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Multiscale modeling approaches and micromechanics of porous rocks
Multiscale modeling approaches and micromechanics of porous rocks
10
Wanqing Shen and Jianfu Shao Lille University of Science and Technology, Villeneuve d’Ascq, France
10.1
Introduction
Most rocks are heterogeneous materials and their macroscopic behaviors are inherently related to their microstructures such as mineral compositions and porosity. Based on extensive experimental evidences and within the framework of irreversible thermodynamics, many phenomenological plastic and damage models have been developed. These models are generally able to capture various features of mechanical behaviors of rock-like materials, but they are not providing explicit relationships between macroscopic responses and microstructures. In order to develop an alternative modeling method and to improve phenomenological models, significant efforts have been undertaken during the last decades on the development of micromechanical models. In particular, different analytical and numerical homogenization techniques for composite materials have been successfully adapted for rock-like materials. Different micromechanical models have then formulated for the description of crack-induced damage and plastic deformation. It is not the objective of this chapter to give a comprehensive review of all micromechanical models so far developed for rock-like materials. As an example, we propose to present here a micromechanical model for modeling elastic-plastic behavior of porous rocks with two families of pores at two different scales. The kind of microstructure can be representative for a wide class of rocks, for instance clayey rocks, chalk, etc. It is possible to identify intraparticle pores inside mineral grains (plates) and interparticle pores between mineral grains. These two families of pores affect in a different way the macroscopic properties of material such as elastic modulus, plastic yield stress, and failure strength (Homand and Shao, 2000; Papamichos et al., 1997; Schroeder, 2003; De Gennaro et al., 2004; Alam et al., 2010; Xie and Shao, 2006). Further, due to the presence of pores, two plastic deformation processes can be generally identified, the plastic pore collapse under mean effective stress and the plastic shearing under deviatoric stress. The objective of micromechanical approach is to establish a macroscopic plastic criterion explicitly taking into account the effects of pores. As a pioneer work, Gurson (1977) proposed an analytical yield criterion using a kinematical limit analysis approach, for porous metal materials constituted of a pressure-independent Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00010-5 © 2017 Elsevier Ltd. All rights reserved.
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von Mises type solid containing a spherical void. A large number of extended criteria have been developed by various authors for different kinds of engineering materials including rocks like materials. For example, considering a Drucker-Prager type pressure-sensitive solid matrix with spherical pores, different macroscopic plastic yield criteria have been formulated for instance (Jeong, 2002; Guo et al., 2008; Lee and Oung, 2000; Durban et al., 2010; Shen et al., 2015). On the other hand, the voids shape effects have been studied by Gologanu et al. (1997), Pardoen and Hutchinson (2003), Monchiet et al. (2014). More recently, Vincent et al. (2009) formulated a semianalytical macroscopic yield criterion for porous materials with two populations of pores and a von Mises solid phase at the microscopic scale. The same material has been studied in Shen et al. (2012a). However, it is found that the obtained elliptic plastic yield surface did not fit well a number of experimental data observed in laboratory tests (Homand and Shao, 2000; Papamichos et al., 1997; Schroeder, 2003; Xie and Shao, 2006). A new closed-form macroscopic criterion has been derived by Shen et al. (2014) for porous materials with a DruckerPrager type solid phase and two populations of pores using a limit-analysis approach. In this chapter, the principle of the limit analysis method is first shortly recalled. Using this method and based on the previous work of Shen et al. (2014), the formulation of the macroscopic yield criterion for porous rocks is presented. After introducing a nonassociated plastic potential and plastic hardening function, the micromechanics-based plastic model is formulated and then applied to describe the mechanical behaviors of a typical chalk.
10.2
Principle of the limit analysis method
In this section, we first recall some basic principles of the limit analysis method for porous materials. In order to derive an explicit expression of macroscopic yield criterion, a hollow sphere Ω with an internal radius a and an external one b is usually chosen as the studied representative volume element (RVE) to consider the influence of porosity f 5 a3 =b3 ; or a cell of a porous material made up of a spheroidal volume Ω containing a confocal spheroidal void ω to account for the effects of pore shape and porosity simultaneously. The RVE is subjected to a homogeneous strain rate D on the outer surface: vðxÞ 5 D x;
’ xA@Ω
(10.1)
The matrix in the RVE obeys to a local convex yield function FðσÞ # 0. The microstress tensor field is statically admissible and plastically admissible. With an associated flow rule, the microscopic strain rate d in the matrix can be calculated: _ @FðσÞ ; d5Λ @σ
_ ΛFðσÞ 5 0;
FðσÞ # 0;
_ $0 Λ
(10.2)
_ is the local plastic multiplier. Associated to the velocity field v, the strain where Λ rate tensor d is kinematically admissible: d 5 ðgrad v1 t grad vÞ=2.
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According to Salenc¸on (1990), the corresponding support function can be expressed πðdÞ 5 supfσ:djFðσÞ # 0. For the uniform strain rate boundary conditions considered in the present study, the following inequality holds for all macroscopic stress Σ and macroscopic strain rate D (HillMandel lemma): X
:D # ΠðDÞ 5 inf vKA
1 jΩj
ð Ω2ω
πðdÞdV
(10.3)
ΠðDÞ represents the macroscopic dissipation, The infimum in (10.3) is taken over all kinematically admissible velocity fields, v. In the framework of limit analysis, the choice of the velocity field v(x) in the matrix complying with the boundary condition (10.1) is one of the key points to derive the macroscopic yield criterion. As classically, the limit stress states of the plastic porous medium at the macroscopic scale are shown to be of the form: X
10.3
5
@Π @D
(10.4)
Macroscopic criterion of double porous rock
In the framework of limit analysis method, an explicit macroscopic criterion will be derived in this section for a class of porous rock with two populations of voids at different scales. A two-step homogenization procedure will be developed. the RVE of the studied double porous material is defined in Fig. 10.1. For the sake of simplicity, we assume that both families of pores are spherical and randomly distributed in a Inter-particle pore (Large pore) Ω2 Solid phase
Ωm Equivalent homogeneous material
Ω
Macroscopic Scale Total porosity f (1–φ) + φ
(A)
r a
Intra-particle pore (Small pore) Ω1
Porous matrix
b
Mesoscopic Scale Ω Meso porosity φ = Ω2
(B)
Microscopic Scale Ω Micro porosity f = Ω –1 Ω
2
(C)
Figure 10.1 The RVE of the studied double porous medium at different scales.
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solid matrix. At the macroscopic scale, the studied material can be seen as an equivalent homogeneous material (see Fig. 10.1A). The interparticle pores (large pores) of the studied porous medium are found at the mesoscopic scale. The matrix in Fig. 10.1B itself is a porous medium that is composed of intraparticle pores (small pores) and the solid phase at the microscopic scale (Fig. 10.1C). The two populations of spherical voids are distributed at two well-separated scales. We denote jΩj the total volume of the RVE, Ω2 the volume of the large voids at the mesoscopic scale, Ω1 and Ωm are the domains occupied by the small voids and the solid phase at the microscopic scale, respectively. With these notations, the porosity at the microscopic scale (intraparticle pores) f, the one at the mesoscopic scale φ (inter-particle pores), and the total porosity Γ at the macroscopic scale can be expressed as f5
jΩ1 j ; jΩ 2 Ω2 j
φ5
jΩ2 j ; jΩj
Γ5
jΩ1 j 1 jΩ2 j 5 f ð1 2 φÞ 1 φ jΩj
(10.5)
A two-step homogenization procedure will be adopted here to derive a macroscopic criterion for the studied material. In the first homogenization from microscale to mesoscale, the effects of microporosity f and the plastic compressibility of the solid phase is taken into account. In the second one from mesoscale to macroscale, a macroscopic plastic criterion will be obtained in the framework of limit analysis theory with considering the mesoporosity φ. The sign conventions of stress and strain are as follows: tensile stress (strain) is positive whereas compressive stress (strain) is negative. For the sake of clarity, the microscopic stress in the solid phase (Fig. 10.1C) is denoted as ~ the mesoscopic and macroscopic ones are σ and Σ, respectively. σ,
10.3.1 Homogenization from microscopic to mesoscopic scale For most geomaterials, the plastic behavior is generally affected by the mean stress and exhibits volumetric compressibility or dilatancy. In the first homogenization step from the microscopic scale to mesoscopic scale (see Fig. 10.1C), the plastic behavior of the solid phase is here assumed to obey to a DruckerPrager criterion: ~ 5 σ~ d 1 Tðσ~ m 2 hÞ # 0 Φm ðσÞ
(10.6)
pffiffiffiffiffiffiffiffiffiffiffiffi ~ the mean stress, and σ~ d 5 σ~ 0 : σ~ 0 the equivalent deviatoric where σ~ m 5 trσ=3 stress with σ~ 0 5 σ~ 2 σ~ m 1. The parameter h represents the hydrostatic tensile strength, whereas T denotes the frictional coefficient. For such a porous medium, Maghous et al. (2009) have derived an effective strength criterion by using the modified secant method (interpreted by Suquet (1995), as equivalent to the variational method of Ponte Casta neda (1991)). F mp ðσ; f ; TÞ 5
1 1 2f =3 2 3f σ 1 2 1 σ2m 1 2ð1 2 f Þhσm 2 ð12f Þ2 h2 5 0 d T2 2T 2 (10.7)
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This criterion explicitly takes into account the effect of the porosity at the microscopic scale f. It also presents a tensioncompression asymmetry that is a manifestation of the pressure sensitive solid phase. This criterion is then adopted here to describe the plastic behavior of the porous matrix.
