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1LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
0 50 100 150 200 250
0
5
10
15
20
25
Stresses (kN/m3)
Dep
th (m
)
Pore Water Pressure, u
Vertical Effective Stress, '
v
Undrained Strength
Profile, su
Uniform, Normally Consolidated Soil Profile
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.600.00 0.50 1.00 1.50 2.00
Settlement (DTOL=1.d-2)Settlement (DTOL=1.d-3)Settlement
(DTOL=1.d-4)
Settl
emen
t at C
ente
rline
/ Foo
ting
Wid
th (%
)
Applied Pressure, p/pa
pa = atmospheric pressure
EXPLICIT ALGORITHMS FOR THE NUMERICALIMPLEMENTATION OF A
NONLINEAR ELASTOPLASTIC
MODEL FOR LIGHTLY OVERCONSOLIDATED CLAYSby
Laurent X. Luccioni and Juan M. Pestana
A report on research sponsored by the National Science
Foundation (NSF)
Geotechnical Engineering Research Report No UCB/GT/99-22
December 1999
GEOTECHNICAL ENGINEERING.Department of Civil and Environmental
Engineering
University of California, Berkeley
-
2LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
TABLE OF CONTENTSTABLE OF CONTENTS
..............................................................................................................................................
2LIST OF
TABLES.........................................................................................................................................................
2LIST OF FIGURES
.......................................................................................................................................................
2ABSTRACT
..................................................................................................................................................................
3INTRODUCTION
.........................................................................................................................................................
4FINITE ELEMENT IMPLEMENTATION
..................................................................................................................
5
Local Stress Integration
Algorithm...........................................................................................................................
7Global Solution
Algorithm......................................................................................................................................
10NUMERICAL
SIMULATIONS................................................................................................................................
12
Single Element
Tests.............................................................................................................................................................
12Patch Tests
............................................................................................................................................................................
13Boundary Value
Problem......................................................................................................................................................
13
EXPLICIT FINITE DIFFERENCE IMPLEMENTATION
........................................................................................
14Explicit Time Marching Algorithm
.........................................................................................................................
14Lagrangian Analysis
...............................................................................................................................................
15Numerical Formulation Used In
Flac.....................................................................................................................
15
The
grid.................................................................................................................................................................................
15Equations of motion
..............................................................................................................................................................
16Dynamic relaxation and
damping..........................................................................................................................................
16Solution stability and mass
scaling........................................................................................................................................
16
Numerical Simulations with Finite Differences-Single Element
Tests....................................................................
17SUMMARY.................................................................................................................................................................
18ACKNOWLEDGEMENT...........................................................................................................................................
18REFERENCES
............................................................................................................................................................
18APPENDIX A: SUMMARY OF BEAR-CLAY MODEL FORMULATION
............................................................ 21
Hypo-Elasticity Formulation
..................................................................................................................................
21Plastic States- Yield Surface
..................................................................................................................................
21Flow Rule and Large Strain Failure Conditions
....................................................................................................
22Hardening
Laws......................................................................................................................................................
22Gradient of Yield Surface and Elastoplastic Modulus
............................................................................................
22
LIST OF TABLESTABLE A.1: BEAR-CLAY INPUT MATERIAL PARAMETERS
USED IN THE ANALYSIS.
.......................................................... 23TABLE
1. SUMMARY OF THE STRAIN SUBINCREMENT FOR OCR =1 SIMULATION.
............................................................
24TABLE 2. RATE OF CONVERGENCE OF LOCAL NEWTON ALGORITHM FOR OCR =
1 SIMULATION...................................... 24TABLE 3. J2
FLOW MODEL PROPERTIES USED FOR TESTING GLOBAL ALGORITHM- PATCH
TEST.................................... 24TABLE 4. SUMMARY OF
GLOBAL SOLUTION ALGORITHM SIMULATION - PATCH TEST
..................................................... 24TABLE 5.
COMPARISON BETWEEN ITERATIVE AND INCREMENTAL GLOBAL SOLUTION
ALGORITHM................................ 25TABLE 6: ITERATIONS
SUMMARY FOR THE GLOBAL ALGORITHMS
...................................................................................
25
LIST OF FIGURESFIGURE 1: SUMMARY OF BEAR-CLAY MODEL
FORMULATION.......................................................................................
26FIGURE 2: PLANE STRAIN FINITE ELEMENT MESH
............................................................................................................
27FIGURE 3: PARAMETRIC STUDY ON LOCAL ALGORITHM FOR NORMALLY
CONSOLIDATED SPECIMEN (OCR=1).............. 27FIGURE 4: PARAMETRIC
STUDY ON LOCAL ALGORITHM FOR OVERCONSOLIDATED SPECIMEN (OCR=2)
........................ 28FIGURE 5. ONE DIMENSIONAL STRESS-STRAIN
LAW FOR THE J2 FLOW MODEL.
...............................................................
29FIGURE 6. FINITE ELEMENT PATCH TEST
MESH...............................................................................................................
29FIGURE 7: COMPARISON OF PERFORMANCE OF GLOBAL INCREMENTAL
ALGORITHM.......................................................
30FIGURE 8. PERFORMANCE OF NUMERICAL INTEGRATION ALGORITHM:
UNDRAINED LOADING OF A FLEXIBLE FOOTING.31FIGURE 9: COMPARISON OF
PREDICTED SETTLEMENT CURVE USING IMPLICIT AND EXPLICIT STRESS
INTEGRATION
ALGORITHM.............................................................................................................................................................
32FIGURE 10: EXPLICIT CALCULATION CYCLE (AFTER ITASCA,
1995).................................................................................
33FIGURE 11: VALIDATION OF THE BEARFISH
SUB-PROGRAM............................................................................................
33
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3LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
EXPLICIT ALGORITHMS FOR THE NUMERICAL IMPLEMENTATIONOF A
NONLINEAR ELASTOPLASTIC MODEL FOR LIGHTLY
OVERCONSOLIDATED CLAYS
Laurent X. Luccioni and Juan M. PestanaDepartment of Civil and
Environmental Engineering
University of California, Berkeley
ABSTRACTImplicit stress integration algorithms have been
demonstrated to provide a robust formulation forfinite element
analyses in computational mechanics, but are difficult and
impractical to apply toincreasingly complex nonlinear constitutive
laws. This report discusses the performance of fullyexplicit local
and global algorithms with automatic error control used to
integrate general nonlinearconstitutive laws into a non-linear
finite element computer code. The local explicit stressintegration
procedure falls under the category of return mapping algorithm with
standard operatorsplit and does not require the determination of
initial yield or the use of any form of stressadjustment to prevent
drift from the yield surface. The global equations are solved using
an explicitload stepping with automatic error control algorithm in
which the convergence criterion is used tocompute automatically the
coarse load increment size. The proposed numerical procedure
isillustrated here through the implementation of a new set of
elastoplastic constitutive relationsincluding isotropic and
kinematic hardening as well as small strain hysteretic
nonlinearity. The newmodel, referred to as BEAR-clay, was
formulated to describe the response of lightlyoverconsolidated
soils in both drained and undrained shearing. A series of numerical
simulationsconfirm the robustness, accuracy and efficiency of the
algorithms at the local and global level. Forcomparison, this
report also documents the implementation of the new set of
constitutive relationsinto the Finite Difference computer code,
FLAC, which is widely used in geotechnical engineeringpractice.
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4LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
INTRODUCTION
Numerical modeling of boundary problems in geomechanics is a
complex process requiring threekey elements: a) the formulation of
the governing equations and constitutive framework describingthe
physical system response, b) the formulation of constitutive
relationships, which accuratelycaptures selected features of
material behavior, and c) the robust implementation of the theory
andconstitutive relations into a numerical framework to ultimately
solve boundary value problems.
If we consider that the practical objective of developing
constitutive laws is to solve boundary valueproblems, it seems
reasonable to think that the ultimate test of constitutive
relations is theboundary value problem test and not the one element
test (or 'elemental response').Nevertheless, the solution of a
boundary value problem is not only a function of the
theoreticalframework and constitutive relations, but it is also a
function of the numerical algorithm used tointegrate the
differential expressions describing the material response. In
addition, uncertainties inthe predicted response arise from
uncertainty in materials properties and in the state variables
(e.g.,initial stresses and previous stress history), as well as
from the inaccurate geometry or othersimplifications among others.
