some micromechanical models of elastoplastic behaviors of porous geomaterials

Accepted Manuscript Some micromechanical models of elastoplastic behaviors of porous geomaterials W.Q Shen, J.F Shao PII: S1674-7755(16)30252-9 DOI: 10.1016/j.jrmge.2016.06.011 Reference: JRMGE 297 To appear in: Journal of Rock Mechanics and Geotechnical Engineering Received Date: 30 March 2016 Revised Date: July 2016 Accepted Date: 12 July 2016 Please cite this article as: Shen WQ, Shao JF, Some micromechanical models of elastoplastic behaviors of porous geomaterials, Journal of Rock Mechanics and Geotechnical Engineering (2017), doi: 10.1016/ j.jrmge.2016.06.011 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT Some micromechanical models of elastoplastic behaviors of porous geomaterials W.Q Shen, J.F Shao* Laboratory of Mechanics of Lille, University of Lille I, Villeneuve d’Ascq, 59655, France Received 30 March 2016; received in revised form July 2016; accepted 12 July 2016 Abstract: Some micromechanics-based constitutive models are presented in this study for porous geomaterials These micro-macro mechanical models focus on the effect of porosity RI PT and the inclusions on the macroscopic elastoplastic behaviors of porous materials In order to consider the effect of pores and the compressibility of the matrix, some macroscopic criteria are presented firstly for ductile porous medium having one population of pores with different types of matrix (von Mises, Green type, Mises-Schleicher and Drucker-Prager) Based on different homogenization techniques, these models are extended to the double porous materials with two populations of pores at different scales and a Drucker-Prager solid phase at the microscale Based on these macroscopic criteria, complete constitutive models are formulated and implemented to describe the overall responses of typical porous geomaterials (sandstone, porous chalk and argillite) Comparisons between the numerical predictions and experimental data with different confining pressures or different mineralogical composites show the capabilities of these micromechanics-based models, which take into account the effects of microstructure on the macroscopic behavior and significantly improve the phenomenological ones SC Keywords: homogenization; porous geomaterials; ductile behavior; double porosity; macroscopic strength; plastic compressibility; Drucker-Prager solids models, micro-macro models have been developed during the last decades Introduction with different homogenization techniques As an example of COx argillite, M AN U Abou-Chakra Guéry et al (2008) proposed a meso-macro model for clayey In many engineering structures, such as chemical and nuclear waste storage and oil boreholes, geomaterials are widely studied and used For example, the hard clayey rock (argillite) has been extensively investigated in the context of feasibility study for geological disposal of radioactive wastes (e.g Chiarelli, 2000; Andra, 2005; Robinet, 2008) Due to their low permeability, relatively high mechanical strength and the absence of major tectonic fractures, these clayey rocks are envisaged as one of potential geological barriers Porous chalks and anisotropic sedimentary rocks have been extensively studied in mining engineering and petroleum industry (Donath, 1961; McLamore and Gray, TE D 1967; Atwell and Sandford, 1974; Lerau et al., 1981; Hoek, 1983; Niandou et al., 1997; Papamichos et al., 1997; Homand and Shao, 2000; Schroeder, 2003; De Gennaro et al., 2004; Alam et al., 2010) Most geomaterials have complex microstructures (pores, mineral inclusions, microcracks, etc.) which induce complex macroscopic behaviors, such as mean stress dependency, microcrackrelated damage, plastic deformation, volumetric compressibility and dilatancy, and dissymmetric responses between tensile and compressive stresses EP Extensive experimental data showed that the pores or inclusions significantly affect the mechanical strength and deformation behaviors of heterogeneous materials Different kinds of constitutive models have been proposed for modeling the effective behaviors of geomaterials In most of them, AC C phenomenological elastoplastic and damage models have been used for the macroscopic behaviors of Callovo-Oxfordian (COx) argillite (e.g Chiarelli et al., 2003; Shao et al., 2006; Hoxha et al., 2007), or for anisotropic materials (Walsh and Brace, 1964; Pariseau, 1968; Hoek and Brown, 1980; Nova, 1980; Pietruszczak and Mroz, 2001; Pietruszczak et al., 2002; Lydzba et al., 2003; Lee and Pietruszczak, 2008) The formulation of models is essentially based on the standard framework of thermodynamics and experimental evidences However, it is admitted that the mechanical behavior of geomaterials is inherently related to the composition and mechanical properties of constituents The macroscopic response directly depends on the evolution of material microstructure such as change of porosity, microcrack initiation and rocks using the Hill’s incremental approach The claystone was considered as a three-phase composite constituted by a clay matrix, calcite and quartz grains The clay matrix is described by a classical Drucker-Prager type plastic model and the porosity inside the clay matrix was neglected This model was improved by Shen et al (2012a, 2013a) by considering the porosity in the clay matrix which is described by a porous medium at the microscale Recently, a micromechanical elastoplastic model was proposed in Shen and Shao (2015a) for anisotropic sedimentary rocks The effects of porosity, inclusions and inherent anisotropy were explicitly taken into account In many engineering applications, geomaterials are subjected not only to mechanical loading, but also to moisture transfer, variation of temperature and chemical degradation The durability analysis of such structures requires the consideration of such multi-physical coupling phenomena The elastic modulus and mechanical strength can significantly vary with the variation of water content during drying or wetting process To this end, the effects of water saturation degree and chemical reaction carbonation are considered for cement-based materials in micromechanics-based models (Chen et al., 2013; Shen et al., 2015a) In this study, we focus on the effects of pores and inclusions on the macroscopic mechanical behavior of porous geomaterials Some micro-macro constitutive models will be presented to describe the elastoplastic behaviors of the studied material The paper is organized as follows In Section 2, some macroscopic criteria will be presented for ductile porous materials with one population of pore Different types of matrices are considered (von Mises, Green type, Mises-Schleicher and Drucker-Prager) The case of double porous materials with a Drucker-Prager solid phase at the microscale is studied in Section Based on different homogenization techniques, two closed-form expressions of strength criteria are established to account for the effects of micro-porosity, meso-porosity and plastic compressibility in the solid phase The proposed criteria are applied in Section to describe the macroscopic plastic behaviors of different porous geomaterials, i.e sandstone, chalk and argillite, respectively growth, and physical and chemical reactions The phenomenological models cannot properly consider such relationships between microstructure and macroscopic behaviors, for example, the porosity and mineralogical Macroscopic criteria of ductile porous materials with one population of pores compositions In order to overcome this weakness of phenomenological In this section, some macroscopic criteria of ductile porous materials with *Corresponding author E-mail address: jian-fu.shao@polytech-lille.fr ACCEPTED MANUSCRIPT one population of voids will be firstly presented For this class of porous variational method: material, the representative volume element (RVE) is usually chosen as the   Σ eq   Σm    + f  − (1 − f ) = 1 + f    σ0   σ0   denoted a and b We denote Ω the volume of the unit