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The influence of temperature dependent properties of thermal rock fragmentation
Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 12, pp. 255-260, Pergamon Press 1975. Printed in Great Britain
The Influence of Temperature Dependent Properties on Thermal Rock Fragmentation T. F. LEHNHOFF? J. D. SCHELLER3; This investigation is concerned with a thermal rock fragmentation system employing heat to create subsurface thermal inclusions. The effectiveness of this system as applied to three different rock types is studied. The rock types are Dresser basalt, Charcoal gray granite, and Sioux quartzite. The influence of temperature dependent material properties is investigated. The actual three-dimensional problem of in situ rock fragmentation involves parallel rows of equally spaced holes drilled to a constant depth. A heat source at the bottom of each hole creates a thermal inclusion resulting in a stress field causing fracture. Axisymmetric models of the fragmentation system were obtained by considering the typical planes of symmetry and nature of the actual three-dimensional problem. These models were used to study the fracture characteristics of the three hard rock types and to investigate the influence of temperature dependent material properties. The temperature and stress solutions were obtained using finite element approximations. Fracture predictions were based on the Griffith and the McClintock-Walsh modified Griffith fracture criteria. It was found that for any given fracture length there was a definite order in critical stress time among the three rock types. The results indicate that the heat transjbr problem is governed by the quartz content of the particular rock type. The complete fracture problem was not governed by any one particular rock characteristic. The dimensionless critical stress time ratio, tf*, was found to be related to the dimensionless fracture length ratio, L*, according to the equation, t * = (L*)3. This suggests that small hole spacings should be used for an optimum fragmentation configuration. Although critical stress times, for a given hole spacilgt, were different for each rock type the fracture patterns were found to be functions of the problem geometry only.
INTRODUCTION The thermal-fragmentation process involves inserting carbon arc or other high intensity heaters in drilled holes (Fig. 1). The thermal inclusions (heated zones) formed at the bottoms of the holes along with mechanical assistance fracture the rock for removal by a loading device and conveyor system [-1-3]. The axisymmetric model of Fig. 1 was used in a thermoelastic fracture analysis to determine a relationship between hole spacing and a condition related to fracture time (tr). In the remainder of this report t r will be referred to as critical stress time. The analysis deals with the effect of treating most of the thermoelastic t Associate Professor, University of Missouri--Rolla, MO, U.S.A. :~Research Engineer, Westinghouse Electric Corporation, Greensburg, PA, U.S.A.
255
properties as functions of temperature which yields a nonlinear mathematical problem. A finite element thermoelastic analysis of the uncoupled temperatures and stresses with the fracture criteria has been used to predict critical stress times for four axisymmetric models and three different materials. The computer codes are discussed in detail by Patel [1], Allen [2-1 and Scheller [3].
MATERIAL PROPERTIES
Due to the characteristics of the rock types they are considered to be homogeneous and isotropic [1, 4]. The thermal properties equations for calculating temperature distributions are least squares polynomial fits of tabular data [2, 3]. The specific heat and conducti-
256
T . F . Lehnhoff and J. D. Scheller
~
A
~E
TRICMODEL
TABLE 1. PROPERTIES USED IN TEMPERATURE ANALYSIS [4-7] Material
0
0
Q
Density~ p,gm/cma
Melt Temperature Tm, *C
SurfaceConvection h,cal/cm2-sec-°C
Dresser basalt Charcoal gray granite
2.97 2.73
1,250 1,250
0.00021 0.00021
Sioux quartzite
2.64
1,610
0.00021
O
k = 0"69 x 10 - 2 - 0"16 x 10 -4 T
L
+ 0"32
0'37 x 10 -1° T 3
× 10 - 7 T z -
+ 0"16 x 10 - 1 3 T 4.
(6)
Table 1 shows additional data used in the temperature analysis. The thermoelastic stress program TRATSA (TRAnsient Thermal Stress Analysis) [1] accepts tabular data as shown in Table 2. For the models where constant properties at ambient temperature were used, the values of the thermal properties were taken as those at 24°C. The fracture initiation analysis in this study is based Fig. 1. Characteristic dimensions for the axisymmetric model. on the Griffith and McClintock-Walsh modified Grifvity for each of the three types of rock are listed as fith criteria [8]. Once the principal stresses have been follows: found in the stress analysis the properties listed in Dresser basalt (or DB): Table 3 are required to predict the initiation of fracture. These properties were assumed to remain constant with c = 0.22exp(0-1l x 1 0 - Z T + 0.10 x 1 0 - ~ T 2 changes in temperature. Temperature dependent data is -0"22 x 1 0 - 7 T 3 + 0"12 x 1 0 - 1 ° T 4) (1) not available and because of the low diffusivity of rock, the bulk of the material in tension remains near k = 0"89 x 10 -2 - 0"31 × 10-2In(T) a +0"91 x 1 0 - 3 I n ( T ) 2 -0"83 x 10-4In(T) 3. (2) m b i e n t conditions. Sioux quartzite (or SQ): c = 0"15 + 0"91
x 10 . 3
T - 0"83
T H E R M O E L A S T I C FRACTION ANALYSIS
x 10 - 6 T 2
- 0 " 3 8 x 10-9T3+0"51
X 10-12T 4
(3)
k = 0"49 x 10 -2 + 0"17 x 10-2In(T) + 0"63 x 10-3In(T) 2 - 0"95 x 10-4In(T) 3 - 0-28 x 10 -5 ln(T) 4 - 0"53 x 10-5 ln(r) 5. + 0'72 x 10-6In(T) 6. (4) Charcoal gray granite (St. Cloud gray granodiorite or CG): c = 0'16exp(0"33 x 10 -2 T - 0.22 x 10 -5 T 2 -0.39 x 1 0 - 8 T 3 +0'35 x 10 - 1 1 T 4) (5)
The two finite element conduction codes used were developed based on the formulation given by Wilson and Nickell [15]. The finite element conduction code $70 which uses constant thermal properties was developed by Keith [16]. The finite element conduction code 2DNLT (Two Dimensional Non-linear Temperature) developed by Allen [2] has the capability of using temperature dependent thermal properties. The boundary conditions on the conduction problem were taken as: ~T/Or