Thus, although appealing, the boundary value problem test seemsto
be inadequate to judge constitutive laws. To address this apparent
shortcoming, the boundaryvalue problem test can be subdivided into
two separate components: (a) the numericalimplementation dealing
with the accuracy, stability and efficiency of the numerical
algorithm, and(b) the overall performance of the model by comparing
measured quantities such as displacementsand stress fields with the
ones in a well controlled physical experiment predicted by the
proposedformulation. In this context, a constitutive law may be
evaluated for its amenability and robustnessfor numerical
implementation, thus addressing the problem of realistic
description of materialresponse separately. Once the numerical
aspect has been investigated, simple boundary valueproblems where
analytical solutions are available may be used to test the overall
performance ofthe model. In this way, uncertainties, which are
irrelevant to the theoretical and numericalformulation, are
minimized.
This report investigates two numerical frameworks, the finite
element and finite difference methods,which are widely used to
solve boundary value problems in geomechanics. Both methods
producea set of algebraic equations to be solved, but these
equations are derived in quite different ways. Inthe finite
difference method, the governing equation is satisfied at the nodes
and every derivative orfunction in the set of governing equations
is replaced directly by algebraic expressions written interms of
nodal values. In contrast, the finite element method assumes that
the field quantities varythroughout each element in a prescribed
fashion using specific functions (i.e., basis functions)controlled
by parameters which are adjusted to minimize some measure of the
error (or energy)term over the domain of integration. Two widely
available codes are used here as numericalplatforms. The finite
element implementation is performed within FEAP, a multi purpose
nonlinearfinite element code developed at the University of
California, Berkeley. The finite differenceimplementation is
performed within FLAC an explicit finite difference code developed
by ItascaConsulting Group, Inc.13 which is widely used in
geotechnical engineering practice. The followingsections discuss in
detail the numerical implementation using explicit integration
algorithms in thetwo frameworks (Finite Element and Finite
Difference methods).
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5LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
FINITE ELEMENT IMPLEMENTATION
There has been a significant effort in computational
geomechanics to describe the discreteconstitutive equations using
fully implicit local and global stress integration algorithms4,5.
Themost popular is, perhaps, the fully implicit return mapping in
which the return directions arecomputed by closest point projection
method4,16,44,20, and has the advantage of being amenable
toconsistent linearization36. Recent advances in classical
computational plasticity have established thesuperiority of fully
implicit algorithms to solve boundary value problems using the
finite elementmethod for relatively simple plasticity models37,39.
However, as the constitutive laws become morecomplex, the
attractiveness of the fully implicit algorithm decreases
significantly. First, theformulation requires the second Frechet
derivative of the yield function with respect to the statevariable
tensors. Second, the solution of the local set of nonlinear
equations for highly nonlinearyield functions is by no means
trivial, as the local Newton scheme may not converge. It is
importantto note that fully implicit algorithms do not guarantee
non-linear stability (or B-stability), eventhough linear stability
is automatically achieved38,39. Thus, as of today none of the
existing stressintegration, either implicit or explicit, algorithms
is able to guarantee stability and convergence forgeneral
incrementally elastoplastic constitutive laws under general
conditions. Finally, an explicitexpression for the algorithmic
consistent tangent may not be available, hence at best, only a
quasi-Newton method may be used and quadratic rate of convergence
can not be expected. For instance,Luccioni et al.18 presented fully
implicit local and global algorithms using a quasi-Newtontechnique
with a numerical tangent computed every load step by finite
difference and optimizedwith iterative updating procedures since an
explicit expression of the algorithmic consistent tangentcould not
be determined.
For classical explicit integration schemes the discrete
constitutive equations are much simpler toformulate, but their
accuracy depends significantly on the selected step size. Sloan40
first proposedthe application of an automatic stepping with error
control numerical algorithm for the prediction ofcollapse load
using a relatively simple constitutive model. This numerical
technique, widely usedin the field of numerical analysis, is based
on extrapolation procedures. The use of automaticsubstepping and
load stepping with error control algorithm overcomes the main
limitation ofexplicit techniques, since the system adapts as the
estimated error changes.
Constitutive modeling and numerical implementation applied to
geomaterials is by no means trivialsince it involves a subtle
balance between the complexity associated with realistically
describingsoil response and the use of both robust and accurate
numerical algorithms. The Cam-Clay familyof models based on the
Critical State Soil Mechanics framework43 is perhaps the most
widely usedplasticity model for clays used to perform geotechnical
(i.e., boundary value problem) analyses.The advantages of this
family of models derive from three main considerations: 1) their
ability tocapture some aspects of soil behavior 34, 2) their
efficient numerical implementation into non-linearfinite element
codes5,6, and 3) the availability of a large database for material
input parameters fordifferent materials22. Nevertheless, these
models do not incorporate key elements of materialresponse, such as
anisotropic stress-strain-strength behavior or small strain
nonlinearity. Inparticular, soils exhibit a high degree on
nonlinearity in the "elastic" (i.e., recoverable strains)regime
which is better described by a Perfectly Hysteretic Formulation.
The main characteristic ofthis formulation is that the tangent
stiffness decreases monotonically with continued straining and isa
function of a (typically dimensionless) stress measure describing
the distance of the current stateof stress (or strain) to the last
reversal state. The Perfectly Hysteretic model predicts
fullyrecoverable strains in a closed unload-reload cycle, but
dissipates energy according to theprescribed stiffness reduction
law (c.f., Gsec/Gmax as a function of shear strain, Figure 1).
Since this
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6LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
nonlinearity is strongly strain level dependent, the proposed
explicit integration algorithm must beable to vary the 'step size'
in order to maintain accuracy while, at the same time, be
computationallyefficient. In contrast, general subincrementation
procedures with no step control must use verysmall incremental
strains in order to accurately capture large changes in material
response (i.e.,change in stiffness) following stress reversal which
becomes computationally inefficient for manyother practical
conditions. The effect of small strain nonlinearity and anisotropy
has been proven tobe of significant importance in the prediction of
deformations in soil-structure interactionproblems12,14, such as
excavations10 and tunneling activities1. This problem is
particularly relevantfor dynamic problems for which soil
nonlinearity and the associated energy dissipation is the
mostimportant aspect of material response.
Previous evaluations of explicit techniques have included
relatively simple models such as linearelastic-perfectly plastic
Mohr-Coulomb or Tresca-strain hardening models2,40. The
followingsections present details of a fully explicit automatic
substepping scheme with error control tointegrate a nonlinear model
with anisotropic plasticity, small strain 'hysteretic' nonlinearity
anddependence of the yield and plastic flow rules on the third
invariant of the stress tensor followingthe Matsuoka-Nakai
generalization (cf., Figure 1a). Hysteretic nonlinearity is
mathematicallyanalogous to the Bounding Surface Plasticity
formulation where the tangent stiffness degrades as afunction of
the proximity of the 'current state' to the Bounding Surface using
appropriate mappingrules. For the particular case of monotonic
loading, this formulation is also similar in concept to
thestiffness degradation resulting from a "damage-related" process
and it is therefore applicable to amore general class of models.
The local stress integration algorithm falls under the category
ofreturn mapping algorithm with standard operator split procedure
and does not require thedetermination of initial yield or drift
correction techniques. The discrete "local" equations areintegrated
using numerical techniques that preserve the incremental nature of
the continuumformulation, while the global system of equations are
solved using an explicit automatic loadstepping with error control
algorithm as proposed by Abbo and Sloan2. In contrast to
previousderivations, the explicit scheme must be used to verify the
integration error even for "elastic states",since there is no
general analytical expression of the elastic stiffness for the
Perfectly Hystereticformulation.
The proposed numerical procedure is illustrated here by
implementing a recently developedconstitutive model for lightly
overconsolidated clays29,31 that has been used in the solution
ofboundary value problems such as the prediction of deformations
around ground openings in softclays19,17. The model, referred to as
Bear-Clay, is based on the theory of incrementally
linearizedplasticity which is extensively documented in the
literature33,15,37. Model formulation includes threeimportant
components to describe the observed clay response: a) an
elastoplastic framework fornormally consolidated clays with a
single anisotropic yield function with dependence on the
thirdinvariant of the stress tensor to represent accurately the
effect of consolidation stress history, b)equations describing the
small strain non-linearity and hysteretic stress-strain response in
unload-reload cycles, and c) non-associated flow and hardening
rules to describe the evolution ofanisotropic stress-strain
properties. The elastic stiffness tensor is assumed isotropic
allowing thedecomposition of the elastoplastic relations into
volumetric and deviatoric components and they arebriefly summarized
in appendix A. It should be emphasized that the numerical algorithm
proposedhere is independent on the particular set of constitutive
expressions used to illustrate the procedureand can be directly
extended to other material models without conceptual changes. The
Numericalsimulations of single element tests as well as a boundary
value problem confirm the robustness,accuracy, and efficiency of
the proposed algorithm at the local and global levels. Finally,
the
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7LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
implicit technique proposed by Luccioni et al.18 and the
explicit technique presented here arecompared in terms
computational effort as reflected by CPU time.
