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Poroelastic solution to the Brazilian test
International Journal of Rock Mechanics & Mining Sciences 126 (2020) 104201
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International Journal of Rock Mechanics and Mining Sciences journal homepage: http://www.elsevier.com/locate/ijrmms
Poroelastic solution to the Brazilian test Amin Mehrabian a, *, Younane N. Abousleiman b a
Pennsylvania State University, Department of Energy and Mineral Engineering, Earth and Mineral Sciences Energy Institute, 102 Hosler Building, University Park, 16802, PA, USA b University of Oklahoma, PoroMechanics Institute, Mewbourne School of Petroleum and Geological Engineering, School of Civil Engineering and Environmental Science, ConocoPhillips School of Geology and Geophysics, 100 E Boyd Street, SEC R210, Norman, OK, 73019, USA
A B S T R A C T
The fully-coupled, poroelastic solution for the time-dependent stress and pore fluid pressure of a hollow or solid, cylindrical specimen of porous, fluid-saturated rock undergoing the diametral compression test is derived and presented. The long-time asymptote of the poroelastic solution for the case of solid specimen recovers the related elastic solution of Hondros.9 Findings reveal the nontrivial trend and substantial effect of the pore fluid pressure disturbances on the rock effective stress. The failure of solid cylinder is predicted to occur instantaneously after the compressive load is applied, yet, at a point away from the specimen center point. Conversely, delayed failure of hollow cylinder may occur after the pore fluid disturbances within the specimen diminish.
1. Introduction The Brazilian test, also known as the indirect tensile strength test, is a standardized laboratory method for measuring the tensile strength of brittle solids including rocks.1,2 The procedure involves diametral compression of a disk-shaped specimen of the tested material until failure. The failure is expected and often observed to occur through tensile crack development along the loading line. The following formula is recommended for estimating the sample tensile strength, TS ¼
2F
πDL
(1)
where TS and F denote the tensile strength and compressive load at failure, while D and L are the sample diameter and thickness. Eq. (1) was suggested after the pioneering works by Carneiro3 and Akazawa4 for determining the tensile strength of concrete.5 A summary of the related analytical solutions is outlined, as follows. The earlier solutions for diametral compression of elastic cylinders can be found in Refs. 6–8 The complete closed-form solution to the elastic stress problem involving diametral compression of disks or cylinders using Airy’s stress function method was reported in Ref. 9 A similar solution approach for the case of hollow, disk-shaped specimens was taken in Ref. 10 A solution for generalized plane-stress configuration of the Brazilian test is re ported in Ref. 11 The solution for the time-dependent stress of linearly viscoelastic rock samples by direct adoption of the elastic solution and by exploiting the principle of correspondence between elasticity and viscoelasticity is presented in Ref. 12 The solution for transversely
isotropic rock samples was presented in Ref. 13 and was later elaborated on in Ref. 14 The tangential traction on the sample boundary arising from friction with loading platens was accounted for in Ref. 15 A poroelastic solution to diametral compression of water-saturated disks is presented in Ref. 16 The Hertzian contact interaction between the loading platen and circular sample is considered in Refs. 17,18 The fully three-dimensional solution that can accommodate arbitrary loading conditions on the boundary of finite-length cylindrical samples is pro vided in Ref. 19 The focus of this paper is on the coupled interaction of pore fluid flow and pressure disturbances with the stress evolution and failure of the rock sample undergoing the Brazilian-type test. Fig. 1 shows the sche matics of the test where a hollow, disk-shaped specimen of porous, fluidsaturated rock is diametrically compressed between two loading platens. The setup can be immersed in a bath of saturating fluid so that during the test the rock close to the sample boundaries remains fluid saturated. The flat faces of the specimen are sealed so that the desired in-plane flow of the pore fluid is upheld.20 In the case of permeable condition at the outer boundary, the platens inner surfaces are perforated to maintain the intended drained conditions. Under the described setup and loading conditions, the stress and failure of porous rock will depend on the coupling between the pore fluid flow and solid deformation. The key assumptions of the solution are listed as follows: � The specimen is homogenous and isotropic. � The solid phase of the specimen follows a linearly elastic constitutive model.
* Corresponding author. E-mail addresses: [email protected] (A. Mehrabian), [email protected] (Y.N. Abousleiman). https://doi.org/10.1016/j.ijrmms.2019.104201 Received 4 April 2019; Received in revised form 29 October 2019; Accepted 27 December 2019 Available online 3 January 2020 1365-1609/© 2019 Elsevier Ltd. All rights reserved.
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
� r2 σkk
� 3ðνu νÞ p ¼0 2Bð1 νÞð1 þ νu Þ
(3)
where (4a)
σ kk ¼ σrr þ σθθ for the plane-stress condition, and
σ kk ¼ ð1 þ νÞðσ rr þ σθθ Þ
3ðνu νÞ p Bð1 þ νu Þ
(4b)
for the plane-strain condition. In Eqs. (3) and (4b) p denotes the pore fluid pressure. The stress tensor and pore fluid pressure in Eqs. (2a), (2b) and (3) are coupled through the poroelastic constitutive relations and by incorpo rating the kinematic variable defined by the divergence of the vector defined by relative displacement of the fluid phase with respect to the solid phase, ζ. The fluid mass conservation equation, along with Darcy’s law of fluid flow within porous media, yields the following equation for ζ. Fig. 1. Schematics of the poroelastic Brazilian-type test configuration.
dζ dt
� The specimen is fully saturated with the saturating fluid. � The stress state of specimen is plane-stress or plane-strain. � Pre-existing flaws or fractures do not exist within the specimen.
where the Laplacian operator r2 is defined as, � � 1 ∂ ∂ 1 ∂ r2 ¼ r þ 2 r ∂r ∂r r ∂θ
Under these assumptions, Biot’s theory of poroelasticity21,22 is used to derive an analytical solution for the time-dependent stress of the hollow and fluid-saturated specimens subjected to loading and setup of the Brazilian test. The long-time asymptote of the solution, corre sponding to complete fluid drainage and re-equilibrium of pore fluid pressure, will recover the related elastic solution.
c¼
2GB2 kð1 νÞð1 þ νu Þ2 9μðνu νÞð1 νu Þ
(7)
where k is the rock permeability and μ is the saturating fluid viscosity. The mixed-stiffness representation of the constitutive equation for volumetric deformation of the fluid and solid phases can be expressed in the following form.
Plane-stress and plane-strain models of the configuration and loading conditions shown in Fig. 1 are considered. If a plane stress model is intended, the out-of-plane components of the stress tensor are assumed to vanish, i.e., σ zz ¼ σ rz ¼ σ θz ¼ 0. The assumption is justified by the lower threshold of the recommended ratio of the sample thickness to diameter L=R ffi 0:2 in Brazilian test.1 Conversely, a plane-strain model would be suitable for diametral compression of a long cylinder (L≫ R). In this case, the out-of-plane components of the strain tensor are assumed to vanish, i.e., εzz ¼ εrz ¼ εθz ¼ 0. Poroelastic formulation of rock deformation can be characterized by four independent constitutive constants. The following poroelastic constants are consistently used in the proceeding derivations23,24; shear modulus, G, drained and un drained Poisson’s ratios, ν and νu , as well as Skempton’s pore fluid pressure coefficient B. The mathematical formulation of quasi-static deformation of the rock sample requires solving Navier’s equations of static stress equilib rium. Formulation of Navier’s equations in polar coordinates takes the following form.25
1 ∂σθθ ∂σ rθ 2σrθ þ þ ¼ 0; r ∂θ ∂r r
(6)
The hydraulic diffusivity in Eq. (5) is defined as follows,
2. Analytical solution
∂σ rr 1 ∂σ rθ σrr σθθ þ þ ¼ 0; ∂r r ∂θ r
(5)
cr2 ζ ¼ 0
σ kk ¼
3 2GBð1 þ νÞð1 þ νu Þ pþ ζ B 3ðνu νÞ
(8)
A solution approach similar to the published work on the generalized poroelastic wellbore problem26 is herein taken. The Fourier series ex pansions of the field variables in terms of the polar angle θ are expressed through the following equations. ζðr* ; θ; t* Þ ¼
∞ X
(9)
ζm ðr* ; t* Þcos mθ
m¼0
p* ðr* ; θ; t* Þ ¼
∞ X
(10)
σ *θθm ðr* ; t* Þcos mθ
m¼0
σ *rr ðr* ; θ; t* Þ ¼
∞ X
σ *rrm ðr* ; t* Þcos mθ
(11)
σ *θθm ðr* ; t* Þcos mθ
(12)
σ*rθm ðr* ; t* Þsin mθ
(13)
m¼0
σ *θθ ðr* ; θ; t* Þ ¼
(2a)
∞ X m¼0
σ *rθ ðr* ; θ; t* Þ ¼
(2b)
∞ X m¼1
The star (*) in Eqs. (9) to (13) indicates a dimensionless variable after scaling the radial distance, time and stresses or pressure by the corre
Eqs. (2a) and (2b) contain three unknown stress components σ rr , σ θθ , and σrθ . Closure of the problem formulation is attained by the stress compatibility equation. It can be written in terms of the first invariant of stress tensor σ kk and pore fluid pressure p, as follows,
2
sponding characteristic quantities ro ¼ rr* , t0 ¼ rco ¼ tt* and σ 0 ¼ 2LðrFo σ ij σ *ij
¼
p ; p*
ri Þ
¼
respectively. Unless otherwise specified, In the proceeding
mathematical derivations the star (*) will be dropped from the symbols
2
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
with the intended connotation that the associated dimensionless quan tity is implied. Substituting Eq. (9) in Eq. (5) and taking the Laplace transform of ζ with respect to the time variable would result in the following ordinary differential equation (ODE), � � ~ � � 2 1 ∂ ∂ζ m ~ζm ¼ 0 þ s (14) r m r ∂r ∂r r2 where tilde (~) denotes the Laplace transform of a function expressed in terms of the transform variable, s. Eq. (14) is a modified Bessel ODE, the solution of which can be written in terms of the mth order modified Bessel functions of first kind Im and second kind Km , as follows. pffiffi � pffiffiffi � ~ζðr; sÞ ¼ A’ð1Þ ðsÞ Im s r þ A’ð2Þ ðsÞ Km s r (15) m m
νÞð1 þ νu Þ2 2 rζ νÞð1 νu Þ
2GB2 ð1 9σ 0 ðνu
;
A’ð2Þ m ¼
m>0
9σ 0 ðνu 2GB2 ð1 9σ 0 ðνu 2GB2 ð1
νÞð1 νu Þ ð1Þ Am νÞð1 þ νu Þ2
(18)
νÞð1 νu Þ ð2Þ Am νÞð1 þ νu Þ2
(19)
Bð1
6ðνu νÞ νÞð1 þ νu Þ
3 B
β2 ¼
(25)
(26)
m
3 Bð1 þ νu Þ
mþ2
(28a) �
þA5 rm
; ðm>0Þ
The stress components are then found by substituting Eq. (28) into Eqs. (23) to (25). The results are expressed, as follows.
