Probabilistic optimization of a continuum mechanics model to predict differential stress-induced damage in claystone Esmaeel Bakhtiary, Hao Xu, Chloé Arson n School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA article info Article history: Received 30 July 2013 Received in revised form 7 January 2014 Accepted 24 February 2014 Keywords: Rock mechanics Damage mechanics Thermodynamics Maximum likelihood method Model calibration Performance assessment abstract Phenomenological modeling of anisotropic damage in rock raises many fundamental thermodynamic and mechanical issues. In this paper, the maximum likelihood method is used to analyze the performance of the Differential Stress Induced Damage (DSID) model recently formulated by Xu and Arson [1]. The stress/strain relationship is a nonlinear function of parameters including unknown constants (i.e., damage constitutive parameters) and known variables (e.g., elastic parameters and controlled stress state). Logarithmic transformation, normalization and forward deletion are employed, in order to find the optimum number of constitutive parameters, as a trade off between accuracy and simplicity. For Eastern France claystone subject to deviatoric stress loading (e.g., triaxial and proportional compression loading), it is found that (1) only one damage parameter (a 2 ) is needed in the expression of the free energy to predict stress/strain curves; (2) a 2 controls the deviation of the current principal directions of stress to the principal directions of damage; (3) model parameters involved in the damage criterion can be related to a 2 . As a result, a 2 is the only parameter needed to model differential-stress induced damage in Eastern France claystone. It is also shown that within the set of assumptions made in this study, the DSID model is not sensitive to the initial damage threshold C 0 , except for C 0 410 6 Pa, a range of values in which only one constitutive parameter becomes insufficient to predict the stress/strain curves of damaged claystone. Coupling probabilistic calibration and optimization methods to numerical codes promises to allow adapting the complexity of anisotropic damage models to different rocks and stress paths. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction At present, 85% of the energy power consumed in the world is produced by fossil fuel combustion [2,3], which has raised increasing interest in renewable energy technologies, non-conventional oil and gas reservoirs, and nuclear power. Innovative nuclear fuels and reactors depend on the economical and environmental impacts of waste management [4]. Disposals in mined geological formations are viewed as potential consolidated storage facilities before final disposi- tion [5]. Rock damage is therefore a core issue in energy production (e.g., hydraulic fracturing [6–8] and geothermal energy extraction [9–11]) , energy storage (e.g., Compressed Air Energy Storage [12–14]) and waste management (e.g., nuclear waste disposals [15–19] and carbon capture [20–23]). Continuum Damage Mechanics (CDM) provides an efficient framework to bridge the failure plane scale with the pore and the crack scale. Damage is a thermodynamic variable used to (1) represent crack initiation, propagation and coalescence in rock; and (2) model the subsequent changes of rock mechanical, physical and chemical properties at the scale of a Representative Elementary Volume (REV) [24,25]. CDM-based models have an important practical interest for engineers, and are based on rigorous closed-form formulations. However, the difficulty to determine the magnitude of material parameters is overwhelming. The maximum likelihood method has been widely used in the past to find the optimum values of unknown parameters in probabilistic models. This method can also be employed to determine the standard error associated with a model, in order to assess the accuracy and reliability of this model. Also, it is possible to establish a procedure to remove unnecessary parameters or combine the ones which are correlated with each other. As a result, simpler models can be obtained, with fewer parameters. The maximum likelihood method also provides some insight into the relative importance of parameters in real physical problems. For instance, Ledesma et al. [26] used this method to find a constitutive model for soft biological tissues. Jung et al. used a Bayesian updating method (based on the maximum likelihood method) to find a constitutive law in a simplified unified compression model for soil deposits [27], and to improve soil classifications [28]. Medina-Cetina and Arson [29] and Arson and Medina-Cetina [30] used the Bayesian paradigm to calibrate a damage mechanics model for rock, and to interpret the mathe- matical independence of the constitutive parameters. Boyce and Chamis [31] used both the maximum entropy principle and the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijrmms International Journal of Rock Mechanics & Mining Sciences http://dx.doi.org/10.1016/j.ijrmms.2014.02.015 1365-1609/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author at: 790 Atlantic Drive Atlanta, GA30332-0355, USA. E-mail address: [email protected](C. Arson). International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149
Transcript
Probabilistic optimization of a continuum mechanics model to predictdifferential stress-induced damage in claystone
Esmaeel Bakhtiary, Hao Xu, Chloé Arson n
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
a r t i c l e i n f o
Article history:Received 30 July 2013Received in revised form7 January 2014Accepted 24 February 2014
Phenomenological modeling of anisotropic damage in rock raises many fundamental thermodynamicand mechanical issues. In this paper, the maximum likelihood method is used to analyze the performanceof the Differential Stress Induced Damage (DSID) model recently formulated by Xu and Arson [1]. Thestress/strain relationship is a nonlinear function of parameters including unknown constants (i.e.,damage constitutive parameters) and known variables (e.g., elastic parameters and controlled stressstate). Logarithmic transformation, normalization and forward deletion are employed, in order to find theoptimum number of constitutive parameters, as a trade off between accuracy and simplicity. For EasternFrance claystone subject to deviatoric stress loading (e.g., triaxial and proportional compression loading),it is found that (1) only one damage parameter (a2) is needed in the expression of the free energy topredict stress/strain curves; (2) a2 controls the deviation of the current principal directions of stress tothe principal directions of damage; (3) model parameters involved in the damage criterion can be relatedto a2. As a result, a2 is the only parameter needed to model differential-stress induced damage in EasternFrance claystone. It is also shown that within the set of assumptions made in this study, the DSID modelis not sensitive to the initial damage threshold C0, except for C04106 Pa, a range of values in which onlyone constitutive parameter becomes insufficient to predict the stress/strain curves of damaged claystone.Coupling probabilistic calibration and optimization methods to numerical codes promises to allowadapting the complexity of anisotropic damage models to different rocks and stress paths.
& 2014 Elsevier Ltd. All rights reserved.