Local Stress Integration Algorithm
The elastoplastic flow problem at the Gauss point level can be
recast in term of a system ofdifferential/algebraic equations that
have to be integrated numerically. From the point of view
ofintegration techniques, this system of differential/algebraic
equations have the property of beinginfinitely stiff11. This type
of stiff system is usually best integrated using methods such
asBackward Differentiation Formula (BDF), where convergence and
accuracy can be formallyestablished11,32. Gear11 shows results
indicating that fully explicit algorithms are, in general,unstable
and do not converge, suggesting that these algorithms should not be
used to integrate thistype of system. However, in the context of
numerical implementation into a computer code one hasto think in
terms of finite precision. Indeed, we do not require to guarantee
yield conditions exactly(i.e., = 0) but only in an approximated
way, || TOL where TOL may be the machine precisionor an even less
restrictive constraint. This concept may be viewed as a penalty
regularization of theyield condition used for viscoplastic
materials9,24.
Following Sloan40, we define two pseudo-time quantities, T and
T, to be used in the substeppingscheme. When entering for the first
time the algorithm, T is set to one, whereas T is set to zero.The
total incremental strain tensor, , is translated into
subincrement:
k+1 = Tk n+1 (1)
where k represents the kth substep and n represents the nth load
or coarse step. For the remainder ofthe paper, strains are assumed
subincremental strains, unless otherwise indicated and
tensorialquantities are represented by boldface italic characters.
The return mapping algorithm is used tointegrate the continuum rate
equations using the first order accurate forward Euler scheme.
Thetrial state, represented by the superscript tr, is characterized
as follows:
1ptrk+1 = pk + Kk (p)k+1 (2a)( ) 11tr1tr1 2 +++ += kkkk G sss
(2b)
where p, s are the mean effective stress and the deviatoric
components of the stress tensor, , p, s are the volumetric and
deviatoric components of the strain tensor, , and the left
superscript"1" denotes the first order integration method. The
model introduces expressions to model the smallstrain non-linearity
through the tangent elastic stiffness parameters Kk and 2Gk
describing the bulkand shear moduli, respectively and summarized in
Appendix A (cf., equations A.1 through A.3).The loading/unloading
conditions are determined by the sign of the yield function at the
trial(predictor) state: (1 tr k+1, qk), where q is a vector
containing the plastic (i.e., memory) variables.This procedure has
been demonstrated to be algorithmically consistent with the forward
Eulerintegration scheme27 while Papadopoulos and Taylor26 discusses
its limitations for implicit stressintegration schemes. For
"loading" states, the (plastic) corrector step is invoked:
1pk+1 = pk + Kk [(p)k+1 (Pp)k ] = 1pktr - Kk (Pp)k (3a)1s k+1 =
s k + 2Gk[(s)k+1 (Ps)k] = 1s ktr - 2Gk (Ps)k (3b)1q k+1 = q k + q
() (3c) k+1 = k+1 (1pk+1, 1s k+1, k+1, 1b k+1) = 0 (3d)
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8LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
where Pp and Ps are the volumetric and deviatoric components of
the flow rule describing thedirection of the plastic strains (cf.,
Appendix A), q represents the change in the plastic variableswith
continued plastic deformation (i.e., hardening) and is the
incremental elastoplasticparameter, also referred to as the
consistency parameter, obtained from the solution of this systemof
equations. For the particular model used to illustrate the
numerical procedure, the plasticvariables are given by and b
representing the size of the yield surface and the traceless
tensordescribing the orientation of the yield surface in a
generalized stress space, respectively. It must benoted that the
numerical algorithm proposed here is not dependent on the
particular set ofconstitutive laws used to illustrate the procedure
and therefore it can be extended to other materialmodels without
conceptual changes.
In contrast to previous models elastoplastic models for clays,
the Bear-Clay model treats theisotropic hardening variable, , as a
dependent variable. The coupling between isotropic andkinematic
hardening existing in the continuum equations can be written as =
(b, p) where isupdated from converged values of anisotropy, b, as
follows:
( )
+
=+
+++ bb
:expe
e1
exp
1k
1k
c
1p1
kkk (4a)
1bk+1 = bk + b () (4b)where e is the void ratio of the soil (=
volume of voids/volume of solids), c is the slope of theHydrostatic
Limiting Compression Curve, H-LCC, for normally consolidated clays
in a log(e)-log(p) space, b represents the change in anisotropy
(i.e., kinematic hardening) resulting fromplastic loading and ":"
represents the double contraction of tensor multiplication. The
term /bdescribes the coupling resulting from the existence of the
Limiting Compression Boundary Surface,LCBS, as described by Pestana
and Luccioni31 or the use of a Spacing Function28,30. The
existenceof the LCBS satisfies the robustness of the drained
response (i.e., consolidation behavior) underconstant shear stress
ratio, (= s/p), conditions for models incorporating both kinematic
and densityhardening as suggested by Pestana28. The system of 12
non-linear implicit equations (cf., eqn. 3a-d)has only one common
unknown, . Then, it follows that only one single scalar equation
needs tobe solved instead of the inversion of a 12x12 Jacobian as
in the case of fully implicit algorithm18.The consistency equation,
k+1 = 0, which insures that the converged state of stress is on the
yieldsurface, can be regarded as a function of only:
k+1 = k+1 ( (), q()) = k+1 () (5)The consistency equation is
then solved using the Newtons method:
( )m
k
mkmm
=+
++
1
11 )( (6a)
where )(
:)()(
1
1
11
1
11
+
=
+
+
++
+
++ k
k
kk
k
kk qq
(6b)
For Bear-Clay, the change in the yield function resulting from
the change in the elastoplasticparameter can be derived making use
of the chain rule as follows:
)(:
)(:
)()(1
1
11
1
11
1
11
+
+
=
+
+
++
+
++
+
++ k
k
kk
k
kk
k
kk pp
bb
ss
(7a)
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9LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
where 1
1
1
1
1
1
1
1
1+
+
+
+
+
+
+
+
+
=
+k
k
k
k
k
k
k
k
kbbb
(7b)
A summary of the constitutive equations for the Bear-Clay model
is presented in appendix A. Inorder to solve equation 6 with
Newtons method, a starting value for is required. Although it isnot
uncommon to use = 0 as an initial guess/value, it is best to start
with a value of as closeas possible to the converged one in order
to recover quickly the asymptotic quadratic rate ofconvergence of
Newtons method. The 'continuum' expression of is used here, since
for smallsteps, this value is very close to the obtained once
Newtons method has converged. Note thatin the context of explicit
integration technique, subincrement steps used to reach a converged
stateare small, which is the basis for the development and validity
of the infinitesimal linearized theoryof plasticity. The initial
value for the elastoplastic parameter, (0) is then given by:
( ) ( )s
ksk
Gp
K
Geep
K
Ps
qq
s
:2P
:2)1(
p
11p)0(
+
+
+
+
+
=++
(8)
where all the quantities, except the strain increment, are
evaluated at kth step. The error controlscheme used in the proposed
formulation is based on local extrapolation11,40, and requires that
theconstitutive laws be reintegrated with a second order method.
The modified Euler (i.e., predictor-corrector) method was chosen
for this purpose. The modified Euler method uses the convergedstate
achieved from the forward Euler method as base values (known
quantities). The discreteequations can be written in the following
form:
Predictor step:2ptrk+1 = pk + 1Kk+1 (p)k+1 (9a)2s trk+1 = sk +
21Gk+1 (s)k+1 (9b)
where the stiffness coefficients, K and G have been evaluated at
the converged state from theforward Euler procedure.
