Finite element implementation of non-linear ...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2001; 50:1191–1212

Finite element implementation of non-linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control‡ Laurent X. Luccioni, Juan M. Pestana∗ ; † and Robert L. Taylor Department of Civil and Environmental Engineering; University of California; Berkeley; CA 94720; U.S.A.

SUMMARY Implicit stress integration algorithms have been demonstrated to provide a robust formulation for nite element analyses in computational mechanics, but are dicult and impractical to apply to increasingly complex nonlinear constitutive laws. This paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non-linear constitutive laws into a non-linear nite element computer code. The local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface. The global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size. The proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non-linearity. A series of numerical simulations con rm the robustness, accuracy and eciency of the algorithms at the local and global level. Published in 2001 by John Wiley & Sons, Ltd. KEY WORDS:

elastoplastic relations; small strain non-linearity; anisotropy; nite element method; explicit integration algorithms

1. INTRODUCTION There has been a signi cant eort in computational geomechanics to describe the discrete constitutive equations using fully implicit local and global stress integration algorithms [1; 2]. The most popular is, perhaps, the fully implicit return mapping algorithm in which the return directions are computed by the closest point projection method [1; 3–5], and has the advantage of being amenable to consistent linearization [6]. Recent advances in classical computational plasticity have ∗ Correspondence

to: Juan M. Pestana. Department of Civil and Environmental Engineering; University of California at Berkeley; 440 Davis Hall 1710; Berkeley; CA 94720-1710; U.S.A. † E-mail: [email protected] ‡ This article is a U.S. Government work and is in the public domain in the U.S.A. Contract=grant sponsor: National Science Foundation; contract=grant number: CMS 9612136

Published in 2001 by John Wiley & Sons, Ltd.

Received 6 September 1999 Revised 19 April 2000

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established the superiority of fully implicit algorithms to solve boundary value problems using the nite element method for relatively simple plasticity models [7; 8]. However, as the constitutive laws become more complex, the attractiveness of the fully implicit algorithm decreases signi cantly. First, the formulation requires the second Frechet derivative of the yield function with respect to the state variable tensors. Second, the solution of the local set of non-linear equations for highly nonlinear yield functions is by no means trivial, as the local Newton scheme may not converge. It is important to note that fully implicit algorithms do not guarantee non-linear stability (or B-stability), even though linear stability is automatically achieved [8; 9]. Thus, as of today, none of the existing stress integration, either implicit or explicit, algorithms is able to guarantee stability and convergence for general incremental elastoplastic constitutive laws under general conditions. Finally, an explicit expression 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. [10] presented fully implicit local and global algorithms for the Bear– Clay model using a quasi-Newton technique with a numerical tangent computed every load step by nite dierence and optimized with iterative updating procedures since an explicit expression of the algorithmic consistent tangent could not be determined. For classical explicit integration schemes the discrete constitutive equations are much simpler to formulate, but their accuracy depends signi cantly on the selected step size. Sloan [11] rst proposed the application of an automatic stepping with error control numerical algorithm for the prediction of collapse load using a relatively simple constitutive model. This numerical technique, widely used in the eld of numerical analysis, is based on extrapolation procedures. The use of automatic substepping and load stepping with error control algorithm overcomes the main limitation of explicit techniques, since the system adapts as the estimated error changes. Constitutive modelling and numerical implementation applied to geomaterials is by no means trivial since it involves a subtle balance between the complexity associated with realistically describing soil response and the use of both robust and accurate numerical algorithms. The Cam– Clay family of models based on the critical state soil mechanics framework [12] is perhaps the most widely used plasticity 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 to capture some aspects of soil behaviour [13], (2) their ecient numerical implementation into non-linear nite element codes [2; 14] and (3) the availability of a large database for material ‘input’ parameters for dierent materials [15]. Nevertheless, these models do not incorporate key elements of material response, such as anisotropic stress–strain–strength behaviour or small strain non-linearity. In particular, soils exhibit a high degree on non-linearity in the ‘elastic’ (i.e. recoverable strains) regime which is better described by a perfectly hysteretic formulation. The main characteristic of this formulation is that the tangent stiness decreases monotonically with continued straining and is a function of a (typically dimensionless) stress (or strain) measure describing the distance of the current state of stress (or strain) to the last reversal state. The perfectly hysteretic model predicts fully recoverable strains in a closed unload–reload cycle, but dissipates energy according to the prescribed stiness reduction law (cf., Gsec =Gmax as a function of shear strain, Figure 1). Since this non-linearity is strongly strain level dependent, the proposed explicit integration algorithm must be able to vary the ‘step size’ in order to maintain accuracy while, at the same time, be computationally ecient. In contrast, general subincrementation procedures with no step control must use very small incremental strains in order to accurately capture large changes in material response (i.e. change in stiness) following stress reversal which becomes computationally inecient for many other practical conditions. The eect of small strain non-linearity Published in 2001 by John Wiley & Sons, Ltd.

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Figure 1. Summary of Bear–Clay model formulation: (a) anisotropic plasticity and generalization of yield surface (after Luccioni et al., 2000); (b) hysteretic elastic model for volumetric and shear response.

and anisotropy has been proven to be of signi cant importance in the prediction of deformations in soil–structure interaction problems [16; 17], such as excavations [18] and tunneling activities [19]. This problem is particularly relevant for dynamic problems for which soil non-linearity and the associated energy dissipation is the most important aspect of material response. Previous evaluations of explicit techniques have included relatively simple models such as linear elastic-perfectly plastic Mohr–Coulomb or Tresca-strain hardening models [11; 19]. The following sections present details of a fully explicit automatic substepping scheme with error control to Published in 2001 by John Wiley & Sons, Ltd.