10.3.2 Homogenization from mesoscale to macroscale During the second homogenization step from the mesoscopic to macroscopic scale, the macroscopic mechanical behavior of the double porous rock is determined by using a limit analysis approach. As shown in Fig. 10.1B, the porous rock is represented by a hollow sphere containing the porosity φ at the mesoscopic scale. Based on (10.7) and for a general case, the effective plastic criterion of the porous matrix can be written in the following general elliptic form: ΦðσÞ 5 βσ2eq 1
9α ðσm Þ2 2 Lσm 2 σ20 # 0 2
(10.8)
The scalars α, β, L, and σ0 are material constants that can be identified from Eq. (10.7) for the porous matrix. The representation of this elliptic criterion (10.7) or (10.8) corresponds to a closed surface. According to (10.8), the mesoscopic strain rate is given by the normality rule: _ 3βσ0 1 3α σm 2 L 1 d5Λ 9α
(10.9)
where 1 is the second-order unit tensor. 2 5 ð2=3Þd0 :d0 and With the definition of the equivalent strain rate deq 2 0 0 _ is defined: Λ _ 5 deq =2βσeq . σeq 5 ð3=2Þσ :σ . The plastic multiplier Λ The plastic dissipation πðdÞ of the porous matrix can be calculated: πðdÞ 5 σ:d 5
deq Lσm 1 2σ20 2βσeq
(10.10)
By using the relationships of (10.9) and (10.8), the expressions of σeq and σm are given in the following form: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2 u σ0 1 L u 18α ; σeq 5 u 2 2 u 2β dm tβ 1 2 α deq
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 L2 u L 2β dm u σ0 1 18α u σm 5 1 2 2 9α 3α deq u tβ 1 2β dm 2 α deq
(10.11)
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Substituting Eq. (10.11) into Eq. (10.10), the mesoscopic plastic dissipation πðdÞ finally readsa L dm 1 πðdÞ 5 3α
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 deq L2 2dm2 1 σ20 1 18α α β
(10.12)
According to (10.3), the macroscopic plastic dissipation ΠðDÞ can be obtained (Suquet, 1985; Buhan, 1986): ΠðDÞ 5 inf vKA
1 jΩj
ð Ω2Ω2
πðdÞdV
(10.13)
For the moment, it is necessary to construct a kinematically admissible velocity fields v. Based on the one used by Gurson (1977), the following trial velocity field is adopted to account for the plastic compressibility of the matrix: v 5 Ax 1
b3 ðDm 2 AÞ e r 1 D0 x r2
(10.14)
in which Dm 5 ð1=3Þtr D and D0 the deviatoric part of the macroscopic strain rate D. The field Ax is homogeneous and allows us to account for matrix plastic compressibility. The two remaining terms are kinematically admissible with (D 2 A1). More precisely, the second term in Eq.(10.14) corresponds to the expansion of the cavity and the outer volume, whereas the third one describes the shape change of the cavity and of the outer boundary without volume change. Hence, for any value of the scalar A, the whole velocity field v complies with the uniform strain rate D applied to the hollow sphere. Due to the presence of A (which remains unknown in the definition of the velocity field), the macroscopic dissipation, ΠðDÞ is computed owing to a minimization procedure with respect to A:
~ ΠðDÞ 5 min ΠðD; AÞ ; A
~ ΠðD; AÞ 5
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 σ20 1 18α Ω
L vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ð 2 3α ð u2dm deq t 1 dV 1 dm dV Ω Ω2Ω2 α β Ω2Ω2
a
More details can be found in Shen et al. (2014).
(10.15)
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The determination of the macroscopic criterion requires then to compute the integrals of deq and dm over the matrix. The strain rate in the matrix can be obtained from (10.14), in spherical coordinates, as
b3 ðDm 2 AÞ 1 2 3e r e r 3 r
d 5 A1 1 D0 1
(10.16)
From which, one can get dm 5 A;
2 deq 5 D2eq 1 4
2
b3 ðDm 2AÞ 4 b3 ðDm 2 AÞ 0 1 D : 1 2 3e r e r 3 3 r 3 r (10.17)
In order to obtain a closed-form expression, the following inequality is classically used: ð
ð Ω2Ω2
deq dV 5
Ω2Ω2
ð b D E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 deq ðr; θ; ϕÞdV # 4π r 2 deq D
2 deq
dr
SðrÞ
a
where SðrÞ is the sphere of radius r and
1=2
E SðrÞ
(10.18)
2 is the average of deq ðr; θ; ϕÞ
over all the orientations: D
2 deq
E SðrÞ
5
1 4π
ð SðrÞ
2 deq ds 5 4
3 2 b ðDm 2AÞ 1 D2eq r3
(10.19)
This eventually yields an upper bound of the macroscopic dissipation by computing: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð u ffi du u L2 1=φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ~ ΠðD; AÞ 5 tσ20 1 Að1 2 φÞ M 2 1 N 2 u2 2 1 u 3α 18α 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ffi #1=φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 L2 uN L M 2 1N 2 u2 t 5 σ0 1 Að1 2 φÞ 1 2 N arcsinh M 3α 18α u 1
(10.20) where M 2 5 2A2 =α 1 D2eq =β , N 2 5 4=β ðDm 2AÞ2 and the change of variable u 5 b3 =r 3 has been introduced. ~ As mentioned before, one has to minimize ΠðD; AÞ over the unknown variable A and to determine the macroscopic yield function by taking advantage of the approx~ imate expression of ΠðD; AÞ. In practice, rather than treating these two steps
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successively, the computation of the criterion may be done by addressing them simultaneously (cf. Monchiet, 2006). It comes X
5
~ @ΠðD; AÞ ; @D
~ @ΠðD; AÞ 50 @A
with
(10.21)
To solve (10.21), it is convenient as in Shen et al. (2012a) to make the following ~ ~ change of variable: ΠðD; P AÞ 5 ΠðM; NÞ with M and N introduced before. The macroscopic stress tensor then reads X
5
~ ~ @M ~ @N @Π @Π @Π 5 1 @D @M @D @N @D
(10.22)
The condition of minimization with respect to A (second relation in (10.21)) becomes ~ @M ~ @N @Π @Π L 1 1 ð1 2 φÞ 5 0 @M @A @N @A 3α
(10.23)
Similarly to the approach used by Gurson, one can then establish the parametric form of the macroscopic yield function: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u u u ~ @Π u 2 L2 4u N2 u 2 N2 5 t1 1 tφ 1 ΣA 5 5 tσ0 1 2 @M 18α M2 M2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u ~ u 2 @Π L2 N N 5 tσ 0 1 ΣB 5 2 arcsinh arcsinh @N φM M 18α
(10.24)
By eliminating the parameter N=M in the above two relationships, one gets 0
12
0
1
B C B C ΣA ΣB Brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 2φcoshBrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiC 2 1 2 φ2 5 0 @ A @ 2 2 A L L σ20 1 σ20 1 18α 18α
(10.25)
The macroscopic yield function (10.25) is of Gurson type with appropriate quantities ΣA and ΣB that needs to be explicit. To this end, noting that M depends only on the deviation D0 and scalar A, whereas N is a function of Dm and A, it comes P m
5
~ ~ 1 @Π 2 @Π ; 5 pffiffiffi 3 @Dm 3 β @N
X
0
5
~ ~ @Π 2D0 @Π 5 0 @D 3βM @M
(10.26)
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223
in which pffiffiffi ~ @Π 3 β 5 Σm @N 2
(10.27)
~ Considering the condition of minimization with respect to A, (10.23), @Π=@M finally reads ~ @Π 5 @M
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 9α L Σm 2 ð12φÞ 1 βΣ2eq 2 9α
(10.28)
The closed-form expression of the macroscopic criterion of the porous medium having a matrix obeying to the general elliptic criterion (10.8) is 2 32 L 2 pffiffiffi ð12φÞ Σ 2 m Σeq 9α 6 3 β Σm 7 9α β 1 2 1 2 φ2 5 0 4 5 1 2φ cosh Σ0 2 Σ0 2 Σ0 (10.29) with Σ0 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi σ20 1 L2 =18α . When L 5 0, α 5 0, and β 5 1, we obviously retrieve
the Gurson (1977) criterion. Let us come back to the problem of the double porous material with a DruckerPrager (10.6) solid phase. The parameters β, α, L, and σ0 can be determined by comparing the two elliptic criteria (10.7) and (10.8). As a final result, the closed-form expression of the macroscopic criterion of the double porous material having a compressible solid phase at the microscale is derived: 32 2 2ð12f ÞhT 2 2 6Σm 1 3f 22T 2 ð12φÞ7 7 6 2 4f Σeq 3f 2 6 7 1 2T 6 Φ5 1 7 3 9 2 Σ0 Σ0 5 4
(10.30)
! sffiffiffiffiffiffiffiffiffiffiffi 3 Σm 1f 1 2φ cosh 2 1 2 φ2 5 0 2 Σ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Σ0 5 ð1 2 f ÞhT 3f =ð3f 2 2T 2 Þ This macroscopic criterion takes into account simultaneously the compressibility of the solid phase, the influences of the microporosity f and the meso one φ. Fig. 10.2 shows the influences of the two porosities f and φ on the yield surface that has a tensioncompression asymmetry. The total macroscopic porosity is the same (43%) in these three cases, but different proportions of microporosity f and
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Figure 10.2 Yield surfaces predicted by (10.30) with different proportions of microporosity f and mesoporosity φ, h 5 32 MPa, T 5 0:3, total porosity Γ 5 0:43.
meso porosity φ are considered. The macroscopic yield surface of the studied double porous material is thus clearly affected by the proportion of f and φ, in particular for the compressive loading.
