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    J. P. BardetDepartment of Cvil Engineering,

    University of Southern California,Los Angeles, CA 90089-0242

    Lode DependencesPressure-SensitiveMaterialsfor IsotropicElastoplastic

    Experimental investigations indicate that the third stress invariant; Lode angle cyaffects significantly the behavior of pressure sensitive materials. The present com-munication presents a formulation to account for cy in isotropic pressure-sensitiveelastoplastic materials. Seven Lode dependences are reviewed. A new one, referredto as LMN, in proposed to generalize Lade and Duncan, and Matsuoka and Nakaifailure surfaces. The formulation is general enough to introduce CYnto the isotropicelastoplastic modes which are only developed in terms of first and second-stressinvariants. As an illustration, several Lode dependences are introduced into Roscoeand Burland model. The performance of the modQiied model is estimated by com-paring experimental and analytical results in the case of true triaxial loadings onnormally consolidated clay.

    1 IntroductionMany conventional plasticity models for pressure-sensitive

    materials, such as Roscoe and Burland (1968), have been for-mulated in terms of the first and second-stress invariants with-out considering the third invariant, sometimes referred to asa Lode angle o(. This simplifying assumption disregards theexperimental observations made during true axial testing whichindicate that the third stress invariant affects significantly thebehavior of pressure sensitive soils, rocks, and concretes (Chenand Saleeb, 1982; Lade and Duncan, 1975; Matsuoka andNakai, 1974).The objective of the present communication is to proposea formulation which accounts for 01 n isotropic pressure-sen-sitive elastoplastic materials. The present work reviews, com-pares, and generalizes the past work of Argyris et al. (1974),Dafalias and Herrmann (1986), Eekelen (1980), Lade (1975),Matsuoka and Nakai (1974), William and Warnke (1975) andZienkiewicz and Pande (1977). The formulation aims at in-troducing the effects of cy nto the isotropic elastoplastic modelswhich are developed in terms of only first and second-stressinvariants.After reviewing the fundamental definitions of isotropicplasticity, the contribution of 01on the direction and amplitudeof incremental plastic strain is isolated from other plastic con-tributions. A particular class of yield and plastic potentialfunctions is introduced for pressure-sensitive materiaIs. Thisclass of functions is illustrated by seven examples of a-de-

    Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETYOF MECHANICAL ENGINEERS for publication in the JOURNAL OF APPLIED ME-CHANICS.

    Discussion on this paper should be addressed to the Technical Editor, LeonM. Keer, the Technological Institute, Northwestern University, Evanston, IL60208, and will be accepted until two months after final publication of the paperitself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASMEApplied Mechanics Division, Mar. 14, 1988; final revision, Oct. 25, 1988.

    498 I Vol. 57, SEPTEMBER 1990

    pendences found in the literature and by a new one referredto as LMN dependence. LMN dependence is shown to describethe failure surfaces of Lade and Duncan (1975) and Matsuokaand Nakai (1974). The last section applies the general for-mulation to introduce (Y into the yield and plastic potentialfunctions of the Roscoe and Burland (1968) model. The per-formance of the modified model is estimated by comparingexperimental and analytical results in the case of true triaxialloadings on normally consolidated clay.

    2 Elastoplastic Constitutive EquationsThe sign convention of soil mechanics is preferred to the

    one of solid mechanics in order to eliminate redundant minussigns for pressure-sensitive materials that experience mostlycompressive stress and strain. Hereafter, stresses and strainsare assumed to be positive in compression and negative intension.2.1 Stress-Strain Relationships. According to one of thefundamental assumptions of the flow theory of plasticity, thestrain increment is the sum of elastic and plastic strain incre-ments:

    de,]= de,,+ de;. (1)The elastic strain increment resulting from a given stress in-crement is generally defined by using isotropic elasticity

    d=2G-3Bt?l 18~~ hrjdakkZ&a,, (2)where the shear modulus G and the bulk modulus B may bestress or strain dependent. Adopting the conventional for-mulation of the flow theory of plasticity, the plastic strainincrement is:

