the role of nonlinear hardening/softening in probabilistic elasto-plasticity

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2007; 31:953–975Published online 3 January 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.571

The role of nonlinear hardening/softeningin probabilistic elasto-plasticity

Kallol Sett, Boris Jeremic*,y and M. Levent Kavvas

Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, U.S.A.

SUMMARY

The elastic–plastic modelling and simulations have been studied extensively in the last century. However,one crucial area of material modelling has received very little attention. The uncertainties in materialproperties probably have the largest influence on many aspects of structural and solids behaviour. Despiteits importance, effects of uncertainties of material properties on overall response of structures and solidshave rarely been studied. Most of the small number of studies on effects of material variabilities have usedrepetitive deterministic models through Monte-Carlo-type simulations. While this approach might appearsound, it cannot be both computationally efficient and statistically accurate (have statistically appropriatenumber of data points).Recently, we have developed a methodology to solve the probabilistic elastic–plastic differential

equations. The methodology is based on Eulerian–Lagrangian form of the Fokker–Planck–Kolmogorovequation and provides for full description of the probability density function (PDF) of stress response for agiven strain.In this paper we describe our development in some details. In particular, we investigate the effects of

nonlinear hardening/softening on predicted PDF of stress. As it will be shown, the nonlinear hardening/softening will create a discrepancy between the most likely stress solution and the deterministic solution.This discrepancy, in fact, means that the deterministic solution is not the most likely outcome of thecorresponding probabilistic solution if material parameters are uncertain (and they always are, we just tendto simplify that fact and use, for example, mean values for deterministic simulations).A number of examples will be presented, illustrating methodology and main results, some of which are

quite surprising as mentioned above. Copyright # 2006 John Wiley & Sons, Ltd.

Received 3 March 2006; Revised 6 September 2006; Accepted 9 September 2006

KEY WORDS: elasto-plasticity; probability theory; Fokker–Planck–Kolmogorov equation

*Correspondence to: Boris Jeremic, Department of Civil and Environmental Engineering, University of California,One Shields Ave., Davis, CA 95616, U.S.A.yE-mail: [email protected]

Copyright # 2006 John Wiley & Sons, Ltd.

1. INTRODUCTION

Numerical simulations of mechanical behaviour of solids and structures made of geomaterials areinherently uncertain. The natural conditions with which geotechnical engineers deal with areunknown and must be inferred from limited and expensive observations (Beacher and Christian[1]). For example Figure 1 shows measured soil properties in Mexico City. Large variability inmaterial properties can be noted as a function of depth. It is important to note that this soilformation is considered very homogeneous (Baecher and Christian [2]). The usual procedure inthis case is to choose either constant properties or fit smooth (usually linear) curves to properties ofinterest (for example, friction angle and/or unconfined compressive strength and/or preconsolida-tion pressure (refer Figure 1)) and use the assumed (much simplified) properties in modelling andsimulations related to this soil profile. This approach will completely neglect the variations whichmight have a large effect on response of this soil profile. The usual remedy is to apply a large factorof safety, which, it is hoped, will cover up for neglected information. Use of large factors of safetyis becoming unacceptable as it leads to design solutions that are not only uneconomical but alsosometimes not even safe (Duncan [2]). More appropriate methods for analysing the random fieldsof material properties have been described, among other researchers, by Ghanem and Spanos [3],Matthies and Keese [4], Roberts and Spanos [5], Zhu et al. [6], and Soize [7].

In recent years, civil engineering practice, and particularly geotechnical engineering practicehas seen an increasing emphasis on quantification of uncertainties. Quantifications ormathematical descriptions of uncertainties are usually done within the framework of probabilitytheory, where random soil parameters are modelled as random variables (if they are specializedto a fixed location in the domain) or random fields (if they are specialized to the function oflocation in the domain). Nice descriptions of different types of soil parameter uncertainties andtheir quantification methods were presented by Phoon and Kulhawy [8, 9] and Fenton [10, 11].The uncertainty in G=Gmax curve was recently quantified by Stokoe et al. [12].

Eventhough there is a widely acknowledged necessity to model soil properties using theory ofprobability, there is very little work on propagation of uncertainties in soil properties throughelastic–plastic constitutive equation. Advanced elasto-plasticity-based constitutive models,when properly calibrated, although very accurate, are, in general, highly sensitive to fluctuationsin model parameters (cf. Borja [13]). In the field of geomechanics, this sensitivity aspect ofconstitutive model is of great importance as the uncertainties associated with soil propertiescould outweigh the advantages gained by advanced and sophisticated models.

First attempt to propagate randomness through the elastic–plastic constitutive equations,considering random Young’s modulus was published only recently by Anders and Hori [14].They based their work on perturbation expansion at the stochastic mean behaviour. Incomputing the mean behaviour they took advantage of bounding media analysis. However,because of the use of Taylor series expansion, this approach is limited to problems with smallcoefficients of variation. According to a recent report by Sudret and Der Kiureghian [15], thecoefficient of variation should not exceed 20% if this approach is to be used. This limitationrestricts the use of perturbation method to general geomechanics problems, where thecoefficients of variation of soil properties are rarely less than 20% (e.g. Phoon and Kulhawy [8]).Another disadvantage of the perturbation method is that it inherits the so-called ‘closureproblem’ (cf. Kavvas [16]), where information on higher-order moments is always neededto calculate lower-order moments. More recently, Fenton and Griffiths [17] used a Monte-Carlo-type method to propagate uncertainties through elastic–plastic c� f soil.

K. SETT, B. JEREMIC AND M. L. KAVVAS954

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2007; 31:953–975

DOI: 10.1002/nag

Figure 1. Measured values of mechanical properties of soil from Mexico city, typical soft spot(after Baecher and Christian [1].

