Indian Journal of Engineering & Materials Sciences Vol. 7, June 2000, pp. 127-135

Rock indentation analysis based on the theories of elasticity

K U M Rao Department of Mining Engineering, Indian Inst itute of Technology, Kharagpur 721302, India

Received 20 September 1999; accepted 17 May 2000

A theoretical analysis based on the Von-Mises fl ow cri teria and extension strain failure criteria, to estimate the extent of fracturing under the static indentation in rocks with and without di scontinuities is presented here. The concept of relating the stress distribution in materials to the ratio of Young's modulus to Shear modulus (ElC) in which a non-linear relation­ship between the elastic parameters (E. C and Poisson ' s ratio j.I) is considered in the form ElC = 2( 1 +j.I) for the rock without discontinuities and ElC > 2(1+j.I) for the rocks with closely spaced discontinuities. The theoretical values are validated by detailed laboratory static indentation experiments in various rock types. Related mechanical properties of these rocks have also been determined in the laboratory. Of the various criterion considered, the extension strain criteri a gave a close estima­tion of the fracture ex tent in almost all the rock types for both conditions of ElC = 2( 1 +j.I) and ElC > 2(1 +j.I) . The theoreti ­cal values by these failure criteria are found to be wi th in the reasonable limits.

Rock fragmentation process by all the mechanical cutting tool s is essentially a rock breakage operation resulting in the formation of chips from the in situ rock mass. The current methods of mechanical fragmentation, regardless of machine design or cutter type, have cutters that travel in a fixed path . Thus, the cutters are ri gidly fixed to a cutter head that itself moves in some fixed curvilinear path . The predominant action in such cutting operation involves the principles of indentation of the cutting tool 1.2.

Based on the type of the cuttrng operation, sometimes the indentation is by impact as in the percussive actions, and sometimes out of a continuous thrust as in the rotary operations. The experimental studies have substantially proved that the rock fracturing in any of the drilling processes is primarily an indentation of the cutting tool due to the applied thrust force . The role of the torque force is secondary and it helps in ploughing out the fragmented rock from the fractured zone'·6.

Wijk7 indicated thatthe bit penetrates the rock with a velocity from about a few mm/sec in rotary drilling to a few m/sec for percussive drilling. However, compared to the stress wave velocities of a few km/sec, the penetration velocities in both the cases are still very low. Therefore, the theories of static indentat ion fracture mechanics can be applied with a close approximation8

. The indentation studies like the concentrated load represented by a conical tip producing a Boussinesque stress field in the specimen9

, lo, as well as a line contact, simulating a

wedge indenter, a hemispherical and other continuous contact surface of distributed loading producing a Hertzian stress distribution9

•11 have been employed in

one way or another as simple methods of predicting the rock cutting forces,

The consideration of rock to be elastic, homogeneous and isotropic, for the stress analysis has been based on the assumption that in drilling, a rigid indenter penetra'tes a half-space which may be considered macroscopicall y homogeneous and isotropic since deformation will be large compared to grain size II, Studies of wedge indentat ion , including the prediction of crater depth and width as a function of the indenter geometry and applied load, are numerous and have been summarized by several researchers l

•2

• How.ever, most of these investigations are confined to a two dimensional analysis and do not deal with the problem of crack initiation and crack propagation under the indentation .

The present study is also based on the elastic theories but emphasis has been laid on the three dimensional analysis of the fracture in rock due to indentation of the cutting tool. The primary consideration in this study is that the cutting tool at the interface offers a concentrated normal loading on the rock. This closely resembles the point load acting normally on the semi-infinite mass, commonly known as Boussinesque 's problem9

.11

, 12,J3. The Bouss inesque 's solutions are adopted in the analys is assuming the rock to be e lastic, isotropic and homogeneous for simplifying the resulting complex

128 INDIAN J ENG. MATER. SCI., JUNE 2000

solutions. However, the analytical investigations are carried out considering rock as intact and near elastic material as well as a material composed of discrete planes of structural features such as joints, faults fissures, cracks which together are considered as discontinuities.

