THE UN I V E R S ITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Civil Engineering Department of Aerospace Engineering Technical Report A NOTE ON THE PENETRATION OF A RIGID WEDGE INTO A NONISOTROPIC BRITTLE MATERIAL Rafael Benjumea David L. Sikarskie ORA Project 01052 supported by: DEPARTMENT OF THE INTERIOR BUREAU OF MINES GRANT NO. MIN-10 WASHINGTON, D. Co administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1968

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ABSTRACT In the present note an existing wedge penetration theory for isotropic brittle materials is extended to explain some "bedding plane" effects observed in a series of experiments on Indiana limestone. The general features of the theory presented in Reference [1] are kept here and in order to extend. the analysis to the anisotropic case Jaeger's modification of the Coulomb-Mohr failure criteria is used. Two specific cases of bedding plane orientation are considered, namely those in which the bedding planes are parallel and perpendicular to the direction of penetrationo The -theory presented gives an insight into the anisotropic effects and. pred.icts, in reasonable agreement with the experimental results, the observed differences in specific energies for the different penetration directionso 1

LIST OF SYMBOLS AND UNITS d = penetration in the i cycle (ino) th d = penetration at the formation of the i chip (in ) i E = energy density for constant rate test (lb/in.2) a E =energy density for constant load test (lb/ino2) k slope of the force-penetration curve during crushing (lb/in,2) K slope of line connecting peak forces (lb/in.2) L length of fracture path (in. ) N =force normal to the fracture surface (lb/in.),th P, wedge force during the i cycle (lb/in.) * th P wedge force at the formation of the i chip (lb/in ) SLS2 =shear constants of the material (lb/ino2) S,S =compressive strength of the material in two directions (lb/ino2) C2 T =force tangential to fracture surface (lb/in.) = failure angle of the chip = one half of the wedge angle y angle the bedding plane makes w.ith the horizontal I= angle of internal friction (material cons'tant) = normal stress on fracture surface (lb/in.2) T =tangential stress on fracture surface (lb/in.2) o= normal stress average along the fracture surface (lb/in.2) T= tangential stress average along the fracture surface (lb/ino2) = related to $ by J. = tan / 2

Io Introduction In the present Note an existing wedge penetration theory for brittle materials [1] is extended in an attempt to explain some "bedding plane" effects observed in a series of experiments on Indiana limestone. Bedding plane effect or rock anisotropy is often neglected. in Rock Mechanics calculations particularly for rocks such as Indiana limestone where there is no discernible bedding plane and further the change in physical properties, e.g., compressive strength, with direction is relatively small. The present theory (substantiated by experiment) indicates, however, that relatively large differences in the specific energy (energy necessary to remove a unit volume of rock) can exist even for relatively small anisotropyo The analysis presented, herein follows that of [1] and contains the same basic assumptions. The material behavior under consideration is typically brittle, i.eo, as the wedge penetrates cycles of crushing followed by the formation of chips are experiencedo To extend the analysis to the anisotropic case Jaeger 's [2,3] modification of the Coulomb-Mohr failure criterion is usedO Two special cases are considered in this Note, namely y = 0~, 90~, see Figure 1, where y represents the angle the bedding plane makes with the horizontalo Intermediate values of y represent a much more difficult problem since symmetry cannot be invoked. This necessitates some description of the wedge kinematics. In the following section theoretical results are derived. A comparison of the theory with some previously unpublished [4] wedge penetration experi 5

ments on Indiana limestone is given in section III followed by a discussion of results in section IV. II. Theoretical Analysis Figure 2 illustrates the assumed. idealized penetration model just at the st formation of the (i+l)S chip. Chip failure is assumed planar and inclined to the horizontal at some as yet unknown angle t. Considering equilibrium just before formation of the chip gives the following for the normal and tangential forces on the failure plane; P* N i+l sin(*+e) 2 sin e (1) +l cos i+l cos(i+n ) T - 2 sin e (2) where P = force per unit length of cutting edge necessary to remove the i +l (i+l) chip 28 = wedge angle and N, T are resultant forces per unit length of cutting edge defined by, N = S a(S)d 0; T = T()d 0 (3) where a(l), T(S) are the normal and shear stress distributions along the failure plane, respectively. L is the chip length and can be expressed; 4

