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Constitutive model for the triaxial behaviour of concreteAuthor(en):William, K.J. / Warnke, E.P.Objekttyp:ArticleZeitschrift:IABSE reports of the working commissions = Rapports des commissions de travail AIPC = IVBH Berichte der ArbeitskommissionenBand(Jahr):19(1974)Persistenter Link:/10.5169/seals-17526Erstellt am:22.08.2011NutzungsbedingungenMit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Dieangebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für dieprivate Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot könnenzusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden.Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorherigerschriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. DieRechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.SEALSEin Dienst des Konsortiums der Schweizer Hochschulbibliothekenc/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweizretro@seals.chhttp://retro.seals.chIABSE AIPC IVBHSEMINAR on:«CONCRETE STRUCTURES SUBJECTED TO TRIAXIAL STRESSES»17th-19th MAY,1974-ISMES-BERGAMO(ITALY)III-lConstitutive Model for the Triaxial Behaviour of Concrete Stoffmodell für das mehrachsiale Verhalten von BetonModlle de Constitution pour le Comportement Triaxial du BitonK.J.WILLAMPh. D.,Project LcaderInstitut für Statik und Dynamik der Luft-und Raumfahrtkonstruktionen University of Stuttgart E.P.WARNKEDipl.-Ing.,Research Associate Institut für Statik und Dynamik der Luft-und Raumfahrtkonstruktionen University of StuttgartSUMMARYThis paper describes different modeis for the failure surface and the constitutive behaviourof concrete under triaxial conditions.The study serves two objectives,the working stressdesign and the ultimate load analysis of three-dimensional concrete components.In the first part a three parameter failure surface is developed for concrete subjected to triaxial loading in the tension and low compression regime.This model is subsequently refined by adding two additional parameters for describing curved meridians,thus extend ing the ränge of appli¬cation to the high compression zone.In the second part two constitutive modeis are formulated for elastic perfectly plastic be¬haviour in compression and elastic perfectly brittle behaviour in tension.Based on the normality principle,explicit ex pressions are developed for the inelastic deformation rate and the correspon¬ding incremental stress-strain relation Thus these modeis can be readily applied to ultimate load analysis using the initial load technique or the tangential stiffness method.Dedicated to the60th birthday of Professor Dr.Drs.h.c.J.H.Argyris.1.INTRODUCTIONOver the last two decades a profound change has taken place with the appearance of digital Computers and recent advances in structural analysis[l],[2],[3].The close symbiosis between Computers and structural theories was instrumental for the development of large scale finite element Software packages[4]which found a wide ränge of application in many fields of eng i nee ring sciences.The high degree of sophistication in structural analysis has clearly left behind many other disciplines,one of them being the field of material science.The proper description of the relevant constitutive phenomena has posed a major limitation on the analysis when applied to complex ope¬rating conditions.In the following a constitutive model is presented for the over load and ultimate load ana¬lysis of three-dimensional concrete structures, e.g.Prestressed Concrete Reactor Vessels and Con¬crete Dams.Considering the size of finite elements in a typical idealization one is clearly deal ing with material behaviour on the continuum level,in which the micro structure of piain and rein¬forced concrete components can be neglected.This scale effect of the analysis allows a macro-scopic point of view according to which material phenomena such as cracking can be simulated bythe behaviour of an equivalent continuum.The objective of this study is twofold:First a mathematical model is developed for the description of initial concrete failure under triaxial conditions.Subsequently,this formulation is applied to construct a constitutive model for the over load and ultimate load analysis of three-dimensional concrete Structures.Alternative ly,the failure surface can be applied to working stress design using relevant safety philosophies.In the first part a three parameter model is developed which defines a conical failure sur¬face with non-circular base section in the principal stress space,thus the strength depends on the hydrostatic as well as deviatoric stress state.The proposed failure surface is convex,continuousand has continuous gradient directions furnishing a close fit of test data in the low compression ränge. In the tension regime the model may be augmented by a tension cut-off criterion.This basic