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CE-STR-80-l2
PLASTICITY MODELS FOR SOILSPART I: THEORY AND CALIBRATIONPART 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 EngineeringPurdue University
West Lafayette, IN 47907
July, 1980
50272 -101
REPORT DOCUMENTATION !_I:.-REPORT NO.
PAGE NSFjRA·-8002084. Title and Subtitle
Plasticity Models for Soils, Part I:Part II: Comparison and Discussion
Theory and Calibration;
3. Recipient's Accession No.
5. Report Date
July 19806.
I--------------------------------~-----t_=__=:__:_---------___i
7. Author(s) 8. Performing Organization Rept. No.
E. Mizuno, W. F. Chen CE-STR-80-129. Performing Organization Name and Address
Purdue UniversitySchool of Civil EngineeringWest Lafayette, IN 47907
12. Sponsoring Organization Name and Address
Engineering and Applied Science (EAS)National Science Foundation1800 G Street, N.W.Washington, D.C. 20550
15. Supplementary Notes
1---------------------- -..---------- -------16. Abstract (Limit: 200 words)
10. Project/Task/Work Unit No.
11. Contract(C) or Grant(G) No.
(C)
(G) PFR780932613. Type of Report & Period Covered
14.
- --- ----------------f
Three types of soil models are described in this report: (1) nonlinear elasticitymaterial model with the Mohr-Coulomb or the Drucker-Prager surface as failure criterion; (2) Mohr-Coulomb type of elastic-plastic material model with two differentsizes of elliptical hardening cap which are defined respectively on the tensile meridian plane and the compressive meridian plane; (3) Mohr-Coulomb type of elasticplastic material model with an elliptical hardening cap whose size depends on the Lodeangle 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 rangefor the isotropic or anisotropic material element; (2) the incremental plasticitytheory is applied to calculate plastic strain increment during loading range. Theconcept of "decomposition" of stress state onto tensile meridian plane and compressive meridian plane is described.
1-------------------------------------------------117. Document Analysis a. Descriptors
SoilsModelsPlastic properties
b. Identifiers/Open·Ended Terms
c. COSATI Field/Group
Elastic propertiesTensile propertiesSoil mechanics
Earthquake Hazards Mitigation
18. Availability Statement 19, Security Class (This Report) 21. No. of Pages
NTIS1--------------+---------
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
E. Mizunol 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
2Professor 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 ••1J
(2)
where 11
, the first invariant of the stress tensor, is the sum of the di
agonal stress componsnts and 6 .. is the Kronecker delta. The hydrostatic1J
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 (11
/3) and the
component BA represents the deviatoric stress Sij. The length of DB and BAare 13 p and p = Isil + S~2 + S~3 ' respectively. If the stress vector-TQA 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-
81 -13 cos
J(313 _3_)
2 J3/22
(3)
where J3
is the third invariant of the deviatoric stress tensor.
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].
F I . "'+ 3(I-sin cj»1 Sl.n 'Y
sin 8 + 13(3+sin cj» cos 8 /--yJ - 3c cos cj>
2 2 o (4)
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
ing it, we obtain
J: -4-
TI/3 into Eq. 4 and rearrang-
ex =
k
2 sin <j>
13 (3-sin <j»
6c cos <j>
13 (3-sin <j»
(6)
These material constants are identical to those given by Zienkiewicz
[13]. For the tensile meridian matching, substituting 6 = 00 in Eq. 4
and we obtain
2 sin <j>ex =
/3 (3+sin <j»
(7)
k 6c cos <J>
13 (3+sin <j> )
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].
_ BIlIJ - (A - C e )
2
:r -5-
(8)
in which A, Band C are material constants. Here, the failure
equation becomes parallel to I axis under large value of I and1 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 changep
dEkk
• The location of the cap x is related to the plasticp
volumetric strain function Ekk
with the material constants W
and D according to
P DxEkk = Wee -1) (10)
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
Ft
o (11)
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 .•1J
(12)
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 element1J
in soil mass and do .. is the principal stress increment after the1-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 = 00 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 deviatoric1J
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
dS ..1J
o 1dS .. + dS ..
1J 1J(13)
owhere dS ..