*X d. i +1 L =(4) sin r where d.+ is the penetration at the (i+l)t chip removal. i +1 The Coulomb-Mohr failure criterion [2,3], modified to account for rock anisotropy, is given by2; I T - tan T a - [S1-S2 cos 2( -7)] = 0 (5) where = angle of internal friction (assumed constant throughout) S1,S2 = material constants. The angle (*-7) measures the angle between the plane of minimum shear strength (bedding plane in this case) and the plane of failure. Note that the maximum and minimum shear strengths are 90~ apart and are S1+S2, Si-S2, respectively. The analysis further assumes that, to a first approximation, the failure criterion is satisfied everywhere along the failure plane. Equation (4) can thus be integrated over the length L and a stress averaged form of the failure criterion used ioe., 71 - tan > a - [S1-S2 cos 2(A-7)] = 0 (6) where N T = -; T = -L L L Using equations (1), (2), and (4), equation (6) can be expressed, 5

p i+l sin r cos(*+0+G) * sin 8 cos > 2di. 1 +1 - [S1-S2 cos 2(*-y)] = 0 (7) Failure will occur at that angle * such that the left-hand. side of equation (. (L. H. Sw ) (7) has a maximum value. This is found by considering d H ) = 0, with the result; tan 2t = pf. i+l i — cos(e+)+4S2 sin 2y cos > sin e d i i+1 pi+l il sin(e8+)+4S2 cos 27 cos ~ sin 9 d i i+1 (8) The ratio P.+l/d.+l is still unknown, however. Substituting equation (8) * * 7 into equation (7), the following quadratic equation in P i+/d.+l results. * Cos2( ) - 8 s e cos sn(+)- sin(2 -i+l cos2 (^+8) - 8 sin e cos ^FS1 sin(y+~)-S2 sin(2y+e+^)] ( ] -t - 16 sin2e cos2F[S2-S2] = 0 (9) Just as in the isotropic case Pi+l/di+l is a constant, K, for all i, dependent only on the parameters and material properties. Note that equation (9) gives the isotropic results for S2 = 0, see [1]. The positive root of equation (9) is the correct one, see Appendix A

* P i+1 4 sin e cos i+l + 4 sin e cos - I S1 sin(GQ+)-S2 sin(27+e+() i+l di+l cos(0e+X) (10) + E[s1 sin(G+z)-S2 sin(2y+G+) ]2+[S2-S2]cos2(+) = K From equation (10) the ratio of load. to penetration at chip formation is now known, however, the individual magnitudes of P+l and, di+ are not. An additional equation is available, however, namely the crushing law.4 i k= [d -d. i (11) i+l = k[ 'i]11 where k is material constant dependent, in addition, on wedge geometry and direction of penetration. At the time of chip formation equation (11) becomes = k[d -d. (12) i+l i+l 1 Following [1], equations (10) and (12) can be solved simultaneously to obtain P, d in terms of d.. Recurrence relations can then be derived which rei+1' i+l i late Pi+n d. to di n i+n \k-K/ 1 (13) d - d. i+n k-K 1 Thus, for i = 1, the force and. penetration levels after n cycles can be related 7

to the penetration at the first chip. Note that geometric similarity exists k (due to linearity) with the similarity variable k o In Figure 3 the actual k-K3 penetration process is shown along with two idealizations (constant rate, constant load. tests) which bound. the actual. Determination of the specific energy (energy required per unit volume of rock removed) for the constant rate, constant load. tests follows the analysis of [1] exactly. It should be noted that only for the special cases included in this Note (7 = 0, 90~) (symmetric cases) will the analysis be the same. The specific energies for the constant rate test and constant load test are obtained by dividing the energy consumed in each test, see Figure 4, by the volume removed. and. are respectively; -kK E tan (14) a 2(2k-K) b 2k(2k-K) Eb -K[(k-K)+k] tan (5) In the following section numerical results are compared. with some experiments on Indiana limestone, III. Experimental and. Numerical Results The pertinent experimental results [4] for Indiana limestone are listed. in Table I. For y = O0~,o 0~ the force is perpendicular and parallel to the bedding plane, respectively, for both the compressive strength and wedge penetration testso The compressive strength tests were conducted on cylinders 1.75 in. in 8

