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The force-penetration characteristic for wedge penetration into rock
Int J R m h Alech Aim, 5ct Vol 4, pp. 165-180 Pergamon Press Ltd 1967. Printed m Great Britain
THE
FORCE-PENETRATION CHARACTERISTIC PENETRATION INTO ROCK
FOR
WEDGE
W . G. PARISEAU* a n d C. FAIRHURST~
(Received 24 August 1966) Abstract--Rock drllhng research studies indicate that the penetration rate of percussion drills and rotary drxlls with roller-cone bRs is determined largely by the relationship between the force applied to the bit and the resulting penetration into the rock. Although the drilling of a hole requires many thousands of Nt loadmgs much of the essential mechanics of penetration can be gamed from a study of a single application of load. Partmular attention has been paid to the penetration of sharp wedges into a rock assumed to behave according to the equations of plasticity. This paper presents a general dtscusslon of the application of plasticity analysts in wedge penetration of rock. The predictmns are compared with the results of experiments
1. PLASTICITY ANALYSIS OF WEDGE BIT PENETRATION THE governing equations o f stress for a h o m o g e n e o u s , isotropic weightless n o n - w o r k h a r d e n i n g m a t e r i a l u n d e r g o i n g a slow plane-strain d e f o r m a t i o n are, using a s t a n d a r d n o t a t i o n [1]: (1) the e q m l i b r i u m equations
~x~ + ~ y = o d~x ~y
O)
8Oxy + ~'ayu = O ?x 6y (il) the yield function
Y (~x, ¢xy, ~y~) = 0.
(2)
F o r the m a n y r o c k a n d rock-like materials which obey the C o u l o m b yield criterion, e q u a h o n (2) m a y be written in the form arx(cos20
- sine) - - %u(cos20 -t- sine) + 2azv sin20 - - 2k cos q~ -~ 0
(3)
where 0 is the angle m e a s u r e d f r o m the x-axis to the direction o f the m a j o r p r i n c i p a l stress, is the angle o f internal friction o f the material, a n d k is the cohesion. The usual M o h r d i a g r a m for a C o u l o m b m a t e r i a l at yield is shown in Fig. 1. Letting 1/2(~-x + auu) + k cot ¢ - - a, a generahzed mean stress,
(3a)
* Department of Mining, The Pennsylvania State University, University Park, Pennsylvania. f School of Mineral and Metallurgical Engineering, Umverslty of Minnesota, Mmneapohs, Minnesota. 165
166
W. G. PARISEAU AND C. FAIRHURST
equations (1) and (3) can be transformed (see Appendix A) into the system d y / d x - tan(0 ~/~)
(,)
(1/2 cot 4,) (In ~) ~. 0 =- constants
(i0
where /, == rr/4 - 4,/2. The two equations of (4 0 define two sets of lines y = f ( x ) known as the first and second families of failure surfaces or the sliplines. The two equations of (4ii) hold along the slipiT
Fro. 1. Mohr diagram for a Coulomb material at yield. lines. The stress components *xx, crxv, and cryUin the plastic region are given by the formulas' obtained by substituting equation (3a) into equation (3) and using the result in the usual equations o f transformation: crxx ---=or(1 + sin $ cos 2 0) - k cot $ % u = or(1 -- sin $ cos 2 0) -- k cot $
(5)
a*u = cr sin 4, sin 20. Ordinarily equations (4) cannot be integrated unless one can guess how 0 varies along the sliplines. Two useful cases that are frequently used arise whenever 0 is constant along one or both families of sliplines. If 0 is constant along both sliplines then, integrating (4i) we find, y -- x tan (0 ~: t~) + constant i.e. the sliplines are straight and intersect at an angle of rr/2 -- 4,. Since in this case log is constant, i.e. cr is constant, the stresses are comtant throughout the region, which is therefore a constant state region. If 0 is constant along one family of sliplines then it may be shown that the other family is composed of exponential spirals. Equations (4) are then more appropriately expressed in polar co-ordinates (r, ~o) in the form to : constant (~0) = exp [(co -- oJ0) tan~] cr =- constant
(6)
qtlE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
167
The resulting stress field is often referred to as a region o f radial shear, in reference to the fact that the family of straight sliplines (~o -----constant) pass through a common point to form a 'centled fan'. The stress fields corresponding to constant state and radial shear regions may be used to estimate the force-penetration characteristics of wedge-shaped bits penetrating a Coulomb plastic rock mass. We will use them in considering the effect of various bit angles and frictional conditions along the bit-rock interfaces. An alternative analysis will then be presented. 1.1 Smooth bit In this and all subsequent analyses the width of the bit contact edge normal to the plane of the paper is b and is assumed to be constant and independent of depth of penetration. The rock is assumed to be in contact with the bit along the entire 'penetrated' sides of the bit. Consider first a 'smooth' bit (i.e. one which cannot sustain any frictional forces at the bit-rock interface). The major principal stress ~1 acts normal to the bit-rock interface and is assumed to be constant along the length of AB (Fig. 2). Similarly, we assume that the free surface BC is yielding under a stress or0 equal to the unconfined compressive strength of the rock. These conditions can be met by assuming the stress fields to consist of two constant state regions (I, III) separated by a region of radial shear (1I) as shown in Fig. 2.
B
C y
FIG. 2. Assumed stress field for s m o o t h bit.
Region 1 is uniformly at yield. The major principal stress is equal to the unconfined compressive strength. The minor principal stress is zero. The angle 0 is :r/2. Successive computations of the stresses in the constant state region I, the radial shear region II, and the constant state region adjacent to the bit (III) enable one to compute the stresses acting normal (crn~,) and tangential (ant) to the bit-rock interface (see Appendix B). The results are ~nn
_
Gnt ~
cro (1 -k sin~) exp (2/3tan~) -- k cot ~ 2sin~
(7)
0.
The resulting upward acting force, F F = 2hb(crnn tan/3 -- c~nt).
(8)
Substituting (7) in (8) we obtain the desired force-penetration characteristic for smooth bits, i.e. F
tan/3
bhcro -- tan~ tantL [exp (2 fltanq~) -- tanS/z]
(9)
168
W
G. PARISEAU AND C. FAIRHURST
where h : depth of penetration /3 = bit half-wedge angle (0 ~ fl =;_ ~r/2) ---- fan angle (~ = t3). Equation (7) is the same as that derived by CHEATHAM[his equation (13)] in has analysts of smooth wedge penetration [2]. As indicated above, equation (9) holds for all bit-wedge angles. In this case the fan angle (~) is equal to the bit half-wedge angle (fl).
1.2 Rough bit We now consider the situation in which friction along the bit-rock interface ,s such that the interface coincides with a failure plane (i.e. the stress state of the rock material adjacent to the bit must be a radial shear region with the interface one of the radii). The interfacial shear stress (ant) is effectively a maximum for a given value of crtt and this is consequently referred to as the 'rough' case. The value of the interface frictional coefficient is not required for the analysis; it is sufficient to specify the direction of the failure plane. If, however, a sliding frictional condition is imposed, then the 'effective' coefficient of friction can be computed from the stress analysis. The centred fan now extends from the constant state region I to the bit (Fig. 3). Region Ili
y
Flo. 3. Assumed stress field for rough bits. is absent. Having established the two regions, the force displacement characteristic for the rough bit case can now be derived in a similar fashion as for the smooth bit, and is F
bhcro
_
tariff {[1 + sin$ (cot/3 tan/* -- 1)] exp (2 ~ tan¢) --- tan2/* : tan$ tan/,
where 0 ~< 13 ~/*;
(lO)
~ ---- ,r/2 - / * +/3.
In this case 0 = (/* --/3) along the bit. Equation (10) corresponds to equation (22) by Cheatham. It is of limited value because of the restriction imposed on/3. It is not possible to fit the same constant state regions on each side of the wedge for/3 >/*. In particular the regions can not be properly matched at the apex A. Since for most rocks ¢ ~ 30 ° the solution is valid for wedge angles 2/3 < 60 °. Many drill bits, and all mining percussion drill bits, have larger angles than this. Such cases can be treated by assuming the presence of a "false nose' ahead of the bit.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
169
1.3 False nose
The assumed stress field about the bit is again composed of two constant state regions separated by a centred fan (Fig. 4). The expression analagous to equations (9) and (10) is F hb~rO
__
tan3 [exp (2~qan4,) -- tanZ/,] tan4, tant~
(11)
where t* ~-- 3 :-~-r¢/2;
~: = ~/2.