1J0° planes,
1and dS .. are the components of the stress in the 8 = 60° and
1Jrespectively. From the geometry, the magnitudes of these
I 0 0components of dJZ
on OO-plane and dJ2
on 60-plane can be calculated
from the total dJ2
as
dJO dJ [cos(n/3-8) - -l sin(n/3-8)]2 1Z Z 13
J(14)
1 1 sin 8)2dJ2dJ (cos 8 - -
2 13
I -8-
Since the deviatoric stresses dSO 1
and dS .. have the following char-ij 1.Jacteristics:
dS? ° 0 0 0 1 0 1 01= (dSll ' dS
22, dS
33) (dS
ll, - 2" dSU ' - "2 dSll)
1.J
J
(15)
1 1 1 1 1 1 1dS .. (dSll ' dS22
, dS 33 ) (dSll
, dSll
, -2dSn )1.J
Therefore, for the case of loading, the components of dSO and dSl can beij ijcalculated from Eqs. 14 and 15 as
0 0 0 EpJ~ JJ~ ff)(dSn , dS22
, dS33
) = -2 -3 3 '
and (16)
1 1 1 (ff. _jdJ~ 2%)(dSll ' dS22
, dS 33 )3 '
-+Similarly, the vector CC I with length dI
lon the CTC-CTE plane can be de-
o 1composed as dIl and dI
lonto the CTC line and CTE line respectively. From
the geometry of Fig. 4, we have
1- 1:..- sin(Tr/3 - 8)J/3
cos 8[
COS(Tr/3 - 8)
dIl 2
(17)
o 1Therefore, the decomposed principal stresses do .. and do .. can be re-
1.J 1.Jwritten as
:t -9-
~IO dIO o )d 0 = -..1. + dS O __1 + dSO dIl 0a .. 3 11' 3 22' -3- + dS
331J
(18)
~I1 dIl
dSl dIl
)1 = -.l + dSl _1+ -..1. + dS ldo ..
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 KO
,Kl
and the shear moduli GO ' Gl
may be determined from the
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 KO
and Kl
are assumed to be functions of theo 1
first invariant of the decomposed stress tensor II and II '
respectively,
and (19)
(20)
and elastic shear moduli GO and Gl are assumed to be functions
of the second invariant of the decomposed deviatoric stresso 1
tensor J2
and J2
, respectively.
GO "GaCM) and G1' G1QJ~)
According to the incremental Hooke's law, the strain increments
d€O and ds 1 are written in terms of the decomposed stresses da~. and do:.~ ~ ~ ~
as
T-10-.:..:,.,
dOl dIO + 1 0E: . . 9 1 6.. 2G dS ..lJ KOlJ 0 lJ
(21)
and the total strain increment dE: .. is the sum of the two components.lJ
dE: ..lJ
o 1dE: " + dE: ..
lJ lJ(22)
It should be noted that the direction of the total strain increment dEij
is not necessary in the same direction as that of the total stress incre-
The following shortcomings are noted in this modelling:ment dO' ..•lJ(i) The decomposition of dI
1into two parts is not unique.
convenience, we use the GTC-CTE planes.
Here, for
(ii) The decomposition of dI1
for the case of isotropic compression
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 KO
and K1
will
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 KO
,Kl
and shear
moduli GO ,Gl
may be considered.
(i) KO
' Kl
, GO and Gl
are constants.
(ii) KO
' Kl
are functions of the first invariant of the decomposed
10 1stress tensor or I and the plastic volumetric strains
PO PillEkk or Ekk , respectively,
Ko I (23)
and G ,G are functions of the second invariant of the decom-o 1 1posed stress tensor .J~ or .J2 and the plastic volumetric
. PO PIstra1ns E
kkor E
kk•
(24)
failure envelope which will not harden.
decreases or increases. The Mohr-Coulomb criterion is used here as
The plastic strain dEP duringij
the loading is derived from the flow rule (Eq. 12). The total plasticP 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
andP
Ekk
the
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 thatij
used previously in the nonlinear elasticity model.
In the plastic range, the two elliptic caps located on the 8 = 0°
60° planes will contract or expand as the plastic volumetric strain
J: -12-
POdE:,. +
1J
PldE: ..