NONLINEAR HARDENING/SOFTENING AND PROBABILISTIC ELASTO-PLASTICITY 955


DOI: 10.1002/nag

Conventional Monte-Carlo simulation, although very accurate, might be computa-tionally expensive for general elastic–plastic simulations. The method typically requires astatistically appropriate number of realizations per random variable (often a very largenumber) in order to satisfy statistical accuracy. This repetitive use of computationallyexpensive deterministic model renders the Monte-Carlo simulation impractical. Monte-Carlosimulations, however, find a great use in verification of analytical developments (cf. Oberkampfet al. [18]).

In order to overcome the drawbacks associated with Monte-Carlo technique and perturbationmethod in dealing with nonlinear stochastic differential equations (SDEs), Kavvas [16] deriveda generic Eulerian–Lagrangian form of Fokker–Planck–Kolmogorov equation (FPKE) forthe second-order exact probabilistic solution of any nonlinear ordinary differentialequations (ODE) with stochastic coefficient and stochastic forcing. The development wasfocusing on applications in water resources engineering, but it is general enough to apply toany nonlinear ODE with stochastic coefficient and stochastic forcing. Using the abovementioned Eulerian–Lagrangian form of FPKE, Jeremic et al. [19] and Sett et al. [20] recentlydeveloped formulation and solution for the general 1-D elastic–plastic constitutive rate equationwith random material properties and random strain rate.

The main advantage of FPKE-based approach is that a complete probabilistic description(probability density function (PDF)) of stress can be obtained exactly to second-order accuracy(to covariance of time), given random material property(ies) and/or random strain. In additionto that, even if the FPKE is not solvable in closed-form solution, the deterministic linearity ofthe FPKE in terms of the state variable, the probability density of stress, considerably simplifiesthe numerical solution process.

In this paper we present a general expression, in the form of a partial differential equation(PDE), for evolution of the probability density of stress with strain for any general 3-D point-location scale/local-average form of elastic–plastic constitutive law with material properties andstrain rate modelled as random variables/fields. The expression for the PDF of stress isdeveloped in a general way, making it usable in any dimension (1-, 2- or 3-D) and for any type ofincremental elastic–plastic material model. Application of the developed methodology isdemonstrated through 1-D point-location scale Cam-Clay shear constitutive law. In particular,the effects of nonlinear hardening and softening on the predicted PDFs of stresses areinvestigated.

2. GENERAL EXPRESSION FOR 3-D PROBABILISTIC CONSTITUTIVE RATEEQUATION IN LOCAL-AVERAGE FORM

The incremental form of spatial-average 3-D elastic–plastic constitutive rate equation can bewritten as

dsijðxt; tÞdt

¼ Depijklðsij ;D

elijkl ; f ;U; q*

; r*; xt; tÞ

deklðxt; tÞdt

ð1Þ

where Depijkl is the random, nonlinear elastic–plastic coefficient tensor which is a function of

random stress tensor ðsijÞ; random elastic moduli tensor ðDelijklÞ; random yield function ðf Þ;

random potential function (U), random internal variables ðq*Þ and random directions of

evolution of internal variables ðr*Þ: The random internal variables ðq

*Þ could be scalar



DOI: 10.1002/nag

(for perfectly plastic and isotropic hardening models), or second-order tensor (for translationaland rotational kinematic hardening models), or fourth-order tensor (for distortional hardeningmodels) or any combinations of the above. The same classification applies to the randomdirection of evolution of internal variables ðr

*Þ: By denoting all the random material parameters

by a parameter tensor as

Dijkl ¼ ½Delijkl ; f ;U; q*

; r*� ð2Þ

and introducing a random operator tensor, Zij ; one can write Equation (1) as

dsijðx; tÞdt

¼ Zijðsij ;Dijkl ; ekl ; x; tÞ ð3Þ

with an initial condition,

sijðx; 0Þ ¼ sij0 ð4Þ

In Equation (3), the stress tensor sij can be considered to represent a point in a 9-D stress(s)-space and hence Equation (3) determines the velocity for the point in that space. It ispossible to visualize this, by imagining an initial point in stress space, given by itsinitial condition sij0 ; with a trajectory starting out that describes the correspondingsolution of the nonlinear stochastic ODE system (Equation (3)). Let us now consider acloud of initial points, described by a density rðsij ; 0Þ in the s-space, and with movementsof these points dictated by Equation (3), the phase density r of sijðx; tÞ varies in timeaccording to a continuity equation which expresses the conservation of all these points inthe s-space.

Expressing this continuity equation in mathematical terms, one obtains Kubo’s stochasticLiouville equation (Kubo [21]):

@rðsijðx; tÞ; tÞ@t

¼ �@

@smnZmnðsmnðx; tÞ;DmnpqðxÞ; epqðx; tÞÞrðsijðx; tÞ; tÞ ð5Þ

with initial condition,

rðsij ; 0Þ ¼ dðsij � sij0 Þ ð6Þ

where dð�Þ is the Dirac delta function. Equation (6) is the probabilistic restatement in the s-phasespace of the original deterministic initial condition (Equation (4)). One can then applyVan Kampen’s Lemma (Van Kampen [22]) to obtain:

hrðsij ; tÞi ¼ Pðsij ; tÞ ð7Þ

where the symbol h�i denotes the expectation operation, and Pðsij ; tÞ denotes evolutionaryprobability density of the state variable tensor sij from the constitutive equations.