Corrector step:2pk+1 = pk + 1Kk+1 [(p)k+1 (1Pp)k+1] = 2ptrk+1 -
1Kk+1 (1Pp)k+1 (10a)2sk+1 = sk + 21Gk+1[(s)k+1 (1Ps)k+1] = 2s trk+1
- 21Gk+1 (1Ps )k+1 (10b)2qk+1 = qk + 1q (,1pk+1, 1sk+1, 1b)
(10c)k+1 = k+1 (2pk+1, 2sk+1, k+1, 2bk+1)= 0 (10d)
where superscript 1 indicates converged state from the forward
Euler method, superscript 2indicates converged state from the
modified Euler method, 1b is the kinematic hardening evaluatedwith
the converged state from the forward Euler procedure, and the
isotropic parameter, , isevaluated as follows:
( )
+
=+
+++
2 bb
1
1k
1k
c
1kp1 :expe
e1
exp kk (11)
The system of equations derived from the corrector step is
solved by using the exact same techniqueas the one proposed above
for the first order integration method. Subtracting equation (10)
from
-
10
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
equation (9), a second order accurate estimate of the local
truncation error for the stress tensor, , isobtained:
( ) 2/1111 +++ = kkk 2222E (12)The relative error for a substep
is written as11,40:
111 / +++ = kkkR E (13)
where ( ) 2/1211 +++ += kkk 1111 , which is second order
accurate. During the integration process, eachsubstep size is
continually updated to insure that Rk+1 STOL. The value of STOL is
problem andconstitutive model dependent. On one hand, if the value
set for STOL is too large, the solutionobtained may be completely
inaccurate and therefore not satisfactory. On the other hand, if
STOL ischosen too small, the problem may not converge. Sloan40
reported values of STOL between 10-1and 10-5 and the same range of
values are investigated here for the Bear-Clay model. A
parametricstudy on the satisfactory values of STOL is reported
through the numerical simulations presented inthe last section.
In the case where Rk+1 STOL, the subincrement is declared
successful and all variables are updatedas follows:
( ) 2/1211 +++ += kkk 1111 (14a)( ) 2/ 12111 +++ += kkk qqq ;
(14b)
T = T + Tk (14c)
The local extrapolation is used to compute the next subincrement
size. Hence, Tk+1 = Tk, where is given by:
[ ] 2/11/ += kRSTOL (15)Heuristic bounds on must be introduced
to prevent the extrapolation to be carried too far, resultingon
unstable results. For the Bear-Clay model, it was found that the
range 0.2 2 gives goodresults which is in agreement with values
reported by Sloan. In the case where Rk+1 STOL, thesubincrement has
failed and the size of the subincrement has to be reduced and the
algorithm isrestarted from the last converged value with a smaller
subincrement size. An extrapolation is usedto compute the next
subincrement size, following the same procedure as shown
previously. In orderto reduce the number of unsuccessful substep
and simultaneously keep Tk 0.01, the magnitude of|| n+1|| has to be
controlled which is achieved by the global solution algorithm as
described in thefollowing paragraphs.
Global Solution Algorithm
The numerical techniques used for solving the global discrete
equations can be broadly classified aseither incremental or
iterative procedures. In the iterative procedure, the
discretization leads to asystem of non-linear equations to be
solved by methods such as Newton-Raphson, or quasi-Newton.These
type of methods have the advantage of satisfying equilibrium
equations at the end of eachconverged time step, and if the
consistent tangent is used an asymptotic rate of convergence
isrecovered. In addition, the stability theorem can be proven for
certain cases23. On the other hand,when the material behavior is
strongly non-linear the iterations may not converge as all of
these
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methods have a finite radius of convergence. Moreover, if the
algorithmic consistent tangent is notused, the inaccuracy of the
substitute tangent may result in intermediate strain increments
that maybe too large, causing the local stress algorithm to either
use too many subincrements or divergealtogether.
Incremental procedures, on the other hand, treat the governing
equations as a system of ordinarydifferential equations (ODE).
Thus, the solution consists of a series of piecewise linear steps
thatattempts to approximate the load-deformation behavior of the
system. This class of methodsguaranties that small load increment
will be used which is an advantage when explicit stressintegration
techniques are used at the Gaussian (i.e., "local") level.
Nevertheless, this class ofmethods tends to drift from equilibrium
as the solution proceeds, leading to very
doubtfulinterpretations.
Abbo and Sloan2 presented an incremental algorithm based on an
automatic load stepping schemewith error control. This algorithm
tries to minimize the drift from equilibrium by calculating
theresidual forces at the end of each load increment and adding
these to the applied forces for the nextincrement. The authors
chose to implement a slightly modified version of the Abbo and
Sloan2algorithm into a non-linear finite element code, referred to
as FEAP. This finite element code wasdeveloped at the University of
California, Berkeley for teaching and research, and
completedocumentation is available41. The elastoplastic continuum
tangent is assembled, and used by theexplicit algorithm to solve
for the displacement. In contrast to the implicit algorithm,
theelastoplastic continuum tangent is not significantly different
from the algorithmic consistent tangentas step increments are much
smaller. The elastoplastic continuum tangent, Cep, is given by:
( )PCQ
PCQCCC e
eeeep
:::/ :
+
+=
Hp (16)
where Q is the gradient to yield surface, P describes the flow
rule indicating the direction of plasticstrain increments, Ce is
the continuum elastic stiffness tensor, and H is the hardening
modulus (cf.,Appendix A). In contrast to the traditional
formulation of incrementally linearized plasticity,equation 16
introduces the rate of change of the shape and/or size of the yield
surface with respectto the total volumetric strain (i.e., void
ratio) as described by Pestana 28.
The proposed modification to the algorithm consists of using a
convergence criterion to computeautomatically the coarse load
increment size, making the global algorithm entirely automatic
fromthe Gauss point level to the coarse load level. If more than
two consecutive unsuccessful substepsare detected at any level, the
current coarse load level is divided by two, whereas if more than
twoconsecutive successful coarse steps are detected, the current
coarse load level is multiplied by 1.2.These bounds are purely
heuristic based on multiple numerical simulations performed using
theBear-Clay model, hence caution should be used in generalizing
these numbers to others model orsituations. The previous remark
showcases some drawbacks in using incremental techniques, but
italso constitutes their strengths as these explicit techniques are
flexible enough to accommodate verystrong nonlinear behavior.
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NUMERICAL SIMULATIONS
This section describes numerical simulations used to validate
the proposed approach. Both local andglobal algorithms are coded
into a series of Fortran90 subroutines and they are linked to the
mainnonlinear finite element code, FEAP18. All computations were
performed with double precisionarithmetic on a 32-bit architecture
DEC 3000 workstation at the University of California atBerkeley.
The convergence criterion used in the acceptance or rejection
strategy of incrementaldisplacements, u, at the global level is
based on the local extrapolation procedure2, where
( ) 2/11121 +++ = nnn uuE (17a)111 / +++ = kkkR uE DTOL
(17b)
Similarly to STOL discussed previously, DTOL is a user-defined
tolerance that is model andproblem dependent. For the Bear-Clay
model and within the context of the numerical simulationspresented
below, values of DTOL ranging between 10-1 and 10-5 have been
investigated.
Single Element TestsThe first numerical example investigates the
local stress integration algorithm and itsimplementation. These
simulations consist of single element (cf., Figure 2) undrained
plane straincompression and extension tests, where the sample is
initially anisotropically (i.e., K0 ='h0/'v0)normally consolidated
or 1-D unloaded to an overconsolidation ratio, OCR, of 2. The
"simulatedsoil" corresponds to Boston Blue Clay a low plasticity
clay widely documented in the geotechnicalliterature. Its model
specific material parameters are described in detail by Luccioni 17
and Pestanaand Luccioni29. The axial strain for the undrained test
is applied in 100 finite steps of size A up toa total axial strain
|A| = 10% (i.e., A = 0.1%). Measure of accuracy is based on the
algorithmicconsistency property, which implies that the solution
accuracy increases as the number of stepsincreased. Figures 3 and 4
show the effective stress path and shear strain response for
normally K0consolidated and overconsolidated (OCR = 2) samples,
respectively. For each past consolidationhistory, a parametric
study of the user defined local tolerance STOL is presented. Figure
3demonstrates that the integration scheme is reasonably accurate
for STOL values ranging from 10-5to 10-3, whereas for STOL equal to
10-1, the algorithm looses accuracy in the compression mode.The
same conclusions are reached from Figure 4 where not only a loss of
accuracy in thecompression mode is observed, but also the extension
test did not converge for STOL equal to 10-1.These results seem to
suggest that values of STOL < 10-3 are acceptable. Once the
value of STOLis fixed, the algorithm computes automatically the
number of substeps needed to achieve thespecified tolerance. Table
1 reports the number of steps, N, associated with a given value of
STOL.Intuitively, the normally consolidated plane strain
compression test is the most critical because theresponse is
entirely elasto-plastic and exhibits strain softening behavior.