σ~rr0 ¼
β1 � ð1Þ pffiffi � A0 I 1 s r r√s
ð2Þ
A0 K1
pffiffi �� β2 ð3Þ Að5Þ s r þ A0 þ 02 2 r
pffiffi � pffiffi ��� β1 � ð1Þ �pffiffi s r Im 1 s r mðm þ 1ÞIm s r A r2 s m �pffiffi pffiffi � pffiffi �� s r Km 1 s r mðm þ 1ÞKm s r þAð2Þ m
(29a)
σ~rrm ¼
β2 � ðm 4
(20b)
Að5Þ m mðm
σ~θθ0 ¼ (21a)
m 2ÞAð3Þ m r
1Þrm
2
ðm þ
m 2ÞAð4Þ m r
Að6Þ m mðm þ 1Þr
ðmþ2Þ
�
(29b)
; ðm > 0Þ
pffiffi �� pffiffi � pffiffi ��� β2 ð3Þ β1 � ð1Þ � pffiffi � ð2Þ � þ A0 A0 I0 s r þ I2 s r þ A0 K0 s r þ K2 s r 2 2 ð5Þ
A0 r2
(30a)
(22a)
3ðνu νÞ νÞð1 þ νu Þ
(27b)
(28b)
for the plane-stress case, and Bð1
1 ∂2 F r ∂r ∂θ
� � ð3Þ mþ2 pffiffi � pffiffi �� ~ m ¼ β1 Að1Þ Im s r þAð2Þ Km s r þ β2 F A r þAð4Þ m m r s m 4ð1 þ mÞ m
(17b)
where
β1 ¼
1 ∂F r2 ∂θ
þ A6 r
� pffiffi � pffiffiffi �� s r þ Að2Þ s r σ~rrm þ σ~θθm ¼ β1 Að1Þ m Im m Km � ð3Þ m � ð4Þ m Am r ; m > 0 þ β 2 Am r
β2 ¼
σ rθ ¼
� pffiffi � pffiffiffi �� β2 ð2Þ ð3Þ ð5Þ ~ 0 ¼ β1 Að1Þ F I0 s r þ A0 K0 s r þ A r2 þ A0 ln r s 0 4ð1 þ mÞ 0
Note that the compatibility of stresses as formulated by Eq. (3) is readily secured by the obtained pore fluid pressure solution in Eq. (17). Substitution of Eqs. (4), (15) and (17) in Eq. (8) gives: � pffiffi � pffiffiffi �� ð2Þ ð3Þ s r þ A0 K 0 s r þ β 2 A0 σ~rr0 þ σ~θθ0 ¼ β1 Að1Þ (20a) 0 I0
β1 ¼
(24)
The particular and homogenous solutions to Eq. (27) can be super posed to obtain the general solution for F~m , as follows.
where A’ð1Þ m ¼
∂2 F ∂r2
� � � pffiffi � pffiffiffi �� 1 ∂ ∂F~ m m2 ~ Fm ¼ β1 Að1Þ s r þ Að2Þ s r r m Im m Km r ∂r ∂r r2 � ð3Þ m � m ; ðm > 0Þ þ β2 Am r Að4Þ m r
(16)
m
σ θθ ¼
By comparing Eqs. (20) and (26) and after applying Fourier series expansion, as well as Laplace transform, the following equation for Biot’s stress function can be obtained. � � � ð1Þ pffiffi � pffiffiffi �� 1 ∂ ∂F~ 0 ð2Þ ð3Þ (27a) r ¼ β1 A0 I0 s r þ A0 K0 s r þ β2 A0 r ∂r ∂r
The general solution for the Laplace transform of the Fourier har monics of p can be obtained after superposing the particular solution of Eq. (16) on the corresponding homogenous solution. The result can be expressed, as follows. � pffiffi � pffiffiffi �� ð2Þ ð3Þ p~0 ðr; sÞ ¼ Að1Þ s r þ A0 K0 s r þ A0 (17a) 0 I0 � pffiffi � pffiffiffi �� m ð4Þ p~m ðr; sÞ ¼ Að1Þ s r þ Að2Þ s r þ Að3Þ m Im m Km m r þ Am r
(23)
ðσ rr þ σ θθ Þ ¼ r2 F
and in Eq. (15) are the arbitrary functions of the solution. In the absence of the body forces, the following relation between the pressure and fluid content increment can be derived by substituting Eq. (8) in the stress compatibility relation of Eq. (3), r2 p ¼
1 ∂F 1 ∂2 F þ r ∂r r2 ∂θ2
The specific forms of Eqs. (23) to (25) offer a general solution to Eqs. (2a) and (2b). Moreover, adding the left and right sides of Eqs. (23) and (24) results in the following equation.
ð2Þ A’ m
ð1Þ A’ m
σ rr ¼
σ~θθm ¼ (21b)
� Að1Þ m β1 Im 4
Að2Þ β � þ m 1 Km 4
2
2
pffiffi � pffiffi � pffiffi �� s r þ 2Im s r þ Imþ2 s r
pffiffi � pffiffi � pffiffi �� s r þ 2Km s r þ Kmþ2 s r
β � m þ 2 ðm þ 2ÞAð3Þ m r 4
(22b)
þAð5Þ m mðm
for the plane-strain case. The stress components may be expressed in terms of Biot’s stress function F, as follows.22
σ~rθ0 ¼ 0 3
1Þrm
2
ðm
þ Að6Þ m mðm þ 1Þr
2ÞAð4Þ m r ðmþ2Þ
(30b)
� m ; ðm > 0Þ (31a)
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
A. Mehrabian and Y.N. Abousleiman
σ~rθm ¼ � Að2Þ m β1 m r2 s
�pffiffi Að1Þ m β1 m s rIm r2 s
pffiffi s rKm
1
þAð5Þ m mðm
1
pffiffi � s r
ðm þ 1ÞIm
pffiffi �� s r
pffiffi � pffiffi �� mβ � ð4Þ m s r þ ðm þ 1ÞKm s r þ 2 Að3Þ m r þ Am r 4 1Þrm
2
Að6Þ m mðm þ 1Þr
ðmþ2Þ
m
; ðm > 0Þ (31b)
Eqs. (17) and (29)–(31) include four arbitrary functions, A0 ði ¼ 1; 2; 3; 5Þ, for zeroth Fourier harmonic and six arbitrary functions, AðiÞ m ði ¼ 1;::;6Þ, for all other Fourier harmonics of the pore fluid pressure and stress components. The solution can be completed by obtaining these arbitrary functions from the problem boundary conditions at r ¼ ri and r ¼ 1. The boundary conditions are outlined, as follows. ðiÞ
Fig. 2. Pore fluid pressure distribution across the cylinder wall.
~m σ~rrm ð1; sÞ ¼ P
(32)
σ~rθm ð1; sÞ ¼ 0
(33)
p~ð1; sÞ ¼ 0
ðPermeableÞ
∂p~ ð1; sÞ ¼ 0 ðImpermeableÞ ∂r
(34a) (34b)
σ~rrm ðri ; sÞ ¼ 0
(35)
σ~rθm ðri ; sÞ ¼ 0
(36)
p~ðri ; sÞ ¼ 0
ðPermeableÞ
∂p~ ðr ; sÞ ¼ 0 ðImpermeableÞ ∂r i
(37a) (37b)
Eqs. (34a) and (37a) denote permeable flow condition at the outer and inner circular boundaries of the tested rock sample while Eqs. (34b) and (37b) formulate the corresponding impermeable conditions. Eqs. (33) and (36) describe zero shear tractions at either boundary. Eq. (35) pertains to the vanishing normal traction at the inner boundary. Eq. (32) pertains to the Laplace transform of the mth Fourier harmonic of the load traction on the sample outer boundary. A sudden, continued, uniform normal traction, PðnÞ ðtÞ ¼ P HðtÞ, over an arc angle 2φ is herein considered. The mathematical expression for ~ m , would find the Fourier series expansion of dimensionless quantity, P following general form.9 8 φ m¼0 > > > < 2ð1 r Þ sin 2 nφ i ~m ¼ P (38) m ¼ 2n πsφ > n > > : 0 m ¼ 2n þ 1
Fig. 3. The ratio of hoop stress at the inner wall of hollow sample to hoop stress at the center of solid sample vs. the inner radius of the hollow sample.
~ where the Laplace transform of Heaviside function, HðsÞ ¼ 1=s, is included in the formulation. Any other loading rate profile could be conveniently formulated instead of Eq. (38). The engineering implication of sudden load is that the time taken by the platens load to reach the peak value, F ¼ 2Pφro L, is negligible compared to the characteristic poroelastic time scale defined by the rock hydraulic diffusivity parameter, t0 ¼ r2o =c. The distribution of contact normal stress and possible tangential stress at the interface of the tested sample and loading platen is indeed nonuniform. However, uniform distribution of the contact stresses is considered here in accordance with earlier formulations of Hondros’s solution9 so that a comparison with the special case of the general solution of this paper is made conveniently. Analytical and experimental evaluation of the effect of nonuniform load distribution on the sample response is made in Refs. 17,18. The resulting inaccuracy in stress prediction is local and at close vicinity of the contact line. Because of the Saint-Venant principle, the effect must diminish at farther points, in particular, at and in close
Fig. 4. Stress concentration around an infinitesimally small hole at the center of the compressed specimen in Brazilian test.
