Crack Initiation and Coalescence Behavior of Two Non-parallel Flaws

Crack initiation and coalescence behavior of rock or rock-like specimens containing artificial flaws under uniaxial compression have been subjects of intensive investigation in the past. Most of these investigations however focused on crack initiation and coalescence between two or more parallel flaws. Although there have been few experimental studies on non-parallel flaws, these studies did not address the influence of geometrical factors such as ligament length and ligament angle on the crack initiation and coalescence behavior of non-parallel flaws. To investigate whether the individual geometrical factors have similar effects on the crack initiation and coalescence behavior of both parallel and non-parallel flaws, we conducted uniaxial compression tests to investigate crack cracking and coalescence processes in rock like material containing two non-parallel flaws. The paper presents the influence of individual geometrical factors on the crack initiation process and coalescence pattern of non-parallel flaws. Initiation of primary first cracks from all the tips of the two flaws did not occur simultaneously in all the flaw configurations. The flaw configuration of the non-parallel flaws influences the crack initiation, crack trajectories and coalescence behavior. The crack coalescence pattern changes with an increasing ligament angle from indirect to shear crack or mixed tensile-shear crack to tensile crack coalescence. The chance of direct coalescence is reduced with an increase in ligament length. In conclusion, the crack initiation and coalescence behavior of prismatic rock-like specimens with non-parallel flaws, as influenced by the geometrical factors, are analogous to the cracking and coalescence pattern observed in specimens with parallel flaws.     

Crack Initiation, Propagation and Coalescence in Rock-Like Material Containing Two Flaws: a Numerical Study Based on Bonded-Particle Model Approach

Cracking and coalescence behavior in a rectangular rock-like specimen containing two parallel (stepped and coplanar) pre-existing open flaws under uniaxial compression load has been numerically studied by a parallel bonded-particle model, which is a type of bonded-particle model. Crack initiation and propagation from two flaws replicate most of the phenomena observed in prior physical experiments, such as the type (tensile/shear) and the initiation stress of the first crack, as well as the coalescence pattern. Eight crack coalescence categories representing different crack types and trajectories are identified. New coalescence categories namely “New 1” and “New 2”, which are first observed in the present simulation, are incorporated into categories 3 and 4, and category 5 previously proposed by the MIT Rock Mechanics Research Group, respectively. The flaw inclination angle (β), the ligament length (L) (spacing between two flaws) and the bridging angle (α) (inclination of a line linking up the inner flaw tips, between two flaws) have different effects on the coalescence patterns, coalescence stresses (before, at or post the peak stress) as well as peak strength of specimens. Some insights on the coalescence processes, such as the initiation of cracks in the intact part of specimens at a distance away from the flaw tips, and coalescence due to the development and linkage of a number of steeply inclined to vertical macro-tensile cracks are revealed by the present numerical study.     

Influence of inclination angles and confining pressures on mechanical behavior of rock materials containing a preexisting crack

To deeply understand the cracking mechanical behavior of brittle rock materials, numerical simulations of a rock specimen containing a single preexisting crack were carried out by the expanded distinct element method (EDEM). Based on the analysis of crack tips and a comparison between stress- and strain-based methods, the strain strength criterion was adopted in the numerical models to simulate the crack initiation and propagation processes under uniaxial and biaxial compression. The simulation results indicated that the crack inclination angle and confining pressure had a great influence on the tensile and shear properties, peak strength, and failure behaviors, which also showed a good agreement with the experimental results. If the specimen was under uniaxial compression, it was found that the initiation stress and peak strength first decreased and then increased with an increasing inclination angle α. Regardless of the size of α, tensile cracks initiated prior to shear cracks. If α was small (such as α ≤ 30°), the tensile cracks dominated the specimen failure, the wing cracks propagated towards the direction of uniaxial compression, and the propagation of shear cracks was inhibited by the high concentration of tensile stress. In contrast, if α was large (such as α ≥ 45°), mixed cracks dominated the specimen failure, and the external loading favored the further propagation of shear cracks. Analyzing the numerical results of the specimen with a 45° inclination angle under biaxial compression, it was revealed that lateral confinement had a significant influence on the initiation sequence and the mechanical properties of new cracks.     