X
FIG. 4. Assumed stress field for 'false nose' s~tuatlon. 1.4 General case
Analysis of the case for arbitrary 0 (constant along the bit face for any given case) results in the expression F tan/3 ~ el[[ [ l + c ° s 2 0 ( l + c ° t f t a n 2 0 ) s i n q a ] ejx p ( 2 ¢ t a n 4 , ) - - t a n 2 t , ~ bh~0 = tan~-tant, (1 + sin4,)
(12)
The particular cases can all be obtained as follows Equation (9) is obtained when 0 :L/3 -~ rr/2; ~: = 3; 0 -= rr/2 - /3. Equation (10) is obtained when 0 ~-/3 ~ t*; ~: =/3 + rr/2 -- t*; 0 : / * --/3. Equation (11) is obtained when/~ ( / 3 ( rr/2; ~: = 7r/2; 0 = O. In general we may write 0 ( / 3 ~ rr/2; ~: = ~r/2 -- 0 where 0 is given by equations (13) or (14) below. 2. ROLE OF INTERFACIAL FRICTION
As defined earher the angle 0 specifies the direction of the major principal stress along the bit face. If the relationship between the normal and shear stresses at the face is governed by the frictional conditions between the bit and rock, then the varmtion of the forcepenetration characteristic with the angle 0, as represented by equation (12), physmally represents a variation with the bit-rock interface frictmnal co-efficient tan4,'. The s~tuatmn can be seen graphically from the Mohr circle representation of stress conditions at the btt, shown m Fig. 5. The Coulomb yield criterion for the rock is represented by the line AD. The fnctmnal slip characteristic of the bit-rock interface is represented by the hne OE. The ratm of
170
w. G. PARISEAUAND (. I.AIRHURSI
normal (~nn) stress to shear (Crnt) stress necessary for frictional sliding at the interface may arise in two ways. In the first way, point H (~nn, ant) is part of the stress system represented by the small circle centre B. In the other way H is part of the larger circle, centre C "F /
-
'--O"
K
FIG. 5. Graphical representation of stresses at b~t-rock interface. As can readily be found from the geometry of Fig. 5; In the first case (circle B) 2(O--a)~Tr-
[
~sinS~] 3 t arcsin~n~lj
~- 7r -- 4' -- arcsm
[(K) 1
(13)
s-in¢:] sine J
in the second case (circle C) 2(0 -- a) ~ 4' + arcsin
l( 51 1 --
, sinai"
(14)
Note that a -- rr/2 - B in this analysis and is the angle from the x-axis (Fig. 2) to the normal to the bit-rock interface. Equations (13) and (14) clearly reveal the influence that the bit-rock angle of friction (4') has upon the angle 0. They also demonstrate that 0 varies with the mean stress a as well as with q~'. In the preceding analysis e is assumed constant along the bit so that 0 vanes with the friction angle 4' only. In such cases, specification of 4' is therefore equivalent to specifying 0 for a given e. All cases are unified in equation (12). Ordinarily, specification of the two conditions of Coulomb yield and Coulomb friction on the three stress components (gas, ~,,t, cru) over the bit-rock interface is not sufficient to uniquely determine the stress boundary conditions there. In the plastic analysis presented above, this difficulty is overcome by assuming the form of the slipline field so as to satisfy sufficient boundary conditions along the free rock surface near the bit rather than at the bit-rock interface. A sliding friction condition is not imposed at the interface although one may accept as a frictional value the ratio of ant/a,~n computed through the plastic analysis. In the friction analysis one in effect assumes sufficient boundary conditions along the bit-rock interface. This is discussed further in the analysis of the parabolic envelope. 3. CASE OF VARIABLE STRESSES A L O N G THE BIT-ROCK INTERFACE
If the stresses are considered variable along the bit-rock interface, then the complete specification of the stresses at the interface is a logical prelude to a computation of the
] H E FORCE-PENETRA'IION CHARACTERISTIC FOR WEDGE PENETRAIION IN'IO ROCK
171
force-penetration characteristic. This is to say that we ought to specify the manner in which 0 (or ~) is expected to vary along the bit-rock boundary in addition to the conditions of yield and friction. We may, for example, assume 0 to vary inversely with depth, such that tan0--(!-.
1) tanfl.
(15)
In this case, we can find by substitution of (15) into (13) and (14) that (i) over the range of applicability of equation (13) a0 tan/~ [ 1 1 -- 2 tan~ l -- (sin~/sin4') sin(~ -- 20 -- 2~ -- ~)
(16a)
O0 and over the range of applicability of equation (14)
.o-n [ cr -- 2 tan~
,
.j
1- ~ (sin~/sin~')sin(~ L 2a -- ~ ) "
(16b)
Having thus computed e one may proceed &rectly to the computation of the forcepenetration characteristic by evaluating the integral. hl~',,.,B
F
bh~o--2
[ ~n~sinfl- ~ c o s f l 3 -bh-~0 ds
(17)
0
where ds is an element of arc length along the bit face and ~n,~, crnt are the stresses normal and tangential to the bit face. 4. BRITTLE FRACTURE IN CRATER FORMATION The use of equations (15), (16) and (17) in computing a force-penetration characteristic leaves open a question as to the extent of the plastic region. Whereas it is implicit m (12), (13) or (14) that the plastic region extends from the bit face to the rock surface, m the present analysis the location of the elastic-plastic boundary is a matter of conjecture. Conditions are specified at the interface only and the extent of the plastic region is unknown. It may extend to the free rock surface or it may be confined to a thin boundary layer along the bit-rock interface or it may be something in between. Now if fracture occurs but the plastic region, though of finite size, does not extend to the rock surface, then brittle failure as well as plastic failure accompanies penetration. The formation of a fracture in an elastic region is by definition brittle behaviour. Fracture in a plastic region is another matter [3]. By itself fracture is not conclusive evidence of brittle behaviour. One must know whether or not the stresses are elastic, that is, related to the strains through Hooke's Law. Hence, one needs to determine the strain field or deformation as well as the stress field for a complete description of the rock behaviour in penetration. This is true whether the stresses are elastic or plastic. In the plastic region it is usual to use strain-rates or velocittes instead of strains to describe the motion. For that purpose, one can regard chip formation as rigid body motions. The translation and rotations of the chips are described by the velocity fields corresponding to the constant state and radial shear regions [4]. Indeed, without velocity fields that are compatible with the stress fields and that satisfy the velocity boundary conditions, the use of equations (1) and (2) is only a limit analysis (lower hound).
172
W. G. PARISEAU AND C. FAIRHURST 5. ' S L I P - S T I C K ' F A I L U R E
In equations (13) and (14) it is assumed that 4,' ~ 4,. An interesting case arises whenever the interracial friction angle exceeds the angle of internal friction so that 4,' ~ 4, In this case the interracial friction line OE. and the yield envelope AD, described by OE : rl =- antan4,' AD: r u -- crntan4, + k
(18)
where ~-f corresponds to a shear stress on OE; r u corresponds to a shear stress on AD: intersect in the a-~- plane. The point of intersection (P) corresponds to a condition of maximum mobihzation of friction along the bit-rock interface. It is a 'shp-stick' condition because for stress states below the mean stress (or) corresponding to the intersection point (circle centre L) frictional resistance to penetration is not fully mobilized so that consequently, sliding (or penetration) cannot occur. At stress levels higher than the mean stress of the intersection point (circle centre N) yield can be expected to occur before sliding.
T
A
0
~
'M~
L ~1
]
N
lot
_FIG.6. Stress conditions for simultaneous frictional sliding and yield at rock interface. As loading occurs we have a condition of yield without sliding so that the rock sticks to the bit and is carried with it during penetration. Failure represented by conditions at the intersection point P is similar to the 'rough tooth case' since the interface coincides with a failure plane. Another interesting feature associated with the intersection point (P) is that all three stress components at the interface can be determined. We readily find for the two of interest that
[
t,
~r.n : : 2 [tan4, -- tan4,J (19) oo [_tant~ tan~ ] ant -= 2 [tan4,' -- tan4,J" From (17) F [ ~ n f l + tan~' ] bh~r0 = tantz [tan4, -- tan4,J
(20)
where 4,' > 4, and 0 : tL -- ft. It should be noted that if a plastic field is assumed to extend from the bit into the rock mass then this solution [equation (20)] is restricted by the condition 0 --<,fl ( / ~ , as in the rough tooth case.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
[ 73
6. NON-LINEAR YIELD F U N C T I O N
The C o u l o m b envelope is for m a n y rocks a linear approximation of what is in fact a non-linear yield envelope. It is instructive therefore to examine a non-linear case. Instead of (3) we may, for example, use the parabola -r~,, = A ~rm + B
(2 l)
where ~m -- (1/2) (ol - - era); ~,, == (1/2) (or1 + rra) A -- (I/2) (cro ? To); B = -- (1/4) croTo ~ro -= uniaxial compressive strength, eo > 0
To = umaxml tensile strength, To - 0. In the range of stresses considered dr.,/d~,n = A/2~'rn = sine
(22)
so that ~osin4o ~'"~ -- 2sine '
~oSm¢o B e"' = 4S~ff-'¢ -- ¢rosln¢o
(23)
where 40 is the inchnatlon of the yield envelope to the ~-axls at the point it touches the uniaxial compressive strength stress circle. ¢ is varmble in general. Smce crosin¢o ==- (1/2) (cro + To) and To/~0 = 0-05 for m a n y rocks, we have approximately ¢o = 29 °. F r o m the geometry of the penetrating wedge ~,~, = ~,, -- ~,n cos 2 (0 + / 3 ) ¢~t =
-- ~-n~sin 2 (0 + / 3 ) .