Therefore, in order to obtain the multivariate PDF, Pðsij ; tÞ; of the state variable tensor sij ; itis necessary to obtain the deterministic PDE of the s-space mean phase density hrðsij ; tÞi fromthe linear stochastic PDE system (Equations (5) and (6)). This necessitates the derivationof the ensemble average form of Equation (5) for hrðsij ; tÞi: The ensemble average form of



DOI: 10.1002/nag

Equation (5) was derived by Kavvas [16] as follows:

@hrðsijðxt; tÞ; tÞi@t

¼ �@

@smn

�hZmnðsmnðxt; tÞ;DmnrsðxtÞ; ersðxt; tÞÞi

�

�Z t

0

dtCov0

�Zmnðsmnðxt; tÞ;DmnrsðxtÞ; ersðxt; tÞÞ;

@Zabðsabðxt�t; t� tÞ;Dabcdðxt�tÞ; ecd ðxt�t; t� tÞÞ@sab

��hrðsijðxt; tÞ; tÞi

�

þ@

@smn

Z t

0

dtCov0½Zmnðsmnðxt; tÞ;DmnrsðxtÞ; ersðxt; tÞÞ;��

Zabðsabðxt�t; t� tÞ;Dabcd ðxt�tÞ; ecd ðxt�t; t� tÞÞ��@hrðsijðxt; tÞ; tÞi

@sab

�ð8Þ

to exact second order (to the order of the covariance time of Z). In Equation (8), Cov0½�� is thetime ordered covariance function defined by

Cov0½Zmnðx; t1Þ; Zabðx; t2Þ� ¼ hZmnðx; t1ÞZabðx; t2Þi � hZmnðx; t1Þi � hZabðx; t2Þi ð9Þ

By combining Equation (8) with Equation (7) and rearranging the terms yields the followingEulerian–Lagrangian form of FPKE (cf. Kavvas [16]):

@Pðsijðxt; tÞ; tÞ@t

¼@

@smn

�hZmnðsmnðxt; tÞ;DmnrsðxtÞ; ersðxt; tÞÞi

�

þZ t

0

dtCov0@Zmnðsmnðxt; tÞ;DmnrsðxtÞ; ersðxt; tÞÞ

@sab;

�

Zabðsabðxt�t; t� tÞ;Dabcd ðxt�tÞ; ecd ðxt�t; t� tÞ��

Pðsijðxt; tÞ; tÞ�

þ@2

@smn@sab

Z t

0

dtCov0 Zmnðsmnðxt; tÞ;DmnrsðxtÞ; ersðxt; tÞÞ;��

Zabðsabðxt�t; t� tÞ;Dabcd ðxt�tÞ; ecd ðxt�t; t� tÞÞ��

Pðsijðxt; tÞ; tÞ�

ð10Þ

to exact second order. The solution of this deterministic linear FPKE (Equation (10)) in terms ofPðsij ; tÞ under appropriate initial and boundary conditions will yield the PDF of the statevariable tensor sij of the original equation (Equation (1)). It is important to note that while theoriginal equation (Equation (1)) is nonlinear, the FPKE (Equation (10)) is linear in terms of itsunknown, the probability density Pðsij ; tÞ of the state variable tensor sij : This linearity, in turn,provides significant advantages in probabilistic solution of the constitutive rate equation.

One should also note that Equation (10) is a mixed Eulerian–Lagrangian equation, since,while the real space location xt at time t is known, the location xt�t is an unknown. If oneassumes, for example, small strain theory, one can relate the unknown location xt�t from theknown location xt by using the strain rate, ’e ð¼ de=dtÞ as

de ¼ ’et ¼xt � xt�t

xtð11Þ



DOI: 10.1002/nag

or, by rearranging

xt�t ¼ ð1� ’etÞxt ð12Þ

Another interesting aspect to note, possibly of theoretical interest, is the Ito SDE form of theconstitutive rate equation with random material properties and random strain rate (Equation(3)). Utilizing the equivalence between FPKE and Ito SDE (cf. Gardiner [23]), one can writeEquation (10) as

dsijðx; tÞ ¼ hZijðsijðxt; tÞ;DijklðxtÞ; eklðxt; tÞÞi þZ t

0

dtCov0@Zijðsijðxt; tÞ;DijklðxtÞ; eklðxt; tÞÞ

@sab;

��

Zabðsabðxt�t; t� tÞ;Dabcd ðxt�tÞ; ecd ðxt�t; t� tÞÞ��

dtþ CijmðsK ; tÞ dWmðtÞ ð13Þ

where

Cijmðsij ; tÞCabmðsab; tÞ ¼ 2

Z t

0

dtCov0½Zijðsijðxt; tÞ;DijklðxtÞ; eklðxt; tÞÞ;

Zabðsabðxt�t; t� tÞ;Dabcd ðxt�tÞ; ecd ðxt�t; t� tÞÞ� ð14Þ

and, dWiðtÞ is an increment of the vector Wiener process Wi having the following properties:

hdWiðtÞi ¼ 0 ð15Þ

and

hdWiðtÞ dWjðtÞi ¼ dij dt for t ¼ t

¼ 0 for t=t ð16Þ

It is also interesting to note that all the stochasticities of the original system of equations(Equation (3)) are lumped together in the last term (Wiener increment term) of the r.h.sof Equation (13).

In theory, one could use the Ito form of the constitutive rate equation (Equation (13)) inobtaining the mean of stress tensor (sij) as (cf. Kavvas [16]):

hdsijðx; tÞidt

¼hZijðsijðxt; tÞ;DijklðxtÞ; eklðxt; tÞÞi þZ t

0

dtCov0@Zijðsijðxt; tÞ;DijklðxtÞ; eklðxt; tÞÞ

@sab;

�

Zabðsabðxt�t; t� tÞ;Dabcd ðxt�tÞ; ecd ðxt�t; t� tÞÞ�

ð17Þ

but the difficulty is that the stress tensor appearing within Zijð�Þ in the covariance term on ther.h.s of Equation (17) is random and needs to be treated accordingly. One possible way isperturbation with respect to mean, however the ‘closure problem’ (cf. Kavvas [16]) will makethis type of solution problematic. Hence in this study the mean of stress tensor ðsijÞ will becomputed by standard operation on the PDF of stress tensor ðsijÞ; which will be obtained bysolving the Fokker–Planck–Kolmogorov (FPK) PDE (Equation (10)).