This intuitive argument isvalidated by numerical simulations, as
for a given STOL, this mode of shearing uses the largestnumber of
steps. Table 2 shows the rate of convergence of the local
Newton-Raphson algorithmfor two typical iterations. The results
indicate that for a strict tolerance (STOL = 10-5) theasymptotic
rate of convergence is quickly achieved, attesting of the quality
of the initial guess forthe consistency parameter, (0). Loosening
the tolerance (STOL= 10-1) results in larger strainsubincrement,
which leads to a progressive breakdown of the explicit integration
technique.Comparisons of the stress-paths and stress-strain curves
between the explicit and implicitintegration techniques, shows
excellent agreement in term of accuracy. The explicit technique
is
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more costly than the implicit one in term of number of
iterations (10,000 vs. 1500), but each explicititeration requires
at the most the solution of a single scalar nonlinear equation
compared to theinversion of a 12 12 Jacobian in the case of the
implicit algorithm18. As a result, the twotechniques are in the
same order of magnitude in terms of the computational effort (i.e.,
CPU time).
Patch TestsThe second series of numerical simulations focus on
the performance of the global solutionalgorithm. Abbo and Sloan2
applied this algorithm to predict collapse load for various
boundaryvalue problems with a cohesive-frictional constitutive law
based on a rounded Mohr-Coulomb yieldsurface. They integrated the
constitutive law using an explicit technique developed by
Sloan40.Herein, we evaluated the global explicit algorithm
performance separately, for a different qualityof the Jacobian
(i.e., continuum versus consistent). For this purpose, the J2 flow
model withisotropic and kinematic hardening laws was selected as
the constitutive law (cf. Figure 5). The J2flow model is a popular
associative model in the structure mechanics field, which is simple
enoughto be consistently linearized under the return map algorithm
such that both continuum andconsistent tangent are available39. A
four-element patch test, shown in Figure 6, is used to comparethe
performance of the different algorithms. A summary of the material
properties for the J2 flowmodel is presented in Table 3. The
surface load applied on the external boundary has a magnitudeof 15
kN, which is higher that the value of the yield stress, Y = 10 kN.
The local stress integrationfor the J2 flow model consists of the
efficient implicit return mapping algorithm. In the
followingnumerical simulations the global solution algorithm is
varied between the proposed incrementalalgorithm and the iterative
algorithm based on Newton-Raphson technique, for the continuum
andconsistent Jacobian. A summary of the first simulation program
is presented in Table 4. Theparameter RES represents the residual
force vector whereas Ux and Uy are the displacements ofnode 5 (cf.,
Figure 6) in the horizontal and vertical direction, respectively.
From Table 4, it is clearthat the global incremental algorithm is
accurate for DTOL smaller than 10-3. One important andinteresting
feature is that for values of DTOL between 10-3 and 10-5, the
algorithm performedequally well using the consistent or continuum
tangent. This result is further illustrated throughFigure 7 where
CPU time and the number of coarse iterations are plotted as a
function of DTOL.Note that as the tolerance DTOL is becoming
looser, using the consistent tangent leads to a moreaccurate and
efficient solution algorithm. These results demonstrate that in the
context of theincremental technique, continuum tangent may be used
as long as the user defined tolerance is keptstrict enough (less
than 10-3 in this case). Table 5 reports a comparison between the
iterative andincremental global solution algorithm using both
consistent and continuum tangent. The iterativealgorithm is ten and
five times faster than the incremental technique using the
consistent andcontinuum tangent, respectively. These results point
out the superior efficiency of the iterativetechnique for this
simple case, but also illustrates the sensitivity of the iterative
technique to thequality of the Jacobian used in the solution.
Results also show that both algorithms are accurateregardless of
the Jacobian used.
Boundary Value ProblemFinally, the third numerical example focus
on the performance of the overall implementationtechnique into the
non-linear finite element code, FEAP. To allow a direct comparison
betweenexplicit and implicit techniques, the application problem
presented here is identical to that presentedby Luccioni and
Pestana18 for the implicit technique. In this example, we consider
the plane strain
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problem of a vertically loaded flexible strip footing of width B
on a uniform 'finite' deposit ofnormally consolidated clay. The
initial geostatic stresses were generated within FEAP using a
bodyforce command to simulate the gravity load at each Gauss point
and a prescribed value of K0 of 0.53describing the ratio of
horizontal to vertical stress. The finite element mesh is shown in
Figure 8,and it is composed of 200, four-noded quadrilateral soil
elements with a 22 Gaussian integrationrule employed for each
element. The soil profile is underlying by a rigid and rough
bedrock. Theload is applied using different global tolerance DTOL,
whereas the local user tolerance is fixed to avalue of 10-3. As
described previously, for each value of DTOL, the algorithm
computesautomatically the number of coarse load increments as well
as the number of subincrementsnecessary to achieve the prescribed
user tolerance. Figure 8 shows the load settlement curve at
thecenterline of the flexible foundation for different values of
DTOL. Again, the results indicate thatthe algorithm is accurate for
values of DTOL less than 10-3. The lack of accuracy observed
whenusing DTOL equal to 10-2 leads to a 25 % overestimation of the
settlement. Table 6 summarizes thecost related variables, showing
that although the difference in accuracy for DTOL values of 10-3
and10-4 is less than 1%, the CPU time is multiplied by a factor
larger than two and the number of coarsesteps quadruple. The
load-settlement curve corresponding to DTOL equal to 10-3 compares
verywell with the one computed using the global implicit-iterative
algorithm. A maximum error of lessthan 5% is observed for a
pressure of p/pa ~ 2 (where pa is the atmospheric pressure). Figure
9compares the load settlement curve for the proposed explicit
algorithm with DTOL= 10-4 and thecorresponding implicit algorithm
with 20 steps18. The results from both procedures are in verygood
agreement and for most practical purposes identical. As can be
seen, the cost for the twotechniques is quite similar with 432
seconds CPU time for the 5 steps iterative solution and 375seconds
for the incremental solution with DTOL equal to 10-3.
EXPLICIT FINITE DIFFERENCE IMPLEMENTATION
The finite difference method is perhaps the oldest numerical
technique used for the solution of setsof differential equations,
given initial values and/or boundary values. In the finite
differencemethod, every derivative in the set of governing
equations is replaced directly by an algebraicexpression written in
terms of the field variables (e.g., stress or displacement) at
discrete points inspace. In contrast to the finite element method,
these variables are only defined at the nodes. Amethod for deriving
difference equations for element of any shape is available in the
literature42 andis outside the scope of this report. The finite
difference code used in this analysis is FLAC (FastLagrangian
Analysis of Continua), an explicit finite difference code developed
by Itasca ConsultingGroup, Inc13 and extensively used in
geotechnical engineering practice.
Explicit Time Marching Algorithm
FLAC uses an explicit time marching algorithm to solve the
system of algebraic equations. Thegeneral calculation sequence
embodied in FLAC is illustrated in Figure 10. This procedure
firstinvokes the equations of motion to derive new velocities and
displacements from stresses andforces. Then, strain rates are
derived from velocities, and new stresses from strain rates. It
takesone time step for every cycle around the loop. The important
thing to realize is that each box inFigure 10 updates all of its
grid variables from known values that remain fixed while the
control isin the box. For example, the lower box takes the set of
velocities already calculated and, for each
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grid point compute new stresses. The velocities are assumed
frozen for the operation of the boxmeaning that new calculated
stresses do not affect the velocities. This may seem
unreasonable,because if a stress changes somewhere, it should
influence its neighbors and change their velocities.However, the
chosen timestep is so small that information cannot physically pass
from one gridpoint to another in that interval, as all materials
have some maximum speed at which informationcan propagate. Of
course, after several cycles of loop, disturbances can propagate
across many gridpoints, just as it would propagate in the physical
world. Hence, the central concept of an explicitalgorithm is that
"calculation wave speed" always keeps ahead of the "physical wave
speed".
Lagrangian Analysis
In the Lagrangian formulation, the incremental displacements are
added to the coordinates so thatthe grid moves and deforms with the
material it represents. This is termed Lagrangianformulation, in
contrast to Eulerian formulation, in which the material moves and
deformsrelative to a fixed grid. Eulerian formulation is often used
in fluid mechanics where tracking agiven particle is impossible.
