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Analysis of fractures formed around cavities in rocks
Z’ectonophysics, 34 (1976) T9-T15 o Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Letter
Section
Analysis of fractures
JAYASRI Department
formed
around
cavities in rocks
DATTA and SACHINATH MITRA of Geological
Sciences,
(Submitted January 23,1976;
Jadavpur
University,
Calcutta-700032
(India)
revised version accepted May 21, 1976)
ABSTRACT Datta, J. and Mitra, S., 1976. Analysis of fractures formed around cavities in rocks. Tectonophysics, 34: T9-Tl5. In this work the authors investigated the generation of fractures under stress around large cavities, naturally formed or artificially created. The rheological properties of the rocks have been assumed to be non-linear and the cavities studied are spherical or cylindrical in shape. The solution comes out to be non-linear. With this, the relationship between the fracture stress and the hydrostatic stress has been quantitatively determined.
INTRODUCTION
Considerable advances have been made in recent years in the study of the mechanical behaviour of rocks under different stress and environmental conditions. Due to the growing interest in earthquakes and in certain other tectonic phenomena, the fracture and flow of rock materials have been investigated considerably by many workers. In recent years Le Tirant and Baron (1968), Bombolakis (1973), and Mogi (1973) have studied and reviewed critically the laboratory studies of the strength and ductility of rock materials under various stress systems, based on stress-strain relations. In most of these studies the rocks were assumed to possess linear mechanical (stress-strain law) properties. But in nature the mechanical properties of rock materials are not strictly linear and to a considerable extent they are non-linear. Some workers, viz. Brace (1964), Brady (1969), Vesic (1972), and Biot (1974), however, brought out some fundamental aspects of the non-linear behaviour of rocks. In the present paper the non-linear aspect of rock properties affected only by stress have been discussed through a case-study of fractures formed around cylindrical and spherical cavities developed in massive rocks. A quantitative (graphical) non-linear relationship between the hydrostatic and failure pressure, as obtained, will be discussed in the following paragraphs.
TlO
PROBLEMS
AND SOLUTfONS
Fracture stress around a cylindrical cavity Massive rocks (they may be igneous or metamorphic) without any major linear structure behave as elastically isotropic. Such materials possess timeindependent non-linear flow properties. We can study the nature of the fracture stress generated in such a rock-mass having a cavity within it. The form of the cavity is assumed to be approaching either a cylinder or a sphere. First we study the case when it is cylindrical. The deformation around the z-axis of the cylindrical cavity would be axially symme~~ so that the state of plane strain is the same in all planes normal to this axis. The cylindricaf co-ordinate system (1; 8, z) is chosen as the convenient co-ordinate system for this problem with r as the radial distance of a material point from the axis of the cylinder. It has been further assumed that the only velocity component ti of the material is along the radial distance r; the other two components ti and r.i~being zero. Let the principal components of the finite strain-rate & and & along the radial and circumferential direction respectively be defined as* : .
i,
=-$and(;2=-~
(1)
and the respective principal stresses 71 and 72 as the normal forces acting at the displaced point per unit initial area of the material. To represent the physical properties of the time-independent non-linear material, let them be related by the non-linear relation:
4 = fl (71,0)
g2= f2 (7190)
(2)
under the single stress r1 with r2 = 0 with the assumption that the material behaves isotropically. The equation of equilibrium for the stress field r1r2is expressed as:
and the compatibility condition as: d; El-E. 1=_-.._2_ dr
r
(4)
Both the equations (3) and (4) lead to the relation: (5)
*For compression the positive sign of the stress and strain-rate has been chosen.