DOI: 10.1002/nag

3. APPLICATION TO PROBABILISTIC BEHAVIOUR OF 1-D CAM-CLAYELASTIC–PLASTIC CONSTITUTIVE EQUATION

In this section the applicability of the presented general expression in obtaining PDF andsubsequently mean and variance behaviours of stress will be demonstrated for 1-D, elastic–plastic, Cam-Clay constitutive rate equation. The general incremental form of constitutiveequation with single scalar internal variable can be written as

dsijðtÞdt

¼ DepijklðtÞ

deklðtÞdt

ð18Þ

where the stiffness tensor Depijkl is given as

Depijkl ¼

Delijkl when f50_ ðf ¼ 0^ df50Þ

Delijkl �

Delijmn

@f

@smn

@f

@spqDel

pqkl

@f

@srsDel

rstu

@f

@stu�@f

@p0%p0

when f ¼ 0_ df ¼ 0

8>>>>><>>>>>:

ð19Þ

where Delijkl is the elastic stiffness tensor and Dep

ijkl is the elastic–plastic stiffness tensor. Theinternal variable p0 in this case depends on plastic volumetric strain and has a physical meaning(for Cam-Clay) as the maximum hydrostatic stress the soil has experienced during previousloading history). It was assumed that the material obeys associative flow rule (as is it is usuallyassumed for Cam-Clay material model). Hence, both the yield and the plastic potentialfunctions f are given by (Cam-Clay material model):

f ¼ #f ðp; qÞ ¼ p2 � p0pþq2

M2¼ 0 ð20Þ

Material parameter M represents the slope of the critical state line in the p–q space. The stressinvariants p and q are given as

p ¼ �skk3¼ �

s11 þ s22 þ s333

ð21Þ

q ¼

ffiffiffiffiffiffiffiffiffiffiffiffi3

2sijsij

r¼

1ffiffiffi2

p ½ðs11 � s22Þ2 þ ðs22 � s33Þ

2 þ ðs33 � s11Þ2 þ 6s212 þ 6s223 þ 6s231�

1=2 ð22Þ

One can work on components of the Equation (19) to obtain:

Akl ¼@f

@spqDel

pqkl

¼@f

@p2G

@p

@s11d1ld1k þ

@p

@s22d2ld2k þ

@p

@s33d3ld3k

� �þ K �

2

3G

� �@p

@scddcddkl

� �

þ@f

@q2G

@q

@sijdikdjl þ K �

2

3G

� �@q

@sabdabdkl

� �ð23Þ



DOI: 10.1002/nag

and

B ¼@f

@srsDel

rstu

@f

@stu

¼@f

@p

� �2

2G@p

@s11

� �2

þ@p

@s22

� �2

þ@p

@s33

� �2( )

þ K �2

3G

� �@p

@sijdij

� �2" #

þ@f

@q

� �2

2G@q

@sij

@q

@sijþ K �

2

3G

� �@q

@sijdij

� �2" #

ð24Þ

where K and G are the elastic bulk and shear modulus, respectively. For the Cam-Clayformulation the material parameter p0 represents an internal variable and its evolution dependson the plastic volumetric strain. Hence, the plastic modulus KP becomes:

KP ¼ �@f

@p0%p0 ¼ ð1þ e0Þ

p0

l� k@f

@pð25Þ

One may note that the differentiations appearing in Equations (23)–(25), involve stochasticprocesses and hence cannot be carried out in a deterministic sense. Substituting Equations (23)–(25) into (19) one may obtain the random operator tensor (Zij) specialized to Cam-Clay model as

Zij ¼

2Gdikdjl þ K �2

3G

� �dijdkl

� �dekldt

when f50_ ðf ¼ 0^ df50Þ

2Gdikdjl þ K �2

3G

� �dijdkl �

AijAkl

Bþ KP

� �dekldt

when f ¼ 0_ df ¼ 0

8>>><>>>:

ð26Þ

where Aij ; B and KP are defined by Equations (23)–(25), respectively.Previously developed random operator tensor Zij for Cam-Clay material model will be now

specialized to 1-D shear behaviour, resulting in the following equation for the PDF of shear stress:

@Pðs12ðtÞ; tÞ@t

¼ �@

@s12

��hZðs12ðtÞ;D1212; e12ðtÞÞi

þZ t

0

dtCov0@Zðs12ðtÞ;D1212; e12ðtÞÞ

@s12; Zðs12ðt�tÞ;D1212; e12ðt�tÞÞ

� ��Pðs12ðtÞ; tÞ

�

þ@2

@s212

Z t

0

dtCov0½Zðs12ðtÞ;D1212; e12ðtÞÞ; Zs12ðt� tÞ;D1212; e12ðt� tÞ��

� Pðs12ðtÞ; tÞ�

ð27Þ

One may note that while solving Equation (27) for probability density (Pðs12Þ) of each fixed valueof s12 in its domain, the s12 terms appearing within Zð�Þ are deterministic and hence thedifferentiations involving s12 can be carried out in a usual sense. In Equation (27) the randomoperator function Z is different for pre-yield (elastic) and post-yield (elastic–plastic) responses. Forpre-yield, linear elastic response, one can obtain the random operator function Z in the form:

Z ¼ 2Gde12dt

ð28Þ



DOI: 10.1002/nag

while for post-yield, non-linear, elastic–plastic response Z obtains the form:

Z ¼ 2G�36

G2

M4

� �s212

ð1þ e0Þp p�3s212pM2

� �2

kþ 18

G

M4

� �s212 þ

1þ e0

l� kpp0 p�

3s212pM2

� �

2666666664

3777777775de12dt

ð29Þ

4. RESULTS AND DISCUSSIONS

In this section the FPK equation is solved using appropriate initial and boundary conditions. TheFPK equation is specialized to 1-D (shear) Cam-Clay constitutive relationship (Equations(27)–(29)) with different degrees of uncertainties and the solution will be in terms of the evolutionof PDF of shear stress (s12) with time (and/or strain), given random material property(ies) anddeterministic shear strain. It should be noted that the strain can be random as well, and theequations can be solved in that case too, but for the sake of simplicity, strain is preset todeterministic value. In other words, the solution is driven by deterministic strain.