10.4
Formulation of a nonassociated elastoplastic model for double porous material
In this section, a complete micromechanics-based constitutive model is proposed for double porous materials. The elastic behavior is firstly considered. Due to the microstructure of the studied REV, the macroscopic elastic stiffness tensor ℂhom can be calculated by the classical upper bound of Hashin and Shtrikman (1963) for the two steps of homogenization procedure. Knowing the values of the bulk and shear modulus κs and μs of the solid phase at the microscopic scale and the intraparticle porosity f, the effective bulk and shear modulus (κp and μp ) of the porous matrix at the mesoscale (Fig. 10.1B) can first be calculated by κp 5
4ð1 2 f Þκs μs ; 4μs 1 3f κs
μp 5
ð1 2 f Þμs κs 1 2μs 1 1 6f 9κs 1 8μs
(10.31)
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Figure 10.3 Influence of the porosity ratio f =φ on the effective elastic properties of double porous material with a total porosity Γ 5 0:43.
In the second homogenization step, the effective elastic properties of the double porous matrix can be determined by considering the influence of the interparticle porosity φ. Based on the result (10.31), the homogenized bulk and shear modulus κhom and μhom of the double porous medium are given by κhom 5
4ð1 2 φÞκp μp ; 4μp 1 3φκp
μhom 5
ð1 2 φÞμp κp 1 2μp 1 1 6φ 9κp 1 8μp
(10.32)
The influences of the mesoporosity φ and the microporosity f on the effective elastic properties are shown in Fig. 10.3. Young’s modulus and Poisson’s ratio of the solid phase are taken as Es 5 12 GPa and υs 5 0:2, respectively. The total porosity of the studied double porous material is Γ 5 0:43. The homogenized bulk and shear modulus κhom and μhom are affected by the ratio of f and φ. When the proportion of the microporosity f is low (the ratio f =φ is small), the values of κhom and μhom are big. In the domain 0 , f =φ , 5, the influence of this ratio on the effective elastic properties is especially important. For the elastoplastic behavior, the macroscopic plastic criterion (10.30) presented in Section 10.3 will be applied. Most geomaterials exhibit a plastic hardening. In the present study, this effect on the macroscopic behavior is taken into account via the evolution of the frictional coefficient T as a function of the equivalent plastic strain εpeq in the solid phase: Tðεpeq Þ.
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In most rock-like materials, it is generally necessary to develop a nonassociated plastic flow rule to more accurately describe plastic volumetric deformation. Inspired by the work of Maghous et al. (2009), Shen et al. (2012b, 2013), the following function is proposed as the macroscopic plastic potential by taking a similar form as the yield function (10.30) and introducing the dilatancy coefficient t: 2
32 2ð12f ÞhT 2 6Σm 1 3f 22Tt ð12φÞ7 2 6 7 2 4f Σeq 3f 7 1 2 Tt 6 1 G5 6 7 Σ0 3 9 2 Σ0 4 5
(10.33)
0sffiffiffiffiffiffiffiffiffiffiffi 1 3 Σm A 1f 1 2φ cosh@ 2 Σ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Σ0 5 ð1 2 f ÞhT 3f =ð3f 2 2TtÞ. The dilatancy coefficient t controls the transition between volumetric contractancy and dilatancy during plastic deformation. The macroscopic plastic potential depends also on the two porosities (f and φ) and the properties of the solid phase (T, t). According to the macroscopic potential (10.33), the plastic flow rule is given by @G ðΣ; f ; φ; T; tÞ Dp 5 λ_ @Σ
(10.34)
The equivalent plastic strain of the solid phase can be computed: ε_ peq 5
Σ:Dp ð1 2 f Þð1 2 φÞ Th 1 ðt 2 TÞ
Σm ð1 2 f Þð1 2 φÞ
(10.35)
The evolutions of microporosity f and mesoporosity φ are respectively evaluated
by f_ 5 d=dt Ω1 =ðΩ1 1 Ωm Þ and φ_ 5 d=dt Ω2 =Ω : _1 _m1Ω _m Ω Ω 2 ð1 2 f Þ f_ 5 ð1 2 f Þ Ωm 1 Ω1 Ωm _11Ω _2 _1 _m 1Ω _m 1Ω Ω Ω 2 ð1 2 φÞ φ_ 5 ð1 2 φÞ Ωm 1 Ω1 1 Ω2 Ωm 1 Ω1
(10.36)
With the increment of strain ΔE, the corresponding stress ΔΣ can be computed by using the macroscopic elastic stiffness tensor ℂhom : ΔΣ 5 ℂhom : ðΔE 2 ΔEp Þ;
ℂhom 5 3κhom J 1 2μhom K
(10.37)
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in which K 5 I 2 J, I is the fourth-order unity tensor and J 5 1=3 1 1. ΔEp is the plastic part of the incremental strain ΔE, which can be calculated by the macroscopic potential (10.33): ΔEp 5 dλ
@G @Σ
(10.38)
where the plastic multiplier dλ can be obtained from the consistency condition: @Φ _ @Φ _ @Φ _ _ 1 _ ðΣ; f ; φ; T Þ 5 @Φ :Σ f1 T 50 Φ φ1 @Σ @f @φ @T
(10.39)
Substituting (10.35), (10.36) and (10.37) into (10.39), one has:
@Φ :ℂ:ΔE @Σ dλ5 3 2 @G Σ: @Φ @G @Φ @G @Φ @T 7 @Φ 6 @Σ :ℂ: 2 ð12f Þ4dv2t R 2dv 2 5 2 ð12φÞ ð12f Þð12φÞTh1ðt2TÞΣm @Σ @Σ @f @φ @Σm @T @εp
(10.40) in which 3f 2Σm @G 1 2ð1 2 φÞh Σ:@G 2 1 Σ: 2T 2 12φ @Σ @Σ : R5 ; dv 5 ~m Σm 12φ ð1 2 f Þð1 2 φÞTh 1 ðt 2 TÞΣ 2 2 2ð12φÞ h 2 2ð1 2 φÞh 12φ
Then the proposed micromacro constitutive model is implemented in a standard finite element code (Abaqus) via a subroutine UMAT. Fig. 10.4 illustrates the stressstrain curves predicted by this model in the case of associated flow rule without plastic hardening. For the solid phase, Young’s modulus Es 5 12 GPa and Poisson’s ratio υs 5 0:2, h 5 32 MPa, t0 5 T0 5 0:3. The effects of proportions of microporosity f and mesoporosity φ on the overall strength of the double porous material with a total porosity Γ 5 0:43 can be obviously seen, for uniaxial and triaxial compression tests.
10.5
Application to a typical porous rock
The proposed model is applied in this section to describe the mechanical behavior of the porous chalk with a nonassociated plastic flow rule. The studied so-called Lixhe chalk is from the Upper Campanian age and was drilled in the CBR quarry near Lie`ge (Belgium). It is composed of more than 98% of CaCO3 and less than
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(A)
(B) Σ11 (MPa)
Σ11–Σ33 (MPa)
–7,5
–7,5
–6,5
–6,5
–5,5 –4,5 –3,5 –2,5
–5,5
f = 0.1, φ =0.3667 f = 0.2, φ =0.2875
–4,5
f = 0.3, φ =0.1857 f = 0.4, φ =0.0500 Total porosity Γ =0.43
–1,5
1,5
E11 (%)
–0,5
E 33 0,5
–0,5
–1,5
–2,5
–3,5
f = 0.1, φ =0.3667 f = 0.2, φ =0.2875
–2,5
f = 0.3, φ =0.1857 f = 0.4, φ =0.0500
–1,5
Total porosity Γ =0.43
E11 (%)
–0,5
E 33 1,5
–3,5
0,5
–0,5
–1,5
–2,5
–3,5
Figure 10.4 Stress—strain curves predicted by the proposed model under compression test with different proportions of f and φ, total porosity Γ 5 0:43. (A) Uniaxial compression test, (B) triaxial compression test (5 MPa).