一种测量岩石压缩裂纹扩展过程声波波速的连续测量方法

中低围压或单轴压缩荷载条件下,可以将脆性岩石材料的裂纹扩展过程划分为原生裂纹压密、新生裂纹起裂并稳定扩展、不稳定裂纹扩展和交互贯通以及峰后5个主要发展阶段。含预制裂纹试样的直接观测法、应变监测、声发射（AE）监测、声波波速测试、CT（computerized tomography）扫描、微电镜观察等间接监测方法被用来研究裂纹起裂和扩展过程。本文基于在中、低围压以及单轴压缩荷载条件下,脆性岩石材料首先产生张拉裂纹,并优先沿加载方向发育的规律,提出采用声波波速连续测量方法进行岩石加载过程波速的连续观测,有助于分析裂纹扩展过程中声波波速的变化,为研究裂纹扩展过程提供一种间接测量手段。     

An Experimental Study of Crack Coalescence Behaviour in Rock-Like Materials Containing Multiple Flaws Under Uniaxial Compression

Experiments on man-made flawed rock-like materials are applied extensively to study the mechanical behaviour of rock masses as well as crack initiation modes and crack coalescence types. A large number of experiments on specimens containing two or three pre-existing flaws were previously conducted. In the present work, experiments on rock-like materials (formed from a mixture of sand, plaster, limestone and water at mass ratio of 126:9:9:16) containing multiple flaws subjected to uniaxial compression were conducted to further research the effects of the layout of pre-existing flaws on mechanical properties, crack initiation modes and crack coalescence types. Compared with previous experiments in which only three types of cracks were found, the present experiments on specimens containing multiple flaws under uniaxial compression revealed five types of cracks, including wing cracks, quasi-coplanar secondary cracks, oblique secondary cracks, out-of-plane tensile cracks and out-of-plane shear cracks. Ten types of crack coalescence occurred through linkage among wing cracks, quasi-coplanar secondary cracks, oblique secondary cracks, out-of-plane shear cracks and out-of-plane tensile cracks. Moreover, the effects of the non-overlapping length and flaw angle on the complete stress–strain curves, the stress of crack initiation, the peak strength, the peak strain and the elastic modulus were also investigated in detail.     

岩石材料裂纹扩展的改进近场动力学方法模拟

在传统键基近场动力学模型的基础上,提出可以反映岩石类材料应力-应变变化特点的改进型模型,弥补了传统近场动力学模型在模拟岩石类材料时无法反映岩石内应力随应变先增加再减小最后破坏这一特性的不足。运用改进型模型,对含有单裂纹试件在不同预制裂纹角度及拉伸荷载条件下的裂纹扩展过程进行了数值模拟,分析了开裂角度在不同预制裂纹角度及拉伸荷载条件下的变化。结果表明：预制角度及拉伸荷载对开裂角度都有影响,在试件周围拉荷载接近的情况下会获得较小的开裂角,并且在拉荷载一定的条件下开裂角会有先增加后减少的规律。同时模拟了含裂纹岩石在单轴压缩条件下的裂纹扩展情况,将数值模拟的结果与室内试验的结果对比,验证了所提出新模型的有效性。所提出的新模型可以较好地模拟岩石类材料在拉荷载条件下的力学特性,在岩石数值模拟方面有着较为广泛的应用前景。     

模拟岩石中裂纹扩展连接的近场动力学方法

脆性岩石材料在压应力作用下常出现两类裂纹：翼型张拉裂纹和次生剪切裂纹。近场动力学是一种新型的无网格数值计算方法。在近场动力学理论中,采用积分形式的控制方程代替微分形式的控制方程使得该数值算法在断裂问题上具有独特的优势。将Mohr-Coulomb准则和最大主应力准则引入非普通“态”基近场动力学理论中,分别用于模拟材料常见的压剪和张拉破坏。这种扩展的非普通“态”基近场动力学可以有效地模拟脆性岩石材料在多种受力状态下的裂纹起裂、扩展和连接问题。通过5个不同的数值算例说明该数值算法在处理脆性岩石材料断裂问题的有效性和准确性。首先,通过模拟含圆孔的弹性板拉伸数值试验说明该数值算法的有效性和准确性。其次,数值模拟了简单三点弯曲试验以及不使用其他外部准则条件下动荷载作用下裂纹的分叉试验,所得结果与其他试验结果或数值结果相吻合,从而验证了该理论的有效性。然后,模拟了包含斜裂纹的巴西圆盘试验,裂纹扩展路径和计算所得的断裂韧度同样吻合于试验结果。最后,模拟了单轴压缩状态下,预制裂纹试样的裂纹扩展和连接问题。将该数值算法与试验结果对比表明,所提出的数值方法可以模拟和预测岩石类材料的张拉和压剪裂纹的起裂、扩展和连接行为。     