4.1. Initial and boundary conditions

The FPK PDE specialized to 1-D Cam-Clay shear constitutive rate equation (Equation (27))can be written in the following simplified form:

@Pðs12; tÞ@t

¼ �@

@s12fPðs12; tÞNð1Þg þ

@2

@s212fPðs12; tÞNð2Þg

¼ �@

@s12Pðs12; tÞNð1Þ �

@

@s12fPðs12; tÞNð2Þg

� �ð30Þ

¼ �@z@s12

ð31Þ

where Nð1Þ and Nð2Þ are coefficients of the FPK PDE (Equation (27)) and represent theexpressions within the curly braces of the first and second term, respectively, on the r.h.s. ofEquation (27). They are called the advection and diffusion coefficients, respectively, as the formof Equation (27) closely resembles advection–diffusion equation (cf. Gardiner [23]). The symbolz in Equation (31) can be considered to be the probability current (cf. Gardiner [23]), sinceEquation (31) is a continuity equation and the state variable of the equation (Equation (31)) isprobability density.

The initial condition for pre-yield (elastic) solution for PDF of shear stress will bedeterministic i.e. at time t ¼ 0 all the probability mass is concentrated at s12 ¼ 0 or at some fixedvalue of s12 (if there was some initial deterministic stress to begin with). Mathematically, this isrepresented using Dirac delta function dð�Þ

Pðs12; 0Þ ¼ dðs12Þ ð32Þ



DOI: 10.1002/nag

On the other hand, for solving for post-yield (elastic–plastic) PDF of shear stress, the initialcondition will be represented by the PDF of shear stress corresponding to the pre-yield solution(PDF of shear stress at yield).

The initial probability mass, dictated by Equation (27), will advect and diffuse into thedomain of the system throughout the evolution of the simulation. As for the boundarycondition, a reflecting barrier boundary is chosen since conservation of the probability masswithin the system is required (i.e. no leaking is allowed at the boundaries). Mathematically thiscondition is expressed as (cf. Gardiner [23])

zðs12; tÞjAtBoundaries ¼ 0 ð33Þ

By making the assumption that the material parameters are normally distributed,theoretically the domain of stress could be from �1 to þ1 and hence one can write theboundary conditions as

zð�1; tÞ ¼ zð1; tÞ ¼ 0 ð34Þ

It should also be noted that any other probabilistic distribution for material parameters can behandled. For example, by assuming that material parameters obey exponential distribution,theoretically the domain of stress could be from 0 to þ1 and hence one can write the boundaryconditions as

zð0; tÞ ¼ zð1; tÞ ¼ 0 ð35Þ

4.2. Numerical scheme

The FPKE specialized to 1-D Cam-Clay constitutive equation (Equation (27) with Equations(28) or (29)) with initial and boundary conditions (Equations (32) and (34), respectively) wassolved numerically by method of lines (using Mathematica [24]). The FPK PDE is first semi-discretized into a set of simultaneous ODE systems and then solved by using an ODE solver.

To this end, differentiating by parts the r.h.s. of Equation (31) yields

@P

@t¼ �Nð1Þ

@P

@s12� P

@Nð1Þ@s12

þNð2Þ@2P

@s212þ 2

@P

@s12

@Nð2Þ@s12

þ P@2Nð2Þ@s212

¼P@2Nð2Þ@s12

�@Nð1Þ@s12

� �þ@P

@s122@Nð2Þ@s12

�Nð1Þ

� �þ@2P

@s212Nð2Þ ð36Þ

then, using central difference discretization, as shown in Figure 2 one can write semi-discretizedform of Equation (36) at any intermediate node i to as

@PðiÞ

@t¼Pði�1Þ

NðiÞð1Þ

2Ds12þ

NðiÞð2Þ

Ds212�

1

Ds12

@NðiÞð2Þ@s12

" #� PðiÞ

@NðiÞð1Þ@s12

þ 2NðiÞð2Þ

Ds212�@2NðiÞð2Þ@s212

" #

þ Pðiþ1Þ �NðiÞð1Þ

2Ds12þ

NðiÞð2Þ

Ds212þ

1

Ds12

@NðiÞð2Þ@s12

" #ð37Þ

which forms an initial value problem in the time dimension. It is worth noting that centralfinite difference technique might not be the most efficient in solving n-dimensional FPK PDE.Work is underway to improve the efficiency of solution method through adaptivity and reducedorder modelling.



DOI: 10.1002/nag

It is also noted that, eventhough in theory the shear stress (s12) ranges from �1 to þ1;for simulation purpose the domain between �0:1 and þ0:1 MPa is chosen. This choiceis based upon the material properties of the example problem, and by considering thepractical range of shear stress (s12). In addition to that, the Dirac delta function used ininitial condition of pre-yield (elastic) solution (Equation (32)) is numerically approximatedusing a Gaussian function with zero mean and standard deviation of 0:00001 MPa; as shownin Figure 3.

While this approximation introduces some error in solution (PDF of shear stress) the errorquickly disappears as soon as the solution moves away from the initial condition.

Boundary

Boundary

12

ii+1

i−1

n−1 n

∆σ12∆σ12

σ12

σ12

t

Figure 2. Spatial discretization of the Fokker–Planck–Kolmogorov PDE.