4
�
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
Fig. 5. Temporal evolution of Terzaghi’s effective hoop stress, as well as the pore fluid pressure of hollow cylinder. Positive stress values imply tension. ri ¼ 0:5.
140 of.30
vicinity of the inner boundary where the failure initiates. This matter is illustrated, e.g., in Figs. 5 and 6 of 27 In the proceeding discussion of the results, the obtained solution in Laplace transform space is numerically converted back to time domain using the Stehfest algorithm.28
3. Special-case solutions The generalized Lam�e’s solution for poroelastic hollow cylinders31 can be considered as a special case corresponding to the zeroth harmonic of the solution presented in section 2 when the only nonzero solution ðiÞ ~0 6¼ 0, while letting all other harmonics vanish, i. terms are A 6¼ 0 and P
2.1. Validity and uniqueness of the solution
0
~ e., AðiÞ m ¼ 0 and Pm ¼ 0, ðm > 0Þ. The classical Brazilian test uses solid, disk-shaped rock specimens. A solution to this case is provided in Ref. 16 The general solution presented in this paper must recover the solution of16 for the special case of ri ¼ 0. Consequently, the long-time asymptote of the same special-case solution must recover the Hondros’s solution to the Brazilian test.9 The described special cases will be discussed in the following subsections. A comparison between the solutions for hollow
The validity of the obtained solution may be conveniently verified by substituting Eqs. (17) and (29)–(31) in the static equilibrium and stress compatibility governing equations of poroelasticity, i.e., Eqs. (2a), (2b), and (3), as well as the boundary conditions outlined in Eqs. (32) to (37). The presented solution is indeed unique owing to the uniqueness theo rem of well-posed problems of poroelasticity. Details of the theorem proof may be found on pages 214 and 215 of,29 as well as pages 139 and 5
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
Fig. 6. Temporal evolution of Terzaghi’s effective hoop stress, as well as the pore fluid pressure of hollow cylinder. Positive stress values imply tension. ri ¼ 0:25.
and solid cylinder and discussion of the compatibility of the two solu tions when ri →0 is made in subsection 3.4.
pressure solution from Eqs. (17) and (10) of this work, along with the pore pressure solution from Eqs. ((21) and (23)–(26) and (32)–(34) of,31 is displayed in Fig. 2. It is assumed that ri ¼ 0:5 and the poroelastic parameters values are B ¼ 0:85, ν ¼ 0:25, and νu ¼ 0:4. The results perfectly match.
3.1. Generalized Lam�e’s solution Axisymmetric deformation of a hollow, elastic cylinder with inner and outer radii ri and ro and subjected to internal and external uniform normal tractions Pi and Po is known as Lam�e’s problem of elasticity.25 Lam�e’s problem is generalized by considering the time-dependent deformation of poroelastic cylinders, while assuming additional boundary conditions due to the pore fluid pressures pi and po at the inner and outer walls of the cylinder, as well as the imposed axial strain.ε*zz 31 The case of nonzero normal traction Po 6¼ 0 on the outer boundary and Pi ¼ pi ¼ po ¼ ε*zz ¼ 0 corresponds to the zeroth harmonic of the solution presented in this work, i.e., Po ¼ P0 when the permeable boundary conditions of Eqs. (34a) and (37a) are applied. The resulting pore fluid
3.2. Solid poroelastic cylinder solution Solid disk-shaped samples of rock are tested in the classical Brazilian test. The solution to this case can be recovered from the presented general solution by setting ri ¼ 0. Therefore, the coefficients of Im ðrÞ and r
m
Að6Þ m
ð4Þ functions in Eqs. (17) and (29)–(31) must vanish, i.e., Að2Þ m ¼ Am ¼
¼ 0, so that the solution to this special case at ri ¼ 0 remains finite.
ð3Þ ð5Þ The resulting solution after solving for Að1Þ m ; Am and Am from Eqs. (32) to (34a) is given in below lines.
6
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
Fig. 7. Temporal evolution of Terzaghi’s effective hoop stress, as well as the pore fluid pressure of solid cylinder. Positive stress values imply tension.
7
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
� � � pffiffi � pffiffi � 1 ~ 2n β1 2nð2n 1ÞI2n s r p ffi ffi I s r P 2nþ1 r2 s β2 sr n¼0 � pffiffi pffiffi � ð2n 2ÞI2n ð s Þ 2n β1 pffiffi I2nþ1 s r 2n 4 β s � �2 � � pffiffi � ð2n 1Þβ1 1 cos 2 nθ pffiffi þ I2n s r2n 2 (43) 4 sβ2 f2n ð s Þ ∞ X
σ~rr ðr; θ; sÞ ¼
σ~θθ ðr; θ; sÞ ¼
σ~rθ ðr; θ; sÞ ¼
Fig. 8. Terzaghi’s effective hoop stress distribution along the solid sample’s vertical axis of symmetry. Positive values imply tension.
p~m ðr; θ; sÞ ¼
Elastic
Poroelastic
Elastic
Location
r ¼ ri
r ¼ ri
r > ri
r ¼0
Time
t→∞
t ¼ 0þ
t ¼ 0þ
t ¼ 0þ
Failure load
Felastic
Felastic
F > Felastic
Felastic
�
~ 2n 2nP
∞ � pffiffi � 1 X ~ 2n I2n s rm P β2 n¼0
I2n
pffiffi �� cos 2 nθ pffiffi s r f2n ð s Þ
(45)
(46)
2m Im ðxÞ x
Imþ1 ðxÞ ¼ Im 1 ðxÞ
Solid disk
Poroelastic
� pffiffi pffiffi � β1 ð2n 1ÞI2n ð s rÞ I2nþ1 ð s rÞ p ffiffi sr2 β2 sr n¼1 � pffiffi pffiffi � Im ð s Þ 2n β1 pffiffi I2nþ1 s r þ þ 4 β2 s � � � � pffiffi � ð2n 1Þβ1 1 sin 2 nθ pffiffi þ I2n s r2n 2 4 f2n ð s Þ sβ2
∞ X
Eqs. (43)–(46) are identical with Eqs. (30)–(33) of.16 Note that re cursions of the following identity are used in deriving Eqs. (39) to (45).
Table 1 Comparison between predictions of failure location, time and load by the poroelastic and elastic solutions. Hollow disk
� � � � � pffiffi pffiffi � 2nð2n 1Þ ~ 2n β1 I2nþ1pð ffiffi s rÞ s r P þ 1 I 2n sr2 β2 sr n¼0 � pffiffi pffiffi � ð2n þ 2ÞI2n ð s Þ 2n β1 pffiffi I2nþ1 s þ r þ 2n 4 β2 s � � � � pffiffi � ð2n 1Þβ1 1 cos 2 nθ pffiffi þ I2n s r2n 2 (44) 4 sβ2 f2n ð s Þ ∞ X
(47)
parameter μ in Ref. 16 is same as β1 =β2 in this work. 3.3. Elastic solution of the Brazilian test The instantaneous (t ¼ 0þ ) response and long-time (t→∞) asymp tote of the solution presented in section 3.2 can be obtained using the initial and final value theorems of Laplace transform. That is, if ~fðsÞ is the
A0 ¼
~ 0 =β2 P pffiffi f0 ð s Þ
(39a)
Að1Þ m ¼
~ m =β2 P pffiffi fm ð s Þ
(39b)
limf ðtÞ ¼ lim s~f ðsÞ
(48a)
ð3Þ
pffiffi ~ I0 ð s ÞP 0 =β2 pffiffi f0 ð s Þ
(40a)
t→∞
limf ðtÞ ¼ lims~f ðsÞ
(48b)
pffiffi ~ Im ð s ÞP =β pffiffim 2 Að3Þ m ¼ fm ð s Þ
(40b)
ð1Þ
A0 ¼
� β1 pffi β2 s
pffiffi Imþ1 ð s Þ þ ðm
� ðm 1Þβ1 sβ2
1 4
� � pffiffi ~ Im ð s Þ P m
pffiffi 1Þfm ð s Þ
ð1 þ mÞβ1 Imþ1 ðxÞ β2 x
s→0
~2n terms Applying Eq. (48b) on Eqs. (43)-(45), after substitution for P from Eq. (38), results in the following relations for the pore fluid pres sure and stress components of Brazilian test on a solid disk.
