科学计算_Annual Maximums of Daily Rainfall in Sydney(悉尼日降水量的年度最大值)

3维Copula函数在降雨特征多变量频率分析中的应用刘成林;周玉文;隋军;高琳【摘要】There is an underlying assumption that run⁃off and rainfall in a given urban catchment are equivalent and, further, to use design rainfall depth as a proxy for run⁃off in hydrological analyses and calculations. However, when employing this approach, it is difficult to accurately and fully reflect the variability in rainfall characteristics. To address this issue, a method for the copula⁃based multivariate frequency analysis of rainfall characteristics was proposed by using historical rainfall data (1961-2012) from Guangzhou city. First, continuous rainfall time series were divided into individual rainfall events using the rainfall intensity method. Then the characteristic variables of rainfall were calculated by sampling using the annual maximum method. Finally, a three⁃dimensional copula was introduced to build a multivariate joint probability distribution model of rainfall characteristics. The results show that the copula⁃based multivariate analysis is easy to implement and provides reliable results. This approach can be used to analyse the conditional probabilities of variables for different orders of magnitude. It can fully reflect rainfall characteristics, which serve an important reference for urban flood control and drainage planning.%鉴于径流数据缺乏且难以长期监测而降雨数据相对完整，通常假定降雨和径流同频率，采用设计降雨进行水文分析计算，但此方法很难真实全面地反映降雨变化特征。

室内外计算参数一般术语第1.0.1条为统一采暖通风与空气调节工程的术语及其释义，实现专业术语的标准化，以利于国内外技术交流，促进采暖通风与空气调节技术的发展，特制订本标准。

第1.0.2条本标准适用于采暖通风与空气调节及其制冷工程的设计、科研、施工、验收、教学及维护管理等方面。

第1.0.3条本标准主要选取采暖通风与空气调节工程中的常用术语。

执行本标准时，尚应遵守国家现行的有关标准的规定。

第2.1.1条计算参数design conditions特指设计计算过程中所采用的表征空气状态或变化过程及太阳辐射的物理量。

常用的计算参数有干球温度、湿球温度、含湿量、比焓、风速和压力等。

第2.1.2条室内外计算参数indoor and outdoor design conditions设计计算过程中所采用的室内空气计算参数、室外空气计算参数和太阳辐射照度等参数的统称。

第2.1.3条空气温度air temperature暴露于空气中但又不受直接辐射的温度表所指示的温度。

一般指干球温度。

第2.1.4条干球温度dry-bulb temperature干球温度表所指示的温度。

第2.1.5条湿球温度wet-bulb temperature湿球温度表所指示的温度。

第2.1.6条黑球温度black globe temperature黑球温度表所指示的温度。

第2.1.7条露点温度dew-point temperature在大气压力一定、某含湿量下的未饱和空气因冷却达到饱和状态时的温度。

第2.1.8条空气湿度air humidity表征空气中水蒸汽含量多少或干湿程度的物理量。

第2.1.9条绝对湿度absolute humidity单位体积的湿空气中所含水蒸汽的质量。

第2.1.10条相对湿度relative humidity空气实际的水蒸汽分压力与同温度下饱和状态空气的水蒸汽分压力之比，用百分率表示。

第2.1.11条历年值annual(value)逐年值。

高考英语一轮总复习必修第一册提能训练：Unit 4 Natural DisastersⅠ.阅读理解A(2024·浙江1月高考题) On September 7, 1991, the costliest hailstorm (雹暴) in Canadian history hit Calgary's southern suburbs. As a result, since 1996 a group of insurance companies have spent about $2 million per year on the Alberta Hail Suppression Project. Airplanes seed threatening storm cells with a chemical to make small ice crystals fall as rain before they can grow into dangerous hailstones. But farmers in east-central Alberta—downwind of the hail project flights—worry that precious moisture (水分) is being stolen from their thirsty land by the cloud seeding.Norman Stienwand, who farms in that area, has been addressing public meetings on this issue for years. “Basically, the provincial government is letti ng the insurance companies protect the Calgary-Edmonton urban area from hail，” Mr. Stienwand says, “but they're increasing drought risk as far east as Saskatchewan.”The Alberta hail project is managed by Terry Krauss, a cloud physicist who works for Weath er Modification Inc. of Fargo, North Dakota. “We affect only a very small percentage of the total moisture in the air, so we cannot be causing drought，” Dr. Krauss says. “In fact, we may be helping increase the moisture downwind by creating wetter ground.”One doubter about the safety of cloud seeding is Chuck Doswell, a research scientist who just retired from the University of Oklahoma. “In 1999, I personally saw significant tornadoes (龙卷风) form from a seeded storm cell in Kansas，” Dr. Doswell says. “Doe s cloud seeding create killer storms or reduce moisture downwind? No one really knows, of course, but the seeding goes on.”Given the degree of doubt, Mr. Stienwand suggests, “it would be wise to stop cloud seeding.” In practice, doubt has had the opposite effect. Due to the lack of scientific proof concerning their impacts, no one has succeeded in winning a lawsuit against cloud-seeding companies. Hence, private climate engineering can proceed in relative legal safety.语篇导读：本文是一篇说明文。

