Stress-Strain In. Plasticity and Generalized. Limit Equilibrium, Geotechnical Engineering Z. P. BAIANT. Proceedings of the Workshop on

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1 Baiant, Z.P. Ansal, A. M., and Krizek, R. J. (1981). Critical appraisal f endchrnic thery fr sils." Prc., Wrlcshp n Umit Equilibrium, Plasticity and Generalized Stress-strain in Getechnical Engineering, held at McGill Univ., May 1980, ed. by R. K Yng, and H.-Y. K, ASCE, ew Yrk, Z. P. BAIAT Prceedings f the Wrkshp n Limit Equilibrium, Plasticity and Generalized Stress-Strain In Getechnical Engineering McGill University May 28-30, 1980 Spnsred by: atinal Science Fundatin, U.S.A. atinal Science and Engineering Research Cuncil, Canada McGill University Univ. f Clrad Raymnd K. Yng and Hn-Yim K, Wrkshp C-Chairmen Published by the American Sciety f Civil Engineers 345 East 47th Street ew Yrk, ew Yrk 10017

2 53R STRESS STRAI FOR SOILS in "- b", 2" 40 "() Prprtinal Lading Path (,6 O'x!::.O'y!:: ) I7 : 10 p~i c CRITICAL APPRAISAL OF EDOCHROIC THEORY FOR SOILS by Zdenek P. Bazant 1 M.ASCE, Atilla M. Ansal 2, and Raymnd J. Krizek l M.ASCE FIG 10 '" u c 2 '" is Axial Strain Effect f Changes In Behavir in Prprtinal Lading Withut Change In Materia! Cnstants With Change In Mater ia! Cns tan ts Material Cnstants n the Mdel Path fr Ottawa Sand ITRODUCTIO Althugh endchrnic (end a internal + chrn05 = time) thery may nt fllw strictly frm the basic principles f cntinuum mechanics and thermdynamics, this nvel apprach t mdeling the nnlinear cnstitutive behavir f sils and ther similar materials is quite effective and versatile. Therefre, it is wrthwhile t examine clsely the capabilities and limitatins f this thery and t determine its relatinship t ther established cnstitutive theries with emphasis n the advantages it ffers. Accrdingly, the basic features f the thery will be summarized and explained, and, after calibrating the mdel by use f a limited set f test data, predictins f the respnse t certain specified types f lading will be presented and critically interpreted in a cmpanin paper included in Sectin 4 f Vlume 1. GEERAL FORM OF THE THEORY Endchrnic thery is best regarded as a special frm f viscplasticity in which the viscsity cefficients depend n the strain rate, as prpsed by Schapery (1968). Thus, the general frm f the resulting cnstitutive equatin i, (1) in which! represents a (6 x 1) clumn matrix f the cmpnents f the strain tensr, ~ represents a similar clumn matrix f the stress cmpnents, C : S(S!,,,) is a (6 x 6) incremental elastic stiffness matrix, F,. F(~'E) is the lading functin (Figure la), and ~ is a nn-decreasing independent scalar variable, termed the intrinsic time (Valanis. 1971), that depends n the Prfessr f Civil Engineering, rthwestern University, Evanstn, Illinis, USA 2Assistant Prfessr f Getechnical Engineering, Macks Civil Engineering Faculty, Istanbul Technical University, Istanbul, Turkey 539

3 540 STRESS STRAI FOR SOILS EDOCHROIC SOILS THEORY 541 G"12 df de-... defrmatin. It is the use f this latter variable which distinguishes endchrnic thery frm ther cnstitutive theries, such as incremental plasticity r hypelasticity. The intrinsic time may be defined as the length f the path (Figure lb) traced by successive states f the material in strain space and it may be expressed as (2) in which represents a (6x6) matrix f the cefficients characterizing the prper metric f the strain space and the superscript T indicates a transpse f the matrix In applicatins t sils cntaining pre water, the stress cr must be interpreted in accrdance with a Bit-type tw-phase medium mdel and may be assciated with the effective stress cmmnly used in sil mechanics. (a) The frms f the metric matrix,, stiffness matrix,, and lading functin, F, may be simplified cnsiderably by evking the cnditins f initial istrpy f the material. In this case, the functin F must be f the frm: (3 ) in which 1, J, and J are the first invariant, secnd deviatr invariant, and third deviatr invariant, respectively, f either the stress tensr r the strain tensr, as indicated by superscripts cr and. In all practical frms f endchrnic thery applied thus far t sils and cncrete, the dependence f F n the third invariants, ~ and J;, has nt been cnsidered; hence, equatin (3) simplifies t (4) ( b) Figure 1 II Despite missin f the third invariants, the failure envelpe btained frm the peak pints f the respnse curves fr simulated triaxial tests with prprtinal lading at varius stress ratis manifests the frm illustrated in Figure 2. Thus, a runded triangular shape f the failure surface in the ctahedral plane des nt necessarily indicate an influence f the third invariants. Such a shape f the lading surface may be interpreted as an indicatin f the simultaneus influence f the first and secnd invariants f bth stress and strain. Indeed, when F depends nly n the first and secnd stress