cell, |Ω| = 4πb3/3, whereas ωe the volume of the mesoscopic voids, |ωe| = 4πa3/3 The porosity corresponding to this hollow sphere is f = a3/b3 Uniform strain rate boundary conditions are applied on the outer surface of the hollow sphere: v = Dx for x = ber , where D represents a uniform macroscopic strain rate In Fig 1, different types of matrices will be studied in this paper (von Mises, Green type, Mises-Schleicher and Drucker-Prager) For simplicity, the incompressible matrix that obeys the von Mises criterion will be firstly considered (4) This criterion was improved in Michel and Suquet (1992) by modifying the term Σm:   Σ eq  1− f   +  1 + f    σ0   ln f  2  Σm   − (1 − f ) =   σ0  (5) Recently, the Gurson’s criterion was improved by Cheng et al (2014) and Shen et al (2015b) using the stress variational homogenization (SVH) We know that the von Mises criterion is suitable for the metallic materials which are incompressible This is not the case of geomaterial whose tensile behavior RI PT one in Fig 1, a hollow sphere with the internal and external radii respectively is very different from the compressive one The plastic compression in the v = Dx z matrix should be taken into account For this purpose, compressible matrix (Green type, Mises-Schleicher type and Drucker-Prager type) will be studied here 2.2 Porous medium with a Green type matrix As shown in Eq (1), the von Mises criterion is independent of the b a SC hydrostatic stress σm In order to consider the compressibility of the matrix, the matrix in Fig is assumed to obey a general elliptic criterion (Shen et al., y 2014a): 9α σ m − Lσ m − σ 02 ≤ M AN U + Φ (σ ) = βσ eq (6) where σm is the mean stress of the microscopic stress σ in the matrix Scalars α, β and L are the material constants (a von Mises matrix is obtained for L = x Fig Hollow sphere with uniform strain rate boundary conditions together with α = 0) The yield function of Eq (6) generalizes the one used in Shen et al (2012b) The limit analysis approach used in Gurson’s model will be adopted here to 2.1 Porous medium with a von Mises type matrix For a ductile porous material with one population of voids, the matrix in Fig is assumed to obey von Mises criterion: establish a macroscopic criterion for the studied porous medium having a Φ m = σ eq − σ ≤ takes the following form: (1) where = (3 / 2)σ ′ : σ ′ with σ ′ representing the deviatoric part of the microscopic stress tensor σ, and σ0 is related to the shear strength TE D σ eq Green type matrix The local plastic dissipation π(d), corresponding to Eq (6), π (d ) = L L2 d m + σ 02 + 3α 18α 2d m2 α + d eq β (7) There are many macroscopic criteria for porous media with a von Mises where deq = (2 / 3)d ′ : d ′ is the equivalent strain rate with d ′ representing matrix The micromechanics-based model proposed by Gurson (1977) is one the deviatoric part of the mesoscopic strain rate d, and dm = trd/3 is the of the most frequently used methods Within the framework of kinematical volumetric strain rate Considering the RVE presented in Fig with the uniform strain rate material containing a spherical or cylindrical void with a uniform macroscopic boundary conditions, the following inequality holds for all macroscopic stress strain rate boundary condition: Σ and macroscopic strain rate D (Suquet, 1985; De Buhan, 1986):   Σ : D ≤ Π ( D) = inf  ∫ π (d )dV  v KA  | Ω | Ω −ωe  Φ = Σ eq2 σ 02 EP limit analysis theory, a macroscopic criterion is obtained for a von Mises  3Σ  + f cosh  m  − − f =  2σ  (2) where Σeq and Σm denote the macroscopic equivalent stress and the AC C Based on this famous model, a huge number of extensions have been proposed (e.g considering the void shape effect (Gologanu et al., 1993, 1994, 1997; Pardoen and Hutchinson, 2000, 2003; Monchiet et al., 2008, 2014; Shen et al., 2011), taking into account the tension-compression asymmetry and the anisotropy of the matrix (Benzerga et al., 1999; Monchiet et al., 2008; Cazacu and Stewart, 2009; Stewart and Cazacu, 2011) To better reproduce unit-cell simulations, Tvergaard (1981) and Tvergaard and Needleman (1984) (see also Tvergaard (1989)) proposed a heuristic and phenomenological extension of the Gurson model, known as the GTN model which is widely used in structural computations: σ 02 3 Σ  + 2q1 f cosh  q2 m  − − q3 f =  σ0  where Π(D) represents the macroscopic dissipation The infimum in Eq (8) is taken over all kinematically admissible (KA) velocity fields, v Classically, macroscopic hydrostatic one, respectively Σ eq2 (8) the limit stress states at the macroscopic scale are given by ∂Π ( D ) Σ = ∂D (9) The determination of Π (D) requires the choice of a trial velocity field which must be plastically admissible Based on the velocity used in Gurson’s model ( v G ), the following velocity field is chosen for the elliptic matrix: v = Ax + v G = Ax + b ( Dm − A) er + D ′x r2 (10) where Dm = trD/3, and D′ is the deviatoric part of the macroscopic strain rate D The homogenous field Ax allows to account for matrix plastic compressibility The two remaining terms (which are of Rice-Tracey type (3) already considered by Gurson) are kinematically admissible with (D − A1) where is the second-order identity tensor More precisely, the second term where q1, q2 and q3 are the model parameters which have great influences on on the right side of Eq (10) corresponds to the expansion of the cavity and the the yield surface outer volume, while the third one describes the shape change of the cavity and On the other hand, Ponte Castaneda (1991) proposed a macroscopic of the outer boundary without volume change Hence, for any value of the criterion for an isotropic porous medium with a von Mises matrix by using a scalar A, the whole velocity field v complies with the uniform strain rate D ACCEPTED MANUSCRIPT hydromechanical case with saturated pore by using Eshelby-like velocity applied to the hollow sphere Due to the presence of A (which remains unknown in the definition of the fields (Shen et al., 2013b, 2014b) velocity field), the macroscopic dissipation, Π(D), is computed owing to a 2.3 Porous medium with a Mises-Schleicher type matrix minimization procedure with respect to A: Π ( D ) = min[Π% ( D, A)] difference between uniaxial tension and uniaxial compression can be For many polymers and geomaterials (Aubertin and Li, 2004), the strength (11) A represented by the Mises-Schleicher criterion (Schleicher, 1926) Many works where    L Ω ∫  Ω −ωe 18α  2d m2 α + deq dV + β L / (3α ) Ω have been done for this pressure- sensitive criterion (e.g Raghava et al., 1973; ∫Ω −ω e d m dV Lubliner, 1990; Lee and Oung, 2000; Kovrizhnykh, 2004; Zhang et al., 2008; Durban et al., 2010) In this section, the isotropic and pressure-sensitive matrix The determination of the macroscopic criterion requires to compute the integrals of deq and dm over the matrix (see Eq (11)) The strain rate in the solid matrix can be obtained from Eq (10), in spherical coordinates, as b ( Dm − A) (1 − er ⊗ er ) r3 (12) e π (d )dV ≤ 4π ∫ r (< π (d ) > S ( r ) )1/2 dr (13) a e where S(r) is the sphere with the radius of r, and S(r) is the average of x(r, According to Eqs (11)–(13), the macroscopic dissipation Π% ( D, A) can be computed: L2  M + N 2u  uN   N ⋅ arcsinh  − 18α  M  u   1/ f  L  + A(1 − f ) 3α  (14) where α β , N2 = β In the special case of purely hydrostatic loading (Σeq = 0), the exact values have been obtained in Monchiet and Kondo (2012): Σ m − α 2W02 ( fp+ ) − 2α 2W0 ( fp+ )  =  σ0 3α  M AN U Π% ( D, A) = σ 02 + (1) Case of purely hydrostatic loading of Σm/σ0 of the hollow sphere with a Mises-Schleicher compressible matrix θ, ϕ) over all the orientations Deq used in Section 2.2 cannot be adopted A heuristic macroscopic criterion was special conditions b ∫Ω −ω π (d )dV =∫Ω −ω + suitable kinematically admissible velocity field, the upper bound approach established in Shen et al (2015c) for this porous material by considering some classically used: A2 (19) Due to the compressibility in the matrix and the difficulty in