doi:10.11676/qxxb2024.20230033气象学报10—20 d和30—60 d低频振荡对华南前汛期持续性暴雨的影响差异及机制研究*臧钰歆1 徐邦琪1 高迎侠2ZANG Yuxin1 HSU Pang-Chi1 GAO Yingxia21. 南京信息工程大学气象灾害教育部重点实验室/气象灾害预报预警与评估协同创新中心，南京，2100442. 内蒙古大学生态与环境学院，呼和浩特，0100211. Key Laboratory of Meteorological Disaster of Ministry of Education/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters，Nanjing University of Information Science and Technology，Nanjing 210044，China2. School of Ecology and Environment，Inner Mongolia University，Hohhot 010021，China2023-03-02收稿，2023-08-09改回.臧钰歆，徐邦琪，高迎侠. 2024. 10—20 d和30—60 d低频振荡对华南前汛期持续性暴雨的影响差异及机制研究. 气象学报，82（2）：137-154Zang Yuxin， Hsu Pang-Chi， Gao Yingxia. 2024. The impacts of 10—20 d vs. 30—60 d low-frequency oscillations on South China pre-flood season persistent heavy rainfall: Comparison and associated mechanisms. Acta Meteorologica Sinica， 82（2）:137-154Abstract Based on the persistent heavy rainfall dataset produced by China Meteorological Administration (CMA), the National Oceanic and Atmospheric Administration (NOAA) Climate Prediction Center daily precipitation and Outgoing Longwave Radiation (OLR) dataset, and the ERA-Interim reanalysis product, this study analyzes relative importance of two types of low-frequency oscillations, i.e., 10—20 d and 30—60 d oscillations, for persistent heavy rainfall events during the pre-flood period over southern China. The associated mechanisms are diagnosed using the scale decomposed moisture equation and vertical velocity equation. The results reveal that the 10—20 d quasi-biweekly oscillation has a more significant impact on the intensity of persistent heavy rainfall events, while the 30—60 d intra-seasonal oscillation shows a higher correlation with the duration of heavy rainfall events. This result suggests that persistent heavy rainfall events are closely related to the occurrence and evolution of low-frequency precipitation anomalies. Based on the moisture budget diagnosis, it is found that the 10—20 d precipitation anomaly mainly comes from horizontal moisture advection induced by the interaction between wind perturbation and background moisture. For the 30—60 d precipitation anomaly, both the moisture advection and convergence processes make positive contributions to the accumulation of moisture anomalies, although advection process related to the interaction between background wind fields and 30—60 d moisture perturbation is the primary contributor. The upward motion anomaly, which results from vertical gradient of vorticity advection by the background wind–vorticity perturbation interaction, provides a favorable dynamic condition for the occurrences of both 10—20 d and 30—60 d precipitation anomalies. The above results suggest that better understanding of the scale interaction processes between low-frequency oscillations and background mean state in numerical models is the basis for accurate forecast of persistent heavy rainfall.Key words Pre-flood period， Low-frequency oscillation， Scale interaction， Persistent heavy precipitation摘 要 利用中国气象局发布的《区域性重要过程检测和评价业务规定》中对省级区域性暴雨过程的统计数据，NOAA 气候预测中心降水、向外长波辐射（OLR）资料以及ERA-Interim再分析资料，探讨了10—20 d和30—60 d两类低频振荡对华南前汛期持续性暴雨的相对重要性，并基于尺度分离的水汽方程和垂直速度方程诊断了相关物理机制。