TABLE 2. PROPERTY DATA USED IN STRESS ANALYSIS [4, Dresser Basalt Temp
T, *t
Young's Poisson's Modulus Ratio
E,IO5
~
kg/cm2
Charcoal gray granite Coeff. of Thermal
Young's Modulus
Expansion E,IO5 ~,I0-5/*C
Poisson's Coeff. of Ratio Thermal
v
kg/cm2
Expansion
= O, r = O, r = c/2
6, 9, 10] Sioux quartzite
Young's Modulus
E,IO5
~,I0"5/*C
kg/cm2
Polsson's Ratio
v
Coeff. of Thermal
Expansion ~,I0"5/'0
24
I0.268
0.24
0.290
7.872
0.20
0.125
8.015
0.02
122
10.054
0.24
0.500
6.873
O.15
0.762
7.729
0.0l
0.417 1.066
260
9.697
0.22
0.770
3.997
0.03
1.165
6.842
-O.Ol
1.512
371
8.933
0.19
0.940
2.366
-0.04
1.491
5,302
-0.04
1.806
482
8. 504
0.18
1.020
1.601
-0.06
1.784
3.763
-0.07
2.178
538
8.147
D.16
1.080
l.ll2
-0.1O
2.305
2.305
-0.22
2.714
593
6.781
0.II
1.160
0.592
-0.07
2.496
2.468
-0.40
2.816
700
0.010
O.10
1.160
0,010
O.00
2.500
O.Ol0
0.00
2.820
(7)
Thermal Rock Fragmentation TABLE 3. PROPERTIES USED FOR FRACTURE PREDICTIONS [8, Material
Uniaxial Tensile Strength ot,kg/cm2
Dresser basalt lOB) 154.32 Charcoal gray granite (CG) 124.50 Sioux quartzite (SQ) 256.00
Unlaxial Compressive Strength~ Oc,kg/cmL
11-14]
Fracture Surface Coefficient of Friction ~f
2980.81
0.9
2342.11 3420.00
0.9 0.9
OT/Or + h T = O, r = d/2, z > B + a ; z = O ; z = A + B T=
(8)
T,,O
(9)
T, denotes the melt temperature of the material and h is the coefficient of convective heat transfer. The finite element code, 'TRATSA' 1-1], used for the stress analysis was developed for plane or axisymmetric bodies with temperature-dependent material properties. The code is based on the formulations given by Zienkiewicz [18] and Jones and Crose [19] which assume linear displacements between nodes, resulting in constant stress elements of both triangular and quadrilateral shape. The boundary conditions for the stress problem were taken as:
a= = 0 , 0
u, = O , r = O , z < B ; r = e l 2
(10)
a, = O , r = d/2, z > B
(11)
<_ r < d/2, z = B ;
d/2 < r < c/2, z = A + B ; O <_ r <_ c/2, z = O.
(12)
The fracture initiation predictions made in this investigation are based on the stress field obtained by assuming that the material remains a continuum. In the actual case initiation of a crack will change the geometry and temperature distribution which will result in a change in the stress field. A detailed analysis of crack propagation would require progressive restructuring of the problem. At each time step, the creation of new fracture surfaces would require the introduction of additional boundary conditions. It is evident that following the propagation of a crack would require a large amount of computing time. In actual processes, nondynamic crack propagation is largely governed by the orientation of pre-existing microcracks and the interactions between them, for which no theory has been developed. The fracture initiation predictions in this investigation were performed using the program FRACTR [1], which is based on the Griffith and McClintock-Walsh modified Griffith criteria. These theories are derived from an energy formulation and neglect the effects of stress concentrations and the interactions between cracks. Even though the theories are themselves approximate they predict theoretical fracture fields (probable fracture paths)which have a statistically good correlation to those obtained experimentally.This correlation is seen in the results of Lauriello [20], Bieniawski [21], and Clark et at. [17]. FRACTR uses the stress
257
data obtained from TRATSA to plot the fracture intensity level of each element in the finite element grid. The fracture intensity level is defined in terms of the stress magnitude in excess of that necessary for the initiation of fracture. The plotted fracture intensity levels are used to predict the approximate fracture zones. The thermal fragmentation analysis was performed in three steps. First, the transient temperature distribution was obtained at various times using one of the two conduction codes $70 or 2DNLT. The temperature distributions were then used as input to the stress code (TRATSA) and the stresses were in turn used as input to the fracture initiation code (FRACTR). Once the transient temperature distribution was obtained, the last two steps above had to be repeated until the critical stress time (tl) was found. The critical stress time refers to the time, in the transient solution, at which the fracture initiation zone was complete across the entire length of the structure or model. The term critical stress time as obtained here is not a true fracture time. Rather, it is simply a characteristic time (the time at which all elements along some continuous path have reached the fracture initiation point) which has been used as a basis for studying several factors of the fragmentation process [1, 2]. AXISYMMETRIC MODEL ANALYSIS Four models of different lengths (hole spacings) and three types of rock were used. The effect of temperature-dependent material properties was obtained by comparing the solutions using variable properties to the solutions obtained using various combinations of constant (at ambient temperature) thermal and thermoelastic material properties. A description of the models is given in Table 4. (A) V a r i a b l e properties analysis Since the material properties of the three rock types are highly temperature dependent, an analysis using variable thermal and thermoelastic material properties is the most accurate solution possible within the approximations previously stated. Therefore, these results were used as a base for comparison to the analyses involving constant ambient material properties. The variable property analysis was also used to study the fracture characteristics, and develop the fracture length-critical stress time relation. Figure 2 shows the 50°C temperature contour of model 33A for each material at the critical stress time. This type of distribution, where the contour for SQ has moved the greatest distance and for DB the least, is typical of all the models at the critical stress time. TABLE 4. PARAMETRIC DESCRIPTION OF MODELS Model ]2A 33A 32B 33B
Hole Diam Spacing d, cm.
c, sin.
5.0 3.8 5.0 3.8
IO.O lO.O 20.0 20.0
Fracture Length L = c-d 5.0 6.2 15.0 16.2
Melt Depth a, cm. 1.3 1.3 1.3 1.3
Cony. Depth a = A-a Ccm 2.5 3.8 7.6 7.6
258
T . F . Lehnhoff and J. D. Scheller TABLE
Materia]
Model
5.
