中:σx、σr分别为撞击方向和径向方向应力, Y0为一维应力状态下的屈服应力。σx、σr满足以下关系:
σr=ν1−νσx (6) 代入式(5)可以确定一维应变条件下屈服极限YH为
YH=1−ν1−2νY0 (7) 式中:ν为泊松比, YH称为侧限屈服极限或Hugoniot弹性极限。显然它高于单向应力下的屈服极限Y0, 例如ν = 1/3时, YH=2Y0。当材料的屈服极限一定时, 体现在自由面的弹性前驱波的速度为
ue=2YH/ρ0cL (8) 式中:cL为纵波声速。
4.2 预应力下的弹性前驱波速度
对于本文的实验来说, 当给径向方向施加初始压应力后, 根据单一曲线假设, Mises条件仍为公式(5), 但是, 此时的σr已经叠加了初始预应力σ0, r,
σr=ν1−νσx+σ0,r (9) 整理得
σx=1−ν1−2ν(Y0+σ0,r) (10) 在初始径向预压应力下, 对轴向应力σx而言的屈服极限YH*为:
Y∗H=YH+1−ν1−2νσ0,r (11) 预应力下的弹性前驱波的速度幅值ue*为:
u∗e=ue+1ρ0cL2(1−ν)(1−2ν)σ0 (12) 预应力越大, 后自由面的弹性前驱波的速度越大。
5. 讨论
弹性前驱波作为一个衡量材料屈服极限大小标志的参数非常重要, 一维应变下的弹性前驱波具有明确的物理意义, 通过它可以计算出一维应变的屈服应力或称之为弹性Hugoniot极限, 它也可通过泊松比转化为一维应力下的屈服应力。非一维应变下的弹性前驱波不但体现了材料本身的属性, 也包含了复杂的应力状态。从实验和数值模拟以及理论推导可以看出, 即使采用同一个屈服应力值, 随着预应力的增加, 弹性前驱波幅值也相应提高。这给我们一个启示, 即在霍普金森围压实验中获得的本构是否同样包含应力状态的影响。李庆明等[16-17]也认为, 霍普金森杆实验中撞击速度对应力状态的影响明显, 撞击速度越高, 试件越偏离一维应力状态。的确, 混凝土在围压下材料屈服应力会增加, 但这一增加量必须扣除复杂应力状态的影响。本文中经过对样品施加预应力的探索, 成功实现了对样品施加径向预应力, 并计算了预应力对样品后自由面速度弹性前驱波的影响, 还从理论上给出了解释。这种影响根本没有改变铝合金材料在该应变率下屈服应力, 是一种“虚假影响”。
弹性前驱波幅值的准确测量同样是一个需要解决的问题, 在实验中也发现了有些弹性前驱波拐点不明显, 在一些速度测试曲线中, 或者弹性前驱波不明显, 且重复性一般并不理想, 有时甚至测不到弹性前驱波, 这既与靶板的安装有关(比如靶板安装倾斜), 也与测试系统本身有关, 文献[12]对此进行了解释。如何从这些看似毫无规律的结果中提取体现物理本征属性, 需要通过更精细的实验设计和测试技术来完成。另外, 本文中针对研究对象的特点进行的模型提取以及预应力施加技术, 对于研究结构初应力状态对它的动力学响应, 提供了新的分析思路和技术可行性。
感谢范诚对撰写本文英文摘要的帮助。 -
图 2 压扭超材料的几何参数约束
Figure 2. Geometric parameter constraints of the compression-torsion metamaterials
图 3 压扭超材料的准静态有限元模型示例
Figure 3. Example of the quasi-static finite element model of the compression-torsion metamaterial
图 4 准静态有限元计算结果分析示例
Figure 4. Examples of quasi-static finite element calculation results analysis
图 5 承载能力和压扭系数与斜杆数量关系曲线
Figure 5. Relation of bearing capacity and compression-torsion coefficient with respect to the number of inclined rods
图 6 承载能力和压扭系数与斜杆倾斜角度关系
Figure 6. Relation of bearing capacity and compression-torsion coefficient with the tilt angle of inclined rods
图 7 承载能力和压扭系数与斜杆直径关系
Figure 7. Relation of bearing capacity and compression-torsion coefficient with respect to the diameter of inclined rods
图 8 承载能力和压扭系数与斜杆长度关系
Figure 8. Relation of bearing capacity and compression-torsion coefficient with respect to the length of inclined rods
图 9 不同斜杆倾斜角度下压扭系数与斜杆长度关系（d=2 mm）
Figure 9. Relationship curves between the compression-torsion coefficient and the length of inclined rods under different tilt angles of inclined rods (d=2 mm)
图 10 压扭超材料和SHPB有限元模型示例
Figure 10. Example of the finite element model of compression-torsion metamaterial and SHPB
图 11 Ti6Al4V钛合金的准静态拉伸实验曲线及其拟合曲线
Figure 11. Quasi-static tensile experiment curve and its fitting curve of Ti6Al4V titanium alloy
图 12 SHPB系统的有限元模拟与实验结果对比 （D=20 mm）
Figure 12. Comparison between FEM and experimental results in SHPB system (D=20 mm)
图 13 不同斜杆直径和透射杆直径的SHPB系统有限元分析结果