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    d+= mij (3)where the symbol < > represents the following function ofany scalar 2:

    2L

    if z>O = 0 if ~50 (4)mij is the normalized gradient of the plastic potential functiong(%j):

    agmij = aa0

    C

    (5)ag ag-

    auk/ aakl

    The normalized gradient nij of the yield function f(o& is sim-ilarly defined by using the flow function in equation (5). Ac-cording to the terminology of plasticity, the flow rule is referredto as associative when nti and mti coincide. The tensors mij andnij are normalized so that they have only directional effects onthe plastic flow, the amplitude of which is directly related tothe plastic modulus H. The modulus His generally calculatedby enforcing the stress state to stay on the yield surface f(qij) = 0during plastic flow (consistency condition). The strain mcre-ment resulting from a given stress increment is obtained byadding equations (2) and (3).

    de, = cijk/ dUk/,where the elastoplastic constitutive matrix is

    (6)

    cijkl = z &,dk,+& (&kh,,+$fijk ) +$kIrnu. (7)2.2 Inverted Stress-Strain Relationship. In many in-stances, such as the displacement formulation of nonlinearfinite element methods, stresses need to be computed as resultsof strain loading. Therefore, equation (6) must be inverted to

    provide the stress increments resulting from given strain in-crements.doll = Di,k, dEk,. (8)

    It can be shown that the inverted elastoplastic constitutivematrix DiJkl s(9)

    where the inverse elastic constitutive matrix Etik, is6,6ki+GBkS /+6,~,k). (10)

    The inversion of equation (6) poses no problem as long as thedenominator of equation (9) is different from zero, which isgenerally satisfied except for some particular stress states.Equation (9) may be rewritten in a simpler formDukt = D,/ - f Mipkl (11

    whereMjj = Eoklmkl N = E,klh (12)

    K=H+N,m,=H+ (n,,)(mbb)+2GnCdmC&(13)

    I& and Nti defines the direction and existence, respectively,of the stress relaxation which results from plastic strain. K isrelated to the amplitude of stress relaxation caused by plasticstrain. In contrast to nii and m,,, Nij and Mi, are not normalized.

    2.3 Istropic Elastoplastic Constitutive Relation-ships. The hypothesis of isotropy is a convenient assumptionwhich simplifies the mathematical formulation of plasticity.By definition, isotropic elastoplastic models have yield andplastic potential functions which depend only on three stressinvariants, instead of six independent stress componentsf(uti) =f(Z, J, U) and g(a,,) =g(Z, J, 4, (14)

    where I, J, and (Yare the first and second-stress invariants andLode angle, respectively,I= ak k

    The deviator stress is

    (15)

    (16)and the third stress invariant S is

    (17)

    cy is preferred to S since cu can be represented in the principalstress space. According to equation (15), CI varies between7r/6 and +*/6. The differentials of I, J, and (Y with respectto stress components aO arear- = 6,ao,aJ-= 3aao 25

    where S, is

    and has the following propertiessp,, = 0s,;i, = 0SCS,,= 1

    (18)

    (19)

    (20)S, is undefined for (Y= f 7r/6 when the gradient of (Y s equalto zero in (18).By invoking the chain rule of differential calculus, the gra-dient of isotropic yields functions is a linear combination off,, fJ and f,, the differential of the yield function with respectto I, J and CX, espectively:

    1 .s.. JzfE=f&,,+;f$++s,,. (21)IJ

    The terms on the right-hand side of equation (21) defines thevolumetric, deviatoric, and Lode contributions, respectively,on the gradient of yield function. The direction and amplitudef,of Lode contribution are characterized by S,j and 7, respec-

    tively. By using equations (20) and (21), the gradient norm isJournal of Applied Mechanics SEPTEMBER 1990, Vol. 57 I499

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    Fig. 1 Typical failure surface for isotropic pressuresensitive materialshown in principal stress space and on deviatorfc plane

    As represented in Fig. 1, x is the angle between the radialdirection and the gradient of the yield function. According to(21) and (26), x characterizes the amplitude of the Lode con-tribution acting in the direction of S,,.As it will be ilIustrated in a later section, (25) is useful tointroduce the effects of (Y nto the isotropic yield functions f (I,J) that are defined only in terms of first and second invariants.