(24)
The force-penetration characteristic is as before F 2 -(onn tan/3 -- o,t). bh~o cro
(25)
Using (23) in (24) and (24) in (25) we obtain F sin¢o tan/3 [ 1 sin(20 + / 3 ) ] Totan/3 bh~o = sine 2sine + sin/3 ~-2aosin¢o"
(26)
F r o m the geometry we also have ~: = rr/2 -- 0 + 1/2(¢o -- 4)
(27)
where again ~: is the fan angle in the region of radial shear. F r o m the slipline field ~: = 1/2(cot¢ -- cote0)
(28)
0 = 7r/2 + 1/2(¢o -- ¢) + l/2(cot4o -- cot¢).
(29)
so that
174
W. G. PARISEAU AND C. FAIRHURST
Equations (23) and (29) taken together constitute a rr~ 0 relationship which must hold throughout the yielding region. If now we specify Oat the bit interface we can compute the force-penetration characteristic through the combination of (23) and (29). Or if we have an additional ~'ra -- 0 relationship, as with an interfacial friction condition, we can compute the force-penetration characteristic through a simultaneous solution of the t~o r,n -- 0 equations. Taking the first alternative of specifying 0, we obtain the three special cases noted previously. Thus, we have (a) Smooth case: 0 : rr/2 - /3; 8 --/3 -~- 1/2(40 -- 4) f (b) Rough case: 0 ~-- t~ -/3; ~ ~ / 3 -t-- ~r/4 -~- 40/2) ( (c) False nose case: O = 0; ~ = rr/2 -f- 1/2(40 -- 4) }
(30)
Equations (30) all reduce to the corresponding linear relations when 40 -- 4 ~- constant. Taking the second alternative, we can obtain a second ~-m -- 0 equation by the specification of a frictional condition at the bit-rock interface. We then have r,,~ =
Asin(2# ÷ 2/3 q- 4') A [sin2(20 + 2,fl ÷ 4') 4B] 1,'~ -2sin~'q - 2 L . . . . . si-n2~ -~- ~21 .
(31)
Equivalently, one can obtain an additional O -- 4 equation, i.e. 1 sin(20 + 2/3 + 4') [sinZ(2O+ 2,fl + 4')_~_ 4B] x/z sin~ : sin4;+ [--~n~- A~]
(32)
then solve (29) and (32) for 0 and 4 and use the results in (26). A graphical solution presents no difficulties. In the above, 4' is defined by o'nt : ann tan4', or since the stresses are constant on the bit face, by T = Ntan4' where N and T are the forces acting normal and tangential to the interface, 4' can, therefore, be measured experimentally. The slip--stick condition referred to previously always obtains now independent of friction angle in the sense that the graphs of the (non-linear) yield function and (linear) friction condition intersect. As the friction angle 4' tends to zero, the stresses on the bit tend to infinity and the angle 0 tends to rr/2 -/3. But as 0 -~ ~r/2 --/3, the 'smooth' case is approached where 0 : 7r/2 --/3. The smooth case, however, does not imply infinite stresses. The contradiction and physical implications lead us to reject the slip-stick condition as being unrealistic when high hydrostatic stresses prevail. Moreover experimental evidence indicates that Coulomb friction does not obtain under high stresses. It seems more likely that a boundary layer 'flow' occurs rather than simple sliding. Nevertheless since the experiment used to determine 'friction' approximates interracial contact conditions during penetration, one may use such an experimental value of 4' to compute the force-penetration characteristic. The result is F
1
[
tan/3](
I )
bh~r-o : 2tan4' 1 . ~n-~] 1 + cos4"
(33)
where the ratio To~go has been neglected. According to (33), the force-penetration characteristie does not depend on 4 or on the strength of the material unless 4' is considered to be an index of strength.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
175
7. EXPERIMENTAL INVESTIGATIONS
Experiments were conducted on two porous rocks, Indiana Limestone and Tennessee Marble, to determine the value of the theoretical results in predicting the force-displacement relationship. The interfacial coefficient of friction between the bit and the rock was measured using the apparatus shown in Fig. 7. The split wedge bit (angle 2fl) was forced to 'penetrate' a
;/ Strain
gouges_~ --
Bit
. . . .
dl
°*e
0
=o o
Gouges e
~a 0
~
o o D 0 t-~
~
ock ° * o o o
o •
0 0
H
Steel housing
o gO o;
FIG 7. Apparatus for frictnon determination between rock-bit interface
spht rock specimen cemented into a rigid steel housing. The horizontal H, and vertical F, components of the force developed across the interface were measured by resistance stratagauge load-transducers mounted as shown. Each test was repeated at least ten times for each of three bit angles. The coefficient of friction was then determined as the ratio between components of the total force tangential and normal to the interface. The influence of a fluid environment on the coefficient of friction was determined by conducting tests on dry and water-saturated rock samples and also in a pressure chamber filled with light oll under 6000 ps~ pressure. The results of tests on Indiana Limestone, Tennessee Marble and Charcoal Gramte a local rock, are shown in Fig, 8. It is interesting to see that the rock type and the environment appear to have no s~gnificant influence on the frictional characteristics of the interface. The latter is probably a consequence of the very high stress state generated at the interface causing any lubricating film to be penetrated by the surface asperities of the materials. The unconfined compression strength (~0) of 1 -- ~ in. × ~ in. cylindrical cores was determined in earlier tests. Results are presented in Table 1.
176
W. G. PARISEAU AND C. FAIRHURST
Force penetration data for the Limestone and Marble were also obtained on dry specimens and saturated specimens subjected to 2000 psi, 4000 psi, and 6000 psi fluid environments. The apparatus is shown in Fig. 9. The force on the bit is recorded by the resistance straingauge load-cell and the penetration by a cantilever which measures the relative movement (i.e. penetration) between the fixed bit and the rock surface as the specimen was raised further into the chamber. Since the specimens were unjacketed there was essentially no difference between the sample pore fluid pressure and the fluid chamber pressure so that the specimens were all tested m the effectively unconfined condition. This is borne out by the similar values obtained for the force penetration relationship under all pressure conditions. The only observable difference was the absence in the pressure tests of flying chips TABLE l. UNIAXIAL COMPRESSIVE STRENGTHS (6,0)
Specimens 1½-in. long., ~-m. dla, cyhnders Strength (oo) (psi) Indiana Limestone (dry) Indiana Limestone (saturated) Tennessee Marble
7700 7500 15,350
Dry
1
Standard deviation of mean (%)
No, of tests 13 45
9'0 10 8
ll
In oil
In oil at 6 0 0 0 ps~
1.00
0.55 0.50
. . . .
=
-
-
- ~
-
-
L
-;,
---T~
-
~-
rndJana Limestone
-
- ~
-
-
- ~
-
I Indiana Limestone f
0,55 0.50
c:
_ _ w~ . . . .
J
t
Charcoal Gramte
0.00 1,00
0,55 0.50
/
....
~ ....
"
....
~_---~
60 °
.........
90 °
- -
.1- -
r
-
~ . . . .
I --
Tndiano Limestone I
INo dotal
....
f
-
/
-
....
i----m
Charcoal Granite
'
t
~___-}
Marble ] 0.00
3"
~____~
Charcoal Granite
f
E -
~_-
4I
g
o
-
1
0,00 1.00
&
-
(No dotal
.....
Marble
Marble
120 °
60 °
90 °
120 °
60 °
90 o
i20 °
B~t ongte FIG. 8. Effect of bit angle, lubrication and pressure on the coefficient of friction between rock and bit interface.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
y Channel To X Y Recorder X Channel
~
Cable gland block
"i--,~ t ~-'~1 1]11 I~,~,~'L
'X
canhlever
......
