Influence of location of large-scale asperity on shear strength of concrete-rock interface under eccentric load

The location and geometry of large-scale asperity present at the foundation of concrete gravity dams and buttress dams affect the shear resistance of the concrete-rock interface.However,the parameters describing the frictional resistance of the interface usually do not account for these asperities.This could result in an underestimate of the peak shear stre ngth,which leads to significantly conservative design for new dams or unnecessary stability enhancing measures for existing ones.The aim of this work was to investigate the effect of the location of first-order asperity on the peak shear strength of a concrete-rock interface under eccentric load and the model discrepancy associated with the commonly used rigid body methods for calculating the factor of safety(FS)against sliding.For this,a series of direct and eccentric shear tests under constant normal load(CNL)was carried out on concrete-rock samples.The peak shear strengths measured in the tests were compared in terms of asperity location and with the predicted values from analytical rigid body methods.The results showed that the large-scale asperity under eccentric load significantly affected the peak shear strength.Furthermore,unlike the conventional assumption of sliding or shear failure of an asperity in direct shear,under the effect of eccentric shear load,a tensile failure in the rock or in the concrete could occur,resulting in a lower shear strength compared with that of direct shear tests.These results could have important implications for assessment of the FS against sliding failure in the concrete-rock interface.

A Comparative Study on Asperity Peak Modeling Methods

The peak identification scheme based method(three-point definition)and the spectral moments based method(spectral moment approach)are both widely used for asperity peak modeling in tribology.To discover the differences between the two methods,a great number of rough surface profile samples with various statistical distributions are first randomly generated using FFT.Then the distribution parameters of asperity peaks are calculated for the generated samples with both methods.The obtained results are compared and verified by experiment.The variation rules of the differences between the two methods with statistical characteristics of rough surfaces are investigated.To explain for the discovered differences,the assumptions by spectral moment approach that the joint distribution of surface height,slope and curvature is normal and that the height distribution of asperities is Gaussian,are examined.The results show that it is unreasonable to assume a joint normal distribution without inspecting the correlation pattern of[z],[z′]and[z′′],and that the height distribution of asperities is not exactly Gaussian before correlation length of rough surface increases to a certain extent,20 for instance.

Dual threshold search method for asperity boundary determination based on geodetic and seismic catalog data

As an important model for explaining the seismic rupture mode,the asperity model plays an important role in studying the stress accumulation of faults and the location of earthquake initiation.Taking Qilian-Haiyuan fault as an example,this paper combines geodetic method and b-value method to propose a multi-source observation data fusion detection method that accurately determines the asperity boundary named dual threshold search method.The method is based on the criterion that the b-value asperity boundary should be most consistent with the slip deficit rate asperity boundary.Then the optimal threshold combination of slip deficit rate and b-value is obtained through threshold search,which can be used to determine the boundary of the asperity.Based on this method,the study finds that there are four potential asperities on the Qilian-Haiyuan fault:two asperities(A1 and A2)are on the Tuolaishan segment and the other two asperities(B and C)are on Lenglongling segment and Jinqianghe segment,respectively.Among them,the lengths of asperities A1 and A2 on Tuolaishan segment are 17.0 km and 64.8 km,respectively.And the lower boundaries are 5.5 km and 15.5 km,respectively;The length of asperity B on Lenglongling segment is 70.7 km,and the lower boundary is 10.2 km.The length of asperity C on Jinqianghe segment is 42.3 km,and the lower boundary is 8.3 km.

Elastohydrodynamic Lubrication Interface Stiffness and Damping Considering Asperity Lateral Contact

Elastohydrodynamic lubrication(EHL)point contact occurs between two rough surfaces at the mesoscopic level,while the interaction of rough surfaces involves contact between asperities at the microscale level.In most cases,the contact between asperities within an interface takes the form of lateral contact rather than peak contact.Regions devoid of contact asperities are filled with lubricating oil.However,conventional models often oversimplify lateral contact forms as interactions between asperities and a smooth,rigid plane.These simplifications fail to accurately represent the true contact conditions and can lead to inaccuracies in the analysis of EHL’s contact performance.To address this issue,we have developed a novel EHL interface model comprising two rough surfaces.This model allows us to explore the influence of asperity height,contact angle,and contact azimuth angle on EHL interface performance.