1. Introduction
At present, 85% of the energy power consumed in the world isproduced by fossil fuel combustion [2,3], which has raised increasinginterest in renewable energy technologies, non-conventional oil andgas reservoirs, and nuclear power. Innovative nuclear fuels andreactors depend on the economical and environmental impacts ofwaste management [4]. Disposals in mined geological formations areviewed as potential consolidated storage facilities before final disposi-tion [5]. Rock damage is therefore a core issue in energy production(e.g., hydraulic fracturing [6–8] and geothermal energy extraction[9–11]) , energy storage (e.g., Compressed Air Energy Storage [12–14])and waste management (e.g., nuclear waste disposals [15–19] andcarbon capture [20–23]). Continuum Damage Mechanics (CDM)provides an efficient framework to bridge the failure plane scale withthe pore and the crack scale. Damage is a thermodynamic variableused to (1) represent crack initiation, propagation and coalescence inrock; and (2) model the subsequent changes of rock mechanical,physical and chemical properties at the scale of a RepresentativeElementary Volume (REV) [24,25].
CDM-based models have an important practical interest forengineers, and are based on rigorous closed-form formulations.However, the difficulty to determine the magnitude of materialparameters is overwhelming. The maximum likelihood methodhas been widely used in the past to find the optimum values ofunknown parameters in probabilistic models. This method canalso be employed to determine the standard error associated witha model, in order to assess the accuracy and reliability of thismodel. Also, it is possible to establish a procedure to removeunnecessary parameters or combine the ones which are correlatedwith each other. As a result, simpler models can be obtained, withfewer parameters. The maximum likelihood method also providessome insight into the relative importance of parameters in realphysical problems. For instance, Ledesma et al. [26] used thismethod to find a constitutive model for soft biological tissues. Junget al. used a Bayesian updating method (based on the maximumlikelihood method) to find a constitutive law in a simplifiedunified compression model for soil deposits [27], and to improvesoil classifications [28]. Medina-Cetina and Arson [29] and Arsonand Medina-Cetina [30] used the Bayesian paradigm to calibrate adamage mechanics model for rock, and to interpret the mathe-matical independence of the constitutive parameters. Boyce andChamis [31] used both the maximum entropy principle and the
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ijrmms
International Journal ofRock Mechanics & Mining Sciences
http://dx.doi.org/10.1016/j.ijrmms.2014.02.0151365-1609/& 2014 Elsevier Ltd. All rights reserved.
maximum likelihood method to establish probabilistic constitutiverelationships for cyclic material strength models. Gardoni et al.[32,33] used a total of 384 strand test specimens to study self-consolidated concrete exposed to various void, moisture, andchloride concentration conditions. Using experimental resultsand the maximum likelihood method, a probabilistic model wasdeveloped. Tsuchiya et al. [34] estimated the Weibull modulus ofbrittle materials using the maximum likelihood method. Instead ofusing a linear regression method, Huang et al. [35] employed themaximum likelihood method to predict concrete compressivestrength using ultrasonic pulse velocity and rebound number.Trejo et al. [36] and Pillai et al. [37] identified and quantifiedimportant parameters influencing the corrosion and tension capacityof strands in post-tensioned bridges.
In the work presented in the following, the maximum likelihoodmethod is used to analyze the performance of the Differential StressInduced Damage (DSID) model recently formulated by the authors[1], with a particular focus on the stress/strain response of EasternFrance claystone subjected to deviatoric stress loading. The mainmineral and mechanical properties of the claystone under study aresummarized in Section 2, along with the main constitutive modelsproposed so far. Section 3 outlines the thermodynamic frameworkof the DSID model, and provides the state-of-the-art of the methodsavailable to calibrate the related damage constitutive parameters.The proposed probabilistic model is presented in Section 4: knownvariables and unknown parameters are identified first, the imple-mentation of the maximum likelihood method is explained then,and a probabilistic strategy is finally established for the use of theDSID model. Section 5 highlights the need for a parameter calibra-tion, and methodically presents the optimization procedure used inthis study. Section 6 discusses the significance of the results, andprovides further performance assessment of the DSID model.
2. Overview of the characterization and modeling of claystone
2.1. Mineral composition and physical properties
Claystone is a mudrock – a class of fine grained siliciclasticsedimentary rocks. More than 50% of the composition of claystoneis clay-sized particles, less than 4 μm in size. Claystones containquartz, feldspar, iron oxides, and carbonate minerals (in variableproportion, depending on the geological formation). In general,claystones tend to have low permeability but high mechanicalstrength. Clay minerals such as smectite and illite are verysensitive to the saturation degree, which can result in pronouncedplastic deformation [38]. The behavior of claystone is more brittlewhen calcite content increases, and inversely becomes moreductile when the quantity of clay elements increases [39]. Thereis also a strong dependence of the mechanical behavior on theconfining pressure, marked by a transition from a fragile towards aductile behavior [40]. Table 1 summarizes the mineral and physicalcharacteristics of claystones.
2.2. Experimental characterization
Claystones are sedimentary rocks: they are structured in layersby the process of deposition. At the microscopic scale, anisotropyis manifested by the sliding of clay sheets and the twilling in a fewlarge calcite grains – two phenomena which are related to thedistribution of voids in the clay matrix. At the scale of thelaboratory sample (Representative Elementary Volume, REV),claystone anisotropy can be seen during a hydrostatic compressionloading (e.g., [39]): the response of the material to the appliedloading exhibits a different deformation in the axial and radialdirections. In order to capture the resulting intrinsic anisotropy of Ta
ble
1Su
mmaryof
themineral
andphy
sicalch
aracteristicsof
clay
ston
e.
