CE-STR-80-l2
PLASTICITY MODELS FOR SOILS PART I: THEORY AND CALIBRATION PART II: COMPARISON AND DISCUSSION
by E. Mizuno and H. F. Chen
Presented at the May 28-30, 1980 North American Workshop on Limit Equilibrium, Plasticity and Generalized Stress-Strain in Geotechnical Engi- neering - Jointly Sponsored by NSF/NSERC.
This material is based upon work supported by the National Science Foundation under Grant No. PFR-7809326 to Purdue University.
School of Civil Engineering Purdue University West Lafayette, IN 47907
July, 1980
50272 -101 REPORT DOCUMENTATION !_I:.-REPORT NO. 3. Recipient's Accession No. PAGE NSF jRA·-800208 4. Title and Subtitle 5. Report Date Plasticity Models for Soils, Part I: Theory and Calibration; July 1980 Part II: Comparison and Discussion 6.
I------~-----t_=__=:__:_------___i 7. Author(s) 8. Performing Organization Rept. No. E. Mizuno, W. F. Chen CE-STR-80-12 9. Performing Organization Name and Address 10. Project/Task/Work Unit No. Purdue University School of Civil Engineering 11. Contract(C) or Grant(G) No. West Lafayette, IN 47907 (C)
(G) PFR7809326
12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered Engineering and Applied Science (EAS) National Science Foundation 1800 G Street, N.W. 14. Washington, D.C. 20550 15. Supplementary Notes
1------.. ------f -16. Abstract (Limit: 200 words) Three types of soil models are described in this report: (1) nonlinear elasticity material model with the Mohr-Coulomb or the Drucker-Prager surface as failure cri terion; (2) Mohr-Coulomb type of elastic-plastic material model with two different sizes of elliptical hardening cap which are defined respectively on the tensile me ridian plane and the compressive meridian plane; (3) Mohr-Coulomb type of elastic plastic material model with an elliptical hardening cap whose size depends on the Lode angle theta. The following assumptions are made for the three types of models: (1) the linear elastic, hypoelastic or hyperelastic function is used in the elastic range for the isotropic or anisotropic material element; (2) the incremental plasticity theory is applied to calculate plastic strain increment during loading range. The concept of "decomposition" of stress state onto tensile meridian plane and compres sive meridian plane is described.
1------1 17. Document Analysis a. Descriptors Soils Elastic properties Models Tensile properties Plastic properties Soil mechanics
b. Identifiers/Open·Ended Terms
Earthquake Hazards Mitigation
c. COSATI Field/Group
18. Availability Statement 19, Security Class (This Report) 21. No. of Pages 1------+------NTIS 20. Security Class (This Page)
(See ANSI-ZS9.18) See Instructions on Reverse OPTIONAL FORM 272 (4-77) (Formerly NTlS-35) Department of Commerce
PLASTICITY MODELS FOR SOILS
Theory and Calibration
by
l E. Mizuno and W. F. Chen2 , M. ASCE
1. Introduction The mechanical behavior of soil and rock is complicated and they can not be modelled accurately as a continuum. At present, however, the con cept of continuum mechanics has been used extensively in the mathematical modelling of these materials. These include the applications of linear elastic models, nonlinear elastic models, and elastic-plastic models to geotechnical engineering problems. Although the models such as hyper elastic or hypoelastic can represent the phenomena such as dilitancy and hardening or softening of soil behavior, the effect of plastic strain induced during loading can not be predicted within the framework of an incremental Hooke's law with variable moduli which are functions of the stress and/or strain levels. Current research in soil constitutive modelling is moving toward the development of three-dimensional stress-strain relations based on the principles of plasticity as well as elasticity. Herein, three types of soil models are described. The first type was used for prediction before the workshop was held, thus without the bene fit of the test results. The second and third types are subsequently developed and used after the workshop. (i) Nonlinear elasticity material model with the Mohr-Coulomb or the Drucker-Prager surface as failure criterion. (ii) Mohr-Coulomb type of elastic-plastic material model with two different sizes of elliptical hardening cap which are defined respectively on the tensile meridian plane (fi = 0°) and the compressive meridian plane (8 = 60°). (Cap Model I) lResearch Assistant, School of Civil Engineering, Purdue University, West Lafayette, IN 47907 2 Professor of Structural Engineering, School of Civil Engineering, Purdue University, West Lafayette, IN 47907
.:r -1- (iii) Mohr-Coulomb type of elastic-plastic material model with an elliptical hardening cap whose size depends on the Lode angle 8. (Cap Model II).
2. A Brief Historical Review The Mohr-Coulomb criterion of failure is certainly the best known in soil mechanics. This criterion states that failure occurs when the shear stress l and the normal stress 0 acting on any element in the material satisfy the linear equation.
l + 0 tan - c a (1)
where c and denote the cohesion and the angle of internal friction, respectively. Although this criterion has in the past been used by necessity and simplicity to obtain reasonable solutions to important, practical problems in geotechnical engineering, the following limitations should be noticed: (1) \This criterion neglects the influence of intermediate principal stress on shear strength; and (2) the failure surface of the Mohr-Coulomb criterion exhibits corners or singularities in the three dimensional principal stress space. From the second limitation, these singularities are difficult to handle in a numerical analysis. The Drucker-Prager surface [7] can then be considered as a three di mensional approximation to the Mohr-Coulomb failure criterion with a simple smooth surface. This criterion is expressed as a linear combination of the first invariant of stress tensor II and the square root of the second invar iant of the deviatoric stress tensor IJ2 together with two material con stants a and k. The material constants a, k can be related to the Coulomb's c and constants in several ways. The Drucker-Prager yield surface with an associated flow rule, however, can not predict the plastic volumetric strain observed in experiments. To improve this, extended von Mises model with convex end cap was proposed by Drucker, Gibson and Kenkel [6]. Following the concept of Drucker et a1., subsequent strain hardening plasticity models using the critical state concept were developed by re searchers at Cambridge [12], and a specific Cam-Clay model based on normally consolidated or lightly overconsolidated clay was suggested by Roscoe,
I -2- Schofield, and Thurairajah [9]. However, failure surface used in this model is still the Drucker-Prager type which resultsin a much greater dila~ tancy prediction than that observed in experiments. As a result, modified failure or yield criterion with an elliptical hardening cap which controls the dila~ancy was subsequently proposed by DiMaggio and Sandler [5]. In recent years, the cap model has been further modified and refined by Sandler [10,11] and Baladi [1]. Various advanced version of this model including such features as kinematic hardening, double hardening etc. have recently been proposed. The historical review of the plasticity modelling of geotechnical materials has been given in a recent paper by Chen [3].
3. Notations The state of stress at a point inside a soil medium can be completely determined by the stress tensor o .. in a three dimensional space. In gener- 1J al, the stress tensor can be decomposed into two parts: (1) the hydrostatic pressure part, where the off-diagonal terms are identically zeros and the diagonal terms are equal to mean normal stress; and (2) the deviatoric part S..• Thus, 1J
0 •• (2) 1J where 1 , the first invariant of the stress tensor, is the sum of the di 1 agonal stress componsnts and 6 .. is the Kronecker delta. The hydrostatic 1J pressure and the deviatoric stress, respectively, cause volumetric change and shape change of the material element. Fig. 1 shows the view of the state of stress in the principal stress --+ coordinate system (01' 02' 03). Stress vector OA can be decomposed into --+ --+ OB in the ~-axis which is called hydrostatic axis (01 = 02 = 03) and BA in the deviatoric plane (n-plane) which is perpendicular to the s-axis. The component vector DB represents the mean normal stress p (1 /3) and the 1 component BA represents the deviatoric stress Sij. The length of DB and BA are 13 p and p = Isil + S~2 + S~3 ' respectively. If the stress vector -T QA is viewed from the hydrostatic ~-axis, the actual length and direction of it can be represented respectively by r and the Lode angle 8. The Lode angle 8 is given by
I -3- J 1 -1 (313 _3_) 8 3 cos (3) 2 J3/2 2 where J is the third invariant of the deviatoric stress tensor. 3 In this paper, the typical continuum mechanics sign convention (tensile stress positive) is utilized in the theoretical development.
4. Failure and Yield Functions The following failure and yield functions are used in the three pro posed models to be described in the subsequent sections.
Mohr-Coulomb Criterion The Mohr-Coulomb criterion given by Eq. I can now be written more generally in terms of stress invariants £8,14].