formu¬lation is refined in AppendixII by a five parameter model with curved meridians which provides a close fit of test data also in the high compression regime.Subsequently,a material model is constructed based on an elastic perfectly plastic formu¬lation which is augmented by a brittle failure condition in the tensile regime.In this context equivalent constitutive constraint conditions are developed,based on the"normality"principle, which can be readily applied to the finite element analysis via the concept of initial loads.In the past considerable experimental evidence has been gathered which could be used for the construction of a triaxial failure envelope of concrete.However,most of the data were ob-tained from tests with proportional loading and uniform stress or strain conditions which were distorted by unknown boundary layer effects.For the ultimate load analysis via finite elements these two assumptions are clearly invalid.The non-linearity is responsible for local unloading even if the structure is subjected to monotonically increasing stresses.Moreover,the action of a curved thick-walled structure is control led by non-uniform stress distributions,even if global bending effects and local stress concentrations are neglected for the time being.However,for obvious reasons it is customary to assume that test results from uniform stress-or strain experiments can be used to predict the failure behaviour of structural components subjected to non-uniform stress or strain conditions. One should be aware that this fundamental hypothesis has little justification,except that it is at present the only realistic approach for construct ing a phenomenological constitutive law.The actual mechanism of crack initiation and crack propagation could in fact differ fundamental ly between uniform and non-uniform stress distributions.Considerable test data has accumulated on the multiaxial failure behaviour of mortar and concrete specimens subjected to short term loads.The experimental results can be classified into tests in which either two or three stress components are varied independently.To the first category belong the classical triaxial compression tests on cylindrical specimens(triaxial cell experiments) [5],[6],[7],[8],[9]and the biaxial tension-compression tests on hollow cylinders[lO],[l1J In addition,there is the class of biaxial compression and tension-compression tests on slabs [12]/D3L t14L L15L L16l E17"L t18L[191-The second category contains experi¬ments in which cubic specimens are subjected to arbitrary load combinations[20J,[21J Some of these types of tests are present ly still being processed[22],[23],[24JSo far few attempts have been made to utilize this experimental evidence for construct inga mathematical model of the triaxial failure behaviour of concrete.A comprehensive study of this problem was undertaken in[25],for which similar conclusions were reached in[26J,[27].All three modeis fall into the class of pyramidal failure envelopes which have been examined extensive ly within the context of brittle material modeis as general izations of the Mohr-Coulomb criterion[28].In the same publication different modifications of the Griffith criterion are discussed,which have also been applied in[20]to model the failue surface of cubic mortar specimens in the tension-compression regime.None of these previous studies on failure envelopes was directed towards the non-linear analysis of concrete structures.To this end a number of rather simple material formulations were reviewed in[29],[30],[31]and applied to the ultimate load analysis of different concrete structures.TRIAXIAL FAILURE SURFACEIn the following a mathematical model is developed for the triaxial failure surface of con¬crete type materials.Assuming isotropic behaviour the initial failure envelope is fully describedin the principal stress space.Figure1shows the triaxial envelope of concrete type materials.The failure surface is basically a cone with curved meridians and a non-circular base section.The limited tension capacity is responsible for the tetrahedral shape in the tensile regime,while in compression a cylindrical form is ultimate ly reached.For the mathematical model only a sextant of the principal stress space has to be considered, if the stress components are ordered according to S,>Ct>Gs The surface is conveniently represented by hydrostatic and deviatoric sections where the first one forms a meridianal plane which contains the equisectrix S.