4 542 STRESS STRAI FOR SOILS EDOCHROIC SOILS THEORY 543 t vn Mises invariants, the prjectin f the failure surface in the ctahedral plane is a circle, but it cannt remain a circle when F depends als n the first and secnd strain invariants because the strain cmpnents are generally nt prprtinal t the stress cmpnents. One imprtant aspect wrth nting is the fact that endchrnic thery can nt be brught t the incrementally linear frm: (5) in which D is the incremental stiffness matrix fr the ttal (elastic plus inelastic) strains. This simple fact distinguishes endchrnic thery frm mst ther cnstitutive theries, such as hypelasticity, incremental hardening plasticity, plastic-fracturing thery, and ttal strain thery. Hwever, if lading directins in the vicinity f a certain fixed directin are restricted, it is pssible t linearize endchrnic thery and btain equatin (5). In such a case a different incremental stiffness matrix, ~, is btained fr each chice f the straining directin in whse vicinity the thery is t be linearized. TREATMET OF ULOADIG AD RELOADIG Figure 2 Much f the effectiveness and flexibility f endchrnic thery is due t the fact that it can mdel the irreversibility assciated with unlading withut the use f any inequalities. The feature which renders this pssible is the use f the square rt in a quadratic frm in the definitin f intrinsic time (equatin 2). In the case f shear straining, this expressin reduces t Idyl and the increment f shear stress, d~, may be expressed as (6) in which y is the shear angle, G is the shear mdulus, and Fl is a certain functin f stress and strain. If lading (dy > 0) changes t unlading (dy < 0), the first (elastic) term f equatin (6) changes sign while the secnd (inelastic) part f the stress increment retains the same sign, as indicated by the vertical dwnward arrws in Figure 3a. This illustrates visually that the unlading slpe given by endchrnic thery must be smaller than the previus lading slpe. In the case f alternating lads, this prperty enables endchrnic thery t describe hysteresis lps withut any use f an unlading criterin (Figure 3b).

5 544 STRESS STRAI FOR SOILS EDOCHROIC SOILS THEORY 545 CI : 1:1 : ::::> The freging attractive prperty, hwever, wrks t the disadvantage f endchrnic thery when a small unlading is fllwed by relading. In this case, ne btains a relading slpe that is smaller than the previus unlading slpe and the respnse des nt frm a clsed lp (Figure 3c). Althugh this is nt strictly unacceptable in all situatins, a relading slpe that is steeper than the unlading slpe shuld be btained in mst cases (fr example, pure deviatric straining). The nly way t achieve such behavir with endchrnic thery is t avid relying n the intrinsic time alne t mdel the irreversibility phenmenn and t intrduce an unlading criterin. One methd t btain clsed hysteresis lps fr small unlad-relad cycles and assure fulfillment f Drucker's pstulate was described by Ba~ant (1978) and applied by Ba~ant, Krizek, and Shieh (1980). This methd cnsists f tw relatively simple crrectins. First, the intrinsic time increment, dr, must be replaced by c dr, in which c is a crrectin cefficient which is taken as unity fr virgin lading and less than unity fr unlading and relading. The fulfillment f a certain cnditin n the values f c fr unlading, relading, and virgin lading is als necessary t satisfy Drucker's stability pstulate (Bazant, 1978). Furthermre, a three-way lading-unladingrelading criterin is needed (Figure 4a). Unlading is characterized by dw < 0, in which W is sme lading functin (fr example, the wrk stred in the material), and virgin lading, as well as relading, are characterized by dw > O. The distinctin between the latter tw cnditins may be made n the basis f the maximum energy, W ' stred in the material up t the current time. If W is larger than the current W, we have relading, and, if W is equal t W, we have virgin lading. The secnd crrectin which must be intrduced in endchrnic thery is a certain particular frm f kinematic hardening, called jump-kinematic hardening (Ba~ant, 1978) which invlves mving the center f the lading surface t the last extreme stress pint whenever lading changes t unlading r unlading t relading. The pints t which the center f the lading surface is "jumped" are indicated in the stress-strain diagram in Figure 4b, and in the stress-space plts f the subsequent lading surfaces in Figures 4c, 4d, and 4e. With the use f kinematic hardening it is impssible fr the current state pint t mve inward frm the lading surface; nly utward mvement frm the current lading surface can be btained.