. "'+ 3(I-sin cj» sin 8 + 13(3+sin cj» cos 8 /-- F I yJ - 3c cos cj> (4) 1 Sl.n 'Y 2 2 o where 6 is the Lode angle (Eq. 3). The cross sectional shape of the Mohr-Coulomb surface on the deviator ic plane is an irregular hexagon as shown in Fig. 2.
Drucker-Prager Criterion This criterion has the simple form:
o (5) where a and k are material constants which can be related to cohesion c and the angle of internal friction cj> of the Mohr-Coulomb criterion in several ways. For example, if the Drucker-Prager criterion is matched with the Mohr-Coulomb criterion in three dimensional principal stress space (Fig. 2) along the compressive meridian (point A) or tensile meridian (point B), the two sets of familiar material constants a and k can be obtained. For the compressive meridian matching, substituting 8 TI/3 into Eq. 4 and rearrang- ing it, we obtain
J: -4- 2 sin (6) k 6c cos These material constants are identical to those given by Zienkiewicz [13]. For the tensile meridian matching, substituting 6 = 0 0 in Eq. 4 and we obtain 2 sin (7) k 6c cos Various matchings between the Drucker-Prager surface and the Mohr Coulomb surface for material constants are given by Chen and Mizuno [4]. In general, if ex is zero, Eq. 5 reduces to the well known von Mises yield condition for metals. The Drucker-Prager surface is used here as the fail ure surface for cap models described in what follows. Eq. 5 represents an axisymmetric cone-shaped surface with respect to 01 = 02 = 03 axis in principal stress space (Fig. 2). 5. Conventional Cap Models The loading functions are usually assumed to be isotropic and to con sist of the following three parts: (i) An ultimate failure envelope can be either of the simple linear Drucker-Prager form or the nonlinear form assumed by Sandler [11]. _ BI l IJ - (A -C e ) (8) 2 :r -5- in which A, Band C are material constants. Here, the failure equation becomes parallel to I axis under large value of I and 1 l' this results in a limited dilatancy under high pressure II. (ii) Strain-hardening cap function has the form of a quarter of an ellipse (Fig. 3) (9) in which x is the intersection of cap with II axis and x is also a hardening function which depends on plastic volumetric change p dE • The location of the cap x is related to the plastic kk p volumetric strain function E with the material constants W kk and D according to P Dx E Wee -1) (10) kk = and R is the ratio of the major to the minor axis of the cap ellipse which may be a function of L and the Lode angle 6, and L is the value of II at the center of the elliptic cap; and (iii) Tension cutoff limit plane is introduced F o (11) t where T is tension cutoff limit. The cap can control the dilatancy of soils under hydrostatic pressure II. Although the cap can predict not only strain-hardening of soils, but also strain-softening, this type of model can not predict exactly the hysteresis loop under shear loading. This is because the hardening function in this model is assumed to be controlled by plastic volumetric strain. Each of these models mentioned above contains several material con stants which can be determined from data of standard simple shear test, iso tropic consolidation test, uniaxial strain test, and triaxial compression, tension tests. The determination of these material constants will be given in the part on model calibration. 1: -6- 6. Basic Concepts of Models Developments The following assumptions are made for the three types of models considered here [2] (i) Linear elastic, hypoelastic or hyperelastic function is used in the elastic range for the isotropic or anisotropic material element, (ii) Incremental plasticity theory is applied to calculate plastic strain increment during loading range, (iii) The Mohr-Coulomb or the Drucker-Prager criterion is used as failure criterion. Effect of strain hardening on this portion of surface is not considered, (iv) Associated flow rule is assumed for the cap hardening portion of the surface. dF __c_ dA dO .• (12) 1J where dA is positive scalar function. In the following, the concept of "decomposition" of stress state onto tensile meridian plane and compressive meridian plane is described. Suppose that 0 .. is the principal stress state acting on an element 1J in soil mass and do .. is the principal stress increment after the 1-J application of an external load increment. The representation of the state of stress as viewed in 11 - ~ space is shown in Fig. 4. The eTE line in Fig. 4 is on the 8 = 0 0 plane and represents the conventional triaxial extension test. The eTe line is on the 8 = 60 0 plane and repre sents the conventional triaxial compression test. If stress path is along 11 axis, it represents the isotropic consolidation test. These three tests are commonly performed tests in geotechnical field. The strengths ob tained by the compression test and tension test for soils are different, and the bulk moduli K and shear moduli G determined from these tests are also different. Thus, in the proposed modelling, different material constants (bulk modulus K and shear modulus G) for CTE and CTe tests are introduced. The following items are taken into consideration in the present developments. :r -7- (i) The state of stress 0ij as represented in II - JJ; space lies in the range of Lode angle from 8 = 0° and 8 = 60° . (ii) The behaviors of material corresponding to the paths lying on the tensile and compressive meridian planes (8 = 0° and 8= 60°) are determined first from CTE and CTC tests, respectively. (iii) The behavior of material corresponding to a stress path lying on a plane making an angle Oo~ 8 ~ 60° is determined from the combined CTE and CTC tests. Herein, the combined concept of CTE and CTC tests for item (iii) is explained. The points A and A' in Fig. 4 denote the present state of principal stress ° and the subsequent state of principal stress (0 .. + dO ..), ij 1J 1J respectively. The vector AX' or do .. is now projected onto the deviatoric 1J stress plane (n-plane) as the vector BB', and onto the CTC-CTE plane as the vector CC'. The vector CC' is on the intersecting line, which is the intersection between the GTC-CTE plane and the plane passing through the -+ hydrostatic axis II and the stress vector AA'. The deviatoric stress vector BB\ on n-plane can now be further de composed into two parts: EE' and DD' along the DE' and OD' axes respectively, as o 1 dS .. dS .. + dS .. (13) 1J 1J 1J o 1 where dS .. and dS .. are the components of the stress in the 8 = 60° and 1J 1J 0° planes, respectively. From the geometry, the magnitudes of these I 0 0 components of dJ on OO-plane and dJ on 60-plane can be calculated Z 2 from the total dJ as 2 dJO dJ [cos(n/3-8) - -l sin(n/3-8)]2 Z Z 13 1 (14) 1 1 dJ dJ (cos 8 - - sin 8)2 2 2 13 J I -8- O 1 Since the deviatoric stresses dS and dS .. have the following char- ij 1.J acteristics: 0 0 0 1 0 1 0 dS? = (dS ' dS , dS ) (dS , - 2" dS ' - "2 dS ) 1.J °ll 22 33 ll U ll 1 (15) 1 1 1 1 1 1 1 dS .. (dS ' dS , dS ) (dS , dS , -2dS ) 1.J ll 22 33 ll ll n J O l Therefore, for the case of loading, the components of dS and dS can be ij ij calculated from Eqs. 14 and 15 as 0 0 0 JJ~ (dS , dS , dS ) = EpJ~-2 - n 22 33 3 3 ' ff) and (16) _jdJ~ 1 1 1 (dS ' dS , dS ) ll 22 33 (ff. 3 ' 2%) -+ Similarly, the vector CC I with length dI on the CTC-CTE plane can be de- l o 1 composed as dI and dI onto the CTC line and CTE line respectively. From l l the geometry of Fig. 4, we have COS(Tr/3 - 8) - 1:..