1J(25)
Each of the plastic strain increments can be written as
POdA
O3FO__c_
ds, . =°1J 30. ,1.J
(26)
P1dAl
3F1
dE: , , =__c_
1.J 3 1°ij
scalar functions, FO, Fl are thec c
the decomposed stresses, respec-
where dAO
, dAl are different positive
11 " f . dOle 1.pt1.C cap unct10ns an cr", 0., are1.J 1.J
tively. From Eqs. 25 and 26, we have
POdE:. ,
1.J
(27)
[
3F.l
nl 3F1
3g 3Jl
.]c 1 c ~J2 21 ----+-- -- --
dA n l 30:, IT 3J1 30~ .1 1.J 3~J2 2 1.J _
The total strain increment dE: is the sum of the elastic strain and plasticij
strain increments.
dE: ..1.J
+ 0 .. +1.J
:r -13-~
r°° aFdA ~
. aI~o..~J
oij
S~J1J
S: '.]1J
+
(28)
°de .. is the strain corresponding to the decomposed deviatoric stress1J 1
Also, d>" has the same form as that of Eq. 29 except changing the
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
° 1consistency condition dF = ° during loading and Eq. 12, d>" ,d>" can bec
derived in a straightforward manner as
aFO o GO dFOs~ . °3KO -6" dE: +-- c
de ..
all kk g d~ 1J 1J
0dA
[aFO~2 [~r - dFO dFO(29)
9KO --% +G3 __c c
aI 0 dIO d PO1 1
E:kk
where
dSOij·
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:
PdE: ..1J
1: -14-
(30)
where FL is the Mohr-Coulomb failure function, cr,. is the total stress and1J
dA is a positive scalar function.
The main characteristics of this model are:
(i) The direction of the strain increment dE .. is not necessary in1J
the same direction as that of the stress increment dcr" even1J
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:k
that may exceed
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 KO,01
or K is now a funct10n of 11 or II and the total plastic volumetric strain
E~k ~ 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 as1J
dEij
I-15-
+[
(IF
+ dA (liC
0 ..1 1.J
+_1_
2JJ2(31)
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
cos e (2-m) (l-2m) (l+m)2 3/2
2 (m -m+ 1)(33)
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 KO
and \' shear moduli GO and Gl
• The bulk modulus K is a function
of 11
and is determined by an isotropic compression test. The shear moduli
GO and Gl
are functions of ~ and ~ and are determined by the stress
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 G1
can be determined respec
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 G1
can be determined from the slopes of the unloading curves of an
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 El
and
E: , or E and E:. These curves are fitted by a function using nonlinear2 2 3
regression analysis. In general, the relation between stress difference
and strain difference can be expressed as
Taking derivative with respect to the strain difference, we have
(34)
or (35)
Each pair of GO and Gl
functions are utilized for the present prediction
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 dall
, d022
and
d033
can be written in the form as
2m -12 -m
(36)
I-18-
from which dJ2
can be expressed in terms of dOll. Thus t for a given dal1tthe corresponding strain increment d£ .. can be computed. The process con
lJtinues until the stress path reaches the Mohr-Coulomb or Drucker-Prager
failure surface.
oThe stress-strain relation can be regarded as linear up to dOll
of 5 psi or dail of 2 psi, respectively. Thus, the following relation is
assumed in the leastic range, for m =0
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.
d 0 0.0014 0 0 -0.001560 0 0 (37)£11 = dallt d£22 dOll and d£33 = 0.0002 dOll
and for m = 1,
d 1 0.00325 dail'1 -0.00267 1 and
1 1(8)£n = d£22 = dOll d£33 = -0.00055 dOll
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 ap
(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
m =1 in
for the
shows the location of hardening caps for the planes of m=O andP
II - ~ space. The plastic strain increments d£ij are calculated
cases of Cap Model I and Cap Model II. As the experiments have
been conducted under stress control condition t 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,
1and dA (40)
and for cap model II,
dA
pTherefore, the plastic strain increment dE is obtained from Eq. 12
ijrespectively.