«6,.»Gh as an axis of revolution The deviatoric section lies in a plane normal to the equisectrix,the deviatoric trace being described by the polar coordinatesr,ösee Fig. 2.Basically,there are four aspects to the mathematical model of the failure surface:1Close fit of experimental data in the operating ränge.2.Simple identification of model parameters from Standard test data.3.Smoothness-continuous surface with continuously varying tangent planes.4.Convexity-monotonically curved surface without inflection points.Close approximation of concrete data is reached if the failure surface depends on the hydrostatic as well as the deviatoric state,whereby the latter should distinguish different strength values according to the direction of deviatoric stress.Therefore,the failure envelope must be basically a conical surface with curved meridians and a non-circular base section.In addition,in the tensile regime the failure suface could be augmented by a tension cut-off criterion in the form of a pyramid with triangulär section in the deviatoric plane.Simple identification means that the mathematical model of the failure surface is definedby a very small number of parameters which can be determined from Standard test data, e.g. uniaxial tension,uniaxial compression,biaxial compression tests,etc.The description of the failure surface should also encompass simple failure envelopes for specific model parameters.In other words,the cylindrical von Mises and the conical Drucker-Prager model should be special cases of the sophisticated failure formula tion.Continuity is an important property for two reasons:From a computational point of view,it is very convenient if a single description of the failure surface is valid within the stress space under consideration.From the theoretical point of view the proposed failure surface should havea unique gradient for defining the direction of the inelastic deformations according to the1normolity principle1.The actual nature of concrete failure mechanisms also supports the conceptof a gradual change of strength for small variations in loading.Geometrical ly,the smoothness condition implies that the failure surface is continuous and has continuous derivatives.Therefore,the deviatoric trace of the failure surface must pass through r,and ru with the tangents4:,and tr at9-O*and0-CO°,see Fig. 2.Recall thatfor Isotropie conditions only a sextant of the stress space has to be considered,O^O*60*.Convexity is an important property since it assures stable matenal behaviour according to the postulate of Drucker[32],if the"normolity"principle determi es the direction of inelastic deformations.Stability infers positive dissipation of inelastic work during a loading cycle according to the coneepts of thermodynamics Figure3indicates that convexity of the overall deviatoric trace can be assured only if there are no inflection points end if the position vector satisfies the basic convexity conditionJL>J_where r;r(0.:*,i2o",z4o#)Ti.*r(ö*£3,iSo^soo)W Continuity infers compatibility of the position vectors and the slopes et0«O*and0¦GOD. Consequently,there are at least four conditions for curve-fitting the deviatoric trace withinO**0*GOt In addition,the convexity condition implies that the curve should have no inflection points in this interval,thus the approximation can not be based on trigonometric functions[30]or Hermitian Interpolation.If the curve should also degenerate to a circle for r,-r^then an elliptic approximation has to be used for the functional Variation of the deviatoric trace.The ellipsoidal surface assures smoothness and convexity for all position vectors r satisfy ing|rl4r<rL(2)The geometric construction of the ellipse is shown in Fig.4,the details of the derivation are given in the Appendix I.The half axes of the ellipse a,b are defined in terms of the position vectorsErt-4.r,(3)a.r,1'-Sr.ty*Lr%¦4.r,-BrtThe elliptic trace is expressed in terms of the polar coordinates r,B by(r>-r.*)cos9*rt(ir.-r«)&(tf-tf)o»9?Sif-Ar.rj(4a) rf8x1ft,with the angle of similarity9^,^gt"2-g3>(4b) aIn the following the deviatoric trace is used as base section of a conical failure surface with the equisectrix as axis of revolution.A linear Variation with hydrostatic stress generates a cone with straight line meridians.In this case the failure surface is defined in principal stress space by a homogeneous expansion in the"average11stress components$>Ä«:«.and the angle of similarity9IdW^MJ.i^^J-i(5)The average stress components6«,t«.represent the mean distribution of normal and shear stresses on an infinitesimal spherical surface.These values are normolized in the failure condition eq.(5)by the uniaxial compressive strength f^The stress components are defined in terms of principal stresses by(6)^-jL[(».-*£-(*-*£>+t*»-**]*These scalar representations of the state of stress at a point are related to the stress components on the"octahedral"plane60^o by**s*o(7a)**«ff t0The average stress components also correspond to the first principal stress invariant I,and the second deviatoric stress invariant I%according to**-T*.(7b)».