finding a In order to obtain a closed-form expression, the following inequality is M2 = f (σ ) = σ eq + 3ασ 0σ m − σ 02 ≤ SC d = A1 + D ′ + is assumed to obey the Mises-Schleicher criterion: RI PT  Π% ( D, A) =  σ 02 + p+ = z+ exp( z+ ) z+ = −α + α + α      (20) for hydrostatic tensile loading and ( Dm − A) We need to minimize Π% ( D, A) over the unknown variable A and to Σ m − α 2W−21 ( fp− ) − 2α 2W−1 ( fp− )  =  σ0 3α  p− = z− exp( z− )      (21) for hydrostatic compressive loading, where W denotes the “Lambert W” Similarly to the approach used by Gurson, one can then establish the function which satisfies W(x)eW(x) = x W(x) has two branches: upper branch TE D determine the macroscopic yield function by taking advantage of the approximate expression of Π% ( D, A) : ∂Π% ( D, A) ∂Π% ( D, A) Σ= , =0 (15) ∂D ∂A parametric form of the macroscopic yield function: ΣB = ∂Π% L  N  1+ = σ 02 + ∂M 18α  M  ∂Π% L2 = σ 02 + ∂N 18α    arcsinh     N   − f2+   M   N   N   − arcsinh     fM   M    EP ΣA = (16) (17) The macroscopic yield function (Eq (17)) is of Gurson type with appropriate quantities ΣA and ΣB which need to be explicit Noticing that M depends only on the deviation D′ and scalar A, while N is a function of Dm and A, the expressions of ∂Π% / ∂N and ∂Π% / ∂M can be calculated with the condition of minimization with respect to A As a final result, the closed-form expression of the macroscopic criterion for the porous medium having a matrix obeying the general elliptic criterion (Eq (6)) is (more details about the calculation can be found in Shen et al (2014a)): Σ eq2 9α + β 2 σ + L / (18α ) 3 β f cosh     Σ − L(1 − f ) / (9α )   m  +  σ + L2 / (18α )      −1− f = σ + L / (18α )  Σm 2 W0 ( x) ≥ −1 (for x ≥ −1/e) and lower branch W−1(x) < −1 (for −1/e ≤ x < 0) In the case of tension, W(px) in Eq (21) is W0(p+x), whereas in the case of compression, W(px) = W−1(p−x) For a general case, the exact solution of purely hydrostatic loading (Eq (21)) can be rewritten as the following form (the value of α2W2(fp) + 2α2W(fp) is (22) In Eq (22), the parameter A does not affect the accuracy of the exact solution in the case of purely hydrostatic loading and will be determined in the second special case of an incompressible matrix (2) Case of the compressible parameter α → and Σeq = For an incompressible matrix (α → 0), the Mises-Schleicher criterion (Eq (19)) reduces to the one of von Mises and there is non-compressibility in the matrix The famous Gurson’s criterion (Eq (2)) should be retrieved from the searched macroscopic criterion when α → To this end, the following mathematical property is used for the appearance of the “cosh()” term as the one in Gurson’s criterion: γ = e −ϑ ⇔ 2γ cosh ϑ − − γ = (23) According to the relationship between Eqs (23) and (22), the searched macroscopic criterion has the following form when Σeq = 0: (18) The Gurson’s criterion can be obviously retrieved when L = 0, α = and β = α [α 2W ( fp ) + 2α 2W ( fp )] A = exp[ A ln(1 − 3αΣ m / σ )] AC C     ΣA ΣB   + f cosh   −1− f =  σ + L2 / (18α )   σ + L2 / (18α )      −α − α + positive): By eliminating the parameter M in the above two equations, one obtains z− = The macroscopic criterion (Eq (18)) has been extended to   Σ  2Γ cosh  A ln  − 3α m   − − Γ = σ     (24) where Γ = [α 2W ( fp ) + 2α 2W ( fp )] A In order to determine the parameter A, the following conditions should be verified when α → 0, both for tension and compression: ACCEPTED MANUSCRIPT    Σm  Σ  lim cosh  A ln  − 3α m   = cosh   , lim Γ = f α →0 σ     σ  α →0  (2 5) compressible parameter α, the porosity f, and also the type of loading, sign(Σm) The proposed macroscopic criterion (Eq (33)) was assessed and validated Finally, an expression of parameter A is obtained: − [1 − α 2W ( fp) − 2α 2W ( fp )] / A = sign(Σ m ) 2α (26) The parameter A depends not only on the compressible parameter α and porosity f, but also on the sign of Σm by comparing it with numerical results (upper and lower bounds) by Pastor et al (2013) for strong and weak plastic compressibility of the Mises-Schleicher type matrix and for different porosities This criterion (Eq (33)) improves the existing ones proposed by Lee and Oung (2000) and Durban et al (2010), Compared with the Gurson’s criterion, the deviatoric part Σeq/σ0 should be added in Eq (24) This will be considered in the next special case of f → (3) Case of the porosity f → respectively 2.4 Porous medium with a Drucker-Prager type matrix Drucker-Prager criterion is widely used in geomaterials to capture the plastic compressibility and the asymmetry between tension and compression 0), the searched macroscopic criterion should reduce to the Mises-Schleicher For a porous material with a Drucker-Prager type matrix, the matrix in Fig criterion (Eq (19)): obeys the following criterion: σ 02 Σ + 3α m − = σ0 (27) some special cases (hydrostatic loadings, the porosity f → 0, and the     Σ  lim  2Γ cosh  A ln 1 − 3α m   − − Γ  = −1 f →0 σ        (28) With this property, the expression of the searched macroscopic criterion is proposed as follows: (2 9) and the compressibility of the matrix α on the deviatoric stress Σeq When f → f →0 (4) Case of Σm = for the determination of the parameters B and C The expressions of the parameters B and C are determined by the property of continuity between tension and compression Unlike the case of purely hydrostatic loadings, we not have the exact solution under the deviatoric loading which is difficult to be obtained analytically To this end, the result of Σeq/σ0 = − f proposed by Gurson’s criterion will be adopted here as an = B (1 − Γ ) TE D approximation The expression of Σeq/σ0 obtained from the macroscopic criterion (Eq (29)) when Σm = is (30) The above special cases (1, and 3) should be considered at the same time during the determination of the parameters B and C: EP  Σ  lim  B − 3α C m  = 1, lim B = 1, lim C = α →0 f →0 f →0 σ0   (31) With Eqs (30) and (31), considering the continuity between tension and AC C (1 − f )2 1− f , C= (1 − Γ )2 (1 − Γ )2 macroscopic criterion for a hollow sphere with a Drucker-Prager type matrix by means of limit analysis techniques In order to obtain a closed-form expression, the implicit criterion was     1 − 3α  Σ eq σ0 Σm / σ0 1− s /2 (1 − f )      + f cosh γ −1 ln  − 3α Σ m   − − f =  σ       (36) where γ= 2α , s = + 2α sign(Σ m ) 2α + sign(Σ m ) On the other hand, a porous material with a Drucker-Prager matrix was studied in Maghous et al (2009) by the modified secant moduli procedure The Drucker-Prager criterion reads Φ m = σ d + T (σ m − h) ≤ (37) where σ d = σ ′ : σ ′ is the equivalent stress (with σ% ′ = σ% − σ% m ), h represents the hydrostatic tensile strength, and T denotes the frictional coefficient By using the modified secant moduli method, interpreted by Suquet (1995) (see also Ponte Castaneda and Suquet (1998)) as equivalent to the variational compression, the expressions of the parameter B and C can be obtained: B=  Σ eq2  1 + 2α sign(Σ m )  Σ   Σ  +  − 3α m   f cosh  ln  − 3α m   − − f  =0 2 σ α σ σ0          (35) Recently, Guo et al (2008) obtained an implicit expression of the approximated by 0, we should find lim B = and lim C = f →0 compressible parameter α → 0): M AN U   Σ m   A ln  − 3α  − − Γ = σ     The parameters B and C were introduced to consider the effects of porosity f Σ eq2 σ 02 (34) Jeong (2002) proposed a heuristic macroscopic criterion by considering When f → 0, the limitation of the left hand side of Eq (24) is Σ eq2 / σ 02 + 2Γ B − 3α C Σ m / σ Φ m = σ eq + 3ασ m − σ ≤ SC Σ eq2 RI PT When the void volume fraction of the porous material becomes zero (f → (32) method of Ponte Castaneda (1991), Maghous et al (2009) succeeded in deriving a macroscopic criterion in the following form: 1+ f /  f  F (σ , f , T ) = σ d +  − 1 σ m2 + 2(1 − f )hσ m − (1 − f )2 h = (38) T2  2T  Taking into account these special conditions and requirements, the searched The representation of this elliptic criterion corresponds to a closed surface macroscopic yield function