(41)
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 KO
' Kl
and shear moduli
GO' Gl
are obtained using the curve fitting procedure similar to that of
clay X. The principal stress increments dOll' d022 and d033
with the
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 G1
for m = 1 is assumed to be a function of ~or constant, respectively. The bulk moduli K
Oand K
1are assumed to be two
different constants respectively. The material constants Wand Dare
estimated to be 0.135 and 0.009. Therefore, Eq. 10 is
PE:kk
0.135 [l - e-O•009 (x-46. 5)1 (43)
where the preconso1idation pressure a is assumed to be the same as that inp
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 G1
which are functions of J2
•
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 da11
by
dOll = dan' dOn = 0 and d033 = m~i dan
for Test No.2, 3, 7, 11 and 12 and
]:-21-
(44)
1 - 2m do d dm- 2 11 an °33
m+1m - 2 dOll (45)
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 the1)
inclusion of hydrostatic pressure increment dIl
causes a volumetric change.
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 Gl
for the loading case, the experimental data
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
GL L
analysis. Thus, the shear moduli ° and Gl
for the loading cases are given
by
and
°1"-22--
(46)
where a~ and a~ are the stress differences at failure in CTC and CTE tests.
The stress difference at a failure is given by
oaf
6 teeDS ~ + am sin cj»
3 - sin 1>
(47-a)
for the modulus GL
ando6(c cos 1> + a sin cj»
m
3 + sin 1>
(47-b)
Lfor the modulus Gl
, where a is the mean normal stress. In this case, the
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 dottedur Gurcurves. Thus, the shear moduli GO and 1 have the same form as that in
Eq. 46.
The bulk modulus KO
and Kl
are determined from the conventional tri
axial compression and tension tests. Although the data on the isotropic
Fig. 20.. KurpOlnt, a
unloading,
The bulk
andKur1
and KNLa
previous unloading
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 inL L 1
moduli KO and Kl
for the loading up to the unloading
for the unloading, reloading up to the point of previous
and K~L for the loading starting from the point of the
are determined from the figure. These bulk moduli
appear to depend on the initial confining pressure ac
general form for K may be written as
Therefore, the
K = K(a )c
(48)
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 preconsolidationmax
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-
H I N 0\ I
Tab
le1
Mate
rial
Co
nst
an
tsc,
¢,a
and
k
Mat
eria
lC
onst
ants
Cla
yX
Cla
yY
Kao
linit
eC
lay
Otta
wa
Sand
C(p
si)
2.0
4.0
13.7
52
.0
ep(0
)2
6.5
71
6.8
81
7.0
44
2.3
4
2si
np
0.2
02
30.
1221
0.1
25
00.
2117
J3
{3-s
ine
:p)
6c
cos
¢2
.42
75
4.8
92
16
.82
31
.39
4J3
(3-
sin
cP)
2si
np
0.1
49
80.
100
.10
27
0.3
34
j3(3
tsin
¢)
6c
cosP
1.7
97
64
.30
41
3.8
30
2.2
01
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"2.
Fig. 2
0",
Shape of Yield Criterion on TI-Plane
'1--28-
T
Failure EnveloP~. F~ ................ /
t, ............................ / Horizontol Tangent
x- L EllipticR Strain Hardening
Cap Fe
1: -----x_X_-Q--J -I I
Fig. 3 Elliptic Cap Model
1-29-
i---'1 I W o I
~~
8'
y:;;
oF
dI.
(Ojj
..dC
Jjj)
F'
CT
CL
ine e=
60
°p
lan
e
8=
0°
Pla
ne
-I.
Fig
.4
Gen
era
lS
tress
Path
in1
1-~
Sp
ace
e= 600 plane
o
Hardening Cap
-I,
a) Cap Model I
\0"''0 --_co\)7
e= 600 plane
,-11 /...- ...- '7"'- / \..ine_j / \) Failure ~--'----- _.... ...-j Coulo,"
-/Hardening Cap
plane
-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-
Dat
a
IIExp
eri
me
nta
I
Cur
veU
sed
in
Pre
dict
ion
o
m=
I
eTc=
\0ps
i
CLA
YX
/..----------
//
q I I ¢ I I Q I
5.0
f-J , , Q I ~ II
10.0
~
Exp
eri
me
nta
lD
ata
Cur
veU
sed
in
Pre
dic
tion
o
CLA
YX
oc=
\0ps
i
m=
O
,,-
".....