0.0005 0.001 0.0015 0.002

Stress (MPa)

10000

20000

30000

40000

Pro

bDen

sity

(S

tres

s)

Figure 3. Approximation of Dirac delta initial condition (Equation (32)) with a Gaussian function of zeromean and standard deviation of 0:00001 MPa:



DOI: 10.1002/nag

4.3. Simulation plots

We begin by presenting the results of elastic and elastic–plastic PDFs for shear stress withonly one random material parameter. Figure 4(a) shows the evolution of PDF of shearstress with time/shear strain for a low over-consolidation ratio (OCR) Cam-Clay materialwith normally distributed random shear modulus G (mean value of 2:5 MPa and a standarddeviation of 0:5 MPa). The other material properties are considered deterministic and

0.1

0.2

0.3 0

0.01

0.02

0.03

0.04

0

200

400

Time (Sec)

Stre

ss (M

Pa)Pr

obD

ensi

ty

Strain (%)0

1.62

0.1

0.2

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.01

0.02

0.03

0.04

Time (s)

Strain (%) 1.620

Stre

ss (

MPa

)

0.1

0.005

1

Mean

Std. Deviations

Mode

Deterministic

(b)

Figure 4. Low OCR Cam-Clay response with random normally distributed shear modulus (G):(a) evolution of PDF; and (b) evolution of contours of PDF, mean, mode, standard deviations

and deterministic solution of shear stress (s12) with time (t)/shear strain (e12).



DOI: 10.1002/nag

are given as follows: slope of critical state line M ¼ 0:6; overconsolidation pressure p0 ¼0:2 MPa; applied confinement pressure p ¼ 0:1 MPa; slopes of the unloading–reloading andnormal compression lines in e� lnðp0Þ space k ¼ 0:05 and l ¼ 0:25; respectively; and initial voidratio e0 ¼ 2:18:

As mentioned earlier, the strain is assumed deterministic (strain-driven algorithm) while thestrain rate is assumed constant (de12=dt in Equations (28) and (29)). The numerical value ofassumed constant strain rate can be substituted in Equations (28) and (29) for the entiresimulation of the evolution of PDF. As mentioned earlier at the beginning of this section, theFPKE (Equation (27)) describes the evolution of PDF of shear stress (s12) with time (t), whilethe shear strain rate (de12=dt) describes the evolution of shear strain (e12) with time (t) also.Combining the two, the evolution of PDF of shear stress (s12) with shear strain (e12) is easilyobtained. Time has been brought in this simulation as an intermediate dimension to help insolution process, and hence, the numerical value of shear strain rate (de12=dt) could be anyarbitrary value. This value of shear strain rate will cancel out once one converts the evolution ofPDF of shear stress (s12) as a function of time, to evolution of PDF of shear stress (s12) as afunction of shear strain (e12). For simulation purpose we assumed the arbitrary value of shearstrain rate (de12=dt) as 0:054=s:

Once the evolution of PDF of shear stress (s12) is obtained, the evolution of mean, mode andstandard deviations was calculated by standard operations on the PDF. The results are shown inFigure 4. In particular, Figure 4(a) shows the evolution (surface) of PDF of shear stress (s12Þwith time (t)/shear strain (e12). The evolution of PDF of shear stress is given in more detail inFigure 4(b) showing contours of the PDF of shear stress (s12) and also the mean, mode andstandard deviations. The deterministic solution, obtained using the mean value of shearmodulus (G) as fixed, deterministic input is also shown. As can be seen from Figure 4(b), thestandard deviation of shear stress increases in elastic regime (before approx. 0:056 s) untilthe yield point and then decreases as the solution approaches the critical state line. In otherwords, close to critical state the uncertainty in shear modulus (G) did not have much effect onthe uncertainty in shear stress as it had close to yielding. But on the other hand if one follows theevolution of deterministic solution one could see that eventhough the deterministic solutioncoincides with mean and mode for pre-yield (elastic) behaviour, it deviates for post-yieldbehaviour, suggesting that the deterministic solution is neither the mean nor the most probablesolution for post-yield response.

When compared with Monte-Carlo simulation, shown in Figure 5, it can be seen thatFPKE approach predicted the mean behaviour exactly but it slightly over-predicted thestandard deviation behaviour. This difference is attributed to the approximation used torepresent the initial deterministic condition s12 ¼ 0 (Dirac delta function). One may note that ate12 ¼ 0; the probability of shear stress (s12) at s12 ¼ 0 should theoretically be 1 i.e. all theprobability mass should theoretically be concentrated at s12 ¼ 0 and would be best describedby a Dirac delta condition. However, for numerical simulation of FPKE, Dirac deltainitial condition was approximated with a Gaussian function of mean zero and standarddeviation of 0:00001 MPa as shown in Figure 3. This initial error in standard deviationadvected and diffused into the domain during the simulation of the evolution process. This errorcould be minimized by better approximating the Dirac delta initial condition (but at the expenseof computational cost).

Next few examples examine results (PDF of shear stress) by the introduction of uncertaintiesin other material parameters (other than shear modulus G). In examples where we use more



DOI: 10.1002/nag

than one random material property (see next examples) they are assumed uncorrelatedand independent. The presented development can also deal with correlated random variables,as described by FPK equation (10). The advection and diffusion coefficients Nð1Þ and Nð2Þ (inEquation (30)) will be changed accordingly for any correlation between random soilproperties.

Additional uncertainty is introduced for the slope of critical state line M in terms of a normaldistribution with mean value of 0.6 and a standard deviation of 0.1. The shear modulus G isagain treated as random, as before. The resulting PDF of shear stress s12 is now exhibiting anon-symmetric distribution in the post-yield (elastic–plastic) region. This non-symmetry isevident from different post-yield evolution of mean and mode of shear stress s12 as seenin Figure 6(b). The non-symmetry is also evident from the trace of the surface of PDFof shear stress at time ðtÞ ¼ 0:3 s (or shear strain (e12Þ ¼ 1:62%) in Figure 6(a). Anotherinteresting aspect to note (refer to Figure 6(b)) is that for this assumed combination of materialproperties (random G and random M) the post-yield deterministic shear stress (s12) under-predicted the post-yield most probable (mode) shear stress (s12) but over-predicted the meanshear stress (s12).