0.8% of SiO2 and 0.15% of Al2 O3 . The average porosity is 43%. According to Talukdar et al. (2004) and Alam et al. (2010), two populations of pores can be found in porous chalks: the microporosity f 5 40% and the meso one φ 5 5%. Based on works of Homand and Shao (2000) and Xie and Shao (2006), the following isotropic plastic hardening law is adopted: p m T 5 T0 1 1 b εpeq 1 c enεeq 2 1
(10.41)
in which T0 is the initial plastic yield threshold of the solid phase. b, m, c, and n are hardening parameters, which can be determined from hydrostatic compression tests. εpeq is the equivalent plastic deformation in the solid phase at the microscopic scale. As the rate of volumetric dilatancy generally varies with plastic deformation history, it is assumed that the dilatancy coefficient is also a function of the equivalent plastic strain in the solid phase. For the simplicity, the same form as (10.41) is adopted for the evolution of t: p m t 5 t0 1 1 b εpeq 1 c enεeq 2 1
(10.42)
in which the values of the parameters b, m, c, and n are the same as the ones used in Eq. (10.41).
10.5.1 Identification of model’s parameters Before applying the micromacro model to describe the macroscopic behavior of the Lixhe chalk, both the elastic and plastic parameters of the proposed model are identified. The macroscopic elastic properties (Young’s modulus E, Poisson’s ratio υ, or the bulk and shear modulus κhom , μhom , respectively) are obtained from the
Multiscale modeling approaches and micromechanics of porous rocks
–40
229
Hydrostatic stress (MPa)
–35 –30 –25 –20 Experiment
–15 Simulation
–10 –5
Volumetric strain (%) 0 –0,5
–1,5
–2,5
–3,5
–4,5
–5,5
–6,5
Figure 10.5 Simulation of a hydrostatic compression test of chalk.
Table 10.1
Typical values of parameters for the non-associated
model Young’s modulus
Poisson’s ratio
Frictional coefficient
Dilatancy coefficient
Hydrostatic tensile strength
Plastic hardening
Es
vs
T0
t0
h
b
m
c
n
12 GPa
0.2
0.2
0.16
32 MPa
1.3
0.03
0.018
38
initial linear part of stressstrain curve during a conventional triaxial compression test. Then the modulus (κp , μp ) of the porous matrix at the mesoscale and the ones (κs , μs ) of the solid phase at the microscale can calculated by inverting Eqs. (10.32) and (10.31), respectively. The plastic parameters are then determined from the numerical fitting of a hydrostatic compression test (see Fig. 10.5). The typical values obtained are given in Table 10.1. We can see that the two plastic strain stages are correctly described in Fig. 10.5 by the model. In the first stage, a high plastic strain rate is obtained due to important plastic collapse of interparticle pores. In the second stage, due to the progressive increase of contacts between particles and also to the decrease of two porosities, the plastic strain rate is decreasing like the consolidation mechanism in granular materials. Further, the good agreement between the numerical results and experimental data indicates that the model’s parameters are well identified.
10.5.2 Experimental validation of micromacro model The proposed nonassociated micromacro model is now used to describe the mechanical behavior of the Lixhe chalk with the values given in Table 10.1. Using the same set
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Porous Rock Failure Mechanics
(B)
(A) –20 –18
–25
Σ11–Σ33(MPa)
–16
E33
Σ11–Σ33(MPa)
–20
E11
–14
E33
–12
E11
–15
–10 –10
–8 Experiment
–6
Experiment
Simulation
–4
Simulation
–5
–2 (%)
0 3
1
–1
–3
(%)
0
–7 2
–5
1
0
–1
–2
–3
–4
–5
–6
Figure 10.6 Comparison of stressstrain curves between numerical results and experimental data—triaxial compression test on Lixhe chalk with different confining pressures. (A) 14 MPa, (B) 17 MPa.
(B)
(A) 0,41
0,0505
f
0,4
14 MPa 17 MPa
0,39
φ
0,05
14 MPa 17 MPa
0,0495
0,38
0,049
0,37
0,0485
0,36
0,048 0,0475
0,35 E11
0,34
0
–0,02
–0,04
–0,06
E11
0,047
0
–0,02
–0,04
–0,06
Figure 10.7 Evolution of f and φ predicted by the proposed model with different confining pressures: (A) microporosity f, (B) mesoporosity φ.
of parameters, numerical simulations are then performed for triaxial compression tests with different confining pressures in order to verify the capacity of the proposed model. Fig. 10.6 illustrates the comparisons of stressstrain curves between experimental data and numerical results with 14 and 17 MPa confining pressures. Generally, an overall good agreement is observed. Both the axial and lateral strains are well predicted by the proposed model. The proposed model is able to capture the main features of the mechanical behaviors of Lixhe chalk, such as the dependence of confining pressure, effect of inter-particle and intra-particle pores, compressibility of the solid phase. Fig. 10.7 shows the evolutions of microporosity f in the matrix and of mesoporosity φ predicted by the proposed model as functions of axial strain E11 during triaxial compression tests with different confining pressures (14 and 17 MPa). It is found that the evolutions of f and φ are different. The changes of the microstructure affect the macroscopic plastic criterion (10.30) and the potential (10.33) that control the macroscopic behavior. With the decrease of the porosity, the plastic yield
Multiscale modeling approaches and micromechanics of porous rocks
231
surface expand and the strength of the studied chalk increases. The mechanical behavior of the Lixhe chalk is clearly sensitive to these two families of porosities. This is the main difference between the micromechanics based model and the phenomenological one.
10.6
Concluding remark
In this chapter, as an example of multiscale approaches, we have presented a micromechanical elastic-plastic model for a class of porous rocks with two populations of pores and a pressure sensitive solid matrix. Using a two-step homogenization procedure and a limit analysis method, an analytical macroscopic yield criterion has been established. This criterion explicitly takes into account the different effects of two populations of pores at different scales. Completed with a plastic potential and hardening law, a micromechanical elastic-plastic model has been formulated. This model can be as easily as any phenomenological models implemented into a standard computer code and then applied to engineering applications. The micromacro model has been applied to describe the macroscopic behavior of a typical porous rock, the Lixhe chalk. It was found that the mean features of the mechanical behavior of the studied material were correctly described by the micromechanical model. In the future work, the micromechanical model can be extended to saturated and partially saturated porous materials by considering effects of fluid pressure in two populations of pores.
References Alam, M.M., Borre, M., Fabricius, I., Hedegaard, K., Røgen, B., Hossain, Z., et al., 2010. Biot’s coefficient as an indicator of strength and porosity reduction: calcareous sediments from Kerguelen plateau. J. Petrol. Sci. Eng. 70, 282297. Buhan, P.D., 1986. A fundamental approach to the yield design of reinforced soil structures— chap. 2: yield design homogenization theory for periodic media. The`se d’e´tat. Universite´ Pierre et Marie Curie, Paris VI, France. De Gennaro, V., Delage, P., Priol, G., Collin, F., Cui, Y., 2004. On the collapse behaviour of oil reservoir chalk. Ge´otechnique. 54 (6), 415420. Durban, D., Cohen, T., Hollander, Y., 2010. Plastic response of porous solids with pressure sensitive matrix. Mech. Res. Commun. 37, 636641. Gologanu, M., Leblond, J., Perrin, G., Devaux, J., 1997. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.), Continuum Micromechanics. Springer Verlag. Guo, T., Faleskog, J., Shih, C., 2008. Continuum modeling of a porous solid with pressure sensitive dilatant matrix. J. Mech. Phys. Solids. 56, 21882212. Gurson, A., 1977. Continuum theory of ductile rupture by void nucleation and growth: part1yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 215. Hashin, Z., Shtrikman, S., 1963. A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids. 11, 127140. Homand, S., Shao, J., 2000. Mechanical behavior of a porous chalk and water/chalk interaction. Part I: Experimental study. Oil Gas Sci. Technol. 55 (6), 591598.