Origin
Physical
prope
rties
Minerals
Den
sity
Porosity
Perm
eability
Water
content
Saturation
Grain
size
Meu
se/H
aute
Marne
(Paris
Basin)
Calcite:27
79%
;qu
artz:23
74%
;clay
matrix:
457
7%.S
omeaccessorymineralsare
subo
rdinatefeldsp
ars,
pyrite,a
ndiron
oxides,a
boutavo
lumetricfraction
of5%
.Theclay
mineral
compositionisrelative
lyco
nstan
tat
65%I/S(illite/smectite
interstratified
minerals),
30%illitean
d5%
kaolinitean
dch
lorite
Thebu
lk,d
ryan
dgrain
den
sity
areresp
ective
ly2.41
70.06
,2.277
0.03
and
2.65
g/cm
3
11.87
1.6%
10�19�10
�20m
26.27
1.38
%95
71.1%
Theav
erag
esize
ofcalcitean
dqu
artz
grainsisge
nerally
less
than
200μm
[39]
EasternFran
ceQuartz:52
%;calcite:
abou
t28
%;clay
s(smectite,illite,k
aolin
itean
dch
lorite)
Intrinsic
permea
bility
10�20m
2
Calcite:20
–40
%,q
uartz
20–30
%,c
lays
40–55
%11.5–12
%4–
5.7%
Calcite:25
–55
%,q
uartz
20–30
%,c
lays
35–55
%11
–13
.5%
4–7%
Calcite:25
–35
%,q
uartz
15–20
%,c
lays
45–60
%12
%4–
7%[41]
EasternFran
ceIntrinsic
permea
bility
10�21
m2
Intrinsic
permea
bility
k¼10
�21m
2,
μ¼10
�3Pa
s
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149 137
claystone mechanical behavior, loading tests have to be performedin directions parallel and perpendicular to the bedding planes.Fig. 1 shows examples of typical stress/strain curves obtainedduring triaxial compression tests, for various confining pressures.The plots (reported from [41]) highlight the non-linear response ofclaystone under deviatoric stress loading. It is worth noticing thatnon-linearities occur early during the loading path, which impliesthat when damage occurs, micro-cracks start propagating at lowstress and low deformation.
2.3. Constitutive models
Due to their favorable strength and permeability properties,Callovo-Oxfordian claystones are seen as a possible host rock forradioactive waste geological barriers. As such, claystones have beenstudied thoroughly by the French National Agency for RadioactiveWaste Management (ANDRA). Constitutive models coupling plasticityor viscoplasticity to damage mechanics were proposed in [39–41].The effect of the water content and structural anisotropy on themechanical properties of claystone was explained in [38]. Anelastoplastic damage model was also formulated within the frame-work of poromechanics [42] in order to account for hydromecha-nical couplings in the prediction of the evolution of the ExcavationDamaged Zone (EDZ) [43,44]. Thermal heating generates importantvolume changes and pore pressure variations, which significantlyaffect the hydraulic and mechanical behavior of claystone [45].A homogenization approach was proposed, in which the clay matrixwas assumed to be a solid (described by a pressure sensitive plasticmodel) containing spherical micropores [46,47]. Continuum DamageMechanics (CDM) models were also proposed in order to predictthermal, hydraulic, and mechanical crack propagation in unsaturatedrock surrounding nuclear waste disposals [48–51]. The advantage ofsuch CDM models (compared to phenomenological models couplingdamage and plasticity) is that only one dissipation potential isrequired to close the formulation [52]. The DSID model used in thefollowing is based on a general, unifying mathematical frameworkand requires few constitutive assumptions [1]. The number of modelparameters needed is then reduced by probabilistic calibration.
3. The DSID model: theoretical framework, calibration issues
3.1. Outline of the Differential Stress-Induced Damage (DSID) model
In most anisotropic damage models proposed for geomaterials,the free energy of the solid skeleton is expressed in terms ofdeformation [53–60]. As a result, the energy release rate Y (also
called damage driving force) that is work-conjugate to damage isalso a function of deformation. In most rock mechanics problemsof interest in engineering, the REV is subject to known conditionsof stress – not deformation. That is the reason why the free energypotential used in the proposed anisotropic damage model isexpressed in terms of stress (Gibbs free energy, Gs). Macroscopicdeformation and stiffness evolve with the propagation of cracksdue to “splitting effects” (i.e., Griffith cracks) and due to “crossingeffects” (i.e., equivalent cracks linking wing shear micro-cracks[61]). In order to capture both of these dissipative phenomena, thedamage criterion is expressed in terms of a damage driving forcedepending on the major principal stress. Because rock strengthincreases with confining pressure, the damage criterion is soughtin the form of a function of stress difference, therefore the nameof the model: Differential Stress-Induced Damage (DSID) model.To stay within the framework of linear elasticity in the absence ofdamage, the expression of the free energy is sought in the form ofa polynomial quadratic in stress (r) [53,62]. The thermodynamicframework of the DSID model is summarized in Table 2. Stress/strain relationships are derived from the expression of the freeenergy, which accounts for the damaged elastic deformationstored in the material and for the surface energy dissipated byresidual crack opening. The damage criterion is similar to Drucker–Prager plastic yield function, but depends on the energy releaserate instead of stress. A projection operator is used to distinguishtension and compression damage. The positivity of dissipation isensured by introducing an ad hoc damage potential, which makesthe damage flow rule non-associate. The irreversible deformationdue to damage follows an associate flow rule, so as to ensure theco-axiality between crack opening vectors and the principaldirections of damage-induced deformation. A more completepresentation and justification of the DSID model is available in [1].
3.2. Experimental determination of the DSID model parameters
Damage models proposed for rock that are based on expres-sions of the free energy similar to the one adopted in the DSIDmodel have a damage criterion and dissipation flow rules differentfrom the ones used in the DSID model: the definition of thedamage driving force is unique to the DSID model. Parameters a1,a2, a3, a4, C0 and C1 were calibrated by Halm and Dragon [54] andby Shao et al. [62]. However, none of the two papers containscalibrated values for the entire set of parameters (a1, a2, a3, a4, C0,C1). One may argue that the set of values for a1, a2, a3, a4 found in[62] may be combined with the set of values for C0 and C1 found in[54], since the two papers deal with the same rock material.However, the two studies are based on different experimentaldata, so the set of model parameters could be inconsistent, orcontain redundant parameters.