CRITICAL
Material Properties
STRESS TIMES
F r a c t u r e Critical Length Stress Time L, cm
tf,sec
Percent Deviation
of t f from Variable Properties
tf
E
\
SIOUXQUARTZITIf:E, 24.2 i=
5 . 0 cm
Fig. 2. Typical 50°C isotherms at the critical stress time.
The curves of critical stress time as a function of fracture length for the three rock types are shown in Fig. 3. It can be seen from this curve that, for a given fracture length, CG has the highest critical stress time and DB the lowest.
(B) Ambient properties analysis Constant thermal and thermoelastic material properties at ambient temperature are used in the ambient property analysis. From the critical stress times in Table 5, it is seen that the use of ambient material properties leads to results which differ significantly from the results in the variable properties analysis. Critical stress times are higher for CG and lower for DB and SQ in the ambient properties analysis as compared to the variable properties results. Also of importance is the order in which different rocks attain critical stresses which differs from the order found in the variable properties analysis. This would suggest that tun-
500
=
I
=
=
i
I
I
I
• 400
32A 33A 32B 33B
Variable
5.0 6.2 15.0 16.2
6.9 24.0 254.0 365.0
Granite
32A 33A 32B 33B
Variable
5.0 6.2 15.0 16.2
]3.0 33.7 327.0 460.0
Quartzite
32A 33A 32B 330
Variable
5.0 6.2 15.0 16.2
8.2 24.2 280.0 450.0
Basalt
32A 33A 32B
Ambient
5.0 6.2 15.0
5.8 20.0 252.5
Granite
32A 33A 320
Ambient
S.O 6.2 15.O
]5.8 50.0 S.S,*
Quartzite
32A 33A 32B
Ambient
5.0 6.2 15.0
4.9 15.0 218.0
40.3 38.0 22.2
Basalt Granite Quartzite
32B 32B 320
A.T.E.**
15.0
500.0 S.S. S.S.
49.3
Basalt Granite Quartzite
32B 320 32B
A.T.***
15.0
140.0 137.0 78.0
44.9 58. l 72.l
* Steady State
** Ambient Thermoelastic
16.0 16.7 0.6 21.5 5.0
*** Ambient Ther~lal
neling and excavation cost estimates, on a site of some particular rock type, performed on the basis of previous experience with another different rock type, should be gaged by the critical stress order given by a variable properties analysis. Applying the critical stress order from a simpler ambient properties analysis would yield incorrect estimates. The critical stress times for DB using ambient properties show the smallest mean deviation from the variable property results of all the rock types. This is due to the fact that, for DB, the thermal conductivity and diffusivity have only small variations with temperature. Also, the thermoelastic properties of DB show a smaller degree of temperature dependence than the properties of the other two rocks. Model 32B CG failed to reach a critical stress. The use of ambient material properties so drastically changed the problem that the system reached the point of steady state heat transfer before the thermal stresses could build to the critical level.
(C) Ambient thermoelastic properties analysis
uS ~-
Basalt
50 0
¢~
CHARCOAL GRAY G R A N I T E
200
_
s oox
77/
OOA.TZTE
D R E S S ~ I00
O
I 2
~ 4
I 6
I 8
I tO
I 12
FRACTURE LENGTH t L
cm
I 14
I 16
18
Fig. 3. Critical stress vs fracture length for three rocks.
This analysis involves temperature-dependent thermal properties and constant thermoelastic properties. From Table 5 it is seen that only the DB model achieved fracture completion. This result suggests that the temperature dependence of the thermoelastic material properties is very significant in the thermal rock fragmentation system. The fact that DB was the only rock to reach a critical stress would suggest that the coefficient of thermal expansion is the most significant of the thermoelastic properties. The thermoelastic properties for all three rocks exhibit about the same degree of variation with
Thermal Rock Fragmentation temperature with the exception of the coefficient of thermal expansion. Relative to the coefficients of thermal expansion for CG and SQ, the coefficient for DB shows only slight variation with temperature. (D) Ambient thermal properties analysis Constant thermal properties and temperature-dependent thermoelastic properties are used in this analysis. As seen in Table 5, the critical stress times are significantly lower than those found in the variable properties analysis. It is important to note that the order in which the critical stress is attained is different from that of the variable properties analysis. This can be attributed to the fact that, for the three rock types, SQ has the highest thermal diffusivity and DB has the lowest. In the variable properties analysis at higher temperatures the differences in thermal diffusivity among the rocks is small and thus has less effect in the overall thermal fragmentation problem [6]. However, these differences in diffusivity are a governing parameter in the heat transfer problem as illustrated by the temperature contours of Fig. 2. The per cent deviation of the critical stress time found in the ambient thermal properties analysis from the times found in the variable properties analysis suggest the degree of temperature dependence of the diffusivity of each of the materials. The diffusivity increases in the degree of nonlinearity, with each material, in the same order as the per cent deviation of Table 5. (E) Fracture pattern characteristics The zones of high stress intensity and probable fracture initiation were obtained from the variable property analysis. Although the critical stress times differ significantly for different materials, the fracture initiation patterns show the same general configuration for every material. However, the fracture initiation zones do differ for each of the four models ([2, 3], Fig. 4).
TABLE 6.
259
FRACTURE
L E N G T H A N D F R A C T U R E TIME R A T I O S
Length
Ll cm
Time-Ratio t f
L2 L* = IT
Dresser
Charcoal
Sioux
basalt
gray granite
quartzite
15.0
1.08
1.44
1.41
5.0
1.24
3.4B
2.59
1.61 2.96
6.2
2.42
10.60
9.70
11.60
6,2
2.60
15.20
13.60
18.60
5.0
3.00
36.80
25.20
34.20
5.0
3.24
52.90
35.40
54.90
(F) Fracture length-Critical stress relation Critical stress times (as noted earlier) for the different models can be expressed as a function of the single variable, the fracture length. Critical stress times for the variable properties analysis models, listed in Table 5, were used to calculate the fracture length ratios and the corresponding critical stress time ratios given in Table 6 and shown on Fig. 5. These ratios were computed by forming all possible ratios of fracture length and corresponding critical stress times greater than unity for each rock type. From this data the following relation was obtained: t? = (L*)3.
(13)
GENERAL TRENDS There is one well defined trend which occurs as a result of the differences in quartz content of the rock
SC tf=(
SO
411
2C
-.. ....
o.