. ..~
, ' ~ 0 , 1 seal [~'\'\,'\\,,\,\\\~',==\C~\ ~,A~E~-Lood celt
C\\\,~, \V
Deflect,Onsens,n-------" g
177
\',
/
\"',''
5
','"
~
~,~'~
,
Dr,II bit
.S
~' Rock specJmen
t
~ ~
0,1 10 g,ve confining
Ram chomber
Ram FI~. 9. Static bit penetration under pressure.
0
• ---
40
Experimental data Linear calculated Nan-linear y,eld calculated
¢
.c 30
. \,~e°~ ~''~
.l:1
20
/
..~-50o~ - " " f
--
]0--
0 0.4
0.6
0.8
1,0
1,2
1.4
1,6
1.8
tan ,B
FIG. 10. Plasticity predictions for various values of the angle ~.
characteristic of tests under normal atmospheric conditions. This is due to the restraint offered by the chamber fluid. Results of these tests are shown, together with the curves for the various theoretical results in Fig. 10. The angle of internal friction was not determined. Instead the angle which gave the best fit of the data to equation (20) was calculated for 2fl = 90 °. The values for if, --27 ° for Indiana Limestone and 26 ° for Tennessee Marble thus obtained are in close agreement with the values usually quoted, but the extreme sensitivity of equation (20) to variations in ~ is further evidence of the unrealistic nature of the slip-stick condition. The various plasticity predictions are shown for a range of ~ values in Fig. 10. M
178
W. G, PARISEAUAND C. FAIRHURST 8. DISCUSSION
Plasticity analysis appears to pre&ct the force-displacement characteristtc r e a s o n a b l y well for wedge p e n e t r a t i o n of saturated, unconfined, porous rock. F u r t h e r tests are required before a definite statement can be made. The theoretical prediction of a 'false nose' region is strikingly confirmed in practice. A d e a r l y defined zone of intensely crushed rock is always observed i n the false nose region b e c o m i n g wider as the bit angle increases.
Acknowledgement--The financial support of this work by the American Petroleum Institute is gratefully acknowledged. REFERENCES 1. HILL R. Mathematical Theory of Plasticity, p. 130, Clarendon Press, Oxford (1950). 2. CHV.ATHAMJ. B., JR. An Analytical Study of Rock Penetration by a Single Bit Tooth, Proceedings of the Eighth Annual Drilling and Blasting Symposium, University of Minnesota, Minn. (1958). 3. THOMAST. Y. Plastic Flow and Fracture in Solids, Academic Press Inc., N.Y. (1961). 4. Srm~D R. T. Mixed boundary value problems in soil mechanics, Quart. appL Math. 11, 61-65 (1953). 5. C o ~ R. and FamvRICnS K. O. Supersonic Flow and Shock Waves, Interscience, Inc., N.Y. (1948).
APPENDIX A Translbrmation of the Stress Equations Letting o -----l/2(o~x + ~ ) + kcot(~ the yield equation (3) becomes osin~ -----1/2(ol - o~). The stress components are then expressed by the equations (5). When used in the equilibrium equations the result is the system (1 + sinqtcos20)Oo/Ox - 2 osind,sin20 ~O/~x + sin~sin20 0o/~v Jr 2 osin4~cos2000/Oy = 0 (A.1) sin~sin20 Oo/Ox + 2 osin4cos20 OO/Ox + (1 - singccos20)Oo/Oy + 2 osm~sin20 OO/~y = O. The transformation of this system to the system (4) depends on the fact that equations (A.I) are of the hyperbolic type and have two real and distinct families of characteristic curves associated with them. The first two equations of (4) define the two famih'es of characteristics. A tan,.~nt element along any characteristic curve is orientated in the direction of the charaeter~c. The c h ~ r i s t i c direction is such that the partial derivatives of the stre-~ variables (o, 0) combine to form a total derivative in the direction of the characteristic. The combination of derivatives that is required for this can he found by multiplying the first of the above equations by a number Nx and the ~eond by a number N~ then adding and further invoking the requirement that the coefficients of the stress derivatives must be in the same ratio as dy(s)/dx(s) which divines the characteristic direction (here s is some monotonic parameter) [5]. One thus obtains two equations for the determination of Nt and N~.
dy/dx
Nt(sin~in20) + N2(1 -- sin~teos20)
Nx(1 + sin4c.os20) + N2(sin4~in20) (A.2) Nt(2 osin4w,os20) + N~(2 osin4~in20) dy/dx = N t ( - 2 osin~sin20) + N2(2 osin4cos2O)" Two additional equations for the dot~a~ation of Nt and N2 may be obtained by multiplyingthe original combination of derivatives by dx/ds and dy/ds and using (A.2). The result is a homoset~ous system of four equations in two unknowns so that all second-order det~minants must vanish if a solution is to be found. One result of this requirement is that
dy/dsl = tan(0 + t~) dx/d~l and
(A.3) dy/ds~ =tan(O -- t~) dx/ds2.
THF FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
179
The independent parameters Sl and s2 may be thought of as arc length along the curves defined by (A.3). Equations (A.3) are just the first two equations of the system (4) which define the characteristic curves (shphnes) of the stress system of equations. A second result of the requirement above is that the determinant ( 1 + singx:os20)dy/ds -- sin~bsm20 dx/ds, sindesin2Ody/ds -- (1 -- sindecos20)dx/ds
]
(1 + sindpcos20)do/ds + ( - 2 asindesin20)dO/ds, sin4sin20 da/ds + 2 oslndecos20)dO/ds
I
must vanish. Th~s implies that da/dst + 2 atandpdO/dsl = 0 and
(A.4) da/ds2 -- 2 atanqgdO/ds2 = O.
Equations (A.4) can be integrated immediately to (1/2 cotff) (lno) + 0 = constant and
(A.5) (1/2 cot~) (lno) -- 0 = constant.
These equations are just the second two equations of the system (4). At this point the original objective of determining N1 and N~ is abandoned. Instead the solution of the system (4) is taken as the primary goal. Any solution of (4) is a solution of (A.1) and vice-versa provided the transformation is possible. The advantage of the new system (4) is that the derivatives are total instead of partial, and hence integration is made s~mpler. In fact two equations of the new system have been integrated (A.5). The new system also shows that knowledge of the sliphne field (of the characteristics) is entirely equivalent to a solution of the stress equations. In a complete analysis of a plane-strain plastic problem some account of the motion of the material as well as of the stress would be required. Since the deformation must be related to the stress, the shpline field is considered fundamental m conventional plasuc problems involwng plane deformatmns. The procedure for solving a complete plastic problem is then to guess the shpline field, compute the stresses to see if the stress boundary conditions are satisfied, and then to compute the velocity field to see ff the velocity boundary conditions are satisfied. (In plastic problems the velocity field characterizes the motion.) If the velocity field is found compatible with the stress field and if the stress field computed from the sliphne field satisfies the stress boundary conditions then a complete solution has been obtained. It is a feature of plane strain problems, however, that the stress equations by themselves (under statable stress boundary conditions) are statically determinant. As a result the velocity field is frequently left unexamined in plane strain problems.
APPENDIX
B
Computation o f Stresses on the Bit Consider the situation depicted in Fig. 1 for the smooth bit. In region I, a constant-state region, the stress variables (o, 0) can be computed from the boundary conditions gtven along BC. These are that or1 : a0, o3 = 0, 0 = ~r/2, where a0 is the unconfined compressive strength of the rock and the confining pressure normal to BC is for convenience taken to be zero. Thus throughout region I and in particular along BD a = o0/2sin~b and 0 = 7r/2.
(B.1)
W~th this result the stress variables can be computed along BE with the aid of the third of equations (4). One has then on BE 0 = 7r/2 -- ~: and ~r = (o0/2sinff)exp (2~tan~b)
(B.2)
where ~ is the included fan angle. From the geometry of Fig. 1, it must be that ~ = /3. Since region III is a constant-state region the stresses acting on the bit face can be computed from (B.2) with the aid of formulas (5). A more useful computatmn, however, is that of the normal and tangential stresses acting over the b~t face BA. This is a standard computation and results in an = a(1 + sin4) -- kcot4 ant ~
0.
180
w . G . PARISEAU AND C. FAIRHURST
Thus along the bit-rook interface an ~- (ao/2sin~) (1 + sinp) ¢xp (2/3 tan~) -- kcotp ant ~ 0.
~B.3~
The computation of the force-penetration chaxactcristic is now straightforward. The same sort of procedure is used to compute the lotto-penetration ¢tmractcxistic when th¢ bit is not 'smooth', and when as ~=0 on BC, though the form of th© equations change.