The Material Property Concept

In almost all the areas of stress analysis, the constants E (Young's modulus) and /1 (Poisson's Ratio) are the most widely recognized elastic properties used to define the engineering behavior of the material. However, there are a few published results available concerning the influence of these material properties (E, C and /1) on the distribution of the stresses. In the c1assieal elastic approach, the relation between E, C and /1 when taken as E/C = 2( I +/1), for the pressure distribution in the homogeneous medium, the end results though not very significant, were found to be well with in the acceptable limits. The above relationship between E, C and /1, firstly, failed to incorporate all the types of materials, and secondly, in the process of mathematical simplifications, the mechanical properties of the medium completely cancel out inferring that the stress equations are free of the medium properties (E, C and 11) . This has led one to reconsider the realistic effect of the ratio E/C of the medium in terms of the stress transmittal. The concept of relating the stress distribution in materials to the ratio of Young's modulus to Shear modulus (£IC) in which a non-linear relationship between the elastic parameters (E, C and Poisson's ratio /1) is considered in the form E/C > 2(1+/1) was first considered by Weiskopfl4. The Weiskopf's model has been based on the theory proposed by Fopplls which states that upon appiication of loading, the granular materials have the tendencies of slipping on each other and the resistance to shear is much less compared to the solid materials and the resulting shear displacement is much greater. This characteristic of granular material is accounted in the model £IC > 2( J + /1) and the term E/C has been considered as the material constant which takes into account not only the elastic behavior of such materials under compression and shear, but also the slipping of the granules on each. The attempt to transform the real material bodies composed of discrete particles into a form such that useful deduction can be made through the exact process of

mathematics is accomplished by reducing the medium to a macroscopic equi " 3.~ e ' lt through the introduction of material properties, represented by the ratio E/C I5

.

Fracture analysis

A detailed theoretical analysis to predict the fracture extent in the intact rocks and the rocks with discontinuities was undertaken. The Boussinesque

expression for stressO'x. O'r ' O'Z' "(X Y, TI'Z,TZX at a

poi n! in the media, under the point load were used in the Von-Mises flow criteria l6 and the extension strain ::riteria of failure l7 (the detailed expressions for both the models £IC = 2(1+/1) and E/G > 2(1+/1) are given in the Appendix):

Model E/G = 2(1 +11)

Von-Mises yield criteria

Von-Mises yield criteria are based on the assumption that the hydrostatic stress alone does not cause appreciable plastic deformation in brittle materials like rocks I6

, 19, Thus, in developing criteria for yielding, it is usual to subtract the hydrostatic part from the actual stresses, calling the remainder a stress deviation, assuming that this quantity alone produces yield and the conditions for yield may be expressed in terms of this stress deviation alone, The resulting failure condition is as given below:

S~ +S~ +S~ +2T~y +2T~ +2T~ -2K2 =0 ", (I)

where,

k=O'T1..[j

crT = permissible tensile strength of the rock, and Sx, Sy, Sz are the components of stress de­viation,

Using Sx. S\,o Sz, "(Xy. T yz' T zx, as obtained from Expressions (i .) to (vi) described 10 the Appendix, Eq.( 1) can be written as: 8 2 4 2

1t R crT = 2(1 _ )2 3P2 11

[(I-COSq»2 cos(I-cos<p) I 1

. 4 - . 2 + -cosq> SIl1 q> SII1 q> 2

- 6(1 - 211 cos(l - cos <p)

+9sin4 <pcos 2 <p+ 18sin2 q>cos4 <p

6 4 ? 2 + 9cos <p - -(1 + 11)- cos <p

3 . .. (2)

RAO : ROCK INDENTATION ANALYSIS BASED ON THE THEORIES OF ELASTICITY 129

Eq. 2 gives the extent of the failure zone defined by R( cp) in the rock mass for the applied point load P.

Extension strain failure criteria

The well-known hypothesis widely used in rock mechanics is Griffith's and Mohr's criteria. As pointed out by Bieniawski 17. 18, clear distinction has been drawn, in that Griffith's theory provides a formula enabling prediction of the level at which failure occurs, where as the Mohr's criteria describes the processes taking place in the material in the course of loading and eventually leading to failure. Many extensions or modifications have been developed to explain the anomalies of .these two theories. In contrast, a very successful prediction of both orientation and extent of fracturing was achieved using a simple extension strain criterion. This criteria was stated simply as fracture of the rock will occur in indirect tension when the tensile strain exceeds a limiting value which is dependent on the properties of the rock.