Calibration methods are rarely proposed in damage rockmechanics: only Halm and Dragon [54] and Hayakawa andMurakami [63] provided mathematical measurement strategiesfor calibration, which were later followed by other authors [64,65].Halm and Dragon's method is based on an iterative process, whichreduces its applicability to models that have a limited number ofparameters. Hayakawa and Murakami proposed different strate-gies for different experimental tests. In both Halm and Dragon'sand Hayakawa and Murakami's techniques, parameters are alldetermined from one type of experiment, which implies that thecalibrated parameters may provide erroneous predictions forstress paths other than the ones tested. Therefore, calibrationmethods employed so far are not sufficient to determine the DSIDparameters relevant for claystone: a more comprehensive analysisis needed, based on datasets obtained for different types of experi-ments (i.e., stress paths), with a sufficient number of experiments foreach stress path.
0
10
20
30
40
-0.6 -0.3 0 0.3 0.6 0.9 1.2 1.5
σ 1-σ
3 (M
Pa)
ε1(%)ε3(%)
Triaxial compression testsE0=5.8GPa, ν0=0.14
Yield Pc = 2 MPaPc = 5 MPa
Pc = 20 MPa
Fig. 1. Typical stress/strain curves of claystone during triaxial compression tests(replotted, after [41]): effect of the confining pressure (pc) on the initiation ofdamage and on damage-induced anisotropy.
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149138
In rock mechanics, experimental tests are mainly the triaxialcompression test, the uniaxial compression test, the uniaxialtension test and the Brazilian test. In the DSID model, the damagedriving force and damage variable cannot be measured directly:they are back-calculated from constitutive parameters and othervariables. Components of the damage tensor have to be derivedfirst, which then allows determining the DSID model parameters.Constitutive parameters of the DSID models include Young'smodulus (E0) and Poisson's ratio (ν0) of the pristine (undamaged)rock, the four constitutive damage parameters (a1, a2, a3 and a4)involved in the expression of the free energy, the initial damagethreshold C0, the damage hardening parameter C1, and a damageparameter α, related to rock dilatancy angle [66,67].
The probabilistic calibration of the DSID model needs to be basedon a sufficient and consistent set of experimental data (i.e., on a largeenough number of experiments performed on the same type ofrock). In this paper, it is proposed to focus on the calibration of thedamage parameters of the DSID model (a1, a2, a3, a4, C0, C1, α) forclaystone, based on the given stress–strain curves obtained during
triaxial compression and proportional tests reported in [39–41]. Theprobabilistic calibration is therefore based on the same material andon similar deviatoric stress paths. Table 3 explains the experimentaldata used in this study.
4. Construction of a probabilistic model for damagemechanics
4.1. Definition of the known variables and unknown parameters
Reported stress/strain curves obtained from experiments arecompared to predictions of strain made with the DSID model,for known states of stress, and for known elastic parameters.A probabilistic model for one component of the total strain can bewritten as
ϵðiÞðx;BÞ ¼ γðx;βÞþsξ ð1Þ
where ϵðiÞðx;BÞ is the predicted total cumulated strain at increment(i). Selected explanatory functions γðx;βÞ provide a way to relatethe total strains' component ϵðiÞ to loading measures and rockproperties (e.g., stress tensor and elastic material parameters). x isa vector of basic variables, assumed to be known or measurable,and that do not need calibration, such as the stress tensor and theelastic material parameters. B¼ ðβ; sÞ is the vector of unknownmodel parameters, in which s is the standard deviation of themodel error, and ξ is a normal random variable with zero meanand unit variance. In this particular study, β¼ fa1; a2; a3; a4;C0;C1;αgis the vector of unknown model parameters that can be optimized.
Table 2Thermodynamic framework of the DSID model.
DSID model
1: Free energy Gsðr;ΩÞ ¼ 12r : S0 : rþa1 TrΩðTr rÞ2þa2 Trðr � r �ΩÞþa3 Tr r TrðΩ � rÞþa4 TrΩ Trðr � rÞ
Table 3Experimental results used as reference datasets in the probabilistic calibration.
References Elastic parameters
Number of tests E0 (GPa) ν0
Chiarelli et al. [39] 12 7.6 0.14Bourgeois et al. [41] 3 5.8 0.14Souley et al. [40] 1 4 0.3
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149 139
Two assumptions are made in assessing the total strain inEq. (1): (1) the homoskedasticity assumption (i.e., s is assumed tobe a constant independent of x) and (2) the normality assumption(i.e., ξ is assumed to have a normal distribution). Usually, bothassumptions can be satisfied by performing transformations tostabilize the variance of the quantities of interest [68]. In thispaper, a natural logarithmic transformation of total strain isadopted, and after transformation, the model writes
lnfϵðiÞðx;BÞg ¼ lnfγðx;βÞgþsξ ð2ÞNote that the model error is written in the same way in bothEqs. (1) and (2), to indicate the generality of the approach:predicted strain (or the logarithm of predicted strain) is the sumof an explanatory function (or the logarithm of this function), anda function of error.
4.2. Application of the maximum likelihood method
The purpose of the probabilistic calibration is to optimize theestimation of the vector of unknown model parameters B¼ ðβ; sÞ.Two types of error may arise when predicting the total strain withan estimator of B¼ ðβ; sÞ instead of the actual value of B¼ ðβ; sÞ:aleatory uncertainties (known as inherent variability or random-ness) and epistemic uncertainties. The former are those that areinherent in nature: they are not related to the way the data or thepredictions are observed, and they are not influenced by theobserver. This kind of uncertainty is accounted for in the variablesx and partly in the error term ξ. The epistemic uncertainties arethose that are due to a lack of knowledge of processes, a deliberatechoice to simplify models, or due to the finite size of experimentalobservation samples or to errors in measuring observations. Thiskind of uncertainty is present in the model parameters B andpartly in the error term ξ. The fundamental difference between thetwo types of uncertainties is that, whereas aleatory uncertaintiesare irreducible, epistemic uncertainties are reducible (e.g., byimproving models, using more accurate measurements or collect-ing additional samples [69]).