J
C'~2/;-~ ~, ,... s~¢p/sls I , "~ ,'1/I / s , • ~ s / " , s s ¢ e"~"*"/ss/ ///'Sl.''ls
I.~"
IC
~
s
.. ;:; -..;; ,',
m
!
o
]
McCLiNTOCK-WALSH
]
GRIFFITHCOMPRESSION
]
GRIFFiTH TENSION
i
-+ 2
/
O DRESSER BASALT A SIOUX QUARTZITE O CHARCOALGRAy GRANITE
I
IO.O
em
Fig. 4. Typical potential fracture zones.
I
I
2 4 6 FRACTURE LENGTH RATIO, L"
I
6
I0
Fig. 5. Critical stress-fracture length relationship.
260
T . F . Lehnhoff and J. D. Scheller
types (DB contains no quartz, CG is composed of 5 per cent quartz, and SQ contains 95 per cent quartz). The properties which increase with increasing quartz content are the thermal diffusivity and the coefficient of thermal expansion. The relation of thermal diffusivity to quartz content supports the result showing increasing penetration of temperature field with increasing quartz content. Also, the critical stress times from the ambient thermal properties analysis are seen to decrease with increasing quartz content as would be expected from the relation of diffusivity to quartz content. However, this is due to the large differences in diffusivity between the rock types at lower temperatures and does not occur in the variable properties analysis. From these general trends it can be seen that only the heat transfer problem is related, to quartz content of the rock types. The mechanical properties of a particular rock type are not related to quartz content exclusively, but are determined by structural characteristics, such as microcracks, grain size, porosity and cleavage planes as well. Therefore, the stress analysis and/ or fracture criterion alter the trend toward a relation of critical stress time to quartz content or any one characteristic of a particular rock type. For these reasons, the thermal fragmentation of rocks becomes a complex problem involving many diverse parameters which restrict the method of solution to a complete fracture analysis, including the influence of temperature. A close correlation exists between the fracture patterns and fracture length-critical stress time relationships predicted in this theoretical analysis and those experimentally observed in field tests [-3, 17]. The results of field tests show the fracture initiating at the top of the thermal inclusion as predicted in this analysis. Also, the observation that critical stress times increase rapidly with increasing fracture length is experimentally confirmed. The approximate critical stress time corresponding to a given fracture length can be predicted from the data for a single test using the relationship t j* = (L*) 3 between the dimensionless fracture completion time t? and the dimensionless fracture length [3] for hard rocks. While equation (13) implies lower critical stress times for smaller hole spacings, the use of smaller spacings would impose higher drilling costs as well as smaller volumes of rock removed with every cycle of operation. Therefore, the choice of optimum hole spacings will involve several factors associated with the actual rock fragmentation and removal processes. Even though the power relationship is inde-
pendent of specific hard rock types, adjustments should be made in cost estimates to account for the particular rock being removed. The curve of Fig. 3 indicates that, for a given hole spacing, CG requires the highest critical stress time and DB the lowest of the three hard rock types studied in this investigation. Received 18 December 1974.
REFERENCES 1. Patel M. R. Rock fragmentation by subsurface thermal inclusions--a finite element study. Ph.D. Dissertation, University of Missouri--Rolla, Missouri, (1973). 2. Allen V. Direct and inverse heat conduction in rock materials using the finite element method. Ph.D. Dissertation, University of Missouri--Rolla, Missouri, (1974). 3. Scheller J. D. Thermal fragmentation of rock--the influence of temperature dependent material properties. M.S Thesis, University of Missouri--Rolla, Missouri, (1974). 4. Thirumalai K. Potential of internal heating method of rock fragmentation. Dynamic Rock Mechanics, Proc. 12th Syrup. Rock Mechanics (edited by Clark G. B.), pp. 697 719 (1971). 5. Maurer W. C; Novel Drilling Techniques, Pergamon Press, Oxford (1.968). 6. Wingquist C. F. Elastic moduli of rock at elevated temperatures. Bureau of Mines, RI 7269 (1970). 7. Baumeister T. and Marks L. S. Standard Handbook for Mechanical Engineers, 7th edition, McGraw-Hill, New York (1967). 8. McClintock F. A. and Walsh J. B. Friction on Griffith cracks under pressure. Proceedings 4th U.S. Congress Applied Mechanics, Berkeley 1962, pp. 1015-1021, ASME, New York (1963). 9. Thirumalai K. Rock fragmentation by creating a thermal inclusion with dielectric heating. Bureau of Mines, RI 7424 (1970). 10. Schumacher B. W. and Holdbrook R. G. Use of electron beam gun for hard rock excavation. Westinghouse Electric Corporation, Rept. 73-8C2-ROCUT-PI (1972). 11. Walsh J. B. The effect of cracks on the uniaxial elastic compression of rocks. J. Geophys. Res. 70(3) (1965). 12. Brace W. F. Brittle Fracture of Rocks. Proceedings International Conference (edited by Judd W. R.), Elsevier, New York (1963). 13. Hoek E. and Bieniawski Z. T. Brittle fracture propagation in rock under compression. Int. J. Fract. Mech., 1, 139--155 (1965). 14. Paone J. and Bruce W. E. Drillability studies; diamond drilling. Bureau of Mines, RI 6324 (1963). 15. Wilson E. L. and Nickell R. E. Application of the finite element method to heat conduction analysis. Nucl. Engng Design 4, 276286 (1966). 16. Keith H. D. Personal Communication, University of Missouri Rolla, MO. 17. Clark G. B., Lehnhoff T. F., Patel M. and Allen V. An investigation of thermal-mechanical fragmentation of hard rock. Final Report, ARPA, University of Missouri Rolla (October 1973). 18. Zienkiewicz O. E. The Finite Element Method in Structural and Continuum Mechanics. McGraw-Hill, New York (1967). 19. Jones R. N. and Crose J. G. SAAS II-Finite element stress analysis of axisymmetric solids with orthotropic, temperature dependent material properties. Aerospace Corporation, CA (1968). 20. Lauriello P. J. J. Thermal fracturing of hard crystalline rocks. Ph.D. Dissertation, Rutgers University ( 1971). 21. Bieniawski Z. T. Mechanism of brittle fracture of rock--Part IL Experimental studies. Int. J. Rock Mech. Min. Sci. 4, 407 423 (1967).