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Jeong, H., 2002. A new yield function and a hydrostatic stress-controlled model for porous solids with pressure-sensitive matrices. Int. J. Solids Struct. 39, 13851403. Lee, J., Oung, J., 2000. Yield functions and flow rules for porous pressure-dependent strainhardening polymeric materials. J. Appl. Mech. 67, 288297. Maghous, S., Dormieux, L., Barthe´le´my, J., 2009. Micromechanical approach to the strength properties of frictional geomaterials. Eur. J. Mech. A/Solid. 28, 179188. Monchiet, V., 2006. Contributions a` la mode´lisation microme´canique de l’endommagement et de la fatigue des me´taux ductiles. The`se. Universite´ des sciences et Technologies de Lille, France. Monchiet, V., Charkaluk, E., Kondo, D., 2014. Macroscopic yield criteria for ductile materials containing spheroidal voids: an Eshelby-like velocity fields approach. Mech. Mater. 72, 118. Papamichos, E., Brignoli, M., Santarelli, F., 1997. An experimental and theoretical study partially saturated collapsible rock. Int. J. Mech. Cohesive-Frictional Mater. 2, 251278. Pardoen, T., Hutchinson, J., 2003. Micromechanics-based model for trends in toughness of ductile metals. Acta Mater. 51, 133148. Ponte Castan˜eda, P., 1991. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids. 39, 4571. Salenc¸on, J., 1990. An introduction to the yield theory and its applications to soil mechanics. Eur. J. Mech. A/Solids. 9 (5), 477500. Schroeder, C., 2003. Du coccolithe au re´servoir pe´trolier; approche phe´nome´nologique du comportement me´canique de la craie en vue de sa mode´lisation a` diffe´rentes e´chelles. Phd, thesis. University of Lie`ge. Shen, W.Q., Shao, J.F., Dormieux, L., Kondo, D., 2012a. Approximate criteria for ductile porous materials having a Green type matrix: application to double porous media. Comput. Mater. Sci. 62, 189194. Shen, W.Q., Shao, J.F., Kondo, D., Gatmiri, B., 2012b. A micro-macro model for clayey rocks with a plastic compressible porous matrix. Int. J. Plast. 36, 6485. Shen, W.Q., Kondo, D., Dormieux, L., Shao, J.F., 2013. A closed-form three scale model for ductile rocks with a plastically compressible porous matrix. Mech. Mater. 59, 7386. Shen, W.Q., He, Z., Dormieux, L., Kondo, D., 2014. Effective strength of saturated double porous media with a DruckerPrager solid phase. Int. J. Numer. Anal. Meth. Geomech. 38, 281296. Shen, W.Q., Shao, J.F., Kondo, D., De Saxce, G., 2015. A new macroscopic criterion of porous materials with a Mises-Schleicher compressible matrix. Eur. J. Mech. A/Solids. 49, 531538. Suquet, P., 1985. Homogenization techniques for composite media. In: Sanchez-Palencia, E. (Ed.), Ch. Elements of Homogenization for Inelastic Solid Mechanics. Springer Verlag, pp. 193278. Suquet, P., 1995. Overall properties of nonlinear composites: a modified secant moduli approach and its link with Ponte Castan˜eda’s nonlinear variational procedure. R.R. Acad. Sci. Paris, IIb. 320, 563571. Talukdar, M.S., Torsaeter, O., Howard, J., 2004. Stochastic reconstruction of chalk samples containing vuggy porosity using a conditional simulated annealing technique. Transport Porous Media. 57, 115. Vincent, P.-G., Monerie, Y., Suquet, P., 2009. Porous materials with two populations of voids under internal pressure: I. Instantaneous constitutive relations. Int. J. Solids Struct. 46, 480506. Xie, S., Shao, J., 2006. Elastoplastic deformation of a porous rock and water interaction. Int. J. Plast. 22, 21952225.
10
Wanqing Shen and Jianfu Shao Lille University of Science and Technology, Villeneuve d’Ascq, France
10.1
Introduction
Most rocks are heterogeneous materials and their macroscopic behaviors are inherently related to their microstructures such as mineral compositions and porosity. Based on extensive experimental evidences and within the framework of irreversible thermodynamics, many phenomenological plastic and damage models have been developed. These models are generally able to capture various features of mechanical behaviors of rock-like materials, but they are not providing explicit relationships between macroscopic responses and microstructures. In order to develop an alternative modeling method and to improve phenomenological models, significant efforts have been undertaken during the last decades on the development of micromechanical models. In particular, different analytical and numerical homogenization techniques for composite materials have been successfully adapted for rock-like materials. Different micromechanical models have then formulated for the description of crack-induced damage and plastic deformation. It is not the objective of this chapter to give a comprehensive review of all micromechanical models so far developed for rock-like materials. As an example, we propose to present here a micromechanical model for modeling elastic-plastic behavior of porous rocks with two families of pores at two different scales. The kind of microstructure can be representative for a wide class of rocks, for instance clayey rocks, chalk, etc. It is possible to identify intraparticle pores inside mineral grains (plates) and interparticle pores between mineral grains. These two families of pores affect in a different way the macroscopic properties of material such as elastic modulus, plastic yield stress, and failure strength (Homand and Shao, 2000; Papamichos et al., 1997; Schroeder, 2003; De Gennaro et al., 2004; Alam et al., 2010; Xie and Shao, 2006). Further, due to the presence of pores, two plastic deformation processes can be generally identified, the plastic pore collapse under mean effective stress and the plastic shearing under deviatoric stress. The objective of micromechanical approach is to establish a macroscopic plastic criterion explicitly taking into account the effects of pores. As a pioneer work, Gurson (1977) proposed an analytical yield criterion using a kinematical limit analysis approach, for porous metal materials constituted of a pressure-independent Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00010-5 © 2017 Elsevier Ltd. All rights reserved.
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von Mises type solid containing a spherical void. A large number of extended criteria have been developed by various authors for different kinds of engineering materials including rocks like materials. For example, considering a Drucker-Prager type pressure-sensitive solid matrix with spherical pores, different macroscopic plastic yield criteria have been formulated for instance (Jeong, 2002; Guo et al., 2008; Lee and Oung, 2000; Durban et al., 2010; Shen et al., 2015). On the other hand, the voids shape effects have been studied by Gologanu et al. (1997), Pardoen and Hutchinson (2003), Monchiet et al. (2014). More recently, Vincent et al. (2009) formulated a semianalytical macroscopic yield criterion for porous materials with two populations of pores and a von Mises solid phase at the microscopic scale. The same material has been studied in Shen et al. (2012a). However, it is found that the obtained elliptic plastic yield surface did not fit well a number of experimental data observed in laboratory tests (Homand and Shao, 2000; Papamichos et al., 1997; Schroeder, 2003; Xie and Shao, 2006). A new closed-form macroscopic criterion has been derived by Shen et al. (2014) for porous materials with a DruckerPrager type solid phase and two populations of pores using a limit-analysis approach. In this chapter, the principle of the limit analysis method is first shortly recalled. Using this method and based on the previous work of Shen et al. (2014), the formulation of the macroscopic yield criterion for porous rocks is presented. After introducing a nonassociated plastic potential and plastic hardening function, the micromechanics-based plastic model is formulated and then applied to describe the mechanical behaviors of a typical chalk.
10.2
Principle of the limit analysis method
In this section, we first recall some basic principles of the limit analysis method for porous materials. In order to derive an explicit expression of macroscopic yield criterion, a hollow sphere Ω with an internal radius a and an external one b is usually chosen as the studied representative volume element (RVE) to consider the influence of porosity f 5 a3 =b3 ; or a cell of a porous material made up of a spheroidal volume Ω containing a confocal spheroidal void ω to account for the effects of pore shape and porosity simultaneously. The RVE is subjected to a homogeneous strain rate D on the outer surface: vðxÞ 5 D x;
’ xA@Ω
(10.1)
The matrix in the RVE obeys to a local convex yield function FðσÞ # 0. The microstress tensor field is statically admissible and plastically admissible. With an associated flow rule, the microscopic strain rate d in the matrix can be calculated: _ @FðσÞ ; d5Λ @σ
_ ΛFðσÞ 5 0;
FðσÞ # 0;
_ $0 Λ
(10.2)
_ is the local plastic multiplier. Associated to the velocity field v, the strain where Λ rate tensor d is kinematically admissible: d 5 ðgrad v1 t grad vÞ=2.
Multiscale modeling approaches and micromechanics of porous rocks
217
According to Salenc¸on (1990), the corresponding support function can be expressed πðdÞ 5 supfσ:djFðσÞ # 0. For the uniform strain rate boundary conditions considered in the present study, the following inequality holds for all macroscopic stress Σ and macroscopic strain rate D (HillMandel lemma): X
:D # ΠðDÞ 5 inf vKA
1 jΩj
ð Ω2ω
πðdÞdV
(10.3)
ΠðDÞ represents the macroscopic dissipation, The infimum in (10.3) is taken over all kinematically admissible velocity fields, v. In the framework of limit analysis, the choice of the velocity field v(x) in the matrix complying with the boundary condition (10.1) is one of the key points to derive the macroscopic yield criterion. As classically, the limit stress states of the plastic porous medium at the macroscopic scale are shown to be of the form: X
10.3
5
@Π @D
(10.4)
Macroscopic criterion of double porous rock
In the framework of limit analysis method, an explicit macroscopic criterion will be derived in this section for a class of porous rock with two populations of voids at different scales. A two-step homogenization procedure will be developed. the RVE of the studied double porous material is defined in Fig. 10.1. For the sake of simplicity, we assume that both families of pores are spherical and randomly distributed in a Inter-particle pore (Large pore) Ω2 Solid phase
Ωm Equivalent homogeneous material
Ω
Macroscopic Scale Total porosity f (1–φ) + φ
(A)
r a
Intra-particle pore (Small pore) Ω1
Porous matrix
b
Mesoscopic Scale Ω Meso porosity φ = Ω2
(B)
Microscopic Scale Ω Micro porosity f = Ω –1 Ω
2
(C)
Figure 10.1 The RVE of the studied double porous medium at different scales.