    3.2 Simple Examples of Yield Function for Pressure-Sen-sitive Materials. Drucker-Prager, Tresca, and Mohr-Cou-lomb provide three examples of yield functions that satisfy(25). In the principal stress space, the Drucker-Prager surfaceis a cone with a circular deviatoric section centered on thehydrostatic axis

    f,(Z, J, a)= J-a,Z-b, (28)where a, and b, are two material parameters. Since the yieldfunction is independent of CX,he Lode dependence of Drucker-

    _f* =dG=/W (22) Prager is p,((y)= 1. (29)and n,, may be calculated from (5), (21), and (22). N,, becomes The extended Tresca surface is an angular cone with a regular

    N,, = 3Bf,6,, + G&9 + J;G$S,hexagonal cross-section; its yield function depends on CY:

    (23) fi (I, J, (Y)= Jcoscy - azZ - b2 (30)The definitions and results established for yield functions in where a2 and b2 are two material parameters. Its Lode de-(21), (22), and (23) pertain to plastic potential functions. The pendence Ismodulus K of (13) is related to the plastic modulus H r3

    1K = H f - g af*g* 9Bf,g, i- GfJgJ f I J*p*(ci) = VJ2coscr (31)(24) According to Chen and Saleeb (1982), the yield function ofMohr-Coulomb is

    3 Lode DependenceThe following section introduces a general class of yield f3 (I, J, a) = J[3coscy - &in$sincr] - Isin+ - 3ccos+, (32)

    functions that can be used to account for (Y n isotropic elas- where 4 is the friction angle and c is the cohesion. The yieldtoplastic materials. After reviewing the yields functions of function of Mohr-Coulomb can be rewritten:Diucker-Prager, Tresca, and Mohr-Coulomb requirements arespecified for lode dependences to give differentiable and con- f3(Z, J, a)=*J-

    P3(ol)+Z(P- l)-2cJ(l-2@(1 +P). (33)vex yield functions. Five examples of Lode dependences are

    also presented. Therefore, its Lode dependence is

    3.1 A General Class of Yield Functions for Pressure-Sen-sitive Material. Figure 1 shows a principal stress view oftypical yield surfaces that has been proposed for pressure-dependent soil, rock, and concrete based on true triaxial ex-periments (Lade and Duncan, 1975; Matsuoka and Nakai,1974). In general, such yield functions depend only on Z andthe ratio J/p (cu)where p is a function of CY.They belong tothe following class of isotropic yield functions:

    f(Z, J, cr)=fwhere the Lode dependence p(a) is arbitrarily set to one whency= 7r/6. The yield surfaces f (I, J/p ( (Y) ) = 0 have deviatoricsections in planes of constant Z that look similar but do notnecessarily coincide. The deviatoric sections are circular whenthe yield function does not depend on QL, .e., p(a) = 1. Theyare not circular any longer when p varies with CX,however,they remain periodic curves of period 2~/3 since u varies be-tween - 7r/6 and 7r/6. It can also be shown that the yieldfunctions satisfying equation (25) give

    f, = - JtanXfJ (26)where tanx is related to pu, the differential of p with respectto cy

    tanX=&.P

    P3(4 = Ah(I +p)cos~ r+&- I)sincY (34)