In the present investigation, this criteria has been applied to the normal point load condition to predict the extent of fracturing in the rock mass.

According to the extension strain failure criteria l 6,

fracture of brittle rocks will initiate when the total extension principal strain e" in the rock exceeds the critical value ec of the rock, that is e,>ec which is defined as the ratio of the ultimate tensile stress to Young's modulus, as described by Eq. (3):

.. . (3)

The principal stresses (O'j, 0'2, 0'3) for the point load case can be obtained using the Expressions (i)-(iv) as . given in the Appendix and the resulting principal stresses can be described as:

0"1 =--;[A]+--;~{[B]2 +[Cf} ... (4) 7rR 7rR

0"2 =--;[A]---;~{[B]2 +[C]2} ... (5) 7rR 7rR

(1- 2/1)P z R 0"3=0"(J= 27rR2 [2R-2(R+Z)] ... (6)

where,

A=[3Z _ (1-2/1)R]; 4R 4(R+Z)

B = [3Z(r2

- z 2) _ (1- 2/1)R];

4R 3 4(R+Z)

C = [_~ rz2

]

2 R3

Substituting Eqs (4)-(6) in Eq.(3), we get:

P 4/1 2 + /1 -3 (-2/1 2 - /1 + I)

R2 = --[ COSqJ + ] E7re c 4 4(1 + cosqJ)

P(I+J.l){[3 2 (1-2J.l)]2 + -cos<p.cos <p+----Ertec 4 4(\ + cos <p)

1 9 . ? 4}2 + -sm- <p.cos <p 4

... (7) where,

ec = O"T / E

The expression given by the Eq.(7) represents the surface R( r,Z) , defining the fracture boundary under the action of the vertical thrust force.

Model E/G > 2(1 +J.l)

The stress distribution problems are mostly based on the Boussinesque's classical theory. However, attempts are being made to modify Boussinesque's stress equations to adopt to the granular materials represented by the model E/G > 2(1 + J1.)15 and these expressions for the stresses are given in the Appendix

. as Eqs (vii)-(xii) .

Von-Mises flow Criteria

Applying the Von-Mises flow criteria using the modified Boussinesque stress Eqs (vii)-(xii) as given in the Appendix for a discontinuous rock mass, the resulting failure condition is:

130 INDIAN J ENG. MATER. SCI., JUNE 2000

... (8)

R = ~(r2 + 2 2) ; r = R sin<p; z = R cos<p and the

values of HI . H2, H3 are given in the Appendix. The expression gi ven by Eq.8 represents a surface

R(q» defining the elasto-plastic boundary under the applied concentrated vertical thrust force P.

Extension strain failure criteria

According to the extension strain failure criteria as given by Eq.(3) for the condition £IG > 2(1+f1), the principal stresses obtained from Eqs (vii)-(xii) are:

P H 2 HI H J (J'=4( A) [{-2 --2 }+-2 ]

n a-I-' z r Z

+ [{ P } 2 { H?2 _ Hd _ Hi1 } 2

4n-(a - [3) z- r Z

I

+{ PH,r J }2]2 27[(a. - Rh , ,- / .....

... (9)

P H 2 +H J HI (J 2 = 4 ( [3) [ 2 - -2 J net- z r

P

2n(et - [3) ... (10)

... (II)

Substituting the val~es of Eq.(9)-( II) in Eq.(3), we get:

R2 = P [(H 2 +H,)(I-J..l-2J..l2

)

2n(a - ~)(JT 2cos2 q>

_HI(I+J..l)] 2sin 2 q>

+ P(I+J..l) [{(H 2+HJ )

2n(a - ~(JT) 2cos2 q>

+{H,s~nq>}2]~ cos' q>

... (12)

Static Experimental Indentation Tests

Static experimental indentation tests were conducted on a standard hydraulic operated compression-testing machine of 50 Ton capacity . The compression testing system has a facility of loading at twelve ranges of strain rates, starting from 5 mmlmin to 0 .000 I mmlmin (Fig. I ). The static point attack indentation experiments were conducted at a constant strain rate. Though, the highest strain rate of 5 mmlmin is possi ble with the testing system, it has been seen that such strain rates are too high for brittle materials like rock. Therefore, in the present set of indentation tests, the point indenter was made to penetrate into the rock at a strain rate of 1.25 mmlmin . A dial gauge with a least count of 0.01 mm was mounted on one of the columns of the testing machine and adjusted accordingly to facilitate the measurement of the indentation depth of the cutting tool.