The effect of addition of further uncertainty into the system, in terms of randomnormally distributed overconsolidation pressure (p0) with a mean value of 0:2 MPaand standard deviation of 0:07 MPa; in addition to previously introduced random shearmodulus (G) and random slope of critical state line (M), is shown in Figure 7. It can beseen by comparing Figures 6 and 7 that the additional uncertainty in overconsolidation

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.01

0.02

0.03

0.04

Time (s)

Strain (%)0 1.62

Stre

ss (

MPa

)

Mean (Fokker–Planck)

Mean (Monte–Carlo)

Std. Devations (Monte–Carlo)

Std. Devaitions (Fokker–Planck)

Figure 5. Comparison of FPK approach and Monte-Carlo approach for low OCR Cam-Clay responsewith random normally distributed shear modulus (G) in terms of evolution of mean and standard deviation

of shear stress (s12) with time (t)/shear strain (e12).



DOI: 10.1002/nag

pressure (p0) did not affect much the probabilistic response of shear stress (s12) whencompared to its response to random normally distributed shear modulus (G) and randomnormally distributed slope of critical state line (M) (Figure 6). That is, the responses inFigures 6 and 7 are quite similar, at least qualitatively while quantitatively there are somesmall differences.

The uncertainty in overconsolidation pressure (p0) did not even have much effect onthe probabilistic response (PDF) of shear stress (s12) when considered separately with

0.3 0

0.01

0.02

0.03

0.04

0

200

400

Stre

ss (M

Pa)

Time (Sec)

Strain (%)

0

1.62

Prob

Den

sity

0.1

0.2

(a)

(b)0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.01

0.02

0.03

0.04

Time (s)

Strain (%)

Stre

ss (

MPa

)

1.620

Std. Deviations

Mean

30

50

10Deterministic

Mode

Figure 6. Low OCR Cam-Clay response with random normally distributed shear modulus (G) andrandom normally distributed slope of critical state line (M): (a) evolution of PDF; and (b) evolution ofcontours of PDF, mean, mode, standard deviations and deterministic solution of shear stress (s12) with

Time (t)/shear strain (e12).



DOI: 10.1002/nag

random normally distributed shear modulus (G). This can be seen by comparing Figure 8,where the probabilistic behaviour of shear stress (s12) for low OCR Cam-Clay modelwith randomly distributed shear modulus (G) and random normally distributed overconsoli-dation pressure (p0) was presented, and Figure 4. Similar to the case where the shearmodulus was the only random parameter (Figure 4), here also the post-yield deterministicshear stress (s12) overpredicted the post-yield mean and post-yield most probable (mode)shear stress (s12).

0.1

0.2

0.30

0.01

0.02

0.03

0.04

0

200

400

Stre

ss (M

Pa)

Time (Sec)

Strain (%)

0

1.62

Prob

Den

sity

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.01

0.02

0.03

0.04

Time (s)

Strain (%) 1.620

Std. Deviation Lines

Stre

ss (

MPa

)

10

30

50

MeanDeterministic

Mode

(b)

Figure 7. Low OCR Cam-Clay response with random normally distributed shear modulus (G), randomnormally distributed slope of critical state line (M), and random normally distributed overconsolidationpressure (p0): (a) evolution of PDF; and (b) evolution of contours of PDF, mean, mode, standard

deviations, and deterministic solution of shear stress (s12) with time (t)/shear strain (e12).



DOI: 10.1002/nag

The probabilistic Cam-Clay elastic–plastic responses (low OCR) with different degrees ofrandomnesses (as presented above) are compared in Figure 9. In this figure the PDFs ofshear stress (s12) at shear strain (e12Þ ¼ 1:62% are shown and compared with the deterministicvalue of shear stress obtained by choosing mean value (which in this case is also the mode

0

0.2

0.4

0.6

00.010.020.030.040

200

400

Stress (MPa)

Prob

Den

sity

Tim

e (S

ec)

Stra

in (

%)

0

3.24

(a)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.01

0.02

0.03

0.04

Time (s)

Strain (%) 1.620

Stre

ss (

MPa

)

Std. Deviations

Mean

10

30

50

Deterministic

Mode

(b)

Figure 8. Low OCR Cam-Clay response with random normally distributed shear modulus (G) andrandom normally distributed overconsolidation pressure (p0): (a) evolution of PDF; and (b) evolution ofcontours of PDF, mean, mode, standard deviations and deterministic solution of shear stress (s12) with

time (t)/shear strain (e12).



DOI: 10.1002/nag

we used in normal distribution) of material properties. It is very interesting to note thatthe deterministic value of shear stress (s12) is different from the most probable (mode) value ofshear stress (s12) for all the cases. It either under-predicted or over-predicted the most probable(mode) value.