A three-dimensional numerical investigation of the fracture of rock specimens containing a pre-existing surface flaw

Three-dimensional surface crack initiation and propagation in two kinds of heterogeneous rocks were numerically investigated via parallel finite element analysis using a supercomputer. Numerically simulated rock specimens containing a pre-existing flaw were subjected to uniaxial compression until failure. The initiation and propagation of wing cracks, anti-wing cracks, and shell-like cracks were reproduced by numerical simulations. The numerically simulated results demonstrate that the further propagation of wing cracks and shell-like cracks stop due to their wrapping (curving) behavior in three-dimensional spaces, even if the applied loads continue to increase. Furthermore, rock heterogeneity could significantly influence crack propagation patterns and the peak uniaxial compressive strengths of rock specimens. Moreover, anti-wing cracks only appeared in relatively heterogeneous rocks, and the peak uniaxial compressive strengths of the specimens were observed to depend on the inclination of the pre-existing flaw. Finally, the mechanism of surface crack propagation is discussed in the context of numerically simulated anti-plane loading tests, wherein it was identified that Mode III loading (anti-plane loading) does not lead to Mode III fracture in rocks due to their high ratio of uniaxial compressive strength to tensile strength. This finding could explain the lateral growth of an existing flaw in its own plane, which is a phenomenon that has not been observed in laboratory experiments.     

Numerical Study on Coalescence of Pre-Existing Flaw Pairs in Rock-Like Material

The present numerical study, which is an extension of our previous numerical analysis on cracking processes of a single pre-existing flaw, focuses on the coalescence of two pre-existing parallel open flaws in rock subjected to a uniaxial compressive loading. To facilitate a systematic investigation, the arrangements of the flaw pair are classified into 11 categories. Simulations engaging AUTODYN are conducted on each category. The numerical results are compared with some published physical experimental test results. Eleven typical coalescence patterns are obtained, which are in good agreement with the experimental results, which include two coalescence patterns obtained in flaw pair arrangements (II) and (VIII″) not being reported in previous studies. The information gathered in the simulations helps identify the type (tensile/shear) of each crack segment involved in the coalescence. Most of the coalescence cracks initiate at or around the flaw tips, except those in flaw pair arrangements (II) and (IX′) with a very short ligament length, in which the coalescence cracks initiate on the flaw surfaces away from the flaw tip regions. Based on the numerical simulation results, the properties of the 11 coalescence patterns are obtained. Except those in flaw pair arrangements (II) and (IX′), the other coalescence patterns can be interpreted with respect to the basic crack types—tensile wing crack, horsetail crack and anti-wing crack. In addition, based on the type of crack segments involved in coalescence, namely tensile and shear, the coalescence can be classified into T mode (tensile mode), S mode (shear mode) and TS mode (mixed tensile–shear mode).     

Influence of single flaw on the failure process and energy mechanics of rock-like material

This paper investigates the influence of a flaw on crack initiation, the failure mode, deformation field and energy mechanism of the rock-like material under uniaxial compression. The results of laboratory test and numerical simulation demonstrate the flaw inclination effect can be classified into three groups: 0–30°, 30–60° and 75–90°. The characteristic stresses increase as the flaw angle increases. The tensile cracks initiate from gentle flaws (α ⩽ 30°) and shear cracks appear at tips of steep flaws (α ⩾ 45°). The input energy, strain energy and dissipation energy of a specimen show approximate increasing trends as the flaw angle increases.     