of a porous medium with a Mises-Schleicher It presents a tension-compression asymmetry which is a manifestation of the matrix can be proposed as follows: Drucker-Prager type matrix Note that the criterion (Eq (4)) established by Φ= Σ eq2 / σ 02 1− f Σm  1− f    − 3α (1 − Γ ) σ 1− Γ    Σ  + 2Γ cosh  A ln 1 − 3α m   − − Γ = σ     (33) Compared with the Gurson’s criterion (Eq (2)), the macroscopic criterion Ponte Castaneda (1991) is retrieved by Eq (38) in the limit of a von Mises solid phase Macroscopic criteria of ductile porous materials with two populations of voids (Eq (33)) has a similar expression It is worth to notice that the criterion (Eq (33)) is analogous to the heuristic one used in Nahshon and Hutchinson (2008) Porous materials with one population of pores have been studied in Section (see also Tvergaard, 1981; Tvergaard and Needleman, 1984) The damage In reality, different sizes of pores can be found in porous geomaterials, e.g parameter Γ was introduced, which replaces the porosity f in Gurson’s criterion and plays an important role in the macroscopic behaviors (asymmetry between tensile and compressive loadings) Γ is a function of the plastic chalk, concrete, and argillite Many works focus on this class of porous materials (see for instance Talukdar et al., 2004; Vincent et al., 2008, 2009, 2014a, b; Shen, 2011; Alam et al., 2010; Cariou et al., 2013; Shen and Shao, ACCEPTED MANUSCRIPT 2016a, b) To this end, these double porous materials will be investigated in The macroscopic criterion for a double porous medium having a von Mises this section Explicit macroscopic criteria will be established which take into solid phase at the microscale can be retrieved from Eqs (40) and (41) by account different influences of small and large pores on the macroscopic considering the following properties: T → 0, h → ∞ and Th = / 3σ mechanical behavior In many engineering structures, the pore pressure due to the saturation has a For this purpose, the schematization of the studied double porous material great influence on the macroscopic behaviors of porous geomaterial To containing two populations of spherical voids is illustrated in Fig The voids consider this effect, Eqs (40) and (41) were generalized in Shen et al (2014a) in these two populations are assumed to be spherical and isotropically for the double porous material saturated by two different pore pressures at the distributed Different scales will be considered: macroscale, mesoscale and microscale and mesoscale The effective stress problems are also discussed microscale The separation between the two scales of the voids allows By considering Eshelby-like trial velocity fields in the limit analysis approach performing a two-step homogenization for the transition from mesoscale to macroscale, a further improvement has been made in Shen et al (2014b) RI PT In Fig 2, we denote |Ω| the total volume of the RVE, Ω1 the domain occupied by the voids at the microscale (the smallest scale), and Ω2 the one of 3.2 Macroscopic criterion of double porous material obtained by the the voids at the mesoscale (the intermediate scale) With these notations, the modified secant method micro-porosity f and the meso-porosity φ can be expressed as | Ω1 | |Ω | f = ,φ= | Ω − Ω2 | |Ω | order to account for the effect of porosity φ at the mesoscale In this section, The kinematical limit analysis approach has been adopted in Section 3.1 in (39) an alternative method will be used for this second homogenization procedure to establish a macroscopic criterion of double porous material with a Drucker- SC Prager type matrix at the microscale For the first homogenization shown in Fig 2b, the criterion (Eq (38)) proposed by Maghous et al (2009) will be adopted for the effects of micro- r a Porous matrix porosity f and the compressibility of the solid phase Based on this criterion M AN U and an associated flow rule, one obtains the plastic strain rate of the porous matrix (Fig 2b): b d= (a) Meso-porosity φ (b) Micro-porosity f Fig Double porous media with spherical voids plastic strain rate tensor d For various porous geomaterials, polymers or other pressure- sensitive materials, plastic compressibility is one of the salient features that must be As the elastic domain delimited by the yield function F(σ, f, T) is convex and closed in the stress space, the support function πmp can be readily expressed as π mp = (1 − f ) h TE D taken into account To this end, the solid phase at the microscale is assumed to obey a Drucker-Prager type criterion (Eq (37)) In order to consider the effects of the micro-porosity f and the meso-porosity φ on the effective behavior of double porous material and to derive a closed-form expression of the macroscopic strength criterion, a two-step homogenization procedure (from microscale to mseoscale and from mesoscale to macroscale) will be EP adopted (42) where d d = d ′ : d ′ , and d ′ = d − dm denotes the deviatoric part of the 3.1 Double porous media with a Drucker-Prager solid phase at the microscale d d 1 + f / T2 2(1 − f )h   3f  2σ 2σ ′ +  − 1 m + 1 + f / 2σ d  T  2T   Concerning the first homogenization from microscale to mesoscale (see Fig 2b), the criterion (Eq (38)) obtained by Maghous et al (2009) for one population of pores is adopted directly The influence of micro-porosity f and AC C compressibility in the solid phase at the microscale is considered in Eq (38) Furthermore, this criterion has an elliptic expression which is similar to Eq (6) 3f T2 1+ f / 2T d d2 + d − (1 − f ) h dv 2 v f − 2T + f / 3 f / −T f − 2T (43) where dv = trd is the volumetric deformation in the porous matrix The stress-strain relationship derived from the support function πmp can be put in the following form which requires the secant bulk and shear moduli and introduces a hydrostatic prestress σp1:      mp κ    + f / 3 f T  M = dd2 + d , N = (1 − f ) h v 3f / −T f − 2T + f /  ∂π mp = 2µ mp d ′ + κ mp d v + σ p ∂d 1+ f / N N 2T = , 2µ mp = , σ p = −(1 − f )h M 3f / −T M f − 2T σ mp = (44) used in Section 2.2 Taking advantage of this property, the result (Eq (18)) in Note that the secant moduli µ mp and κmp in Eq (44) are non-uniform Section 2.2 obtained by means of limit analysis approach can be used to because they depend on the non-uniform strain rate tensor d As in Maghous et describe the second homogenization from mesoscale to macroscale (see Fig al (2009), the effective strain rate deff will be considered, which is an 2a) The meso-porosity φ is taken into account in this step Finally, the appropriate average of d over the porous matrix to capture the effect of loading macroscopic criterion for a double porous material with a Drucker-Prager history on the nonlinear plastic properties Therefore, the effective moduli κmp solid phase at the microscale is given by and µ mp depend on the macroscopic strain rate D of the double porous matrix β Σ eq2 σ 02 + L / (18α ) 3 β 2φ cosh    The effective volumetric and deviatoric strain rates of the porous matrix are + 9α  Σ m − L(1 − φ ) / (9α )  +  σ + L2 / (18α )     Σm  −1− φ = 2 σ + L / (18α )  defined as dveff = < d v2 >Ω mp , ddeff = < dd2 >Ω mp (40) the properties, T and h, of the solid phase: + f / 9α f , = − 1, L = −2(1 − f )h, σ = (1 − f )h β= T2 2T from which the state equation can be rewritten as p σ mp = C mp ( d veff , d deff ) : d + σ eq where the parameters β, α, L and σ0 are the functions of micro-porosity f and C mp ( d veff , d deff p =σp σ eq (41) (45) )= mp 3κ eq J + mp 2µeq K      (46) The homogenized stress-strain relations of the double porous medium can ACCEPTED MANUSCRIPT be expressed in the following form: In order to describe the macroscopic behavior of the Vosges sandstone Σ = C hom : D + Σ p   hom eff eff hom mp mp hom mp mp C (d v , dd ) = 3κ (κ eq , µeq )J + 2µ (κ eq , µeq )K  taking into account the effects of porosity and the asymmetric between tension (47) where Σ is the stress tensor at the mesoscale in the double porous matrix, and Σ denotes the mesoscopic prestress which can be deduced