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Porous Rock Failure Mechanics
solid matrix. At the macroscopic scale, the studied material can be seen as an equivalent homogeneous material (see Fig. 10.1A). The interparticle pores (large pores) of the studied porous medium are found at the mesoscopic scale. The matrix in Fig. 10.1B itself is a porous medium that is composed of intraparticle pores (small pores) and the solid phase at the microscopic scale (Fig. 10.1C). The two populations of spherical voids are distributed at two well-separated scales. We denote jΩj the total volume of the RVE, Ω2 the volume of the large voids at the mesoscopic scale, Ω1 and Ωm are the domains occupied by the small voids and the solid phase at the microscopic scale, respectively. With these notations, the porosity at the microscopic scale (intraparticle pores) f, the one at the mesoscopic scale φ (inter-particle pores), and the total porosity Γ at the macroscopic scale can be expressed as f5
jΩ1 j ; jΩ 2 Ω2 j
φ5
jΩ2 j ; jΩj
Γ5
jΩ1 j 1 jΩ2 j 5 f ð1 2 φÞ 1 φ jΩj
(10.5)
A two-step homogenization procedure will be adopted here to derive a macroscopic criterion for the studied material. In the first homogenization from microscale to mesoscale, the effects of microporosity f and the plastic compressibility of the solid phase is taken into account. In the second one from mesoscale to macroscale, a macroscopic plastic criterion will be obtained in the framework of limit analysis theory with considering the mesoporosity φ. The sign conventions of stress and strain are as follows: tensile stress (strain) is positive whereas compressive stress (strain) is negative. For the sake of clarity, the microscopic stress in the solid phase (Fig. 10.1C) is denoted as ~ the mesoscopic and macroscopic ones are σ and Σ, respectively. σ,
10.3.1 Homogenization from microscopic to mesoscopic scale For most geomaterials, the plastic behavior is generally affected by the mean stress and exhibits volumetric compressibility or dilatancy. In the first homogenization step from the microscopic scale to mesoscopic scale (see Fig. 10.1C), the plastic behavior of the solid phase is here assumed to obey to a DruckerPrager criterion: ~ 5 σ~ d 1 Tðσ~ m 2 hÞ # 0 Φm ðσÞ
(10.6)
pffiffiffiffiffiffiffiffiffiffiffiffi ~ the mean stress, and σ~ d 5 σ~ 0 : σ~ 0 the equivalent deviatoric where σ~ m 5 trσ=3 stress with σ~ 0 5 σ~ 2 σ~ m 1. The parameter h represents the hydrostatic tensile strength, whereas T denotes the frictional coefficient. For such a porous medium, Maghous et al. (2009) have derived an effective strength criterion by using the modified secant method (interpreted by Suquet (1995), as equivalent to the variational method of Ponte Casta neda (1991)). F mp ðσ; f ; TÞ 5
1 1 2f =3 2 3f σ 1 2 1 σ2m 1 2ð1 2 f Þhσm 2 ð12f Þ2 h2 5 0 d T2 2T 2 (10.7)
Multiscale modeling approaches and micromechanics of porous rocks
219
This criterion explicitly takes into account the effect of the porosity at the microscopic scale f. It also presents a tensioncompression asymmetry that is a manifestation of the pressure sensitive solid phase. This criterion is then adopted here to describe the plastic behavior of the porous matrix.
10.3.2 Homogenization from mesoscale to macroscale During the second homogenization step from the mesoscopic to macroscopic scale, the macroscopic mechanical behavior of the double porous rock is determined by using a limit analysis approach. As shown in Fig. 10.1B, the porous rock is represented by a hollow sphere containing the porosity φ at the mesoscopic scale. Based on (10.7) and for a general case, the effective plastic criterion of the porous matrix can be written in the following general elliptic form: ΦðσÞ 5 βσ2eq 1
9α ðσm Þ2 2 Lσm 2 σ20 # 0 2
(10.8)
The scalars α, β, L, and σ0 are material constants that can be identified from Eq. (10.7) for the porous matrix. The representation of this elliptic criterion (10.7) or (10.8) corresponds to a closed surface. According to (10.8), the mesoscopic strain rate is given by the normality rule: _ 3βσ0 1 3α σm 2 L 1 d5Λ 9α
(10.9)
where 1 is the second-order unit tensor. 2 5 ð2=3Þd0 :d0 and With the definition of the equivalent strain rate deq 2 0 0 _ is defined: Λ _ 5 deq =2βσeq . σeq 5 ð3=2Þσ :σ . The plastic multiplier Λ The plastic dissipation πðdÞ of the porous matrix can be calculated: πðdÞ 5 σ:d 5
deq Lσm 1 2σ20 2βσeq
(10.10)
By using the relationships of (10.9) and (10.8), the expressions of σeq and σm are given in the following form: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2 u σ0 1 L u 18α ; σeq 5 u 2 2 u 2β dm tβ 1 2 α deq
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 L2 u L 2β dm u σ0 1 18α u σm 5 1 2 2 9α 3α deq u tβ 1 2β dm 2 α deq
(10.11)
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Porous Rock Failure Mechanics
Substituting Eq. (10.11) into Eq. (10.10), the mesoscopic plastic dissipation πðdÞ finally readsa L dm 1 πðdÞ 5 3α
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 deq L2 2dm2 1 σ20 1 18α α β
(10.12)
According to (10.3), the macroscopic plastic dissipation ΠðDÞ can be obtained (Suquet, 1985; Buhan, 1986): ΠðDÞ 5 inf vKA
1 jΩj
ð Ω2Ω2
πðdÞdV
(10.13)
For the moment, it is necessary to construct a kinematically admissible velocity fields v. Based on the one used by Gurson (1977), the following trial velocity field is adopted to account for the plastic compressibility of the matrix: v 5 Ax 1
b3 ðDm 2 AÞ e r 1 D0 x r2
(10.14)
in which Dm 5 ð1=3Þtr D and D0 the deviatoric part of the macroscopic strain rate D. The field Ax is homogeneous and allows us to account for matrix plastic compressibility. The two remaining terms are kinematically admissible with (D 2 A1). More precisely, the second term in Eq.(10.14) corresponds to the expansion of the cavity and the outer volume, whereas the third one describes the shape change of the cavity and of the outer boundary without volume change. Hence, for any value of the scalar A, the whole velocity field v complies with the uniform strain rate D applied to the hollow sphere. Due to the presence of A (which remains unknown in the definition of the velocity field), the macroscopic dissipation, ΠðDÞ is computed owing to a minimization procedure with respect to A:
~ ΠðDÞ 5 min ΠðD; AÞ ; A
~ ΠðD; AÞ 5
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 σ20 1 18α Ω
L vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ð 2 3α ð u2dm deq t 1 dV 1 dm dV Ω Ω2Ω2 α β Ω2Ω2
a
More details can be found in Shen et al. (2014).