6 天溝排水量計算Tonnage of cullis calculation1降雨量I=100mm/hr判定原則 Qp>Qr ---ok.Rainfall2GUTTER SLOPE S=0.002(1/500)計算3雨棚長L=2.7m Qr=I*Ar=I*L*BThe length of canopy4雨棚寬B=53.7m Qr=0.0040275m^3/secThe width of canopy5天溝上方寬度H=350mm A=0.0224m^2The width of cullis at top6天溝下方寬度T=210mm P=T+2*HThe width of cullis at bottom7天溝總深度W=90mm P=0.37mThe depth of cullis8天溝安全深度Dw=80mm Rh=A/PThe safty depth of cullis9铝板之粗造度係數(SUS)n=0.0125Rh=0.060540541mThe rough efficient of AL panel曼寧公式(Manning Formula)Qp=1/n*A*(Rh)^2/3*S^1/2Qp=1/n*A*(Rh)^2/3*S^1/2The tonnage of culis Qp=天溝之單位排水量Qp=0.012241201m^3/secRh=A/P Rh=水力半徑落水管計算The radius water powerful The calculation for rainspoutA=流體流動之斷面積1落水管支數m=3支The area of water flow section The number of rainspoutP=排水斷面周長2落水管外徑d=7mm The parameters of water flow The outer diameter of rainspoutQr=I*Ar Qr=屋面之單位集水量3重力加速度g=9.8m/sec^2The acceleration of gravityI=降雨強度4天溝安全深度Dw=80mmThe strength of rainfullAr=雨棚之單位面積5落水管斷面積Ad=0.022345852m^2The area of canopy The area of rainspout section6落水管排水量Qd=0.083944298m^3/secThe tonnage of rainspoutQd=m*Ad*(2*g*Dw)^1/2判定原則 Qd>Qr ---ok.。