Tll
For the present case, using the stress-strain-rate relation (eq. 2), eq. 5 can be expressed as a first-order differenti~ equation for 71 as a function of the independent variable 72 as follows: a&,,01 dr L=_ dTz
-;
Q-i,a;, +a7,. 71-72 [i 37,
(6)
The general solution of this equation may be written as: 72
=
do (71,
(7)
k)
representing a one-parameter family of curves with parameter k as an integration constant. If with the help of the boundary condition of the stress within the cylindrical cavity, k can be determined, the stress-field may be expressed as a function of r and the failure pressure may be analyzed. Illustration Let the stress--strain-rate relation be represented by: t1 = f, (71, 0) = Ad + Bt + C & = f2 (Q, 0) = -(A+ + B71 + C)
(8)
The quantity &,(7,, O)/a7, of eq. 6 in the neighbourhood of 71 = 0 may be obtained from eq. 8 as:
(~), =(2) o,.
=B
where ( ) o o denotes the value at the point 0,O. The above relation (eq. 9) is justified since the property of the material we have assumed earlier is isotropic in the plane of defo~ation, Within a reasonable accuracy the slqpe (eq. 6) on the line 72 = 0 may then be expressed as:
The general solution of this equation is: 72
=$lnk-
,rl - y
ln7i
(11)
where k is an integration constant. We next find out the value of Ink. The pressure around the cavity is assumed to be essentially hydrostatic in nature. The pressure CIbecomes hydrostatic in the rock masses* at depth *The porosity of the rock masses around the cavity is considered negligible.
T12
because of homogenization in the heterogeneity of stresses by the flow properties of rocks. Since the stresses are assumed to be equivalent to a uniform hydrostatic pressure u, the integral constant Fzof eq. 11 may be eliminated as: Ink = 2Bo + 2C lno
(12)
Evidently the stress for crack initiation is appreciably lower than the fracture stress,. Yet at the pressure r! = pf the crack is initiated and the crack occurs at the tensile value of the stress 72 = -R. The variation of the failure pressure with u is obtained ~~ytic~ly as: 2c pf+-2k: lnpf= R + 20 + B
B
Ino
(13)
The quantitative analysis of this failure pressure and hydrostatic pressure relation has been illustrated in Fig. 1. Numerical constants used are R = 1, C = 1, B = 2.
Fig. 1. F~il~re~p~~~e pf against hydrostatic pressure 0 for cylindrical catity. The diagram shows a non-linear curve.
T13
Failure pressure ar0und.a spherical cavity We can also study the nature of the fracture stress generated within the rock-mass having a spherical cavity. The material of the rock-mass possesses all the properties as mentioned in the preceding section. In this case we assume that the material surrounding the cavity undergoes deformation with spherical symmetry. The convenient co-ordinate system we choose is the spherical polar co-ordinates (r, 8, $). The only velocity component along the radial direction r is assumed as zi, as in the previous case, the other two components fi and LJbeing zero, and the stress-strain-rate relationship is assumed as before. The equation of equilibrium for the stress field 71~2in this case will be: dT1 2(~~-+2) dr+p= r
0
(14)
and the compatibility condition for the strain-rate is: (15) Combining eqs. 14 and 15 and using the stress-strain-rate relation (eq. 2), we obtain the first-order differential equation: d7 --.L =_
(16)
drz
The general solution of this equation also represents a one-parameter family of curves from which we can analyze the failure pressure of the rock material. The equations l’-7 and 14-16 bear resemblance with those of Biot (1974) but we have considered strain-rate in place of his strain considerations. Illustration We assume the stress-strain-rate relation to be as in eq. 8. Within the reasonable accuracy the slope (eq. 16) on the line r2 = 0 becomes: (17) The general solution of this equation is: 72
=$1&z.
+gT:-
gln7,
Using the argument as before, the analytical relation between the failure
(16)
T14
I
5
’ ’ I ’ ’
6 6
7
8
9
Fig. 2. Failure pressure pf against hydros~ti~ shows a non-linear curve.