-nfi^-RW«^For material failure,-f(>)*0the following constraint condition must hold between the average normal stress and the average shear stresst-rt^D-Ht]i«.1(8)The free parameters of the failure surface model fc H and f«.are identified below from typical concrete test data,such as the uniaxial tension test«f t the uniaxial compression testf4a and the biaxial compression test^Introducing the strength ratios ocft,«c0*z*ft/fco(9) the three tests are characterized by6.TEST Wf«,«/?«.er.*,-ft+--(¥->6Ä»feo3K CO***xei*^"£fc"*««R-O*n(10)Substituting these strength values into the failure condition eq.8,the model parametersare readily obtainedn<*o Af«*»-**«Tu**£«r+ä**oätjt*«««?<*.-«z(11)The apex of the conical surface lies on the equisectrix at*»«*The opening angle<P of the cone varies betweenandtan<y,«-^«+0-GO°(12)(13) The proposed three parameter model is illustrated in Fig.5for the strength ratios0(O'and42*o.i The hydrostatic and deviatoric sections indicate the convexity and smoothness failure envelope.The proposed failure surface degenerates to the Drucker-Prager model of a circular cone ifIn this case the conical failure surface is described by the two parameters2and r0±_**^-^-1i.i of the(14)(15)The single parameter von Mises model is obtained/if in addition7.2.-t>oo(16)[21].In this case the Drucker-Prager cone degenerates into a circular cylinder whose radius is defined byr.fo (17)with the strength ratios**m **"¦(18)Figure 6shows a comparison between the failure surface and experimental data reported inClose agreement can be observed in the low pressure regime for the strength ratios *oc'-Äand *i«o.i?In the high compression regime there is considerable disagreement mainly along the compressive branch.Therefore,the three parameter model is refined in the Appendix II by two additional parameters,extend ing the ränge of application to the high compression regime.This five parameter model establishes a failure surface with curved meridians in which the generators are approximated by second order parabolas along 0s O*and 0*Go with a common apex at the equisectrix,see also Fig.11Figure 7shows the biaxial failure envelope of the three parameter model for three differentstrength ratios otu*l&*^»o-U &u~)o,**.*o.o%and *u-l-&j **-o.is A comparison with test data from [18]^2l]indicates that the shear strength is overestimated consi-derably because of the acute intersection with the biaxial stress plane.However,if we consider the dominant influence of the post-failure behaviour on the structural response [30J,there is little reason for further refinements of the initial failure surface model.3.CONSTITUTIVE MODELIn the following the previous model of the failure envelope is utilized for the developmentof an elastic perfectly plastic material formula tion in compression.The constitutive model is sub¬sequently augmented by a tension cut-off criterion to account for cracking in the tension regime.In both cases it is assumed that the normo lity principle determines the direction of the inelastic deformation rates for ductile as well as brittle post failure behaviour.3.1Elastic Plastic FormulationInviscid plasticity is the classical approach for describing inelastic behaviour via incremen-tal stress-strain relations.The constitutive model is based on two fundamental assumptions,an appropriate description of the material failure envelope and the definition of inelastic deformation rates e.g.via the normo lity principle.a.Yield ConditionThe yield surface serves two objectives,it distinguishes linear from non-linear andelastic from inelastic deformations.The failure envelope is defined by a scalar function of stress,$(J5»o indicating plastic flow if the stress path intersects the yield surface.For concrete type of materials the yield condition can be approximated by the three parameter model shown in Fig.5or more accurately by the five parameter model developed in the Appendix II.b.Flow RuleFor perfectly plastic behaviour the yield surface does not change its configurationduring plastic flow,hence the stress path describes a trajectory on the initial yield surface,whilethe inelastic strains increase continuously.In this case the inelastic deformations do not contri-bute to the elastic strain energy,thus the inner product of plastic strain and elastic stressrates must be zeron**-°09)-In other words,the plastic strain rate must be perpendicular to the yield surfaceV^(2°) where the normal n is the unit gradient vector of the yield surface9j/9*(21)mExplicit expressions of2f/d9are developed in Appendix III for different yield surfaces.The normal defines the direction of the plastic strain rate,the length of which determines the loading parameter a The normality condition follows from Drucker's stability postulate which assures non-negative work dissipation during a loading cycle,also infernng convexity of the yield surface.For perfectly plastic behaviour the material stability is"indifferent"in the small, corresponding to the"neutral"loading condition for which initial yield and subsequent flow is governed by*«>-0and fC*>-o(22) The consistency condition implies that?