Presented methodology is applicable to both hardening (as shown in examples above)and softening material response. A high OCR Cam-Clay material sample was analysed.Following parameters were used: random normally distributed shear modulus (G) with meanof 2:5 MPa and standard deviation of 0:5 MPa; random normally distributed slope ofcritical state line (M) with mean of 0.6 and standard deviation of 0.1; deterministicoverconsolidation pressure p0 ¼ 0:8 MPa; deterministic applied confinement pressure p ¼0:02 MPa; deterministic slopes of the unloading–reloading and normal compression lines ine� lnðp0Þ space k ¼ 0:05 and l ¼ 0:25; respectively; deterministic initial void ratio e0 ¼ 2:18:The (arbitrary) value of strain rate is still the same, de12=dt ¼ 0:054=s: The resulting PDF ofshear stress is shown in Figure 10(a). Figure 10(b) gives a more detailed view of results, interms of evolution of contours of PDF, mean, mode, standard deviation and deter-ministic solution of shear stress (s12). Similar to the low OCR simulation with randomG and random M; the high OCR simulation also yielded a symmetric distribution of PDFshear stress (s12) in the pre-yield (elastic) regime and a non-symmetric distribution of PDFshear stress (s12) in the post-yield (elastic–plastic) regime. Similar to probabilistic responsefor low OCR (with random G and random M), the deterministic shear stress (s12)coincided with the mean and most probable (mode) shear stress in the pre-yield (elastic)regime but deviated in the post-yield non-linear elastic–plastic regime. The deterministicshear stress (s12) under-predicted the mean shear stress (s12) in the entire post-yield regimebut over-predicted the most probable (mode) shear stress (s12) in the region close to theyield point, though close to critical state line it under-predicted the most probable (mode)shear stress (s12).

Verification of probabilistic simulations for high OCR sample were done again usingMonte-Carlo approach. Figure 11 compares the evolution of mean and standard deviationof shear stress (s12) obtained using FPKE approach and Monte-Carlo approach. It can beseen that FPKE approach slightly overpredicted the standard deviation behaviour because

0.01 0.02 0.03 0.04

100

200

300

400

500

Random G and Random M

Deterministic Value = 0.02906 MPa

PDF

of S

hear

Str

ess

Random G, Random M, and Random p0

Random G and Random p0Random G only

Shear Stress (MPa)

Figure 9. Comparison of shear stresses at 1.62% shear strain obtained from low OCR Cam-Clay modelwith different degrees of randomnesses.



DOI: 10.1002/nag

of the reasons discussed earlier. That is, the FPKE approach is somewhat ‘wider’ since theinitial condition was not a Dirac delta function, but rather an approximation. Again,this deviation in initial condition (from Dirac delta function) can be controlled by choosingan initial normal distribution for PDF of shear stress with smaller standard deviation, butthis will increase computational cost in solving the resulting finite difference system ofequations.

0.1

0.2

0.3

00.010.020

200

400

Stress (MPa)

Tim

e (s

)

Stra

in (

%)

0

2

0

Prob

Den

sity

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

0.005

0.01

0.015

0.02

0.025

0.03

Strain (%)

Time (s)

0.20

Stre

ss (

MPa

)

Deterministic

Std. Deviations

Mean

100

5030

Mode

(b)

Figure 10. High OCR Cam-Clay response with random normally distributed shear modulus (G) andrandom normally distributed slope of critical state line (M): (a) evolution of PDF; and (b) evolution ofcontours of PDF, mean, mode, standard deviations and deterministic solution of shear stress (s12) with

time ðtÞ/shear strain (e12).



DOI: 10.1002/nag

5. CONCLUSIONS

In this paper we presented methodology to solve the probabilistic elastic–plastic differentialequations. The methodology is based on Eulerian–Lagrangian form of the Fokker–Planck–Kolmogorov equation and provides for full description of the probability density function(PDF) of stress response for given strain. Of particular interest was the modelling of stress–strain constitutive response for materials with uncertain material parameters. Theseuncertainties in material properties are present in most materials, and it is just our simplificationto a single value (say mean) that renders deterministic properties usually used for simulations.The effects of nonlinear hardening/softening on predicted PDF of stress were investigated insome more detail. It was shown that the nonlinear hardening/softening creates a discrepancybetween the most likely stress solution and the deterministic solution, meaning that the usuallyobtained deterministic solution is not the most likely outcome of a constitutive stress–strainsimulation if material parameters are uncertain.

APPENDIX: NOTATION

This is a summary of notations used in mathematical derivations:

Dijkl material parameter tensor, function of Delijkl ; f ; U; q*

and r*

Delijkl elastic stiffness tensor

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.005

0.01

0.015

0.02

0.025

0.03

Strain (%)

Time (s)

0 2.0

Stre

ss (

MPa

)

Std. Deviation (Fokker–Planck)

Mean (Fokker–Planck)

Std. Deviation (Monte–Carlo)

Mean (Monte–Carlo)

Figure 11. Comparison of FPK approach and Monte-Carlo approach for high OCR Cam-Clay response withrandom normally distributed shear modulus (G) and random normally distributed slope of critical state line(M) in terms of evolution of mean and standard deviation of shear stress (s12) with time (t)/shear strain (e12).



DOI: 10.1002/nag

Depijkl elastic–plastic tangent stiffness tensor

sij Cauchy stress tensor, in indicial notatione void ratio of soile0 initial void ratio of soilf yield surfaceG shear modulus of soilK bulk modulus of soilKP plastic modulusCijm coefficient tensor in multivariate Ito equation, in indicial notationM slope of critical state line in p2q spaceNð1Þ advection coefficient of FPK equationNð2Þ diffusion coefficient of FPK equationPð� ; tÞ probability density at time, tp first invariant of stress tensorp0 hydrostatic pressure, the soil under analysis has experienced before (internal variable

in Cam-Clay model)

%p0 direction of evolution of p0q invariant of stress tensor, equal to

ffiffiffiffiffiffiffiffiffiffiffi32 sijsij

qq*

internal variablesr*

directions of evolution of internal variablessij second invariant of the deviatoric stress tensort; t1; t2 timeU potential surfacedWi increment of vector Wiener processxt spatial location at time, tekl strain tensor’e strain ratel slope of normal consolidation line in e� lnðpÞ spacek slope of unloading–reloading line in e� lnðpÞ spaceZij spatial–temporal operator tensor, function of sij ; Dijkl and eklrð� ; tÞ phase density at time, tt timez probability current

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the role of nonlinear hardening/softening in probabilistic elasto-plasticity

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