- sin(Tr/3 - 8)J /3 dIl [ 2 cos 8 1 (17) o 1 Therefore, the decomposed principal stresses do .. and do .. can be re- 1.J 1.J written as :t -9- dIO ~IO O __1 + O dIlo0 ) da 0.. = -..1. + dS dS -3- + dS 1J 3 11' 3 22' 33 (18) l l dI 1 ~I1 l l dI l ) do .. = -.l + dS _1+ dS -..1. + dS 1J 3 11' 3 22' 3 33 In the following, explicit expressions for calculating the strains are developed for the three proposed models: (1) nonlinear elasticity model with the Mohr-Coulomb or the Drucker-Prager failure surface; (2) cap model I; and (3) cap model II. 7. Incremental Constitutive Equations 7.1 Nonlinear Elasticity Model In this modelling, behavior of materials is assumed to be elastic until the state of stress reaches the failure surface. For an isotropic, the bulk moduli K ,K and the shear moduli GO ' G may be determined from the O l l triaxial compression and triaxial tension tests. Two types of bulk moduli and shear moduli are considered. (i) Bulk moduli and shear moduli are constant. This reduces to the linear elastic model. (ii) Elastic bulk moduli K and K are assumed to be functions of the O l o 1 first invariant of the decomposed stress tensor II and II ' respectively, and (19) and elastic shear moduli GO and G are assumed to be functions l of the second invariant of the decomposed deviatoric stress o 1 tensor J and J , respectively. 2 2 GO "GaCM) and G1' G1QJ~) (20) According to the incremental Hooke's law, the strain increments d€O and ds1 are written in terms of the decomposed stresses da~. and do:. ~ ~ ~ ~ as .:..:,.,T -10- dOl dIO + 1 0 E: .. 9 1 6.. 2G dS .. lJ KOlJ 0 lJ (21) and the total strain increment dE: .. is the sum of the two components. lJ o 1 dE: .. dE: " + dE: .. (22) lJ lJ lJ It should be noted that the direction of the total strain increment dE ij is not necessary in the same direction as that of the total stress incre- ment dO' ..• The following shortcomings are noted in this modelling: lJ (i) The decomposition of dI into two parts is not unique. Here, for 1 convenience, we use the GTC-CTE planes. (ii) The decomposition of dI for the case of isotropic compression 1 test can not be made because many combinations of decomposition can be considered. (iii) For two stress paths which are extremely close to each other along the hydrostatic pressure axis but lie on two different planes e = 0° and 60° respectively, the volumetric strains predicted with the corresponding bulk moduli K and K will O 1 have different values at the boundary of the hydrostatic axis. The nonlinear elasticity model described above was used for pre dictions at the workshop. 7.2 Cap Model I The conventional cap model used by Ba1adi [1], among others, has the Drucker-Prager type of yield surface with an elliptic cap. The size of the cap is assumed to have either a constant value or to be a function of the plastic volumetric strain. Since the Mohr-Coulomb failure surface is probably the best among all the failure criteria for soils, it follows that the size of the cap should depend not only on the plastic volumetric strain, but also on the Lode angle e. In the present modelling, therefore, J:' -11- two different caps are used. One is defined in the tensile meridian plane (8 = 0°) and the other is defined in the compressive meridian plane (8 60°). This is illustrated in Fig. 5-a, where the Mohr-Coulomb failure lines on these two planes are also shown. In the elastic range, two types of bulk moduli K ,K and shear O l moduli GO ,G may be considered. l (i) K 'K , GO and G are constants. O l l (ii) K 'K are functions of the first invariant of the decomposed O l 0 1 stress tensor 1 or I and the plastic volumetric strains PO Pill Ekk or Ekk , respectively, Ko I (23) and G ,G are functions of the second invariant of the decom- o 1 1 .J~ posed stress tensor or .J 2 and the plastic volumetric . PO PI stra1ns E or E • kk kk (24) where dE~~ and dE~~ are the plastic volumetric strains induced by the caps on 8 = 60° and 0° planes, respectively. The elastic strains dEe are calculated in the same manner as that ij used previously in the nonlinear elasticity model. In the plastic range, the two elliptic caps located on the 8 = 0° and 60° planes will contract or expand as the plastic volumetric strain P E decreases or increases. The Mohr-Coulomb criterion is used here as kk P the failure envelope which will not harden. The plastic strain dE during ij the loading is derived from the flow rule (Eq. 12). The total plastic P PO strain increment dE .. is the sum of the plastic strain increments of dE .. PI 1J 1J and dE .. which are induced by the two caps: 1J J: -12- PO Pl dE:,. + dE: .. (25) 1J 1J Each of the plastic strain increments can be written as 3FO PO O __c_ ds, . = dA 1J 30. , °1.J (26) 1 P1 __3Fc_ dE: ,, = dAl 1.J 3 1 °ij O l l where dA , dA are different positive scalar functions, FO, F are the c c e111.pt1.C" cap funct10ns . andOlcr", 0., are the decomposed stresses, respec- 1.J 1.J tively. From Eqs. 25 and 26, we have PO dE:. , 1.J (27) l 1 l 3F.c nl1 3Fc 3~J2g 3J2 .] 1 ----+------dA [ n l 30:, IT 3J1 30~ . 1 1.J 3 ~J 2 2 1.J _ The total strain increment dE: is the sum of the elastic strain and plastic ij strain increments. dE: .. + 0 .. + 1.J 1.J -13- :r~ ° aF° dA ~ o.. S~J + r. aI~ ~J 1J o (28) ij S:1J'.] Eq. 28 is the incremental stress-strain relations corresponding to the proposed cap model I. In order to use this relation (Eq. 28) for a stress analysis, the scalar functions dAD and d>..l must be determined. Using the consistency condition dF = during loading and Eq. 12, d>" ,d>"1 can be c ° ° derived in a straightforward manner as aFO dFO o GO c 3K dE: +-- s~ . de .. O -6" 1J °1J all kk g d~ 0 dA (29) dF O dFO [aFO~2 3 __c c 9KO --% +G aI 0 [~r dIO dE: PO 1 - 1 kk where de .. is the strain corresponding to the decomposed deviatoric stress °1J 1 dS O Also, d>" has the same form as that of Eq. 29 except changing the ij· subscript from 0 to 1. For the states of stresses on the Mohr-Coulomb surface, the corresponding plastic strain increment dE:~. is derived from ~J the flow rule: P dE: .. (30) 1J 1: -14- where F is the Mohr-Coulomb failure function, cr,. is the total stress and L 1J dA is a positive scalar function. The main characteristics of this model are: (i) The direction of the strain increment dE .. is not necessary in 1J the same direction as that of the stress increment dcr" even 1J with elastic region. (ii) The hardening caps exist only in the e = 0° and e = 60° planes. Two caps control the hardening of the isotropic materials within the range of 0° to 60°. The limitations of this model are: (i) The same limitations as that of nonlinear elasticity model described previously. (ii) Because this model assumes two independent caps, the model may predict a total plastic volumetric strain E: that may exceed k the maximum plastic strain W. (iii) The intersections of the caps with II-axis are not the same. The model does not satisfy the continuity condition along the hydrostatic axis. 7.3 Cap Model II This model can be considered as a generalization of the cap model I described in the preceding section. The two caps in the e = 0° and 60° planes are now connected by a three-dimensional cap as shown in Fig. 5-b. In the elastic range, the same procedure as that of Cap Model I is used for the elastic strain calculation. However, the bulk modulus K ,01 O or K is now a funct10n of 1 or II and the total plastic volumetric strain E~k ~ 1 ~ ~ while the shear modulus GO or Gl is a function of or and the total plastic volumetric strain S~k' respectively. Similarly, the plastic strain increment is determined from Eq. 12. Thus, the total strain increment dE,. is written as 1J dE ij I-15- (IF C +_1_ + + dA (li 0 .. (31) [ 1 1.J 2J J 2 where dA is derived using the same procedure as in Cap Hodel I. For the stress state on the Mohr-Coulomb failure surface, the plastic strain increment is given by Eq. 30. 