(«>-Tt*"°(23) This Statement is clearly equivalent to the normality principle stated in eq.(19).c.Incremental Stress-Strain RelationsIn the following an elastic perfectly plastic consitutive model is derived using the previous Statements and the kinematic decomposition of the total deformationsy-C+Ylf,and TF"*+V(24) The linear elastic material behaviour is given by the rate formulation of generalized Hooke's law*Ei-£(i-^(25) Substituting the stress rate into the consistency condition,eq.(23),we obtain£«-n4E(JHW(26) This expression yields for the undetermined loading parameter ahtE(T-n\)-o(27)and hereby±i'WTT"E*(28) x The dot indicates the rate of change.9. The plastic strain rate follows from eq.(20)(29) The incremental stress-strain relations are obtained by Substitut ing y*v into the expression ofthe stress rate,eq.(25)Note the linear relationship between the stress and deformation rates in eq.(30)&F Y od The tangential material law Tr is defined byFor perfectly plastic behaviour,T depends only on the elastic properties and the instantaneous stress state via II The second term of eq.(32)represents the degradation of the material Constitution due to plastic flow.3.2Elastic Cracking FormulationSmall tensile strength is the predominant feature of concrete-type materials.In the following a simple constitutive model is developed for perfectly brittle behaviour in the tensile regime.In analogy to the elastic plastic formulation the elastic cracking model is based on two fundamental assumptions,a tension cut-off criterion for the prediction of cracking and an appro-priate description of inelastic deformation rates e.g.via the normality principle.a.Crack ConditionThe tension cut-off criterion distinguishes elastic behaviour from brittle fracture,i.e. Separation of the material constituents due to excess tension.To this end it is assumed that the scale of Observation justifies a continuum approach.For concrete-type materials cracking may be predicted by the single one parameter model based on the major principal stress^t**)<5-.-^e wifh*>i*^i>^i(33) where C\corresponds in general to the uniaxial tensile strength ft The failure surface is shown in Fig.8,which indicates the pyramidal shape and the triangulär base section in the deviatoric plane.Alternative ly,the tension cut-off condition could also be expressed in terms of the three parameter model of the previous section or the five parameter model developed inthe Appendix II.b.Fracture RuleFor ductile behaviour in the post failure ränge the inelastic deformation rate due to cracking is derived exactly along the formulation of an elastic plastic solid.The ductilepost failure behaviour forms an upper bound of the actual soften ing behaviour[30J,which may develop in concrete components due to reinforcements,dowel action and aggregate interlock.In the following/the case of perfectly brittle post-failure behaviour is discussed,since it requires slight modifications of the previous constitutive model for an elastic perfectly plastic solid. In analogy to elasto-plasticity the inelastic deformations due to cracking tlc do not contribute to the elastic strain energytYI.5-°/iü\10.This normality principle corresponds to the flow rule of plasticity stating that the inelastic strain rate due to cracking is perpendicular to the plane of fractureT|c-nX(35) For the maximum stress tension cut-off criterion the normal vector f\is defined by the direction of the major principal stress;thus in the principal stress spacem »f/»««(36)n|9$/9«M^iwhere£,is the unit vector«,-[l,o,o,o,o.°i(37) For perfectly brittle behaviour the loading parameter X is determined from the sofrening condition|C«}-0and£C«^*-*t(360 In this case the consistency condition infers that|t**t(39)c.Inelastic Strain IncrementsIn the following an expression is derived for the inelastic deformation rotes due to cracking.Substituting the stress rate expression eq.(25)into the consistency condition eq.(39)&*-**(*-M w we obtain an expression for the undetermined loading parameter A«*6(*-«.X)--«e.(") and hereby*-«Fg-*,(*,*T+<)(«)Note the equivalence to the elastic plastic formulation in eq.(28)except for the release of^due to brittle softening.The resulting inelastic fracture strain rate follows from eq.(35)(43)The first portion of this expression can be used to construct incremental stress-strain rela-tions in analogy to the elastic plastic formulation,see eq.(30).This part would correspondexactly to a ductile cracking model in which the major principal stress is kept constant at thetensile strength S,^e The corresponding tangential material law would become transversely isotropic with zero stiffness along the major principal axis.Additional cracking in other directions can be considered according ly.The second portion of eq.