单轴压缩条件下含三叉裂隙类岩石试样力学特性的细观研究

三叉裂隙是自然界普遍存在的一种岩体缺陷形式,其对岩体的力学特性有重要影响。对含预制三叉裂隙的水泥砂浆试样进行室内单轴压缩试验,配合使用摄像机拍摄裂纹的起裂、扩展、贯通过程,通过数字图像技术处理获取试样的应变场云图,并结合PFC2D程序研究不同?、? 条件下试样的强度特征、裂纹模式和裂纹演化扩展规律。研究表明：三叉裂隙对试样单轴抗压强度有明显的削弱作用。当? 恒定为120°时,试样在? = 30°时达到最大抗压强度;当? 恒定为90°时,随?增大,试样抗压强度呈先减小后增大的趋势,且当? = 45°时达到最大抗压强度。试样产生的裂纹可分为3类,分别是张拉型裂纹(Ⅰ型裂纹)、剪切型裂纹(Ⅱ型裂纹)、混合型裂纹(Ⅲ型裂纹)。这3类裂纹通常从裂隙尖端开始产生,并且Ⅰ型裂纹沿加载方向扩展,通常未扩展至试样边界;Ⅱ型和Ⅲ型裂纹通常与加载方向呈一定角度扩展至试样边界。通过对裂纹的几何形态和组成宏观裂纹的微裂纹成分的分析,得知导致含三叉裂隙试样在单轴压缩条件下失效的是张拉破坏。数字图像技术得到的应变云图表明,当载荷达到一定阶段,裂隙尖端出现应力集中,微破裂开始发育并聚集成微破裂区,微破裂区扩大产生宏观裂纹。通过对主应变和剪应变云图分析,发现导致试样失效的是张拉破坏,剪应变在裂纹扩展过程中的影响较小。     

含折线型裂隙砂岩试件翼型裂纹起裂与扩展机制研究

岩体内部赋存的裂隙很多表现为折线型,为探究这类岩体的断裂机制,制备含折线型裂隙砂岩试件并对其进行单轴压缩试验。采用数字图像相关(DIC)方法计算加载过程中的变形场演化,根据新生裂纹两侧的位移差异识别裂纹类型;运用扩展有限元法(XFEM)模拟断裂过程,根据应力分布特征解释翼型裂纹起裂与扩展机制。DIC计算结果表明,新生裂纹处出现应变局部化带,裂纹两侧发生相对分离;含直线型和折线型裂隙砂岩试件的翼型裂纹分别萌生于预制裂隙端部以及折角处,这是因为裂隙几何形态会改变拉应力集中位置;含折线型裂隙砂岩试件的起裂应力小于含直线型裂隙砂岩试件,这是因为相同加载条件下前者的最大拉应力值更大;这2类试件的裂纹扩展均是由于裂纹尖端集中的拉应力引起的,裂纹依然呈张开状态;裂隙几何形态未改变试件的最终破坏模式,均表现为对角剪切破坏。     

基于DIC技术的3D打印节理试件破裂机制研究

为了准确表征不同角度预制节理岩石在单轴压缩下的变形破坏模式,基于3D打印技术制作了节理模型用于模拟岩体中的结构面,通过水泥砂浆的浇筑得到含不同角度预制节理的岩石试件并进行单轴压缩试验,同时采用数字图像相关技术（DIC）观测、分析试验过程中试件裂纹产生、扩展以及贯通过程。结果表明：随着预制节理从0°增加到90°,试件强度与峰值应变均呈现先降低后升高的变化趋势,0°和45°试件弹性模量相对于完整试件有所降低。基于DIC检测结果,0°、30°、45°及60°试件裂纹皆从预制节理尖端部位起裂,各角度试件的起裂应力与试件强度变化规律一致。各角度试样起裂时在剪应力控制下以剪切翼型裂纹形式起裂,0°与45°试件裂纹在扩展过程由剪切发展为张拉型裂纹,30°和60°试件以剪切裂纹形式贯穿始终,90°试件从底部起裂并最终表现为张拉破坏。研究还发现,下翼起裂角θ2和上翼起裂角θ1之间存在明显的线性正相关关系,关系式为θ2 =0.828 6θ1 +12.185,且起裂应力大小变化与峰值应力变化一致,皆随节理角度的增加先减小后增大。     