diameter and 4.75 in. long. Sample ends were finished to a tolerance of ~.005 in. and during tests were directly in contact with the platen (unlubricated). All of the wedge-penetration experiments were run with a tungsten carbide wedge having a 1 in. cutting edge, a wedge angle of 20 = 90~, and, a tip radius of.05 in. Maximum penetration depths of the order.15 in. were run. The rock sample was a cube approximately 12 in. on a side. Force, penetration results were taken out directly on an x-y plotter and. measurements of crushing slope, envelope slope, and energy were computed from the graphs. Crater volumes were obtained. using a burette. The craters were first coated to prevent absorption and several readings were then averaged. As input for the numerical results, the following items are needed: k, 20, A, S1, and S2o k will be taken from the experimental results. The wedge angle is, 20 = 90~. The angle of internal friction f is difficult to establish, particularly in the vicinity of the wedge tip where the mean or hydrostatic stress can be extremely high. In any event, numerical values are computed for three different values of 5: (~ = 50 100, 200)~ Si and S2 can be found from the following equations, see Appendix B; 4[S-SS2] + 4[Sl+S2]l S S2 1 C Cl (16) 4Fs2-s2] + 4[Si-S2), S = C2 where 9

S = compressive strength for 7 = 0~ C1 S = compressive strength for y = 90~ C2 p = tan ^ Numerical values for S1, S2 as a function of i are given in Table IIo Using equations (8), (10), (14), and, (15), the chip failure angle, the slope of the envelope, and. the specific energies for both cases can be computed. for 7 = 0~, 7 90~e The results are presented in Table III. These results are discussed, and. compared. with the experimental data in the following section. IVo Discussion of Results The experimental results for Indiana limestone illustrated, in Table I, indicate a rather interesting behavior; namely, a much larger difference in speific energy in perpendicular directions than in the corresponding material properties. An extension of an existing theory has been outlined herein which predicts at least qualitatively the same trend.s, see Table IIIo Several. problems exist. While the trends are established., a large amount of scatter was evident; in the experimental results. The numbers presented. represent a simple average of available data. Concerning the theory, it should. be pointed. out that not all input data (material properties) is clearly established, e.g., the angle of internal friction ~o Two major drawbacks of the theory are the assumptions made in the analysis and. the inability of the theory to adequately describe the initial phases of the penetration process. These questions are discussed in more detail in Reference [1]. In spite of this, however, the theory is in good agreement qualitatively, ioe, it provides an explanation of 10

the difference in specific energies in the two directions. It is also felt that quantitative agreement is reasonable in view of the statistical nature of the material, and the simplifying assumptions of the analysiso From Table III it is evident that the numerical results are very sensitive to the angle of internal friction. Comparing Tables I and. III for -10~ indicates that the computed envelope slope is in good agreement with the experimental value. This is some what inconclusive, however, due to the limited test data. Concerning the specific energies, although the same trends exist in the numerical data for y = 0, 90~, the numerical results are uniformly higher than the experimental. For f = 10~, the experimental value of the specific energy lies outside the "bounds. " It is felt that this is due to the assumption of the failure criterion being satisfied. along the entire failure plane. Several interesting features appear in the analysis. As anisotropy increases, i.e., as S -S increases, it can be shown that the differences in i1 C2 the failure angle 4 and the specific energies for y = 0~, y = 90~ also increases, as is expectedo It is also interesting to note that the slope of the envelope is larger in the direction parallel to the bedding plane. This indicates the possibility of wedge forces larger for penetrations parallel to the bedding than perpendicular (for equal penetration distances). This is opposite to the crushing phase where the crushing slope is largest perpendicular to the bedding plane. The physical difference in specific energies is also related to this question, ioe., since K is larger, k is smaller for y = 90~, more energy is consumed in the penetration. Also, the failure angle is larger 11

for 7 = 90~ leading to smaller volumeso Both of these effects lead to larger specific energies for y = 90~o 12