Coupled evolution of piston asperity and cylinder bore contour of piston/cylinder pair in axial piston pump

The wear condition of the piston/cylinder pair is crucial to the performance and reliability of the axial piston pump.The hard piston surface,the soft cylinder bore surface,and the interface oil film affects each other during the wear process.Specifically,in the mixed lubrication region,the geometry of the hard piston surface asperity directly affects the wear of soft cylinder bore surface,while the asperities may deform or even degrade when penetrating and sliding against the cylinder bore.So far,there is no suitable method to simulate their coupled evolution.This paper proposed a wear process simulation model considering the real-time interaction between the elasto-plastic deformation of the piston surface asperity,the wear contour of the cylinder bore,and the lubrication condition of the interface.An offline library of the elasto-plastic constitutive behavior of the asperity based on the finite element method(FEM)is established as a part of the simulation model to precisely analyze the deformation and degradation of the asperity and quickly invoke them in the numerical wear process simulation.The simulation and experimental results show that the piston asperity and the cylinder bore contour converge to a steady state after running-in for about 0.5 h.The distribution of the simulated asperity degradation and wear depth is also verified by the experiment.

Multiscale study of the dynamic friction coefficient due to asperity plowing

A macroscopically nominal flat surface is rough at the nanoscale level and consists of nanoasperities.Therefore,the frictional properties of the macroscale-level rough surface are determined by the mechanical behaviors of nanoasperity contact pairs under shear.In this work,we first used molecular dynamics simulations to study the non-adhesive shear between single contact pairs.Subsequently,to estimate the friction coefficient of rough surfaces,we implemented the frictional behavior of a single contact pair into a Greenwood-Williamson-type statistical model.By employing the present multiscale approach,we used the size,rate,and orientation effects,which originated from nanoscale dislocation plasticity,to determine the dependence of the macroscale friction coefficient on system parameters,such as the surface roughness,separation,loading velocity,and direction.Our model predicts an unconventional dependence of the friction coefficient on the normal contact load,which has been observed in nanoscale frictional tests.Therefore,this model represents one step toward understanding some of the relevant macroscopic phenomena of surface friction at the nanoscale level.

A Model of Tangential Contact Damping Considering Asperity Interaction and Lateral Contact

A precise tangential contact damping model is proposed,which includes the lateral contact of the upper-lower asperities and the interaction of adjacent asperities.The effects of the normal static preload,frequency,and amplitude of tangential displacement on the tangential contact damping were analyzed by simulation,respectively.Furthermore,the results of simulation are verified by experiment.The tangential contact damping of considering the interaction and lateral contact of asperity is very close to the experimental results.

Shear behaviours and roughness degeneration based on a quantified rock joint surface description

The asperity wear of rock joints significantly affects their shear behaviour.This study discusses the wear damage of the asperities on the joint surface,highlighting the roughness degradation characteristics during the shear process.The direct shear experiment of artificial specimens containing rock joints was conducted under different normal stresses based on three-dimensional scanning technology.These experimental results showed the contribution of joint wear to roughness degeneration,such as the height,zone,and volume of asperity degeneration.The wear coefficient of the rock joint was obtained based on the volume wear of asperities in the laboratory experiment.The functional relationship between the friction coefficient and wear coefficient is subsequently determined.To quantitatively analyse the wear damage of a joint surface,a calculation method for determining the wear depth of the rock joint after shearing was proposed based on wear theory.The relationship between the ultimate dilation and wear depth was analysed.A coefficient m,which can describe the damage degree of the joint surface,and a prediction method of joint surface roughness after shearing are established.Good agreement between analytical predictions and measured values demonstrates the capability of the developed model.Lastly,the sensitivity factors on the wear depth are explored.

经验格林函数方法模拟强地面运动的研究进展

经验格林函数方法(简称EGFM)是当前模拟强地面运动的一种主流方法。主要介绍EGFM方法的发展历程和特点,重点解释了EGFM核心理论的沿革;详细介绍了加入凹凸体模型的改进的经验格林函数方法,并指出一种划分凹凸体区域新方法的可能性。

广州市主要断裂带特征震源模型研究

在广州市目标区内的主要活动断层危险性评价的基础上,综合活断层探测的研究成果和资料,对于具有潜在发震可能的主要断层,基于凹凸体震源模型建模理论进行了特征计算模型的建模。根据断层地震危险性分析确定的断层发震震级和几何参数,进行了断层的宏观参数和微观参数的设定。为通过复核预测方法计算与合成近断层强地震动场和城市危害性评价提供了科学的依据。