10
pressure u for spherical cavity. The diagram
pressure and hydrostatic pressure may be obtained as c =R +,_; u2 +-lno B
(19)
Figure 2 illustrates the quantitative analysis of the failure pressure versus hydrostatic pressure. Numerical constants used are R = 1, A = 1, B = 1, C = 1. Therefore, this non-linear analysis for the state of finite strain-rate and stress around the spherical and cylindrical cavities in rocks reduces the problem to a non-linear solution of a first order differential equation. CONCLUSION
Underground cavities and channels occur in various geological formations as caverns and solution cavities in limestone and other rock beds, as underground mines and as confined cavities formed by explosions, e.g., Pokhran (Rajasthan, India) type PNE. The study brought out the relationship between the failure pressure and the hydrostatic stress present around such cavities as : +g
*f
lnpf=
R + 20 + p
3
for cylindrical cavity and:
lno
T15
A
$lnp&$=R+ for spherical
o---o~
2B
+$I
cavity, where pf is failure pressure,
u is hydrostatic
pressure
and
R is stress for crack initiation. The graphical representations of the above two relations show that the failure pressure pf increases non-linearly with the increase of the hydrostatic stress u. ACKNOWLEDGEMENT
Thanks are due to the University Grants Commission, New Delhi, for financially supporting the research. Thanks are also due to Sm. Bakul Chattopadhyay (Bandyopadhyay) for her valuable suggestions in this work and to the Head of the Department of Geological Sciences, Jadavpur University for the offer of necessary laboratory facilities. REFERENCES Biot, M.A., 1974. Exact simplified non-linear stress and fracture analysis around cavities in rocks. Int. J. Rock. Mech. Min. Sci., 11: 261-266. Bombolakis, E.G., 1973. Study of the brittle fracture process under uniaxial compression. Tectonophysics, 18: 231-248. Brace, W.F., 1964. Brittle fracture of rock. In: State of Stress in the Earth’s Crust. Elsevier, New York, pp. 111-178. Brady, B.T., 1969. The non-linear behaviour of brittle rock. Int. J. Rock Mech. Min. Sci., 6: 301-310. Le Tirant, P. and Baron, G., 1968. Fracturation hydraulique des roches sedimentaires en condition de contraintes de fond, Communication au 3” colloque de A.R.T.F.P., Pau. MO@, K., 1973. Fracture and flow of rocks. Tectonophysics, 13(1-4): 541-568. Vesic, A.S., 1972. Expansion of cavities in infinite soil masses. J. Soil Mech. Found. Div., Proc. ASCE, 3: 265-290.
Letter
Section
Analysis of fractures
JAYASRI Department
formed
around
cavities in rocks
DATTA and SACHINATH MITRA of Geological
Sciences,
(Submitted January 23,1976;
Jadavpur
University,
Calcutta-700032
(India)
revised version accepted May 21, 1976)
ABSTRACT Datta, J. and Mitra, S., 1976. Analysis of fractures formed around cavities in rocks. Tectonophysics, 34: T9-Tl5. In this work the authors investigated the generation of fractures under stress around large cavities, naturally formed or artificially created. The rheological properties of the rocks have been assumed to be non-linear and the cavities studied are spherical or cylindrical in shape. The solution comes out to be non-linear. With this, the relationship between the fracture stress and the hydrostatic stress has been quantitatively determined.
INTRODUCTION
Considerable advances have been made in recent years in the study of the mechanical behaviour of rocks under different stress and environmental conditions. Due to the growing interest in earthquakes and in certain other tectonic phenomena, the fracture and flow of rock materials have been investigated considerably by many workers. In recent years Le Tirant and Baron (1968), Bombolakis (1973), and Mogi (1973) have studied and reviewed critically the laboratory studies of the strength and ductility of rock materials under various stress systems, based on stress-strain relations. In most of these studies the rocks were assumed to possess linear mechanical (stress-strain law) properties. But in nature the mechanical properties of rock materials are not strictly linear and to a considerable extent they are non-linear. Some workers, viz. Brace (1964), Brady (1969), Vesic (1972), and Biot (1974), however, brought out some fundamental aspects of the non-linear behaviour of rocks. In the present paper the non-linear aspect of rock properties affected only by stress have been discussed through a case-study of fractures formed around cylindrical and spherical cavities developed in massive rocks. A quantitative (graphical) non-linear relationship between the hydrostatic and failure pressure, as obtained, will be discussed in the following paragraphs.