THE
FORCE-PENETRATION CHARACTERISTIC PENETRATION INTO ROCK
FOR
WEDGE
W . G. PARISEAU* a n d C. FAIRHURST~
(Received 24 August 1966) Abstract--Rock drllhng research studies indicate that the penetration rate of percussion drills and rotary drxlls with roller-cone bRs is determined largely by the relationship between the force applied to the bit and the resulting penetration into the rock. Although the drilling of a hole requires many thousands of Nt loadmgs much of the essential mechanics of penetration can be gamed from a study of a single application of load. Partmular attention has been paid to the penetration of sharp wedges into a rock assumed to behave according to the equations of plasticity. This paper presents a general dtscusslon of the application of plasticity analysts in wedge penetration of rock. The predictmns are compared with the results of experiments
1. PLASTICITY ANALYSIS OF WEDGE BIT PENETRATION THE governing equations o f stress for a h o m o g e n e o u s , isotropic weightless n o n - w o r k h a r d e n i n g m a t e r i a l u n d e r g o i n g a slow plane-strain d e f o r m a t i o n are, using a s t a n d a r d n o t a t i o n [1]: (1) the e q m l i b r i u m equations
~x~ + ~ y = o d~x ~y
O)
8Oxy + ~'ayu = O ?x 6y (il) the yield function
Y (~x, ¢xy, ~y~) = 0.
(2)
F o r the m a n y r o c k a n d rock-like materials which obey the C o u l o m b yield criterion, e q u a h o n (2) m a y be written in the form arx(cos20
- sine) - - %u(cos20 -t- sine) + 2azv sin20 - - 2k cos q~ -~ 0
(3)
where 0 is the angle m e a s u r e d f r o m the x-axis to the direction o f the m a j o r p r i n c i p a l stress, is the angle o f internal friction o f the material, a n d k is the cohesion. The usual M o h r d i a g r a m for a C o u l o m b m a t e r i a l at yield is shown in Fig. 1. Letting 1/2(~-x + auu) + k cot ¢ - - a, a generahzed mean stress,
(3a)
* Department of Mining, The Pennsylvania State University, University Park, Pennsylvania. f School of Mineral and Metallurgical Engineering, Umverslty of Minnesota, Mmneapohs, Minnesota. 165
166
W. G. PARISEAU AND C. FAIRHURST
equations (1) and (3) can be transformed (see Appendix A) into the system d y / d x - tan(0 ~/~)
(,)
(1/2 cot 4,) (In ~) ~. 0 =- constants
(i0
where /, == rr/4 - 4,/2. The two equations of (4 0 define two sets of lines y = f ( x ) known as the first and second families of failure surfaces or the sliplines. The two equations of (4ii) hold along the slipiT
Fro. 1. Mohr diagram for a Coulomb material at yield. lines. The stress components *xx, crxv, and cryUin the plastic region are given by the formulas' obtained by substituting equation (3a) into equation (3) and using the result in the usual equations o f transformation: crxx ---=or(1 + sin $ cos 2 0) - k cot $ % u = or(1 -- sin $ cos 2 0) -- k cot $
(5)
a*u = cr sin 4, sin 20. Ordinarily equations (4) cannot be integrated unless one can guess how 0 varies along the sliplines. Two useful cases that are frequently used arise whenever 0 is constant along one or both families of sliplines. If 0 is constant along both sliplines then, integrating (4i) we find, y -- x tan (0 ~: t~) + constant i.e. the sliplines are straight and intersect at an angle of rr/2 -- 4,. Since in this case log is constant, i.e. cr is constant, the stresses are comtant throughout the region, which is therefore a constant state region. If 0 is constant along one family of sliplines then it may be shown that the other family is composed of exponential spirals. Equations (4) are then more appropriately expressed in polar co-ordinates (r, ~o) in the form to : constant (~0) = exp [(co -- oJ0) tan~] cr =- constant
(6)
qtlE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
167
The resulting stress field is often referred to as a region o f radial shear, in reference to the fact that the family of straight sliplines (~o -----constant) pass through a common point to form a 'centled fan'. The stress fields corresponding to constant state and radial shear regions may be used to estimate the force-penetration characteristics of wedge-shaped bits penetrating a Coulomb plastic rock mass. We will use them in considering the effect of various bit angles and frictional conditions along the bit-rock interfaces. An alternative analysis will then be presented. 1.1 Smooth bit In this and all subsequent analyses the width of the bit contact edge normal to the plane of the paper is b and is assumed to be constant and independent of depth of penetration. The rock is assumed to be in contact with the bit along the entire 'penetrated' sides of the bit. Consider first a 'smooth' bit (i.e. one which cannot sustain any frictional forces at the bit-rock interface). The major principal stress ~1 acts normal to the bit-rock interface and is assumed to be constant along the length of AB (Fig. 2). Similarly, we assume that the free surface BC is yielding under a stress or0 equal to the unconfined compressive strength of the rock. These conditions can be met by assuming the stress fields to consist of two constant state regions (I, III) separated by a region of radial shear (1I) as shown in Fig. 2.
B
C y
FIG. 2. Assumed stress field for s m o o t h bit.
Region 1 is uniformly at yield. The major principal stress is equal to the unconfined compressive strength. The minor principal stress is zero. The angle 0 is :r/2. Successive computations of the stresses in the constant state region I, the radial shear region II, and the constant state region adjacent to the bit (III) enable one to compute the stresses acting normal (crn~,) and tangential (ant) to the bit-rock interface (see Appendix B). The results are ~nn
_
Gnt ~
cro (1 -k sin~) exp (2/3tan~) -- k cot ~ 2sin~
(7)
0.
The resulting upward acting force, F F = 2hb(crnn tan/3 -- c~nt).
(8)
Substituting (7) in (8) we obtain the desired force-penetration characteristic for smooth bits, i.e. F
tan/3
bhcro -- tan~ tantL [exp (2 fltanq~) -- tanS/z]
(9)
168
W
G. PARISEAU AND C. FAIRHURST
where h : depth of penetration /3 = bit half-wedge angle (0 ~ fl =;_ ~r/2) ---- fan angle (~ = t3). Equation (7) is the same as that derived by CHEATHAM[his equation (13)] in has analysts of smooth wedge penetration [2]. As indicated above, equation (9) holds for all bit-wedge angles. In this case the fan angle (~) is equal to the bit half-wedge angle (fl).
1.2 Rough bit We now consider the situation in which friction along the bit-rock interface ,s such that the interface coincides with a failure plane (i.e. the stress state of the rock material adjacent to the bit must be a radial shear region with the interface one of the radii). The interfacial shear stress (ant) is effectively a maximum for a given value of crtt and this is consequently referred to as the 'rough' case. The value of the interface frictional coefficient is not required for the analysis; it is sufficient to specify the direction of the failure plane. If, however, a sliding frictional condition is imposed, then the 'effective' coefficient of friction can be computed from the stress analysis. The centred fan now extends from the constant state region I to the bit (Fig. 3). Region Ili
y
Flo. 3. Assumed stress field for rough bits. is absent. Having established the two regions, the force displacement characteristic for the rough bit case can now be derived in a similar fashion as for the smooth bit, and is F
bhcro
_
tariff {[1 + sin$ (cot/3 tan/* -- 1)] exp (2 ~ tan¢) --- tan2/* : tan$ tan/,
where 0 ~< 13 ~/*;
(lO)
~ ---- ,r/2 - / * +/3.
In this case 0 = (/* --/3) along the bit. Equation (10) corresponds to equation (22) by Cheatham. It is of limited value because of the restriction imposed on/3. It is not possible to fit the same constant state regions on each side of the wedge for/3 >/*. In particular the regions can not be properly matched at the apex A. Since for most rocks ¢ ~ 30 ° the solution is valid for wedge angles 2/3 < 60 °. Many drill bits, and all mining percussion drill bits, have larger angles than this. Such cases can be treated by assuming the presence of a "false nose' ahead of the bit.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
169
1.3 False nose
The assumed stress field about the bit is again composed of two constant state regions separated by a centred fan (Fig. 4). The expression analagous to equations (9) and (10) is F hb~rO
__
tan3 [exp (2~qan4,) -- tanZ/,] tan4, tant~
(11)
where t* ~-- 3 :-~-r¢/2;
~: = ~/2.