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integrate a non-linear model with anisotropic plasticity, small strain ‘hysteretic’ non-linearity and dependence of the yield and plastic ow rules on the third invariant of the stress tensor following the Matsuoka–Nakai generalization (cf. Figure 1(a)). Hysteretic non-linearity is mathematically analogous to the bounding surface plasticity formulation where the tangent stiness degrades as a function of the proximity of the ‘current state’ to the bounding surface using appropriate mapping rules. For the particular case of monotonic loading, this formulation is also similar in concept to the stiness degradation resulting from a ‘damage-related’ process and it is therefore applicable to a more general class of models. The local stress integration algorithm falls under the category of return mapping algorithm with standard operator split procedure and does not require the determination of initial yield or drift correction techniques. The discrete ‘local’ equations are integrated using numerical techniques that preserve the incremental nature of the continuum formulation, while the global system of equations are solved using an explicit automatic load stepping with error control algorithm as proposed by Abbo and Sloan [20]. In contrast to previous derivations, 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 stiness for the perfectly hysteretic formulation. The proposed numerical procedure is illustrated here by implementing a recently developed constitutive model for lightly overconsolidated clays [21; 22] that has been used in the solution of boundary value problems such as the prediction of deformations around ground openings in soft clays [23; 24]. The model, referred to as Bear–Clay, is based on the theory of incrementally linearized plasticity which is extensively documented in the literature [7; 25; 26]. Model formulation includes three important components to describe the observed clay response: (a) an elastoplastic framework for normally consolidated clays with a single anisotropic yield function with dependence on the third invariant of the stress tensor to represent accurately the eect 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 ow and hardening rules to describe the evolution of anisotropic stress–strain properties. The elastic stiness tensor is assumed isotropic allowing the decomposition of the elastoplastic relations into volumetric and deviatoric components and they are brie y summarized in Appendix A. It should be emphasized that the numerical algorithm proposed here is independent on the particular set of constitutive expressions used to illustrate the procedure and can be directly extended to other material models without conceptual changes. The numerical simulations of single element tests as well as a boundary value problem con rm the robustness, accuracy, and eciency of the proposed algorithm at the local and global levels. Finally, the implicit technique proposed by Luccioni et al. [10] and the explicit technique presented here are compared in terms computational eort as re ected by CPU time.

2. NUMERICAL IMPLEMENTATION 2.1. Local stress integration algorithm The elastoplastic ow problem at the Gauss point level can be recast in term of a system of dierential=algebraic equations that have to be integrated numerically. From the point of view of integration techniques, this system of dierential=algebraic equations has the property of being ‘in nitely sti’ [27]. This type of sti system is usually best integrated using methods such as backward dierentiation formula (BDF), where convergence and accuracy can be formally established [27; 28]. Gear [27] shows results indicating that fully explicit algorithms are, in general, Published in 2001 by John Wiley & Sons, Ltd.

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unstable and do not converge, suggesting that these algorithms should not be used to integrate this type of system. However, in the context of numerical implementation into a computer code one has to think in terms of nite precision. Indeed, we do not require to guarantee yield conditions exactly (i.e. ’ = 0) but only in an approximated way, |’| 6 TOL where TOL may be the machine precision or an even less restrictive constraint. This concept may be viewed as a penalty regularization of the yield condition used for viscoplastic materials [29; 30]. Following Sloan [11], we de ne two pseudo-time quantities,
T and T , to be used in the substepping scheme. When entering for the rst 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 of the paper, strains are assumed subincremental strains, unless otherwise indicated and tensorial quantities are represented by boldface roman characters. The return mapping algorithm is used to integrate the continuum rate equations using the rst-order accurate forward Euler scheme. The trial state, represented by the superscript tr, is characterized as follows: 1 tr pk+1

= pk + Kk (
p )k+1

(2a)

1 tr sk+1

tr = sk+1 + 2Gk (
Us )k+1

(2b)

where p; s are the mean eective stress and the deviatoric components of the stress tensor, A;
p ;
Us are the volumetric and deviatoric components of the strain tensor,
U, and the left superscript ‘1’ denotes the rst-order integration method. The model introduces expressions to model the small strain non-linearity through the tangent elastic stiness parameters Kk and 2Gk describing the bulk and shear moduli, respectively, and summarized in Appendix A (cf. Equations (A1) – (A3)). The loading=unloading conditions are determined by the sign of the yield function tr ; qk ), where q is a vector containing the plastic (i.e. memory) at the trial (predictor) state: ’(1 Ak+1 variables. This procedure has been demonstrated to be algorithmically consistent with the forward Euler integration scheme [31] while Papadopoulos and Taylor [32] discusses its limitations for implicit stress integration schemes. For ‘loading’ states, the (plastic) corrector step is invoked: 1

pk+1 = pk + Kk [(
p )k+1 −
(Pp )k ] = 1pktr − Kk
(Pp )k

(3a)

1

sk+1 = sk + 2Gk [(
s )k+1 −
(Ps )k ] = 1 sktr − 2Gk
(Ps )k

(3b)

qk+1 = qk +
q(
)

(3c)

’k+1 = ’k+1 (1pk+1 ; 1 sk+1 ; k+1 ; 1 bk+1 ) = 0

(3d)

1

where Pp and Ps are the volumetric and deviatoric components of the ow rule describing the direction of the plastic strains (cf. Appendix A),
q represents the change in the plastic variables with continued plastic deformation (i.e. hardening) and
 is the incremental elastoplastic parameter, also referred to as the consistency parameter, obtained from the solution of this system of equations. For the particular model used to illustrate the numerical procedure, the plastic variables are given by  and b representing the size of the yield surface and the traceless tensor describing the orientation of the yield surface in a generalized stress space, respectively. It must be noted Published in 2001 by John Wiley & Sons, Ltd.