The Influence of Temperature Dependent Properties on Thermal Rock Fragmentation T. F. LEHNHOFF? J. D. SCHELLER3; This investigation is concerned with a thermal rock fragmentation system employing heat to create subsurface thermal inclusions. The effectiveness of this system as applied to three different rock types is studied. The rock types are Dresser basalt, Charcoal gray granite, and Sioux quartzite. The influence of temperature dependent material properties is investigated. The actual three-dimensional problem of in situ rock fragmentation involves parallel rows of equally spaced holes drilled to a constant depth. A heat source at the bottom of each hole creates a thermal inclusion resulting in a stress field causing fracture. Axisymmetric models of the fragmentation system were obtained by considering the typical planes of symmetry and nature of the actual three-dimensional problem. These models were used to study the fracture characteristics of the three hard rock types and to investigate the influence of temperature dependent material properties. The temperature and stress solutions were obtained using finite element approximations. Fracture predictions were based on the Griffith and the McClintock-Walsh modified Griffith fracture criteria. It was found that for any given fracture length there was a definite order in critical stress time among the three rock types. The results indicate that the heat transjbr problem is governed by the quartz content of the particular rock type. The complete fracture problem was not governed by any one particular rock characteristic. The dimensionless critical stress time ratio, tf*, was found to be related to the dimensionless fracture length ratio, L*, according to the equation, t * = (L*)3. This suggests that small hole spacings should be used for an optimum fragmentation configuration. Although critical stress times, for a given hole spacilgt, were different for each rock type the fracture patterns were found to be functions of the problem geometry only.
INTRODUCTION The thermal-fragmentation process involves inserting carbon arc or other high intensity heaters in drilled holes (Fig. 1). The thermal inclusions (heated zones) formed at the bottoms of the holes along with mechanical assistance fracture the rock for removal by a loading device and conveyor system [-1-3]. The axisymmetric model of Fig. 1 was used in a thermoelastic fracture analysis to determine a relationship between hole spacing and a condition related to fracture time (tr). In the remainder of this report t r will be referred to as critical stress time. The analysis deals with the effect of treating most of the thermoelastic t Associate Professor, University of Missouri--Rolla, MO, U.S.A. :~Research Engineer, Westinghouse Electric Corporation, Greensburg, PA, U.S.A.
255
properties as functions of temperature which yields a nonlinear mathematical problem. A finite element thermoelastic analysis of the uncoupled temperatures and stresses with the fracture criteria has been used to predict critical stress times for four axisymmetric models and three different materials. The computer codes are discussed in detail by Patel [1], Allen [2-1 and Scheller [3].
MATERIAL PROPERTIES
Due to the characteristics of the rock types they are considered to be homogeneous and isotropic [1, 4]. The thermal properties equations for calculating temperature distributions are least squares polynomial fits of tabular data [2, 3]. The specific heat and conducti-
256
T . F . Lehnhoff and J. D. Scheller
~
A
~E
TRICMODEL
TABLE 1. PROPERTIES USED IN TEMPERATURE ANALYSIS [4-7] Material
0
0
Q
Density~ p,gm/cma
Melt Temperature Tm, *C
SurfaceConvection h,cal/cm2-sec-°C
Dresser basalt Charcoal gray granite
2.97 2.73
1,250 1,250
0.00021 0.00021
Sioux quartzite
2.64
1,610
0.00021
O
k = 0"69 x 10 - 2 - 0"16 x 10 -4 T
L
+ 0"32
0'37 x 10 -1° T 3
× 10 - 7 T z -
+ 0"16 x 10 - 1 3 T 4.
(6)
Table 1 shows additional data used in the temperature analysis. The thermoelastic stress program TRATSA (TRAnsient Thermal Stress Analysis) [1] accepts tabular data as shown in Table 2. For the models where constant properties at ambient temperature were used, the values of the thermal properties were taken as those at 24°C. The fracture initiation analysis in this study is based Fig. 1. Characteristic dimensions for the axisymmetric model. on the Griffith and McClintock-Walsh modified Grifvity for each of the three types of rock are listed as fith criteria [8]. Once the principal stresses have been follows: found in the stress analysis the properties listed in Dresser basalt (or DB): Table 3 are required to predict the initiation of fracture. These properties were assumed to remain constant with c = 0.22exp(0-1l x 1 0 - Z T + 0.10 x 1 0 - ~ T 2 changes in temperature. Temperature dependent data is -0"22 x 1 0 - 7 T 3 + 0"12 x 1 0 - 1 ° T 4) (1) not available and because of the low diffusivity of rock, the bulk of the material in tension remains near k = 0"89 x 10 -2 - 0"31 × 10-2In(T) a +0"91 x 1 0 - 3 I n ( T ) 2 -0"83 x 10-4In(T) 3. (2) m b i e n t conditions. Sioux quartzite (or SQ): c = 0"15 + 0"91
x 10 . 3
T - 0"83
T H E R M O E L A S T I C FRACTION ANALYSIS
x 10 - 6 T 2
- 0 " 3 8 x 10-9T3+0"51
X 10-12T 4
(3)
k = 0"49 x 10 -2 + 0"17 x 10-2In(T) + 0"63 x 10-3In(T) 2 - 0"95 x 10-4In(T) 3 - 0-28 x 10 -5 ln(T) 4 - 0"53 x 10-5 ln(r) 5. + 0'72 x 10-6In(T) 6. (4) Charcoal gray granite (St. Cloud gray granodiorite or CG): c = 0'16exp(0"33 x 10 -2 T - 0.22 x 10 -5 T 2 -0.39 x 1 0 - 8 T 3 +0'35 x 10 - 1 1 T 4) (5)
The two finite element conduction codes used were developed based on the formulation given by Wilson and Nickell [15]. The finite element conduction code $70 which uses constant thermal properties was developed by Keith [16]. The finite element conduction code 2DNLT (Two Dimensional Non-linear Temperature) developed by Allen [2] has the capability of using temperature dependent thermal properties. The boundary conditions on the conduction problem were taken as: ~T/Or
TABLE 2. PROPERTY DATA USED IN STRESS ANALYSIS [4, Dresser Basalt Temp
T, *t
Young's Poisson's Modulus Ratio
E,IO5
~
kg/cm2