The computer code FLAC is based on a Lagrangian formulation.Note
that a small strain formulation is not equivalent to a large-strain
formulation over many steps.Indeed, the small strain formulation
disregards the objectivity principle essential for a
meaningfullarge strain formulation. Hence, it is not recommended to
use FLAC for a large strain problem.
Numerical Formulation Used In Flac
FLACs formulation is conceptually similar to that of dynamic
relaxation 25, with adaptations forarbitrary grid shapes42 and
different damping. Given that an explicit time marching algorithm
solvesthe discrete equations of motion, the BEAR-Clay constitutive
model is implemented using anexplicit technique. The algorithm
consists of a straightforward Cauchy-Euler integration (BDF1) ofthe
continuum elasto-plastic equations that incorporates an automatic
error control algorithm similarto the one used in the finite
element implementation described earlier. As a result, the
FISHsubprogram, referred to as BearFISH, was developed and
successfully optimized within FLAC.FISH is a programming language
embedded within FLAC that enables the user to define newvariables,
functions, and constitutive models. A FISH model is simply a FISH
function containingspecial statements and references to special
variables that correspond to local entities within a singlezone.
The FISH model is called by FLAC four times per zone for every
global solution step. Oncecompiled successfully, a new FISH model
behaves just like a built-in model as far as the user isconcerned.
However, optimized FISH models will typically run at somewhere
between one-halfand one-third the speed of a built-in model.
The gridThe solid body is divided by the user into a finite
difference mesh composed of quadrilateralelements. Internally, FLAC
subdivides each element into two overlaid sets of
constant-straintriangular elements. In this way, if one pair of
triangles becomes badly distorted (e.g., if the area ofone triangle
becomes much smaller than the area of its companion), then the
correspondingquadrilateral is not used; only nodal forces from the
other quadrilateral are used. In the case whereboth overlaid sets
of triangle are badly distorted, an error message is issued. The
use of triangularelement eliminates the problem of hourglass
deformations, which may occur with constant-strain
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finite difference quadrilaterals. In addition, a mixed
discretization procedure21 is used to overcomemesh-locking. The
isotropic stress and strain components are taken to be constant
over the wholequadrilateral element, while the deviatoric
components are treated separately for each
triangularsub-element.
Equations of motionIn FLAC, the full dynamic system of equations
is always solved, even to find the solution to astatic problem. One
reason for doing this is to ensure that the numerical scheme is
stable whenthe physical system being modeled is unstable, as with
nonlinear materials there is always thepossibility of physical
instability. In the real world, some of the strain energy in the
system isconverted into kinematics energy, which radiates away from
the source and dissipates. FLACmodels this process directly,
because inertial terms are included, kinetic energy is generated
anddissipated. In contrast, schemes that do not include inertial
terms must use some numericalprocedure to treat physical
instabilities (e.g. arc length methods). Even when the procedure
used toprevent instabilities is successful, the path taken may not
be a realistic one3.
Dynamic relaxation and dampingTo solve static problems, the
equation of motion must be damped to provide static or
quasi-staticsolutions. The damping used in standard dynamic
relaxation methods is velocity-proportional,meaning that the
magnitude of the damping is proportional to the velocity of the
nodes. Thisconcept is equivalent to a dashpot fixed to the ground
at each nodal point. The use of velocityproportional damping
involves two main difficulties: a) the damping introduces body
forces, whichare erroneous in flowing regions and may influence the
mode of failure, b) the optimumproportionality constant depends on
the eigenvalues, which are unknown unless a complete modalanalysis
is performed.
One way to overcome these difficulties consists on proposing
alternative forms of damping such ashysteretic damping. For example
in soils and rocks, natural damping is mainly hysteretic.However,
the numerical treatment of hysteretic damping involves at least two
difficulties: first, theprecise nature of the hysteresis curve is
often unknown for complex loading-unloading paths.Second,
ratcheting can occur, for which each cycle in the oscillation of a
body causes irreversiblestrain to be accumulated. The use of this
type of damping has been avoided, since it increases pathdependence
and produces results that are more difficult to interpret.
Cundall8 describes an adaptive global damping, where viscous
damping forces are still used but theviscosity constant is
continuously adjusted in such a way that the power absorbed by
damping is aconstant proportion of the rate of change of kinetic
energy in the system. This form of dampingovercomes the
difficulties mentioned above since as a system approaches
equilibrium the rate ofchange of kinetic energy approaches zero and
consequently the damping forces tend to zero. InFLAC this latter
damping strategy is employed in which the damping force on a node
is such thatenergy is always dissipated.
Solution stability and mass scalingThe explicit-solution
algorithm is conditionally stable if the speed of the calculation
front must begreater that the maximum speed at which information
propagates. Hence, a timestep must bechosen that is smaller than
some critical timestep. The stability condition for an elastic
soliddiscretized into elements of size x is
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t < x/C* ; where C* ~ Cp (18a)
+
=
3/G4KCp (18b)
where C* is the maximum speed at which information can
propagate, and is typically taken as the p-wave speed, Cp , K, G
are the current elastic bulk and shear modulus, respectively, and
is thedensity of the material. When the static solution to a
boundary value problem is needed, a nodalmass or mass density ,,
may be regarded as a relaxation factor in the equation of motion.
Thus,they can be adjusted for optimum speed convergence. Note that
gravitational forces are not affectedby this scaling of the initial
mass.
Numerical Simulations with Finite Differences-Single Element
Tests
Herein, several numerical simulations, developed to validate the
implementation of the Bear-Clayconstitutive model into FLAC, are
presented. Stability and accuracy properties of the
stressintegration algorithm proposed to integrate the continuum
constitutive equations have beeninvestigated in previous sections.
The explicit time marching algorithm built into FLAC solves
theglobal equations of motion. This algorithm has been extensively
verified against closed-formsolutions, physical models, and
field-testing 7, 35. Therefore, the numerical examples are
singleelement tests that focus on validating the implementation
itself. The simulations consist ofundrained plane strain
compression and extension tests of K0 normally consolidated
specimens.These simulations are run with the same material
properties for Boston Blue Clay to allow for directcomparison.
Figure 11 depicts the effective stress path and shear stress strain
response for twovalues of the user defined local tolerance STOL.
Results obtained from FLAC are compared withthe one obtained with
the finite element program FEAP. The stress paths and shear
stress-straincurves obtained from FEAP and FLAC are in excellent
agreement for these single element tests.These results do not come
as a surprise because the same explicit stress integration
algorithm isimplemented in both codes, thus validating the explicit
algorithm implementation in FLAC.
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SUMMARY
Fully implicit stress integration algorithms are attractive from
the computational point of view butthey are typically very
cumbersome, and in many cases impractical, to implement for
complexnonlinear constitutive soil models. For these models, the
choice of an explicit automaticsubstepping with error control
algorithm represents a computationally efficient alternative.
Thisreport has presented detailed information of the application of
fully explicit local and globalalgorithms with automatic error
control for the integration of nonlinear elastoplastic
constitutivelaws into a nonlinear finite element code, FEAP, and an
explicit finite difference code, FLAC, withsimilar results.
The proposed explicit local stress integration algorithm falls
under the category of return mappingalgorithm and does not require
determination of the initial yield or any drift correction
techniques.The iterations for the explicit algorithm require, at
most, the solution of a single scalar non-linearequation compared
to the inversion of a 12x12 Jacobian as in the case of the implicit
algorithm.The proposed global explicit technique is found to be
computationally efficient and accurate usingthe continuum tangent
as long as the user tolerance DTOL is tight enough. The numerical
procedureis illustrated here by the implementing a recently
developed constitutive model for lightlyoverconsolidated clays
included anisotropic behavior with small strain nonlinearity in
shear into afinite element computer code. The numerical algorithm
proposed here can be directly extended toother models without
conceptual changes and is not dependent on the selected set of
constitutiveexpressions used to illustrate the procedure. Numerical
simulations of single element tests as wellas a boundary value
problem confirm the robustness, accuracy, and efficiency of the
proposedalgorithm at the local and global level. Preliminary
comparisons suggest that the proposed explicitand implicit
algorithms have similar computational performances in terms of CPU
time, in spite ofthe fact that the number of steps is 3-4 times
larger for the explicit technique.