B is said to be an unbiased estimator of B if the expected valueof B is equal to B, i.e., if the mean of the probability distribution ofB is equal to B. One of the best methods to obtain a point estimatorof a parameter is the method of maximum likelihood [70–72]. Asthe name implies, the estimation of B is based on the value of Bthat maximizes the likelihood function, which is defined hereinby using experimental data. Data points (noted ϵðiÞexp to referto experimental data) used as reference observation data arepoints of stress/strain curves collected from the literature (Table 3).The vector of unknown model parameters is estimatedðB ¼ fa1; a2; a3; a4; C0; C1; α ; sgÞ, which allows computing the totalstrain ϵðiÞ using Eq. (1): this provides a strain prediction with acertain error.
As discussed earlier, after performing the natural logarithmictransformation, the model error ξ is assumed to have a normaldistribution. Therefore, using the well known Gaussian distribu-tion, the probability distribution corresponding to B is written as
PðBÞ ¼ 1s
ffiffiffiffiffiffi2π
p e�ðϵðiÞexp � ϵ ðiÞ Þ2=ð2s2Þ ð3Þ
The above equation is the probability function for one point on theexperimental stress–strain curve. Usually, not only several experi-ments are used for calibration, but also several points of thestress–strain curve are read and used as observation data, for eachexperiment. Therefore, the likelihood function for a set of nindependent experimental observations is written as
LðBÞ ¼ ∏n
j ¼ 1
1s
ffiffiffiffiffiffi2π
p e�ðϵðiÞexpj � ϵ ðiÞ
j Þ2=ð2s2Þ ð4Þ
For practical reasons in computer programs, the natural logarithmof the likelihood function is determined as
ln LðβÞn o
¼ ∑n
j ¼ 1ln
1s
ffiffiffiffiffiffi2π
p e�ðϵðiÞexpj � ϵ ðiÞ
j Þ2=ð2s2Þ� �
ð5Þ
Because the logarithm is a one-to-one function, Eq. (4) aboveallows determining the likelihood function defined in Eq. (3). Thevalue of the likelihood function is obtained for the assumed set ofvalues B taken by the unknown model parameters. To obtain theoptimum representative values of the components of the vector ofunknown model parameters B, simulations were performed byusing the built-in function fmincon in MATLAB: the likelihoodfunction was computed for several hundreds of sets of valuesassigned to the vector of unknown model parameters. The set ofparameters that maximize the likelihood function is retained toestimate the vector of unknown model parameters B. Otherstatistical properties, such as the standard deviation and thecorrelation coefficient matrix, can also be determined.
4.3. DSID model calibration: probabilistic strategy
As mentioned earlier, the aim of this study is to optimize the set ofmodel parameters β¼ fa1; a2; a3; a4;C0;C1;αg for claystone, based onthe observation dataset presented in Table 3. The one-point estimatoris based on the prediction of the total, cumulated strain ϵðiÞðx;BÞ, fromthe knowledge of stress. From Table 2, we have ϵðiÞ ¼ ϵEðiÞ þϵidðiÞ. Strainestimation thus depends on the estimation of elastic parameters(to get the purely elastic part of ϵEðiÞ), damage (to get the damage-induced part of ϵEðiÞ), and irreversible deformation ϵidðiÞ. The unda-maged elastic parameters E0 and ν0 affect the elastic deformationenergy that can be stored in the material before damage occurrence,which affects the total deformation during damage propagation.Therefore, the undamaged elastic parameters affect damage evolution,but indirectly. This study focuses on the brittle deformation regime,and to simplify the computations, the undamaged elastic parameterswere not calibrated by the maximum likelihood method: instead,E0 and ν0 were taken equal to the values fitted on the experimentalstress/strain curves used as reference data, as recommended by theauthors [39–41]. The estimation of both ϵEðiÞ and ϵidðiÞ requires thecomputation of the damage variable Ω at the point of observation(more details on the incremental constitutive laws of the DSID modelare available in [1]).However, Ω cannot be obtained directly fromthe stress/strain curves used for the reference observation data.Fortunately, it is possible to relateΩ to measurable parameters anda set of unknown constants. Cumulated damage is the sum ofall damage increments from “time 0” to “current time” (i.e., toobservation point (i)):
ΩðiÞ ¼ ∑i
k ¼ 1
_ΩðkÞ ¼ ∑i
k ¼ 1
_λdðkÞ ∂gd
∂r
� �ðkÞð6Þ
Using the chain rule, the derivative ∂gd=∂r in the right of Eq. (6) canbe expressed with unknown model parameters ai and the knownvariable r
The same strategy can be used for irreversible strain:
ϵidðkÞ ¼ ∑i
k ¼ 1
_ϵ idðkÞ ¼ ∑i
k ¼ 1
_λdðkÞ ∂f d
∂r
� �ðkÞð14Þ
As a result, the damage and irreversible strain cumulated at thepoint of observation can be determined from the known variable rand the vector of unknown parameters β¼ fa1; a2; a3; a4;C0;C1;αg,which makes it possible to make a prediction of the cumulated totalstrain from the DSID model for given states of stress. Therefore, it ispossible to use a probabilistic approach to determine the statisticalproperties of all unknown parameters. Corresponding analyses areprovided in the next section.
5. Probabilistic optimization of the DSID model for claystoneunder deviatoric stress loading
5.1. Probabilistic calibration of the DSID model
The probabilistic model explained above is based on theiterative maximization of a likelihood function, in which thealgorithm needs to be initiated with estimates of the unknownparameters. By definition, unknown parameters are not availablein the literature. In the following analysis, the vector of unknownparameters is initialized with damage parameters found in theliterature for granite: a1, a2, a3 and a4 are taken from the work byShao et al. [62] for Lac du Bonnet granite; C0 and C1 are taken fromthe work of Halm and Dragon [54] for Vienne granite. Theoptimization method employed in this paper assumes that theunknown parameters can take any value, and the final result isindependent of the initialization. However, a proper initializationhelps us to converge faster to the optimized results. The initial setof values for the components of the vector of unknown parametersβ is presented in Table 4.
From the high departure from the 1:1 line in Fig. 2a, it can beseen that the stress/strain curves predicted with the equationsderived in Section 4.3 and the set of parameters tabulated in Table 4do not match the experimental stress/strain curves used as refer-ence data (Table 3). Moreover, the standard deviation of the modelincreases for the larger values of strain. Using a natural logarithmictransformation (Eq. (2)) allows stabilizing the standard deviation,but does not improve the performance of the model (Fig. 2b).