    where the coefficient /3 is related to the friction angle 4:3 - sin@p=- 3 + sin4

    When 4 varies between 0 deg and 90 deg, fi varies between 1to 0.5. As shown in Fig. 2(a), the deviatoric sections of theMohr-Coulomb surface are piecewise linear. They coincidewith Tresca sections when /3 is equal to one and are triangularwhen @ is equal to 0.5. The angle x of Fig. 1 varies IinearIyas a function of CX,but changes abruptly when Q is equal to*r/6; the yield function of Mohr-Coulomb is not differen-

    tiable at Q = f 7r/6.3.3 Requirements for Lode Dependence for Pressure-Sen-

    sitive Materials. Besides Drucker-Prager, Tresca, and Mohr-Coulomb, several types of failure surfaces have been proposedfor pressure sensitive materials. They generally satisfy (25) andobey the requirements for aspect ratio, differentiability, andconvexity.

    3.3.1 RequirementforAspect Ratio, The requirement onaspect ratio states that:P and p i = 10 (37)

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    Fig. 2 MohrCoulomb (a) and cubic (b) dependences for various valuesof i3

    where fl varies between 0.5 and 1. Equation (37) controls thegeneral shape for the deviatoric sections of yield surfaces. Thisrequirement comes from the experimental results (Lade andDuncan, 1975; Matsuoka and Nakai, 1974):

    0.5sJ(< 1J(n/6) - at Z constantwhere J(r/6) and J( - ~16) are the values of J that correspondto failure in compression with two identical minor principalstresses and to failure in tension with two identical majorprincipal stresses, respectively.

    3.3.2 Requirement for Differentiability. The yield func-tions of (25) are differentiable at a= &r/6 when=o,

    provided that p(cr) and f(Z, J) are differentiable.3.3.3 Requirement on Convexity. The yield surfaces of

    (25) are convex in deviatoric planes (Lin and Bazant, 1986;Jiang and Pietruszczak, 1988) whend* 2-&P+-P:. (39)P

    The convexity requirement is useful to satisfy the stabilitypostulate of Drucker (1951).The following sections present five Lode dependences whichsatisfy the requirement for aspect ratio and differentiabilityand occasionally the requirement for convexity.

    3.4 Additional Examples of Lode Dependence. The firstdependence is based upon a cubic polynomial that is calibratedto satisfy the requirements of (37) and (38). This cubic de-pendence is:

    p,(a)=l-2(~)3(l-il)(~-a)2(~+~) (40)

    Figure 2(b) shows ~~(0) for various values of p between 0.5and 1. The deviatoric cross-sections are smooth but do notsatisfy the convexity requirement of (39) for p smaller than0.85, which corresponds to a friction angle += 13.8 deg ac-cording to (35). x varies continuously with CY ut not linearlyas in the case of Mohr-Coulomb. Instead of a cubic depend-ence, William and Warnkle (1975) proposed an elliptical de-pendence:

    As shown in Fig. 3(a), the elliptical dependence provides convexcross-sections for all values of fl. Due to its convexity for allvalues of /3, William and Warnke dependence was used toaccount for cx n modeling the elastoplastic behavior of concrete(Lin et al., 1987). When /3 tends to 0.5, the deviatoric sectionbecomes triangular, as for Mohr-Coulomb. Argyris 1974) pro-posed a simpler expression than William and Warnke:

    Pfsff) = 231 +P+(P- l)sin3aHowever, as shown in Fig. 3(b), the deviatoric section becomesnonconvex when p becomes less than 7/9, which correspondsto a friction angle 4 = 22 deg. Argyris dependence was latergeneralized by Eekelen (1980) who introduced an additionalparameter n:

    1p,(a) = 2( 1 + pn + (1 - P)sin3a). (43)Equation (42) is a particular case of (43) when n is equal to- 1. Eekelen dependence is plotted in Fig. 4(a) for the valuen = - 0.299. The surface loses its convexity for a value of pbetween 0.7 and 0.6.