The point indenter used in the experiments is made of conical shape giving a point contact at the tool-rock interface. This cutting tool is pressed into hardened

LOAO CELL -Fr--i-

DIAL GAUG E --,-..,o--.-r",

S P E C I MEN --+-+-<-fl --tt-++ROC K f-K:>LDER

BOTTOM PLATE

---i+~::::::rr:::7~ BLOC K

DIG4TAL METER

Fig. I - Arrangement for rock indentation analysis

RAO: ROCK INDENTATION ANALYSIS BASED ON THE THEORIES OF ELASTICITY 131

steel cylindrical disc, which in turn is threaded to a bit holder of 50 mm diameter and 50 mm long hardened steel cylinder. The attachments such as the cutting tool, disc and the bit holder are made into separate modules for the ease of replacement of a worn out part, particularly with respect to the indenter. In the present studies, totally three indenters have been replaced on being turned unserviceable. The upper surface of the bit holder is curved and the radius of curvature is matching to that of the load cell, as shown in the figure. The curved surface facilitates not only a perfect contact but also transfers the load axially through the bit-holder to the sample.

'The initial static indentation trials were restricted to loading the rock samples until the total projected height of the tool was indented into the rock mass. There were many difficulties' in this mode of loading in that most of the samples failed similar to the mode of failure observed in point loading condition making it difficult to measure the extent of fracturing along the perpendicular axes. In view of these difficulties, the subsequent tests were modified and the tests were conducted on the rock samples with a rectangular cross-section. The normal point indentation load was applied at a prefixed burden from the free edge of the sample, as shown in the Figs 2(a)-(b) . The burden in these tests was a variable too, starting with 3 mm and 5 mm. The failure load is read from the output of the load cell. The capacity of the load cell is 500 KN. The load cell readings are taken at regular intervals and the load at which a complete failure manifests in the formation of chips, has been taken as the ultimate

load. The depth, width arrd the thickness of the chip give the dimensions of the fractured zone.

Results and Discussion

Table 1 gives the properties of the rock type considered in the experimental investigations determined in the laboratory as per the standards of International Society of Rock Mechanics2o

. The values of Poisson's ratio and tensile strength were used to compute the extent of the fracturing using the Eqs (2), (7), (8) and (II) of fracture analysis for the loads applied in the experimental static indentation tests.

The static experimental indentation loads and the corresponding values of the fracture zone are given in the Table 2 along with the fractured zone values obtained from the theoretical analysis. The Eqs (2) and (7) represent the fracture extent in a flawless elastic material. The results obtained by solving these equations are given in the table. Eqs (8) and (I I) are developed for the inelastic material represented by the model E/G > 2( J + J1) and the predicted values of the fracture depth for all the models along with the experimental values are shown in the Fig. 3.

Laboratory investigations

In the absence of direct observations of the formation and the growth of fracture, one is restricted to the observations before and after the indentation . These observations have clearly revealed that two regions exist distinctly with in the fractured zone. A very small completely crushed and apparently

( , j { : '

Fig. 2 -Indentation by Point Loading and Fracture Growth: (a) Side view, and (b) Front view

Table I - Physico-mechanical properties of the rocks

Rock Type Compressive Tensile Strength Young's Modulus Poisson 's Rat.io Strength (MPa) (GPa) (11)

(MPa)

Granite 169.8 1 9.00 92 0.33

Limestone 59.92 6.35 47.5 0.24

Sandstone - 1 44.96 4.99 41.06 0.26

Sandstone-II 29.11 3.56 36.00 0 .25

132 INDIAN J ENG. MATER. SCI., JUNE 2000

Table 2 - Comparison of experimental and theoretical results of the Extent of Fracture in different rock types