Therefore, a rigorous calibration is needed to determine the set ofunknown damage parameters β¼ fa1; a2; a3; a4;C0;C1;αg.
In the process of maximization of the likelihood function, it wasfound that model performance is not sensitive to the initialdamage threshold C0 (this observation is confirmed in Section6.2, Fig. 4). Claystone is indeed a very brittle material, in whichcracks develop early on during triaxial compression and propor-tional tests (Fig. 1). Once damage is initiated, the stress/straincurve is mostly influenced by the damage hardening parameter C1.For high levels of damage, important irreversible deformation isexpected: this behavior trend is herein referred to as “ductiledeformation regime”. To simplify the optimization process, thevalue of C0 was fixed to a low value (C0 ¼ 1:1� 105 Pa, as reportedin Table 4), which made it possible to predict damage even at lowdeformation and stay in the brittle deformation regime – the focusof this study. Table 5 summarizes the values of the remainingconstitutive damage parameters fa1; a2; a3; a4;C1;αg, optimized bythe maximum likelihood method. A comparison between resultsobtained using these values against experimental data are pre-sented in Fig. 2b and c in normal and logarithmic scales, respec-tively. As it can be seen in these figures, predictions are noticeablyimproved after optimization, and the standard deviation in thelogarithmic scale remains constant for different values of strain.The standard deviation for the model is equal to 0.29. The standarddeviation s for the model after logarithmic transformation isapproximately equal to the Coefficient Of Variation (C.O.V. inTable 5) of the model before logarithmic transformation [69].The Coefficient Of Variation (C.O.V. in Table 5), defined as
C:O:V :¼ SD=μ ð15Þprovides an indication on model uncertainty associated with aspecific constitutive parameter. The C.O.V. is representative of thenon-linearities of the DSID model, of the uncertainties involved inthe prediction of the mechanical behavior of rock (which arenatural materials), and of the measurement errors.
5.2. Probabilistic optimization of the DSID model: removingparameters
Deviatoric stress loading (e.g., triaxial compression tests andproportional loading tests) is a representative state of stress formany rock engineering applications. For instance, the rock massundergoes differences of principal stress during the excavation oftunnels and during the pressurization of well bores. Therefore it isinteresting to know whether the DSID model could be simplified forcases where damage is expected to occur due to deviatoric stress. Inthe following, it is proposed to assess the performance of the DSIDmodel for the prediction of compression-induced damage, with areduced number of constitutive parameters. Because the damagefunction is necessary to predict the occurrence of damage itself,related parameters C0, C1 and α are maintained in the modelformulation. Probabilistic optimization is focused on parametersa1, a2, a3 and a4, involved in the expression of the free energy of thedamaged rock (Table 2). It is noteworthy that the polynomial usedin the DSID model is similar to the one used by Shao et al. [62],while Halm and Dragon [53] used only two constitutive parameters.Shao et al. [73] observed experimentally that the damage modelderived with four parameters was not sensitive to a1, and therefore
Table 4Initial set of damage parameters for probabilistic calibration (from [54,62]).
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149 141
assumed a1 ¼ 0. The objective of the following probabilistic opti-mization is to justify the form of the polynomial used in the freeenergy of damaged claystone from a mathematical stand point, fordifferential stress loading. The calibration results summarized in
Table 5 indicate that (1) a4 has the largest C.O.V. and (2) a4 has theminimum mean value. The first statement (1) implies that the DSIDmodel is less sensitive to a4 than to the other constitutive para-meters. The second observation (2) supports this conclusion, since
Fig. 2. Comparison of the deformation predicted with the DSID model with deformation points taken from experimental stress/strain curves reported in Table 3. Plots on theleft (resp. on the right) display the results before (resp. after) performing the logarithmic transformation of strain. (a) and (b) Before model calibration, using modelparameters obtained by curve-fitting in [54,62]. (c) and (d) After calibration based on the maximum likelihood method. (e) and (f) After calibration, without a4. (g) and(h) After calibration, without a4 and a1.
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149142
for a given state of stress, the order of magnitude of the monomialspresent in the expression of the free energy is controlled by theabsolute value of the coefficient ai multiplying them. The maximumlikelihood method is used to optimize the calibration of the set ofparameters β¼ fa1; a2; a3;C0;C1;αg, i.e., to optimize the calibrationof the DSID parameters when a4 is removed from the formulation(i.e., when a4 ¼ 0). Results are presented in Table 6.
Interestingly, the model standard deviation does not change afterthe removal of the a4 parameter: this means that for the stress pathandmaterial under study, a4 does not contribute significantly to stress/strain prediction, and that a4 (and its associated explanatory function)can be safely removed from the model formulation. The comparisonbetween model predictions (without a4) and experimental observa-tions (Fig. 2e and f) confirms that model performance is not sensitiveto a4. By following the same procedure, parameters a1 and a3 aresuccessively removed from themodel formulation. Probabilistic resultsare presented in Tables 7 and 8. Comparisons between the predictionswith a reduced number of model parameters and experimentalobservations show that neither a1 (Fig. 2g and h) nor a3 (Fig. 3c andd) significantly affects the performance of the DSID model. Moreover,predictions obtained with a2 only (Fig. 3c and d) are not significantlydifferent from the ones obtained with the four constitutive parameters(Fig. 2c and d). The performance of the model was assessed for otherchoices of single-parameter based formulations (i.e., a1, a3 or a4 only):results confirm that a2 is the only constitutive parameter needed to
predict differential stress-induced damage in Eastern France claystone(Fig. 3).
5.3. Probabilistic optimization of the DSID model: combiningparameters
The correlation coefficient matrix for the remaining variables isshown in Table 9. If the absolute value of the correlation coeffi-cient between two variables (βi and βj) is close to one, the twovariables have high correlation and may be combined using thefollowing equation obtained from statistical relations [69,74]:
β i ¼ μβiþρβiβj
SDβi
SDβj
ðβj�μβjÞ ð16Þ
where μβi(resp. μβj
) is the mean value of βi (resp. βj); ρβiβjis the
correlation coefficient between βi and βj; SDβi(resp. SDβj
) is thestandard deviation of βi (resp. βj); and β i is the new estimate for βi.According to Table 9, the correlation coefficient between allvariables is almost one. This means that these variables are highlycorrelated with each other, which allows using Eq. (16) to combineparameters and reduce the number of independent model para-meters in the formulation of the DSID model.