from the Levin’s p mp p / κ eq )σ eq the plastic hardening of the solid phase is considered via the evolution of σ0 The thermodynamic potential of the double porous medium reads: W = D : C hom : D + Σ p tr D (48) The corresponding state equations are easily deduced as (see also Eq (47)): Σm p  σ eq = κ hom  Dv + mp  κ eq    , Σ d = µ hom Dd   (49) the double porous medium are related to D and given by p (1 − φ )( ddeff ) = ∂κ mp ∂µeq hom p   σ eq κ hom p  Dv + mp  − mp σ eq Dv   κ eq  (κ eq )   ∂µ hom  + Dd mp  ∂κ eq  p   σ eq ∂µ hom  Dv + mp  + Dd mp   κ ∂µeq eq    +            of the Hashin-Shtrikman upper bounds: κ µ hom mp 4(1 − φ ) µeq   mp mp + 4µeq 3φκ eq   mp mp (1 − φ )(9κ eq + 8µeq )  mp = µeq  mp mp κ eq (9 + 6φ ) + µeq (8 + 12φ )  mp = κ eq (50) +  Σ   2Γ cosh  A ln 1 − 3α m   − − Γ = σ    (54) (55) can be determined by the plastic consistency condition: ∂Φ & ∂Φ & ∂Φ & :Σ + :f + :σ = Φ& ( Σ , f , α , σ ) = ∂Σ ∂f ∂σ (56) compatibility: ∫Ω tr d p dΩ (57) m p where d is the microscopic plastic strain rate tensor According to the yield criterion (Eq (19)) of the matrix, adopting the relationship (1 − f )σ : d p = Σ : D p , the porosity evolution law can be rewritten: TE D  + f /  6T − 13 f − 3/ 2+ f 3f Θ= φ + 1 , ϒ = φ + −1  − − T2 T 12 f T2 2T   This closed-form plastic yield function explicitly depends on the two EP porosities (f and φ) at two different scales Applications 1− f Σm  1− f    − 3α (1 − Γ )2 σ 1− Γ  Ω (52) where Σ eq2 / σ 2 f& = (1 − f ) tr D p − Φ ( Σ , φ , f , T ) = ΘΣ d2 + ϒΣ m2 + 2(1 − f )h (1 − φ )Σ m − (1 − f ) (1 − φ ) h = becomes The variation of the porosity f can be determined from the kinematical (51) different scales takes the following form: (53) where Dp denotes the macroscopic plastic strain rate The plastic multiplier λ& Combining Eqs (44), (49), (50) and (51), the macroscopic yield criterion of σ = σ + [ H (ε eqp )]m Adopting the normality rule, the plastic flow is given by ∂Φ D p = λ& ∂Σ the double porous material with two populations of spherical cavities at two sandstone: Φ (Σ, f , α, σ ) = The homogenized secant moduli κhom and µ hom are evaluated with the help hom phase An isotropic hardening is considered in the following form for the M AN U κ hom   σ eq 1 1 − φ − mp   mp  κ eq   κ eq p used in Eq (33), as a function of the equivalent plastic strain ε eq in the solid With this plastic hardening parameter, the macroscopic criterion (Eq (33)) Following Dormieux et al (2002), the effective strain rates associated with 1 ∂κ hom (1 − φ )(d veff ) = mp 2 ∂κ eq constitutive plastic model For most porous geomaterials, a significant plastic hardening can be observed in experimental phenomenon In this application, RI PT hom 2.3 will be applied We aim now at formulating and implementing a complete SC theorem Σ = (κ p and compression, the macroscopic criterion (Eq (33)) established in Section AC C Some recently established macroscopic strength criteria have been presented in Sections and for different types of porous geomaterials having one or two populations of pores The effects of porosities at different scales and the influence of the matrix compressibility on the macroscopic mechanical behaviors are explicitly taken into account In this section, we will provide some applications of these strength criteria to describe the overall responses of porous geomaterials 4.1 Application of the macroscopic criterion (Eq (33)) to sandstone As a first application, a typical porous geomaterial “Vosges sandstone” will be considered This material comes from the Vosges mountains in France, f& = (1 − f ) tr Dp − 3α Σ : Dp 2σ − 3αΣ m / (1 − f ) (58) p The microscopic equivalent plastic strain rate ε&eq of the matrix is calculated by p = ε&eq Σ : Dp Σ σ − 3ασ m 1− f   (1 − f ) σ − ασΣ m / (1 − f )    (59) Substituting Eqs (58) and (59) into Eq (56) yields          ∂Φ Σ :   ∂Φ ∂Φ ∂Φ  ∂Φ ∂Σ  (1 − f ) − HG = :C : − − 3α Σ m   ∂Σ ∂Σ ∂f  ∂Σ m 2σ − 3α  − f    ∂Φ  Σ: Σ ∂Φ ∂σ  ∂Σ σ − 3ασ m  p ∂σ ∂ε eq 1− f  Σ   (1 − f )  σ − ασ m  1− f    ∂Φ :C : D & λ = ∂Σ G H (60) The tangent elastoplastic operator of the studied porous material can be which has been studied in a series of experimental investigations (Khazraei, determined as follows: 1996; Shao and Khazraei, 1996; Besuelle et al., 2000) The Vosges sandstone typical porous quasi-brittle rock With the increase of confining pressure, its C   ∂Φ   ∂Φ   L = :C  C : ⊗ ∂Σ   ∂Σ C −    HG mechanical behavior exhibits a brittle-ductile transition from low to high where Χ is the macroscopic elastic modulus tensor of the porous material By confining pressures (Menéndez et al., 1996; Besuelle et al., 2000) adopting a Mori-Tanaka homogenization scheme, the macroscopic elastic is mainly composed of quartz grains (93%), with a few percent of feldspar and white mica The porosity is about 22% This sandstone is considered to be a (Φ ≤ 0, Φ& < 0) (61) (Φ = 0, Φ& > 0) ACCEPTED MANUSCRIPT properties (κ, µ) are related to those of the matrix (κs, µ s) by 4(1 − f )κ s µs (1 − f ) µs , µ= κ= µs + f κ s + f (κ s + 2µs ) / (9κ s + 8µs ) ‐80 (62) ‐70 during loading Then the proposed micromechanical constitutive model is ‐50 implemented in a finite element code (ABAQUS) via a subroutine UMAT ‐40 For the purpose of applying to sandstone porous geomaterial, all the model ‐30 parameters should be identified According to the experimental data, the ‐10 of the porous sandstone With the relationship of Eq (62) and the porosity f = E33 (10-3) yield stress σ0 are identified from a triaxial compression test All calibrated ‐1 ‐120 m 0.27 SC ‐80 ‐60 experimental data and numerical simulations with different confining M AN U agreements can be found for both low and high confining pressures The axial and lateral strains are well predicted by the proposed associated model which 4.2 Application of the macroscopic criterion (Eq (40)) to chalk In the second application, a complete constitutive model will be established for a double porous material and applied to describing the mechanical behavior of the porous “Lixhe chalk” which is from the Upper Campanian age and drilled in the CBR quarry near Liège (Belgium) This material has been studied in a series of previous experimental investigations because its TE D mechanical behavior is qualitatively close to that of North Sea reservoir chalks (Schroeder, 2003) It is composed of more than 98% of CaCO3, less than 0.8% of SiO2 and 0.15% of Al2O3 The porosity of Lixhe chalk ranges from 42% to 44%, which is a highly porous rock According to Talukdar et al (2004) and Alam et al (2010), two kinds of porosity are observed in chalk The large porosity φ at the mesocale is about 5% and the small f is about 40% in the matrix The macroscopic plastic yield function (Eq (40)) presented in Section EP 3.1 will be applied to describing the elastoplastic behavior of this porous chalk The plastic hardening of the solid phase at the microscale is taken into account via the evolution of the frictional coefficient T as a function of the AC C − 1)] (63) where b, m, c and n are the parameters of the hardening law, which will be determined by simulating a hydrostatic compression test In order to better describe the plastic volumetric deformation, a nonassociated plastic flow rule is proposed Inspired by the work of Maghous et al (2009) and Shen et al (2012a, 2013a), the following function is adopted as the macroscopic plastic potential by taking a similar form as the yield function: Simulation Experiment ‐40 pressures (10 MPa and 40 MPa) are illustrated in Figs and Good is able to capture the main mechanical features of the porous sandstone Σ11-Σ33 (MPa) ‐100 Using the same set of values given in Table 1, comparisons between p nε eq ‐7 ‐140 Table Typical values of parameters for the proposed model Es (GPa) H νs σ0 (MPa) α 30 0.3 50 p m T = T0 [1 + b(ε eq ) + c (e ‐5 Fig Simulation of a triaxial compression test on Vosges sandstone with 10 MPa confining pressure parameters are given in Table equivalent plastic strain: ‐3 E11 (10-3) RI PT be calculated The corresponding