σ rr ðr; θ; t → ∞Þ ¼ 2ð1
ri Þ
πφ (41b)
� ∞ � X 1 φþ
2ð1
πφ (42)
ri Þ
� φ
σ rθ ðr; θ; t → ∞Þ ¼
Substitution of Eqs. (39) to (41) back in Eq. (29), to (31), as well as Eq. (17), and subsequently, in Eqs. (10) to (13) yields:
� 1
� � 1 2 2n r r n
2
sin 2 nφ cos 2 nθ
� � � 1 1 þ r2 r2n n
2
sin 2 nφ cos 2 nθ
n¼1
σ θθ ðr; θ; t → ∞Þ ¼
where the function fm ðxÞ is defined, as follows. 1 fm ðxÞ ¼ Im ðxÞ 2
s→∞
t→0
(41a)
ð5Þ
A0 ¼ 0
Að5Þ m ¼
Laplace transform of an arbitrary function of time fðtÞ, the following conditions must hold.24
∞ � X 1 n¼1
2ð1
πφ
∞ ri Þ X
1
� r2 r2n
2
� (49)
�
sin 2 nφ sin 2 nθ
(50)
(51)
n¼1
Eqs. (49)–(51) are identical with the equations presented in Appen dix I of9 on page 267 for “σr “, “σ θ “, and “τrθ “. Note that in Ref. 9 the symbol α is used for the arc half-angle of the load distribution while in this study φ refers to the same quantity so that possible confusion with the Biot-Willis effective stress coefficient of poroelasticity is avoided. 8
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
3.4. Elastic solution to diametral compression of hollow cylinders
(r ¼ ri ; θ ¼ 0), the solid rock sample shows the highest tensile effective stress instantaneously after the load is applied (t ¼ 0þ ) and at a distance away from the center point along the sample’s vertical axis of symmetry. Figs. 5 and 6 indicate that the hollow sample is expected to exhibit a delayed failure effect due to the poroelastic coupling between the evolving pore fluid pressure and effective stress. The dilating pore space at point A induces negative pore fluid pressure shortly after the load is applied causing a reduction in the rock effective stress. The effect di minishes after the disturbances in the pore fluid pressure are dissipated and the elastic response is recovered at times long enough compared to the poroelastic time scale of the specimen. The resulting delayed in crease in the maximum tensile effective stress of the hollow sample is ð3 1:8Þ=1:8 ¼ 67% for the case of ri ¼ 0:5 and ð3:45 2:6Þ=2:6 ¼ 33% for the case of ri ¼ 0:25. Conversely, Fig. 5 indicates that the poroelastic coupling effect causes the solid sample to exhibit a ð0:9 0:305Þ=0:9 ¼ 67% decrease in the maximum tensile effective stress between the instantons and long-time responses. Therefore, the failure of the solid, fluid-saturated sample is expected to initiate immediately after the compressive load is applied and at a point away from the center of the sample. To further elucidate the matter, the instantaneous and long-time responses of the solid disk are derived from Eqs. (48a) and (48b) and expressed in terms of elementary algebraic and trigonometric functions, as follows.
The elastic solution to diametral compression of solid cylinders cannot be recovered from the general solution presented in this paper simply by allowing the limiting case of ri →0 for hollow cylinders. To further elucidate the matter, σ*θθ ðθ; ri Þ is defined as the ratio of long-time limit of the hoop stress at the inner boundary of hollow cylinder to the corresponding value of hoop stress at the center of solid disk. hollow
σ *θθ ðθ; ri Þ ¼
s~ σθθ ðr ¼ ri ; θ; s→0Þ σ hollow ðr ¼ ri ; θ; t→∞Þ lim s→0 θθ ¼ σsolid ðr ¼ 0; θ; t→∞Þ lim s~ σ solid θθ θθ ðr ¼ 0; θ; s→0Þ
(52)
s→0
The plot of σ*θθ ðθ; ri Þ versus the radius of the sample inner hole ri is shown in Fig. 3. Results closely agree with the reported elastic solutions presented in Fig. 11 of.32 The following finding is of particular interest: 8 < 6 θ¼0 * limσθθ ðθ; ri Þ ¼ 10 (53) π ri →0 : θ¼ 3 2 An implication of Eq. (53) is that the hoop stress values along and perpendicular to the diametral loading line on an infinitesimally small hole at the center of the tested sample (points A and B in Fig. 4) are 6 and 10/3 times greater than the corresponding values for solid cylinder. This finding can be explained using Kirsch solution33 for stress concentration around a circular hole subjected to far-field stresses. The hoop stress values along and perpendicular to the loading line at the center of a solid elastic disk (Brazilian test) are found as follows.9
π ro L F
π ro L F
�π � 2
¼ σ? ¼
2ð1
2ð1
πφ
The stresses outlined in Eqs. (54a) and (54b) are observed as far-field stresses from the viewpoint of an infinitesimally small hole at the center of the solid disk. Under such conditions, the Kirsch solution for stress concentration around the small hole would be obtained as follows.33
π ro L F
π ro L F
σθθ ð0; ri → 0Þ ¼ σA ¼ 3σ k σθθ
�π 2
� ; ri → 0 ¼ σ B ¼ 3 σ ?
σk ¼
10
(55b)
Results of Eqs. (55a) and (55b) precisely match the reported results of32 in accordance with the illustrations of Fig. 3 and Eq. (53). 4. Results and discussion Terzaghi’s effective stress tensor is defined, as follows.24
σ ’ij ¼ σ ij þ pδij
ð1 r2 Þsin 2 φ 1 2r2 cos 2 φ þ r4
ri Þ
�
ð1 r2 Þsin 2 φ 1 2r2 cos 2 φ þ r4
tan
1
�� � 1 þ r2 tan φ 2 1 r
(57)
� � �� 2 1 þ r2 tan 1 tan φ β2 1 r2
(58)
� 1þ
Details regarding conversion of the related Fourier series expansion to the functional forms of Eq. (57) can be found in Ref. 9 A similar approach is herein taken to derive Eq. (58). Fig. 8 shows the plots of Eqs. (57) and (58) for plane-stress and planestrain solutions. The plot clearly shows higher tensile effective stress throughout the sample at t ¼ 0þ . Points A indicate the occurrence of maximum effective tensile hoop stress along the vertical axis of sym metry. The theoretical value of the distance between point A and the sample center point, rf , can be conveniently obtained by finding the extremum of the function given in Eq. (58). The result is expressed, as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiffi rffiffiffihffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq � � 2 ð2 þ β2 Þcos 2 φ þ 2 β2 ð4 β2 Þ ð2 þ β2 Þ2 cos 2 φ 4 sin 2 φ β2 rf ¼ 2 (59)
(55a)
σ? ¼ 6
�
σ ’θθ ðr; θ ¼ 0; t ¼ 0þ Þ ¼
(54b)
3
ri Þ
πφ
(54a)
σθθ ð0Þ ¼ σ k ¼ þ 1 σθθ
σ ’θθ ðr; θ ¼ 0; t → ∞Þ ¼
(56)
A summary of the key findings regarding the location, time and required load of failure is presented in Table 1. Felastic in this context implies the failure load predicted from the elastic solution.
The tensile failure of the rock sample is assumed to be governed by the larger principal component of Terzaghi’s effective stress. Figs. 5–7 illustrate the temporal evolution of Terzaghi’s effective hoop stress, as well as pore fluid pressure of hollow and solid rock samples under planestrain conditions. The selected values for poroelastic parameters are ν ¼ 0:25; νu ¼ 0:45 and B ¼ 0:9. It is further assumed that the external load is uniformly distributed over an arc angle of 2φ ¼ 20� . The outside boundaries of the rock samples in both cases are considered permeable while the inner boundary of the hollow sample is assumed to be impermeable. Poroelasticity imposes contrasting effects on the failure tendency of rock samples in the considered cases. The time and location of the maximum effective tensile stress for each case is shown by point A in Figs. 5–7. While the maximum tensile effective stress for the hollow sample is triggered by the long-time asymptotic value (t→ ∞) and at
5. Conclusion The classical solution to the elastic problem defined by the geometry and loading setup of the Brazilian test is generalized to the timedependent poroelastic solution for diametral compression of hollow and fluid-saturated, cylindrical samples. Application of a constant diametral compressive load at t ¼ 0 would instantaneously bring the solid sample to failure. That is, the failure is expected to initiate at t ¼ 0þ . The failure, however, is not expected to initiate from the sample center due to occurrence of maximum tensile effective stress at a point away from the center point. Delayed failure of hollow samples with large-enough inner diameter may occur after complete dissipation of the resulting disturbances in pore fluid pressure. That is, the maximum 9
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
tensile effective stress is predicted to occur by the long-time asymptote (t →∞Þ of the poroelastic solution to the considered problem.
15 Lavrov A, Vervoort A. Theoretical treatment of tangential loading effects on the Brazilian test stress distribution. Int J Rock Mech Min Sci. 2002;39(2):275–283. 16 Mikami Y, Kurashige M, Imai K. Mechanical response of a water-saturated core sample under opposite diametrical loadings. Acta Mech. 2002;158(1-2):15–32. 17 Markides CF, Pazis DN, Kourkoulis SK. The Brazilian disc under non-uniform distribution of radial pressure and friction. Int J Rock Mech Min Sci. 2012;50:47–55. 18 Markides CF, Kourkoulis SK. The influence of jaw’s curvature on the results of the Brazilian disc test. J. Rock Mech. Geotech. Eng. 2016;8(2):127–146. 19 Serati M, Alehossein H, Williams DJ. s. Rock Mech Rock Eng. 2014;47(4):1087–1101. 20 Zhang J, Scherer GW. Permeability of shale by the beam-bending method. Int J Rock Mech Min Sci. 2012;53:179–191. 21 Biot MA. General theory of three-dimensional consolidation. J Appl Phys. 1941;12(2): 155–164. 22 Biot MA. General solutions of the equations of elasticity and consolidation for a porous material. J Appl Mech. 1956;23(1):91–96. 23 Rice JR, Cleary MP. Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev Geophys. 1976;14(2):227–241. 24 Wang HF. Theory of Linear Poroelasticity. Princeton Unversity Press; 2000. 25 Boresi AP, Chong K, Lee JD. Elasticity in Engineering Mechanics. John Wiley & Sons; 2010. 26 Mehrabian A, Abousleiman YN. Generalized poroelastic wellbore problem. Int J Numer Anal Methods Geomech. 2013;37(16):2727–2754, 2013. 27 Li D, Wong LNY. The Brazilian disc test for rock mechanics applications: review and new insights. Rock Mech Rock Eng. 2013;46:269–287. 28 Cheng AH, Sidauruk P, Abousleiman Y. Approximate inversion of the Laplace transform. Math J. 1994;4(2):76–82. 29 Coussy O. Poromechanics. John Wiley & Sons; 2004. 30 Cheng AHD. Poroelasticity. Springer; 2016. 31 Kanj MY, Abousleiman YN. The generalized lame problem—Part I: coupled poromechanical solutions. J Appl Mech. 2004;71(2):168–179. 32 Mellor M, Hawkes I. Measurement of tensile strength by diametral compression of discs and annuli. Eng Geol. 1971;5(3):173–225. 33 Mehrabian A. The stability of inclined and fractured wellbores. Soc Pet Eng J. 2016; 21(5):1178–1188.