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that the numerical algorithm proposed here is not dependent on the particular set of constitutive laws used to illustrate the procedure and therefore it can be extended to other material models without conceptual changes. In contrast to previous models elastoplastic models for clays, the Bear–Clay model treats the isotropic hardening variable, , as a dependent variable. The coupling between isotropic and kinematic hardening existing in the continuum equations can be written as  = (b; p ) where  is updated from converged values of anisotropy, b, as follows: 

(
p )k+1 k+1 = k exp c 1



1 + ek+1 ek+1



 exp

@ :
b @b



bk+1 = bk +
b(
)

(4a) (4b)

where e is the void ratio of the soil (= volume of voids=volume of solids), c is the slope of the hydrostatic 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 from plastic loading and ‘:’ represents the double contraction of tensor multiplication. The term @=@b describes the coupling resulting from the existence of the limiting compression boundary surface, LCBS, as described by Pestana and Luccioni [22] or the use of a spacing function [33; 34]. The existence of the LCBS satis es the robustness of the drained response (i.e. consolidation behaviour) under constant shear stress ratio, W( = s=p), conditions for models incorporating both kinematic and density hardening as suggested by Pestana [33]. The system of 12 non-linear implicit equations (cf. Equations (3a)–(3d)) has only one common unknown,
. Then, it follows that only one single scalar equation needs to be solved instead of the inversion of a 12 × 12 Jacobian as in the case of fully implicit algorithm [10]. The consistency equation, ’k+1 = 0, which insures that the converged state of stress is on the yield surface, can be regarded as a function of
 only: ’k+1 = ’k+1 (A(
); q(
)) = ’k+1 (
)

(5)

The consistency equation is then solved using the Newton’s method
m+1 =
m −

’k+1 (
m ) (@’k+1 \@(
))|m

(6a)

where @’k+1 @Ak+1 @’k+1 @qk+1 @’k+1 = + : @(
) @Ak+1 @(
) @qk+1 @(
)

(6b)

For Bear–Clay, the change in the yield function ’ resulting from the change in the elastoplastic parameter can be derived making use of the chain rule as follows:

where

@’k+1 @pk+1 @’k+1 @sk+1 @’k+1 @bk+1 @’k+1 = + + : : @(
) @pk+1 @(
) @sk+1 @(
) @bk+1 @(
)

(7a)

@’k+1 @’k+1 @k+1 @’k+1 = + @bk+1 @bk+1 k+1 @k+1 @bk+1

(7b)

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A summary of the constitutive equations for the Bear–Clay model is presented in Appendix A. In order to solve Equation (6) with Newton’s method, a starting value for
 is required. Although it is not uncommon to use
 = 0 as an initial guess=value, it is best to start with a value of
 as close as possible to the converged one in order to recover quickly the asymptotic quadratic rate of convergence of Newton’s method. The ‘continuum’ expression of
 is used here, since for small steps, this value is very close to the
 obtained once Newton’s method has converged. Note that in the context of explicit integration techniques, subincrement steps used to reach a converged state are small, which is the basis for the development and validity of the in nitesimal linearized theory of plasticity. The initial value for the elastoplastic parameter,
(0) is then given by
(0) =

(K(@’=@p) + (@’=@e)(1 + e)) (
p )k+1 + 2G(@’=@s) : (
Us )k+1 −(@’=@q)(@q=@) + K(@’=@p)Pp + 2G(@’=@s) : Ps

(8)

where all the quantities, except the strain increment, are evaluated at kth step. The error control scheme used in the proposed formulation is based on local extrapolation [11; 27], and requires that the constitutive laws be reintegrated with a second-order method. The modi ed Euler (i.e. predictor–corrector) method was chosen for this purpose. The modi ed Euler method uses the converged state achieved from the forward Euler method as base values (known quantities). The discrete equations can be written in the following form: Predictor step: 2 tr pk+1 2 tr sk+1

= pk + 1 Kk+1 (
p )k+1 1

= sk + 2 Gk+1 (
Us )k+1

(9a) (9b)

where the stiness coecients, K and G have been evaluated at the converged state from the forward Euler procedure. Corrector step: 


tr pk+1 = pk + 1 Kk+1
p k+1 −
 1 Pp k+1 = 2pk+1 − 1 Kk+1
 1 Pp k+1 


2 tr sk+1 = sk + 21 Gk+1
Us k+1 −
 1 Ps k+1 = 2 sk+1 − 21 Gk+1
 1 Ps k+1
2 qk+1 = qk + 1
q
; 1pk+1 ; 1 sk+1; 1 b
’k+1 = ’k+1 2pk+1 ; 2 sk+1 ; k+1 ; 2 bk+1 = 0

2

(10a) (10b) (10c) (10d)

where superscript 1 indicates converged state from the forward Euler method, superscript 2 indicates converged state from the modi ed Euler method, 1
b is the kinematic hardening evaluated with the converged state from the forward Euler procedure, and the isotropic parameter, , is evaluated as follows:      (
p )k+1 1 + ek+1 @ 1 2 :
b k+1 = k exp (11) exp c ek+1 @b The system of equations derived from the corrector step is solved by using the exact same technique as the one proposed above for the rst-order integration method. Subtracting Equation (10) from Published in 2001 by John Wiley & Sons, Ltd.