Unsaturated Soil Mechanics in Engineering PracticeDelwyn G.Fredlund1Abstract:Unsaturated soil mechanics has rapidly become a part of geotechnical engineering practice as a result of solutions that have emerged to a number of key problems͑or challenges͒.The solutions have emerged from numerous research studies focusing on issues that have a hindrance to the usage of unsaturated soil mechanics.The primary challenges to the implementation of unsaturated soil mechanics can be stated as follows:͑1͒The need to understand the fundamental,theoretical behavior of an unsaturated soil;͑2͒the formulation of suitable constitutive equations and the testing for uniqueness of proposed constitutive relationships;͑3͒the ability to formulate and solve one or more nonlinear partial differential equations using numerical methods;͑4͒the determination of indirect techniques for the estimation of unsaturated soil property functions,and͑5͒in situ and laboratory devices for the measurement of a wide range of soil suctions.This paper explains the nature of each of the previous challenges and describes the solutions that have emerged from research puter technology has played a major role in achieving practical geotechnical engineering puter technology has played an important role with regard to the estimation of unsaturated soil property functions and the solution of nonlinear partial differential equations.Breakthroughs in the in situ and laboratory measurement of soil suction are allowing unsaturated soil theories and formulations to be verified through use of the“observational method.”DOI:10.1061/͑ASCE͒1090-0241͑2006͒132:3͑286͒CE Database subject headings:Unsaturated soils;Soil mechanics;Geotechnical engineering;Research.PreambleKarl Terzaghi is remembered most for providing the“effective stress”variable,͑␴−u w͒,that became the key to describing the mechanical behavior of saturated soils;where␴=total stress and u w=pore–water pressure.The effective stress variable became the unifying discovery that elevated geotechnical engineering to a science basis and context.As a graduate student I was asked to purchase and study the textbook,Theoretical Soil Mechanics,by Karl Terzaghi͑1943͒.I had already selected the subject of unsaturated soil behavior as myfield of research and was surprised tofind considerable infor-mation on this subject in this textbook.Two of the19chapters of the textbook contribute extensively toward understanding unsat-urated soil behavior;namely,Chapter14on“Capillary Forces,”and Chapter15,on“Mechanics of Drainage”͑with special atten-tion to drainage by desiccation͒.These chapters emphasize the importance of the unsaturated soil portion of the profile and in particular provide an insight into the fundamental nature and importance of the air–water interface͑i.e.,contractile skin͒. Considerable attention was given to soils with negative pore–water pressures.Fig.1shows an earth dam illustrating how waterflowed above the phreatic line through the capillary zone ͑Terzaghi1943͒.The contributions of Karl Terzaghi toward unsaturated soil behavior are truly commendable and still worthy of study.Subsequent reference to the textbook Theoretical Soil Mechan-ics during my career,has caused me to ask the question,“Why did unsaturated soil mechanics not emerge simultaneously with saturated soil mechanics?”Pondering this question has led me to realize that there were several theoretical and practical challenges associated with unsaturated soil behavior that needed further re-search.Unsaturated soil mechanics would need to wait several decades before it would take on the character of a science that could be used in routine geotechnical engineering practice.I am not aware that Karl Terzaghi ever proposed a special description of the stress state in an unsaturated soil;however, his contemporary,Biot͑1941͒,was one of thefirst to suggest the use of two independent stress state variables when formulating the theory of consolidation for an unsaturated soil.This paper will review a series of key theoretical extensions that were required for a more thorough representation and formulation of unsaturated soil behavior.Research within the agriculture-related disciplines strongly influenced the physical and hydraulic model that Terzaghi developed for soil mechanics͑Baver1940͒.With time,further significant contributions have come from the agriculture-related disciplines͑i.e.,soil science,soil physics,and agronomy͒to geo-technical engineering.It can be said that geotechnical engineers tended to test soils by applying total stresses to soils through the use of oedometers and triaxial cells.On the other hand, agriculture-related counterparts tended to apply stresses to the water phase͑i.e.,tensions͒through use of pressure plate cells. Eventually,geotechnical engineers would realize the wealth of information that had accumulated in the agriculture-related disciplines;information of value to geotechnical engineering. Careful consideration would need to be given to the test proce-dures and testing techniques when transferring the technology into geotechnical engineering.1Professor Emeritus,Dept.of Civil and Geological Engineering,Univ. of Saskatchewan,Saskatoon SK,Canada S7N5A9.Note.Discussion open until August1,2006.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must befiled with the ASCE Managing Editor.The manuscript for this paper was submitted for review and pos-sible publication on February16,2005;approved on May1,2005.This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering,V ol.132,No.3,March1,2006.