X
FIG. 4. Assumed stress field for 'false nose' s~tuatlon. 1.4 General case
Analysis of the case for arbitrary 0 (constant along the bit face for any given case) results in the expression F tan/3 ~ el[[ [ l + c ° s 2 0 ( l + c ° t f t a n 2 0 ) s i n q a ] ejx p ( 2 ¢ t a n 4 , ) - - t a n 2 t , ~ bh~0 = tan~-tant, (1 + sin4,)
(12)
The particular cases can all be obtained as follows Equation (9) is obtained when 0 :L/3 -~ rr/2; ~: = 3; 0 -= rr/2 - /3. Equation (10) is obtained when 0 ~-/3 ~ t*; ~: =/3 + rr/2 -- t*; 0 : / * --/3. Equation (11) is obtained when/~ ( / 3 ( rr/2; ~: = 7r/2; 0 = O. In general we may write 0 ( / 3 ~ rr/2; ~: = ~r/2 -- 0 where 0 is given by equations (13) or (14) below. 2. ROLE OF INTERFACIAL FRICTION
As defined earher the angle 0 specifies the direction of the major principal stress along the bit face. If the relationship between the normal and shear stresses at the face is governed by the frictional conditions between the bit and rock, then the varmtion of the forcepenetration characteristic with the angle 0, as represented by equation (12), physmally represents a variation with the bit-rock interface frictmnal co-efficient tan4,'. The s~tuatmn can be seen graphically from the Mohr circle representation of stress conditions at the btt, shown m Fig. 5. The Coulomb yield criterion for the rock is represented by the line AD. The fnctmnal slip characteristic of the bit-rock interface is represented by the hne OE. The ratm of
170
w. G. PARISEAUAND (. I.AIRHURSI
normal (~nn) stress to shear (Crnt) stress necessary for frictional sliding at the interface may arise in two ways. In the first way, point H (~nn, ant) is part of the stress system represented by the small circle centre B. In the other way H is part of the larger circle, centre C "F /
-
'--O"
K
FIG. 5. Graphical representation of stresses at b~t-rock interface. As can readily be found from the geometry of Fig. 5; In the first case (circle B) 2(O--a)~Tr-
[
~sinS~] 3 t arcsin~n~lj
~- 7r -- 4' -- arcsm
[(K) 1
(13)
s-in¢:] sine J
in the second case (circle C) 2(0 -- a) ~ 4' + arcsin
l( 51 1 --
, sinai"
(14)
Note that a -- rr/2 - B in this analysis and is the angle from the x-axis (Fig. 2) to the normal to the bit-rock interface. Equations (13) and (14) clearly reveal the influence that the bit-rock angle of friction (4') has upon the angle 0. They also demonstrate that 0 varies with the mean stress a as well as with q~'. In the preceding analysis e is assumed constant along the bit so that 0 vanes with the friction angle 4' only. In such cases, specification of 4' is therefore equivalent to specifying 0 for a given e. All cases are unified in equation (12). Ordinarily, specification of the two conditions of Coulomb yield and Coulomb friction on the three stress components (gas, ~,,t, cru) over the bit-rock interface is not sufficient to uniquely determine the stress boundary conditions there. In the plastic analysis presented above, this difficulty is overcome by assuming the form of the slipline field so as to satisfy sufficient boundary conditions along the free rock surface near the bit rather than at the bit-rock interface. A sliding friction condition is not imposed at the interface although one may accept as a frictional value the ratio of ant/a,~n computed through the plastic analysis. In the friction analysis one in effect assumes sufficient boundary conditions along the bit-rock interface. This is discussed further in the analysis of the parabolic envelope. 3. CASE OF VARIABLE STRESSES A L O N G THE BIT-ROCK INTERFACE
If the stresses are considered variable along the bit-rock interface, then the complete specification of the stresses at the interface is a logical prelude to a computation of the
] H E FORCE-PENETRA'IION CHARACTERISTIC FOR WEDGE PENETRAIION IN'IO ROCK
171
force-penetration characteristic. This is to say that we ought to specify the manner in which 0 (or ~) is expected to vary along the bit-rock boundary in addition to the conditions of yield and friction. We may, for example, assume 0 to vary inversely with depth, such that tan0--(!-.
1) tanfl.
(15)
In this case, we can find by substitution of (15) into (13) and (14) that (i) over the range of applicability of equation (13) a0 tan/~ [ 1 1 -- 2 tan~ l -- (sin~/sin4') sin(~ -- 20 -- 2~ -- ~)
(16a)
O0 and over the range of applicability of equation (14)
.o-n [ cr -- 2 tan~
,
.j
1- ~ (sin~/sin~')sin(~ L 2a -- ~ ) "
(16b)
Having thus computed e one may proceed &rectly to the computation of the forcepenetration characteristic by evaluating the integral. hl~',,.,B
F
bh~o--2
[ ~n~sinfl- ~ c o s f l 3 -bh-~0 ds
(17)
0
where ds is an element of arc length along the bit face and ~n,~, crnt are the stresses normal and tangential to the bit face. 4. BRITTLE FRACTURE IN CRATER FORMATION The use of equations (15), (16) and (17) in computing a force-penetration characteristic leaves open a question as to the extent of the plastic region. Whereas it is implicit m (12), (13) or (14) that the plastic region extends from the bit face to the rock surface, m the present analysis the location of the elastic-plastic boundary is a matter of conjecture. Conditions are specified at the interface only and the extent of the plastic region is unknown. It may extend to the free rock surface or it may be confined to a thin boundary layer along the bit-rock interface or it may be something in between. Now if fracture occurs but the plastic region, though of finite size, does not extend to the rock surface, then brittle failure as well as plastic failure accompanies penetration. The formation of a fracture in an elastic region is by definition brittle behaviour. Fracture in a plastic region is another matter [3]. By itself fracture is not conclusive evidence of brittle behaviour. One must know whether or not the stresses are elastic, that is, related to the strains through Hooke's Law. Hence, one needs to determine the strain field or deformation as well as the stress field for a complete description of the rock behaviour in penetration. This is true whether the stresses are elastic or plastic. In the plastic region it is usual to use strain-rates or velocittes instead of strains to describe the motion. For that purpose, one can regard chip formation as rigid body motions. The translation and rotations of the chips are described by the velocity fields corresponding to the constant state and radial shear regions [4]. Indeed, without velocity fields that are compatible with the stress fields and that satisfy the velocity boundary conditions, the use of equations (1) and (2) is only a limit analysis (lower hound).
172
W. G. PARISEAU AND C. FAIRHURST 5. ' S L I P - S T I C K ' F A I L U R E
In equations (13) and (14) it is assumed that 4,' ~ 4,. An interesting case arises whenever the interracial friction angle exceeds the angle of internal friction so that 4,' ~ 4, In this case the interracial friction line OE. and the yield envelope AD, described by OE : rl =- antan4,' AD: r u -- crntan4, + k
(18)
where ~-f corresponds to a shear stress on OE; r u corresponds to a shear stress on AD: intersect in the a-~- plane. The point of intersection (P) corresponds to a condition of maximum mobihzation of friction along the bit-rock interface. It is a 'shp-stick' condition because for stress states below the mean stress (or) corresponding to the intersection point (circle centre L) frictional resistance to penetration is not fully mobilized so that consequently, sliding (or penetration) cannot occur. At stress levels higher than the mean stress of the intersection point (circle centre N) yield can be expected to occur before sliding.
T
A
0
~
'M~
L ~1
]
N
lot
_FIG.6. Stress conditions for simultaneous frictional sliding and yield at rock interface. As loading occurs we have a condition of yield without sliding so that the rock sticks to the bit and is carried with it during penetration. Failure represented by conditions at the intersection point P is similar to the 'rough tooth case' since the interface coincides with a failure plane. Another interesting feature associated with the intersection point (P) is that all three stress components at the interface can be determined. We readily find for the two of interest that
[
t,
~r.n : : 2 [tan4, -- tan4,J (19) oo [_tant~ tan~ ] ant -= 2 [tan4,' -- tan4,J" From (17) F [ ~ n f l + tan~' ] bh~r0 = tantz [tan4, -- tan4,J
(20)
where 4,' > 4, and 0 : tL -- ft. It should be noted that if a plastic field is assumed to extend from the bit into the rock mass then this solution [equation (20)] is restricted by the condition 0 --<,fl ( / ~ , as in the rough tooth case.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
[ 73
6. NON-LINEAR YIELD F U N C T I O N
The C o u l o m b envelope is for m a n y rocks a linear approximation of what is in fact a non-linear yield envelope. It is instructive therefore to examine a non-linear case. Instead of (3) we may, for example, use the parabola -r~,, = A ~rm + B
(2 l)
where ~m -- (1/2) (ol - - era); ~,, == (1/2) (or1 + rra) A -- (I/2) (cro ? To); B = -- (1/4) croTo ~ro -= uniaxial compressive strength, eo > 0
To = umaxml tensile strength, To - 0. In the range of stresses considered dr.,/d~,n = A/2~'rn = sine
(22)
so that ~osin4o ~'"~ -- 2sine '
~oSm¢o B e"' = 4S~ff-'¢ -- ¢rosln¢o
(23)
where 40 is the inchnatlon of the yield envelope to the ~-axls at the point it touches the uniaxial compressive strength stress circle. ¢ is varmble in general. Smce crosin¢o ==- (1/2) (cro + To) and To/~0 = 0-05 for m a n y rocks, we have approximately ¢o = 29 °. F r o m the geometry of the penetrating wedge ~,~, = ~,, -- ~,n cos 2 (0 + / 3 ) ¢~t =
-- ~-n~sin 2 (0 + / 3 ) .