,,"J:J
'//
//
all
//
/
c;f
I I I¢ I I P I I 6 I I I o I I I ¢ I I ¢ I v'
5.0~
10.0~
(/)
(-1
0-
I..
W-
Wbit
)I
I 0- -
O.
1.0
2.0
3.0
O.1.
02
.03.
0
(€
I-€;)
t01
0(€~-
€3
)t
0/0
Fig
.7
Cali
bra
tio
no
fS
hea
rM
od
uli
for
Cla
yX
(ae
=10
psi
.N
on
lin
ear
Ela
sti
cit
yM
odel
)
CLA
YX
CLA
YX
20
I-m
=O
eTc=
20
psi
o
20
I-m
=,
ac=
20
psi
Dat
a
Cur
veU
sed
InP
red
icti
on
Exp
erim
enta
lo
/"..
,---------Q
.
/'
//'
0/ /0 / / }5 /
0/ I I
:)1 I 0/
o
10I
Exp
erim
enta
lD
ata
Cur
veU
sed
10
Pre
dic
tio
n
o
..,-
."".
----
---
.,,'"
/"..
'"/
/0
II /) I I o I
0/ I c;) I o I ~ I'
'u; 0.
10I-
~ - t:r I b- -
Ii I Vol~
I
o.1 10
.O
.I 10
.
Fig
.8
(€,
-€
;),0
/0
Cali
bra
tio
no
fS
hea
rM
od
uli
for
Cla
yX
(0c
(Er
-E
3),
0/0
20p
si,
No
nli
nea
rE
lasti
cit
yM
odel
)
Cur
veU
sed
in
Pre
dic
tion
Exp
erim
enta
lD
ata
Io
CLA
YX
m=I
ac=
30
psi
o
o-------
--.",-
.;'"
,/
/0/
//
//0
/
//C
II / I 1 I I 0/ 1 1 9 1 d
10.1
-
20.1
-
Dat
a
I
Cur
veU
sed
in
Pre
dic
tion
Ex.
perim
enta
lo
m=
O
CLA
YX
).:.
J".
.".
.,/
,/
,/
/0
//
/.
/a:
=3
0p
SI
/0
c/
1 I I 10
I I I I '0 I I I IQ I I 1
01 I I 1
\0.1
-I
01 I / /
0/ I / W I I
20.1
-
.~
t-J0
-
I W U1
~
I~ I b -
0.10
.10
.
(E,-
E3
')%
(Er-
E3)
0/0
Fig
.9
Cali
bra
tio
no
fS
hear
Mo
du
lifo
rC
lay
X(O
c=
30
psi,
No
nli
near
Ela
sti
cit
yM
od
el)
CLA
YX
m=
Op
IC
LAY
Xm
=I
I1.
2~
at=
10ps
iJ
1.2I
CTc
=10
psi
I I0
, /
\.0I-
0E
1/
1.0
f-0
E1
I /I:':
.I:':
./
IP
I:':.
-E2
-E9'
I2
//
//
/0
.8I-
0-E
318
-../.
b""J
!.0
.8c;
fD
-E3
/I
/H
I,
...,I
I Vol
OSr
/I0
.6C
'\ I.-
..-
.~
~0
0.....
.-
IVIV
'C0
.4,
(/
Pc
0.4
'0/
'0~
/~
....9'
enen
02~
/,
/JY
0.2
I-./
//
'"I
dP
'"I
24
II
Il
II
68
0I
23
~ Stre
ssIn
crem
ent
60
jt
psi
Str
ess
Incr
emen
t6
CT 1
tps
i
Fig
.1
0C
ali
bra
tio
no
fO
rth
otr
op
icM
ate
rial
15
CLAY X
\0
5
m=O
m= I
o 10 20 30 40 50 60
Fig. 11 Location of Hardening Caps for Clay X
12.-37-
Curve Used in
Prediction
Experimental Datao
_-"0(:) --),it-
,- ...09"'-
;:fJY
10'12"''''(;:5'-
20.-
II
m=O
CLAY Y
DC = 2.5 psi,/
o/o
I::i 10.~ ¢
bI
¢P
6I
~, I I
O. 2. 4. O. 0.5
III0.