References 1 ASTM D 3967-08. Standard Test Method for Splitting Tensile Strength of Intact Rock Core Specimens. ASTM International; 2008. 2 ISRM. Suggested methods for determining tensile strength of rock materials. Int J Rock Mech Min Sci. 1978;15:99–103. 3 Carneiro FLLB. A new method to determine the tensile strength of concrete. In: Proceedings of the 5th Meeting of the Brazilian Association for Technical Rules. vol 3. September 1943:126–129, 16. 4 Akazawa T. New test method for evaluating internal stress due to compression of concrete (the splitting tension test) (part 1). J. Japan Soc. Civ. Eng. 1943;29:777–787. 5 Fairbairn EM, Ulm FJ. A tribute to Fernando LLB Carneiro (1913–2001) engineer and scientist who invented the Brazilian test. Mater Struct. 2002;35(3):195–196. 6 Hertz H. On the contact of elastic solids. J für die Reine Angewandte Math (Crelle’s J). 1882;92:156–171. 7 Mitchell JH. Elementary distributions of plane stress. Proc Lond Math Soc. 1900;32: 35–61. 8 Timoshenko S. The Theory of Elasticity. New York: McGraw-Hill; 1934. 9 Hondros G. The evaluation of Poisson’s ratio and the modulus of materials of low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete. Aust J Appl Sci. 1959;10(3):243–268. 10 Jaeger JC, Hoskins ER. Rock failure under the confined Brazilian test. J Geophys Res. 1966;71(10):2651–2659. 11 Wijk G. Some new theoretical aspects of indirect measurements of the tensile strength of rocks. Int J Rock Mech Min Sci Geomech Abstr. 1978;15(4):149–160. 12 Zhang W, Drescher A, Newcomb DE. Viscoelastic analysis of diametral compression of asphalt concrete. J Eng Mech. 1997;123(6):596–603. 13 Chen CS, Pan E, Amadei B. Determination of deformability and tensile strength of anisotropic rock using Brazilian tests. Int J Rock Mech Min Sci. 1988;35(1):43–61. 14 Claesson J, Bohloli B. Brazilian test: stress field and tensile strength of anisotropic rocks using an analytical solution. Int J Rock Mech Min Sci. 2002;39(8):991–1004.
10
Contents lists available at ScienceDirect
International Journal of Rock Mechanics and Mining Sciences journal homepage: http://www.elsevier.com/locate/ijrmms
Poroelastic solution to the Brazilian test Amin Mehrabian a, *, Younane N. Abousleiman b a
Pennsylvania State University, Department of Energy and Mineral Engineering, Earth and Mineral Sciences Energy Institute, 102 Hosler Building, University Park, 16802, PA, USA b University of Oklahoma, PoroMechanics Institute, Mewbourne School of Petroleum and Geological Engineering, School of Civil Engineering and Environmental Science, ConocoPhillips School of Geology and Geophysics, 100 E Boyd Street, SEC R210, Norman, OK, 73019, USA
A B S T R A C T
The fully-coupled, poroelastic solution for the time-dependent stress and pore fluid pressure of a hollow or solid, cylindrical specimen of porous, fluid-saturated rock undergoing the diametral compression test is derived and presented. The long-time asymptote of the poroelastic solution for the case of solid specimen recovers the related elastic solution of Hondros.9 Findings reveal the nontrivial trend and substantial effect of the pore fluid pressure disturbances on the rock effective stress. The failure of solid cylinder is predicted to occur instantaneously after the compressive load is applied, yet, at a point away from the specimen center point. Conversely, delayed failure of hollow cylinder may occur after the pore fluid disturbances within the specimen diminish.
1. Introduction The Brazilian test, also known as the indirect tensile strength test, is a standardized laboratory method for measuring the tensile strength of brittle solids including rocks.1,2 The procedure involves diametral compression of a disk-shaped specimen of the tested material until failure. The failure is expected and often observed to occur through tensile crack development along the loading line. The following formula is recommended for estimating the sample tensile strength, TS ¼
2F
πDL
(1)
where TS and F denote the tensile strength and compressive load at failure, while D and L are the sample diameter and thickness. Eq. (1) was suggested after the pioneering works by Carneiro3 and Akazawa4 for determining the tensile strength of concrete.5 A summary of the related analytical solutions is outlined, as follows. The earlier solutions for diametral compression of elastic cylinders can be found in Refs. 6–8 The complete closed-form solution to the elastic stress problem involving diametral compression of disks or cylinders using Airy’s stress function method was reported in Ref. 9 A similar solution approach for the case of hollow, disk-shaped specimens was taken in Ref. 10 A solution for generalized plane-stress configuration of the Brazilian test is re ported in Ref. 11 The solution for the time-dependent stress of linearly viscoelastic rock samples by direct adoption of the elastic solution and by exploiting the principle of correspondence between elasticity and viscoelasticity is presented in Ref. 12 The solution for transversely
isotropic rock samples was presented in Ref. 13 and was later elaborated on in Ref. 14 The tangential traction on the sample boundary arising from friction with loading platens was accounted for in Ref. 15 A poroelastic solution to diametral compression of water-saturated disks is presented in Ref. 16 The Hertzian contact interaction between the loading platen and circular sample is considered in Refs. 17,18 The fully three-dimensional solution that can accommodate arbitrary loading conditions on the boundary of finite-length cylindrical samples is pro vided in Ref. 19 The focus of this paper is on the coupled interaction of pore fluid flow and pressure disturbances with the stress evolution and failure of the rock sample undergoing the Brazilian-type test. Fig. 1 shows the sche matics of the test where a hollow, disk-shaped specimen of porous, fluidsaturated rock is diametrically compressed between two loading platens. The setup can be immersed in a bath of saturating fluid so that during the test the rock close to the sample boundaries remains fluid saturated. The flat faces of the specimen are sealed so that the desired in-plane flow of the pore fluid is upheld.20 In the case of permeable condition at the outer boundary, the platens inner surfaces are perforated to maintain the intended drained conditions. Under the described setup and loading conditions, the stress and failure of porous rock will depend on the coupling between the pore fluid flow and solid deformation. The key assumptions of the solution are listed as follows: � The specimen is homogenous and isotropic. � The solid phase of the specimen follows a linearly elastic constitutive model.
* Corresponding author. E-mail addresses: [email protected] (A. Mehrabian), [email protected] (Y.N. Abousleiman). https://doi.org/10.1016/j.ijrmms.2019.104201 Received 4 April 2019; Received in revised form 29 October 2019; Accepted 27 December 2019 Available online 3 January 2020 1365-1609/© 2019 Elsevier Ltd. All rights reserved.
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
� r2 σkk
� 3ðνu νÞ p ¼0 2Bð1 νÞð1 þ νu Þ
(3)
where (4a)
σ kk ¼ σrr þ σθθ for the plane-stress condition, and
σ kk ¼ ð1 þ νÞðσ rr þ σθθ Þ
3ðνu νÞ p Bð1 þ νu Þ
(4b)
for the plane-strain condition. In Eqs. (3) and (4b) p denotes the pore fluid pressure. The stress tensor and pore fluid pressure in Eqs. (2a), (2b) and (3) are coupled through the poroelastic constitutive relations and by incorpo rating the kinematic variable defined by the divergence of the vector defined by relative displacement of the fluid phase with respect to the solid phase, ζ. The fluid mass conservation equation, along with Darcy’s law of fluid flow within porous media, yields the following equation for ζ. Fig. 1. Schematics of the poroelastic Brazilian-type test configuration.
dζ dt
� The specimen is fully saturated with the saturating fluid. � The stress state of specimen is plane-stress or plane-strain. � Pre-existing flaws or fractures do not exist within the specimen.
where the Laplacian operator r2 is defined as, � � 1 ∂ ∂ 1 ∂ r2 ¼ r þ 2 r ∂r ∂r r ∂θ
Under these assumptions, Biot’s theory of poroelasticity21,22 is used to derive an analytical solution for the time-dependent stress of the hollow and fluid-saturated specimens subjected to loading and setup of the Brazilian test. The long-time asymptote of the solution, corre sponding to complete fluid drainage and re-equilibrium of pore fluid pressure, will recover the related elastic solution.
c¼
2GB2 kð1 νÞð1 þ νu Þ2 9μðνu νÞð1 νu Þ
(7)
where k is the rock permeability and μ is the saturating fluid viscosity. The mixed-stiffness representation of the constitutive equation for volumetric deformation of the fluid and solid phases can be expressed in the following form.