TlO
PROBLEMS
AND SOLUTfONS
Fracture stress around a cylindrical cavity Massive rocks (they may be igneous or metamorphic) without any major linear structure behave as elastically isotropic. Such materials possess timeindependent non-linear flow properties. We can study the nature of the fracture stress generated in such a rock-mass having a cavity within it. The form of the cavity is assumed to be approaching either a cylinder or a sphere. First we study the case when it is cylindrical. The deformation around the z-axis of the cylindrical cavity would be axially symme~~ so that the state of plane strain is the same in all planes normal to this axis. The cylindricaf co-ordinate system (1; 8, z) is chosen as the convenient co-ordinate system for this problem with r as the radial distance of a material point from the axis of the cylinder. It has been further assumed that the only velocity component ti of the material is along the radial distance r; the other two components ti and r.i~being zero. Let the principal components of the finite strain-rate & and & along the radial and circumferential direction respectively be defined as* : .
i,
=-$and(;2=-~
(1)
and the respective principal stresses 71 and 72 as the normal forces acting at the displaced point per unit initial area of the material. To represent the physical properties of the time-independent non-linear material, let them be related by the non-linear relation:
4 = fl (71,0)
g2= f2 (7190)
(2)
under the single stress r1 with r2 = 0 with the assumption that the material behaves isotropically. The equation of equilibrium for the stress field r1r2is expressed as:
and the compatibility condition as: d; El-E. 1=_-.._2_ dr
r
(4)
Both the equations (3) and (4) lead to the relation: (5)
*For compression the positive sign of the stress and strain-rate has been chosen.
Tll
For the present case, using the stress-strain-rate relation (eq. 2), eq. 5 can be expressed as a first-order differenti~ equation for 71 as a function of the independent variable 72 as follows: a&,,01 dr L=_ dTz
-;
Q-i,a;, +a7,. 71-72 [i 37,
(6)
The general solution of this equation may be written as: 72
=
do (71,
(7)
k)
representing a one-parameter family of curves with parameter k as an integration constant. If with the help of the boundary condition of the stress within the cylindrical cavity, k can be determined, the stress-field may be expressed as a function of r and the failure pressure may be analyzed. Illustration Let the stress--strain-rate relation be represented by: t1 = f, (71, 0) = Ad + Bt + C & = f2 (Q, 0) = -(A+ + B71 + C)
(8)
The quantity &,(7,, O)/a7, of eq. 6 in the neighbourhood of 71 = 0 may be obtained from eq. 8 as:
(~), =(2) o,.
=B
where ( ) o o denotes the value at the point 0,O. The above relation (eq. 9) is justified since the property of the material we have assumed earlier is isotropic in the plane of defo~ation, Within a reasonable accuracy the slqpe (eq. 6) on the line 72 = 0 may then be expressed as:
The general solution of this equation is: 72
=$lnk-
,rl - y
ln7i
(11)
where k is an integration constant. We next find out the value of Ink. The pressure around the cavity is assumed to be essentially hydrostatic in nature. The pressure CIbecomes hydrostatic in the rock masses* at depth *The porosity of the rock masses around the cavity is considered negligible.
T12
because of homogenization in the heterogeneity of stresses by the flow properties of rocks. Since the stresses are assumed to be equivalent to a uniform hydrostatic pressure u, the integral constant Fzof eq. 11 may be eliminated as: Ink = 2Bo + 2C lno
(12)
Evidently the stress for crack initiation is appreciably lower than the fracture stress,. Yet at the pressure r! = pf the crack is initiated and the crack occurs at the tensile value of the stress 72 = -R. The variation of the failure pressure with u is obtained ~~ytic~ly as: 2c pf+-2k: lnpf= R + 20 + B
B
Ino
(13)
The quantitative analysis of this failure pressure and hydrostatic pressure relation has been illustrated in Fig. 1. Numerical constants used are R = 1, C = 1, B = 2.
Fig. 1. F~il~re~p~~~e pf against hydrostatic pressure 0 for cylindrical catity. The diagram shows a non-linear curve.