(10.15)
Multiscale modeling approaches and micromechanics of porous rocks
221
The determination of the macroscopic criterion requires then to compute the integrals of deq and dm over the matrix. The strain rate in the matrix can be obtained from (10.14), in spherical coordinates, as
b3 ðDm 2 AÞ 1 2 3e r e r 3 r
d 5 A1 1 D0 1
(10.16)
From which, one can get dm 5 A;
2 deq 5 D2eq 1 4
2
b3 ðDm 2AÞ 4 b3 ðDm 2 AÞ 0 1 D : 1 2 3e r e r 3 3 r 3 r (10.17)
In order to obtain a closed-form expression, the following inequality is classically used: ð
ð Ω2Ω2
deq dV 5
Ω2Ω2
ð b D E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 deq ðr; θ; ϕÞdV # 4π r 2 deq D
2 deq
dr
SðrÞ
a
where SðrÞ is the sphere of radius r and
1=2
E SðrÞ
(10.18)
2 is the average of deq ðr; θ; ϕÞ
over all the orientations: D
2 deq
E SðrÞ
5
1 4π
ð SðrÞ
2 deq ds 5 4
3 2 b ðDm 2AÞ 1 D2eq r3
(10.19)
This eventually yields an upper bound of the macroscopic dissipation by computing: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð u ffi du u L2 1=φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ~ ΠðD; AÞ 5 tσ20 1 Að1 2 φÞ M 2 1 N 2 u2 2 1 u 3α 18α 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ffi #1=φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 L2 uN L M 2 1N 2 u2 t 5 σ0 1 Að1 2 φÞ 1 2 N arcsinh M 3α 18α u 1
(10.20) where M 2 5 2A2 =α 1 D2eq =β , N 2 5 4=β ðDm 2AÞ2 and the change of variable u 5 b3 =r 3 has been introduced. ~ As mentioned before, one has to minimize ΠðD; AÞ over the unknown variable A and to determine the macroscopic yield function by taking advantage of the approx~ imate expression of ΠðD; AÞ. In practice, rather than treating these two steps
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Porous Rock Failure Mechanics
successively, the computation of the criterion may be done by addressing them simultaneously (cf. Monchiet, 2006). It comes X
5
~ @ΠðD; AÞ ; @D
~ @ΠðD; AÞ 50 @A
with
(10.21)
To solve (10.21), it is convenient as in Shen et al. (2012a) to make the following ~ ~ change of variable: ΠðD; P AÞ 5 ΠðM; NÞ with M and N introduced before. The macroscopic stress tensor then reads X
5
~ ~ @M ~ @N @Π @Π @Π 5 1 @D @M @D @N @D
(10.22)
The condition of minimization with respect to A (second relation in (10.21)) becomes ~ @M ~ @N @Π @Π L 1 1 ð1 2 φÞ 5 0 @M @A @N @A 3α
(10.23)
Similarly to the approach used by Gurson, one can then establish the parametric form of the macroscopic yield function: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u u u ~ @Π u 2 L2 4u N2 u 2 N2 5 t1 1 tφ 1 ΣA 5 5 tσ0 1 2 @M 18α M2 M2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u ~ u 2 @Π L2 N N 5 tσ 0 1 ΣB 5 2 arcsinh arcsinh @N φM M 18α
(10.24)
By eliminating the parameter N=M in the above two relationships, one gets 0
12
0
1
B C B C ΣA ΣB Brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 2φcoshBrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiC 2 1 2 φ2 5 0 @ A @ 2 2 A L L σ20 1 σ20 1 18α 18α
(10.25)
The macroscopic yield function (10.25) is of Gurson type with appropriate quantities ΣA and ΣB that needs to be explicit. To this end, noting that M depends only on the deviation D0 and scalar A, whereas N is a function of Dm and A, it comes P m
5
~ ~ 1 @Π 2 @Π ; 5 pffiffiffi 3 @Dm 3 β @N
X
0
5
~ ~ @Π 2D0 @Π 5 0 @D 3βM @M
(10.26)
Multiscale modeling approaches and micromechanics of porous rocks
223
in which pffiffiffi ~ @Π 3 β 5 Σm @N 2
(10.27)
~ Considering the condition of minimization with respect to A, (10.23), @Π=@M finally reads ~ @Π 5 @M
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 9α L Σm 2 ð12φÞ 1 βΣ2eq 2 9α
(10.28)
The closed-form expression of the macroscopic criterion of the porous medium having a matrix obeying to the general elliptic criterion (10.8) is 2 32 L 2 pffiffiffi ð12φÞ Σ 2 m Σeq 9α 6 3 β Σm 7 9α β 1 2 1 2 φ2 5 0 4 5 1 2φ cosh Σ0 2 Σ0 2 Σ0 (10.29) with Σ0 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi σ20 1 L2 =18α . When L 5 0, α 5 0, and β 5 1, we obviously retrieve
the Gurson (1977) criterion. Let us come back to the problem of the double porous material with a DruckerPrager (10.6) solid phase. The parameters β, α, L, and σ0 can be determined by comparing the two elliptic criteria (10.7) and (10.8). As a final result, the closed-form expression of the macroscopic criterion of the double porous material having a compressible solid phase at the microscale is derived: 32 2 2ð12f ÞhT 2 2 6Σm 1 3f 22T 2 ð12φÞ7 7 6 2 4f Σeq 3f 2 6 7 1 2T 6 Φ5 1 7 3 9 2 Σ0 Σ0 5 4
(10.30)
! sffiffiffiffiffiffiffiffiffiffiffi 3 Σm 1f 1 2φ cosh 2 1 2 φ2 5 0 2 Σ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Σ0 5 ð1 2 f ÞhT 3f =ð3f 2 2T 2 Þ This macroscopic criterion takes into account simultaneously the compressibility of the solid phase, the influences of the microporosity f and the meso one φ. Fig. 10.2 shows the influences of the two porosities f and φ on the yield surface that has a tensioncompression asymmetry. The total macroscopic porosity is the same (43%) in these three cases, but different proportions of microporosity f and
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Porous Rock Failure Mechanics
Figure 10.2 Yield surfaces predicted by (10.30) with different proportions of microporosity f and mesoporosity φ, h 5 32 MPa, T 5 0:3, total porosity Γ 5 0:43.
meso porosity φ are considered. The macroscopic yield surface of the studied double porous material is thus clearly affected by the proportion of f and φ, in particular for the compressive loading.
10.4
Formulation of a nonassociated elastoplastic model for double porous material
In this section, a complete micromechanics-based constitutive model is proposed for double porous materials. The elastic behavior is firstly considered. Due to the microstructure of the studied REV, the macroscopic elastic stiffness tensor ℂhom can be calculated by the classical upper bound of Hashin and Shtrikman (1963) for the two steps of homogenization procedure. Knowing the values of the bulk and shear modulus κs and μs of the solid phase at the microscopic scale and the intraparticle porosity f, the effective bulk and shear modulus (κp and μp ) of the porous matrix at the mesoscale (Fig. 10.1B) can first be calculated by κp 5
4ð1 2 f Þκs μs ; 4μs 1 3f κs
μp 5
ð1 2 f Þμs κs 1 2μs 1 1 6f 9κs 1 8μs
(10.31)
Multiscale modeling approaches and micromechanics of porous rocks
225
Figure 10.3 Influence of the porosity ratio f =φ on the effective elastic properties of double porous material with a total porosity Γ 5 0:43.
In the second homogenization step, the effective elastic properties of the double porous matrix can be determined by considering the influence of the interparticle porosity φ. Based on the result (10.31), the homogenized bulk and shear modulus κhom and μhom of the double porous medium are given by κhom 5
4ð1 2 φÞκp μp ; 4μp 1 3φκp
μhom 5
ð1 2 φÞμp κp 1 2μp 1 1 6φ 9κp 1 8μp
(10.32)
The influences of the mesoporosity φ and the microporosity f on the effective elastic properties are shown in Fig. 10.3. Young’s modulus and Poisson’s ratio of the solid phase are taken as Es 5 12 GPa and υs 5 0:2, respectively. The total porosity of the studied double porous material is Γ 5 0:43. The homogenized bulk and shear modulus κhom and μhom are affected by the ratio of f and φ. When the proportion of the microporosity f is low (the ratio f =φ is small), the values of κhom and μhom are big. In the domain 0 , f =φ , 5, the influence of this ratio on the effective elastic properties is especially important. For the elastoplastic behavior, the macroscopic plastic criterion (10.30) presented in Section 10.3 will be applied. Most geomaterials exhibit a plastic hardening. In the present study, this effect on the macroscopic behavior is taken into account via the evolution of the frictional coefficient T as a function of the equivalent plastic strain εpeq in the solid phase: Tðεpeq Þ.
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Porous Rock Failure Mechanics
In most rock-like materials, it is generally necessary to develop a nonassociated plastic flow rule to more accurately describe plastic volumetric deformation. Inspired by the work of Maghous et al. (2009), Shen et al. (2012b, 2013), the following function is proposed as the macroscopic plastic potential by taking a similar form as the yield function (10.30) and introducing the dilatancy coefficient t: 2
32 2ð12f ÞhT 2 6Σm 1 3f 22Tt ð12φÞ7 2 6 7 2 4f Σeq 3f 7 1 2 Tt 6 1 G5 6 7 Σ0 3 9 2 Σ0 4 5
(10.33)
0sffiffiffiffiffiffiffiffiffiffiffi 1 3 Σm A 1f 1 2φ cosh@ 2 Σ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Σ0 5 ð1 2 f ÞhT 3f =ð3f 2 2TtÞ. The dilatancy coefficient t controls the transition between volumetric contractancy and dilatancy during plastic deformation. The macroscopic plastic potential depends also on the two porosities (f and φ) and the properties of the solid phase (T, t). According to the macroscopic potential (10.33), the plastic flow rule is given by @G ðΣ; f ; φ; T; tÞ Dp 5 λ_ @Σ
(10.34)
The equivalent plastic strain of the solid phase can be computed: ε_ peq 5
Σ:Dp ð1 2 f Þð1 2 φÞ Th 1 ðt 2 TÞ
Σm ð1 2 f Þð1 2 φÞ
(10.35)
The evolutions of microporosity f and mesoporosity φ are respectively evaluated
by f_ 5 d=dt Ω1 =ðΩ1 1 Ωm Þ and φ_ 5 d=dt Ω2 =Ω : _1 _m1Ω _m Ω Ω 2 ð1 2 f Þ f_ 5 ð1 2 f Þ Ωm 1 Ω1 Ωm _11Ω _2 _1 _m 1Ω _m 1Ω Ω Ω 2 ð1 2 φÞ φ_ 5 ð1 2 φÞ Ωm 1 Ω1 1 Ω2 Ωm 1 Ω1
(10.36)
With the increment of strain ΔE, the corresponding stress ΔΣ can be computed by using the macroscopic elastic stiffness tensor ℂhom : ΔΣ 5 ℂhom : ðΔE 2 ΔEp Þ;
ℂhom 5 3κhom J 1 2μhom K
(10.37)
Multiscale modeling approaches and micromechanics of porous rocks
227
in which K 5 I 2 J, I is the fourth-order unity tensor and J 5 1=3 1 1. ΔEp is the plastic part of the incremental strain ΔE, which can be calculated by the macroscopic potential (10.33): ΔEp 5 dλ
@G @Σ
(10.38)
where the plastic multiplier dλ can be obtained from the consistency condition: @Φ _ @Φ _ @Φ _ _ 1 _ ðΣ; f ; φ; T Þ 5 @Φ :Σ f1 T 50 Φ φ1 @Σ @f @φ @T
(10.39)
Substituting (10.35), (10.36) and (10.37) into (10.39), one has:
@Φ :ℂ:ΔE @Σ dλ5 3 2 @G Σ: @Φ @G @Φ @G @Φ @T 7 @Φ 6 @Σ :ℂ: 2 ð12f Þ4dv2t R 2dv 2 5 2 ð12φÞ ð12f Þð12φÞTh1ðt2TÞΣm @Σ @Σ @f @φ @Σm @T @εp
(10.40) in which 3f 2Σm @G 1 2ð1 2 φÞh Σ:@G 2 1 Σ: 2T 2 12φ @Σ @Σ : R5 ; dv 5 ~m Σm 12φ ð1 2 f Þð1 2 φÞTh 1 ðt 2 TÞΣ 2 2 2ð12φÞ h 2 2ð1 2 φÞh 12φ
Then the proposed micromacro constitutive model is implemented in a standard finite element code (Abaqus) via a subroutine UMAT. Fig. 10.4 illustrates the stressstrain curves predicted by this model in the case of associated flow rule without plastic hardening. For the solid phase, Young’s modulus Es 5 12 GPa and Poisson’s ratio υs 5 0:2, h 5 32 MPa, t0 5 T0 5 0:3. The effects of proportions of microporosity f and mesoporosity φ on the overall strength of the double porous material with a total porosity Γ 5 0:43 can be obviously seen, for uniaxial and triaxial compression tests.