沈阳市F6断层的强地面运动分布模拟

目的研究沈阳市F6断层在设定地震6.0级时的强地面运动分布,为避免城市活动断层的地震破坏和为重要建筑合理的抗震设防提供依据.方法通过断裂所处应力环境、形变特征及活动性、结构、规模、地震活动、深部构造等综合分析,确定F6断层的地震危险性.以深、浅层人工地震、钻探、多道直流电法、探地雷达和地脉动数据为基础,建立了沈阳市接近真实的三维地下速度结构模型,强地面运动计算采用了三维有限差分法和统计学格林函数法的合成方法.结果 F6断层为中更新世活动断裂,最大潜在地震震级为6.0级;F6断层的震源模型为具有两个Asperity(凹凸体)的矩形断层:一个最大18km2的Asperity和一个7km2的Asperity.在设定地震作用下,F6断层强地震动PGA的影响范围集中于F6断层两侧,最大值达到285gal;PGV、PGD的分布形态与PGA相似.结论 F6断层发生6.0级时在沈阳市能够形成带状的强地面运动分布,其幅值甚至超出大震设防水准.

大震近场地震动数值模拟的不确定性研究

地震动数值模拟是研究地震发生、发展特征和规律性的基础研究,涉及震源、传播路径、场地三方面的一系列基本问题。研究结果可以直接用于强地震动基本参数的标定和地震动的工程预测以及地震危险性分析,是防震减灾中一项很重要的基础工作。当前考虑断层模型中的几个重要震源参数模拟未来地震动成为热点。例如,不管是随机有限断层方法,还是经验格林函数方法,都特别重视震源参数的选取,但当前对于强地面运动的数值模拟大多是对已发生地震的反演,反演所需的震源参数多是根据地震发生后的观测资料、经验关系推断或者人为的假定得到的。地震动特征预测的好坏,关键也在于震源参数的选取是否可靠,针对工程应用,对未来近断层地面运动的预测应当考虑参数选取的不确定性。强震动数值模拟方法已经比较成熟,比如随机有限断层法、经验格林函数法、混合方法等。但这些模拟地震动的方法多是确定性方法,对模拟过程中各个环节的不确定性因素考虑较少,尤其是断层震源参数的选取都是精确的数值,模拟结果自然不能综合表征未来地震发生时的地震动特征。另外一种地震动特性表征形式是利用地震动预测方程来表现参数的衰减关系,但是预测方程也会受观测数据、模型选取、随机因素的影响,不确定性无法避免。在地震动预测中每个步骤都带有很大的不确定性,主要包括认知的不确定性和随机的不确定性。震源模型的建立依赖各种专家的观点和意见,这种观点和意见往往相互差别很大,有很大的人为不确定性。地震资料的完整性、可靠性,以及理论模型等的建立也会造成很大的不确定性。因此,本论文的研究内容重点在预测未来大震近场地震动时震源环节的不确定性因素,系统地研究影响大震近场地震动数值模拟不确定性的影响因素,以期建立描述影响大震近场地震动数值模拟不确定性影响因素的模型,并将考虑不确定性因素的震源模型应用在破坏性大地震的地震动特征预测中去。针对地震发生时震源参数具有的随机不确定性,通过统计学方法得到震源参数间的经验关系。同时由于地球内部的不可入性以及观测技术手段的限制,地震震源模型还存在很多的认知不确定性因素,我们采用逻辑树方法处理认知不确定性因素,引入强震生成区的概念,重点分析Asperity的认知不确定性因素并在此基础上建立Asperity震源模型,取多种方案结果的最优值作为地震动的预测结果。论文具体的研究内容有以下5点:(1)在Hiroe Miyake工作的基础上,用经验格林函数法模拟了1997年日本九州鹿儿岛县的MJMA6.5地震,验证了经验格林函数法模拟强地面运动的有效性。数据使用的是日本K-NET强震台站数据。选择此次地震以及用日本K-NET数据的原因是日本的地震记录比较丰富,台站收集到的地震记录质量相对较好。同时分析了地震动模拟值中在工程领域使用较多的参数,表明模拟结果较好地反映了真实的地震动记录。证明经验格林函数预测未来强地震动是切实可行的。(2)本论文在大量地震记录及文献调研的基础上,运用统计学方法,通过统计回归分析得到地震密集区域震源参数间的经验关系,重点统计了包含龙门山断裂带在内的中国西南地区震源参数间的经验关系。为了得到更加适应于中国大陆区域的局部震源参数的经验关系,我们将地震样本局限在一定的区域内,缩小统计区域的范围,同时得到多个Asperity等局部震源参数的经验关系。(3)研究和分析各不确定性因素对地震动结果的影响程度对提高地震动预测的可靠性和准确性意义重大。本论文以2016年发生的日本熊本7.3级地震为例,通过参数敏感性分析技术筛选了对拟合结果有决定性影响的关键模型参数。未来地震动预测时会考虑在这几个不确定性因素中加入更高的权重,以提高地震动预测的可信度和准确性。(4)基于汶川地震资料,研究分析Asperity的认知不确定性对地震动预测的影响,运用逻辑树方法建立了考虑不确定性因素的Asperity震源模型。采用Hiroe Miyake提出的强震生成区的概念,将Asperity区域等同于强震生成区域,主要从Asperity的面积、数量、应力降、上升时间等方面进行了认知不确定性的分析研究,得到了针对强震的可能性较大的Asperity的面积和数量的最佳搭配形式。本文的创新点将概率地震危险性分析中不确定性因素的处理方法应用到未来大震近场地震动数值模拟中去。随机不确定性因素用统计学方法处理,认知不确定性用逻辑树方法处理。模拟结果能更好地表征未来大震近场的地震动特征,地震动参数的模拟结果在未来抗震设防及灾害预防中会有更好、更高的参考价值。考虑不确定性因素预测未来破坏性大地震,从而定量地评价发生地震时近场地面运动的综合特征,为防震减灾提供科学的依据,为工程设计决策提供参考依据。