Charcoal gray granite Coeff. of Thermal
Young's Modulus
Expansion E,IO5 ~,I0-5/*C
Poisson's Coeff. of Ratio Thermal
v
kg/cm2
Expansion
= O, r = O, r = c/2
6, 9, 10] Sioux quartzite
Young's Modulus
E,IO5
~,I0"5/*C
kg/cm2
Polsson's Ratio
v
Coeff. of Thermal
Expansion ~,I0"5/'0
24
I0.268
0.24
0.290
7.872
0.20
0.125
8.015
0.02
122
10.054
0.24
0.500
6.873
O.15
0.762
7.729
0.0l
0.417 1.066
260
9.697
0.22
0.770
3.997
0.03
1.165
6.842
-O.Ol
1.512
371
8.933
0.19
0.940
2.366
-0.04
1.491
5,302
-0.04
1.806
482
8. 504
0.18
1.020
1.601
-0.06
1.784
3.763
-0.07
2.178
538
8.147
D.16
1.080
l.ll2
-0.1O
2.305
2.305
-0.22
2.714
593
6.781
0.II
1.160
0.592
-0.07
2.496
2.468
-0.40
2.816
700
0.010
O.10
1.160
0,010
O.00
2.500
O.Ol0
0.00
2.820
(7)
Thermal Rock Fragmentation TABLE 3. PROPERTIES USED FOR FRACTURE PREDICTIONS [8, Material
Uniaxial Tensile Strength ot,kg/cm2
Dresser basalt lOB) 154.32 Charcoal gray granite (CG) 124.50 Sioux quartzite (SQ) 256.00
Unlaxial Compressive Strength~ Oc,kg/cmL
11-14]
Fracture Surface Coefficient of Friction ~f
2980.81
0.9
2342.11 3420.00
0.9 0.9
OT/Or + h T = O, r = d/2, z > B + a ; z = O ; z = A + B T=
(8)
T,,O
(9)
T, denotes the melt temperature of the material and h is the coefficient of convective heat transfer. The finite element code, 'TRATSA' 1-1], used for the stress analysis was developed for plane or axisymmetric bodies with temperature-dependent material properties. The code is based on the formulations given by Zienkiewicz [18] and Jones and Crose [19] which assume linear displacements between nodes, resulting in constant stress elements of both triangular and quadrilateral shape. The boundary conditions for the stress problem were taken as:
a= = 0 , 0
u, = O , r = O , z < B ; r = e l 2
(10)
a, = O , r = d/2, z > B
(11)
<_ r < d/2, z = B ;
d/2 < r < c/2, z = A + B ; O <_ r <_ c/2, z = O.
(12)
The fracture initiation predictions made in this investigation are based on the stress field obtained by assuming that the material remains a continuum. In the actual case initiation of a crack will change the geometry and temperature distribution which will result in a change in the stress field. A detailed analysis of crack propagation would require progressive restructuring of the problem. At each time step, the creation of new fracture surfaces would require the introduction of additional boundary conditions. It is evident that following the propagation of a crack would require a large amount of computing time. In actual processes, nondynamic crack propagation is largely governed by the orientation of pre-existing microcracks and the interactions between them, for which no theory has been developed. The fracture initiation predictions in this investigation were performed using the program FRACTR [1], which is based on the Griffith and McClintock-Walsh modified Griffith criteria. These theories are derived from an energy formulation and neglect the effects of stress concentrations and the interactions between cracks. Even though the theories are themselves approximate they predict theoretical fracture fields (probable fracture paths)which have a statistically good correlation to those obtained experimentally.This correlation is seen in the results of Lauriello [20], Bieniawski [21], and Clark et at. [17]. FRACTR uses the stress
257
data obtained from TRATSA to plot the fracture intensity level of each element in the finite element grid. The fracture intensity level is defined in terms of the stress magnitude in excess of that necessary for the initiation of fracture. The plotted fracture intensity levels are used to predict the approximate fracture zones. The thermal fragmentation analysis was performed in three steps. First, the transient temperature distribution was obtained at various times using one of the two conduction codes $70 or 2DNLT. The temperature distributions were then used as input to the stress code (TRATSA) and the stresses were in turn used as input to the fracture initiation code (FRACTR). Once the transient temperature distribution was obtained, the last two steps above had to be repeated until the critical stress time (tl) was found. The critical stress time refers to the time, in the transient solution, at which the fracture initiation zone was complete across the entire length of the structure or model. The term critical stress time as obtained here is not a true fracture time. Rather, it is simply a characteristic time (the time at which all elements along some continuous path have reached the fracture initiation point) which has been used as a basis for studying several factors of the fragmentation process [1, 2]. AXISYMMETRIC MODEL ANALYSIS Four models of different lengths (hole spacings) and three types of rock were used. The effect of temperature-dependent material properties was obtained by comparing the solutions using variable properties to the solutions obtained using various combinations of constant (at ambient temperature) thermal and thermoelastic material properties. A description of the models is given in Table 4. (A) V a r i a b l e properties analysis Since the material properties of the three rock types are highly temperature dependent, an analysis using variable thermal and thermoelastic material properties is the most accurate solution possible within the approximations previously stated. Therefore, these results were used as a base for comparison to the analyses involving constant ambient material properties. The variable property analysis was also used to study the fracture characteristics, and develop the fracture length-critical stress time relation. Figure 2 shows the 50°C temperature contour of model 33A for each material at the critical stress time. This type of distribution, where the contour for SQ has moved the greatest distance and for DB the least, is typical of all the models at the critical stress time. TABLE 4. PARAMETRIC DESCRIPTION OF MODELS Model ]2A 33A 32B 33B
Hole Diam Spacing d, cm.
c, sin.
5.0 3.8 5.0 3.8
IO.O lO.O 20.0 20.0
Fracture Length L = c-d 5.0 6.2 15.0 16.2
Melt Depth a, cm. 1.3 1.3 1.3 1.3
Cony. Depth a = A-a Ccm 2.5 3.8 7.6 7.6
258
T . F . Lehnhoff and J. D. Scheller TABLE
Materia]
Model
5.