(24)
The force-penetration characteristic is as before F 2 -(onn tan/3 -- o,t). bh~o cro
(25)
Using (23) in (24) and (24) in (25) we obtain F sin¢o tan/3 [ 1 sin(20 + / 3 ) ] Totan/3 bh~o = sine 2sine + sin/3 ~-2aosin¢o"
(26)
F r o m the geometry we also have ~: = rr/2 -- 0 + 1/2(¢o -- 4)
(27)
where again ~: is the fan angle in the region of radial shear. F r o m the slipline field ~: = 1/2(cot¢ -- cote0)
(28)
0 = 7r/2 + 1/2(¢o -- ¢) + l/2(cot4o -- cot¢).
(29)
so that
174
W. G. PARISEAU AND C. FAIRHURST
Equations (23) and (29) taken together constitute a rr~ 0 relationship which must hold throughout the yielding region. If now we specify Oat the bit interface we can compute the force-penetration characteristic through the combination of (23) and (29). Or if we have an additional ~'ra -- 0 relationship, as with an interfacial friction condition, we can compute the force-penetration characteristic through a simultaneous solution of the t~o r,n -- 0 equations. Taking the first alternative of specifying 0, we obtain the three special cases noted previously. Thus, we have (a) Smooth case: 0 : rr/2 - /3; 8 --/3 -~- 1/2(40 -- 4) f (b) Rough case: 0 ~-- t~ -/3; ~ ~ / 3 -t-- ~r/4 -~- 40/2) ( (c) False nose case: O = 0; ~ = rr/2 -f- 1/2(40 -- 4) }
(30)
Equations (30) all reduce to the corresponding linear relations when 40 -- 4 ~- constant. Taking the second alternative, we can obtain a second ~-m -- 0 equation by the specification of a frictional condition at the bit-rock interface. We then have r,,~ =
Asin(2# ÷ 2/3 q- 4') A [sin2(20 + 2,fl ÷ 4') 4B] 1,'~ -2sin~'q - 2 L . . . . . si-n2~ -~- ~21 .
(31)
Equivalently, one can obtain an additional O -- 4 equation, i.e. 1 sin(20 + 2/3 + 4') [sinZ(2O+ 2,fl + 4')_~_ 4B] x/z sin~ : sin4;+ [--~n~- A~]
(32)
then solve (29) and (32) for 0 and 4 and use the results in (26). A graphical solution presents no difficulties. In the above, 4' is defined by o'nt : ann tan4', or since the stresses are constant on the bit face, by T = Ntan4' where N and T are the forces acting normal and tangential to the interface, 4' can, therefore, be measured experimentally. The slip--stick condition referred to previously always obtains now independent of friction angle in the sense that the graphs of the (non-linear) yield function and (linear) friction condition intersect. As the friction angle 4' tends to zero, the stresses on the bit tend to infinity and the angle 0 tends to rr/2 -/3. But as 0 -~ ~r/2 --/3, the 'smooth' case is approached where 0 : 7r/2 --/3. The smooth case, however, does not imply infinite stresses. The contradiction and physical implications lead us to reject the slip-stick condition as being unrealistic when high hydrostatic stresses prevail. Moreover experimental evidence indicates that Coulomb friction does not obtain under high stresses. It seems more likely that a boundary layer 'flow' occurs rather than simple sliding. Nevertheless since the experiment used to determine 'friction' approximates interracial contact conditions during penetration, one may use such an experimental value of 4' to compute the force-penetration characteristic. The result is F
1
[
tan/3](
I )
bh~r-o : 2tan4' 1 . ~n-~] 1 + cos4"
(33)
where the ratio To~go has been neglected. According to (33), the force-penetration characteristie does not depend on 4 or on the strength of the material unless 4' is considered to be an index of strength.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
175
7. EXPERIMENTAL INVESTIGATIONS
Experiments were conducted on two porous rocks, Indiana Limestone and Tennessee Marble, to determine the value of the theoretical results in predicting the force-displacement relationship. The interfacial coefficient of friction between the bit and the rock was measured using the apparatus shown in Fig. 7. The split wedge bit (angle 2fl) was forced to 'penetrate' a
;/ Strain
gouges_~ --
Bit
. . . .
dl
°*e
0
=o o
Gouges e
~a 0
~
o o D 0 t-~
~
ock ° * o o o
o •
0 0
H
Steel housing
o gO o;
FIG 7. Apparatus for frictnon determination between rock-bit interface
spht rock specimen cemented into a rigid steel housing. The horizontal H, and vertical F, components of the force developed across the interface were measured by resistance stratagauge load-transducers mounted as shown. Each test was repeated at least ten times for each of three bit angles. The coefficient of friction was then determined as the ratio between components of the total force tangential and normal to the interface. The influence of a fluid environment on the coefficient of friction was determined by conducting tests on dry and water-saturated rock samples and also in a pressure chamber filled with light oll under 6000 ps~ pressure. The results of tests on Indiana Limestone, Tennessee Marble and Charcoal Gramte a local rock, are shown in Fig, 8. It is interesting to see that the rock type and the environment appear to have no s~gnificant influence on the frictional characteristics of the interface. The latter is probably a consequence of the very high stress state generated at the interface causing any lubricating film to be penetrated by the surface asperities of the materials. The unconfined compression strength (~0) of 1 -- ~ in. × ~ in. cylindrical cores was determined in earlier tests. Results are presented in Table 1.
176
W. G. PARISEAU AND C. FAIRHURST
Force penetration data for the Limestone and Marble were also obtained on dry specimens and saturated specimens subjected to 2000 psi, 4000 psi, and 6000 psi fluid environments. The apparatus is shown in Fig. 9. The force on the bit is recorded by the resistance straingauge load-cell and the penetration by a cantilever which measures the relative movement (i.e. penetration) between the fixed bit and the rock surface as the specimen was raised further into the chamber. Since the specimens were unjacketed there was essentially no difference between the sample pore fluid pressure and the fluid chamber pressure so that the specimens were all tested m the effectively unconfined condition. This is borne out by the similar values obtained for the force penetration relationship under all pressure conditions. The only observable difference was the absence in the pressure tests of flying chips TABLE l. UNIAXIAL COMPRESSIVE STRENGTHS (6,0)
Specimens 1½-in. long., ~-m. dla, cyhnders Strength (oo) (psi) Indiana Limestone (dry) Indiana Limestone (saturated) Tennessee Marble
7700 7500 15,350
Dry
1
Standard deviation of mean (%)
No, of tests 13 45
9'0 10 8
ll
In oil
In oil at 6 0 0 0 ps~
1.00
0.55 0.50
. . . .
=
-
-
- ~
-
-
L
-;,
---T~
-
~-
rndJana Limestone
-
- ~
-
-
- ~
-
I Indiana Limestone f
0,55 0.50
c:
_ _ w~ . . . .
J
t
Charcoal Gramte
0.00 1,00
0,55 0.50
/
....
~ ....
"
....
~_---~
60 °
.........
90 °
- -
.1- -
r
-
~ . . . .
I --
Tndiano Limestone I
INo dotal
....
f
-
/
-
....
i----m
Charcoal Granite
'
t
~___-}
Marble ] 0.00
3"
~____~
Charcoal Granite
f
E -
~_-
4I
g
o
-
1
0,00 1.00
&
-
(No dotal
.....
Marble
Marble
120 °
60 °
90 °
120 °
60 °
90 o
i20 °
B~t ongte FIG. 8. Effect of bit angle, lubrication and pressure on the coefficient of friction between rock and bit interface.
THE FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
y Channel To X Y Recorder X Channel
~
Cable gland block
"i--,~ t ~-'~1 1]11 I~,~,~'L
'X
canhlever
......