镉胁迫对续断菊Sonchus asper L.Hill.根系分泌物的影响

通过盆栽试验,研究了不同Cd质量分数(0、50、100、200 mg.kg-1)对续断菊Sonchus asper L.Hill.根系分泌总有机酸、游离氨基酸、可溶性糖的影响,旨在探明根系分泌物对续断菊超积累Cd的影响。结果表明:Cd胁迫下续断菊根系分泌总有机酸、游离氨基酸和可溶性糖的质量浓度显著增加,同时,总有机酸、可溶性糖和游离氨基酸又促进了植株对Cd的吸收。随着Cd处理质量分数的增加,续断菊地上部和根部镉质量分数显著增加,90 d时续断菊地上部镉质量分数与可溶性糖、游离氨基酸的质量浓度呈极显著正相关,相关系数分别为0.999(P<0.01)和0.995(P<0.01),根部镉质量分数与可溶性糖、游离氨基酸的质量浓度也呈显著正相关,相关系数分别为0.998(P<0.01)和0.987(P<0.05);Cd对续断菊根系可溶性糖的分泌、游离氨基酸的合成有刺激作用,根系分泌的可溶性糖和游离氨基酸可能在续断菊累积镉的过程中有重要作用。

Assessing current faulting behaviors and seismic risk of the Anninghe-Zemuhe fault zone from seismicity parameters

Using the data of regional seismic network, this paper analyzes the current faulting behaviors of different segments of the Anninghe-Zemuhe fault zone, western Sichuan, and identifies the likely risky segments for potential large earthquakes. The authors map the probable asperities from the abnormally low b-value distribution, develop and employ a method for identifying current faulting behaviors of individual fault segment from the combinations of multiple seismicity parameter values, and make an effort to estimate the average recurrence intervals of character-istic earthquakes by using the parameters of magnitude-frequency relationship of the asperity segment. The result suggests that the studied fault zone contains 5 segments of different current faulting behaviors. Among them, the Mianning-Xichang segment of the Anninghe fault has been locked under high stress, its central part is probably an asperity with a relatively large scale. The Xichang-Puge segment of the Zemuhe fault displays very low seismicity under low stress. Both the locked segment and the low-seismicity segment can be outlined on the across-profile of relocated hypocenter depths. The Mianning-Xichang segment is identified to be the one with potential large earth-quake risk, for which the average recurrence interval between the latest M = 6.7 earthquake in 1952 and the next characteristic event is estimated to be 55 to 67 years, and the magnitude of the potential earthquake between 7.0 and 7.5. Also, it has been preliminarily suggested that for a certain fault segment, its faulting behaviors may change and evolve with time gradually.