    3.5 LMN Dependence. Based on the failure surfaces ofLade and Duncan (1975) and Matsuoka and Nakai (1974), thefollowing Lode dependence is proposed:(44)

    whereif ~50

    0=> if cr>O.

    (45)This dependence is hereafter referred to as LMN, after thefirst letter of Lade, Matsuoka, and Nakai. 0 varies continuouslywith cy since (45) gives 0 = 7r/6 for cr = 0. The deviatoric sectionis shown in Fig. 4(b). By comparing Figs. 3(a) and 4(b), it isconcluded that LMN and elliptical dependences are similar.

    In summary, Lode contribution can be accounted for inisotropic elastoplastic constitutive models by using cubic,

    P&4 =2( 1 - /3)cos(*/6 - Q) + (20 - 1)2/4( I- /32)cos2(7r/6 - (Y)+ /3(5@ 4)

    4(1- fi2)cos2(7r/6 - (u) + (2fi - 1)2 (41)

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    Fig. 3 Elliptical (a) and Argyris (b) dependences for various values of 6

    Fig. 4 Eekelen (8) and LMN (6) dependences for various values of B

    Argyris, elliptical, Eekelen or LMN dependence. Cubic, Ar-gyris, and Eekelen functions have simple mathematical expres-sions but lose convexity for values of parameter 0, typical tofrictional materials. Elliptical and LMN dependences, whichhave a more complicated analytical expression than the otherdependences, satisfy the convexity requirement and are almostidentical for all values of parameter p.

    4 Application of LMN Dependence to Failure SurfacesOne of the advantages of LMN over elliptical dependence

    is that it describes exactly the failure surfaces of Lade-Duncanand Matsuoka-Nakai.

    4.1 Application to Lade-Duncan Surface. Lade andDuncan (1975) proposed the following failure criterion:P-l),z,=o (46)

    where ?, is a dimensionless material constant, Z is the firststress invariant and Z3 is the third stress invariant

    Z, = det(u,,) = ~1~2~3. (47)(Jl, a29 and u3 are the principal stresses. Using the relationbetween Z,, I, J, and (Y:

    I3 = 2 1 1-J3sin3cu - -ZJ* + -Z3.36 3 27 (48)

    By using (46) and (48), the ratio J/Z obeys the following cubicequation:

    $(g3sin3cy- (g*+ (k-i) =O. (49)

    It can be shown that J=p(cu)J,,,,, is the acceptable solution of(49) when p(a) is given by (44) and J,,,,X is(50)

    Lade constant 7, is related to p through:1 (3 - sind)3

    II = l pz(l + p)(l - p*)* = (1 - sind)cos*4 (51)27- 4(1 +p3)3

    4 is the friction angle measured during triaxial compression,i.e., for (Y= n/6. When fi tends toward one, which correspondsto 4 = 0 deg, vI is equal to 27. When fi tends toward 0.5, whichcorresponds to 4 = 90 deg, 7, is infinite. It is worth noting thatLade and Duncan give a larger value for J in tensile failure(CY - p/6) than Mohr-Coulomb.

    4.2 Application to Matsuoka-Nakai Surface. Based alsoupon experimental results, Matsuoka and Nakai (1974) pro-posed a failure surface that slightly differs from a Lade-Duncansurface

    z,z - l/* z3 = 0, (52)where v2 is a dimensionless material constant and I, is thesecond stress invariant

    1z2=- (z*-u#7~).2 (53)

    The ratio I: obeys the following equation:$(g3sin3a- (1-i) (g*+ (k-i) =O. (54)

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    It can be shown that J=p(a)J,, is the acceptable solutionof (54) when p(u) is given by (44) and J,,,, is the same as forMohr-Coulomb:

    J,,,~ = 18++-4 (55)The constant q2 is related to fl and $J through:

    90 9 - sin+*= (2p - 1)(2 - /3) =p1 - sin@ (56)

    In summary, LMN dependence describes the failure surfacesof Lade-Duncan and Matsuoka-Nakai by using two differentvalues of 0. In contrast to the Lade-Duncan surface, the sur-faces of Matsuoka-Nakai, and Mohr-Coulomb coincide atQ= &t/6.