(43)represents the sudden stress release due to brittle fracture,*\, which is projeeted onto the structural level by a single initial load step in the analysis.11.3.3Transition ProblemThe previous rate formulation for elastic plastic and brittle behaviour is valid in a diffe¬rential sense only.In a numericaj environment clearly finite increments prevail during numerical Integration of the rate equations[33J,[34J.This approximation problem is magnified by the sudden transition from elastic to plastic or elastic to brittle behaviour.In the latter case the dis-continuity of the process is further increased due to the immediate stress release if the failure condition has been reached.Clearly,the success of the numerical technique depends primärily on the proper treatment of the transition problem for finite increments.Consider the most general case of a finite load step shown in Fig.9.At the outset we assume that the stress path has reached point A for which^C«*V°indicates an elastic state. Due to the finite load increment a fully elastic stress path would reach point B penetrating the yield surface at C for proportional loading.The condition$C^O>°violates Tne constitutive constraint condition^«^°(45) and suggests two strategies for numerical implementation.a.^P^^^J.^6.^^*J£n MethodAssuming proportional loading the load increment is subdivided into two parts,an elastic portion for the path A-C and an inelastic portion governing the behaviour after the failure surface has been reached at C.The evaluation of the penetration point C reduces to the geometric problem of intersecting a surface with a line,a task which is non-linear for curved failure envelopes.The computation of the stress trajectory on the yield surface involves the numerical integration of5^1F if(46)since the tangential material law varies with the current state of stress.In addition we have to assume that the inelastic strains increase proportional ly from ycto<jf£.In numerical calculations additional corrections are required at each iteration step to place the stress path back onto the yield surface[37]b.Normal Penetration MethodIn this scheme we assume that the elastic path reaches the yield surface at the inter-section with the normal ns The evaluation of the foot point D reduces to the geometric-prob¬lem of minimizing the distance between B and the failure envelope,see Fig.9<A-(«^«^(«^«J"*Minimum(47)The extremum condition is used to determine the components of C^by solving the linear system of equations.subjected to the constraint conditionft«*)*ö(49)Note that the loading parameter X is proportional to the distance d,thus the length of the inelastic deformation increment is determined from。
高中英语世界著名科学家单选题50题1. Albert Einstein was born in ____.A. the United StatesB. GermanyC. FranceD. England答案：B。
解析：Albert Einstein（阿尔伯特·爱因斯坦）出生于德国。
本题主要考查对著名科学家爱因斯坦国籍相关的词汇知识。
在这几个选项中，the United States是美国，France是法国，England是英国，而爱因斯坦出生于德国，所以选B。
2. Isaac Newton is famous for his discovery of ____.A. electricityB. gravityC. radioactivityD. relativity答案：B。
解析：Isaac Newton 艾萨克·牛顿）以发现万有引力gravity）而闻名。
electricity是电，radioactivity是放射性，relativity 是相对论，这些都不是牛顿的主要发现，所以根据对牛顿主要成就的了解，选择B。
3. Marie Curie was the first woman to win ____ Nobel Prizes.A. oneB. twoC. threeD. four答案：B。
解析：Marie Curie 居里夫人）是第一位获得两项诺贝尔奖的女性。
这题主要考查数字相关的词汇以及对居里夫人成就的了解，她在放射性研究等方面的贡献使她两次获得诺贝尔奖，所以选B。
4. Thomas Edison is well - known for his invention of ____.A. the telephoneB. the light bulbC. the steam engineD. the computer答案：B。
解析：Thomas Edison（ 托马斯·爱迪生）以发明电灯（the light bulb）而闻名。
(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。
0+||zero-dagger; 读作零正。
1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。
AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。
BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。
B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。
C0 类函数||function of class C0; 又称“连续函数类”。
CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。
Cp统计量||Cp-statisticC。
21-centimeter line, 21厘米线AAbsorption, 吸收Addition of angular momenta, 角动量叠加Adiabatic approximation, 绝热近似Adiabatic process, 绝热过程Adjoint, 自伴的Agnostic position, 不可知论立场Aharonov-Bohm effect, 阿哈罗诺夫-玻姆效应Airy equation, 艾里方程；Airy function, 艾里函数Allowed energy, 允许能量Allowed transition, 允许跃迁Alpha decay, 衰变；Alpha particle, 粒子Angular equation, 角向方程Angular momentum, 角动量Anomalous magnetic moment, 反常磁矩Antibonding, 反键Anti-hermitian operator, 反厄米算符Associated Laguerre polynomial, 连带拉盖尔多项式Associated Legendre function, 连带勒让德多项式Atoms, 原子Average value, 平均值Azimuthal angle, 方位角Azimuthal quantum number, 角量子数BBalmer series, 巴尔末线系Band structure, 能带结构Baryon, 重子Berry's phase, 贝利相位Bessel functions, 贝塞尔函数Binding energy, 束缚能Binomial coefficient, 二项式系数Biot-Savart law, 毕奥-沙法尔定律Blackbody spectrum, 黑体谱Bloch's theorem, 布洛赫定理Bohr energies, 玻尔能量；Bohr magneton, 玻尔磁子；Bohr radius, 玻尔半径Boltzmann constant, 玻尔兹曼常数Bond, 化学键Born approximation, 玻恩近似Born's statistical