8. Hodel Calibration General In the workshop, three sets of soil materials are available for prediction under different stress paths. These are "Clay X", "Clay y", "Kaolinite Clay" and "Ottawa Sand". Herein, nonlinear elasticity model is applied to predict the behavior of all three materials. Further, plasticity models (Cap Model I and Cap Model II) are applied only to "Clay X" and "Clay Y". In the following, the stress paths used in the experiments and in the predictions are first defined. Then, the general procedure for determining the material constants for the three models is explained, and the stress strain relations corresponding to particular stress paths are derived. Herein,typical soil mechanics sign convention (compressive stress positive) is used in the model calibration. 8.1 Stress Paths in Experiments The stress path used in the experiments can be described by the stress ratio m. m (32) where 01' 02 and 03 are the principal stresses applied to the cylindrical or the cubic soil spedimen and 01 is the stress in the vertical or axial direction. 'I-16- The stress path with the ratio m =0 or m =1 is on the plane e = 60° or e = 0° respectively. The stress path corresponding to a simple shear test as viewed in the n-plane is shown in Fig. 6-a. If the Lode angle e is defined from the plane, m =0, then, the relation between e and m is given by (2-m) (l-2m) (l+m) cos e (33) 2 3/2 2 (m -m+ 1) The stress path corresponding to conventional triaxial test lies between the stress path CTC test and the path CTE test as shown in Fig. 6-b in 11 - ,r; space. 8.2 Determination of Material Constants and Analysis General For the nonlinear elasticity model, we need to determine the bulk moduli K and \' shear moduli GO and G • The bulk modulus K is a function O l of 1 and is determined by an isotropic compression test. The shear moduli 1 GO and G are functions of ~ and ~ and are determined by the stress l difference-strain difference curves from drained triaxial compression and tension tests conducted at different levels of confining pressure. If the model is applied to problems involved cyclic and reversed loading, bulk modulus K, and the shear moduli GO and G can be determined respec 1 tively by the unloading curves of the tests mentioned above. Hence, the variable moduli model is used. For the plasticity models, the bulk modulus K and the shear moduli GO and G can be determined from the slopes of the unloading curves of an 1 isotropic compression test; and from the slopes of the unloading stress difference-strain difference curves of a drained triaxial compression tests and tension tests at different levels of confining pressure, respec tively. In the plastic range, the values of a and k are obtained from c and ¢ values associated with the Mohr-Coulomb failure envelope, which is constructed through simple shear tests. The material constants Wand D in Eq. 10 associated with the hardening function are determined by isotropic compression and unloading tests. The constant R associated with cap shape is determined from simple shear and uniaxial strain tests. The choice I-17- of material constants R from experimental data requires a considerable experience. Clay X The experimental data on the simple shear tests with the stress ratio m=O and 1 under confining pressure a 10, 20 and 30 psi are available. c The nonlinear elasticity model, cap model I and cap model II are used for prediction. In the case of nonlinear elasticity model, the stress difference-strain difference curves are drawn for each confining pressure as shown in Figs. 7 through 9 where we have used the average values of E and l E: , or E and E:. These curves are fitted by a function using nonlinear 2 2 3 regression analysis. In general, the relation between stress difference and strain difference can be expressed as (34) Taking derivative with respect to the strain difference, we have or (35) Each pair of GO and G functions are utilized for the present prediction l under different confining pressure. The matchings of the Mohr-Coulomb constants (c,~) and the Drucker-Prager constants (a,k) are listed in Table 1 for all three materials. The principal stress increments da , d0 and ll 22 d0 can be written in the form as 33 2m -1 (36) 2 -m I-18- from which dJ can be expressed in terms of dOll. Thus for a given da 2 t l1t the corresponding strain increment d£ .. can be computed. The process con lJ tinues until the stress path reaches the Mohr-Coulomb or Drucker-Prager failure surface. In the case of plasticity models, clay X is assumed to be an anisotropic material. Fig. 10 shows the relation between the stress increment da~l or da~l and each strain increment. The stress-strain relation can be regarded as linear up to dOllo of 5 psi or dail of 2 psi, respectively. Thus, the following relation is assumed in the leastic range, for m =0 d 0 0 0 0 0 0 £11 = 0.0014 dallt d£22 -0.00156 dOll and d£33 = 0.0002 dOll (37) and for m = 1, d 1 1 1 1 1 £n = 0.00325 dail' d£22 = -0.00267 dOll and d£33 = -0.00055 dOll (8) The constants Wand D are estimated to be 0.3174 and 0.0087, respectively. Therefore, Eq. 10 is = 0.3174 [1 _ e-0.0087(x-46.5)] (39) where the value 46.5 is three times of the preconsolidation pressure a p (compressive stress taken as positive). The material constants R (cap shape) are determined to be 4.7 from the simple shear test data with the stress ratio m = 0 and to be 5.7 from those with the stress ratio m = 1. Fig. 11 shows the location of hardening caps for the planes of m=O and P m =1 in II - ~ space. The plastic strain increments d£ij are calculated for the cases of Cap Model I and Cap Model II. As the experiments have been conducted under stress control conditiont the plastic volumetric strain increment d£~k can be calculated from Eq. 39 after the value of x is obtained from the subsequent state of stress and elliptic cap equation. Therefore, dA is obtained from Eq. 12. .....T-19- For cap model I, 1 and dA (40) and for cap model II, dA (41) p Therefore, the plastic strain increment dE is obtained from Eq. 12 ij respectively. Clay Y The experimental data are obtained from the triaxial tests with the stress ratio m =0 and m =1 under the initial confining pressure er =2.5, c 5.0 and 10 psi. The property of clay Y appears to be similar to that of clay X. As the initial location of