    5 Application of Lode Dependence to the Roscoe-Bur-land Model

    One of the main applications of Lode dependence is tointroduce the effects of (Y into isotropic elastoplastic modelsthat are formulated only in terms of first and second-stressinvariants. Following the suggestion of Dafalias and Hermann(1986), the modified yield functions f (Z, J, CY) re obtained byreplacing J by J/p( (Y) in the yield functionf(Z, J) as indicated in (25). The following section illustrates theintroduction of o( into a particular elastoplastic model.

    5.1 Generalization of Stress-Strain Relation-ships. Among all the material models of soil mechanics, theRoscoe and Burland (1968) model is certainly the most com-monly used to describe the behavior of normally and over-consolidated clays. This model was developed based upontriaxial tests. In spite of the appellation triaxial, these lattertests apply axisymmetric stress states to the soil samples

    022=(r33, and uij=O if i#j. (57)According to (57), stresses are described by two independentcomponents, such as mean pressure p and deviator stress q

    1P=-(%+2%) q=u,,-J33.3When (57) applies, p and q are simply related to the stressinvariants Z and J

    p=; q=&J (59)and

    1~16 if q>O(Y= -?r/6 if q

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    The incremental strain can be integrated analytically when Jvaries from 0 to J,,,,,. Since Z and CY re constant, the stress isa linear function of J:sin(i?-a) if i=j= 1,2,3 (74)

    if i#jand the direction of Lode contribution is constant:

    -?&Los3cu ( _J;_!3 2 sin3++$$ >2 if i=j= 1 2 1s,= 0 if i#j(75)

    After integrating (71) and (73), the elastic and plastic strainsbecomes

    Fig. 6 Comparison between theory and experiment: failure stress indeviatoric stress plane for various true triaxial tests (experimental dataafter Nakai et al., 1966)

    ISuch stress-controlled loadings were carried out in the labo-ratory by Nakai et al. (1986). With the mean of a true triaxialapparatus, they applied five different values of cx ((Y= 30 deg,15 deg, 0 deg, - 15 deg, and -30 deg) to cubical materialsamples made of normally consolidated Fujinomori clay. The-ory and experiment can be compared by examining (1) failurestress states and (2) stress-strain responses. The values of themodel constants are provided by Nakai et al. (1986):h- = 0.0508,1 +e, *=0.0112,+=33.7degand v=O. Thevalueof p=O.688 for yiid function corresponds to 4= 33.7 deg.The value of &, for plastic potential function is chosen to beequal to 0.688 and 1 in the case of associative and nonasso-ciative flow rules, respectively.

    5.2.1 Comparison of Predicted and Measured FailureStresses. The maximum value JmaXor Jcorresponds to H= 0:Jmax= 3&ZMg (a). (70)

    Figure 6 compares the experimental and theoretical failurestresses obtained for five different values of cy. LMN andelliptical dependences provide the best agreement with exper-imental observations. The Mohr-Coulomb dependence givesconservative failure stresses whereas cubic, elliptical, Argyris,and Eekelen dependences produce nonconservative failurestresses. The original Roscoe-Burland model that does notaccount for cy(/3 = 1) clearly overestimates the material strengthwhen LYs different of 7r/6.