©ASCE,ISSN1090-0241/ 2006/3-286–321/$25.00.An attempt is made in this paper to give the theory of unsat-urated soil mechanics its rightful position.Terzaghi ͑1943͒stated that “the theories of soil mechanics provide us only with the working hypothesis,because our knowledge of the average physical soil properties of the subsoil and the orientation of the boundaries between the individual strata is always incomplete and often utterly inadequate.”Terzaghi ͑1943͒also emphasized the importance of clearly stating all assumptions upon which the theories are based and pointed out that almost every “alleged contradiction between theory and practice can be traced back to some misconception regarding the conditions for the validity of the theory.”And so his advice from the early days of soil mechan-ics is extremely relevant as the theories for unsaturated soil be-havior are brought to the “implementation”stage in geotechnical engineering.IntroductionFundamental principles pivotal to understanding the behavior of saturated soils emerged with the concept of effective stress in the 1930s ͑Terzaghi 1943͒.There appeared to be considerable interest in the behavior of unsaturated soil at the First International Conference on Soil Mechanics and Foundation Engineering in 1936,but the fundamental principles required for formulating unsaturated soil mechanics would take more than another 30years to be forthcoming.Eventually,a theoretically based set of stress state variables for an unsaturated soil would be proposed within the context of multiphase continuum mechanics ͑Fredlund and Morgenstern 1977͒.There have been a number of challenges ͑i.e.,problems or difficulties ͒that have slowed the development and implement-ation of unsaturated soil mechanics ͑Fredlund 2000͒.Each of these challenges has provided an opportunity to develop new and innovative solutions that allow unsaturated soil mechanics to become part of geotechnical engineering practice.It has been necessary for geotechnical engineers to adopt a new “mindset”toward soil property assessment for unsaturated soils ͑Fredlund et al.1996͒.The primary objective of this paper is to illustrate the progres-sion from the development of theories and formulations to practical engineering protocols for a variety of unsaturated soil mechanics problems ͑e.g.,seepage,shear strength,and volume change ͒.The use of “direct”and “indirect”means of characteriz-ing unsaturated soil property functions has been central to the emergence of unsaturated soil mechanics.The key challenges faced in the development of unsaturated soil mechanics are described and research findings are presented that have made it possible to implement unsaturated soil mechanics into geotech-nical engineering practice.A series of unsaturated soil mechanics problems are presented to illustrate the procedures and methodology required to obtain meaningful solutions to plete and detailed case histories will not be presented but sufficient information is pro-vided to illustrate the types of engineering solutions that are feasible.Gradual Emergence of Unsaturated Soil Mechanics Experimental laboratory studies in the late 1950s ͑Bishop et al.1960͒showed that it was possible to independently measure ͑or control ͒the pore–water and pore–air pressures through the use of high air entry ceramic boratory studies were reported over the next decade that revealed fundamental differences be-tween the behavior of saturated and unsaturated soils.The studies also revealed that there were significant challenges that needed to be addressed.The laboratory testing of unsaturated soils proved to be time consuming and demanding from a technique standpoint.The usual focus on soil property constants was diverted toward the study of nonlinear unsaturated soil property functions.The increased complexity of unsaturated soil behavior extended from the laboratory to theoretical formulations and solutions.Originally,there was a search for a single-valued effective stress equation for unsaturated soils but by the late 1960s,there was increasing awareness that the use of two independent stress state variables would provide an approach more consistent with the principles of continuum mechanics ͑Fredlund and Morgenstern 1977͒.The 1970s was a period when constitutive relations for the classic areas of soil mechanics were proposed and studied with respect to uniqueness ͑Fredlund and Rahardjo 1993͒.Initially,constitutive behavior focused primarily on the study of seepage,shear strength,and volume change problems.Gradually it became apparent that the behavior of unsaturated soils could be viewed as a natural extension of saturated soil behavior ͑Fredlund and Morgenstern 1976͒.Later,numerous studies attempted to combine volume change and shear strength in the form of elasto-plastic models that were an extension from the saturated soil range to unsaturated soil conditions ͑Alonso et al.1990;Wheeler and Sivakumar 1995;Blatz and Graham 2003͒.The study of con-taminant transport and thermal soil properties for unsaturated soils also took on the form of nonlinear soil property functions ͑Newman 1996;Lim et al.1998;Pentland et al.2001͒.The 1980s was a period when boundary-value problems were solved using numerical,finite element,and finite difference mod-eling methods.Digital computers were required and iterative,numerical solutions became the norm.The challenge was to find techniques that would ensure convergence of highly nonlinear partial differential equations on a routine basis ͑Thieu et al.2001;Fig.1.An earth dam shown by Terzaghi ͑1943͒illustrating that water can flow above the phreatic line through the capillary zone ͑reprinted with permission of ErLC Terzaghi ͒Fredlund et al.2002a,b,c͒.Saturated–unsaturated seepage model-ing became thefirst of the unsaturated soils problems to comeinto common engineering practice.Concern for stewardshiptoward the environment further promoted interest in seepage andgeoenvironmental,advection-dispersion modeling.The1990s and beyond have become a period where therehas been an emphasis on the implementation of unsaturated soilmechanics into routine geotechnical engineering practice.A seriesof international conferences have been dedicated to the exchangeof information on the engineering behavior of unsaturated soilsand it has become apparent that the time had come for increasedusage of unsaturated soil mechanics in engineering practice.Implementation can be defined as“a unique and important stepthat brings theories and analytical solutions into engineeringpractice”͑Fredlund2000͒.There are several stages in the devel-opment of a science that must be brought together in an efficientand appropriate manner in order for implementation to becomea reality.The primary stages suggested by Fredlund͑2000͒,areas follows:͑1͒State variable;͑2͒constitutive;͑3͒formulation;͑4͒solution;͑5͒design;͑6͒verification and monitoring;and ͑7͒implementation.Research is required for all of the above-mentioned stages in order that practical,efficient,cost-effective,and appropriate technologies emerge.Primary Challenges to the Implementationof Unsaturated Soil MechanicsThere are a number of primary challenges that needed to beaddressed before unsaturated soil mechanics could become a partof routine geotechnical engineering practice.Several of thechallenges are identified here.Each challenge has an associatedsolution that