6 546 STRESS STRAI FOR SOILS EDOCHROIC SOILS THEORY 547 ESSETIAL DIFFERECES FROM OTHER THEORIES The mst significant difference between varius cnstitutive mdels, such as plastic, endchrnic, and ttal strain (defrmatin) mdels, is the behavir fr s-called lading t the side (that is, lading which is parallel t the lading surface and apprximately nrmal t the prprtinal lading path in.c Q) stress space). The stiffness f the inelastic respnse fr lading t the side (and generally the stiffness f the respnse fr lading f any directin) may be characterized and instructively visualized by means f the inelastic stiffness lcus (Bazant, 1978). This lcus is defined as the lcus f all strain increment vectrs, de ij (that is, a nine-dimensinal vectr frmed f the cmpnents f de.), that give the same magnitude \lde'.'.\1 f the inelastic ij ~J strain increments, de~j' The distance f a pint n the lcus fr any lading directin is prprtinal t the stiffness mdulus fr inelastic strain in that directin. b 0'1 "0 0 3: II 3: C 0 0'10 0 1\ V "0 3: C 3: 0'1 "0 0 "0... > ::> :: <l 3: V 3: 0'1 0 "0 1\ C 3: 0 "0 Q)... a::: >. C 01 ~ Q)... ~ Q) :: Ul.E <l E ::). :: )( 0 0" <l ::E II II :: <l ; b 0" (,) It can be shwn (Bazant, 1978) fr incremental plasticity that the inelastic stiffness lcus is a straight line parallel t the tangent t the lading surface (Figure Sa). The fact that this lcus must be a straight line is als evident frm the linearity prperty f incremental plasticity. Since the lading directin parallel t the lading surface intersects the elastic stiffness lcus at infinity, the respnse t such lading is btained as perfectly elastic. Recently, hwever, it has becme widely accepted that the actual respnse resulting frm lading t the side is nt r shuld nt be perfectly elastic, but sfter than elastic. Fr this reasn, varius theries which intrduce inelastic strain due t lading t the side are being develped. This trend is reflected in the vertex-hardening mdels fr the plasticity f metals and ther materials. In general, the vertex cncept des nt invlve a fixed vertex (crner) at a predetermined place n the lading surface, but a vertex which is always superimpsed n the current stress pint f the lading surface and mves jintly with this pint. Cnceptually, the mst simple and effective vertex mdel appears t be that f Rudnicki and Rice (1975) fr which the inelastic stiffness lcus is btained as the smth curve shwn in Figure 5b. Fr endchrnic thery it can be shwn that the inelastic stiffness lcus is either a circle centered arund the current stress pint n the lading

7 548 STRESS STRAI~ FOR SOILS EDOCHROIC SOILS THEORY 549 0> Q> "t:>... c J: )( Q> -... Q> > '"... C c. E u surface r an ellipse (Figure 5c). Since fr this shape f the inelastic stiffness lcus the tip f the vectr dg fr lading t the side r fr unfinite distance frm the lading surface, the respnse fr ij lading is at a these lading directins is inelastic. Accrdingly, given that ne accepts the widespread pinin that the respnse fr lading tangent t the lading surface is nt perfectly elastic, endchrnic thery appears t be mre realistic. It shuld als be nted that the curved frm f the inelastic stiffness lcus fr endchrnic thery is similar t that fr vertex-hardening plasticity, 8S well as fr the ttal strain (defrmatin) thery_ In view f the curved frm f the inelastic stiffness lcus, endchrnic thery represents 8 develpment in the same directin as the intrductin f vertex-hardening mdels in plasticity. It has been mentined that endchrnic thery can nt be expressed in an.. 15 I:) incrementally linear frm (equatin 5), even if unlading is excluded. In this respect, endchrnic thery differs frm mst existing nnlinear cnstitutive theries, including incremental plasticity. hypelasticity, and the ttal strain (defrmatin) thery. evertheless, it is pssible t linearize endchrnic thery in the vicinity f a chsen fixed lading directin fr the given stress state under cnsideratin. and this brings the thery t the frm f equatin (5). This is achieved by replacing the increment f intrinsic time, dc, given by equatin (2), by the linearized expressin T.ed~ d( --r.i (7).. I:),., () - '" C 0.. C I:) () 0....&: () 0 "t:> lj.j () ~ This linearizatin transfrms endchrnic thery t a stress-strain relatin f essentially the same frm as in incremental plasticity. Gemetrically, this linearizatin crrespnds t the replacement f the circular r curved inelastic stiffness lcus by a tangent straight line. The respnse f this linearized thery is then very clse t the respnse indicated by endchrnic thery fr all straining directins that deviate by less than 30 0 frm the chsen stress directin abut which the linearizatin is made (Figure Sd). Within this fan f directins it is bviusly impssible t find much difference between the varius theries which lead t a different frm f the inelastic stiffness lcus. The freging cnsideratin shws that, in rder t btain experimental infrmatin t prve r refute the validity f endchrnic thery r any ther