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Equation (9), a second-order accurate estimate of the local truncation error for the stress tensor, A, is obtained:
(12) Ek+1 = 2 Ak+1 −1Ak+1 =2 The relative error for a substep is written as [27; 11] Rk+1 = kEk+1 k=kAk+1 k

(13)

where Ak+1 = (1 Ak+1 + 2 Ak+1 )=2, which is second-order accurate. During the integration process, each substep size is continually updated to insure that Rk+1 6STOL. The value of STOL is problem and constitutive model dependent. On one hand, if the value set for STOL is too large, the solution obtained may be completely inaccurate and therefore not satisfactory. On the other hand, if STOL is chosen too small, the problem may not converge. Sloan [11] reported values of STOL between 10−1 and 10−5 and the same range of values are investigated here for the Bear–Clay model. A parametric study on the satisfactory values of STOL is reported through the numerical simulations presented in the last section. In the case where Rk+1 6STOL, the subincrement is declared successful and all variables are updated as follows:
Ak+1 + 2 Ak+1 =2
qk+1 = 1 qk+1 + 2 qk+1 =2

Ak+1 =

1

(14a) (14b) (14c)

T = T +
Tk

The local extrapolation is used to compute the next subincrement size. Hence,
Tk+1 = Â
Tk , where  is given by  = [STOL=Rk+1 ]1=2

(15)

Heuristic bounds on  must be introduced to prevent the extrapolation to be carried too far, resulting on unstable results. For the Bear–Clay model, it was found that the range 0:26Â62 gives good results which is in agreement with values reported by Sloan. In the case where Rk+1 ¿STOL, the subincrement has failed and the size of the subincrement has to be reduced and the algorithm is restarted from the last converged value with a smaller subincrement size. An extrapolation is used to compute the next subincrement size, following the same procedure as shown previously. In order to reduce the number of unsuccessful substep and simultaneously keep
Tk ¿0:01, the magnitude of k
n+1 k has to be controlled which is achieved by the global solution algorithm as described in the following paragraphs. 2.2. Global solution algorithm The numerical techniques used for solving the global discrete equations can be broadly classi ed as either incremental or iterative procedures. In the iterative procedure, the discretization leads to a system of non-linear equations to be solved by methods such as Newton–Raphson, or quasiNewton. These type of methods have the advantage of satisfying equilibrium equations at the end of each converged time step, and if the consistent tangent is used an asymptotic rate of convergence is recovered. In addition, the stability theorem can be proven for certain cases [35]. On the other hand, when the material behaviour is strongly non-linear the iterations may not converge as all of Published in 2001 by John Wiley & Sons, Ltd.

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these methods have a nite radius of convergence. Moreover, if the algorithmic consistent tangent is not used, the inaccuracy of the substitute tangent may result in intermediate strain increments that may be too large, causing the local stress algorithm to either use too many subincrements or diverge altogether. Incremental procedures, on the other hand, treat the governing equations as a system of ordinary dierential equations (ODE). Thus, the solution consists of a series of piecewise linear steps that attempts to approximate the load–deformation behaviour of the system. This class of methods guaranties that small load increment will be used which is an advantage when explicit stress integration techniques are used at the Gaussian (i.e. ‘local’) level. Nevertheless, this class of methods tends to ‘drift’ from equilibrium as the solution proceeds, leading to very doubtful interpretations. Abbo and Sloan [20] presented an incremental algorithm based on an automatic load stepping scheme with error control. This algorithm tries to minimize the ‘drift’ from equilibrium by calculating the residual forces at the end of each load increment and adding these to the applied forces for the next increment. The authors choose to implement a slightly modi ed version of the Abbo and Sloan [20] algorithm into a non-linear nite element code, referred to as FEAP. This nite element code was developed at the University of California, Berkeley for teaching and research, and complete documentation is available [36]. The elastoplastic continuum tangent is assembled, and used by the explicit algorithm to solve for the displacement. In contrast to the implicit algorithm, the elastoplastic continuum tangent is not signi cantly dierent from the algorithmic consistent tangent as step increments are much smaller. The elastoplastic continuum tangent, C ep , is given by Cep = Ce −

(Ce : Q + @’=@p ) ⊗ Ce : P H + Q : Ce : P

(16)

where Q is the gradient to yield surface, P describes the ow rule indicating the direction of plastic strain increments, C e is the continuum elastic stiness 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 respect to the total volumetric strain (i.e. void ratio) as described by Pestana [33]. The proposed modi cation to the algorithm consists of using a convergence criterion to compute automatically the coarse load increment size, making the global algorithm entirely automatic from the Gauss point level to the coarse load level. If more than two consecutive unsuccessful substeps are detected at any level, the current coarse load level is divided by two, whereas if more than two consecutive 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 the Bear–Clay model, hence caution should be used in generalizing these numbers to others model or situations. The previous remark showcases some drawbacks in using incremental techniques, but it also constitutes their strengths as these explicit techniques are exible enough to accommodate very strong non-linear behaviour.

3. NUMERICAL SIMULATIONS This section describes numerical simulations used to validate the proposed approach. Both local and global algorithms are coded into a series of Fortran90 subroutines and they are linked to the Published in 2001 by John Wiley & Sons, Ltd.

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Figure 2. Plane strain nite element mesh.

main non-linear nite element code, FEAP [36]. All computations were performed with doubleprecision arithmetic on a 32-bit architecture DEC 3000 workstation at the University of California at Berkeley. The convergence criterion used in the acceptance or rejection strategy of incremental displacements, u, at the global level is based on the local extrapolation procedure [20], where En+1 =

2



un+1 −1
un+1 =2

Rk+1 = kEk+1 k=kuk+1 k6DTOL

(17a) (17b)

Similarly to STOL discussed previously, DTOL is a user-de ned tolerance that is model and problem dependent. For the Bear–Clay model and within the context of the numerical simulations presented below, values of DTOL ranging between 10−1 and 10−5 have been investigated. 3.1. Single-element tests The rst numerical example investigates the local stress integration algorithm and its implementation. These simulations consist of single element (cf. Figure 2) undrained plane strain compression and extension tests, where the sample is initially anisotropically (i.e. K0 = h0 0 =v0 0) normally consolidated or 1-D unloaded to an overconsolidation ratio, OCR, of 2. The ‘simulated soil’ corresponds to Boston Blue Clay a low-plasticity clay widely documented in the geotechnical literature. Its model speci c material parameters are described in detail by Luccioni [24] and Pestana and Luccioni [22]. The axial strain for the undrained test is applied in 100 nite steps of size
A up to a total axial strain |A | = 10 per cent (i.e.
A = 0:1 per cent). Measure of accuracy is based on the algorithmic consistency property, which implies that the solution accuracy increases as the number of steps increased. Figures 3 and 4 show the eective stress path and shear strain response for normally K0 consolidated and overconsolidated (OCR = 2) samples, respectively. For each past consolidation history, a parametric study of the user-de ned local tolerance STOL is presented. Figure 3 demonstrates that the integration scheme is reasonably accurate for STOL values ranging from 10−5 to 10−3 , whereas for STOL equal to 10−1 , the algorithm looses accuracy Published in 2001 by John Wiley & Sons, Ltd.