Young’s modulus Es = 30 GPa and the Poisson’s ratio νs = 0.3 The plastic compressible parameter α and the initial Simulation Experiment ‐20 macroscopic elastic properties (κ, µ) can be calculated from the elastic regime 0.22, the compressible modulus κs and the shear one µ s of the solid matrix can Σ11-Σ33 (MPa) ‐60 The macroscopic elastic properties depend on the evolution of porosity E33 (10-3) ‐20 ‐2 ‐7 E11 (10-3) ‐12 Fig Simulation of a triaxial compression test on Vosges sandstone with 40 MPa confining pressure 2   2(1 − f ) hT    Σ eq   f    − Tt    Σ m + (1 − φ )  Σ  +   +  f − 2Tt  Σ0            Σ  + f m  2φ cosh   (64) Σ     3f  Σ = (1 − f ) hT  f − 2Tt  2 4f G= + 3 where the parameter t is the dilatancy coefficient which controls the transition between volumetric contractance and dilatancy under deviatoric loading As the rate of volumetric dilatancy generally varies with plastic deformation history, for simplicity, the same form as Eq (63) is adopted for t: p m t = t0 [1 + b (ε eq ) + c (e p nε eq − 1)] (65) According to the macroscopic potential (Eq (64)), the plastic flow rule is given by ∂G D p = λ& ( Σ , f , φ , T , t) ∂Σ (66) The equivalent plastic strain of the solid phase is given by p ε&eq = Σ : Dp  (t − T )Σ m  (1 − f )(1 − φ ) Th + (1 − f )(1 − φ )   (67) The evolutions of micro-porosity f and meso-porosity φ are determined as & & &  &f = (1 − f ) Ω m + Ω1 − (1 − f ) Ω m  Ω m + Ω1 Ωm  (68)  Ω& m + Ω&1 + Ω& Ω& m + Ω&1  & φ = (1 − φ ) − (1 − φ )  Ω m + Ω1 + Ω Ω m + Ω1  Then the proposed micromechanical constitutive model is implemented in a finite element code (ABAQUS) via a subroutine UMAT Before it is applied to describing the macroscopic behavior of the Lixhe chalk, we need to determine the elastic and plastic parameters of the proposed non-associated model The plastic parameters are determined by simulating the hydrostatic ACCEPTED MANUSCRIPT compression test (see Fig 5) The typical values obtained are given in Table ‐40 techniques (Abou-Chakra Guéry et al., 2008; Shen et al., 2012a, 2013a; Shen ‐35 and Shao, 2015b) have been proposed for COx argillite In this study, the ‐30 effects of two populations in the clay matrix at the microscale and the ones of ‐25 minerals grains (calcite and quartz) at the mesoscale will be taken into account The RVE of the studied heterogeneous material is schematized in Fig 11 ‐20 Experiment Simulation ‐15 ‐14 ‐10 ‐12 ‐5 -1.5 -2.5 -3.5 -4.5 Volumetric strain (%) -5.5 ‐8 -6.5 ‐6 Fig Simulation of a hydrostatic compression test on Lixhe chalk ‐4 ‐2 n 38 0.5 E33 (%) The proposed non-associated model now is used to describe the mechanical behaviors of the Lixhe chalk The simulation of a hydrostatic compression test M AN U simulations are now performed for triaxial compression tests with low and ‐16 high confining pressures in order to verify the capacity of the proposed model ‐14 Fig shows the comparison between experimental data and numerical ‐12 results for low confining pressure of MPa The material fails with a small ‐10 value of the axial strain The cases of high confining pressures (14 MPa and ‐8 17 MPa) are shown in Figs and 8, respectively One can see a good ‐6 agreement between the numerical simulation and experimental data The ‐4 TE D ‐2 Experiment Simulation ‐2 proposed non-associated model is able to capture the main aspects of the meso-porosity φ as functions of axial strain during triaxial compression tests ‐1.5 ‐18 to the confining pressure With the same values given in Table 2, numerical Figs and 10 show the evolutions of micro-porosity f in the matrix and of ‐1 E11 (%) ‐20 simulation and experimental data The chalk behavior is significantly sensitive microstructure ‐0.5 Fig Simulation of a triaxial compression test on Lixhe chalk with MPa confining pressure is shown in Fig A good agreement is found between the numerical mechanical behaviors of Lixhe chalk by considering the effects of Experiment Simulation SC Es (GPa) 12 Table Typical values of parameters for the non-associated model t0 h b m c T0 νs 0.2 0.2 0.16 32 1.3 0.03 0.018 Σ11-Σ33 (MPa) -0.5 RI PT ‐10 Σ11-Σ33 (MPa) Hydrostatic stress (MPa) models (for instance Chiarelli et al., 2003; Shao et al., 2006; Hoxha et al., 2007) and micromechanicals constitutive models based on different homogenization E33 (%) -1 -3 -5 -7 E11 (%) Fig Simulation of a triaxial compression test on Lixhe chalk with 14 MPa confining pressure with different confining pressures (4 MPa, 14 MPa and 17 MPa) We can see ‐25 that the evolutions of f and φ are different These changes of the microstructure affect the macro-scopic plastic criterion (Eq (40)) The macroscopic ‐20 ‐15 4.3 Application of the macroscopic criterion (Eq (52)) to COx argillite with the effects of inclusions ‐10 Σ11-Σ33 (MPa) of porosities EP mechanical behavior of the Lixhe chalk is clearly sensitive to these two types AC C In the above sections, the effects of porosity have been taken into account on Experiment ‐5 the macroscopic behavior of porous geomaterial For many materials used in engineering structure, for instance concrete and rocks, complex microstructure with mineral inclusions, granular aggregates and pores can be found Taking the COx argillite for example, the effects of pores at different scales and the influences of inclusions will be both considered Because of its low permeability and high mechanical strength, the COx argillite is considered as a potential geological barrier According to Andra (2005) and Robinet (2008), the COx claystone is composed of 40%–50% of clay minerals, 20%–27% of calcite grains, 23%–25% of quartz grains and 5%–10% of minor minerals The porosity of COx claystone varies from 11.04% to13.84% The average pore size (tens of nm) is significantly smaller than that of calcite and quartz grains The majority of pores are located inside the clay matrix At the microscale, the porous clay matrix can be treated as an assembly of clay particles and inter-particular pores Furthermore, the clay particle contains also small voids (intra-particle pores) between clay sheets In order to describe elastoplastic and damage behaviors, some phenomenological Simulation E33 (%) -1 -3 E11 (%) -5 -7 Fig Simulation of a triaxial compression test on Lixhe chalk with 17 MPa confining pressure ACCEPTED MANUSCRIPT 0.41 inclusion phase and inclusion phase 2, respectively With these notations, the porosity f at the intra-particular scale (Fig 11c) and the one φ at the inter- MPa 14 MPa 17 MPa 0.40 0.39 particular scale (Fig 11b) of the porous matrix, as well as the volumetric fractions of inclusions (ρ1, ρ2) can be calculated: 0.38 f Ω I1 Ω  , ρ = I2  Ω Ω   Ωb φ=  Ωm + Ωs + Ωb   Ωs f =  Ω m + Ωs  ρ1 = 0.37 0.36 0.35 0.34 -0.02 -0.04 -0.06 E11 RI PT As shown in Fig 11a, the COx argillite is made of a porous clay matrix and mineral grains of calcite and quartz at the mesoscale The criterion (Eq (52)) obtained in Section 3.2 will be applied to describing the effective behaviors of the clay matrix considering the two populations of pores (Fig 11b and c) and the compressibility The influences of calcite and quartz grains on the macroscopic plastic behavior will be estimated by using an incremental approach proposed by Hill (1965) The advantage of this approach is the possibility to consider several families of mineral inclusions and also complex SC MPa 14 MPa 17 MPa loading path due to its incremental form 4.3.1 Principle of incremental approach In the incremental approach, the local stress rate σ& of each constituent -0.01 -0.02 -0.03 E11 -0.04 -0.05 -0.06 phase is related to the local strain rate ε& by the local tangent stiffness operator L ( z ) : σ& ( z ) = L ( z ) : ε&( z ) M AN U φ Fig Evolution of micro-porosity f with different confining pressures 0.0505 0.05 0.0495 0.049 0.0485 0.048 0.0475 0.047 0.0465 0.046 0.0455 0.045 (69) Fig 10 Evolution of meso-porosity φ with different confining pressures We denote Ω as the total volume of RVE (Fig 11), Ωs the domain (70) At each iteration step, it is possible to introduce a tangent