. ..~
, ' ~ 0 , 1 seal [~'\'\,'\\,,\,\\\~',==\C~\ ~,A~E~-Lood celt
C\\\,~, \V
Deflect,Onsens,n-------" g
177
\',
/
\"',''
5
','"
~
~,~'~
,
Dr,II bit
.S
~' Rock specJmen
t
~ ~
0,1 10 g,ve confining
Ram chomber
Ram FI~. 9. Static bit penetration under pressure.
0
• ---
40
Experimental data Linear calculated Nan-linear y,eld calculated
¢
.c 30
. \,~e°~ ~''~
.l:1
20
/
..~-50o~ - " " f
--
]0--
0 0.4
0.6
0.8
1,0
1,2
1.4
1,6
1.8
tan ,B
FIG. 10. Plasticity predictions for various values of the angle ~.
characteristic of tests under normal atmospheric conditions. This is due to the restraint offered by the chamber fluid. Results of these tests are shown, together with the curves for the various theoretical results in Fig. 10. The angle of internal friction was not determined. Instead the angle which gave the best fit of the data to equation (20) was calculated for 2fl = 90 °. The values for if, --27 ° for Indiana Limestone and 26 ° for Tennessee Marble thus obtained are in close agreement with the values usually quoted, but the extreme sensitivity of equation (20) to variations in ~ is further evidence of the unrealistic nature of the slip-stick condition. The various plasticity predictions are shown for a range of ~ values in Fig. 10. M
178
W. G, PARISEAUAND C. FAIRHURST 8. DISCUSSION
Plasticity analysis appears to pre&ct the force-displacement characteristtc r e a s o n a b l y well for wedge p e n e t r a t i o n of saturated, unconfined, porous rock. F u r t h e r tests are required before a definite statement can be made. The theoretical prediction of a 'false nose' region is strikingly confirmed in practice. A d e a r l y defined zone of intensely crushed rock is always observed i n the false nose region b e c o m i n g wider as the bit angle increases.
Acknowledgement--The financial support of this work by the American Petroleum Institute is gratefully acknowledged. REFERENCES 1. HILL R. Mathematical Theory of Plasticity, p. 130, Clarendon Press, Oxford (1950). 2. CHV.ATHAMJ. B., JR. An Analytical Study of Rock Penetration by a Single Bit Tooth, Proceedings of the Eighth Annual Drilling and Blasting Symposium, University of Minnesota, Minn. (1958). 3. THOMAST. Y. Plastic Flow and Fracture in Solids, Academic Press Inc., N.Y. (1961). 4. Srm~D R. T. Mixed boundary value problems in soil mechanics, Quart. appL Math. 11, 61-65 (1953). 5. C o ~ R. and FamvRICnS K. O. Supersonic Flow and Shock Waves, Interscience, Inc., N.Y. (1948).
APPENDIX A Translbrmation of the Stress Equations Letting o -----l/2(o~x + ~ ) + kcot(~ the yield equation (3) becomes osin~ -----1/2(ol - o~). The stress components are then expressed by the equations (5). When used in the equilibrium equations the result is the system (1 + sinqtcos20)Oo/Ox - 2 osind,sin20 ~O/~x + sin~sin20 0o/~v Jr 2 osin4~cos2000/Oy = 0 (A.1) sin~sin20 Oo/Ox + 2 osin4cos20 OO/Ox + (1 - singccos20)Oo/Oy + 2 osm~sin20 OO/~y = O. The transformation of this system to the system (4) depends on the fact that equations (A.I) are of the hyperbolic type and have two real and distinct families of characteristic curves associated with them. The first two equations of (4) define the two famih'es of characteristics. A tan,.~nt element along any characteristic curve is orientated in the direction of the charaeter~c. The c h ~ r i s t i c direction is such that the partial derivatives of the stre-~ variables (o, 0) combine to form a total derivative in the direction of the characteristic. The combination of derivatives that is required for this can he found by multiplying the first of the above equations by a number Nx and the ~eond by a number N~ then adding and further invoking the requirement that the coefficients of the stress derivatives must be in the same ratio as dy(s)/dx(s) which divines the characteristic direction (here s is some monotonic parameter) [5]. One thus obtains two equations for the determination of Nt and N~.
dy/dx
Nt(sin~in20) + N2(1 -- sin~teos20)
Nx(1 + sin4c.os20) + N2(sin4~in20) (A.2) Nt(2 osin4w,os20) + N~(2 osin4~in20) dy/dx = N t ( - 2 osin~sin20) + N2(2 osin4cos2O)" Two additional equations for the dot~a~ation of Nt and N2 may be obtained by multiplyingthe original combination of derivatives by dx/ds and dy/ds and using (A.2). The result is a homoset~ous system of four equations in two unknowns so that all second-order det~minants must vanish if a solution is to be found. One result of this requirement is that
dy/dsl = tan(0 + t~) dx/d~l and
(A.3) dy/ds~ =tan(O -- t~) dx/ds2.
THF FORCE-PENETRATION CHARACTERISTIC FOR WEDGE PENETRATION INTO ROCK
179
The independent parameters Sl and s2 may be thought of as arc length along the curves defined by (A.3). Equations (A.3) are just the first two equations of the system (4) which define the characteristic curves (shphnes) of the stress system of equations. A second result of the requirement above is that the determinant ( 1 + singx:os20)dy/ds -- sin~bsm20 dx/ds, sindesin2Ody/ds -- (1 -- sindecos20)dx/ds
]
(1 + sindpcos20)do/ds + ( - 2 asindesin20)dO/ds, sin4sin20 da/ds + 2 oslndecos20)dO/ds
I
must vanish. Th~s implies that da/dst + 2 atandpdO/dsl = 0 and
(A.4) da/ds2 -- 2 atanqgdO/ds2 = O.
Equations (A.4) can be integrated immediately to (1/2 cotff) (lno) + 0 = constant and
(A.5) (1/2 cot~) (lno) -- 0 = constant.
These equations are just the second two equations of the system (4). At this point the original objective of determining N1 and N~ is abandoned. Instead the solution of the system (4) is taken as the primary goal. Any solution of (4) is a solution of (A.1) and vice-versa provided the transformation is possible. The advantage of the new system (4) is that the derivatives are total instead of partial, and hence integration is made s~mpler. In fact two equations of the new system have been integrated (A.5). The new system also shows that knowledge of the sliphne field (of the characteristics) is entirely equivalent to a solution of the stress equations. In a complete analysis of a plane-strain plastic problem some account of the motion of the material as well as of the stress would be required. Since the deformation must be related to the stress, the shpline field is considered fundamental m conventional plasuc problems involwng plane deformatmns. The procedure for solving a complete plastic problem is then to guess the shpline field, compute the stresses to see if the stress boundary conditions are satisfied, and then to compute the velocity field to see ff the velocity boundary conditions are satisfied. (In plastic problems the velocity field characterizes the motion.) If the velocity field is found compatible with the stress field and if the stress field computed from the sliphne field satisfies the stress boundary conditions then a complete solution has been obtained. It is a feature of plane strain problems, however, that the stress equations by themselves (under statable stress boundary conditions) are statically determinant. As a result the velocity field is frequently left unexamined in plane strain problems.
APPENDIX
B
Computation o f Stresses on the Bit Consider the situation depicted in Fig. 1 for the smooth bit. In region I, a constant-state region, the stress variables (o, 0) can be computed from the boundary conditions gtven along BC. These are that or1 : a0, o3 = 0, 0 = ~r/2, where a0 is the unconfined compressive strength of the rock and the confining pressure normal to BC is for convenience taken to be zero. Thus throughout region I and in particular along BD a = o0/2sin~b and 0 = 7r/2.
(B.1)
W~th this result the stress variables can be computed along BE with the aid of the third of equations (4). One has then on BE 0 = 7r/2 -- ~: and ~r = (o0/2sinff)exp (2~tan~b)
(B.2)
where ~ is the included fan angle. From the geometry of Fig. 1, it must be that ~ = /3. Since region III is a constant-state region the stresses acting on the bit face can be computed from (B.2) with the aid of formulas (5). A more useful computatmn, however, is that of the normal and tangential stresses acting over the b~t face BA. This is a standard computation and results in an = a(1 + sin4) -- kcot4 ant ~
0.
180
w . G . PARISEAU AND C. FAIRHURST
Thus along the bit-rook interface an ~- (ao/2sin~) (1 + sinp) ¢xp (2/3 tan~) -- kcotp ant ~ 0.
~B.3~
The computation of the force-penetration chaxactcristic is now straightforward. The same sort of procedure is used to compute the lotto-penetration ¢tmractcxistic when th¢ bit is not 'smooth', and when as ~=0 on BC, though the form of th© equations change.