    5.2.2 Comparison of Predicted and Measured Stress-StrainResponse. For the particular loading of (69), the elastic strainincrement is

    where the shear modulus G is constant since Z is constant(Roscoe and Burland, 1968):1-2~ l+eoG=- -I.2(1 +V) K (72)

    The plastic strain increment ish--K 1&+--- 1 +eo fJ %dJ27J2 3g, au0 l+- M;p*Z

    (73)

    (76)where

    In (76), p and tanx depend on fl and flp, respectively. Thestress-strain responses of (76) have been specified for the ma-terial constants of Nakai et al. (1986) and plotted in Figs. 7,8, and 9 for various values of a. The material responses arereported as in Nakai et al. (1986) by plotting the ratio betweenthe major and minor principal stresses, i.e., u1/u3, versus ma-jor, intermediate, and minor principal strains, noted t,, t2,and e3, respectively. The volumetric strain E,= t, + t2 + t3 is alsoplotted versus the major principal strain t,. In the triaxialcompression test of Fig. 7(a), all Lode dependences give iden-tical results since p(7r/6) = 1 and p,(r/6) = 0 for any values ofp and 6,. The stress-strain response is identical to the onepredicted by the original Roscoe-Burland model. In the tensiontest of Fig. 7(b), all Lode dependences with 0 = 0.688 give thesame stress-strain response independently of Pp, sincep( - 7r/6) =/3 and p,( - 7r/6)=0 for any 0,. The stress-strainresponse of the original Roscoe and Burland model that isplotted in a dashed line in Fig. 7(b) largely overestimates thematerial strength. Similar results are obtained for the loadingscorresponding to cy= f 15 deg which are shown in Figs. 7(c)and 9(d). Figure 8 compares the associative and nonassociativeresponses resulting from LMN dependences during the par-ticular loading at CX=O deg; the intermediate principal strainEd s simulated more accurately with a nonassociative flow rule.Figure 9 compares the responses of six different Lode de-pendences for p=O.688 and &= 1. Figure 9 indicates majordifferences for the maximum value of a,/a3, but not for theintermediate strain c2, which is to be expected since pa = 0 when/3,= 1.In summary, it is concluded that both LMN and ellipiticaldependences give identical stress-strain responses and failurestresses for the generalized Roscoe and Burland model. TheMohr-Coulomb dependence gives the most conservative failurestress and the softest stress-strain response. The other Lodedependences-cubic, Argyis, and Eekelen-can be used alter-natively to LMN or elliptical dependences, when the frictionangle is small or when yield and plastic potential functions arenot required to be convex.

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    Fig. 7(a)

    6-0!= -30 \ 0 \ -I5- \ -a,,\ I\ I4- E3 \- I E,A A \ I

    2-

    I- , , , , ,- 1 0 r f I I1 0

    o o A +----_

    Fig. 7(c)6 CY=-15 \\ a,; I

    /

    Fig. 7(b) Fig. 7(d)Fig. 7 Comparison between theory and experiment: principal strainsversus principal stress ratio and volumetric strain in true triaxial testsfor (a) a=30 deg, (b) CI= -30 deg, (c) a=15 deg, and (do U= -15 deg(experimental data after Nakai et al., 1966)

    Fig. 6 Comparison between experiment and theory for associative andnonassociative flow rules: principal strains versus principal stress ratioand volumetric strain in true triaxial tests for u=O deg (experimentaldata after Nakai et al., 1966).

    6 ConclusionA particular class of yield and plastic potential functions

    has been introduced in order to account for the effects of Lodeangle on isotropic pressure-sensitive materials. This class offunctions is illustrated by seven Lode dependences found inthe literature and by a new one, referred to as LMN, that issimilar to the William and Warnke (1975) dependence. The

    6cl!= 0 (T -I /5 \

    Fig. 9 Comparison between experimental and theory for six differenttypes of lode dependence: principal strains versus principal stress ratioand volumetric strain in true triaxial tests for a=0 deg (experimentaldata after Nakai et al., 1966)

    LMN dependence gives a convex surface for all values of fric-tional angle and generalizes the failure surfaces of Lade andDuncan (1975) and Matsuoka and Nakai (1974). The proposedformulation has been applied to introduce the Lode angle (Yinto the Roscoe and Burland model. The modified model iscapable of reproducing the experimental observations madeon true triaxial tests of clay. The formulation is general enoughto introduce CY nto the isotropic elastoplastic models which

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    are only developed in terms of first and second-stress invar-iants.