confining pressure lies on "Dry of Critical" from viewpoint of the critical state soil mechanics, the Coulomb constants c and ~ (in Table 1) are different from those constants for clay X. The experimental data indicate that the behavior of clay Y appears to be isotropic. Therefore, the stress difference-strain difference relation, and the stress invariant II - the volumetric strain relation are checked for three sets of data as shown in Fig. 12 through 14. For nonlinear elasticity model, the functions of bulk moduli K ' K and shear moduli O l GO' G are obtained using the curve fitting procedure similar to that of l clay X. The principal stress increments dOll' d0 and d0 with the 22 33 stress ratio m is expressed in terms of dOll by (42) <1:-20- The corresponding strain increment can be calculated as that of clay x. For the plasticity models, the stress-strain relations are again checked for the data with m = 0 and 1 as shown in Fig. 15. The shear modulus GO for m = 0 or G for m = 1 is assumed to be a function of ~ 1 or constant, respectively. The bulk moduli K and K are assumed to be two O 1 different constants respectively. The material constants Wand Dare estimated to be 0.135 and 0.009. Therefore, Eq. 10 is P [l - O• E: 0.135 e- 009 (x-46. 5)1 (43) kk where the preconso1idation pressure a is assumed to be the same as that in p clay x. The material constants R (cap shape) are determined to be 2.39 from the triaxial test data with m = 0 and to be 0.87 from those with m = 1. The location of hardening caps are shown in Fig. 16. The calculation of the strain increment is the same as that of clay X. Kaolinite Clay The available experimental data are the triaxial tests (No. 1 and 10) and the simple shear tests (No. 4 and 13) with m = 0 or m = 1 under an un drained condition. The nonlinear elasticity model is utilized for the prediction. The experimental data are plotted in the stress difference-the strain .' difference space as shown in Fig. 17. Here, as in the previous case, the stress difference-the strain difference curves are fitted by functions in order to determine the shear moduli GO and G which are functions of J • 1 2 For these tests to be predicted by the model, the major principal stress is inclined at an angle to the vertical axis of the specimen, while the inter mediate principal stress remains horizontal. Since the material is assumed to be isotropic for present case, the direction of the applied principal stresses will not affect the results of prediction. The increments of principal stresses with stress ratio m are expressed in terms of da by 11 ~i (44) dOll = dan' dOn = 0 and d033 = m dan for Test No.2, 3, 7, 11 and 12 and ]:-21- 1 - 2m do d d m+1 (45) m- 2 11 an °33 m - 2 dOll for Test No.5, 6, 8 and 9. It should be noted that those principal stresses are the total princi pal stresses because the tests are conducted under an undrained condition. Therefore, the effective principal stress increments must be known in order to calculate the principal strain increments. In this case, the effective stress increments are the deviatoric stress increments dS .. because the 1) inclusion of hydrostatic pressure increment dI causes a volumetric change. l Thus, the effective stress increments are calculated from Eqs. 44 and 45, and the corresponding strain increments can be calculated in a same manner as that described above. Ottawa Sand The experimental data available for predictions are the conventional triaxial compression tests under the initial confining pressure of 5, 10 psi, the conventional triaxial tension test under the initial confining pressure of 5 psi, and the simple shear tests with the stress ratio of m=O, 1 under the mean normal stresses of 5, 10 and 20 psi. Here, as the experimental data for the loading, unloading and reload ing cases are given, the variable moduli model is used. In order to deter mine the shear moduli GO and G for the loading case, the experimental data l for the loading parts in the simple shear tests are plotted in the space of the strain difference and the stress difference divided by the stress difference at each failure (Fig. 18). The curves used in the prediction are shown by the dotted curves which are obtained by a nonlinear regression LL analysis. Thus, the shear moduli G and G for the loading cases are given ° l by and (46) °1"-22-- where a~ and a~ are the stress differences at failure in CTC and CTE tests. The stress difference at a failure is given by o 6 teeDS ~ + am sin cj» a (47-a) f 3 - sin 1> L for the modulus Go and 6(c cos 1> + a sin cj» m (47-b) 3 + sin 1> L for the modulus G , where a is the mean normal stress. In this case, the l shear moduli are functions ~f II and J~ or J~. To determine the shear moduli G~r, G~r for the unloading and reloading cases, the experimental data for the unloading and reloading parts in the simple shear tests are plotted in Fig. 19. In this case, the stress dif ference and the strain difference are measured from the unloading and re loading points. The curves used in the prediction are shown by the dotted ur Gur curves. Thus, the shear moduli GO and 1 have the same form as that in Eq. 46. The bulk modulus K and K are determined from the conventional tri O l axial compression and tension tests. Although the data on the isotropic consolidation test are given, they are not used here to determine the bulk moduli. The experimental data are plotted in I and £kk space as shown in LL 1 Fig. 20. The bulk moduli K and K for the loading up to the unloading ur O l pOlnt,.Kur andK for the unloading, reloading up to the point of previous a 1 and KNL K~L unloading, a and for the loading starting from the point of the previous unloading are determined from the figure. These bulk moduli appear to depend on the initial confining pressure a Therefore, the c general form for K may be written as K = K(a ) (48) c In general, the bulk modulus of Ottawa sand may be expressed by K =K(a ) max where a is a maximum confining pressure similar to the preconsolidation max pressure. 'I-23- The calculation of the strain can be carried out in the same way as mentioned previously. 9. Acknolwedgments This material is based upon work supported by the National Science Foundation under Grant No. PFR-7809326 to Purdue University. References [1] Baldi, G. Y. and Rohani, B., "An Elastic-Plastic Constitutive Model for Saturated Sand Subjected to Monotonic and/or Cyclic Loadings," Third International Conference on Numerical Method in Geomechanics, Aachen, 2-6 April 1979, pp. 389-404. [2] Chen, W. F., "Limit Analysis and Soil Plasticity," Elsevier, Amsterdam, The Netherlands, 1975. [3] Chen, W. F., "Plasticity in Soil Mechanics and Landslides," Engineer ing Mechanics Division, ASCE, Vol. 106, No. EM3, June, 1980, pp. 443 464. [4] Chen, W. F. and Mizuno, E., "On Material Constants for Soil and Concrete Models," Third ASCE/F~ Specialty Conference, 1979, pp. 539 542. [5] DiMaggio, F. L. and Sandler, 1. S., "Material Models for Granular Soils," Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM3, 1971, pp. 936-950. [6] Drucker, D. C., Gibson, R. E., and Henkel, D. J., "Soil Mechanics and Work-Hardening Theories of Plasticity," Transactions, ASCE, Vol. 122, 1957, pp. 338-346. [7] Drucker, D. C. and Prager, W.