T13
Failure pressure ar0und.a spherical cavity We can also study the nature of the fracture stress generated within the rock-mass having a spherical cavity. The material of the rock-mass possesses all the properties as mentioned in the preceding section. In this case we assume that the material surrounding the cavity undergoes deformation with spherical symmetry. The convenient co-ordinate system we choose is the spherical polar co-ordinates (r, 8, $). The only velocity component along the radial direction r is assumed as zi, as in the previous case, the other two components fi and LJbeing zero, and the stress-strain-rate relationship is assumed as before. The equation of equilibrium for the stress field 71~2in this case will be: dT1 2(~~-+2) dr+p= r
0
(14)
and the compatibility condition for the strain-rate is: (15) Combining eqs. 14 and 15 and using the stress-strain-rate relation (eq. 2), we obtain the first-order differential equation: d7 --.L =_
(16)
drz
The general solution of this equation also represents a one-parameter family of curves from which we can analyze the failure pressure of the rock material. The equations l’-7 and 14-16 bear resemblance with those of Biot (1974) but we have considered strain-rate in place of his strain considerations. Illustration We assume the stress-strain-rate relation to be as in eq. 8. Within the reasonable accuracy the slope (eq. 16) on the line r2 = 0 becomes: (17) The general solution of this equation is: 72
=$1&z.
+gT:-
gln7,
Using the argument as before, the analytical relation between the failure
(16)
T14
I
5
’ ’ I ’ ’
6 6
7
8
9
Fig. 2. Failure pressure pf against hydros~ti~ shows a non-linear curve.
10
pressure u for spherical cavity. The diagram
pressure and hydrostatic pressure may be obtained as c =R +,_; u2 +-lno B
(19)
Figure 2 illustrates the quantitative analysis of the failure pressure versus hydrostatic pressure. Numerical constants used are R = 1, A = 1, B = 1, C = 1. Therefore, this non-linear analysis for the state of finite strain-rate and stress around the spherical and cylindrical cavities in rocks reduces the problem to a non-linear solution of a first order differential equation. CONCLUSION
Underground cavities and channels occur in various geological formations as caverns and solution cavities in limestone and other rock beds, as underground mines and as confined cavities formed by explosions, e.g., Pokhran (Rajasthan, India) type PNE. The study brought out the relationship between the failure pressure and the hydrostatic stress present around such cavities as : +g
*f
lnpf=
R + 20 + p
3
for cylindrical cavity and:
lno
T15
A
$lnp&$=R+ for spherical
o---o~
2B
+$I
cavity, where pf is failure pressure,
u is hydrostatic
pressure
and
R is stress for crack initiation. The graphical representations of the above two relations show that the failure pressure pf increases non-linearly with the increase of the hydrostatic stress u. ACKNOWLEDGEMENT
Thanks are due to the University Grants Commission, New Delhi, for financially supporting the research. Thanks are also due to Sm. Bakul Chattopadhyay (Bandyopadhyay) for her valuable suggestions in this work and to the Head of the Department of Geological Sciences, Jadavpur University for the offer of necessary laboratory facilities. REFERENCES Biot, M.A., 1974. Exact simplified non-linear stress and fracture analysis around cavities in rocks. Int. J. Rock. Mech. Min. Sci., 11: 261-266. Bombolakis, E.G., 1973. Study of the brittle fracture process under uniaxial compression. Tectonophysics, 18: 231-248. Brace, W.F., 1964. Brittle fracture of rock. In: State of Stress in the Earth’s Crust. Elsevier, New York, pp. 111-178. Brady, B.T., 1969. The non-linear behaviour of brittle rock. Int. J. Rock Mech. Min. Sci., 6: 301-310. Le Tirant, P. and Baron, G., 1968. Fracturation hydraulique des roches sedimentaires en condition de contraintes de fond, Communication au 3” colloque de A.R.T.F.P., Pau. MO@, K., 1973. Fracture and flow of rocks. Tectonophysics, 13(1-4): 541-568. Vesic, A.S., 1972. Expansion of cavities in infinite soil masses. J. Soil Mech. Found. Div., Proc. ASCE, 3: 265-290.