is further developed throughout the manuscript.Insome cases it has been necessary to adopt a new approach tosolving problems involving unsaturated soils.In this paper,anattempt is made to describe the techniques and procedures thathave been used to overcome the obstacles to implementation;thuspreparing the way for more widespread application of unsaturatedsoil mechanics.Challenge1:The development of a theoretically sound basisfor describing the physical behavior of unsaturated soils,startingwith appropriate state variables.Solution1:The adoption of independent stress state variablesbased on multiphase continuum mechanics has formed the basisfor describing the stress state independent of soil properties.The stress state variables can then be used to develop suitableconstitutive models.Challenge2:Constitutive relations commonly accepted forsaturated soil behavior needed to be extended to also describeunsaturated soil behavior.Solution2:Gradually it became apparent that all constitutiverelations for saturated soil behavior could be extended to embraceunsaturated soil behavior and thereby form a smooth transitionbetween saturated and unsaturated soil conditions.In each case,research studies needed to be undertaken to verify the uniqueness of the extended constitutive relations.Challenge3:Nonlinearity associated with the partial differen-tial equations formulated for unsaturated soil behavior resulted in iterative procedures in order to arrive at a solution.The conver-gence of highly nonlinear partial differential equations proved to be a serious challenge.Solution3:Computer solutions for numerical models have em-braced automatic mesh generation,automatic mesh optimization,and automatic mesh refinement͓known as adaptive grid refine-ment͑AGR͔͒,and these techniques have proved to be of greatassistance in obtaining convergence when solving nonlinear par-tial differential equations.Solution procedures were forthcomingfrom the mathematics and computer science disciplines.Challenge4:Greatly increased costs and time were required for the testing of unsaturated soils.As well,laboratory equipmentfor measuring unsaturated soil properties has proven to be tech-nically demanding and quite complex to operate.Solution4:Indirect,estimation procedures for the character-ization of unsaturated soil property functions were related to thesoil–water characteristic curve͑SWCC͒and the saturated soilproperties.Several estimation procedures have emerged for eachof the unsaturated soil property functions.The computer has alsoplayed an important role in computing unsaturated soil propertyfunctions.Challenge5:Highly negative pore–water pressures͑i.e., matric suctions greater than100kPa͒,have proven to be difficultto measure,particularly in thefield.Solution5:New instrumentation such as the direct,high suc-tion tensiometer,and the indirect thermal conductivity suctionsensor,have provided new measurement techniques for thelaboratory and thefield.Other measurement systems are alsoshowing promise.These devices allow suctions to be measuredover a considerable range of matric suctions.The null type,axis-translation technique remains a laboratory reference procedure forthe measurement of matric suction.Challenge6:New technologies such as those proposed for unsaturated soil mechanics are not always easy to incorporate intoengineering practice.The implementation of unsaturated soilmechanicsfindings into engineering practice has proven to be achallenge.Solution6:Educational materials and visualization systems have been assembled to assist in effective technology transfer ͑Fredlund and Fredlund2003͒.These are a part of teaching and demonstrating the concepts of unsaturated soil behavior;information that needs to be incorporated into the undergraduateand graduate curriculum at universities.Protocols for engineeringpractice are being developed for all application areas of geotech-nical engineering.Changes are necessary in geotechnical engineering practicein order for unsaturated soil mechanics to be implemented.Eachchallenge has been met with a definitive and practical solution.In the case of the determination of unsaturated soil propertyfunctions a significant paradigm shift has been required͑Houston2002͒.The new approaches that have been developed appearto provide cost-effective procedures for the determination ofunsaturated soil property functions for all classes of problems ͑Fredlund2002͒.Laboratory and Field Visualizationof Varying Degrees of SaturationClimatic conditions around the world range from very humid to arid,and dry.Climatic classification is based on the average net moistureflux at the ground surface͓i.e.,precipitation minus potential evaporation͑Thornthwaite1948͔͒.The ground surface climate is a prime factor controlling the depth to the groundwater table and therefore,the thickness of the unsaturated soil zone ͑Fig.2͒.The zone between the ground surface and the water table is generally referred to as the unsaturated soil zone.This is some-what of a misnomer since the capillary fringe is essentially saturated.A more correct term for the entire zone above the water table is the vadose zone ͑Bouwer 1978͒.The entire zone sub-jected to negative pore–water pressures is commonly referred to as the unsaturated zone in geotechnical engineering.The unsaturated zone becomes the transition between the water in the atmosphere and the groundwater ͑i.e.,positive pore–water pressure zone ͒.The pore–water pressures in the unsaturated soil zone can range from zero at the water table to a maximum tension of approximately 1,000,000kPa ͑i.e.,soil suction of 1,000,000kPa ͒under dry soil conditions ͑Croney et al.1958͒.The water degree of saturation of the soil can range from 100%to zero.The changes in soil suction result in distinct zones of saturation.The zones of saturation have been defined in situ as well as in the laboratory ͓i.e.,through the soil–water characteristic curve ͑Fig.3͔͒.Table 1illustrates the terminologies commonly used to describe saturation conditions in situ and in the laboratory.Soils in situ start at saturation at the water table and tend to become unsaturated toward the ground surface.Soils near to the ground surface are often classified as “prob-lematic”soils.It is the changes in the negative pore–water pressures that can result in adverse changes in shear strength and volume mon problematic soils are:expansive orswelling soils,collapsible soils,and residual soils.Any of the above-mentioned