Cumulative damage effect and stability analysis of the rock slope with a locked segment under cyclic blasting
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摘要: 岩石中存在许多微裂纹和微孔洞，这些微裂纹和微孔洞在动荷载作用下会萌生、扩展和聚并，导致岩石失稳和破坏。在进行爆破开挖时，预留岩体会受到循环爆破产生的动荷载影响，产生累积损伤，从而导致岩体强度降低，甚至破坏。为了模拟这一物理过程，将现有的能够较好地描述岩石动态损伤的岩石动态损伤本构模型通过二次开发嵌入到FLAC中，用于分析锁固型岩质边坡在循环爆破作用下的损伤效应及稳定性。结果表明：考虑岩质边坡累积损伤效应后，随着循环爆破次数的增加，边坡稳定性逐渐降低。对于锁固型岩质边坡，锁固段的破坏首先发生在两端，然后向中间扩散，岩体在其中呈现递进破坏模式。由于考虑了岩质边坡的累积损伤，每次爆破后边坡的安全系数都会减小。当不考虑累积损伤时，边坡的安全系数基本不变。另外，锁固段在软弱夹层中的位置影响边坡的破坏模式和稳定性。因此，在进行类似工程活动时，应考虑岩体的累积损伤效应，避免工程事故的发生。Abstract: There are many microcracks and micropores in the rock, which will initiate, propagate, and coalescence under dynamic loading, leading to rock instability and failure. When blasting excavation is carried out, the retained rock mass will be subjected to the dynamic loading generated by cyclic blasting, resulting in cumulative damage, which will lead to the reduction of the rock mass strength, and even failure. In order to simulate this physical process, the existing rock dynamic damage constitutive model, which could perfectly describe the rock dynamic damage induced by blasting, was embedded into FLAC through secondary development to analyze the cumulative damage of rock mass under cyclic blasting. And then it was adopted to simulate the damage effect and stability of the rock slope with the locked segment under cyclic blasting. The stability of the slope under cyclic blasting was determined by the displacement criterion method, and the safety factor of the slope after each blasting was obtained by the strength reduction method. Finally, the relationship between the failure mode and stability of the slope and the location of the locked segment was discussed by analyzing the damage, displacement field, and safety factor of the numerical models for different locations of the locked segment in the soft interlayer. The results show that the slope stability gradually decreases with increasing the number of cyclic blasting after considering the cumulative damage effect of the rock slope. For the rock slope with the locked segment, the damage of the locked segment firstly occurs at both ends, and then propagates to the middle, in which the rock mass shows a progressive failure mode. Because the cumulative damage of the rock slope is considered, the safety factor of the slope will decrease after each blasting. When the cumulative damage is not considered, the safety factor of the slope is basically unchanged. The failure mode of the rock slope with a locked segment under cyclic blasting is the combination of dynamic tensile failure and shear failure caused by rock mass slip. The location of the locked segment in the weak interlayer affects the failure mode and stability of the slope. Therefore, when carrying out similar engineering activities, the cumulative damage effect of rock mass should be considered to avoid engineering accidents.