Effect of rock joint roughness on its cyclic shear behavior

Rock joints are often subjected to dynamic loads induced by earthquake and blasting during mining and rock cutting. Hence, cyclic shear load can be induced along the joints and it is important to evaluate the shear behavior of rock joint under this condition. In the present study, synthetic rock joints were prepared with plaster of Paris(Po P). Regular joints were simulated by keeping regular asperity with asperity angles of 15°-15° and 30°-30°, and irregular rock joints which are closer to natural joints were replicated by keeping the asperity angles of 15°-30° and 15°-45°. The sample size and amplitude of roughness were kept the same for both regular and irregular joints which were 298 mm×298 mm×125 mm and 5 mm, respectively. Shear test was performed on these joints using a large-scale direct shear testing machine by keeping the frequency and amplitude of shear load under constant cyclic condition with different normal stress values. As expected, the shear strength of rock joints increased with the increases in the asperity angle and normal load during the first cycle of shearing or static load. With the increase of the number of shear cycles, the shear strength decreased for all the asperity angles but the rate of reduction was more in case of high asperity angles. Test results indicated that shear strength of irregular joints was higher than that of regular joints at different cycles of shearing at low normal stress. Shearing and degradation of joint asperities on regular joints were the same between loading and unloading, but different for irregular joints. Shear strength and joint degradation were more significant on the slope of asperity with higher angles on the irregular joint until two angles of asperities became equal during the cycle of shearing and it started behaving like regular joints for subsequent cycles.

Numerical manifold method modeling of coupled processes in fractured geological media at multiple scales

The greatest challenges of rigorously modeling coupled hydro-mechanical(HM)processes in fractured geological media at different scales are associated with computational geometry.These challenges include dynamic shearing and opening of intersecting fractures at discrete fracture scales as a result of coupled processes,and contact alteration along rough fracture surfaces that triggers structural and physical changes of fractures at micro-asperity scale.In this paper,these challenges are tackled by developing a comprehensive modeling approach for coupled processes in fractured geological media based on numerical manifold method(NMM)at multiple scales.Based on their distinct geometric features,fractures are categorized into three different scales:dominant fracture,discrete fracture,and discontinuum asperity scales.Here the scale is relative,that of the fracture relative to that of the research interest or domain.Different geometric representations of fractures at different scales are used,and different governing equations and constitutive relationships are applied.For dominant fractures,a finite thickness zone model is developed to treat a fracture as a porous nonlinear domain.Nonlinear fracture mechanical behavior is accurately modeled with an implicit approach based on strain energy.For discrete fractures,a zero-dimensional model was developed for analyzing fluid flow and mechanics in fractures that are geometrically treated as boundaries of the rock matrix.With the zero-dimensional model,these fractures can be modeled with arbitrary orientations and intersections.They can be fluid conduits or seals,and can be open,bonded or sliding.For the discontinuum asperity scale,the geometry of rough fracture surfaces is explicitly represented and contacts involving dynamic alteration of contacts among asperities are rigorously calculated.Using this approach,fracture alteration caused by deformation,re-arrangement and sliding of rough surfaces can be captured.Our comprehensive model is able to handle the computational challenges with accurate representation of intersections and shearing of fractures at the discrete fracture scale and rigorously treats contacts along rough fracture surfaces at the discontinuum asperity scale.With future development of three-dimensional(3D)geometric representation of discrete fracture networks in porous rock and contacts among multi-body systems,this model is promising as a basis of 3D fully coupled analysis of fractures at multiple scales,for advancing understanding and optimizing energy recovery and storage in fractured geological media.

Effect of source parameters on forward-directivity velocity pulse for vertical strike slip fault in half space

It has been found that the large velocity pulse is one of the most important characteristics of near-fault strong ground motions. Some statistical relationships between pulse period and the moment magnitude for near-fault strong ground motions have been established by Somerville （1998）; Alavi and Krawinkler （2000）; and Mavroeidis and Papageorgiou （2003）, where no variety of rupture velocity, fault depth, and fault distance, etc. were considered. Since near-fault ground motions are significantly influenced by the rupture process and source parameters, the effects of some source parameters on the amplitude and the period ofa forward-directivity velocity pulse in a half space are analyzed by the finite difference method combined with the kinematic source model in this paper. The study shows that the rupture velocity, fault depth, position of the initial rupture point and distribution of asperities are the most important parameters to the velocity pulse. Generally, the pulse period decreases and the pulse amplitude increases as the rupture velocity increases for shallow crustal earthquakes. In a definite region besides the fault trace, the pulse period increases as the fault depth increases. For a uniform strike slip fault, rupture initiating from one end of a fault and propagating to the other always generates a higher pulse amplitude and longer pulse period than in other cases.