Rock Applied Experimental Values Von - Mises Flow Criteria Extension Strain Criteria

Type Load Depth of Width of Depth of Width of Frac- Depth of Width of (kg) Fracture Fracture Fracture ture Fracture Fracture

(mm) (mm) (mm) (mm) (mm) (mm)

Granite 750 22 18 63.50 25.98 22.80 17.9

800 24 20

850 32 15

1220 41 25

1450 48 27

Lime- 650 18 18.5 stone 1·200 20 24

1350 32 24

Sand- 400 16 15 stone-I 750 30 20

Sand- 50 12 15 stone-II 75 22 20

90 28 18

compacted innermost zone lying below the indenter enveloped by a relatively bigger zone from where near lenticular-shaped chips fell out. The shape of the chips indicate the generation of non-linear lateral cracks radiating from the zone of compression, that is the inner most zone, to the free end of the surface. A similar observation has been explained as a development of a median crack which initiates from the tip of the inner most crusned zone and a Hertzian 'cone' crack from a surface flaw by the elastic contact of a normally loaded indenter2 1

. Further, it has been explained that both types of ~racks expand radially outward on near-circular fronts. The geometry of the fracture is in fact dependent on the shape and size of the indenter and also on the elastic behavior of the brittle material 22

.

Fig. 4 indicates that the fracture width and depth increase linearly with the increase of applied load in all the rock types considered in the present investigations, and for brevity the trend has been shown only for limestone and granite. It is also noted that the load at which failure occurred increases with the increase in the burden distance and is different for different rock types. Granite, in the present investigation, is the strongest rock while the coarse grained sandstone is the softest, and so are the magnitudes of the failure loads in these rock types. For a given geometry of the indenter, the fracture dimensions are controlled by the strength of the rock and the failure load . The prediction of which is being attempted through the Von-Mises yield criteria and the extension strain criteria.

70.44 28.85 25.35 19.89

72.60 29.73 26.14 20.00

86.04 35.66 31.30 24.50

94.80 38.80 34.10 26.70

75.8 40.00 33.6 23 .4

105.80 54.40 45.7 31.70

109.92 57.77 40.50 34.80

32.73 66.97 27.85 19.80

44.80 91.70 38.14 27.90

14.30 29.0 12.20 8.67

25 .73 28.20 14.94 10.10

28 .1 8 39.44 16.36 11 .64

~ FRACTURE EXTENT ON X-AXIS I FRACTURE EXTENT ON X-AX IS-

0.3 0.2 0.1 t 0. 1 0. 2 0.3

\ i I

i \ ~--f-.. -___ ""' .. rL--"'="-lft " 2Skg / {m'

~/G >: ( UG ,. , , ."

o 0.005'111 ..........

.. " .

ELASTO · PLASTIC APPROACH (VON-MISES FLOW CRITERIA)

w 0. 2~

~ ... ... o

O.3~ .. w o

•

_ .• - PR£!oI!NT ~ACHf'G) 2(1.plCOHCE~l

(O't =n~" t ."l)

-EI151IHGAHfttJ ACM I!/G=l(l.jJl

EXTENSION STRAIN CRITERIA

Fig. 3 - Failure envelopes as obtained by different approaches for Jl = 0.3

Theoretical Analysis

The theoretically predicted values by the Von­Mises yield criteria is 2.8 times more than the measured values in granites and maximum over­prediction was observed in softer rock where it is up to 5 times the measured values. The extension failure criterion values are 1.036 times higher than the measured values for lower failure loads and is under predicted by 1.5 times the measured values for higher failure loads in granites and around 1.5 to 2.25 times over predicted in limestone. In gen ral, the predicted values of fracture depth based on the extension strain criteria of Eq.7 are well with in reasonable limits .