CRITICAL
Material Properties
STRESS TIMES
F r a c t u r e Critical Length Stress Time L, cm
tf,sec
Percent Deviation
of t f from Variable Properties
tf
E
\
SIOUXQUARTZITIf:E, 24.2 i=
5 . 0 cm
Fig. 2. Typical 50°C isotherms at the critical stress time.
The curves of critical stress time as a function of fracture length for the three rock types are shown in Fig. 3. It can be seen from this curve that, for a given fracture length, CG has the highest critical stress time and DB the lowest.
(B) Ambient properties analysis Constant thermal and thermoelastic material properties at ambient temperature are used in the ambient property analysis. From the critical stress times in Table 5, it is seen that the use of ambient material properties leads to results which differ significantly from the results in the variable properties analysis. Critical stress times are higher for CG and lower for DB and SQ in the ambient properties analysis as compared to the variable properties results. Also of importance is the order in which different rocks attain critical stresses which differs from the order found in the variable properties analysis. This would suggest that tun-
500
=
I
=
=
i
I
I
I
• 400
32A 33A 32B 33B
Variable
5.0 6.2 15.0 16.2
6.9 24.0 254.0 365.0
Granite
32A 33A 32B 33B
Variable
5.0 6.2 15.0 16.2
]3.0 33.7 327.0 460.0
Quartzite
32A 33A 32B 330
Variable
5.0 6.2 15.0 16.2
8.2 24.2 280.0 450.0
Basalt
32A 33A 32B
Ambient
5.0 6.2 15.0
5.8 20.0 252.5
Granite
32A 33A 320
Ambient
S.O 6.2 15.O
]5.8 50.0 S.S,*
Quartzite
32A 33A 32B
Ambient
5.0 6.2 15.0
4.9 15.0 218.0
40.3 38.0 22.2
Basalt Granite Quartzite
32B 32B 320
A.T.E.**
15.0
500.0 S.S. S.S.
49.3
Basalt Granite Quartzite
32B 320 32B
A.T.***
15.0
140.0 137.0 78.0
44.9 58. l 72.l
* Steady State
** Ambient Thermoelastic
16.0 16.7 0.6 21.5 5.0
*** Ambient Ther~lal
neling and excavation cost estimates, on a site of some particular rock type, performed on the basis of previous experience with another different rock type, should be gaged by the critical stress order given by a variable properties analysis. Applying the critical stress order from a simpler ambient properties analysis would yield incorrect estimates. The critical stress times for DB using ambient properties show the smallest mean deviation from the variable property results of all the rock types. This is due to the fact that, for DB, the thermal conductivity and diffusivity have only small variations with temperature. Also, the thermoelastic properties of DB show a smaller degree of temperature dependence than the properties of the other two rocks. Model 32B CG failed to reach a critical stress. The use of ambient material properties so drastically changed the problem that the system reached the point of steady state heat transfer before the thermal stresses could build to the critical level.
(C) Ambient thermoelastic properties analysis
uS ~-
Basalt
50 0
¢~
CHARCOAL GRAY G R A N I T E
200
_
s oox
77/
OOA.TZTE
D R E S S ~ I00
O
I 2
~ 4
I 6
I 8
I tO
I 12
FRACTURE LENGTH t L
cm
I 14
I 16
18
Fig. 3. Critical stress vs fracture length for three rocks.
This analysis involves temperature-dependent thermal properties and constant thermoelastic properties. From Table 5 it is seen that only the DB model achieved fracture completion. This result suggests that the temperature dependence of the thermoelastic material properties is very significant in the thermal rock fragmentation system. The fact that DB was the only rock to reach a critical stress would suggest that the coefficient of thermal expansion is the most significant of the thermoelastic properties. The thermoelastic properties for all three rocks exhibit about the same degree of variation with
Thermal Rock Fragmentation temperature with the exception of the coefficient of thermal expansion. Relative to the coefficients of thermal expansion for CG and SQ, the coefficient for DB shows only slight variation with temperature. (D) Ambient thermal properties analysis Constant thermal properties and temperature-dependent thermoelastic properties are used in this analysis. As seen in Table 5, the critical stress times are significantly lower than those found in the variable properties analysis. It is important to note that the order in which the critical stress is attained is different from that of the variable properties analysis. This can be attributed to the fact that, for the three rock types, SQ has the highest thermal diffusivity and DB has the lowest. In the variable properties analysis at higher temperatures the differences in thermal diffusivity among the rocks is small and thus has less effect in the overall thermal fragmentation problem [6]. However, these differences in diffusivity are a governing parameter in the heat transfer problem as illustrated by the temperature contours of Fig. 2. The per cent deviation of the critical stress time found in the ambient thermal properties analysis from the times found in the variable properties analysis suggest the degree of temperature dependence of the diffusivity of each of the materials. The diffusivity increases in the degree of nonlinearity, with each material, in the same order as the per cent deviation of Table 5. (E) Fracture pattern characteristics The zones of high stress intensity and probable fracture initiation were obtained from the variable property analysis. Although the critical stress times differ significantly for different materials, the fracture initiation patterns show the same general configuration for every material. However, the fracture initiation zones do differ for each of the four models ([2, 3], Fig. 4).
TABLE 6.
259
FRACTURE
L E N G T H A N D F R A C T U R E TIME R A T I O S
Length
Ll cm
Time-Ratio t f
L2 L* = IT
Dresser
Charcoal
Sioux
basalt
gray granite
quartzite
15.0
1.08
1.44
1.41
5.0
1.24
3.4B
2.59
1.61 2.96
6.2
2.42
10.60
9.70
11.60
6,2
2.60
15.20
13.60
18.60
5.0
3.00
36.80
25.20
34.20
5.0
3.24
52.90
35.40
54.90
(F) Fracture length-Critical stress relation Critical stress times (as noted earlier) for the different models can be expressed as a function of the single variable, the fracture length. Critical stress times for the variable properties analysis models, listed in Table 5, were used to calculate the fracture length ratios and the corresponding critical stress time ratios given in Table 6 and shown on Fig. 5. These ratios were computed by forming all possible ratios of fracture length and corresponding critical stress times greater than unity for each rock type. From this data the following relation was obtained: t? = (L*)3.
(13)
GENERAL TRENDS There is one well defined trend which occurs as a result of the differences in quartz content of the rock
SC tf=(
SO
411
2C
-.. ....
o.