Plane-stress and plane-strain models of the configuration and loading conditions shown in Fig. 1 are considered. If a plane stress model is intended, the out-of-plane components of the stress tensor are assumed to vanish, i.e., σ zz ¼ σ rz ¼ σ θz ¼ 0. The assumption is justified by the lower threshold of the recommended ratio of the sample thickness to diameter L=R ffi 0:2 in Brazilian test.1 Conversely, a plane-strain model would be suitable for diametral compression of a long cylinder (L≫ R). In this case, the out-of-plane components of the strain tensor are assumed to vanish, i.e., εzz ¼ εrz ¼ εθz ¼ 0. Poroelastic formulation of rock deformation can be characterized by four independent constitutive constants. The following poroelastic constants are consistently used in the proceeding derivations23,24; shear modulus, G, drained and un drained Poisson’s ratios, ν and νu , as well as Skempton’s pore fluid pressure coefficient B. The mathematical formulation of quasi-static deformation of the rock sample requires solving Navier’s equations of static stress equilib rium. Formulation of Navier’s equations in polar coordinates takes the following form.25
1 ∂σθθ ∂σ rθ 2σrθ þ þ ¼ 0; r ∂θ ∂r r
(6)
The hydraulic diffusivity in Eq. (5) is defined as follows,
2. Analytical solution
∂σ rr 1 ∂σ rθ σrr σθθ þ þ ¼ 0; ∂r r ∂θ r
(5)
cr2 ζ ¼ 0
σ kk ¼
3 2GBð1 þ νÞð1 þ νu Þ pþ ζ B 3ðνu νÞ
(8)
A solution approach similar to the published work on the generalized poroelastic wellbore problem26 is herein taken. The Fourier series ex pansions of the field variables in terms of the polar angle θ are expressed through the following equations. ζðr* ; θ; t* Þ ¼
∞ X
(9)
ζm ðr* ; t* Þcos mθ
m¼0
p* ðr* ; θ; t* Þ ¼
∞ X
(10)
σ *θθm ðr* ; t* Þcos mθ
m¼0
σ *rr ðr* ; θ; t* Þ ¼
∞ X
σ *rrm ðr* ; t* Þcos mθ
(11)
σ *θθm ðr* ; t* Þcos mθ
(12)
σ*rθm ðr* ; t* Þsin mθ
(13)
m¼0
σ *θθ ðr* ; θ; t* Þ ¼
(2a)
∞ X m¼0
σ *rθ ðr* ; θ; t* Þ ¼
(2b)
∞ X m¼1
The star (*) in Eqs. (9) to (13) indicates a dimensionless variable after scaling the radial distance, time and stresses or pressure by the corre
Eqs. (2a) and (2b) contain three unknown stress components σ rr , σ θθ , and σrθ . Closure of the problem formulation is attained by the stress compatibility equation. It can be written in terms of the first invariant of stress tensor σ kk and pore fluid pressure p, as follows,
2
sponding characteristic quantities ro ¼ rr* , t0 ¼ rco ¼ tt* and σ 0 ¼ 2LðrFo σ ij σ *ij
¼
p ; p*
ri Þ
¼
respectively. Unless otherwise specified, In the proceeding
mathematical derivations the star (*) will be dropped from the symbols
2
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
with the intended connotation that the associated dimensionless quan tity is implied. Substituting Eq. (9) in Eq. (5) and taking the Laplace transform of ζ with respect to the time variable would result in the following ordinary differential equation (ODE), � � ~ � � 2 1 ∂ ∂ζ m ~ζm ¼ 0 þ s (14) r m r ∂r ∂r r2 where tilde (~) denotes the Laplace transform of a function expressed in terms of the transform variable, s. Eq. (14) is a modified Bessel ODE, the solution of which can be written in terms of the mth order modified Bessel functions of first kind Im and second kind Km , as follows. pffiffi � pffiffiffi � ~ζðr; sÞ ¼ A’ð1Þ ðsÞ Im s r þ A’ð2Þ ðsÞ Km s r (15) m m
νÞð1 þ νu Þ2 2 rζ νÞð1 νu Þ
2GB2 ð1 9σ 0 ðνu
;
A’ð2Þ m ¼
m>0
9σ 0 ðνu 2GB2 ð1 9σ 0 ðνu 2GB2 ð1
νÞð1 νu Þ ð1Þ Am νÞð1 þ νu Þ2
(18)
νÞð1 νu Þ ð2Þ Am νÞð1 þ νu Þ2
(19)
Bð1
6ðνu νÞ νÞð1 þ νu Þ
3 B
β2 ¼
(25)
(26)
m
3 Bð1 þ νu Þ
mþ2
(28a) �
þA5 rm
; ðm>0Þ
The stress components are then found by substituting Eq. (28) into Eqs. (23) to (25). The results are expressed, as follows.
σ~rr0 ¼
β1 � ð1Þ pffiffi � A0 I 1 s r r√s
ð2Þ
A0 K1
pffiffi �� β2 ð3Þ Að5Þ s r þ A0 þ 02 2 r
pffiffi � pffiffi ��� β1 � ð1Þ �pffiffi s r Im 1 s r mðm þ 1ÞIm s r A r2 s m �pffiffi pffiffi � pffiffi �� s r Km 1 s r mðm þ 1ÞKm s r þAð2Þ m
(29a)
σ~rrm ¼
β2 � ðm 4
(20b)
Að5Þ m mðm
σ~θθ0 ¼ (21a)
m 2ÞAð3Þ m r
1Þrm
2
ðm þ
m 2ÞAð4Þ m r
Að6Þ m mðm þ 1Þr
ðmþ2Þ
�
(29b)
; ðm > 0Þ
pffiffi �� pffiffi � pffiffi ��� β2 ð3Þ β1 � ð1Þ � pffiffi � ð2Þ � þ A0 A0 I0 s r þ I2 s r þ A0 K0 s r þ K2 s r 2 2 ð5Þ
A0 r2
(30a)
(22a)
3ðνu νÞ νÞð1 þ νu Þ
(27b)
(28b)
for the plane-stress case, and Bð1
1 ∂2 F r ∂r ∂θ
� � ð3Þ mþ2 pffiffi � pffiffi �� ~ m ¼ β1 Að1Þ Im s r þAð2Þ Km s r þ β2 F A r þAð4Þ m m r s m 4ð1 þ mÞ m
(17b)
where
β1 ¼
1 ∂F r2 ∂θ
þ A6 r
� pffiffi � pffiffiffi �� s r þ Að2Þ s r σ~rrm þ σ~θθm ¼ β1 Að1Þ m Im m Km � ð3Þ m � ð4Þ m Am r ; m > 0 þ β 2 Am r
β2 ¼
σ rθ ¼
� pffiffi � pffiffiffi �� β2 ð2Þ ð3Þ ð5Þ ~ 0 ¼ β1 Að1Þ F I0 s r þ A0 K0 s r þ A r2 þ A0 ln r s 0 4ð1 þ mÞ 0
Note that the compatibility of stresses as formulated by Eq. (3) is readily secured by the obtained pore fluid pressure solution in Eq. (17). Substitution of Eqs. (4), (15) and (17) in Eq. (8) gives: � pffiffi � pffiffiffi �� ð2Þ ð3Þ s r þ A0 K 0 s r þ β 2 A0 σ~rr0 þ σ~θθ0 ¼ β1 Að1Þ (20a) 0 I0
β1 ¼
(24)
The particular and homogenous solutions to Eq. (27) can be super posed to obtain the general solution for F~m , as follows.
where A’ð1Þ m ¼
∂2 F ∂r2
� � � pffiffi � pffiffiffi �� 1 ∂ ∂F~ m m2 ~ Fm ¼ β1 Að1Þ s r þ Að2Þ s r r m Im m Km r ∂r ∂r r2 � ð3Þ m � m ; ðm > 0Þ þ β2 Am r Að4Þ m r
(16)
m
σ θθ ¼
By comparing Eqs. (20) and (26) and after applying Fourier series expansion, as well as Laplace transform, the following equation for Biot’s stress function can be obtained. � � � ð1Þ pffiffi � pffiffiffi �� 1 ∂ ∂F~ 0 ð2Þ ð3Þ (27a) r ¼ β1 A0 I0 s r þ A0 K0 s r þ β2 A0 r ∂r ∂r
The general solution for the Laplace transform of the Fourier har monics of p can be obtained after superposing the particular solution of Eq. (16) on the corresponding homogenous solution. The result can be expressed, as follows. � pffiffi � pffiffiffi �� ð2Þ ð3Þ p~0 ðr; sÞ ¼ Að1Þ s r þ A0 K0 s r þ A0 (17a) 0 I0 � pffiffi � pffiffiffi �� m ð4Þ p~m ðr; sÞ ¼ Að1Þ s r þ Að2Þ s r þ Að3Þ m Im m Km m r þ Am r
(23)
ðσ rr þ σ θθ Þ ¼ r2 F
and in Eq. (15) are the arbitrary functions of the solution. In the absence of the body forces, the following relation between the pressure and fluid content increment can be derived by substituting Eq. (8) in the stress compatibility relation of Eq. (3), r2 p ¼
1 ∂F 1 ∂2 F þ r ∂r r2 ∂θ2
The specific forms of Eqs. (23) to (25) offer a general solution to Eqs. (2a) and (2b). Moreover, adding the left and right sides of Eqs. (23) and (24) results in the following equation.
ð2Þ A’ m
ð1Þ A’ m
σ rr ¼
σ~θθm ¼ (21b)
� Að1Þ m β1 Im 4
Að2Þ β � þ m 1 Km 4
2
2
pffiffi � pffiffi � pffiffi �� s r þ 2Im s r þ Imþ2 s r
pffiffi � pffiffi � pffiffi �� s r þ 2Km s r þ Kmþ2 s r
β � m þ 2 ðm þ 2ÞAð3Þ m r 4
(22b)
þAð5Þ m mðm
for the plane-strain case. The stress components may be expressed in terms of Biot’s stress function F, as follows.22
σ~rθ0 ¼ 0 3
1Þrm
2
ðm
þ Að6Þ m mðm þ 1Þr
2ÞAð4Þ m r ðmþ2Þ
(30b)
� m ; ðm > 0Þ (31a)
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
A. Mehrabian and Y.N. Abousleiman
σ~rθm ¼ � Að2Þ m β1 m r2 s
�pffiffi Að1Þ m β1 m s rIm r2 s
pffiffi s rKm
1
þAð5Þ m mðm
1
pffiffi � s r
ðm þ 1ÞIm
pffiffi �� s r
pffiffi � pffiffi �� mβ � ð4Þ m s r þ ðm þ 1ÞKm s r þ 2 Að3Þ m r þ Am r 4 1Þrm
2
Að6Þ m mðm þ 1Þr
ðmþ2Þ
m
; ðm > 0Þ (31b)
Eqs. (17) and (29)–(31) include four arbitrary functions, A0 ði ¼ 1; 2; 3; 5Þ, for zeroth Fourier harmonic and six arbitrary functions, AðiÞ m ði ¼ 1;::;6Þ, for all other Fourier harmonics of the pore fluid pressure and stress components. The solution can be completed by obtaining these arbitrary functions from the problem boundary conditions at r ¼ ri and r ¼ 1. The boundary conditions are outlined, as follows. ðiÞ
Fig. 2. Pore fluid pressure distribution across the cylinder wall.