Application of Fractal Contact Model in Dynamic Performance Analysis of Gas Face Seals

Fractal theory provides scale?independent asperity contact loads and assumes variable curvature radii in the contact analyses of rough surfaces, the current research for which mainly focuses on the mechanism study. The present study introduces the fractal theory into the dynamic research of gas face seals under face?contacting conditions. Structure?Function method is adopted to handle the surface profiles of typical carbon?graphite rings, proving the fractal con?tact model can be used in the field of gas face seals. Using a numerical model established for the dynamic analyses of a spiral groove gas face seal with a flexibly mounted stator, a comparison of dynamic performance between the Majumdar?Bhushan(MB) fractal model and the Chang?Etsion?Bogy(CEB) statistical model is performed. The result shows that the two approaches induce differences in terms of the occurrence and the level of face contact. Although the approach distinctions in film thickness and leakage rate can be tiny, the distinctions in contact mechanism and end face damage are obvious. An investigation of fractal parameters D and G shows that a proper D(nearly 1.5) and a small G are helpful in raising the proportion of elastic deformation to weaken the adhesive wear in the sealing dynamic performance. The proposed research provides a fractal approach to design gas face seals.

Instantaneous stress release in fault surface asperities during mining-induced fault-slip

Fault-slip taking place in underground mines occasionally causes severe damage to mine openings as a result of strong ground motion induced by seismic waves arising from fault-slip. It is indicated from previous studies that intense seismic waves could be generated with the shock unloading of fault surface asperities during fault-slip. This study investigates the shock unloading with numerical simulation. A three-dimensional （3D） numerical model with idealized asperities is constructed with the help of discrete element code 3DEC. The idealization is conducted to particularly focus on simulating the shock unloading that previous numerical models, which replicate asperity degradation and crack development during the shear behavior of a joint surface in previous studies, fail to capture and simulate. With the numerical model, static and dynamic analyses are carried out to simulate unloading of asperities in the course of fault-slip. The results obtained from the dynamic analysis show that gradual stress release takes place around the center of the asperity tip at a rate of 45 MPa/ms for the base case, while an instantaneous stress release greater than 80 MPa occurs near the periphery of the asperity tip when the contact between the upper and lower asperities is lost. The instantaneous stress release becomes more intense in the vicinity of the asperity tip, causing tensile stress more than 20 MPa. It is deduced that the tensile stress could further increase if the numerical model is discretized more densely and analysis is carried out under stress conditions at a great depth. A model parametric study shows that in-situ stress state has a significant influence on the magnitude of the generated tensile stress. The results imply that the rapid stress release generating extremely high tensile stress on the asperity tip can cause intense seismic waves when it occurs at a great depth.

A Normal Contact Stiffness Model of Joint Surface Based on Fractal Theory

Based on the fractal theory,a normal contact stiffness model is established.In the model,the asperity is initially in elastic deformation under contact interference.As the interference is increased,a transition from elastic to elastoplastic to full plastic deformation occurs in this order.The critical elastic interference,the first elastoplastic critical interference and the second elastoplastic critical interference are scale-dependent.According to the truncated asperity size distribution function,the relations between the total normal contact stiffness and the total contact load are obtained.The results show the total normal contact stiffness depends on the range of frequency indexes of asperities.The normal contact stiffness in elastic deformation is major contribution to the total normal contact stiffness.When the first six frequency indexes are less than the critical elastic frequency index,the dimensionless load-stiffness relation approximately isF^*r^(K^*r)^3.When the initial frequency index is greater than the critical elastic frequency index,the dimensionless load-stiffness relation approximately isF^*r^K^*r.The comparison between the theoretical results and the experimental results indicates that the theoretical results are consistent with the experimental data;therefore,the present fractal model of contact stiffness is reasonable.