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NiTi 合金具有形状记忆效应、超弹性效应、高阻尼（吸能、减震）能力和生物相容性等优异性能，在工业领域具有广泛应用[1]。近年来，随着NiTi合金应用范围的不断拓展，其在部分应用环境中可能遭受瞬态（微秒、纳秒量级）动态载荷加载，如NiTi合金用于飞机的吸能器件、航天器机构收缩重复动作部件时，可能遭受平均速度为数千米每秒高速空间碎片的撞击[2-3]；用于盔甲等防护系统[4-6]时，可能受到子弹等的撞击。虽然针对准静态加载下NiTi合金的力学性能已有较多研究，但是由于惯性效应和应变率效应，材料在高应变率下的力学特性、变形机理与准静态下存在显著差异[7-9]。因此，开展高应变率下NiTi合金的力学特性和动态响应规律研究具有迫切需求和重要价值。
NiTi合金的形状记忆效应、伪弹性特性由其在一定应力、温度条件下发生奥氏体（austenite，B2）相与马氏体（martensite，B19ˊ）相之间的转变引起。NiTi合金奥氏体-马氏体相变对应力状态、温度非常敏感[10]。在中、低应变率加载下，随着加载应变率升高，由于变形等耗散生热使马氏体相变阻力增大，导致应力诱导马氏体相变压力随应变率的升高而增大[11-12]。当初始温度达到临界相转变温度时，变形物理机制发生转变，NiTi 合金在应力加载后不再发生应力诱导奥氏体-马氏体相变，而是发生母相奥氏体相的弹塑性变形[13-14]。准静态实验研究表明，马氏体相NiTi合金与奥氏体相NiTi合金的屈服强度差异较大[15]，而高应变率加载接近于绝热过程，由于绝热温升以及材料变形的非瞬态响应，必将影响材料的相变驱动力、相变条件及塑性屈服条件等基本物性[9, 16-17]。鉴于NiTi合金相态及其相应力学性质对温度具有较强的敏感性，解耦获得不同初始相态在高应变率下的屈服强度等对应用设计至关重要[18-19]。为此，本文中，通过控制样品温度实现材料初始相态改变，获得高应变率下不同初始相变温度NiTi合金屈服应力等基本物理特性和力学响应规律，以期为完善NiTi合金物理模型和NiTi合金关键部件在极端环境下的安全设计提供参考。
1. 实验方法与技术
1.1 样品静态性能表征
实验选择了4种不同初始相变温度的近等原子比NiTi合金。首先表征获得了材料在常压下的基本物性参数，其中相变温度采用差热分析测量，组分采用化学分析和气体分析，声速测量采用5 MHz石英传感器的超声波测量仪，相变温度、组分等参数见表1。表中：TMs为马氏体相变起始温度，TMf为马氏体相变结束温度，TAs为奥氏体相变起始温度，TAf为奥氏体相变结束温度，cL0为常压下的纵波声速，cs为常压下的横波声速。所选材料相变温度均处于室温附近，根据相变温度，NiTi-1在室温（293 K）下完全处于马氏体，NiTi-2、NiTi-3和NiTi-4在室温下完全处于奥氏体。
表 1 常压下NiTi合金的物性参数Table 1. Physical parameters of NiTi alloys at normal conditions编号 密度/(g·cm–3) 组分 TMs/K TMf/K TAs/K TAf/K cL0/(km·s–1) cs/(km·s–1) NiTi-1 6.40 Ni55Ti45 342.0 295.0 349.0 391.0 5.376 1.731 NiTi-2 6.40 Ni56Ti44 244.0 227.0 259.0 281.0 NiTi-3 6.40 Ni56Ti44 227.0 196.0 243.0 262.0 NiTi-4 6.42 Ni52Ti46-48 258.4 253.3 261.6 272.3 5.434 1.775 采用材料试验机分别开展了准静态压缩和拉伸实验，测量了NiTi-1、NiTi-2、NiTi-3和NiTi-4在10−3 s−1应变率下室温压缩和拉伸的应力-应变曲线，实验结果如图1所示。拉伸实验采用“狗骨”试样，测试段圆柱直径为5 mm、长30 mm；压缩实验采用直径为10 mm、高10 mm的圆柱样品。应力-应变曲线中获得的典型参数如表2所示。NiTi-1样品在压缩和拉伸过程中的应力-应变曲线均出现2个拐点，意味着在此过程中材料模量发生了显著变化。结合材料初始相态和相近实验条件下的结果[20]，NiTi-1样品中模量变化分别由初始马氏体相中晶体重定向和马氏体相塑性变形引起。NiTi合金马氏体结构由具有不同晶向的畴域组成，而在每个畴域内还存在孪晶亚结构，通常第1个拐点（σPH处，表示马氏体相畴域结构内晶体重定向相变应力）是由于畴域及孪晶边界的移动引起畴域结构内晶体重定向造成的。由于位错的运动与增殖，就发生不可逆的非弹性塑性变形（σP处，表示非弹性塑性变形应力）。尽管NiTi-2、NiTi-3和NiTi-4合金在常压下的初始相变温度不同，但在室温下均处于奥氏体相，从图1的压缩和拉伸应力-应变曲线中也可以看出，其具有相同的响应规律，即先经过奥氏体相弹性阶段，当应力达到奥氏体-马氏体相变应力（σA-M处，表示奥氏体-马氏体相变应力）后，在应力诱发下材料开始进入奥氏体向马氏体相变阶段，到达σP处时达到马氏体相弹性极限，然后开始发生位错屈服，材料进入塑性阶段。