ACKNOWLEDGEMENT
Support for this research was provided by the National Science
Foundation Grant No. CMS9612136 with the University of California,
Berkeley and by the National Science FoundationCAREER award to the
second author. This support is gratefully acknowledged. The authors
thankProfessor Taylor from the Civil Engineering Department for
invaluable help with theimplementation of the explicit algorithm in
the computer code FEAP.
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
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Prentice-
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15. Hashigushi, K. "Constitutive Equations of elastoplastic
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16. Lee, J.H. and Zhang, Y. On the Numerical Integration of a
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17. Luccioni, L.X. Numerical Development And Implementation Of A
Constitutive Model ForClays With Application To Deformations Around
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18. Luccioni, X.L., Pestana, J.M. and Rodriguez-Marek, A. An
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19. Luccioni, L., Pestana, J.M. and Koutsoftas, D. Modeling of
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20. Manzari, M.T. and Nour, M.A. "On Implicit Integration of
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21. Marti, J. and Cundall, P.A. Mixed Discretization Procedure
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
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29. Pestana, J.M. and Luccioni, L. "Description of Drained and
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model for Clays and Sands: I. ModelFormulation", Int. J. Num. Anal.
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32. Petzold, L.R. Recent Developments in the Numerical Solutions
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34. Randolph, M.F., and Wroth, C.P. Application of the Failure
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Rate Independent Elasto-Plasticity, Comput. Meths. Appl. Mech.
Engrg., 48, (1985), 79-116
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(1985) 221-245.
38. Simo, J.C. and Govindjee, S. Non-linear B-stability and
Symmetry Preserving Return MappingAlgorithms for Plasticity and
Viscoplasticity, Int. J. Numer. Meths. Engrg., 31, (1991)
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39. Simo, J.C. and Hughes T.J.R. Computational Inelasticity,
Interdisciplinary AppliedMathematics (IAM) Springer (1998)
40. Sloan, S.W. Substepping Schemes for the Numerical
Integration of Elastoplastic Stress-StrainRelations, Intl. Journal.
For Num. Meth. In Engr., 24, (1987), 893-911.
41. Taylor, R.T. "FEAP manual, Dep. Civil & Env. Engng.,
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42. Wilkins, M.L. Calculation of Elasto-Plastic Flow, Methods of
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Cambridge University Press,462pp (1990)
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Mapping Algorithm forPressure-Dependent Elastoplasticity Models,
Comput. Meths. Appl. Mech. Engrg., 121,(1995), 29-44.
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
APPENDIX A: SUMMARY OF BEAR-CLAY MODEL FORMULATIONThe following
sections summarize the equations defining the formulation of the
BEAR-CLAYmodel for lightly overconsolidated soils. Tensorial
quantities are represented by boldface italiccharacters.
Hypo-Elasticity FormulationThe tangent elastic stiffness tensor
is assumed isotropic and it is decomposed into the volumetricand
deviatoric components, K and 2G respectively, describing the bulk
and shear moduli:
( )( )
( )( )a
ss
a
r
a pGee
pppK
/1
121
23
1/1
max
01
+
+
+
+
=
(A.1)
)1()21(32
+
=
KG (A.2)
c
a
b
a pp
eG
pG
3.11
3.1max
= (A.3)
where Gb is a constant parameter describing the magnitude of the
small strain shear modulus, Gmax,c is the slope of the compression
curve of normally consolidated clays in a log(e)-log(p) space, r0is
the slope of the compression curve at OCR > 10, parameter s
controls the small strain non-linearity in shear, is a constant
elastic Poissons ratio chosen to match the measured unloadingfrom
normally consolidated states to moderately overconsolidated states
(OCR 3-4) and pa is theatmospheric pressure. The parameters and s
are memory parameters describing the amount ofunloading from the
reversal state:s = {(- rev): (- rev) } ; = min (prev/p, p/prev) (
only activated for drained testing)where (= s/p) is the current
shear stress ratio tensor, rev is the shear stress ratio
corresponding tothe last reversal point, prev is the mean effective
stress at last reversal, and ":" represents the doublecontraction
of tensor multiplication.
Plastic States- Yield SurfaceThe yield surface prescribing
"plastic state conditions" is described by a function :
0)/( - : 22 =+= mpc (A.4a)where bbb : )2(: )1(22 aac +=
(A.4b)
where controls the size of the yield surface, b is a second
order tensor describing the orientationof the yield surface in
effective stress space, parameter m controls the slenderness of the
yieldsurface, parameter a defines the shape of the yield surface
near the tip (i.e., p~) and controls theratio of horizontal to
vertical stress (i.e., K0) for one dimensional strain
consolidation. Theparameter c2 describes the aperture of the yield
surface at the origin (i.e., p ~ 0) and follows theMatsuoka-Nakai
generalization given by the maximum angle 'm:( ) 1m2m22a2a2a2
)sin3)(sin8(c ; 2/c3cc 3 +=+= J (A.5)where J3 is the third
invariant of the stress ratio tensor (i.e., J3 = det [])
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
Flow Rule and Large Strain Failure ConditionsFailure conditions
are represented by an isotropic function of the form proposed by
Matsuoka andNakai (1974):
0:kh 2f == ; )sin3/(sin8k ; )2/k3(kk cs2cs22a2a2a2 +=+= 3J
(A.6)The tensor P defines the direction of the plastic strain
increment with volumetric and deviatoriccomponents Pp and Ps,
respectively (i.e., P = Pp I + Ps , where I is the identity
tensor):
Pp = 0.5 (k2 :); ( ) sPs / 2
=
app (A.7)
Hardening LawsThe isotropic and kinematic hardening are
described by the following expressions:
( ) bb
dde
ed pc
:1
+=
(A.8)
( ) dd .bb = (A.9)where dp is the incremental change in total
volumetric strain and /b represents the couplingbetween the density
and kinematic hardening controlled by the Limiting Compression
BoundarySurface, LCBS (after Pestana and Luccioni25):
=
bb
bbbbbb.3cos.
:6
8
): (3
2
2 bJd
d(A.10a)
bb :333cos ;
)3cossin3(sin24 32 b
cs
cs Jwhered =
=
(A.10b)
where is a material parameter describing the shape of LCBS and
J3b is the third invariant of theanisotropy tensor (i.e., J3b = det
[b]).
Gradient of Yield Surface and Elastoplastic ModulusHerein, all
quantities are evaluated at k+1 unless specified otherwise. The
gradient of the yieldsurface is decomposed into the volumetric and
deviatoric components, respectively
( )
+
+=
=
m
b
m pJcpampp
13: 2: )2(1Q 322p b (A.11)
=
=
m
b
m pJcpap
1. )2( 21 32
bs
Qs (A.12)
[ ] TT JJ ==
..det 33
(A.13)
The change in the yield surface with respect to changes in the
memory parameters, and b, can bedetermined:
( )mpm
/./
2
= ; { }( )mpaa /. )2().1(2 / = bb (A.14)As a result the
elastoplastic modulus can be determined as:
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
pppb
bbb
b
+
=
+
= : 1:1
dd
H (A.15)
where p is the plastic strain tensor and d is the elastoplastic
parameter obtained from:
ss
s
PQ
Q
: 2PKQH
: 2de)(1e
KQ
pp
pp
G
dGd
s
++
+
+
+
=
(A.16)
Table A.1: Bear-Clay Input Material Parameters used in the
Analysis.
Test Type InputParameter
Physical Interpretation Boston BlueClay
1. Hydrostatic or 1-D c Compressibility of clays LCC
0.178compression test r0 Unloading behavior in 1D compression
0.035
2. 1-D compression withlateral stress measurement
Average Poissons ratio 0.26
m
Geometry of the yield surfaceGeometry of LC Boundary surface
1.01.25
cs Critical State friction angle 33.50 Rate of evolution of
anisotropy 10.0
3.Undrained triaxialcompression and extension
s Small strain non-linearity in shear 4.04. Elastic shear wave
velocity Gb Small strain shear stiffness 250
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
Table 1. Summary of the strain subincrement for OCR =1
simulation.
Average # of Strain Subincrements per StepSuccessful Failed
Tolerance STOL Compression Extension Compression Extension10-5
100 80 1 010-3 20 14 2 310-1 4 3 4 n/a
n/a: The algorithm did not converge.