• "Soil Mechanics and Plastic Analysis or Limit Design," Quarterly of Applied Mathematics, Vol. 10, No.2, July 1952, pp. 157-175. [8] Mizuno, E. and Chen, W. F., "Analysis of Soil Response with Different Plasticity Models," ASCE Symposium in Florida, 1980. [9] Roscoe, K. H.• Schofield, A. N., and Thurairajah, A., "Yielding of Clays in State Wetter than Critical," Geotechnique, Vol. 13, No.3, 1963, pp. 211-240. i. -24- [10] Sandler, 1. S., DiMaggio, F. L., and Baladi, G. Y., "Generalized Cap Model for Geological Materials," Geotechnical Division, ASCE, VoL 102, No. GT. 7, 1976, pp. 683-699. [11] Sandler, 1. S. and Melvin, L. B., "Material Models of Geological Materials in Ground Shock," Numerical Method in Geomechanics. Edited by C. S. Desai, Vol. 1, 1976, pp. 219-231. [12] Schofield, A. N. and Wroth, P., "Critical State Soil Mechanics," McGraw-Hill, New York, 1968. [13] Zienkiewicz, o. C., "The Finite Element Nethod," McGraw-Hill, 1978 (Chap. 18). [14] Zienkiewicz, O. C., Humpheson, C., and Lewis, R. W., "Associated and Non-Associated Visco-Plasticity and Plasticity in Soil Mechanics," Geotechnique, 25, No.4, 1975, pp. 671-689. ::t -25- Table 1 Material Constants c, ¢, a and k Material Constants Clay X Clay Y Kaolini te Clay Ottawa Sand C (psi) 2.0 4.0 13.75 2.0 ep ( 0) 26.57 16.88 17.04 42.34 H 2 sin p I 0.2023 0.1221 0.1250 0.2117 N J3{3-sine:p) 0\ I 6c cos ¢ 2.4275 4.892 16.823 1.394 J3 (3 - sin cP ) 2 sin p 0.1498 0.10 0.1027 0.334 j3(3tsin¢) 6c cos P 1.7976 4.304 13.830 2.201 y3< 3+ sIn ep) --OA = (OJ, CTz, CT;s ) --.-. OB = (p,p,p) --BA = (5" Sa'S) Fig. 1 Stress State in Principal Str~ss Space :r -27- 0", 0" 2. Fig. 2 Shape of Yield Criterion on TI-Plane '1--28- Failure EnveloP~. F~ ...... / t, ...... ...... / Horizontol Tangent x- L Elliptic R Strain Hardening Cap Fe T -II 1: -----x_X_-Q--J Fig. 3 Elliptic Cap Model 1-29- y:;; 8' (Ojj ..dCJjj) CT CLine e= 60° plane ~~ i---'1 I oW I 8 = 0° Plane o F F' -I. d I. Fig. 4 General Stress Path in 1 - ~ Space 1 e= 600 plane Hardening Cap o -I, a) Cap Model I e= 600 plane co\)\0"''0 --_ 7 plane ,-11 / ...- ...- '7"'- / \..ine _j / \) Failure ~--'----- _ Coul o," -/...... -j Hardening Cap -I I b) Cap Model ]I Fig. 5 Scheme of Cap Model I and Cap Model II ,\ -31- m=O a} Stress Path on .". - Plane o b) Stress Path in II-jJ;, Space Fig. 6 Stress Path in Experiments T -32- ,,- ,," " ..... CLAY X J:J' CLAY X // m=O // m= I oc= \0 psi all eTc = \0 psi / / / c;f /..------10.0~ I ~ I 10.0 I / ¢ q / (/) I I 0- I I (-1 .. ¢ I P W -bit) I W I o Experimental Data o Experimenta I Data I I I I 0- 6 Curve Used in Q Curve Used in I I - I Prediction Prediction ~ 5.0 oI 5.0 f-J I , I I , ¢ Q I I I ¢ ~ I II v' I I O. 1.0 2.0 3.0 O. 1.0 2.0 3.0 ( € I- € ;) t 010 ( €~- €3) t 0/0 Fig. 7 Calibration of Shear Moduli for Clay X (a 10 psi. Nonlinear Elasticity Model) e = CLAY X CLAY X m=O m= , 20 I- 20 I- eTc = 20 psi ac = 20 psi o ..,-."".------.,,'" /".. o / '" / 0 /"..,------Q. I I /' /) //' 0 Ii I I I / Vol 'u; /0 ~ 0. 10 I- 10 I / I o ~ I o Experimental Data / o Experimental Data -t:r 0/ }5 I Curve Used 10 / Curve Used In b- I c;) Prediction 0/ Prediction - I I I o :)1 I I ~ 0/ I' 1 I o. 10. O. 10. ( €, - €;) , 0/0 (Er - E ) , 3 0/0 Fig. 8 Calibration of Shear Moduli for Clay X (0 20 psi, Nonlinear Elasticity Model) c ).:.J ".. ".. ,/ CLAY X ,/ CLAY X ,/ /0 / m=O / m =I . / / a: = 30 pSI / 0 ac = 30 psi c / o 1 I I 1 I 0 o ------ 20.1- I 20.1- .",- I .;'" -- I ,/ '0 /0 I / I // .~ I I / t-J 0- Q /0 I / W I U1 ~ I // I ~ 1 I 01 CI I I b I / 1 - \0.1- I o Ex.perimental Data 10.1- I o Experimental Data 01 1 I Curve Used in I Curve Used in / I / Prediction Prediction 0/ 0/ I 1 / 1 W 9 I 1 I I d I 0. 10. 10. (E,-E3') % (Er- E3) 0/0 Fig. 9 Calibration of Shear Moduli for Clay X (Oc = 30 psi, Nonlinear Elasticity Model) p CLAY X m=O I I CLAY X m = I 1.2 ~ J 1.2 at = 10 psi I I CTc = 10 psi I 0, / / \.0 I- 0 E 1 I 1.0 f- 0 E 1 / I:':. / I I:':. -E -E 2 I P I:':. 2 9' / / / / / 0.8 I- 0 -E3 0.8 c;f D -E3 18-../.b""J!. / I / I , ..., H I I Vol C'\ 0.6 I .-. .-. ~ OSr ~ 0...... -0 IV /I IV 'C P c 0.4 '0 0.4, / '0 ....~ (/ / ~ en 9' en , / / JY 0.2 I- .// I 02~ d '" P / '" I 2 4 I I I l I I 6 8 0 I 2 3 ~ Stress Increment 60j t psi Stress Increment 6CT1 t psi Fig. 10 Calibration of Orthotropic Material CLAY X 15 m=O \0 m= I 5 o 10 20 30 40 50 60 Fig. 11 Location of Hardening Caps for Clay X 12.-37- CLAY Y II o Experimental Data DC = 2.5 psi, m=O / o Curve Used in / Prediction Io ::i 10.~ ¢ 20.- _-"0 (:)),it--- b ,- ...0 I 9"'- ;:f ¢ JY P 10'12"'''' 6 (;:5'- I ~ , I I O. 2. 4. O. 0.5 CLAY Y 1 o / 1 / I / I / I 0:=c 2.5 psi / / 0 m = I / ? / I / I / 0;/ III 10 r- ¢ 20. - 0. / I / I / I L. /0 c1:J / I / I 10. r- / 0 Experimental Data I ~/ ¢ ( Curve Used In I --- I Prediction I I , I o 2 4 0.5 (E~-E 3)' % E kk' % Fig. 12 Calibration of Shear and Bulk Moduli for Clay Y (0 2.5 psi, Nonlinear Elasticity Model) c T -38- CLAY Y ~ = 5.0 psi, m = 0 .,.,- _----0--- .0-- 0 .,.,- - I / tJ / I / , / I / 10. 25.f- 0/ 9 I "v; I / 0- I 0 Experimental Data I ~ 1 ,-... / r<'l cd b I Curve Used In I /J 0- Prediction I ~ I J 01 I I I I I I 15' t o. 5.0 10,0 0.5 / / CLAY Y ac = 5.0 psi / / m= I / / / / J / / 0 ¢ 1 I / , / J I / 10. f-¢ 25.- / I / 'v; , / Q. 0 Experimental Data 1 I / ~ ,-... 10 Q 1 tf I Curve Used in I 1 I Prediction / 0- / ~ I I I 1 I / I 15 ill o. 5.0 10,0 0.5 (€~-€3)' .% Fig. 13 Calibration of Shear and Bulk Moduli for Clay Y (0 = 5.0 psi, Nonlinear Elasticity Model) c I: -39- CLAY Y oc = 10 psi, m = 0 ---(5". ".----- ".------/" ,,'" 0 /'0 / / / I / I (/) 10.~ c/ 40.~ q a. I I I , o Experimental Data I I Q t1 I Curve Used in I I I Prediction I J ,6 (. ,, I 30 I O. 5. 0. 1.0 20, .----~------.,-----'------., __I, CLAY Y CTc = 10 psi, m = I ...... ----0 ".- 9 /0 p'" / / / I / I / o / P / I / I I I I .~ 10. ... 0 50. I o Experimental Data / P I / / ' ___ Curve Used in / ~ o / , Prediction / - / I / I (. / I / / 30. / I I I O. O. I. 2. Fig. 14 Calibration of Shear and Bulk Moduli for Clay Y (0 = 10 psi, Nonlinear Elasticity Model) c T -40- CLAY Y 40.1- ~ 15 ./ ./ ro/ / 30.1- r;f / 'Vi 10. a. 4J:f / ~ L /0 20.1- -b I I ~ 0" / 5,.. b o ac = 2.5 psi .~ ac =5.0 psi P o oc = \0.0 psi I I I o. o. 0.5 1.0 Ekk , 0/0 CLAY Y I I 40.- I 15.- I /0 ~ I 30..... I P 10 (/) I a. I \0.... t¥J I ~ I I ...... I q 20. tf I I I ~ 0 ()c = 2.5 psi / 0- --- I f!> 5. ~ I 8 ()c = 5.0 psi / I 10.:"'/ tb, 0 oc= 10.0 psi 8 I / I I I I o. I. 2. 0. 0.5 1.0 Ekk' 0/0 Fig. 15 Calibration of Shear and Bulk Moduli for Clay Y (Cap Model I and Cap Model II) .r -41- .JJ;. CLAY Y 15 m=O 10 m =I " ", 5 /', \y?R1= 0.87 Ro= 2.39 ", \ \ \ \ \ \, 0 10 20 30 40 50 60 II' psi Fig. 16 Location of Hardening Caps for Clay Y 1: -42- 70, 70' I Kaolinite Clay m=O Kaolinite Clay m = I 60 I- 60 f- ,J:!"~....o---O'-!-Q._tl- ~ - L - d· • 50 I- 1':· 50 f- g. el· .. .. ~ . 40 I- f- I • Test No. I 40 ...... • ",...." ... 0-- 0 0 (--1 d· I VI • " ... 0 0 .I:'- a. I o Test No.4 w • • /",,"'0 0 I / • "" 0 ~ 30 I- '1 Curve Used in 30 l- • A",,-t> -b , • p"" 0 I Prediction " 0 1 0·,,-0. b- , .v --- I ~ 20 ~., 20 6.0 • Test No. \0 ,I ~~ , ~ o Test No. 13 d 10~, 10~, I , II 'f i I I I I I I I I o 2. 4. 6. 8. o 5. 10. 15. 