图 1 NiTi-1、NiTi-2、NiTi-3和NiTi-4在10−3 s−1应变率下室温压缩和拉伸的应力-应变曲线Figure 1. Stress-strain curves of NiTi-1, NiTi-2, NiTi-3, and NiTi-4 at room temperature and under compression and tension at the strain rate of 10−3 s−1表 2 准静态加载下应力-应变曲线拐点应力Table 2. Stress value of inflection point on stress-strain curve under quasi-static loading编号 拉伸应力/MPa 压缩应力/MPa σPH或σA-M σP σPH或σA-M σP NiTi-1 230 657 266 1 528 NiTi-2 415 445 1 377 NiTi-3 400 534 1 198 NiTi-4 405 615 1 280 1.2 高应变率加载实验方法及条件
高应变率加载实验在小型脉冲功率装置CQ-4上完成。该装置在充电75～85 kV的短路放电情况下，电流峰值可以达到3～4 MA，电流上升前沿在470～600 ns，可用于应变率104～107 s−1、压力百吉帕范围内的材料动力学行为研究[21-22]。高应变率实验采用磁驱动准等熵加载和磁驱动飞片冲击加载2种技术方法覆盖不同应变率范围。
磁驱动冲击加载和准等熵加载实验原理分别如图2(a)和(b)所示。装置短路放电产生的强电流从2个平行的正负电极板的内表面流过（趋肤效应），极板上流过的电流与另一极板上电流产生的磁场相互作用，在极板内表面产生大小与电流密度的平方成正比的磁压力。随着放电电流的逐渐增大，在电极板内表面形成一个压力平滑上升的压缩波向样品方向传播。磁驱动加载电极板上磁压力的均匀性主要由电流和磁场在极板加载面的均匀性决定。采用优化设计后的电极负载构型[22-23]，能实现在样品区极板加载面磁压力的不均匀性小于1%，所以磁驱动加载在一发实验中可实现多个样品同时测量，且各样品对应电极板加载面位置的加载磁压力基本相同。典型实验负载区照片如图3所示，实验中利用陶瓷加热片实现样品初始升温，升控温装置采用比例积分微分温度控制系统，功率为50 W，可实现控温精度在0.5 K以内。采用热电偶测量样品温度，选用的铠装pt100热电偶的测温精度为0.1 K。采用激光干涉速度仪测量加载后样品响应的速度波剖面。基于双光源外差测速技术将干涉频率向上变换使干涉条纹细分，以提高短时低速测量的时间分辨率和速度测量精度，测速精度优于1%[24-25]。实验条件如表3所示，表中S1表示样品1，S2表示样品2。根据初始样品表征结果，NiTi-1和NiTi-4是其中典型的2种初始不同相态材料，因此，选择这2种材料开展高应变率下的冲击和准等熵加载实验。
表 3 高应变率实验条件Table 3. Conditions of high strain rate experiments实验编号 实验加载方式 样品编号 样品材料 样品尺寸/mm 样品初始温度/K Shot-518 准等熵压缩 S1 NiTi-4 ∅12×2.010 300 S2 NiTi-4 ∅12×2.305 300 Shot-522 准等熵压缩 S1 NiTi-4 ∅12×2.012 300 S2 NiTi-4 ∅12×2.295 300 Shot-1036 准等熵压缩 S1 NiTi-1 ∅8×1.500 300 S2 NiTi-1 ∅8×1.802 300 Shot-1037 准等熵压缩 S1 NiTi-1 ∅8×1.498 346 S2 NiTi-1 ∅8×1.800 346 Shot-1040 准等熵压缩 S1 NiTi-1 ∅8×1.504 383 S2 NiTi-1 ∅8×1.804 383 Shot-653 冲击加载 NiTi-4 ∅8×1.010 300 Shot-654 冲击加载 NiTi-4 ∅8×1.004 300 Shot-1035 冲击加载 NiTi-1 8×8×0.809 302 Shot-1038 冲击加载 NiTi-1 8×8×0.800 402 Shot-1039 冲击加载 NiTi-1 8×8×0.800 302 2. 实验结果与分析