10
CLAY Y 1 1o /
I //
I /I 0:= 2.5 psi /c
? / 0m = I /
/I /I /
r- ¢ 20. - 0;//I /
I /I L.
c1:J/0
/I /I 10. r- / 0 Experimental DataI ~/¢ (
--- Curve Used InII PredictionI
II ,o 2 4 0.5
(E~-E3)' %Fig. 12 Calibration of
(0 2.5 psi,c
E kk' %
Shear and Bulk Moduli for Clay YNonlinear Elasticity Model)
T -38-
Experimental Data
t
0.5
~ = 5.0 psi, m = 0
.0---.,.,-tJ/
//
/I
/0/
I/
I1
/
/JI
I01II
15'
10,0
25.f-
I
5.0
Curve Used In
Prediction
CLAY Y
_----0---.,.,-
o.
0II,
10. 9"v; I0- I 0~
cd,-...r<'l
b II
~0-JIII
Experimental Data
Curve Used in
Prediction
0.5
m= I
ac = 5.0 psi/
//
//
//
//
/ 01
//
I/
//
/1
/10
11
//
II
1/
15 ill
10,0
25.-
I
5.0
CLAY Y
o.
J
¢I,J
10. f-¢'v; I,Q.
I 0~ Q,-...
tf II I0-
~II
(€~-€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
I
Curve Used in
Prediction
Experimental Datao
---(5"-- .--".-
/"
/'0/
II
40.~ qI,
It1II
J(.
I 30
----".--,,'" 0
//
/(/) 10.~ c/a. I
II
QIII6,,,
O. 5. 0. 1.0
20,.----~-------.,-----'-----------.,
II
CTc = 10 psi, m = I
o
9/0
//
//
o //
/I
I
P/
//
//
//
/I
/I
//
30. /I
50. o Experimental Data
CLAY Y
(.
__I,...... ..----0
".-
p'"/
II
PIII
.~ 10. ... 0II
~' ___ Curve Used in
, Prediction
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-
40.1-
I
1.0I
0.5Ekk , 0/0
o.
30.1-
20.1-
ac =5.0 psi
ac = 2.5 psi
oc = \0.0 psi
o
oI
o.
CLAY Y
././ro
//
r;f/
4J:f/
L/0
I~
/5,.. b.~
P
15 ~
'Vi 10.a.~-b
I
0"
CLAY Y
20.()c = 2.5 psi
II
40.-
30.....
III
~P
10I
II
qI
/
f!>5.0 psi /10.:"'/
10.0 psi 8
I
()c =
oc=
15.- I
/0II
(/) Ia. \0.... t¥J~ I
....... Itf II
~ 00---- I
5. ~ I 8I
tb 0,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= 0.87Ro= 2.39 ", \
\ \\ \\,
0 10 20 30 40 50 60
II' psi
Fig. 16 Location of Hardening Caps for Clay Y
1: -42-
I7
0,
70
'
Ka
olin
ite
Cla
ym
=O
Ka
olin
iteC
lay
m=
I
60
I-6
0f-
20.
I 15.
Tes
tN
o.\0
Test
No.
13
I• o 10.
I 5.o
30
l-
40
f-
50
f-
....
~.
....
...--
•",
....
"..
.0
00
•"
...0
0•
/",,
"'0
0•
""0
•A
",,
-t>
•p
""0
."0·
,,-0
.v
20~
6.0
~~
~ d10~ , , , 'f I
I 8.I 6.
Cur
veU
sed
in
Tes
tN
o.4
Tes
tN
o.I
Pre
dic
tion
I• o 4.I 2.
o
50
I-
,J:!
"~..
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T. -45-
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and
PLASTICITY MODELS FOR SOILS
Comparison and Discussion
by
E. Mizunol
and W. F. Chen2
, 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 = 20c
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 assumedc 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 thev
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. Inc
the prediction (0 = 10 psi), the cap models show some plastic strains inc
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 isc
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, thec
predictions for the tests with 0 10 psi, m = 0.5 and 0 = 10 psi, mc 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 psic
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 Toct
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.
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lr -11-
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':IT'. -12-
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::rt -13-
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