soils,as well as other soil types,can also be compacted,once again giving rise to a material with negative pore–water pressures.Unsaturated Soil as a Four-Phase MixtureAn unsaturated soil is commonly referred to as a three-phase mixture ͑i.e.,solids,air,and water ͒but there is strong justification for including a fourth independent phase called the contractile skin or the air–water interface.The contractile skin acts like a thin membrane interwoven throughout the voids of the soil,acting as a partition between the air and water phases.It is the interaction of the contractile skin with the soil structure that causes an unsatur-ated soil to change in volume and shear strength.The unsaturated soil properties change in response to the position of the contrac-tile skin ͑i.e.,water degree of saturation ͒.It is important to viewTable parison of Terminology Used to Describe In Situ and Laboratory Degrees of Saturation In situ zones of saturation Zones of saturation on the soil-watercharacteristic curveCapillary fringeBoundary effect Two phase fluid flowTransition Dry ͑vapor transport of water ͒ResidualFig.2.Illustration of the unsaturated soil zone ͑vadose zone ͒on a regional and localbasisFig. 3.Illustration of the in situ zones of desaturation defined by a soil–water characteristic curvean unsaturated soil as a four-phase mixture for purposes of stress analysis,within the context of multiphase continuum mechanics.Consequently,an unsaturated soil has two phases that flow under the influence of a stress gradient ͑i.e.,air and water ͒and two phases that come to equilibrium under the influence of a stress gradient ͑i.e.,soil particles forming a structural arrangement and the contractile skin forming a partition between the fluid phases ͒͑Fredlund and Rahardjo 1993͒.The contractile skin has physical properties differing from the contiguous air and water phases and interacts with the soil structure to influence soil behavior.The contractile skin can be considered as part of the water phase with regard to changes in volume–mass soil properties but must be considered as an independent phase when describing the stress state and phenom-enological behavior of an unsaturated soil.Terzaghi ͑1943͒emphasized the important role played by surface tension effects associated with the air–water interface ͑i.e.,contractile skin ͒.Distinctive Features of the Contractile Skin :Numerous research studies on the nature of the contractile skin point toward its important,independent role in unsaturated soil mechanics.Terzaghi ͑1943͒suggested that the contractile skin might be in the order of 10−6mm in thickness.More recent studies suggest that the thickness of the contractile skin is in the order of 1.5–2water molecular diameters ͑i.e.,5Å͒͑Israelachvili 1991;Townsend and Rice 1991͒.A surface tension of approximately 75mN/m translates into a unit stress in the order of 140,000kPa.Lyklema ͑2000͒showed that the distribution of water molecules across the contractile skin takes the form of a hyperbolic tangent function as shown in Fig.4.Properties of the contractile skin are different from that of ordinary water and have a water molecular structure similar to that of ice ͑Derjaguin and Churaev 1981;Matsumoto and Kataoka 1988͒.The Young–Laplace and Kelvin equations describe fundamen-tal behavioral aspects of the contractile skin but both equations have limitations.The Young–Laplace equation is not able to explain why an air bubble can gradually dissolve in water without any apparent difference between the air pressure and the water pressure.The validity of the Kelvin equation becomes suspect as the radius of curvature reduces to the molecular scale ͑Adamson and Gast 1997;Christenson 1988͒.Terzaghi ͑1943͒recognized the limitations of the Kelvin equa-tion and stated that if the radius of a gas bubble “approaches zero,the gas pressure …approaches infinity.However,within the range of molecular dimensions,”the equation “loses its validity.”Although Terzaghi recognized this limitation,later researchers would attempt to incorporate the Kelvin equation into formula-tions for the compressibility of air–water mixtures,to no avail ͑Schuurman 1966͒.The details of the laws describing the behav-ior of the contractile skin are not fully understood but the contractile skin is known to play a dominant role in unsaturated soil behavior.Terzaghi ͑1943͒stated that surface tension “is valid regardless of the physical causes.…The views concerning the molecular mechanism which produces the surface tension are still controversial.Yet the existence of the surface film was established during the last century beyond any doubt.”Designation of the Stress StateState variables can be defined within the context of continuum mechanics as variables independent of soil properties required for the characterization of a system ͑Fung 1965͒.The stress state variables associated with an unsaturated soil are related to equilibrium considerations ͑i.e.,conservation of energy ͒of a multiphase system.The stress state variables form one or more tensors ͑i.e.,3ϫ3matrix ͒because of the three-dimensional Cartesian coordinate system generally used for the formulation of engineering problems ͑i.e.,a three-dimensional world ͒.The description of the state variables for an unsaturated soil becomes the fundamental building block for an applied engineering science.The universal acceptance of unsaturated soil mechanics depends largely upon how satisfactorily the stress state variables can be defined,justified,and measured.Historically,it has been the lack of certainty regarding the description of the stress state for an unsaturated soil that has been largely responsible for the slow emergence of unsaturated soil mechanics.Biot ͑1941͒was probably the first to suggest the need for two independent stress state variables for an unsaturated soil.This is evidenced from the stress versus deformation relations used in the derivation of the consolidation theory for unsaturated soils.Other researchers began recognizing the need to use two independent stress state variables for an unsaturated soil as early as the 1950s.This realization can be observed through the three-dimensional plots of the volume change constitutive surfaces for an unsatur-ated soil ͑Bishop and Blight 1963;Matyas and Radakrishna 1968͒.It was during the 1970s that a theoretical basis and justi-fication was provided for the use of two independent stress state variables ͑Fredlund and Morgenstern 1977͒.The