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Figure 3. Parametric study on local algorithm for normally consolidated specimen. (OCR = 1).

Figure 4. Parametric study on local algorithm for overconsolidated specimen (OCR = 2).

in the compression mode. The same conclusions are reached from Figure 4 where not only a loss of accuracy in the compression 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 STOL is xed, the algorithm computes automatically the number of substeps needed to achieve the speci ed tolerance. Table I reports the number of steps, N , associated with Published in 2001 by John Wiley & Sons, Ltd.

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Table I. Summary of the strain subincrement for OCR = 1 simulation. Average No. of strain subincrements per step Tolerance STOL 10−5 10−3 10−1

Successful Compression Extension 100 20 4

80 14 3

Failed Compression Extension 1 2 4

0 3 n=a

n=a: The algorithm did not converge.

Table II. Rate of convergence of local Newton algorithm for OCR = 1 simulation. Tolerance STOL

Rate of convergence 10−5 10−1

Step 2

1:186 × 10−3 6:283 × 10−6 1:799 × 10−10

Step 100

9:459 × 10−4 5:009 × 10−6 1:462 × 10−10

0.3168 0.1331 5:978 × 10−2 1:245 × 10−3 3:452 × 10−4 5:888 × 10−2 8:560 × 10−3 3:600 × 10−4 7:687 × 10−7

a given value of STOL. Intuitively, the normally consolidated plane strain compression test is the most critical because the response is entirely elasto-plastic and exhibits strain softening behaviour. This intuitive argument is validated by numerical simulations, as for a given STOL, this mode of shearing uses the largest number of steps. Table II shows the rate of convergence of the local Newton–Raphson algorithm for two typical iterations. The results indicate that for a strict tolerance (STOL = 10−5 ) the asymptotic rate of convergence is quickly achieved, attesting of the quality of the initial guess for the consistency parameter,
(0) . Loosening the tolerance (STOL = 10−1 ) results in larger strain subincrement, which leads to a progressive breakdown of the explicit integration technique. Comparisons of the stress–paths and stress–strain curves between the explicit and implicit integration techniques, shows excellent agreement in terms of accuracy. The explicit technique is more costly than the implicit one in term of number of iterations (10 000 vs 1500), but each explicit iteration requires at the most the solution of a single scalar non-linear equation compared to the inversion of a 12 × 12 Jacobian in the case of the implicit algorithm [10]. As a result, the two techniques are in the same order of magnitude in terms of the computational eort (i.e. CPU time). 3.2. Patch tests The second series of numerical simulations focus on the performance of the global solution algorithm. Abbo and Sloan [20] applied this algorithm to predict collapse load for various boundary value problems with a cohesive-frictional constitutive law based on a rounded Mohr–Coulomb Published in 2001 by John Wiley & Sons, Ltd.

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Figure 5. One-dimensional stress–strain law for the J2 ow model.

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Figure 6. Finite element patch test mesh.

Table III. J2 ow model properties used for testing global algorithm—patch test. Parameter Shear modulus Yield stress Yield increment Exponent isotropic Hardening modulus

Value 100 000 10 10 300 30

yield surface. They integrated the constitutive law using an explicit technique developed by Sloan [11]. Herein, we evaluated the global explicit algorithm performance separately, for a dierent ‘quality’ of the Jacobian (i.e. continuum vs consistent). For this purpose, the J2 ow model with isotropic and kinematic hardening laws was selected as the constitutive law (cf. Figure 5). The J2 ow model is a popular associative model in the structure mechanics eld, which is simple enough to be consistently linearized under the return map algorithm such that both continuum and consistent tangent are available [8]. A four-element patch test, shown in Figure 6, is used to compare the performance of the dierent algorithms. A summary of the material properties for the J2 ow model is presented in Table III. The surface load applied on the external boundary has a magnitude of 15 kN, which is higher that the value of the yield stress, Y = 10 kN. The local stress integration for the J2 ow model consists of the ecient implicit return mapping algorithm. In the following numerical simulations the global solution algorithm is varied between the proposed incremental algorithm and the iterative algorithm based on Newton–Raphson technique, for the continuum and consistent Jacobian. A summary of the rst simulation program is presented in Table IV. The parameter RES represents the residual force vector whereas Ux and Uy are the displacements of node 5 (cf. Figure 6) in the horizontal and vertical direction, respectively. From Table IV, it is clear that the global incremental algorithm is accurate for DTOL smaller than 10−3 . One important and interesting feature is that for values of DTOL between 10−3 and 10−5 , the Published in 2001 by John Wiley & Sons, Ltd.