20. (E\- E3), % (E,-E3), % Fig. 17 Calibration of Shear Moduli for Kaolinite Clay 1.01 I ~_. x ----- ."._'"0 0_'1:'""--- 0 ----o ------0 .... ------0---- x'" Simple Shear Test b...... • DC = 5 psi blf) - 0.5 x. /0 I "I OTTAWA SAND m= I x o-c = 10 psi b- xb - .J ---- Curve used in prediction o o-c = 20 psi o 2 3 4 (-1 I (Ez- Ex), % -I:' -I:' I \.0 I I ---~----- .....-0__...... _0xx------0 x x 0 ------x• s ...... Simple Shear Test -tf 0.5 X /0 • o-c = 5 psi I OTTAWA SAND m=O b- • If x9 x o-c = 10 psi - ·1 ---- Curve used in Prediction o o-c = 20 psi o 2 3 4 (Ez-Ex), % Fig. 18 Calibration of Shear Moduli for Ottawa Sand (Loading Case) 1.0 r------. _------x-- 5.5. TEST ------",/"- X. ./ / / m:O ~~C1iJ • OTTAWA SAND y../tJ Unloading Reloading ,ft6 • o-c: 5 psi )( C'c: 5 psi I I• o DC: 10 psi t:. ac: 10 psi o CTc: 20 psi v DC: 20 psi It Curve Used In Prediction O. 0.5 1.0 1.5 1.0 r------, ------S.S. TEST ------,- " -- -- !:¥~ '" t:. x. / I et:./ 0 I 1"fJ OTTAWA SAND m : I 0 5 /v b.... . /iJ.J Unloading Reloading '- ~t.9 I • a;c : 5 psi x CTc : 5 psi I I 0 CTc : 10 psi t:. OC: 10 psi 0 DC: 20 psi V DC: 20 psi ____ 0 Curve Used In Prediction L.. J..I '-- ...J o 0.5 \.0 1.5 Fig. 19 Calibration of Shear Moduli for Ottawa Sand (Unloading and Reloading Cases) T. -45- & Experimental Data I, OTTAWA SAND o Experimental Data x Unloading Data I10 ~------~------A.. -- ...... A.., ...... ~ 70 --fI ,/""" 0- --.r\ - 1';'\ -- --<."\ -0------& --' ~------p------.~o----..b''='--'-- I . ,/ --0-,. . -tK----- I ,/" , " .o(JP--tls-- . j,,/~ CTE TEST r-I ,/" .... ~ 0" O ~. I r 1Y~ 0-", +' t *'"~ 30 CTC TEST (o-c 10 pSI) 0\ *' __ ro)t" = I ( o-c =10 psi) 40 ------0------o-----~. ---o--o-~ 20 0-"oO"o/C~ TE TEST. .•{}(T {o-c -_ 5 pSI ) 0.2 0.3 0.4 -0.1 0.0 0.1 -0.4 -0.3 -0.2 €kk (%) Fig. 20 Calibration of Bulk Moduli for Ottawa Sand PLASTICITY MODELS FOR SOILS Comparison and Discussion by l 2 E. Mizuno and W. F. Chen , M. ASCE Introduction Herein, the predictions are made using the following three material models: (i) Nonlinear elasticity model with the Mohr-Coulomb or the Drucker Prager criterion as failure surface. This model is applied to all predictions. (ii) The Mohr-Coulomb type of elastic-plastic material model with two different elliptical hardening caps. One on tensile meridian plane (8 = 0°) and the other on compressive meridian plane (8 = 60°). (Cap Model I). This model is applied to "Clay X" and "Clay Y". (iii) The Mohr-Coulomb type of elastic-plastic material model with a three-dimensional elliptical cap whose shape depends on the Lode angle 8 (Cap Model II). This model is applied to "Clay X" and "Clay Y". The comparison and discussion are given in the forthcoming. Clay X The comparison of the predictions with the experimental data on the simple shear tests with constant mean stresses at 10 psi, 20 psi and lResearch Assistant, School of Civil Engineering, Purdue University, West Lafayette, IN 47907 2professor of Structural Engineering, School of Civil Engineering, Purdue University, West Lafayette, IN 47907 l!: -1- 30 psi are presented in Figs. 1 through 4 for each of the stress ratio m = 0.25, 0.5, 0.75. In these figures, the stress difference-axial strain relation, and the axial strain-volumetric strain relation are presented. The experimental data are shown by the open circles and the predictions are denoted by lines. As can be seen from the figures, the predictions by the nonlinear elasticity model are off somewhat from the experimental data with 0 = 20 c psi and 30 psi. The model, however, gives a good prediction for the tests with 0 = 10 psi and for 0 = 30 psi, m = 0.75. Since the model is assumed c c to be isotropic, it is difficult to predict the behavior of the anisotropic clay X. Also, it should be noted that the volumetric strain e on the v simple shear tests can not be predicted by this model. Cap models are seen to predict the axial strain-stress difference relation and the axial strain-volumetric strain relation well for all experiments shown in figures. The tests at 0 = 10 psi show elastic. In c the prediction (0 = 10 psi), the cap models show some plastic strains in c the region near failure surface. For the experiment with a 10 psi, c m = 0.5, the cap model I predicts only elastic strains because the de- composed stresses do not reach the hardening caps. Therefore, there is no plastic strains. Clay Y The comparison of prediction with experimental data on the triaxial tests under initial confining pressures a = 2.5, 5.0 and 10 psi is c presented in Figs. 5 through 7 for each of the stress ratio m = 0.25, 0.5, 0.75. As the initial confining pressures lie on the swelling line, clay Y behaves first like an elastic material in the tests. Nonlinear elasticity model predicts well the experiments with 0 2.5 and 5.0 psi. However, the c predictions for the tests with 0 10 psi, m = 0.5 and 0 = 10 psi, m c c 0.75 are off somewhat from the actual data. This implies that plastic strains must occur in the tests. Cap models reflect this as shown in Fig. 7. The predictions by cap models for the triaxial tests with 0 2.5 psi c and 5 psi are almost the same as that of nonlinear elasticity model. For these cases, either the stress paths have not reached the hardening caps, or the plastic strains are small at this stage of loading. 1t -2- Kaolinite Clay The comparison of the predictions with the data on the triaxial or simple shear tests under the undrained condition is presented in Figs. 8 through 12. The total axial stress vs. axial strain curves are shown in the figures. As can be seen. the nonlinear elasticity model predicts the experiments well. In particular. the predictions for Test NO.7. 8. 9, 11 and 12 are very good. In this model. the pore water pressure increment is taken to be equivalent to the increment of the first invari ant of the stress tensor II' Ottawa Sand The comparison of predictions with experimental data on various tests is presented in Figs. 13 through 21. In each figure. the strains in x. y and z directions are plotted against the octahedral shear stress T oct Except the circular stress path. all predictions are good. In particular. the initial slope of the actual behavior of Ottawa sand and the failure load are accurately predicted by the nonlinear elasticity model combined with the Mohr-Coulomb criterion. As for the circular stress path. the model prediction is different significantly from the actual data in the region of Lode angle e between 60 0 and 420 0 (Fig. 21). Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. PFR-7809326 to Purdue University. IT -3- CLAY X -_._- Non-linear Elasticity 0 Experimental Data Cap Model I 8 Mohr - Coulomb Failure OC=IO psi ----- Cap Model II 0 Drucker - Prager Failure 20 I m= 0.25 I m= 0.50 I m=0.75 en 0. ~ ~IO -- ...... ------10 I R .....0- I ~ I 5 0' "'" .... I" l.o.ooo-oo ,5 , == 1 I o \ " \ " o \ " .... \ " \ €v " o \ " \ " "- \ "- 0/0 "- o " , "- " "- ...... " 2 2 2 "", " "" € I % E: 1 0/0 €, 0/0 10 psi Fig. 1 Stress-Strain Relations for Clay X, ac CLAY X --- Non-linear Elasticity 0 Experimental Data Cap Model I /). Mohr - Coulomb Fai lure CTc. = 20 psi --- Cap Model ]I 0 Drucker - Prager Failure 30 I m= 0.25 I m= 0.5 I m=0.75 . ~ 20~ /'J ~o __0_:0 / --2~f ~--- o -- I /_.... ---- o _o - -- /~~ .... - ,/ ~b ~ //j>-// - 0/ -- I / b- \0 I /' 10 , / 10 FJ I Vl I ~- 15 110 15 dO 15 ,10 0 r r ! :-... ! 