By substituting the values from Table 9 into Eq. (16), the newestimates of C1 and α can be written as functions of a2:
C1 ¼ 64:34� 106þ 6:30� 106
35:80� 10�11 a2�704:94� 10�11
ð17Þ
α ¼ 3:29� 10�1þ 0:26� 10�1
35:80� 10�11 a2�704:94� 10�11
ð18Þ
According to the preceding probabilistic optimization, the freeenergy of damaged claystone can be written as
Gsðr;ΩÞ ¼ 12r : S0 : rþa2 Trðr � r �ΩÞ ð19Þ
and the damage function writes
f d ¼ffiffiffiffiJn
q� αIn�C0� C1 TrΩ ð20Þ
in which C1 and α are functions of a2. Optimized values of a2, C1and α, obtained by using the combinations above in the explana-tory functions used in the maximum likelihood method, areprovided in Table 10.
6. Discussion
6.1. Physical interpretation of the model optimization
The main finding of the preceding probabilistic analysis is thatonly two damage parameters are needed to predict damagedstress/strain curves of claystone subjected to deviatoric stressloading: C0 and a2. As mentioned earlier, the DSID model is notsensitive to C0 in the brittle deformation regime: it may beconcluded that the only constitutive parameter needed to predict
Table 5Damage parameters calibrated with the maximum likelihood method, with
μ: mean value. SD: standard deviation. C.O.V.: Coefficient Of Variation.
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149 143
differential stress-induced damage in claystone is a2, the coeffi-cient multiplying the monomial Trðr � r �ΩÞ in the expression ofthe damaged free energy. The optimized formulation of the DSID
model obtained after applying the maximum likelihood methodis therefore much simpler than the one dictated by elasticityand thermodynamic principles, in Table 2. The most general
Fig. 3. Performance of the DSID model to predict experimental stress/strain curves reported in Table 3, when only one constitutive model parameter is used in theformulation. Plots on the left (resp. on the right) display the results before (resp. after) performing the logarithmic transformation of strain. (a) and (b) Only a1. (c) and(d) Only a2. (e) and (f) Only a3. (g) and (h) Only a4.
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149144
formulation of the DSID model needs however to be used as is,when no information is available on the type of rock materialtested or stress path expected.
In triaxial compression and proportional tests, damage is drivenby a deviatoric stress (in compression). By design, the DSID modelcaptures anisotropic damage induced by differential stress. As aresult, it is expected that explanatory functions depending on stress,differences of principal stresses in particular, should be the mostaffected by the stress path. Table 11 summarizes the relation of eachai parameter to the free energy Gs, the total elastic deformation ϵE
and the damage driving force Y. In the expression of the freeenergy, every ai multiplies a trace of stress, damage or products ofstress and damage. Therefore it is impossible to conclude on therelative importance of the constitutive parameters for the predic-tion of anisotropic damage induced by stress difference.
In the expression of total elastic strain, a1 and a3 multiplytraces. Although the term in a3 allows quantifying the deviation ofstress from the principal directions of damage (i.e., from the “past”principal directions of stress), the relation to the anisotropic stresspath is expected to be better captured by the terms in a2 and a4,which indeed contain a non-volumetric term of stress. In theexpression of the damage driving force, only a2 and a3 multiply anon-volumetric stress. Hence, a2 influences stress-induceddamage and the subsequent anisotropy of both the elastic defor-mation and the damage driving force. Conceptually, it could beexpected that a2 would play the most important role in thedamage model for the tests.
However, the order of magnitude of constitutive parameters isalso critical in the analysis. From Table 11, we have
ja4jo ja1jo ja3jo ja2j ð21Þa2 is two orders of magnitude larger than a3, which is one order ofmagnitude larger than a1 and a4, a1 being slightly larger than a4.
From this analysis, it can be recommended to simplify the DSIDmodel by removing a4 first, then a1, and finally a3. a2 turns out tobe again the most significant parameter in the model. All of theseanalyses concur with the conclusions raised in the probabilisticoptimization of the damage model.
6.2. Probabilistic assessment of the optimized DSID model
The model optimization explained in Section 5.1 assumes thatC0 does not play a significant role in the evolution of damage, sincecracking is expected to occur for low stress and low deformation inclaystone. In order to check this assumption, the influence ofparameter C0 on the performance of the optimized damage modelis assessed, after removing and combining parameters (Eqs. (17)–(20)).Values of C0 are varied between 0 and 107 Pa. Fig. 4 showsthat model predictions are not affected by the initial damagethreshold when 0oC0o105 Pa. For C04106 Pa, the optimizeddamage model obtained previously tends to underestimate thetotal strain. This could be expected, since higher values of C0“delay” the occurrence of damage until higher stress differencesare reached in the sample. Therefore, increasing the value of C0tends to under-estimate damage and the related damage-induceddeformation components, which results in under-estimated totaldeformation. Therefore, it is recommended to use the optimizedDSID model (based on a2 only) for initial damage thresholdsbetween 0 and 106 Pa.
The probabilistic calibration presented above is based on theoptimization of the estimation of axial strain only. In order toproperly assess the model performance, both axial (ϵ11) and lateral(ϵ33) strains predicted by the calibrated and optimized DSID modelwere compared to experimental stress/strain curves, in Figs. 5 and 6.It appears that after simplification and calibration, the DSID modeldoes not perform equally well for all the tests used as reference data.In summary, the results of tests 5, 9, 10, 12, 13 and 15 provide closeprediction for vertical strains (Figs. 5(e), and 6(a), (b), (d), (e) and (g)),while the simulations of tests 3, 4 ,11, 13, 15, and 16 show good matchfor lateral strains (Figs. 5(c) and (d), and 6(c), (e), (g) and (h)). Themain reason for these differences is that the model was optimized onthe basis of the entire experimental dataset, and the plots inFigs. 5 and 6 only show the results for one test at a time, whichmay deviate from the average response measured from all thereference tests.