高水压下岩体裂纹扩展的渗流-断裂耦合机制与数值实现

采用渗流力学、断裂力学理论结合Monte Carlo方法描述岩体裂纹的随机分布,研究高水压作用下岩体原生裂纹的变形和翼形裂纹的萌生、扩展、贯通的渗流-断裂耦合作用机制,建立高水压作用下岩体裂纹的渗流-断裂耦合数学模型,给出该数学模型的求解策略与方法,在Fortran95平台下开发高水压下岩体裂纹扩展的渗流-断裂耦合分析程序HWFSC.for。高水压下岩体裂纹扩展的渗流-断裂耦合体现在岩体裂纹网络和渗流初始条件都随渗流时步变化。对高压注水岩体裂纹扩展过程进行渗流-断裂耦合分析。结果表明,高压注水条件下,岩体裂纹扩展存在起动水压力,当水压力大于起动水压力时,裂纹尖端开始萌生翼形裂纹,随着裂纹水压力的增加,翼形裂纹扩展,进而与其他裂纹搭接贯通,停止扩展。渗流-断裂耦合分析考虑了裂纹动、静水压力对裂纹产生的法向扩张效应及翼形裂纹的扩展而形成新的渗流通道两方面的影响,连通裂纹数随渗流的发展而增加。岩体裂纹的渗流-断裂耦合分析,能较真实地再现岩体裂纹的水力劈裂现象,描述岩体裂纹的扩展、贯通过程及与之相耦合的渗流响应。     

基于3D-DIC的砂岩裂纹扩展及损伤监测试验研究

裂纹监测对岩石损伤演化的认识至关重要。为研究岩石裂纹扩展及损伤变形特性,开展了含不同倾角（0°～90°）预制裂隙的标准细黄砂岩样的单轴压缩试验。利用三维数字图像相关技术（3D-DIC）获取岩样三维空间坐标下的应变分布,并结合声发射从光学与声学的角度监测了裂纹的扩展演化。由此提出了一种裂纹主应变的计算方法,定量表征岩石劣化的损伤变量D值。最后,探讨了由声发射与损伤变量D值确定岩样特征强度的影响因素。结论如下：（1）裂纹主应变反映了岩样受荷过程中同源裂纹在时间上的变化速率与空间上的扩展趋势,能较好地表征岩石的开裂行为;（2）声发射适用于确定岩样的起裂应力,不适用于损伤应力的确定,损伤变量D值所确定的起裂应力滞后于声发射,但适用于损伤应力特征值的确定;（3）结合声发射与DIC技术确定的归一化起裂应力范围为0.63～0.94、归一化损伤应力的范围为0.83～0.99;（4）预有裂隙会影响岩石的材料力学性能。随着倾角的增加,岩石的起裂应力、损伤应力及峰值应力呈增长的趋势,由于难以形成局部应变场聚集,裂纹的萌生与起裂更加困难。结果表明,3D-DIC技术的利用可以提高对岩石开裂行为的理解,对岩石的损伤监测与判识更有重要的意义。     

Simulating damage evolution and fracture propagation in sandstone containing a preexisting 3-D surface flaw under uniaxial compression

Preexisting flaws and rock heterogeneity have important ramifications on the process of rock fracturing and on rock stability in many applications. Therefore, there is great interest in numerical modelling of rock fracture and the underlying mechanisms. We simulated damage evolution and fracture propagation in sandstone specimens containing a preexisting 3-D surface flaw under uniaxial compression. We applied the linear elastic damage model based on the unified strength theory following the rock failure process analysis code. However, in contrast to the rock failure process analysis code, we used the finite element method with tetrahedron elements on unstructured meshes. It provided higher geometrical flexibility and allowed for a more accurate representation of the disk-shaped flaw with various flaw depths, angles, and lengths through locally adapted meshes. The rock heterogeneity was modelled by sampling the initial local Young's modulus from a Weibull distribution over a cubic grid. The values were then interpolated to the computational finite element method mesh. This method introduced an additional length scale for the rock heterogeneity represented by the cell size in the sampling grid. The generation of three typical surface cracking patterns, called wing cracks, anti-wing cracks, and far-field cracks, were identified in the simulation results. These depend on the geometry of the preexisting surface flaw. The simulated fracture propagation, coalescence types, and failure modes for the specimens with preexisting surface flaw show good agreement with recent experimental studies.     