8 550 STRESS STRAI FOR SOILS EDOCHROIC SOILS THEORY 551 thery, it is necessary t cnduct experiments in which the lading path frms a sharp crner with a sudden and large change in the lading directin (that is, s-called lading t the side). Fr lading paths which d nt invlve such sharp changes in the lading directin and remain clse t a prprtinal lading path, the test data (with experimental scatter) can prpably be represented equally well by varius theries and it will be virtually impssible t evaluate the advantages f any particular thery. Hwever, highly nnprprtinal lading paths with sudden changes in the lading directin are ften typical f final failure mdes; fr example, the stress cnditin in the sil mass directly under a fting prir t failure cnsists essentially f a vertical stress, whereas at the mment f failure the strain cnsists chiefly f shear n the plane f the previus vertical stress. Significant differences between varius theries are als btained in unlading and cyclic lading. Anther prperty fr which varius theries manifest significant differences are the s-called crss effects, such as the effects f the shear strain increment, dy ' n the nrmal stress increment, da, the shear stress increment, 1Z ll dt ' n the nrmal strain increment, dell' r ne shear strain increment, dy1z' 1Z n the shear stress increment, dt ' n anther plane. In the linear r linearized frm f the incremental cnstitutive equatin (equatin 5), this is mani Z3 fested by nn-zer values f the stiffness matrix cefficients crrespnding t these cmpnents (the upper right crner and lwer left crner f the stiffness matrix). In endchrnic thery, as well as certain ther frmulatins, these crss effects are rather significant and their presence is required t mdel varius salient cnstitutive prperties, such as densificatin r dilatancy due t mntnic and cyclic shear strain. This phenmenn leads t nncaxiality f stress and strain and precludes the use f an rthtrpic frm f the incremental stiffness matrix (in particular, an istrpic frm f this matrix). Thus, the differences between endchrnic thery and the s-called incremental rthtrpic r istrpic mdels are rather significant, althugh these differences can be clearly discerned nly fr certain lading paths. ADVATAGES OF EDOCHROIC THEORY Frm a practical pint f view, the fllwing useful features f endchrnic thery may be summarized: 1. The thery is rather cnvenient and effective fr representing unlading irreversibility, and this makes it particularly useful fr cyclic lading. Since the intrinsic time increases with the number f lading cycles, this variable is suitable as a measure f the sftening r hardening prduced in the sil by cyclic lading. This fact, which was first explited by Valanis (1971), enables endchrnic thery t represent rather simply the stiffening and cntractin exhibited by hysteresis lps frm ne cycle t the next. Z. Cmpared t classical plasticity, the fact that the intrinsic time is independent f the lading surface and its evlutin lends the thery cnsiderable flexibility and enables it t represent diverse phenmena mre easily. Furthermre, certain f the characteristic material functins, such as the hardening functin, sftening functin, and dilatancy functin, have a relatively simple, albeit intuitive, physical interpretatin, and this prvides endchrnic thery with an advantage in helping t understand the behavir f the mdel. 3. Frm the fundamental theretical pint f view, the main difference between endchrnic thery and ther theries is btained fr lading t the side. In this case endchrnic thery always exhibits a sfter respnse fr lading than incrementally linear theries, such as classical incremental plasticity. Since a sfter respnse is bviusly safer with regard t failure predictins (particularly because failure ften ccurs with a stress path f this type), endchrnic thery will yield cnservative predictins. At the same time, a sfter r inelastic respnse fr lading t the side is prbably mre crrect, since all micrstructural mdels fr inelastic behavir f materials pint t the lack f purely elastic respnse fr this type f lading. 4. Endchrnic thery is particularly effective fr mdeling crss effects, such as dilatancy due t shear, and crss hardening (fr example, hardening f the vlumetric respnse due t accumulated shear). 5. Finally, endchrnic thery prvides a relatively simple way t intrduce strain-rate effects. There is, hwever, ne majr disadvantage t endchrnic thery. The cnstitutive equatin can nt be integrated explicitly t give the respnse curves fr varius basic types f tests; this cmplicates data fitting and requires the use f trial-and-errr iterative appraches, pssibly cmbined with ptimizatin cmputer techniques. One ntewrthy exceptin is the case f cyclic pure shear straining, fr which endchrnic thery yields relatively simple explicit