Table 9Correlation coefficient matrix for the DSID model parameters, using only a2.
The “reduction” column indicates in which order the model parameters should beremoved, according to their relative importance in the DSID model formulation.
Fig. 4. Performance of the DSID model to predict stress/strain curves during triaxialcompression and proportional tests, using a2 (in Pa�1) as the sole constitutiveparameter, with various values for the initial damage threshold C0 (in Pa).
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149 145
Experimentally, it is observed that claystones generally do notdilate significantly upon deviatoric loading: the volumetric defor-mation is compressive. However, the important increase of radialstrains at higher deviatoric stress results in less compressivevolumetric strains. In proportional tests, experimental measuresindicate that lateral strains are almost zero, especially at lowstress. Some of the predicted stress/strain curves underestimate
radial strains, which tend to decrease upon deviatoric compression(Figs. 5(g) and 6(d)). Based on the simulation results obtained, it isnoted that DSID predictions generally underestimate lateral strainsfor the higher deviatoric stress levels: the model does not capturethe degradation of compressive volumetric strains. Moreover, itcan be concluded that stress path affects the model calibrationresults.
Fig. 5. Comparison between model predictions and experimental data, after model optimization: stress/strain plots for tests 1–8. (a) Test 1 (experimental data from [41]),(b) Test 2 (experimental data from [41]), (c) Test 3 (experimental data from [41]), (d) Test 4 (experimental data from [40]), (e) Test 5 (experimental data from [39]), (f) Test 6(experimental data from [39]), (g) Test 7 (experimental data from [39]) and (h) Test 8 (experimental data from [39]).
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149146
7. Conclusion
Phenomenological modeling of anisotropic damage in rockraises many thermodynamic issues (e.g., ensuring the positivityof dissipation) and mechanical challenges (e.g., differentiatingbetween tension and compression strength, accounting for thepresence of frictional closed cracks). Moreover, rigorous calibra-tion methods are missing: the rare procedures explaining the
determination of damage model parameters from rock mechanicstests are usually limited to one stress path, and supported by a lownumber of physical experiments. In some models, it is evenimpossible to use direct measurements of stress and strain only,because any additional datapoint brings an additional unknown(the damage variable) in the set of equations to solve: an iterativeprocedure has to be followed in order to fit predicted stress/straincurves to experimental data.
Fig. 6. Comparison between model predictions and experimental data, after model optimization: stress/strain plots for tests 9–16. (a) Test 9 (experimental data from [39]),(b) Test 10 (experimental data from [39]), (c) Test 11 (experimental data from [39]), (d) Test 12 (experimental data from [39]), (e) Test 13 (experimental data from [39]),(f) Test 14 (experimental data from [39]), (g) Test 15 (experimental data from [39]) and (h) Test 16 (experimental data from [39]).
E. Bakhtiary et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 136–149 147
In this paper, the maximum likelihood method is used to analyzethe performance of the Differential Stress Induced Damage (DSID)model recently formulated by the authors, in order to predict thestress/strain response of claystone subjected to triaxial compressiontests and proportional tests. The forward deletion approach isemployed to find the optimum number of constitutive parametersas a trade off between accuracy and simplicity. It is found that(1) only one damage parameter (a2) is needed in the expression ofthe free energy to predict stress/strain curves; (2) a2 controls thedeviation of the current principal directions of stress to theprincipal directions of damage (which are co-axial with the cumu-lated deviatoric stress); (3) model parameters involved in thedamage criterion cannot be removed from the simulation, but canbe related to a2. As a result, the DSID model can be simplified whenused for claystone under triaxial compression tests and propor-tional tests: the model can be formulated with only one unknowna2, which can be viewed as the only parameter needed to modeldifferential-stress induced damage in Eastern France claystone.
To simplify the computations, the undamaged elastic para-meters E0 and ν0 were taken equal to the values fitted on theexperimental stress/strain curves used as reference data. Rigor-ously speaking, E0 and ν0 are expected to vary in a specific range ofvalues. But for low levels of stress difference, claystone is a verybrittle rock, so that both elastic parameters and the initial damagethreshold (C0) do not significantly affect the stress/strain curve. Forhigher levels of stress difference however, the DSID model isexpected to be sensitive to C0. The present study focuses on thebrittle deformation regime of claystone, and as a result, it waschosen not to include the undamaged elastic moduli and the initialdamage threshold in the vector of unknown variables to becalibrated. The performance of the DSID model was assessed afteroptimization, i.e., after transforming the constitutive equations toexpress them in terms of a2 only. It is shown that within the set ofassumptions made in this study, the DSID model indeed is notsensitive to C0, except for higher values, above 106 Pa. In fact highdamage thresholds delay the occurrence of damage under givenstress conditions, so that only one constitutive parameter becomesinsufficient to predict the stress/strain curves of damage claystone.
The advantage of the DSID model is that it provides a generalmathematical framework to model the effects of crack-opening onstiffness and deformation, for complex stress including changes ofprincipal directions, deviatoric stress in compression, and devia-toric stress in tension. The polynomials used in the expressions ofthe energy potentials are expressed in terms of independentinvariants of stress and damage, and the formulation ensuresthermodynamic consistency. The generality of the model isexpected to allow its use for a wide range of brittle and quasi-brittle materials – rocks in particular. The number of damageparameters needed is rather limited (seven in total), but notstraightforward to determine from laboratory tests. Probabilisticcalibration and optimization promises to be an efficient tool thatcan be coupled to numerical codes in order to adapt the complex-ity of the DSID model to different rocks and stress paths. Furtherdevelopments are currently on-going to calibrate, optimize andassess the DSID model for different types of rock, in order tohighlight the potential need to enrich the continuum-basedformulation of the DSID model with micro-structure descriptors.Another study will focus on the performance of damage modelswhen the stress path used for calibration does not match exactlythe stress path undergone by the rock in the field.
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