Figure 13. FEM results of SHPB systems with different diameters of inclined rods and different diameters of transmission bars
图 14 SHPB系统的有限元模拟与实验结果对比（D=50 mm）
Figure 14. Comparison between FEM and experimental results in SHPB system (D=50 mm)
图 16 动态压缩-扭转实验下弹性应变范围内（压缩应变0.002 8）薄壁圆筒钛试件的DIC压缩应变云图和剪切应变云图
Figure 16. Cloud maps of compression strain and shear strain obtained from DIC for a thin-walled cylinder titanium specimen in elastic strain range (compression strain 0.002 8) under dynamic compression-torsion experiments
图 17 动态压缩-扭转实验下塑性应变范围内（压缩应变0.004 9）薄壁圆筒钛试件的DIC压缩应变和剪切应变云图
Figure 17. Cloud maps of compression strain and shear strain obtained from DIC for a thin-walled cylinder titanium specimen in plastic strain range (compression strain 0.004 9) under dynamic compression-torsion experiments
图 18 钛动态压缩-扭转实验的应变率-应变曲线
Figure 18. Strain rate-strain curves of titanium under dynamic compression-torsion experiments
图 19 钛动态压缩-扭转实验的应力-应变曲线
Figure 19. Stress-strain curves of titanium under dynamic compression-torsion experiments
图 20 准静态条件下304不锈钢的压缩应力-应变曲线
Figure 20. Compressive stress-strain curves of 304 stainless steel under quasi-static conditions
图 21 304不锈钢动态压缩-扭转实验的应力-应变曲线
Figure 21. Stress-strain curves of 304 stainless steel under dynamic compression-torsion experiments
图 22 316L不锈钢动态压缩-扭转实验的应力-应变曲线
Figure 22. Stress-strain curves of 316L stainless steel under dynamic compression-torsion experiments
图 23 压扭超材料能提供的扭矩和轴向载荷范围
Figure 23. Range of torque and axial loads that compression-torsion metamaterials can provide
图 24 基于压扭超材料的动态复合加载技术应用范围
Figure 24. Application range of the dynamic combined loading technique based on the compression-torsion metamaterials
表 1 有限元模型采用的材料参数
Table 1. Material parameters used in the finite element model
材料 密度/
(g∙cm−3)弹性模量/
GPa屈服应力/
MPa泊松比 7075铝合金 2.85 71 0.33 Ti6Al4V钛合金 4.40 120 910 0.33 
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表 2 钛的动态压缩-扭转实验结果
Table 2. Dynamic compression-torsion experiment results of titanium
试件编号 θ/(°) σs/MPa τs/MPa σy/MPa 1 70 453.58 117.01 496.80 2 70 429.49 126.49 482.15 3 60 374.31 174.72 481.34 4 60 382.43 180.40 493.84 5 50 346.23 198.34 487.74
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表 3 304不锈钢的动态压缩-扭转实验结果
Table 3. Dynamic compression-torsion experiment results of 304 stainless steel
试件编号 θ/(°) σs/MPa τs/MPa σy/MPa 1 70 350.08 125.87 412.42 2 60 305.43 160.12 412.56
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表 4 316L不锈钢的动态压缩-扭转实验结果
Table 4. Dynamic compression-torsion experiment results of 316L stainless steel
试件编号 θ/(°) σs/MPa τs/MPa σy/MPa 1 70 471.19 172.86 558.27 2 60 440.02 221.21 583.45 3 50 402.85 237.92 576.28
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