~m σ~rrm ð1; sÞ ¼ P
(32)
σ~rθm ð1; sÞ ¼ 0
(33)
p~ð1; sÞ ¼ 0
ðPermeableÞ
∂p~ ð1; sÞ ¼ 0 ðImpermeableÞ ∂r
(34a) (34b)
σ~rrm ðri ; sÞ ¼ 0
(35)
σ~rθm ðri ; sÞ ¼ 0
(36)
p~ðri ; sÞ ¼ 0
ðPermeableÞ
∂p~ ðr ; sÞ ¼ 0 ðImpermeableÞ ∂r i
(37a) (37b)
Eqs. (34a) and (37a) denote permeable flow condition at the outer and inner circular boundaries of the tested rock sample while Eqs. (34b) and (37b) formulate the corresponding impermeable conditions. Eqs. (33) and (36) describe zero shear tractions at either boundary. Eq. (35) pertains to the vanishing normal traction at the inner boundary. Eq. (32) pertains to the Laplace transform of the mth Fourier harmonic of the load traction on the sample outer boundary. A sudden, continued, uniform normal traction, PðnÞ ðtÞ ¼ P HðtÞ, over an arc angle 2φ is herein considered. The mathematical expression for ~ m , would find the Fourier series expansion of dimensionless quantity, P following general form.9 8 φ m¼0 > > > < 2ð1 r Þ sin 2 nφ i ~m ¼ P (38) m ¼ 2n πsφ > n > > : 0 m ¼ 2n þ 1
Fig. 3. The ratio of hoop stress at the inner wall of hollow sample to hoop stress at the center of solid sample vs. the inner radius of the hollow sample.
~ where the Laplace transform of Heaviside function, HðsÞ ¼ 1=s, is included in the formulation. Any other loading rate profile could be conveniently formulated instead of Eq. (38). The engineering implication of sudden load is that the time taken by the platens load to reach the peak value, F ¼ 2Pφro L, is negligible compared to the characteristic poroelastic time scale defined by the rock hydraulic diffusivity parameter, t0 ¼ r2o =c. The distribution of contact normal stress and possible tangential stress at the interface of the tested sample and loading platen is indeed nonuniform. However, uniform distribution of the contact stresses is considered here in accordance with earlier formulations of Hondros’s solution9 so that a comparison with the special case of the general solution of this paper is made conveniently. Analytical and experimental evaluation of the effect of nonuniform load distribution on the sample response is made in Refs. 17,18. The resulting inaccuracy in stress prediction is local and at close vicinity of the contact line. Because of the Saint-Venant principle, the effect must diminish at farther points, in particular, at and in close
Fig. 4. Stress concentration around an infinitesimally small hole at the center of the compressed specimen in Brazilian test.
4
�
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
Fig. 5. Temporal evolution of Terzaghi’s effective hoop stress, as well as the pore fluid pressure of hollow cylinder. Positive stress values imply tension. ri ¼ 0:5.
140 of.30
vicinity of the inner boundary where the failure initiates. This matter is illustrated, e.g., in Figs. 5 and 6 of 27 In the proceeding discussion of the results, the obtained solution in Laplace transform space is numerically converted back to time domain using the Stehfest algorithm.28
3. Special-case solutions The generalized Lam�e’s solution for poroelastic hollow cylinders31 can be considered as a special case corresponding to the zeroth harmonic of the solution presented in section 2 when the only nonzero solution ðiÞ ~0 6¼ 0, while letting all other harmonics vanish, i. terms are A 6¼ 0 and P
2.1. Validity and uniqueness of the solution
0
~ e., AðiÞ m ¼ 0 and Pm ¼ 0, ðm > 0Þ. The classical Brazilian test uses solid, disk-shaped rock specimens. A solution to this case is provided in Ref. 16 The general solution presented in this paper must recover the solution of16 for the special case of ri ¼ 0. Consequently, the long-time asymptote of the same special-case solution must recover the Hondros’s solution to the Brazilian test.9 The described special cases will be discussed in the following subsections. A comparison between the solutions for hollow
The validity of the obtained solution may be conveniently verified by substituting Eqs. (17) and (29)–(31) in the static equilibrium and stress compatibility governing equations of poroelasticity, i.e., Eqs. (2a), (2b), and (3), as well as the boundary conditions outlined in Eqs. (32) to (37). The presented solution is indeed unique owing to the uniqueness theo rem of well-posed problems of poroelasticity. Details of the theorem proof may be found on pages 214 and 215 of,29 as well as pages 139 and 5
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
Fig. 6. Temporal evolution of Terzaghi’s effective hoop stress, as well as the pore fluid pressure of hollow cylinder. Positive stress values imply tension. ri ¼ 0:25.
and solid cylinder and discussion of the compatibility of the two solu tions when ri →0 is made in subsection 3.4.
pressure solution from Eqs. (17) and (10) of this work, along with the pore pressure solution from Eqs. ((21) and (23)–(26) and (32)–(34) of,31 is displayed in Fig. 2. It is assumed that ri ¼ 0:5 and the poroelastic parameters values are B ¼ 0:85, ν ¼ 0:25, and νu ¼ 0:4. The results perfectly match.
3.1. Generalized Lam�e’s solution Axisymmetric deformation of a hollow, elastic cylinder with inner and outer radii ri and ro and subjected to internal and external uniform normal tractions Pi and Po is known as Lam�e’s problem of elasticity.25 Lam�e’s problem is generalized by considering the time-dependent deformation of poroelastic cylinders, while assuming additional boundary conditions due to the pore fluid pressures pi and po at the inner and outer walls of the cylinder, as well as the imposed axial strain.ε*zz 31 The case of nonzero normal traction Po 6¼ 0 on the outer boundary and Pi ¼ pi ¼ po ¼ ε*zz ¼ 0 corresponds to the zeroth harmonic of the solution presented in this work, i.e., Po ¼ P0 when the permeable boundary conditions of Eqs. (34a) and (37a) are applied. The resulting pore fluid
3.2. Solid poroelastic cylinder solution Solid disk-shaped samples of rock are tested in the classical Brazilian test. The solution to this case can be recovered from the presented general solution by setting ri ¼ 0. Therefore, the coefficients of Im ðrÞ and r
m
Að6Þ m
ð4Þ functions in Eqs. (17) and (29)–(31) must vanish, i.e., Að2Þ m ¼ Am ¼
¼ 0, so that the solution to this special case at ri ¼ 0 remains finite.
ð3Þ ð5Þ The resulting solution after solving for Að1Þ m ; Am and Am from Eqs. (32) to (34a) is given in below lines.
6
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
Fig. 7. Temporal evolution of Terzaghi’s effective hoop stress, as well as the pore fluid pressure of solid cylinder. Positive stress values imply tension.
7
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
� � � pffiffi � pffiffi � 1 ~ 2n β1 2nð2n 1ÞI2n s r p ffi ffi I s r P 2nþ1 r2 s β2 sr n¼0 � pffiffi pffiffi � ð2n 2ÞI2n ð s Þ 2n β1 pffiffi I2nþ1 s r 2n 4 β s � �2 � � pffiffi � ð2n 1Þβ1 1 cos 2 nθ pffiffi þ I2n s r2n 2 (43) 4 sβ2 f2n ð s Þ ∞ X
σ~rr ðr; θ; sÞ ¼
σ~θθ ðr; θ; sÞ ¼
σ~rθ ðr; θ; sÞ ¼
Fig. 8. Terzaghi’s effective hoop stress distribution along the solid sample’s vertical axis of symmetry. Positive values imply tension.
p~m ðr; θ; sÞ ¼
Elastic
Poroelastic
Elastic
Location
r ¼ ri
r ¼ ri
r > ri
r ¼0
Time
t→∞
t ¼ 0þ
t ¼ 0þ
t ¼ 0þ
Failure load
Felastic
Felastic
F > Felastic
Felastic
�
~ 2n 2nP
∞ � pffiffi � 1 X ~ 2n I2n s rm P β2 n¼0
I2n
pffiffi �� cos 2 nθ pffiffi s r f2n ð s Þ
(45)
(46)
2m Im ðxÞ x
Imþ1 ðxÞ ¼ Im 1 ðxÞ
Solid disk
Poroelastic
� pffiffi pffiffi � β1 ð2n 1ÞI2n ð s rÞ I2nþ1 ð s rÞ p ffiffi sr2 β2 sr n¼1 � pffiffi pffiffi � Im ð s Þ 2n β1 pffiffi I2nþ1 s r þ þ 4 β2 s � � � � pffiffi � ð2n 1Þβ1 1 sin 2 nθ pffiffi þ I2n s r2n 2 4 f2n ð s Þ sβ2
∞ X
Eqs. (43)–(46) are identical with Eqs. (30)–(33) of.16 Note that re cursions of the following identity are used in deriving Eqs. (39) to (45).