interpretation, 玻恩统计诠释Bose condensation, 玻色凝聚Bose-Einstein distribution, 玻色-爱因斯坦分布Boson, 玻色子Bound state, 束缚态Boundary conditions, 边界条件Bra, 左矢Bulk modulus, 体积模量CCanonical commutation relations, 正则对易关系Canonical momentum, 正则动量Cauchy's integral formula, 柯西积分公式Centrifugal term, 离心项Chandrasekhar limit, 钱德拉赛卡极限Chemical potential, 化学势Classical electron radius, 经典电子半径Clebsch-Gordan coefficients, 克-高系数Coherent States, 相干态Collapse of wave function, 波函数塌缩Commutator, 对易子Compatible observables, 对易的可观测量Complete inner product space, 完备内积空间Completeness, 完备性Conductor, 导体Configuration, 位形Connection formulas, 连接公式Conservation, 守恒Conservative systems, 保守系Continuity equation, 连续性方程Continuous spectrum, 连续谱Continuous variables, 连续变量Contour integral, 围道积分Copenhagen interpretation, 哥本哈根诠释Coulomb barrier, 库仑势垒Coulomb potential, 库仑势Covalent bond, 共价键Critical temperature, 临界温度Cross-section, 截面Crystal, 晶体Cubic symmetry, 立方对称性Cyclotron motion, 螺旋运动DDarwin term, 达尔文项de Broglie formula, 德布罗意公式de Broglie wavelength, 德布罗意波长Decay mode, 衰变模式Degeneracy, 简并度Degeneracy pressure, 简并压Degenerate perturbation theory, 简并微扰论Degenerate states, 简并态Degrees of freedom, 自由度Delta-function barrier, 势垒Delta-function well, 势阱Derivative operator, 求导算符Determinant, 行列式Determinate state, 确定的态Deuterium, 氘Deuteron, 氘核Diagonal matrix, 对角矩阵Diagonalizable matrix, 对角化Differential cross-section, 微分截面Dipole moment, 偶极矩Dirac delta function, 狄拉克函数Dirac equation, 狄拉克方程Dirac notation, 狄拉克记号Dirac orthonormality, 狄拉克正交归一性Direct integral, 直接积分Discrete spectrum, 分立谱Discrete variable, 离散变量Dispersion relation, 色散关系Displacement operator, 位移算符Distinguishable particles, 可分辨粒子Distribution, 分布Doping, 掺杂Double well, 双势阱Dual space, 对偶空间Dynamic phase, 动力学相位EEffective nuclear charge, 有效核电荷Effective potential, 有效势Ehrenfest's theorem, 厄伦费斯特定理Eigenfunction, 本征函数Eigenvalue, 本征值Eigenvector, 本征矢Einstein's A and B coefficients, 爱因斯坦A,B系数；Einstein's mass-energy formula, 爱因斯坦质能公式Electric dipole, 电偶极Electric dipole moment, 电偶极矩Electric dipole radiation, 电偶极辐射Electric dipole transition, 电偶极跃迁Electric quadrupole transition, 电四极跃迁Electric field, 电场Electromagnetic wave, 电磁波Electron, 电子Emission, 发射Energy, 能量Energy-time uncertainty principle, 能量-时间不确定性关系Ensemble, 系综Equilibrium, 平衡Equipartition theorem, 配分函数Euler's formula, 欧拉公式Even function, 偶函数Exchange force, 交换力Exchange integral, 交换积分Exchange operator, 交换算符Excited state, 激发态Exclusion principle, 不相容原理Expectation value, 期待值FFermi-Dirac distribution, 费米-狄拉克分布Fermi energy, 费米能Fermi surface, 费米面Fermi temperature, 费米温度Fermi's golden rule, 费米黄金规则Fermion, 费米子Feynman diagram, 费曼图Feynman-Hellman theorem, 费曼-海尔曼定理Fine structure, 精细结构Fine structure constant, 精细结构常数Finite square well, 有限深方势阱First-order correction, 一级修正Flux quantization, 磁通量子化Forbidden transition, 禁戒跃迁Foucault pendulum, 傅科摆Fourier series, 傅里叶级数Fourier transform, 傅里叶变换Free electron, 自由电子Free electron density, 自由电子密度Free electron gas, 自由电子气Free particle, 自由粒子Function space, 函数空间Fusion, 聚变Gg-factor, g-因子Gamma function, 函数Gap, 能隙Gauge invariance, 规范不变性Gauge transformation, 规范变换Gaussian wave packet, 高斯波包Generalized function, 广义函数Generating function, 生成函数Generator, 生成元Geometric phase, 几何相位Geometric series, 几何级数Golden rule, 黄金规则"Good" quantum number, "好"量子数"Good" states, "好"的态Gradient, 梯度Gram-Schmidt orthogonalization, 格莱姆-施密特正交化法Graphical solution, 图解法Green's function, 格林函数Ground state, 基态Group theory, 群论Group velocity, 群速Gyromagnetic railo, 回转磁比值HHalf-integer angular momentum, 半整数角动量Half-life, 半衰期Hamiltonian, 哈密顿量Hankel functions, 汉克尔函数Hannay's angle, 哈内角Hard-sphere scattering, 硬球散射Harmonic oscillator, 谐振子Heisenberg picture, 海森堡绘景Heisenberg uncertainty principle, 海森堡不确定性关系Helium, 氦Helmholtz equation, 亥姆霍兹方程Hermite polynomials, 厄米多项式Hermitian conjugate, 厄米共轭Hermitian matrix, 厄米矩阵Hidden variables, 隐变量Hilbert space, 希尔伯特空间Hole, 空穴Hooke's law, 胡克定律Hund's rules, 洪特规则Hydrogen atom, 氢原子Hydrogen ion, 氢离子Hydrogen molecule, 氢分子Hydrogen molecule ion, 氢分子离子Hydrogenic atom, 类氢原子Hyperfine splitting, 超精细分裂IIdea gas, 理想气体Idempotent operaror, 幂等算符Identical particles, 全同粒子Identity operator, 恒等算符Impact parameter, 碰撞参数Impulse approximation, 脉冲近似Incident wave, 入射波Incoherent perturbation, 非相干微扰Incompatible observables, 不对易的可观测量Incompleteness, 不完备性Indeterminacy, 非确定性Indistinguishable particles, 不可分辨粒子Infinite spherical well, 无限深球势阱Infinite square well, 无限深方势阱Inner product, 内积Insulator, 绝缘体Integration by parts, 分部积分Intrinsic angular momentum, 内禀角动量Inverse beta decay, 逆衰变Inverse Fourier transform, 傅里叶逆变换KKet, 右矢Kinetic energy, 动能Kramers' relation, 克莱默斯关系Kronecker delta, 克劳尼克LLCAO technique, 原子轨道线性组合法Ladder operators, 阶梯算符Lagrange multiplier, 拉格朗日乘子Laguerre polynomial, 拉盖尔多项式Lamb shift, 兰姆移动Lande g-factor, 朗德g-因子Laplacian, 拉普拉斯的Larmor formula, 拉摩公式Larmor frequency, 拉摩频率Larmor precession, 拉摩进动Laser, 激光Legendre polynomial, 勒让德多项式Levi-Civita symbol, 列维-西维塔符号Lifetime, 寿命Linear algebra, 线性代数Linear combination, 线性组合Linear