模拟岩石压剪状态下主次裂纹萌生开裂的扩展有限元法

压剪应力状态下,岩石类材料中常见两类裂纹：翼型张拉裂纹和次生压剪裂纹。基于扩展有限元方法（XFEM）,提出了模拟压剪裂纹面作用机制的扩展有限元位移增强方案,并给出了扩展有限元法分叉裂纹处理方法,分别用最大周向拉应力准则和Mohr-Coulomb准则判断张拉裂纹和压剪裂纹的萌生和扩展。基于Matlab平台编写了数值计算程序Betaxfem 2D,通过两个算例对所提方案进行了验证,所得结果与有限元法（FEM）计算结果吻合很好。模拟了单轴压缩载荷下含预制闭合裂纹试件的裂纹分叉、扩展过程,与试验结果的对比表明,所提方案可以模拟和预测岩石类材料张拉、压剪交互分叉裂纹的萌生和扩展行为。     

双边非对称裂纹平板试样拉伸试验研究

采用自制的压–拉转换装置,配合RMT 150C岩石力学试验系统及数字散斑相关方法,对双边非对称裂纹类岩石平板试样进行直接拉伸试验,得到类岩石试样的拉应力–应变曲线、试样表面应变场演化过程和裂纹扩展模式。研究发现,类岩石试样直接拉伸的拉应力–应变曲线大致可以分为4个阶段：（1）近似线性阶段,预制裂纹基本不起作用,应力随应变增加较快,试样表面应变场的分布主要受试样内部的孔隙及颗粒的影响;（2）整体缓慢增加阶段,两预制裂纹和试样内部的孔隙及颗粒共同影响试样表面应变场的分布,整体上应力随应变呈增加趋势;（3）短暂峰值过渡阶段,试样中某个预制裂纹对试样表面应变场的分布起决定性作用;（4）破坏阶段,裂纹起裂位置在应变场相对集中区域,并扩展导致试样破坏。对于直接拉伸条件下的双边非对称裂纹平板试样,其中某条预制裂纹会率先扩展,先向远离前方裂纹的方向扩展,再向靠近前方裂纹的方向扩展,对采用数值模拟方法研究张拉应力状态下裂纹相互作用扩展规律具有重要意义。     

Crack Coalescence in Molded Gypsum and Carrara Marble: Part 2—Microscopic Observations and Interpretation

Experimental uniaxial compression loading tests were conducted on molded gypsum and Carrara marble prismatic specimens to study the cracking and coalescence processes between pre-existing artificial flaws. The study showed that material had an influence on the cracking and coalescence processes (see the companion paper in this issue). As reported in the companion paper, one of the pronounced features as observed in the high-speed video recordings was the development of macroscopic white patches prior to the development of observable cracks in marble, but not in gypsum. This paper (part 2) deals with the microscopic aspects of the study. Specifically, the scanning electron microscope (SEM) and the environmental scanning electron microscope (ESEM) imaging techniques were used to study the microscopic development of white patches and their evolution into macroscopic tensile cracks and shear cracks in marble, and the microscopic initiation of hair-line tensile cracks and their evolution into macroscopic tensile cracks in gypsum. The microscopic imaging study in marble showed that the white patches were associated with extensive microcracking zones (process zones), while the extent of process zone development in gypsum was limited. The comparison of the macroscopic and microscopic results indicates that the different extent of microcracking zone development, related to the material textural properties, is a key factor leading to different macroscopic cracking behavior in gypsum and marble.

Mechanical Behaviour of Reservoir Rock Under Brine Saturation

Acoustic emissions (AE) and stress–strain curve analysis are well accepted ways of analysing crack propagation and monitoring the various failure stages (such as crack closure, crack initiation level during rock failure under compression) of rocks and rock-like materials. This paper presents details and results of experimental investigations conducted for characterizing the brittle failure processes induced in a rock due to monocyclic uniaxial compression on loading of two types of sandstone core samples saturated in NaCl brines of varying concentration (0, 2, 5, 10 and 15 % NaCl by weight). The two types of sandstone samples were saturated under vacuum for more than 45 days with the respective pore fluid to allow them to interact with the rocks. It was observed that the uniaxial compressive strength and stress–strain behaviour of the rock specimens changed with increasing NaCl concentration in the saturating fluid. The acoustic emission patterns also varied considerably for increasing ionic strength of the saturating brines. These observations can be attributed to the deposition of NaCl crystals in the rock’s pore spaces as well some minor geo-chemical interactions between the rock minerals and the brine. The AE pattern variations could also be partly related to the higher conductivity of the ionic strength of the high-NaCl concentration brine as it is able to transfer more acoustic energy from the cracks to the AE sensors.