localization tensor A ( z ) to relate the local strain rate ε&( z ) to the macroscopic strain rate E& : occupied by the intra-particle pores, and Ωb the one occupied by the interparticle pores Ωm is the volume of the solid phase in particles At the mesoscopic scale, ΩI1 and ΩI2 denote the volumetric domains of mineral Inter-particle pores TE D matrix (a) Intra-particle pores Particle (c) (b) ε&( z ) = A ( z ) : E& EP Fig 11 RVE of studied material with double porosity and inclusions (71) The rate form of the macroscopic constitutive relations can be written in the AC C general form: Σ& = L hom : E& , L hom =< L ( z ) : A ( z ) > where Αr is the tangent localization operator corresponding to the average strain per phase ε&r With the matrix-inclusion morphology at the mesoscale, the Mori-Tanaka (72) scheme (Mori and Tanaka, 1973) is considered for the evaluation of the where Λhom denotes the effective tangent stiffness operator which is obtained tangent strain localization operator: as the average over the RVE of the product of the local tangent stiffness and  N  A r = [I + P Ir0 : (L r − L )]−1 :  ∑ f s A s0   s =0  tangent strain localization tensor on all constituent phases In general, the local tangent stiffness is not uniform in each phase due to the material heterogeneity As a consequence, it is not possible to provide a closed-form expression of its average Some simplifications are then needed to −1 (75) where Ι denotes the fourth-order unit tensor, I ijkl = (δ ik δ jl + δ il δ jk ) / with δij the Kronecher’s symbol; and fs is the volumetric fraction of the phase (s) It is worth recalling that the Hill tensor P Ir0 in Eq (75) depends on the make the homogenization procedure computationally operable For this inclusion form Ir, its orientation as well as the tangent stiffness of the matrix purpose, at any point, z , of a given phase (r), the local constitutive relation is phase in the linear comparison composite Λ0 As the calcite and quartz grains approximated by ∀z ∈ (r ), σ& ( z ) = L r : ε&( z ) have the same morphology, one notices that P I01 = P I02 = PI0 PI0 is related to (73) The local tangent operator Λr is evaluated at a suitable reference strain state, Elshelby tensor ΣE through: P I0 = S E (L ) : L 0−1 (76) which is classically taken as the average value of local strain field in the phase As the tangent stiffness depends on the direction of plastic deformation, the (r) This implies that the tangent stiffness is uniform in each phase local tangent operator Λ0 is inherently anisotropic in nature This implies that Accordingly, the incremental form of the strain localization relation is in the general case Hill tensor P Ir0 can be estimated only numerically Once simplified as follows: ε& = A : E& the Hill tensor and strain localization operator are calculated, the macroscopic r r (74) tangent stiffness tensor can be easily deduced: ACCEPTED MANUSCRIPT and and evaluate the error R: N L hom = ∑ f r L r : Ar (r=0, 1, 2) (77) r =0 where fr is the volumetric fraction of the phase (r) R1i := A1i : ∆E − ∆ε1i (88) R2i := A 2i : ∆E − ∆ε 2i (89) The incremental method has been applied to modeling nonlinear behaviors If || R1i ||< tolerance and || R2i ||< tolerance 2, then compatibility is reached of various heterogeneous materials (see for example Doghri and Ouaar, 2003; Otherwise, an additional iteration is needed until the convergence criterion is Chaboche et al., 2005; Abou-Chakra Guéry et al., 2008; Shen et al., 2012a), verified and one obtains this method generally leads to a too stiff macroscopic response Various ∆ε1i +1 = ∆ε1i + R1i (90) techniques have been proposed to improve numerical predictions and to obtain ∆ε 2i +1 (91) technique has been introduced and largely used by various authors The efficiency of this technique has also been confirmed for clayey rocks by AbouChakra Guéry et al (2008), Shen et al (2012a) and Chen et al (2013) Therefore, for the present work, we adopt again the general scheme proposed by Bornert et al (2001) to perform the isotropization of a fourth-order tensor = + R2i (10) The use of Mori-Tanaka scheme leads to the determination of macroscopic tangent stiffness tensor: L hom := [ f L + f1L : A10 + f L : A 20 ] : A RI PT softer macroscopic responses Among those techniques, the isotropization ∆ε 2i (92) so that the macroscopic stress tensor can be calculated as ∆Σ n+1 = L hom : ∆E n+1 (93) Like most geomaterials, the COx claystone exhibits a transition from The isotropic part of the tangent stiffness tensor of the clay matrix is given by (78) L iso ( K :: L ) K = 3κ t J + µ t K = ( J :: L ) J + volumetric compressibility to dilatancy during plastic deformation In order to Due to the isotropic character of L iso and the fact that the calcite and quartz plastic flow rule Inspired by the previous work of Maghous et al (2009) and grains are assumed to be of spherical form, a closed-form expression of the based on the plastic criterion (Eq (52)), we propose the following non- Eshelby tensor can be obtained: 3κ t κ t + 2µ t S E (L iso J + K ) = 3κ t + µ t 3κ t + µ t associated plastic potential for the double porous clay matrix: −1 P Ir0 = S E ( L iso ) : L0 ( r = 1, 2) (80) 4.3.2 Numerical implementation SC Accordingly, the Hill tensor P Ir0 in Eq (76) is replaced by 3f   3/ 2+ f  G ( Σ% , φ , f , T , t ) = ΘΣ%d2 +  φ+ − 1 Σ%m2 + 2(1 − f ) h(1 − φ )Σ%m  2Tt   Tt    + f /  6Tt − 13 f −  Θ= φ + 1   Tt  4Tt − 12 f −   (94) M AN U (79) capture such a transition, it is generally necessary to use a non-associated where the dilatancy coefficient t defines the current plastic volumetric strain The proposed micro-macro model is implemented in a standard finite element code (ABAQUS) as a UMAT subroutine We present here the numerical scheme for the local integration of the model at each Gauss point rate As for most porous materials, the porous clay matrix also exhibits a plastic hardening process, firstly due to the evolution of porosities However, according to some previous studies (Abou-Chakra Guéry et al., 2008; Shen et al., 2012a), the current friction coefficient of the solid phase T and the En + ∆E, while the strain at the step n is known and the strain increment ∆E is dilatancy coefficient t vary with the plastic deformation A complementary given The problem to be solved here is to find the corresponding macroscopic plastic hardening law is necessary for the solid phase of the clay matrix to stress state at the end of the loading step by using the incremental consider the strength and the volumetric compressibility-dilatancy transition homogenization method presented above The following numerical scheme is Thus we consider here the frictional coefficient T and the dilatancy one t as adopted: functions of the equivalent plastic strain εp: TE D The loading path is divided into a limit number of steps At the step n + 1, the material point at the macroscale is subjected to a macroscopic strain En+1 = (1) Input data: En, ∆E; Phase clay matrix: ε0 n , εnp and ε0pn ; Phase calcite grains: ε1n ; Phase quartz grains: ε2n macroscopic strain increment: ∆ε10 = ∆E , ∆ε 20 = ∆E EP (2) Initially, the local strain increments in phases and are set equal to the (3) In phases and 2, the values of ∆ε1i and ∆ε2i AC C t = tm − (tm − t0 )e p (95) − b2ε p (96) where T0 (or t0) is the initial yield threshold, and Tm (or tm) is the asymptotic value of the frictional coefficient (81) Using the non-associated plastic potential defined above, the plastic flow are known, so one obtains rule in the clay matrix is given by ∂G % D% p = λ& (Σ, f , φ, T , t ) (97) ∂Σ% where D% p denotes the plastic strain rate in the clay matrix at the mesoscale ε1n+1 , ε n+1 and L 1i , L i2 which are the local stiffness tensors of phases and 2, respectively (4) The average local strain in the clay matrix is given by ∆ε0i := T = Tm − (Tm − T0 )e − b1ε ∆E − f1ε1i − f ε2i − f1 − f (82) (5) At the iteration i for the phase 0, the values of ∆ε0i , ε0 n , ε np , ε0pn are known, so one can compute ε0 n+1 , εnp+1 , ε0pn+1 and L