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Table IV. Summary of global solution algorithm simulation—patch test. Global incremental algorithm Consistent Jacobian Continuum Jacobian DTOL 10−5 10−4 10−3 10−2 10−1

RES

No. of Coarse increment

CPU (s)

Ux (cm)

Uy (cm)

No. of Coarse increment

CPU (s)

Ux (cm)

Uy (cm)

6 × 10−7 1 × 10−7 1 × 10−7 5 × 10−2 10−1

1980 890 310 50 10

49.7 25.3 9.8 4.1 1.9

1.720 1.720 1.720 2.857 1.980

2.007 2.007 2.007 3.333 2.310

1990 920 330 120 35

50.2 26.4 11.3 6.4 5.1

1.720 1.720 1.720 3.012 3.450

2.007 2.007 2.007 2.998 3.237

Figure 7. Comparison of performance of global incremental algorithm.

algorithm performed equally well using the consistent or continuum tangent. This result is further illustrated through Figure 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 more accurate and ecient solution algorithm. These results demonstrate that Published in 2001 by John Wiley & Sons, Ltd.

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Table V. Comparison between iterative and incremental global solution algorithm. Iterative algorithm Jacobian Consistent Continuum

(RES¡10−8 )

Incremental algorithm

(DTOL = 10−3 )

No. of Iteration

CPU (s)

Ux (cm)

Uy (cm)

No. of coarse increment

CPU (s)

Ux (cm)

Uy (cm)

8 13

0.87 2.13

1.720 1.720

2.007 2.007

310 330

10.03 11.12

1.720 1.720

2.007 2.007

in the context of the incremental technique, continuum tangent may be used as long as the user de ned tolerance is kept strict enough (less than 10−3 in this case). Table V reports a comparison between the iterative and incremental global solution algorithm using both consistent and continuum tangent. The iterative algorithm is 10 and ve times faster than the incremental technique using the consistent and continuum tangent, respectively. These results point out the superior eciency of the iterative technique for this simple case, but also illustrates the sensitivity of the iterative technique to the ‘quality’ of the Jacobian used in the solution. Results also show that both algorithms are accurate regardless of the Jacobian used. 3.3. Boundary value problem Finally, the third numerical example focuses on the performance of the overall implementation technique into the non-linear nite element code, FEAP. To allow a direct comparison between explicit and implicit techniques, the application problem presented here is identical to that shown by Luccioni and Pestana [10] for the implicit technique. In this example, we consider the plane strain problem of a vertically loaded exible strip footing of width B on a uniform ‘ nite’ deposit of normally consolidated clay. The initial geostatic stresses were generated within FEAP using a body force command to simulate the gravity load at each Gauss point and a prescribed value of K0 of 0.53 describing the ratio of horizontal to vertical stress. The nite element mesh is shown in Figure 8, and it is composed of 200, four-noded quadrilateral ‘soil’ elements with a 2 × 2 Gaussian integration rule employed for each element. The soil pro le is underlying by a rigid and rough bedrock. The load is applied using dierent global tolerance DTOL, whereas the local user tolerance is xed to a value of 10−3 . As described previously, for each value of DTOL, the algorithm computes automatically the number of coarse load increments as well as the number of subincrements necessary to achieve the prescribed user tolerance. Figure 8 shows the loadsettlement curve at the centerline of the exible foundation for dierent values of DTOL. Again, the results indicate that the algorithm is accurate for values of DTOL less than 10−3 . The lack of accuracy observed when using DTOL equal to 10−2 leads to a 25 per cent overestimation of the settlement. Table VI summarizes the cost-related variables, showing that although the dierence in accuracy for DTOL values of 10−3 and 10−4 is less than 1 per cent, the CPU time is multiplied by a factor larger than two and the number of coarse steps quadruple. The load–settlement curve corresponding to DTOL equal to 10−3 compares very well with the one computed using the global implicit-iterative algorithm. A maximum error of less than 5 per cent is observed for a pressure of p=pa ∼ 2 (where pa is the atmospheric pressure). Figure 9 compares the load–settlement curve for the proposed explicit algorithm with DTOL = 10−4 and the corresponding implicit algorithm with 20 steps [10]. The results from both procedures are in very good agreement and for most practical purposes identical. As can be seen, the cost for the two techniques is quite similar with Published in 2001 by John Wiley & Sons, Ltd.

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Figure 8. Performance of numerical integration algorithm: undrained loading of a exible footing: (a) undeformed and deformed mesh con guration; (b) soil pro le and load settlement curves. Table VI. Iterations summary for the global algorithms. Tolerance DTOL 10−2 10−3 10−4

No. of coarse steps

CPU time (s)

50 345 1210

65 375 780

432 s CPU time for the 5 steps iterative solution and 375 s for the incremental solution with DTOL equal to 10−3 . 4. CONCLUSIONS Fully implicit stress integration algorithms are attractive from the computational point of view but they are typically very cumbersome, and in many cases impractical, to implement for complex Published in 2001 by John Wiley & Sons, Ltd.

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Figure 9. Comparison of predicted settlement curve using implicit and explicit stress integration algortihm.

non-linear constitutive soil models. For these models, the choice of an explicit automatic substepping with error control algorithm represents a computationally ecient alternative. The paper has presented detailed information of the application of fully explicit local and global algorithms with automatic error control for the integration of non-linear elastoplastic constitutive laws. The proposed explicit local stress integration algorithm falls under the category of return mapping algorithm 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-linear equation compared to the inversion of a 12 × 12 Jacobian as in the case of the implicit algorithm. The proposed global explicit technique is found to be computationally ecient and accurate using the continuum tangent as long as the user tolerance DTOL is tight enough. The numerical procedure is illustrated here by the implementing a recently developed constitutive model for lightly overconsolidated clays including anisotropic behaviour with small strain non-linearity in shear into a nite element computer code. The numerical algorithm proposed here can be directly extended to other models without conceptual changes and is not dependent on the selected set of constitutive expressions used to illustrate the procedure. Numerical simulations of single-element tests as well as a boundary value problem con rm the robustness, accuracy, and eciency of the proposed algorithm at the local and global level. Preliminary comparisons suggest that the proposed explicit and implicit algorithms have similar computational performances in terms of CPU time, in spite of the fact that the number of steps is 3–4 times larger for the explicit technique.

