UCBGT99-22 Explicit_algorithm BEARCLAY

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


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


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.


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).


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


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


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)


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)


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

11


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.

12


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

13


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

14


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

15


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

16


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

17


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.

18


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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21


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 [])

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:

23


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

24


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

25


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

26


.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.

27


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

28


-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


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


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


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


b) Soil Profile and Load Settlement Curves

Figure 8. Performance of Numerical Integration Algorithm:Undrained Loading of a Flexible Footing.

Uniform Load

32


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


Figure 9: Comparison of Predicted Settlement Curve Using Implicit andExplicit Stress Integration Algorithm

33


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

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