准等熵压缩实验获得的速度时程曲线如图4所示，由图4(a)和(b)中的速度时程曲线可以看出，材料NiTi-1和NiTi-4在低速段约150 m/s处存在显著差异。NiTi-4实验中在自由面速度（vfs）约为150 m/s处存在显著的双波结构，NiTi-1实验在自由面速度为150 m/s附近未观察到明显的双波结构。由于显著速度变化是由材料压缩过程中材料模量变化出现显著体积变化引起，因此，推测在加载过程中NiTi-1和NiTi-4的物理响应过程存在本质差异。
采用Lagrange 数据处理方法对准等熵压缩实验进行数据处理，获得的准等熵压缩Lagrangian声速如图5(a)所示，声速在粒子速度75 m/s处存在显著差异，该拐点与速度剖面上自由面速度拐点是对应的。由Lagrangian声速结果得到的弹性波声速（5.4 km/s）与常温常压下声速测量结果基本一致，由此也可确定加载初期为材料弹性变形阶段。另外，过拐点后声速随粒子速度线性增大，Shot-518和Shot-522实验的声速结果规律与初始奥氏体相声速测量结果响应规律一致[23]。由此确定NiTi-4样品在实验Shot-518和Shot-522中的双波结构对应发生初始奥氏体相的弹塑性转变。实验Shot-1036和Shot-1037中初始样品为马氏体相NiTi合金，实验中声速值由初始横波声速逐渐降低至体波声速后再线性增大。由于马氏体相强度低及实验中受低速段测速精度限制，未观察到明显的马氏体相材料弹塑性转变点。准等熵压缩应力、应变及应变率分别为：
σ=ρ0∫u0cLdu (1) ε=∫u01cL(u)du (2) ˙ε=˙ucL(u) (3) 式中：
σ 为应力，ε 为应变，u 为粒子速度，cL为 Lagrangian声速。由声速结果计算获得的准等熵压缩实验的应变率结果见图5(b)，实验中准等熵加载的应变率在105 s−1量级。计算获得的准等熵加载实验应力应变曲线见图6，实验Shot-518和Shot-522中样品准等熵压缩下弹性极限分别为2.2和2.0 GPa。从应力-应变曲线的压缩性比较，初始奥氏体相的NiTi-4比初始马氏体相的NiTi-1更硬。冲击加载实验的样品后自由面速度历史如图7所示，室温下，初始奥氏体相NiTi-4样品在冲击加载下表现出显著的弹塑性转变双波结构（见图7(a)），实验Shot-653和Shot-654中弹塑性转变粒子速度分别为130和110 m/s，冲击Hugoniot弹性极限分别为4.5和3.8 GPa。虽然冲击实验中压缩波上升时间很难准确测量，但鉴于实验中冲击波加载弹性段速度上升沿约为数纳秒，样品弹塑性转变时的应变约为0.02，则估算获得该冲击实验中应变率在107 s−1量级。实验Shot-1035和Shot-1039开展了初始马氏体相NiTi-1样品在初始温度为302 K的冲击实验，由样品后自由面曲线结果（见图7(b)）可以看出，初始样品为马氏体相NiTi在冲击加载后速度约34 m/s处存在双波结构，为马氏体相的弹塑性转变点。实验Shot-1038开展了NiTi-1样品在初始温度为402 K的冲击实验，根据常压下样品初始相变温度结果，此初始温度下样品为奥氏体相，由样品后自由面曲线结果（见图7(b)）可以看出，由于初始温度的影响，导致样品在冲击加载后动力学响应存在较大差异。在自由面速度约100 m/s时存在明显的双波结构，此时对应加载压力为1.72 GPa，为NiTi-1样品奥氏体相高温下的弹塑性转变点。
图 7 不同条件冲击实验中样品速度剖面Figure 7. Velocity profiles of shock compression experiments under different conditions实验获得的不同初始温度、不同应变率条件下NiTi合金的弹性极限如图8所示。由图中相同组分、相同初始样品温度、不同应变率下NiTi合金的弹性极限可以看出，随着应变率从约105 s−1升高至107 s−1，弹性极限由约2 GPa增大至4 GPa，说明NiTi合金的弹性极限存在很强的应变率效应。在高于奥氏体相完成温度、相同加载应变率下，NiTi合金的弹性极限随着初始样品温度的升高而降低。与文献实验结果比较，冲击加载下，温度为402 K样品的弹性极限与文献[26]中413 K样品的结果相近。改变初始样品温度使得实验样品处于不同的初始相，由于奥氏体相NiTi合金与马氏体相NiTi合金在高应变率下的弹性极限差异较大，造成实验速度剖面也存在较大差异。此外，奥氏体NiTi合金在初始温度升高后，样品的弹性极限比常温下低，该现象与温度对材料强度热软化规律是一致的。