APPENDIX A: SUMMARY OF BEAR–CLAY MODEL FORMULATION The following sections summarize the equations de ning the formulation of the Bear–Clay model for lightly overconsolidated soils. Tensorial quantities are represented by boldface roman characters. Published in 2001 by John Wiley & Sons, Ltd.

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A.1. Hypo-elasticity formulation The tangent elastic stiness tensor is assumed isotropic and it is decomposed into the volumetric and deviatoric components, K and 2G, respectively, describing the bulk and shear moduli: 

K pa

−1

r0 (1 − ) = (p=pa )



e 1+e



3 + 2



1 − 2 1+



(1 + !s s ) (Gmax =pa )

2G 3(1 − 2) = K (1 + ) Gb Gmax = 1:3 pa e



p pa

(A1a)

(A1b)

1−1:3c

(A1c)

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, r0 is the slope of the compression curve at OCR¿10, parameter !s controls the small strain non-linearity in shear,  is a constant elastic Poisson’s ratio chosen to match the measured unloading from normally consolidated states to moderately overconsolidated states (OCR 3–4) and pa is the atmospheric pressure. The parameters  and s are memory parameters describing the amount of unloading from the reversal state: s ={(W − Wrev ): (W − Wrev )}1=2 ;

= min(prev =p; p=prev ) ( only activated for drained testing)

where W (= s=p) is the current shear stress ratio tensor, Wrev is the shear stress ratio corresponding to the last reversal point, prev is the mean eective stress at last reversal, and ‘:’ represents the double contraction of tensor multiplication.

A.2. Plastic states—yield surface The yield surface prescribing ‘plastic state conditions’ is described by a function ’: ’ = W : W − c2 + 2 (p=)m = 0

(A4a)

2 = c2 + (1 − a)b : b − (2 − a)W : b

(A4b)

where

where  controls the size of the yield surface, b is a second-order tensor describing the orientation of the yield surface in eective stress space, parameter m controls the slenderness of the yield Published in 2001 by John Wiley & Sons, Ltd.

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surface, parameter ‘a’ de nes the shape of the yield surface near the tip (i.e. p ∼ ) and controls the ratio of horizontal to vertical stress (i.e. K0 ) for one-dimensional strain consolidation. The parameter c2 describes the aperture of the yield surface at the origin (i.e. p ∼ 0) and follows the Matsuoka–Nakai generalization given by the maximum angle 0m :
c2 = ca2 + 3 − ca2 =2 J3Á ;

ca2 = 8 sin2 m




3 + sin2 m


−1

(A5)

where J3Á is the third invariant of the stress ratio tensor (i.e. J3Á = det[W]) A.3. Flow rule and large strain failure conditions Failure conditions are represented by an isotropic function of the form proposed by Matsuoka and Nakai (1974): hf = k 2 − W : W = 0;

k 2 = ka2 + (3 − ka2 =2)J3Á ;

ka2 = 8 sin2 cs =(3 + sin2 cs )

(A6)

The tensor P de nes the direction of the plastic strain increment with volumetric and deviatoric components Pp and Ps , respectively (i.e. P = Pp I + Ps , where I is the identity tensor): Pp = 0:5 (k2 − Á : Á);

Ps = 2(p=pa )

@’ @s

(A7)

A.4. Hardening laws The isotropic and kinematic hardening are described by the following expressions:

d =

@  (1 + e) dp − : db c e @b

(A8)

db = (W − b) d

(A9)

where dp is the incremental change in total volumetric strain and @=@b represents the coupling between the density and kinematic hardening controlled by the limiting compression boundary surface, LCBS (after Pestana and Luccioni [22]): s  " #  3d2 r 6 @J   @ 3b = : − cos 3Â:b − b  @b (ÿd2 − b : b)  8 b : b @b d2 =

24 (3= sin cs − cos 3Â)2

√ where cos 3Â = 3 6

J3b (b : b)1:5

(A10a)

(A10b)

where ÿ is a material parameter describing the shape of LCBS and J3b is the third invariant of the anisotropy tensor (i.e. J3b = det[b]). Published in 2001 by John Wiley & Sons, Ltd.

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Table A1. Bear–Clay input material parameters used in the analysis. Input parameter

Physical interpretation

c r0

Compressibility of Clays LCC Unloading behaviour in 1D compression

0.178 0.035

2. 1-D compression with lateral stress measurement



Average Poisson’s ratio

0.26

3. Undrained triaxial compression and extension

m ÿ cs !s

Geometry of the yield surface Geometry of LC boundary surface Critical state friction angle Rate of evolution of anisotropy Small strain non-linearity in shear

Gb

Small strain shear stiness

Test type 1. Hydrostatic or 1-D compression test

4. Elastic shear wave velocity

Boston blue Clay

1.0 1.25 33:50 10.0 4.0 250

A.5. Gradient of yield surface and elastoplastic modulus Herein, all quantities are evaluated at k + 1 unless speci ed otherwise. The gradient of the yield surface is decomposed into the volumetric and deviatoric components, respectively  p m   p m o @’ 1 n = (m2 + (2 − a)W : b) − 2W : W + 3cb2 J3Á 1 − @p p      p m   p m  @J3Á  @’ 1 2W − (2 − a) 1− :b − cb2 = Qs = @s p  @W 

Qp =

@J3Á = det[W]:W−T = J3Á :W−T @W

(A11) (A12) (A13)

The change in the yield surface with respect to changes in the memory parameters,  and be determined as m2 :(p=)m ; @’=@b = {2(1 − a):b − (2 − a)W}:(p=)m @’=@ = −  As a result the elastoplastic modulus can be determined as     @b @’ @b 1 @’ @ @’ 1 @’ @ : p + : + = − H =− p p d @ @ @b @ d @ @b @b @ where p is the plastic strain tensor and d is the elastoplastic parameter obtained from (KQp + @’=@e(1 + e)) dp + 2GQs : dUs d = H + KQp Pp + 2GQs : Ps

b, can (A14)

(A15)

(A16)

ACKNOWLEDGEMENTS

Support for this research was provided by the National Science Foundation Grant No. CMS 9612136 with the University of California, Berkeley and by the National Science Foundation CAREER award to the second Published in 2001 by John Wiley & Sons, Ltd.