justification was based on the superposition of coincident equilibrium stress fields for each of the phases of a multiphase system,within the context of continuum mechanics.From a continuum mechanics stand-point,the representative element volume ͑REV ͒must be suffi-ciently large such that the density function associated with each phase is a constant.It should be noted that it is not necessary for all phases to be continuous but rather that the REV statistically represents the multiphase system.Although the stress analysis had little direct application in solving practical problems,it helped unite researchers on how best to describe the stress state of an unsaturated soil.Three possible combinations of independent stress state vari-ables were shown to be justifiable from the theoretical continuum mechanics analysis.However,it was the net normal stress ͓i.e.,␴−u a ,where ␴=total net normal stress and u a =pore–air pressure ͔and the matric suction ͑i.e.,u a −u w ,where u w =pore–water pres-sure ͒combination of stress state variables that proved to be the easiest to apply in engineering practice.The net normal stress primarily embraces the activities of humans which aredominatedFig.4.Density distribution across the contractile skin reprinted from Liquid–Fluid Interface ,V ol.3of Fundamental of Interface and Colloid Science,J.Lyklema ͑2000͒,with permission from Elsevierby applying and removing total stress͑i.e.,excavations,fills,and applied loads͒.The matric suction stress state variable primarily embraces the impact of the climatic environment above the ground surface.The stress state for an unsaturated soil can be defined in the form of two independent stress tensors͑Fredlund and Morgenstern1977͒.There are three sets of possible stress tensors, of which only two are independent.The stress state variables most often used in the formulation of unsaturated soil problems form the following two tensors:΄͑␴x−u a͒␶yx␶zx␶xy͑␴y−u a͒␶zy␶xz␶yz͑␴z−u a͒΅͑1͒and΄͑u a−u w͒000͑u a−u w͒000͑u a−u w͒΅͑2͒where␴x,␴y,and␴z=total stresses in the x,y,and z directions, respectively;u w=pore–water pressure;and u a=pore–air pressure.The stress tensors contain surface tractions that can be placed on a cube to represent the stress state at a point͑Fig.5͒.The stress tensors provide a fundamental description of the stress state for an unsaturated soil.It has also been shown͑Barbour and Fredlund 1989͒that osmotic suction forms another independent stress tensor when there are changes in salt content of either a saturated or unsaturated soil.All the stress state variables are independent of soil properties and become the“keys”to describing physical phenomenological behavior,as well as defining functional relationships for unsaturated soil properties.The inclusion of soil parameters at the stress state level is unacceptable within the context of continuum mechanics.As a soil approaches saturation,the pore–air pressure,u a, becomes equal to the pore–water pressure,u w.At this point,the two independent stress tensors revert to a single stress tensor that can be used to describe the behavior of saturated soils:΄͑␴x−u w͒␶yx␶zx␶xy͑␴y−u w͒␶zy␶xz␶yz͑␴z−u w͒΅͑3͒Variations in the Description of Stress StateStress tensors containing stress state variables form the basis for developing a behavioral science for particulate materials. The stress tensors make it possible to writefirst,second,and third stress invariants for each stress tensor.The stress invariants associated with thefirst and second stress tensors are shown in Fredlund and Rahardjo͑1993͒.It is not imperative that the stress invariants be used in developing constitutive models;however, the stress invariants are fundamental in the sense that all three Cartesian coordinates are taken into consideration.There have been numerous equations proposed that relate some of the stress variables to other stress variables through the inclusion of soil properties.It is important to differentiate be-tween the role of these equations and the description of the stress state͑at a point͒in an unsaturated soil.It is also important to understand the role that these equations might play in subsequent formulations for practical engineering problems.The oldest and best known single-valued relationship that has been proposed is Bishop’s effective stress equation͑Bishop 1959͒:␴Ј=͑␴−u a͒+␹͑u a−u w͒͑4͒where␴Ј=effective stress and␹=soil parameter related to water degree of saturation,and ranging from0to1.Bishop’s equation relates net normal stress to matric suction through the incorporation of a soil property,␹.Bishop’s equation does not qualify as a fundamental description of stress state in an unsaturated soil since it is constitutive in character.It would be erroneous to elevate this equation to the status of stress state for an unsaturated soil.Morgenstern͑1979͒explained the limitations of Bishop’s effective stress equation as follows:•Bishop’s effective stress equation“…proved to have little impact on practice.The parameter,␹,when determined for volume change behavior was found to differ when determined for shear strength.While originally thought to be a function of degree of saturation and hence bounded by0and1,experi-ments were conducted in which␹was found to go beyond these bounds.•The effective stress is a stress variable and hence related to equilibrium considerations alone.”Morgenstern͑1979͒went on to explain:•Bishop’s effective stress equation“…contains the parameter,␹,that bears on constitutive behavior.This parameter is found by assuming that the behavior of a soil can be expressed uniquely in terms of a single effective stress variable and by matching unsaturated soil behavior with saturated soil be-havior in order to calculate␹.Normally,we link equilibrium considerations to deformations through constitutive behavior and do not introduce constitutive behavior into the stress state.Another form of Bishop’s equation has been used by several researchers in the development of elastoplastic models͑Jommi 2000;Wheeler et al.2003;Gallipoli et al.2003͒.␴ij*=␴ij−͓S w u w+͑1−S w͒u a͔␦ij͑5͒where␴ij=total stress tensor;␦ij=Kroneker delta or substitutiontensor;␴ij*=Bishop’s average soil skeleton stress;and Sw=water degree of saturation.In this case,the water degree of saturation has been substituted for the␹soil parameter.The above-mentioned equation is once again empirical and constitutive in character.Consequently,the Fig.5.Definition of stress state at a point in an unsaturated soil。