3. 总 结
研究不同初始相变温度NiTi合金的力学响应规律对完善其物理模型和指导应用设计具有重要意义。本文中，系统地开展了不同初始相变温度NiTi合金在10−3 s−1应变率准静态压缩-拉伸、105 s−1应变率准等熵压缩和107 s−1应变率的冲击加载实验，获得了不同加载条件下的应力-应变、弹塑性屈服等关键物理数据，取得以下主要认识与结论：
(1) 在准静态压缩和拉伸实验中，尽管初始马氏体相和初始奥氏体相NiTi合金的应力-应变曲线均出现2个模量变化引起的拐折点，但两者对应的应力幅值和物理过程不同。其中初始马氏体相NiTi合金的模量变化分别对应马氏体相畴域结构内的晶体重定向和不可逆塑性变形，而初始奥氏体相NiTi合金中的模量变化则由马氏体相变和相变后马氏体屈服引起。
(2) 获得了2种不同初始相变温度NiTi合金的准等熵压缩Lagrangian声速和准等熵压缩线，其中初始马氏体相NiTi合金的Lagrangian声速随粒子速度增大而增大，未观察到间断等非线性变化；而初始奥氏体相中声速曲线存在间断，声速由初始横波值间断减小至体波声速后再随粒子速度线性增大。
(3) 冲击加载下，初始奥氏体相NiTi合金样品的后自由面速度曲线上升沿存在显著的弹塑性转变双波结构，弹塑性转变粒子速度约为130 m/s。而马氏体相NiTi合金冲击实验中样品的初始温度对其变形机制有较大影响，样品初始温度为302 K时，后自由面速度约34 m/s处出现马氏体相弹塑性屈服引起的双波结构；样品初始温度升至402 K时，在后自由面速度约100 m/s时出现由奥氏体相塑性屈服引起的明显的双波结构。
(4) 随着应变率从约105 s−1升高至107 s−1，相同组分奥氏体相NiTi合金的弹性极限由约2 GPa增大至约4 GPa，107 s−1应变率下，随着初始样品温度升至402 K时，弹性极限降至1.7 GPa。结果表明，NiTi合金的弹性极限存在显著的温度和应变率效应。
感谢中国工程物理研究院流体物理研究所的陈学秒、吴刚、胥超、税荣杰在实验装置运行和测试中提供的帮助！
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图 5 每次爆破结束后锁固型边坡位移场及位移矢量场
Figure 5. Displacement field and displacement vector field of the slope after each blasting cycle
图 6 每次爆破结束后锁固型边坡的损伤分布
Figure 6. Damage distribution of the slope after each blasting cycle
图 8 边坡在一次爆破作用下不同折减系数的位移
Figure 8. Displacements of monitor points with different strength-reduction ratios under the action of once blasting cycle
图 9 不同爆破次数下锁固型边坡的安全系数
Figure 9. Safety factor of the slope with the locked segment after each blasting
图 11 Model Ⅰ和Model Ⅲ边坡的累积损伤和位移云图
Figure 11. Cumulative damage and displacement distributions of Model Ⅰ and Model Ⅲ
密度/(kg·m−3) 弹性模量/GPa 泊松比 黏聚力/MPa 内摩擦角/(°) 抗拉强度/MPa 剪胀角/(°) α β 2700 68.69 0.228 27.7 55 5.6 12 3.15×106 2 
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表 2 模型物理力学参数
Table 2. Physical parameters for the models
岩体类型 密度/(kg·m−3) 弹性模量/GPa 泊松比 黏聚力/MPa 内摩擦角/(°) 抗拉强度/MPa 基岩 2700 20.0 0.24 5.00 42 3.00 风化岩体及锁固段 2500 0.8 0.25 0.20 34 0.20 软弱夹层 2200 0.6 0.25 0.04 18 0.02
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