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author. This support is gratefully acknowledged. The authors thank Miss Lynn Salvati for careful review of the nal manuscript. REFERENCES 1. Borja RI, Lee SR. CAM-CLAY plasticity, Part I: implicit integration of elasto-plastic constitutive relations. Computer Methods in Applied Mechanics and Engineering 1990; 78:49 –72. 2. Borja RI. CAM-CLAY plasticity, Part II: implicit integration of constitutive equations based on a nonlinear stress predictor. Computer Methods in Applied Mechanics and Engineering 1991; 88(2):225 –240. 3. Lee JH, Zhang Y. On the numerical integration of a class of pressure-dependent plasticity models with mixed hardening. International Journal for Numerical Methods in Engineering 1991; 32:419 – 438. 4. Zhang ZL. Explicit consistent tangent moduli with a return mapping algorithm for pressure-dependent elastoplasticity models. Computer Methods in Applied Mechanics and Engineering 1995; 121:29 – 44. 5. Manzari MT, Nour MA. On implicit integration of bounding surface plasticity models. Computers and Structures 1997; 63:385 – 395. 6. Simo JC, Taylor RL. Consistent tangent operator for rate independent elasto-plasticity. Computer Methods in Applied Mechanics and Engineering 1985; 48:79 –116. 7. Simo JC, Ortiz M. A uni ed approach to nite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Computer Methods in Applied Mechanics and Engineering 1985; 49:221– 245. 8. Simo JC, Hughes TJR. Computational Inelasticity, Interdisciplinary Applied Mathematics (IAM). Springer: Berlin, 1998. 9. Simo JC, Govindjee S. Non-linear B-stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity. International Journal for Numerical Methods in Engineering 1991; 31:151–176. 10. Luccioni XL, Pestana JM, Rodriguez-Marek A. An implicit integration algorithm for the nite element implementation of a nonlinear anisotropic material model including hysteretic nonlinearity. Computer Methods in Applied Mechanics and Engineering 2000, to appear. 11. Sloan SW. Substepping schemes for the numerical integration of elastoplastic stress–strain relations. International Journal for Numerical Methods in Engineering 1987; 24:893 – 911. 12. Wood DM. Soil Behavior and Critical State Soil Mechanics. Cambridge University Press: Cambridge, 1990; 462. 13. Randolph MF, Wroth CP. Application of the failure state in undrained simple shear to the shaft capacity of driven piles. GÃeotechnique, London 1981; 7(1):19 – 38. 14. Borja RI, Tamagnini C, Amorosi A. Coupling plasticity and energy-conservating elasticity models for clays. ASCE, Journal of Geotechnical Engineering 1997; 123(10):948 – 956. 15. Nakase A, Kamei T, Kusakabe O. Constitutive parameters estimated by plasticity index. ASCE, Journal of Geotechnical Engineering 1998; 114(7):844 – 858. 16. Hight DW, Higgins KG. An approach to the prediction of ground movements in engineering practice: background and application. In Pre-failure Deformation of Geomaterials, Shibuya S, Mitachi T, Miura S (eds). Balkema: Rotterdam, 1995; 909 – 945. 17. Jardine RJ, Potts DM, Fourie AB, Burland JB. Studies of the in uence of non-linear stress–strain characteristics in soil–structure interaction. GÃeotechnique 1986; 36(3):377 – 396. 18. Finno RJ, Harahap IS. Finite element analysis of HDR-4 excavation. ASCE, Journal of Geotechnical Engineering 1991; 117(10):1590 –1609. 19. Addenbrooke TI, Potts DM, Puzrin AM. The in uence of pre-failure soil stiness on the numerical analysis of tunnel construction. GÃeotechnique 1997; 47(3):693–712. 20. Abbo AJ, Sloan W. An automatic load stepping algorithm with error control. International Journal for Numerical Methods in Engineering 1996; 39:1737 –1759. 21. Pestana JM, Luccioni L. Description of drained and undrained behaviour of soft marine clay deposits. ASCE, 12th Engineering Mechanical Conference, San Diego, CA, 1998; 1013 –1016. 22. Pestana JM, Luccioni LX. A simpli ed constitutive model for lightly overconsolidated clays. Report UCB=GT 99-23, Department of Civil and Environmental Engineering, University of California, Berkeley, 1999. 23. Luccioni LX, Pestana JM, Koutsoftas D. Modeling of deformations around deep excavations in soft soils. In Proceedings of Geo-Engineering for Underground Facilities, Fernandez, Bauer (eds), Geotechnical Special Publication No. 90. ASCE: New York, 1999; 231– 242. 24. Luccioni LX. Numerical development and implementation of a constitutive model for clays with application to deformations around a deep excavation. Ph.D. Thesis, Department of Civil and Environmental Engineering, University of California, Berkeley, 1999. 25. Prevost JH. Plasticity theory for soil stress–strain behaviour. ASCE, Journal of Engineering Mechanics 1978; 104(EM5):1177 –1194. 26. Hashigushi K. Constitutive equations of elastoplastic materials with elastic-plastic transition. Journal of Applied Mechanics, ASME 1980; 47:266 –272. Published in 2001 by John Wiley & Sons, Ltd.

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