In the present analysis, it has been assumed that the microscopic cracks, and other geological structural features make the rock to exhibit the plastic behavior even at lower stress levels and the yield in such

RAO: ROCK INDENTATION ANALYS IS BASED ON THE THEORIES OF ELASTICITY 133

• EXPERIMENTAL VALUES + EXTENSION STRAIN CRHERIA; 'EXTENSION STRAIN

CRITERIA WITH MODEL E/G>2 (1+,u1 i x VON-MISES FLOW CRITERIA

• VON-MISES FLOW CRITERIA WITH MODEL E/G > 2 11+,u I

80 80 ROCK TYPE; LIMESTONE ROCK TYPE : GRANITE

60

+-

~ 1.0 x-

----"--,x~~-r~+ . .

20

( ( )

o 120 160 O~----:'-=----+-:-------=-:!,"=--~

12

10

E 8 E ::I: t-Q.. UJ 6 0

UJ a: ::::::> t-

I. u 4 « a: LL

2 2

( b) (d )

0 o~ ______ ~ ____ ~~ ____ ~~ ____ ~ 0 '0 80 120 160 0 1.0 80 120 160

APPLIED LOAD ( " 10) , kg

Fig. 4 - Extent of fracture (theoretical and experimental) for: limestone - (a) fracture width , and (b) fracture depth ; granite­(c) fracture width, and Cd) fracture depth

134 INDIAN J ENG. MATER. SCI., JUNE 2000

material could best be represented by the Von-Mises yield criteria. .

The Von-Mises failure criteria considers that the yie lding can not be effected by the choice of the principal axes and it is assumed that the plastic deformation depends only on the stress deviation rather than the isotropic stress condit ion and further it considers the invariants of the stress deviations for the yield criteria. It has, however, been observed that this criteria has over-predicted the fracture ex tent by 2.7 times the extension strain criteria. First ly, the over­prediction may be due to the limitations in the scope of experimental measurements of the fracture zone, which could have ignored the fine fractures beyond the chipped out zone. Secondly, the assumpti ons of the concepts of plastic flow under the present loadi ng conditions, that is, without the lateral confinement wh ich does not provide a near tri-axial condition, may not complete ly be valid. On the other hand, the mechanism of extension strain fracturing includes the effect of heterogeneity of the rocks in terms of the mismatch of deformation properties of grains within the rock matrix and the Poisson 's ratio effect. It considers that the fracture of the rock is a brittle phenomena and it occurs in indirect tension when the tensile strain exceeds a limiting value, which is dependent on the properties of the rock.

Fig. 3 is drawn fo r a Poisson 's ratio of 3 and a tensile strength of 2.5 MPa. It includes the fractured zone for all the models , that is E/C = 2( I +~l) and E/C > 2(1+p). It is considered that the influence of discontinuities on the rock wou ld be simi lar to that of a granu lar material and the model E/C > 2( I +~l) represents the rock with discontinuities. The resu lts, however, have not c learly represented the above model for the structural discontinuities in the rock . The rock samples taken in the present indentation test were more or less intact and therefore cause the error in the predicted values.

Canclusions The extension strain criteria are found to give a

more realistic estimate of the fractured zone than the Von-Mises flow criteria. The: merits of the extension strain criteria are that they automatically take account of the three-dimensional state of stress and strain and hence no confusion arises on the effect of the intermediate principal stress. It is effective under low stress levels.

The concepts of E/C > 2( 1+ p) are an attempt towards quantifying the discontinuities and their influence on the fracture extent. It, in fac t, has over­predicted for the entire failure criterion .

Nomenclature E Young ' s Modu lus G Ri gidity Modulus p Poi sson's Rati o

0' Normal Stress 1: Shear Stress S,.S\,S , Stress deviators

0''1' Tensile Strength P Unit Normal Load R (<p) Fracture Envelop Function e, Principal Strain ec: Critical Extension Strain

References

Larson D A, Morre ll R J & Mades J F, US Bur Mine Ref! III ' vest, 9106 (1987) 2-28.