combination of atomic orbitals, 原子轨道的线性组合Linear operator, 线性算符Linear transformation, 线性变换Lorentz force law, 洛伦兹力定律Lowering operator, 下降算符Luminoscity, 照度Lyman series, 赖曼线系MMagnetic dipole, 磁偶极Magnetic dipole moment, 磁偶极矩Magnetic dipole transition, 磁偶极跃迁Magnetic field, 磁场Magnetic flux, 磁通量Magnetic quantum number, 磁量子数Magnetic resonance, 磁共振Many worlds interpretation, 多世界诠释Matrix, 矩阵；Matrix element, 矩阵元Maxwell-Boltzmann distribution, 麦克斯韦-玻尔兹曼分布Maxwell's equations, 麦克斯韦方程Mean value, 平均值Measurement, 测量Median value, 中位值Meson, 介子Metastable state, 亚稳态Minimum-uncertainty wave packet, 最小不确定度波包Molecule, 分子Momentum, 动量Momentum operator, 动量算符Momentum space wave function, 动量空间波函数Momentum transfer, 动量转移Most probable value, 最可几值Muon, 子Muon-catalysed fusion, 子催化的聚变Muonic hydrogen, 原子Muonium, 子素NNeumann function, 纽曼函数Neutrino oscillations, 中微子振荡Neutron star, 中子星Node, 节点Nomenclature, 术语Nondegenerate perturbationtheory, 非简并微扰论Non-normalizable function, 不可归一化的函数Normalization, 归一化Nuclear lifetime, 核寿命Nuclear magnetic resonance, 核磁共振Null vector, 零矢量OObservable, 可观测量Observer, 观测者Occupation number, 占有数Odd function, 奇函数Operator, 算符Optical theorem, 光学定理Orbital, 轨道的Orbital angular momentum, 轨道角动量Orthodox position, 正统立场Orthogonality, 正交性Orthogonalization, 正交化Orthohelium, 正氦Orthonormality, 正交归一性Orthorhombic symmetry, 斜方对称Overlap integral, 交叠积分PParahelium, 仲氦Partial wave amplitude, 分波幅Partial wave analysis, 分波法Paschen series, 帕邢线系Pauli exclusion principle, 泡利不相容原理Pauli spin matrices, 泡利自旋矩阵Periodic table, 周期表Perturbation theory, 微扰论Phase, 相位Phase shift, 相移Phase velocity, 相速Photon, 光子Planck's blackbody formula, 普朗克黑体辐射公式Planck's constant, 普朗克常数Polar angle, 极角Polarization, 极化Population inversion, 粒子数反转Position, 位置；Position operator, 位置算符Position-momentum uncertainty principles, 位置-动量不确定性关系Position space wave function, 坐标空间波函数Positronium, 电子偶素Potential energy, 势能Potential well, 势阱Power law potential, 幂律势Power series expansion, 幂级数展开Principal quantum number, 主量子数Probability, 几率Probability current, 几率流Probability density, 几率密度Projection operator, 投影算符Propagator, 传播子Proton, 质子QQuantum dynamics, 量子动力学Quantum electrodynamics, 量子电动力学Quantum number, 量子数Quantum statics, 量子统计Quantum statistical mechanics, 量子统计力学Quark, 夸克RRabi flopping frequency, 拉比翻转频率Radial equation, 径向方程Radial wave function, 径向波函数Radiation, 辐射Radius, 半径Raising operator, 上升算符Rayleigh's formula, 瑞利公式Realist position, 实在论立场Recursion formula, 递推公式Reduced mass, 约化质量Reflected wave, 反射波Reflection coefficient, 反射系数Relativistic correction, 相对论修正Rigid rotor, 刚性转子Rodrigues formula, 罗德里格斯公式Rotating wave approximation, 旋转波近似Rutherford scattering, 卢瑟福散射Rydberg constant, 里德堡常数Rydberg formula, 里德堡公式SScalar potential, 标势Scattering, 散射Scattering amplitude, 散射幅Scattering angle, 散射角Scattering matrix, 散射矩阵Scattering state, 散射态Schrodinger equation, 薛定谔方程Schrodinger picture, 薛定谔绘景Schwarz inequality, 施瓦兹不等式Screening, 屏蔽Second-order correction, 二级修正Selection rules, 选择定则Semiconductor, 半导体Separable solutions, 分离变量解Separation of variables, 变量分离Shell, 壳Simple harmonic oscillator, 简谐振子Simultaneous diagonalization, 同时对角化Singlet state, 单态Slater determinant, 斯拉特行列式Soft-sphere scattering, 软球散射Solenoid, 螺线管Solids, 固体Spectral decomposition, 谱分解Spectrum, 谱Spherical Bessel functions, 球贝塞尔函数Spherical coordinates, 球坐标Spherical Hankel functions, 球汉克尔函数Spherical harmonics, 球谐函数Spherical Neumann functions, 球纽曼函数Spin, 自旋Spin matrices, 自旋矩阵Spin-orbit coupling, 自旋-轨道耦合Spin-orbit interaction, 自旋-轨道相互作用Spinor, 旋量Spin-spin coupling, 自旋-自旋耦合Spontaneous emission, 自发辐射Square-integrable function, 平方可积函数Square well, 方势阱Standard deviation, 标准偏差Stark effect, 斯塔克效应Stationary state, 定态Statistical interpretation, 统计诠释Statistical mechanics, 统计力学Stefan-Boltzmann law, 斯特番-玻尔兹曼定律Step function, 阶跃函数Stem-Gerlach experiment, 斯特恩-盖拉赫实验Stimulated emission, 受激辐射Stirling's approximation, 斯特林近似Superconductor, 超导体Symmetrization, 对称化Symmetry, 对称TTaylor series, 泰勒级数Temperature, 温度Tetragonal symmetry, 正方对称Thermal equilibrium, 热平衡Thomas precession, 托马斯进动Time-dependent perturbation theory, 含时微扰论Time-dependent Schrodinger equation, 含时薛定谔方程Time-independent perturbation theory, 定态微扰论Time-independent Schrodinger equation, 定态薛定谔方程Total cross-section, 总截面Transfer matrix, 转移矩阵Transformation, 变换Transition, 跃迁；Transition probability, 跃迁几率Transition rate, 跃迁速率Translation,平移Transmission coefficient, 透射系数Transmitted wave, 透射波Trial wave function, 试探波函数Triplet state, 三重态Tunneling, 隧穿Turning points, 回转点Two-fold degeneracy , 二重简并Two-level systems, 二能级体系UUncertainty principle, 不确定性关系Unstable particles, 不稳定粒子VValence electron, 价电子Van der Waals interaction, 范德瓦尔斯相互作用Variables, 变量Variance, 方差Variational principle, 变分原理Vector, 矢量Vector potential, 矢势Velocity, 速度Vertex factor, 顶角因子Virial theorem, 维里定理WWave function, 波函数Wavelength, 波长Wave number, 波数Wave packet, 波包Wave vector, 波矢White dwarf, 白矮星Wien's displacement law, 维恩位移定律YYukawa potential, 汤川势ZZeeman effect, 塞曼效应。