Table 2. Rate of convergence of local Newton algorithm for OCR =
1 Simulation
Rate of ConvergenceTolerance STOL 10-5 10-1
Step 2 1.186 x10-36.283 x10-61.799 x10-10
0.31680.1331
5.978 x10-21.245 x10-33.452 x10-4
Step 100 9.459 x10-45.009 x10-61.462 x10-10
5.888 x10-28.560 x10-33.600 x10-47.687 x10-7
Table 3. J2 Flow Model Properties Used for Testing Global
Algorithm- Patch Test
Parameter ValueShear Modulus 100,000
Yield Stress 10Yield Increment 10
Exponent Isotropic 300Hardening Modulus 30
Table 4. Summary of Global Solution Algorithm Simulation - Patch
Test
Global Incremental AlgorithmConsistent Jacobian Continuum
Jacobian
DTOL RES # CoarseIncrement
CPU(sec)
Ux(cm)
Uy(cm)
# CoarseIncrement
CPU(sec)
Ux(cm)
Uy(cm)
10-5 6x10-7 1980 49.7 1.720 2.007 1990 50.2 1.720 2.00710-4
1x10-7 890 25.3 1.720 2.007 920 26.4 1.720 2.00710-3 1x10-7 310 9.8
1.720 2.007 330 11.3 1.720 2.00710-2 5x10-2 50 4.1 2.857 3.333 120
6.4 3.012 2.99810-1 10-1 10 1.9 1.980 2.310 35 5.1 3.450 3.237
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
Table 5. Comparison Between Iterative and Incremental Global
Solution Algorithm
Iterative Algorithm (RES< 10-8) Incremental Algorithm (DTOL=
10-3)Jacobian # of
IterationCPU(sec)
Ux(cm)
Uy(cm)
# CoarseIncrement
CPU(sec)
Ux(cm)
Uy(cm)
Consistent 8 0.87 1.720 2.007 310 10.03 1.720 2.007Continuum 13
2.13 1.720 2.007 330 11.12 1.720 2.007
Table 6: Iterations Summary for the Global Algorithms
ToleranceDTOL
# of CoarseSteps
CPU Time(sec.)
10-2 50 6510-3 345 37510-4 1210 780
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
.bLC Boundary
Surface(e=constant)
pe
Shea
r Stre
ss
Mean Effective Stress
Yield Surface
Directions of Anisotropy
Critical StateFailure Envelope
0.950.75
0.50.25
1 2
3
= p/
Maximum Aperture Cone
Plane
a) Anisotropic Plasticity and Generalization of Yield Surface
(after Luccioni and Pestana)
Voi
d R
atio
,e (l
og sc
ale)
Mean Effective Stress, p (log scale)
Hydrostatic LimitingCompression
Curve (H-LCC)
1
r
Hysteretic Volumetric Response
reload
unload
virgin load
e= const.
1
c
pe
0.0
0.2
0.4
0.6
0.8
1.0
Shea
r Mod
ulus
Deg
rada
tion,
Gse
c / G
max
Shear Strain (log scale)
Increasing Small Strain Nonlinearity
Strain
Stress
b) Hysteretic Elastic Model for Volumetric and Shear
Response
Figure 1: Summary of BEAR-CLAY Model Formulation.
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27
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
Figure 2: Plane Strain Finite Element Mesh
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Norm. Confining Stress, ('v+'
h) / 2 '
p
Nor
mal
ized
She
ar S
tress
, (
' v-
' h) /
2
' p
1.d-5
1.d-1
STOL
1.d-3 'cs
= 37.9
InitialState
K0nc
= 'h/'
h = 0.53
0 0.02 0.04 0.06 0.08 0.1
Axial Strain, |a|
Compression
Extension
Figure 3: Parametric Study on Local Algorithm for Normally
Consolidated Specimen (OCR=1)
Fv
Fh
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LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Norm. Confining Stress, ('v+'
h) / 2 '
p
Nor
mal
ized
She
ar S
tress
, (
' v-
' h) /
2
' p
1.d-5
1.d-1
STOL
1.d-3
'cs
= 37.9Compression
Extension
InitialState
Undrained Plane Strain Tests
0 0.02 0.04 0.06 0.08 0.1
Axial Strain, |a|
Figure 4: Parametric Study on Local Algorithm for
Overconsolidated Specimen (OCR=2)
-
29
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
Figure 5. One Dimensional Stress-strain Law for the J2 Flow
Model.
Figure 6. Finite Element Patch Test Mesh
Strain
Stress
YieldStress
0
Unloading
Reloading
5
-
30
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
1
10
100
ConsistentContinuum
Com
puta
tiona
l Eff
ort,
CPU
(s)
10
100
1000
104
10-5 0.0001 0.001 0.01 0.1
Consistent
Continuum
Tolerance, DTOL
Com
puta
tiona
l Effo
rt, #
coa
rse
iter.
Figure 7: Comparison of Performance of Global Incremental
Algorithm
-
31
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
a)Undeformed and Deformed Mesh Configuration
0 50 100 150 200 250
0
5
10
15
20
25
Stresses (kN/m3)
Dep
th (m
)
Pore Water Pressure, u
Vertical Effective Stress, '
v
Undrained Strength
Profile, su
Uniform, Normally Consolidated Soil Profile
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.600.00 0.50 1.00 1.50 2.00
Settlement (DTOL=1.d-2)Settlement (DTOL=1.d-3)Settlement
(DTOL=1.d-4)
Settl
emen
t at C
ente
rline
/ Foo
ting
Wid
th (%
)
Applied Pressure, p/pa
pa = atmospheric pressure
b) Soil Profile and Load Settlement Curves
Figure 8. Performance of Numerical Integration
Algorithm:Undrained Loading of a Flexible Footing.
Uniform Load
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32
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
0.00
0.20
0.40
0.60
0.80
1.00
1.200.00 0.50 1.00 1.50 2.00
Implicit Algorithm (20 steps)Explicit Algorithm (DTOL=1.d-4)
Settl
emen
t at c
ente
rline
/ Foo
ting
wid
th (%
)
Applied Uniform Load, p/pa
pa = atmospheric pressure
Figure 9: Comparison of Predicted Settlement Curve Using
Implicit andExplicit Stress Integration Algorithm
-
33
LUCCIONI & PESTANA (1999). EXPLICIT ALGORITHMS FOR THE
NUMERICAL IMPLEMENTATION OF A NONLINEAR ELASTO-PLASTIC MODEL
FORLIGHTLY OVERCONSOLIDATED CLAYS. UCB/GT/99-22
Equation of Motion
Constitutive Equations
NewVelocities andDisplacements
NewStresses/Force
Figure 10: Explicit Calculation Cycle (after Itasca, 1995)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8('
v+'
h) / 2 '
p
(' v-
' h) /
2
' p 'cs= 37.9
Finite Element
1.d-3
STOL 1.d-5
Bear
FISH
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 0.02 0.04 0.06 0.08 0.1
Axial Strain, |a|
Ko=0.53
Figure 11: Validation of the BearFISH Sub-program
TABLE OF CONTENTSLIST OF TABLESLIST OF
FIGURESABSTRACTINTRODUCTIONFINITE ELEMENT IMPLEMENTATIONLocal
Stress Integration AlgorithmGlobal Solution AlgorithmNUMERICAL
SIMULATIONSSingle Element TestsPatch TestsBoundary Value
Problem
EXPLICIT FINITE DIFFERENCE IMPLEMENTATIONExplicit Time Marching
AlgorithmLagrangian AnalysisNumerical Formulation Used In FlacThe
gridEquations of motionDynamic relaxation and dampingSolution
stability and mass scaling
Numerical Simulations with Finite Differences-Single Element
Tests
SUMMARYACKNOWLEDGEMENTREFERENCESAPPENDIX A: SUMMARY OF BEAR-CLAY
MODEL FORMULATIONHypo-Elasticity FormulationPlastic States- Yield
SurfaceFlow Rule and Large Strain Failure ConditionsHardening
LawsGradient of Yield Surface and Elastoplastic ModulusTolerance
STOL
Uy
Figure 1: Summary of BEAR-CLAY Model Formulation.Figure 2: Plane
Strain Finite Element MeshFigure 3: Parametric Study on Local
Algorithm for Normally Consolidated Specimen (OCR=1)Figure 4:
Parametric Study on Local Algorithm for Overconsolidated Specimen
(OCR=2)Figure 5. One Dimensional Stress-strain Law for the J2 Flow
Model.Figure 6. Finite Element Patch Test MeshFigure 7: Comparison
of Performa