5~ E 0 5 " 5 v o I " ...... I ...... % ...... o ..... _- I 10 o 10 10 E % I E I % E I % 1 Fig. 2 Stress-Strain Relations for Clay X, a = 20 psi c CLAY X Non -Linear Elasticity o Experimental Data Cap Model I 8. Mohr-Coulomb Failure ac = 30 PSI Cap Model n o Drucker - Prager Fai lure 301- 30 o ~-- -".., ,..- ,- -- ,..- - j/ 0 ,,// ,,/ ,/ ,,/ ·-20f/) 20 20 Q. /" /" .. /' / // ~ / I / / !>r ~ 0- I I - 10 10 10 fI I I 0- I ~ hv' m =0.25 m = 0.5 " m= 0.75 o~ ,5 .10 5 10 5 10 '\! ,,~ Ey " , 5 5 , 0/0 o "- " " ...... o 10 0 1 ''-..... , 1 ______1 ..... --::::::i E. % E,% E, 0/0 Fig. 3 Stress-Strain Relations for Clay X, a 30 psi c 50 CLAY X m=O 0;; = 40 psi 0 40 0 0 0 0 30 III 0 Q. ~ -bit) I 0- 20 - 0 Experimental Data Cop Model \0 o 5 (%) 10 o o E, (%) Fig. 4 Stress-Strain Relations for Clay X, a 40 psi c orr: -7- ------'" ------CLAY Y --- Non - Linear Elasticity 0 Experimental Data Cap Model I O"'c = 2.5 psi ---- Cap Model IT I:::. Mohr - Coulomb Failure I 20 I m= 0.25 m= 0.5 m=0.75 0 0 0 I 0 0--- -- I p-==_ =9. - I ~~------ ~ 0 ;J, \0 t- I \0 -b- t=! I ex> I lOOg ,5 o( 15 ,5 l 00 "' I 'r 00 J 0 "" €y % I I 0 0 2~ 2~ 2 € I (%) €, (%) € I (%) Fig. 5 Stress-Strain Relations for Clay Y, a 2.5 psi c o CLAY Y Non - Linear Elasticity Experimental Data Cop Model I 201 I I I m= 0.25 m= 0.5 m =0.75 o o o o 'in 0... 10 b~, 10 10 b- \=l - I \.0 I ·0 5 o lUUOQ 0 I l: I ~o I I \0 , \, 0 ...... --- \0"-.0 \\0 " \ \ \ o €y \ % o 2 2 2 €, % €, 0/0 €, 0/0 Fig. 6 Stress-Strain Relations for Clay Y, 0 = 5.0 psi c CLAY Y --- Non-Linear Elasticity 0 Experimental Data Cap Model 1 DC = 10 psi ---- Cop Model II A Mohr - Coulomb Failure I 20 I AA- ...... ------.,,- r..------.--- 0 "' .... k/~ 0 / ,/ '" / TIl 0 / / / I I II) I a. 10 10 ~ IO~ -tr 0" ~ - m = 0.25 m= 0.5 III m= 0.75 I f-' 0 I [ ok ,5 L- ,5 ,5 ~o , \ \~~ , \ €ov Il- \ ~-- II- \'\ 0 \ 0 0/0 I \ I\\ \\ 0 \ 21- \ 2J- \ 2 \ 0 \ I \ 0 \ 0 I \ \. , 31 "- 3 1 \ 'b 3 1 "" ..... E. 0/0 €o1 0/0 €o. 0/0 Fig. 7 Stress-Strain Relations for Clay Y, a = 10 psi TEST NO. 2 (Kaolinite Clay) 000 0 000 II) 110 o 0 0- 0 II) II) Q) ~ 90 &. Mohr - Coulomb Failure en- "0 o Experimental Data ~ 70 - Prediction 50 L.- -l- -l ---JL-- ....L- -L __ 0. 2. 4. 6. 8. 10. Axial Strain E, (%) TEST NO.3 (Kaolinite Clay) 'iii 110 c- o 000 o o 000 II) o 0 II) Q) o ~ 90 en o - &. Mohr - Coulomb Failure 0 x « o Experimental Data 70 - Prediction 50 L... --l__.__-L ...L- ----'L.-- -..l... ~- o 2. 4. 6. 8. 10. Axial Strain E,(%) Fig. 8 Axial Stress-Strain Relations for Kaolinite Clay, Test No. 2 and 3 lr -11- ---_.•._--_.-- TEST NO.5 (Kaolinite Clay) 'iii c. 110 0- Il) II) Q) ~ en- 0000000 0 >< <{ 8 Mohr - Coulomb Failure o Experimental Data - Prediction 4. 6, 8. Axial Strain E', (%) 110 TEST NO, 6 (Kaolinite Clay) 'iii c... 0- Il) II) Q) ~ en - o 0 00 o o 0 )( <{ 8 Mohr - Coulomb Failure o Experimental Data - Prediction 50 l-- -L ..l.- ~ ..J_ - 2. 4, 6. 8, Axial Strain Fig. 9 Axial Stress-Strain Relations for Kaolinite Clay, Test No.5 and 6 ':IT'. -12- -- '------,------TEST NO. 7 (Kaolinite Clay) (/) 0- 0 90 (/) 0 (/) Q) 0 0 .-.... 0 (f) 8 Mohr- Coulomb Fai lure 0 70 x 0 4: Experimental Data Prediction 50 O. 2. 4. 6. 8. 10. (%) Axial Strain E1 TEST NO.8 (Kaolinite Cloy) (/) a. b- 75 0 000 (/) o III Q) .-.... (f) 8 Mohr - Coulomb Failure 0 )( o Experimental Data 4: Prediction .. 2. 4. 6. 8. 10. Axial Strain E, (%) Fig. 10 Axial Stress-Strain Relations for Kaolinite Clay, Test No_ 7 and 8 ::rt -13- ----_._.-.-.•_-----_...... --.----_._---_.._-_.------_."-_._-----'...... ------TEST NO.9 (Kaolinite Clay) A Mohr - Coulom b Failure o Experimental Data 'in a. Prediction 0 75 II) II) Q)... en- .2 )( 2. 4. 6. 8. Axial Strain TEST NO, II (Kaolinite Clay) II) a. 0 75 A Mqhr - Coulomb Failure II) II) Q) o Experimenta I Data ... 0 00 en- Prediction 15 ')( 65 55 l..- ---1 --J..' --'-,------'-,------i...... O. 2. 4. 6. 8. Axial Strain Fig. 11 Axial Stress-Strain Relations for Kaolinite Clay, Test No. 9 and 11 :::tt -14- TEST NO. 12 (Kaolinite Clay) (f) 0- 61 000 0- 00 (f) (f) 60 0 0 Experimental Data x 58 .)------'------''------'------'-----...... O. 2. 4. 6. 8. Axial Strain (%) Fig. 12 Axial Stress-Strain Relation for Kaolinite Clay, Test No. 12 JI. -15- Ul Ul 0- 0- OTTAWA SAND ft ft u ....-8 ...,0- Simple Shear Test ac =.IOpsi, m=0.2 10 10 &-_- ~------::-:::?-~::""'---=-=:::..::::..:::...::>< _ _=------0 -.... 8 i;.'\------0--,''£l FE I f-' 0\ ~I', I , , VI Mohr - Coulomb Failure I~ , \ o z Direction 9Q (J X Direction A Y Direction H~¢ - Prediction \I 2 d -- Experimental Data \I -0.8 o -2.0 -1.0 0.0 1.0 2.0 Tension E (%) Compression Fig. 13 Stress-Strain Relations for Ottawa Sand, Simple Shear Test (0 = 10 psi, m = 0.2) c If) OTTAWA SAND Y1 Mohr - Coulomb Failure 0: ~ ...., Simple 0 Z Direction ",,0 Shear Test ~ a: = 5 psi, m= 0.5 [] X Direction c Prediction -1-5 Experimental Data 4 t=f ~------Lf--- ~__ I I-' '-J I --, -1.0 -0.5 0.0 0.5 1.0 E (6/ ) Tension 0 Compression Fig. 14 Stress-Strain Relations for Ottawa Sand, Simple Shear Test (0 = 5 psi, m = 0.5) c If) a. u ",,0 14 0--- _ 9-__ ------0 ", '\01 ------£].... 12 ...... / -- ...... , / ~q, P 10 OTTAWA SAND 'V Mohr - Coulomb Failure ef 9 0 Z Direction f;:/ ~ Simple Shear Test I EJ X Direction f-' 00 CTc = 20 psi, m= 0.5 I Prediction Experimental Data -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Tension €(%) CompressIon Fig. 15 Stress~Strain Relations for Ottawa Sand, Simple Shear Test (0 = 20 psi, m = 0.5) c 'V Mohr- Coulomb Foilure OTTAWA SAND 0 Z Direction Simple Shear Test IJ) 0- ac = 10 psi, m = OB 8 Y Direction ~ u 1--0- 0 X Direction Prediction Experimental Data 18 8 I 6 FI I 6 .n - - -s:.---p_--=-:..;:-:;JO-..---- f-' 0- ",.--v ..0-- I ------rlo... .", '" . u-__ " J:f - f-r' ~~ P f{ I~ .. " C!> II" M II III 0.8 -1.0 O. \.0 2.0 E 0/0 Tension Compression Fig. 16 Stress-strain Relations for Ottawa Sand, Simple Shear Test (0 = 10 psi, m = 0.8) c. EI o t:t EI A I oN I 6 4 2 2 -1.0 o -2.0 -1.0 o. 1.0 2.0 E <"0) Tension Compression Fig. 17 Stress-Strain Relations for Ottawa Sand, RTC Test (0 = 20 psi, m =0) c °en a... n .....0 ~ 40 o o 35 30 0 Z Direction ~ 8 X Direction \ L!:J 125 (:4 \ L:EJ I 0 / OTTAWA SAND I A Y Direction N N I Prediction Propotional Loading Test (PL I) ~ = 10 psi, m= 0 -5 -4 -3 -2 -I O. I. 2. 3. 4. 50 6. Tension E (0/0) Compression Fig. 19 Stress-Strain Relations for Ottawa Sand, PL1 (0 = 10 psi, ill = 0) Ow Proportional Loading Test (P L 2) Co.. DC = lOpsi, m= 0 u ...-0 Z Direction X Direction • y mrection 24t 24 24 ---- Experimental Data (;) [) Prediction II 20 20 f- 20 I,ft P,/I d ~, / I I / I , 16 9' 1/ l / d9I, /6 I , ~ I , I N 9 d I J I w I, I I I I 12 ~ 12 ~~ 12 ~ I I I, I ¢ 0 4 I 8L I 8 8 I P ~ 4 I I I I 6 ,I l 4f- I 4 I I 4 I P' if I L O~ I 0 O. 0.5 0.5 0.5 E (%) Compression (OTTAWA SAND) Fig. 20 Stress-Strain Relations for Ottawa Sand, PL2 (0 = 10 psi, m = 0) c o Prediction -o~ - OTTAWA SAND Circular Path 8 Experimental Data >-,.0u Zone l[ ]]I m I m I E I JZIII 1.0 l ~~~ ~~ 8 8 8 ~ ~ ~ r 8 'P\ 8 18 I N .p- 8 I 8 III 8 ]X A I 8 A O. 100. 200. 300. 400. eo Fig. 21 Strain-Degree Relation for Ottawa Sand, Circular Path