    7 AcknowledgmentThe financial support of the U.S. National Science Foun-

    dation (grants CEE-8404315 and MSM 8657999) is acknowl-edged.

    ReferencesArgyris, J. H., et al., 1974, Recent Development in the Finite ElementAnalysis of Pressure Container Reactor Vessel , Nuclear Engineer ing and De-

    sign, Vol. 28, pp. 42-75.Chen, W. F., and Saleeb, A. F., 1982, ConstirutiveEquarionsforEngineedngMaterials: EIosficify ond Modeling, John Wiley and Sons, New York.Dafalias, Y. F., and Herrmann, L. R., 1986, Bounding Surface Plasticity.II Application to Isotropic Cohesive Soils, Journal of Engineering Mechanics,Vol. 112, pp. 1263-1290.

    Drucker, D. C., 1951, A More Fundamental Approach to Plastic-StressStrain Relations, Proceedings 1st U.S. National Congress on Applied Me-chonrcs, ASME, New York, pp. 487-491.Eekelen, H. A. M., 1980, Isotropic Yield Surface in Three Dimensions forUse in Soil Mechanics , Inl. J. for Numerical and Anolytrcal Methods in Geo-mechanics, Vol. 4, pp. 89-101.Gudehus, G., 1973, Elastoplastische stoffgleichungen fur trockenen sand,Ingenieur Archiu, Vol. 42, pp. 151-169.

    Jiang, J., and Pietruszczak, S., 1988, Convexity of Yield Loci for PressureSensitive Materials, submitted to Computers ond Georechnics.Lade, P. V., and Duncan, J. M., Duncan, 1975, Elastoplastic Stress-StrainTheory for Cohesionless Soil, J. Geotechnical Engineering Division, ASCE,

    New York, Vol. lOI, pp. 1037-1053.Lin, F. B., and Bazant, Z. P., 1986, Convexity of Smooth Yield Surface ofFrictional Material, Journal of Engineering Mechanics, ASCE, New York,Vol. 112, pp. 1259-1262.Lin, F. B., Bazant, Z. P., Chern, J. C., and Marchertas, A. H., 1987, Con-crete Model with Normality and Sequential Identification, Computers and

    Stmclures, Vol. 26, pp. 1011-1025.Matsuoka, H., and Nakai, T., 1974, Stress-deformation and Strength Char-acteristics of Soil Under Three Different Principal Stress, Proceedings JapaneseSociety of Civil Engineers, Vol. 232, pp. 59-70.

    Nakai, T., Matsuoka, H., Okuno, N., and Tsuzuki, K., 1986, True TriaxialTests on Normally Consolidated Clay and Analysis of the Observed Shear Be-havior Using Elastoplastic Constitutive Models, Soils ond Foundations. Vol.26, pp. 67-78.Roscoe, K. H., and Borland, J. B., 1968, On the Generalized Stress-StrainBehavior of Wet Clay, Engineering Plasricify, Cambridge Univ. Press, pp.535-609.William, K. J., and Warnke, E. P., 1975, Constitutive Model for the TriaxialBehavior of Concrete, Inlernational Association for Bridge and Structure En-gineering Proceedings, Bergamo, Italy, Vol. 19.Yang, B. L., and Dafalias, Y. F., and Herrmann, L. R., 1985, A BoundingSurface Plasticity Model for Concrete, Journal of Engineerrng Mechanics,ASCE, New York, Vol. I I I, pp. 359-380.Zienkiewicz, 0. C., and Pande, G. N., 1977, Some Useful Forms of IsotropicYield Surfaces for Soil andRock Mechanics, Fimle Elements in Geomechanics,G. Gudehus, ed., John Wiley and Sons, New York, pp. 179-190.

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