Table 1 Comparison between predictions of failure location, time and load by the poroelastic and elastic solutions. Hollow disk
� � � � � pffiffi pffiffi � 2nð2n 1Þ ~ 2n β1 I2nþ1pð ffiffi s rÞ s r P þ 1 I 2n sr2 β2 sr n¼0 � pffiffi pffiffi � ð2n þ 2ÞI2n ð s Þ 2n β1 pffiffi I2nþ1 s þ r þ 2n 4 β2 s � � � � pffiffi � ð2n 1Þβ1 1 cos 2 nθ pffiffi þ I2n s r2n 2 (44) 4 sβ2 f2n ð s Þ ∞ X
(47)
parameter μ in Ref. 16 is same as β1 =β2 in this work. 3.3. Elastic solution of the Brazilian test The instantaneous (t ¼ 0þ ) response and long-time (t→∞) asymp tote of the solution presented in section 3.2 can be obtained using the initial and final value theorems of Laplace transform. That is, if ~fðsÞ is the
A0 ¼
~ 0 =β2 P pffiffi f0 ð s Þ
(39a)
Að1Þ m ¼
~ m =β2 P pffiffi fm ð s Þ
(39b)
limf ðtÞ ¼ lim s~f ðsÞ
(48a)
ð3Þ
pffiffi ~ I0 ð s ÞP 0 =β2 pffiffi f0 ð s Þ
(40a)
t→∞
limf ðtÞ ¼ lims~f ðsÞ
(48b)
pffiffi ~ Im ð s ÞP =β pffiffim 2 Að3Þ m ¼ fm ð s Þ
(40b)
ð1Þ
A0 ¼
� β1 pffi β2 s
pffiffi Imþ1 ð s Þ þ ðm
� ðm 1Þβ1 sβ2
1 4
� � pffiffi ~ Im ð s Þ P m
pffiffi 1Þfm ð s Þ
ð1 þ mÞβ1 Imþ1 ðxÞ β2 x
s→0
~2n terms Applying Eq. (48b) on Eqs. (43)-(45), after substitution for P from Eq. (38), results in the following relations for the pore fluid pres sure and stress components of Brazilian test on a solid disk.
σ rr ðr; θ; t → ∞Þ ¼ 2ð1
ri Þ
πφ (41b)
� ∞ � X 1 φþ
2ð1
πφ (42)
ri Þ
� φ
σ rθ ðr; θ; t → ∞Þ ¼
Substitution of Eqs. (39) to (41) back in Eq. (29), to (31), as well as Eq. (17), and subsequently, in Eqs. (10) to (13) yields:
� 1
� � 1 2 2n r r n
2
sin 2 nφ cos 2 nθ
� � � 1 1 þ r2 r2n n
2
sin 2 nφ cos 2 nθ
n¼1
σ θθ ðr; θ; t → ∞Þ ¼
where the function fm ðxÞ is defined, as follows. 1 fm ðxÞ ¼ Im ðxÞ 2
s→∞
t→0
(41a)
ð5Þ
A0 ¼ 0
Að5Þ m ¼
Laplace transform of an arbitrary function of time fðtÞ, the following conditions must hold.24
∞ � X 1 n¼1
2ð1
πφ
∞ ri Þ X
1
� r2 r2n
2
� (49)
�
sin 2 nφ sin 2 nθ
(50)
(51)
n¼1
Eqs. (49)–(51) are identical with the equations presented in Appen dix I of9 on page 267 for “σr “, “σ θ “, and “τrθ “. Note that in Ref. 9 the symbol α is used for the arc half-angle of the load distribution while in this study φ refers to the same quantity so that possible confusion with the Biot-Willis effective stress coefficient of poroelasticity is avoided. 8
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
3.4. Elastic solution to diametral compression of hollow cylinders
(r ¼ ri ; θ ¼ 0), the solid rock sample shows the highest tensile effective stress instantaneously after the load is applied (t ¼ 0þ ) and at a distance away from the center point along the sample’s vertical axis of symmetry. Figs. 5 and 6 indicate that the hollow sample is expected to exhibit a delayed failure effect due to the poroelastic coupling between the evolving pore fluid pressure and effective stress. The dilating pore space at point A induces negative pore fluid pressure shortly after the load is applied causing a reduction in the rock effective stress. The effect di minishes after the disturbances in the pore fluid pressure are dissipated and the elastic response is recovered at times long enough compared to the poroelastic time scale of the specimen. The resulting delayed in crease in the maximum tensile effective stress of the hollow sample is ð3 1:8Þ=1:8 ¼ 67% for the case of ri ¼ 0:5 and ð3:45 2:6Þ=2:6 ¼ 33% for the case of ri ¼ 0:25. Conversely, Fig. 5 indicates that the poroelastic coupling effect causes the solid sample to exhibit a ð0:9 0:305Þ=0:9 ¼ 67% decrease in the maximum tensile effective stress between the instantons and long-time responses. Therefore, the failure of the solid, fluid-saturated sample is expected to initiate immediately after the compressive load is applied and at a point away from the center of the sample. To further elucidate the matter, the instantaneous and long-time responses of the solid disk are derived from Eqs. (48a) and (48b) and expressed in terms of elementary algebraic and trigonometric functions, as follows.
The elastic solution to diametral compression of solid cylinders cannot be recovered from the general solution presented in this paper simply by allowing the limiting case of ri →0 for hollow cylinders. To further elucidate the matter, σ*θθ ðθ; ri Þ is defined as the ratio of long-time limit of the hoop stress at the inner boundary of hollow cylinder to the corresponding value of hoop stress at the center of solid disk. hollow
σ *θθ ðθ; ri Þ ¼
s~ σθθ ðr ¼ ri ; θ; s→0Þ σ hollow ðr ¼ ri ; θ; t→∞Þ lim s→0 θθ ¼ σsolid ðr ¼ 0; θ; t→∞Þ lim s~ σ solid θθ θθ ðr ¼ 0; θ; s→0Þ
(52)
s→0
The plot of σ*θθ ðθ; ri Þ versus the radius of the sample inner hole ri is shown in Fig. 3. Results closely agree with the reported elastic solutions presented in Fig. 11 of.32 The following finding is of particular interest: 8 < 6 θ¼0 * limσθθ ðθ; ri Þ ¼ 10 (53) π ri →0 : θ¼ 3 2 An implication of Eq. (53) is that the hoop stress values along and perpendicular to the diametral loading line on an infinitesimally small hole at the center of the tested sample (points A and B in Fig. 4) are 6 and 10/3 times greater than the corresponding values for solid cylinder. This finding can be explained using Kirsch solution33 for stress concentration around a circular hole subjected to far-field stresses. The hoop stress values along and perpendicular to the loading line at the center of a solid elastic disk (Brazilian test) are found as follows.9
π ro L F
π ro L F
�π � 2
¼ σ? ¼
2ð1
2ð1
πφ
The stresses outlined in Eqs. (54a) and (54b) are observed as far-field stresses from the viewpoint of an infinitesimally small hole at the center of the solid disk. Under such conditions, the Kirsch solution for stress concentration around the small hole would be obtained as follows.33
π ro L F
π ro L F
σθθ ð0; ri → 0Þ ¼ σA ¼ 3σ k σθθ
�π 2
� ; ri → 0 ¼ σ B ¼ 3 σ ?
σk ¼
10
(55b)
Results of Eqs. (55a) and (55b) precisely match the reported results of32 in accordance with the illustrations of Fig. 3 and Eq. (53). 4. Results and discussion Terzaghi’s effective stress tensor is defined, as follows.24
σ ’ij ¼ σ ij þ pδij
ð1 r2 Þsin 2 φ 1 2r2 cos 2 φ þ r4
ri Þ
�
ð1 r2 Þsin 2 φ 1 2r2 cos 2 φ þ r4
tan
1
�� � 1 þ r2 tan φ 2 1 r
(57)
� � �� 2 1 þ r2 tan 1 tan φ β2 1 r2
(58)
� 1þ
Details regarding conversion of the related Fourier series expansion to the functional forms of Eq. (57) can be found in Ref. 9 A similar approach is herein taken to derive Eq. (58). Fig. 8 shows the plots of Eqs. (57) and (58) for plane-stress and planestrain solutions. The plot clearly shows higher tensile effective stress throughout the sample at t ¼ 0þ . Points A indicate the occurrence of maximum effective tensile hoop stress along the vertical axis of sym metry. The theoretical value of the distance between point A and the sample center point, rf , can be conveniently obtained by finding the extremum of the function given in Eq. (58). The result is expressed, as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiffi rffiffiffihffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq � � 2 ð2 þ β2 Þcos 2 φ þ 2 β2 ð4 β2 Þ ð2 þ β2 Þ2 cos 2 φ 4 sin 2 φ β2 rf ¼ 2 (59)
(55a)
σ? ¼ 6
�
σ ’θθ ðr; θ ¼ 0; t ¼ 0þ Þ ¼
(54b)
3
ri Þ
πφ
(54a)
σθθ ð0Þ ¼ σ k ¼ þ 1 σθθ
σ ’θθ ðr; θ ¼ 0; t → ∞Þ ¼
(56)
A summary of the key findings regarding the location, time and required load of failure is presented in Table 1. Felastic in this context implies the failure load predicted from the elastic solution.
The tensile failure of the rock sample is assumed to be governed by the larger principal component of Terzaghi’s effective stress. Figs. 5–7 illustrate the temporal evolution of Terzaghi’s effective hoop stress, as well as pore fluid pressure of hollow and solid rock samples under planestrain conditions. The selected values for poroelastic parameters are ν ¼ 0:25; νu ¼ 0:45 and B ¼ 0:9. It is further assumed that the external load is uniformly distributed over an arc angle of 2φ ¼ 20� . The outside boundaries of the rock samples in both cases are considered permeable while the inner boundary of the hollow sample is assumed to be impermeable. Poroelasticity imposes contrasting effects on the failure tendency of rock samples in the considered cases. The time and location of the maximum effective tensile stress for each case is shown by point A in Figs. 5–7. While the maximum tensile effective stress for the hollow sample is triggered by the long-time asymptotic value (t→ ∞) and at
5. Conclusion The classical solution to the elastic problem defined by the geometry and loading setup of the Brazilian test is generalized to the timedependent poroelastic solution for diametral compression of hollow and fluid-saturated, cylindrical samples. Application of a constant diametral compressive load at t ¼ 0 would instantaneously bring the solid sample to failure. That is, the failure is expected to initiate at t ¼ 0þ . The failure, however, is not expected to initiate from the sample center due to occurrence of maximum tensile effective stress at a point away from the center point. Delayed failure of hollow samples with large-enough inner diameter may occur after complete dissipation of the resulting disturbances in pore fluid pressure. That is, the maximum 9
A. Mehrabian and Y.N. Abousleiman
International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104201
tensile effective stress is predicted to occur by the long-time asymptote (t →∞Þ of the poroelastic solution to the considered problem.
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