9 552 STRESS STRAI FOR SOILS expressins that can be fitted t test data quite easily (Cuellar et ai, 1977). As a cnsequence f these difficult<ies in the data fitting prcedure, it has thus far been necessary t use a relatively large number f material parameters t represent the hardening, sftening, and dilatancy functins, and this has made it impssible t attach a clear physical significance t each numerical parameter. Hpefully, further wrk will lead t imprvements in this regard. COCLUSIO Endchrnic thery prvides a very flexible and effective apprach fr mdeling the nnlinear behavir f sils. It is able t handle a wide range f phenmena (such as strain rate effects, dilatancy due t shear, hardening and sftening, etc) and seems t predict the prper behavir fr cases invlving highly nn-prprtinal lading with a sharp crner n the stress path. ACKOWLEDGMET Financial supprt by the U.S. atinal Science Fundatin under Grant EG is gratefully acknwledged. REFERECES 1. Bal';ant, Z. P. (1978), "Endchrnic Inelasticity and Incremental Plasticity," Internatinal Jurnal f Slids and Structures, Vlume 14, umber 9, pp Ba~ant, Z. P., Krizek, R. J. and Shieh, C. L. (1979),Hysteretic Endchrnic Thery fr Sand, Technical Reprt, Department f Civil Engineering, rthwestern University, Evanstn, Illinis. 3. Cuellar, V., Ba~ant, Z. P., Krizek, R. J. and Silver, M. L. (1977), '~ensificatin and Hysteresis f Sand under Cyclic Shear," Jurnal f the G"technical and Engineering Divisin, American Sciety f Civil Engineers, Vlume 103, GT5, pp Rudnicki, J. W. and Rice, J. R. (1975), "Cnditins f the Lcalizatin f Defrmatin in Pressure-Sensitive Dilatant Materials," Jurnal f the Mechanics and Physics f Slids, Vlume 23, pp Schapery, R. A. (1968), "On a Thermdynamic Cnstitutive Thery and Its Applicatin t Varius nlinear Materials," prceedings f the Internatinal Unin f Theretical and Applied Mathematics Sympsium, East Kilbride, B. A. Bley, Editr, Springer-Verlag, ew Yrk, ew Yrk, pp Valanis, K. C. (1971), "A Thery f Viscplasticity Withut a Yield Surface; Part I. General Thery; Part II. Applicatin t Mechanical Behavir f Metals," Archives f Mechanics, (Archiwum Mechaniki Stswanej), Vlume 23, pp PLASTICITY MODELS FOR SOILS Thery and Calibratin by E. Mizun l and W. F. Chen 2, M. ASCE 1. Intrductin The mechanical behavir f sil and rck is cmplicated and they can nt be mdelled accurately as a cntinuum. At present, hwever, the cncept f cntinuum mechanics has been used extensively in the mathematical mdelling f these materials. These include the applicatins f linear elastic mdels, nnlinear elastic mdels, and elastic-plastic mdels t getechnical engineering prblems. Althugh the mdels such as hyperelastic r hypelastic can represent the phenmena such as dilitancy and hardening r sftening f sil behavir, the effect f plastic strain induced during lading can nt be predicted within the framewrk f an incremental Hke's law with variable mduli which are functins f the stress and/r strain levels. Current research in sil cnstitutjve mdelling is mving tward the develpment f three-dimensinal stress-strain relatins based n the principles f plasticity as well as elasticity. Herein, three types f sil mdels are described. The first type was used fr predictin befre the wrkshp was held, thus withut the benefit f the test results. The secnd and third types are subsequently develped and used after the wrkshp. (i) nlinear elasticity material mdel with the Mhr-Culmb r the Drucker-Prager surface as failure criterin. (ii) Mhr-Culmb type f elastic-plastic material mdel with tw different sizes f elliptical hardening cap which are defined respectively n the tensile meridian plane (8 = 0 ) and the cmpressive meridian plane (6 = 60 ). (Cap Mdel I) IResearch Assistant, Schl f Civil Engineering, Purdue University, West Lafayette, I Prfessr f Structural Engineering, Schl f Civil Engineering, Purdue University, West Lafayette, I