i0 (6) The Hill tensor is then numerically evaluated by Eq (80) (7) The tensors A10,i an d A 20,i are given by A10,i := [ I + P I01 : ( L 1i − L i0 )]−1 (83) A 20,i := [ I + P I02 : ( L i2 − L i0 )]−1 (84) (8) It is now possible to evaluate the strain localization tensors A ri for each phase: The equivalent plastic strain of the solid phase in the clay matrix is given by Σ% : D% p (98) ε& = (1 − f )(1 − φ ){Th + ( t − T ) Σ% / [(1 − f )(1 − φ )]} p m The void evolution laws can be derived as classically from the energy compatibility condition: Ω& + Ω& s Ω& f& = (1 − f ) m − (1 − f ) m Ω m + Ωs Ωm & + Ω& + Ω& Ω Ω& + Ω& s s b φ& = (1 − φ ) m − (1 − φ ) m Ωm + Ωs + Ωb Ωm + Ωs (99) 4.3.3 Experimental verification The proposed non-associated micromechanical constitutive model is now A 0i := ( f I + f1A10,i + f A 20,i ) −1 (85) applied to reproducing the macroscopic responses of the COx argillite in A1i := A10,i : A 0i (86) uniaxial and triaxial compression tests performed on samples with different (87) mineralogical compositions The elastic and plastic parameters are identified (9) Check the compatibility of local strains between two iterations for phases and given in Table by using the same strategy as that proposed in Shen et al A 2i := A 20,i : A 0i ACCEPTED MANUSCRIPT (2012a) The plastic parameters are identified from a unixial compression test 16 and 17, respectively, for samples at the depth of 482.2 m with different on a sample from the depth of 466.8 m with 51% of clay matrix, 26% of confining pressures It is worth to notice that even if the variation of porosity calcite and 23% of quartz Then the same group of parameters is used for all remains quite small, the plastic behavior of the clay matrix is significantly other uniaxial and triaxial compression tests with different confining pressures affected by the presence of inter-particular and intra-particular porosities (5 MPa and 10 MPa) because of the dependence of the yield function and plastic potential on the Fig 12 shows the comparison between experimental data and numerical both porosities results for uniaxial compression test on the sample with 51% of clay matrix, 26% of calcite and 23% of quartz The cases of triaxial compression tests (5 Conclusions MPa and 10 MPa) with different mineralogical compositions are illustrated in The present study has concerned the micromechanics-based constitutive Figs 13-15, respectively Good agreements are found for both the axial and COx argillite are well predicted, such as pressure dependency, transition from volumetric compressibility to dilatancy with the increase of deviatoric stress, effects of both inter-particular and intra-particular pores, as well as the influence of volume fraction of mineral grains The evolutions of inter- and inclusion on the macroscopic behavior These micro-macro models take into account the effects of microstructure and significantly improve the phenomenological ones For simplicity, porous materials containing one p o p u E11 Σ 11 (MPa) -30 -25 -20 xxxxxxxxxxxxxxxx -15 ep s1 Simulation -10 0.5 TE D -0.5 E (%) -1 -1.5 Fig 12 Comparison between the numerical simulation and the experimental data of uniaxial compression test f0 = 51%, f1 = 26%, and f2 = 23%, where f0, f1 and f2 are the contents of clay matrix, calcite and quartz, respectively -40 -35 E11 AC C -30 Ev EP -45 E33 Σ11-Σ 33 (MPa) -50 r e Inter-porosity, φ (%) Intra-porosity, f (%) 23.75 1.6 s 0.5 -45 Ev E11 -40 -35 -30 -25 -20 -15 xxxxxxx xxx Simulation xxx Série1 Experiment -10 -5 0 -0.5 -1 -1.5 -2 E (%) Fig 14 Comparison between the numerical simulation and the experimental data of triaxial compression test with the confining pressure of 10 MPa f0 = 55%, f1 = 23%, and f2 = 22% with different types of matrices (von Mises, Green type, Mises-Schleicher and implemented to describe the macroscopic responses of typical porous Simulation xxxxxxxxxxxxxx geomaterials (sandstone, porous chalk and argillite) The accuracies of these Experiment Série1 micromechanics-based constitutive models are assessed and validated by comparisons between numerical predictions and experimental data with different confining pressures or different mineralogical composites -1 -1.5 -2 E (%) Fig 13 Comparison between the numerical simulation and the experimental data of triaxial compression test with the confining pressure of MPa f0 = 44%, f1 = 33%, and f2 = 23%, o macroscopic criteria, complete constitutive models are formulated and -0.5 p assumed to obey to a Drucker-Prager type criterion Based on these -5 f populations of pores at different scales The solid phase at the microscale is -10 0.5 o extended to a more complex case of double porous media with two -20 n Drucker-Prager) are studied firstly for the macroscopic criteria Then, it is -25 -15 o -50 Experiment -5 i Σ11-Σ 33 (MPa) E33 t T0=0.05, Tm=0.9, b2=220 t0=-1, tm=0.3, b3=220, h=17 E33 Ev a M AN U -35 Plastic parameters l SC particular and intra-particular porosities in the clay matrix are shown in Figs Table Values of the parameters of the non-associated model for COx argillite Phase Material Elastic parameters E (GPa) ν Clay 5.027 0.33 Calcite 95 0.27 Quartz 101 0.06 -40 RI PT the volumetric strains The main features of the macroscopic behavior of the models for ductile porous geomaterials taking into account the effects of pore ACCEPTED MANUSCRIPT -45 -40 -35 -30 -25 Ev E11 The plastic multiplier The equivalent plastic strain in the solid phase κ, µ E, ν Σ11-Σ33 (MPa) E33 λ& p ε eq x x x Ι -20 v Simulation Série1 Experiment Série3 -15 Χ Λ -10 : -5 0.5 -0.5 -1 -1.5 The bulk and shear moduli, respectively Young's modulus and Poisson's ratio, respectively Scalar Vector Second-order tensor Second-order identity tensor Fourth-order identity tensor The trial velocity field The elastic stiffness tensor The tangent stiffness tensor Simple contraction of two tensors Double contraction of two tensors -2 RI PT -50 References E (%) Fig 15 Comparison between the numerical simulation and the experimental data of triaxial compression test with the confining pressure of 10 MPa f0 = 60%, f1 = 26%, and f2 = 14% Abou-Chakra Guéry A, Cormery F, Shao JF, Kondo D A micromechanical model of elastoplastic and damage behavior of a cohesive geomaterial International Journal of Solids and Structures 2008; 45(5): 1406–29 0.24 Alam MM, Borre MK, Fabricius IL, Hedegaard K, Røgen B, Hossain Z, Krogsbøll AS 482.2 m: MPa 0.239 Biot’s coefficient as an indicator of strength and porosity reduction: Calcareous 482.2 m: MPa sediments from Kerguelen Plateau 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(2011)” from the International Association for Computer Methods and Advances in Geomechanics (IACMAG) Professor Shao has developed a series of theoretical, experimental and numerical studies in mechanics of geomaterials He and his team have brought significant contributions to various topics, in particular, mechanics of saturated and partially saturated porous materials, damage mechanics and plasticity, thermo-hydromechanical and chemical coupling He is an international expert on multi-scale approaches for nonlinear behaviors and thermo-hydromechanical and chemical coupling problems His research results have been widely applied to various engineering fields such as petroleum industry, geological disposal of nuclear waste, sequestration of acid gas, hydraulic power engineering He was and is a principal investigator of more than thirty TE D national and international projects in France and in China He is author and co-author of more than 180 SCI peer-reviewed journal papers He has also edited books and contributed to 16 collective books He has received 2135 citations (1541 without self-citations) with an h-index 25 (Web of Science) and is ranked among the top 1% of authors in the field of engineering for the last ten years He is an editorial member of four top-level international journals (IJP, IJRMMS, COGE, NAG) and an associated editor of the AC C EP European Journal of Environmental and Civil Engineering (EJECE)