2 Tutluoglu L, Mood M & Monton C, Pro!.' 24'}, US SYIIIP Rock Mechanics, Texas A&M University, Texas , 1983,743-48.

3 Miller 0 & Ball A, lilt J Rock Mech Mill Sci, 27(1990) 363-371.

4 Miller 0 & Ball A, J S Afr Ill st Mi Metall. 90(1990) 283-288. 5 Rowl ands 0 , Rock fracture by dialllolld drillillg, PhD Thesi s,

University of Melbourne, 1975. 6 Karan am U M R & Misra B, Prillciples afrock drillillg (A A

Balkema Publishers, Rotherdam), 1998 . 7 Wijk G, In t J Rock Mech Mill Sci, 28( 1991 ) 35-42. 8 Tan Y C, Lindquist P A & Kou S Q, Proc First North Ameri­

can rock mech symp, (1994)335-342. 9 Timoschenko S P & Goodier J N, Theory of elasticity

(McGraw Hill Co. , New York), 1970. 10 Boussinesque J , Theory of elasticity (McGraw Hill Co., New

York) , 1970, 398-402. II Pang S S, Rock Mech Ruck Ellg, ( 1990) 53-63. 12 Braja M 0 , Stresses in. soil mass (Hemisphere Publishers,

New York) , 1985. 13 Pou las H G & Davis E H, Ela.\'tc solUlions fo r soil and rock

mechanics (John Wiley and Sons, New York), 1974. 14 Wei skopf W H, J Frallklilllnst, 239(1945) 445-465. 15 Misra B & Sen B R, 1111 J Ellg Sci , 13 (1975) 743-761 . 16 Prager W & Hodge P G , Th eory of perfectly plastic solids

(John Wiley Publi shers, New York), 1951. 17 Stacey T R, Inl J Rock Mech Mill Sci , 18( 198 1) 469-471. 18 Bieniawski Z T, Inl J Rock Mech Mill Sci, 4( 1967)395-406. 19 Hahn R S, Adv Mach Too l Des Res, ( 1962) 129-154. 20 Vutukuri V S, Lama R D & Saluja S S, Hand book all

mechasnical properties of rocks (Trans Tech Publications, Germany), 1974.

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22 Niihara K, Moren R & Hasselman D P H, J Maler Sci. 10(1978) 97-105.

RAO : ROCK INDENTATION ANALYSIS BASED ON THE THEORIES OF ELASTICITY 135

Appendix

Boussinesque closed form. solution for the stresses in a semi-infinite mass due to point loading in rectangular coordinates are expressed as:

3PZ 3

a z =--2nR5

0" x = 3 P { X 2 Z + (1- 2/l) ] I 21t R5 3 R(R + Z)

(2R +Z)X 2 Z

R\R +Z)2 R3

O"y = 3P {y 2Z + (1- 2/l)] I 21t R5 3 R(R + Z)

(2R + Z)Y2

R3(R + Z)2

1: _ 3P {X YZ _ (1- 2/l) Xy - 21t R5 3

[(2R+Z)X Y]} R\R + Z)2

3P X Z 2 l' =---

xz 2n R 5

3P YZ 2 1'yz =---

2n R 5

... (i)

... (ii)

.. . (iii)

... (iv)

... (v)

.. . (vi)

Modified Boussinesque Solutions to account for the granular materials are as follows:

P HI J.l a 66 = [-? + -2 (H 2 + H 3)]

2n(a - f3) r- Z ... (vii)

P H 2 HI a rr = 2n (a - f3) [7 -7] ... (viii)

PH , a U. = - 2

2n(a - f3)z ... (ix)

PH,r a = -rz 2n(a - f3) Z3

1'r6 = 0

1'&z = 0

where, I

HI = [(L+ K){~[a2 +(_~l(2 } z

I

-(M + K){a-[~2 +(~l]-2 } z

... (x)

.. . (xi)

... (xii)

-' 3

H 2 = [{a 2[a 2 +(.c)2f2 }_{f3 2 _[f32 + (.c)2f2)] Z Z

3 -'

H, = [ (f3 2 + (.c) 2] -2 } _ ( [a 2 + (.c) 2] -2 } ] - Z Z

and,

L= (l-J.l)(a2 -\) .

2 ' a

M = (l_J.l)(f32 -\) .. f3 2 '

K= E [~_(l+J.l)]; (l+J.l) G E

K' = (l + K) (l-J.l)

2 ? . 2 • a and ~- are the roots of the equatIon: X + (K -2)X + I.