J
C'~2/;-~ ~, ,... s~¢p/sls I , "~ ,'1/I / s , • ~ s / " , s s ¢ e"~"*"/ss/ ///'Sl.''ls
I.~"
IC
~
s
.. ;:; -..;; ,',
m
!
o
]
McCLiNTOCK-WALSH
]
GRIFFITHCOMPRESSION
]
GRIFFiTH TENSION
i
-+ 2
/
O DRESSER BASALT A SIOUX QUARTZITE O CHARCOALGRAy GRANITE
I
IO.O
em
Fig. 4. Typical potential fracture zones.
I
I
2 4 6 FRACTURE LENGTH RATIO, L"
I
6
I0
Fig. 5. Critical stress-fracture length relationship.
260
T . F . Lehnhoff and J. D. Scheller
types (DB contains no quartz, CG is composed of 5 per cent quartz, and SQ contains 95 per cent quartz). The properties which increase with increasing quartz content are the thermal diffusivity and the coefficient of thermal expansion. The relation of thermal diffusivity to quartz content supports the result showing increasing penetration of temperature field with increasing quartz content. Also, the critical stress times from the ambient thermal properties analysis are seen to decrease with increasing quartz content as would be expected from the relation of diffusivity to quartz content. However, this is due to the large differences in diffusivity between the rock types at lower temperatures and does not occur in the variable properties analysis. From these general trends it can be seen that only the heat transfer problem is related, to quartz content of the rock types. The mechanical properties of a particular rock type are not related to quartz content exclusively, but are determined by structural characteristics, such as microcracks, grain size, porosity and cleavage planes as well. Therefore, the stress analysis and/ or fracture criterion alter the trend toward a relation of critical stress time to quartz content or any one characteristic of a particular rock type. For these reasons, the thermal fragmentation of rocks becomes a complex problem involving many diverse parameters which restrict the method of solution to a complete fracture analysis, including the influence of temperature. A close correlation exists between the fracture patterns and fracture length-critical stress time relationships predicted in this theoretical analysis and those experimentally observed in field tests [-3, 17]. The results of field tests show the fracture initiating at the top of the thermal inclusion as predicted in this analysis. Also, the observation that critical stress times increase rapidly with increasing fracture length is experimentally confirmed. The approximate critical stress time corresponding to a given fracture length can be predicted from the data for a single test using the relationship t j* = (L*) 3 between the dimensionless fracture completion time t? and the dimensionless fracture length [3] for hard rocks. While equation (13) implies lower critical stress times for smaller hole spacings, the use of smaller spacings would impose higher drilling costs as well as smaller volumes of rock removed with every cycle of operation. Therefore, the choice of optimum hole spacings will involve several factors associated with the actual rock fragmentation and removal processes. Even though the power relationship is inde-
pendent of specific hard rock types, adjustments should be made in cost estimates to account for the particular rock being removed. The curve of Fig. 3 indicates that, for a given hole spacing, CG requires the highest critical stress time and DB the lowest of the three hard rock types studied in this investigation. Received 18 December 1974.
REFERENCES 1. Patel M. R. Rock fragmentation by subsurface thermal inclusions--a finite element study. Ph.D. Dissertation, University of Missouri--Rolla, Missouri, (1973). 2. Allen V. Direct and inverse heat conduction in rock materials using the finite element method. Ph.D. Dissertation, University of Missouri--Rolla, Missouri, (1974). 3. Scheller J. D. Thermal fragmentation of rock--the influence of temperature dependent material properties. M.S Thesis, University of Missouri--Rolla, Missouri, (1974). 4. Thirumalai K. Potential of internal heating method of rock fragmentation. Dynamic Rock Mechanics, Proc. 12th Syrup. Rock Mechanics (edited by Clark G. B.), pp. 697 719 (1971). 5. Maurer W. C; Novel Drilling Techniques, Pergamon Press, Oxford (1.968). 6. Wingquist C. F. Elastic moduli of rock at elevated temperatures. Bureau of Mines, RI 7269 (1970). 7. Baumeister T. and Marks L. S. Standard Handbook for Mechanical Engineers, 7th edition, McGraw-Hill, New York (1967). 8. McClintock F. A. and Walsh J. B. Friction on Griffith cracks under pressure. Proceedings 4th U.S. Congress Applied Mechanics, Berkeley 1962, pp. 1015-1021, ASME, New York (1963). 9. Thirumalai K. Rock fragmentation by creating a thermal inclusion with dielectric heating. Bureau of Mines, RI 7424 (1970). 10. Schumacher B. W. and Holdbrook R. G. Use of electron beam gun for hard rock excavation. Westinghouse Electric Corporation, Rept. 73-8C2-ROCUT-PI (1972). 11. Walsh J. B. The effect of cracks on the uniaxial elastic compression of rocks. J. Geophys. Res. 70(3) (1965). 12. Brace W. F. Brittle Fracture of Rocks. Proceedings International Conference (edited by Judd W. R.), Elsevier, New York (1963). 13. Hoek E. and Bieniawski Z. T. Brittle fracture propagation in rock under compression. Int. J. Fract. Mech., 1, 139--155 (1965). 14. Paone J. and Bruce W. E. Drillability studies; diamond drilling. Bureau of Mines, RI 6324 (1963). 15. Wilson E. L. and Nickell R. E. Application of the finite element method to heat conduction analysis. Nucl. Engng Design 4, 276286 (1966). 16. Keith H. D. Personal Communication, University of Missouri Rolla, MO. 17. Clark G. B., Lehnhoff T. F., Patel M. and Allen V. An investigation of thermal-mechanical fragmentation of hard rock. Final Report, ARPA, University of Missouri Rolla (October 1973). 18. Zienkiewicz O. E. The Finite Element Method in Structural and Continuum Mechanics. McGraw-Hill, New York (1967). 19. Jones R. N. and Crose J. G. SAAS II-Finite element stress analysis of axisymmetric solids with orthotropic, temperature dependent material properties. Aerospace Corporation, CA (1968). 20. Lauriello P. J. J. Thermal fracturing of hard crystalline rocks. Ph.D. Dissertation, Rutgers University ( 1971). 21. Bieniawski Z. T. Mechanism of brittle fracture of rock--Part IL Experimental studies. Int. J. Rock Mech. Min. Sci. 4, 407 423 (1967).