10.5
Application to a typical porous rock
The proposed model is applied in this section to describe the mechanical behavior of the porous chalk with a nonassociated plastic flow rule. The studied so-called Lixhe chalk is from the Upper Campanian age and was drilled in the CBR quarry near Lie`ge (Belgium). It is composed of more than 98% of CaCO3 and less than
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Porous Rock Failure Mechanics
(A)
(B) Σ11 (MPa)
Σ11–Σ33 (MPa)
–7,5
–7,5
–6,5
–6,5
–5,5 –4,5 –3,5 –2,5
–5,5
f = 0.1, φ =0.3667 f = 0.2, φ =0.2875
–4,5
f = 0.3, φ =0.1857 f = 0.4, φ =0.0500 Total porosity Γ =0.43
–1,5
1,5
E11 (%)
–0,5
E 33 0,5
–0,5
–1,5
–2,5
–3,5
f = 0.1, φ =0.3667 f = 0.2, φ =0.2875
–2,5
f = 0.3, φ =0.1857 f = 0.4, φ =0.0500
–1,5
Total porosity Γ =0.43
E11 (%)
–0,5
E 33 1,5
–3,5
0,5
–0,5
–1,5
–2,5
–3,5
Figure 10.4 Stress—strain curves predicted by the proposed model under compression test with different proportions of f and φ, total porosity Γ 5 0:43. (A) Uniaxial compression test, (B) triaxial compression test (5 MPa).
0.8% of SiO2 and 0.15% of Al2 O3 . The average porosity is 43%. According to Talukdar et al. (2004) and Alam et al. (2010), two populations of pores can be found in porous chalks: the microporosity f 5 40% and the meso one φ 5 5%. Based on works of Homand and Shao (2000) and Xie and Shao (2006), the following isotropic plastic hardening law is adopted: p m T 5 T0 1 1 b εpeq 1 c enεeq 2 1
(10.41)
in which T0 is the initial plastic yield threshold of the solid phase. b, m, c, and n are hardening parameters, which can be determined from hydrostatic compression tests. εpeq is the equivalent plastic deformation in the solid phase at the microscopic scale. As the rate of volumetric dilatancy generally varies with plastic deformation history, it is assumed that the dilatancy coefficient is also a function of the equivalent plastic strain in the solid phase. For the simplicity, the same form as (10.41) is adopted for the evolution of t: p m t 5 t0 1 1 b εpeq 1 c enεeq 2 1
(10.42)
in which the values of the parameters b, m, c, and n are the same as the ones used in Eq. (10.41).
10.5.1 Identification of model’s parameters Before applying the micromacro model to describe the macroscopic behavior of the Lixhe chalk, both the elastic and plastic parameters of the proposed model are identified. The macroscopic elastic properties (Young’s modulus E, Poisson’s ratio υ, or the bulk and shear modulus κhom , μhom , respectively) are obtained from the
Multiscale modeling approaches and micromechanics of porous rocks
–40
229
Hydrostatic stress (MPa)
–35 –30 –25 –20 Experiment
–15 Simulation
–10 –5
Volumetric strain (%) 0 –0,5
–1,5
–2,5
–3,5
–4,5
–5,5
–6,5
Figure 10.5 Simulation of a hydrostatic compression test of chalk.
Table 10.1
Typical values of parameters for the non-associated
model Young’s modulus
Poisson’s ratio
Frictional coefficient
Dilatancy coefficient
Hydrostatic tensile strength
Plastic hardening
Es
vs
T0
t0
h
b
m
c
n
12 GPa
0.2
0.2
0.16
32 MPa
1.3
0.03
0.018
38
initial linear part of stressstrain curve during a conventional triaxial compression test. Then the modulus (κp , μp ) of the porous matrix at the mesoscale and the ones (κs , μs ) of the solid phase at the microscale can calculated by inverting Eqs. (10.32) and (10.31), respectively. The plastic parameters are then determined from the numerical fitting of a hydrostatic compression test (see Fig. 10.5). The typical values obtained are given in Table 10.1. We can see that the two plastic strain stages are correctly described in Fig. 10.5 by the model. In the first stage, a high plastic strain rate is obtained due to important plastic collapse of interparticle pores. In the second stage, due to the progressive increase of contacts between particles and also to the decrease of two porosities, the plastic strain rate is decreasing like the consolidation mechanism in granular materials. Further, the good agreement between the numerical results and experimental data indicates that the model’s parameters are well identified.
10.5.2 Experimental validation of micromacro model The proposed nonassociated micromacro model is now used to describe the mechanical behavior of the Lixhe chalk with the values given in Table 10.1. Using the same set
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Porous Rock Failure Mechanics
(B)
(A) –20 –18
–25
Σ11–Σ33(MPa)
–16
E33
Σ11–Σ33(MPa)
–20
E11
–14
E33
–12
E11
–15
–10 –10
–8 Experiment
–6
Experiment
Simulation
–4
Simulation
–5
–2 (%)
0 3
1
–1
–3
(%)
0
–7 2
–5
1
0
–1
–2
–3
–4
–5
–6
Figure 10.6 Comparison of stressstrain curves between numerical results and experimental data—triaxial compression test on Lixhe chalk with different confining pressures. (A) 14 MPa, (B) 17 MPa.
(B)
(A) 0,41
0,0505
f
0,4
14 MPa 17 MPa
0,39
φ
0,05
14 MPa 17 MPa
0,0495
0,38
0,049
0,37
0,0485
0,36
0,048 0,0475
0,35 E11
0,34
0
–0,02
–0,04
–0,06
E11
0,047
0
–0,02
–0,04
–0,06
Figure 10.7 Evolution of f and φ predicted by the proposed model with different confining pressures: (A) microporosity f, (B) mesoporosity φ.
of parameters, numerical simulations are then performed for triaxial compression tests with different confining pressures in order to verify the capacity of the proposed model. Fig. 10.6 illustrates the comparisons of stressstrain curves between experimental data and numerical results with 14 and 17 MPa confining pressures. Generally, an overall good agreement is observed. Both the axial and lateral strains are well predicted by the proposed model. The proposed model is able to capture the main features of the mechanical behaviors of Lixhe chalk, such as the dependence of confining pressure, effect of inter-particle and intra-particle pores, compressibility of the solid phase. Fig. 10.7 shows the evolutions of microporosity f in the matrix and of mesoporosity φ predicted by the proposed model as functions of axial strain E11 during triaxial compression tests with different confining pressures (14 and 17 MPa). It is found that the evolutions of f and φ are different. The changes of the microstructure affect the macroscopic plastic criterion (10.30) and the potential (10.33) that control the macroscopic behavior. With the decrease of the porosity, the plastic yield
Multiscale modeling approaches and micromechanics of porous rocks
231
surface expand and the strength of the studied chalk increases. The mechanical behavior of the Lixhe chalk is clearly sensitive to these two families of porosities. This is the main difference between the micromechanics based model and the phenomenological one.
10.6
Concluding remark
In this chapter, as an example of multiscale approaches, we have presented a micromechanical elastic-plastic model for a class of porous rocks with two populations of pores and a pressure sensitive solid matrix. Using a two-step homogenization procedure and a limit analysis method, an analytical macroscopic yield criterion has been established. This criterion explicitly takes into account the different effects of two populations of pores at different scales. Completed with a plastic potential and hardening law, a micromechanical elastic-plastic model has been formulated. This model can be as easily as any phenomenological models implemented into a standard computer code and then applied to engineering applications. The micromacro model has been applied to describe the macroscopic behavior of a typical porous rock, the Lixhe chalk. It was found that the mean features of the mechanical behavior of the studied material were correctly described by the micromechanical model. In the future work, the micromechanical model can be extended to saturated and partially saturated porous materials by considering effects of fluid pressure in two populations of pores.
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