* i+1 Appendix Ao Derivation of the Quadratic Equation in * i+l The failure criterion is given by equation (7) with the failure angle t defined by equation (8). If we further define, * P i+l 1 K1 = il sin cos (A1) i+l the failure criterion can be written in the following form K1 sin C cos(~+e+r) - [S1-S2 cos 2(Q-y)] = 0 (A2) expanding: Kl[sin 2o cos(~+))-sin(G+~)(1-cos 2t)] -2[S1-S2(cos 2y cos 2t + sin 27 sin 2r)] = 0 (A5) Dividing by cos 2. and collecting terms; FK1 cos(Q+$)+2S2 sin 2y]tan 21r + [K1 sin(Q+z)+2S2 cos 2y] (A4) - [2Si+K1 sin(G+ ) l+tan22* = 0 This has the form A tan 2f + B - C 1 + tan2 2 = 0 (A5) where A, B, and C can be determined from equation (A4). Note also, from 13

equation (8), that; A tan 2t = B B (A6) Substituting (A6) into (A5) we find: A2 + B2- C A+B2 = 0 or6 A2 + B2 = C2 (A7) Substituting back for A, B, C, and K1, and rearranging, the final form resultso 2 i +1 i2+ ( ~2-^-) cos2(@+#) - 8 sin 9 cos X [SI sin(~+z)-S2 sin(27+9+^)](2 ) - 16 sin2 cos2 [s- ] = 0 (A8) This quadratic equation has a positive and a negative rooto The negative root P i+l means the slope * is negative which d., has no physical meaningo

Appendix B. Computation of the Shear Constants S1, S2 S1 and S2 are not directly known but can be obtained, in terms of the compressive strengths in the two directions. This is done by reducing the biaxial failure criterion to two independent uniaxial cases. The biaxial failure criterion given by Jaeger [2] is; 2(C +a)2 - (l+42)(T +b)2 = S - b2(+2) (B1) m m where C = mean or hydrostatic stress m T = maximum shear stress m a = S1/l b = S2[sin 2(90~=y)+k cos 2(90~-7)]/1+ 2 = + S2 4/1+42 for 7 = 0990~ respectively Two special cases of equation (B1) are now considered, namely uniaxial compression perpendicular and parallel to the bedding planeo First Case, y = 0 C T = S /2 m m ci Substituting into equation (B1) +1 S - (l+2) ( - (1 S2 = S2 2 2 p S22 2 2 2 1_12/1+~t2 15

simplifying; 4(S2-s2) + 4(sl+s2)1 s Cl = 2 C1 (B2) Second Case, 7 = 90~ C m T = S /2 m C2 Substituting into equation (Bl) and simplifying; C2 = 2 C2 (B3) Equations (B2), and (B3) are identical to equations (16). 16

LIST OF REFERENCES [1] Paul, Bo, and Sikarskie, Do L., "Preliminary theory of static penetration by a rigid wedge into a brittle material," Transo AIME, 232, 372-383 (1965). F2] Jaeger, J. Co, "Shear failure of anisotropic rocks, " Geological Magazine, 96, 55-72 (1960)o [3] Jaeger, Jo Co, Elasticity, Fracture and. Flow, 2nd. edition, John Wiley and Sons, Inc, 19620 [4] Sikarskie, Do L., "Experiments on wedge indentation in rocks," Ingersoll-Rand T. No-262, Ingersol-Rand Research Center, Princeton, New Jersey, May, 1966o [5] Walsh, Jo B., and Brace, WO Fo, "Fracture criterion for brittle anisotropic rock," Jo Geophyso Res., 69, No. 16, 3449-3456 (1964)o [6] McLamore, Ro, and. Gray, K, E., "The mechanical behavior of anisotropic sedimentary rocks," Jo Basic Eng., Trans. ASME, 88, 62-76 (1967)0 17

ACKNOWLEDGMENT The authors would like to express their sincere appreciation to the Department of the Interior Bureau of Mines for the support of this work under Grant Noo MIN-10, 18

FOOTNOTES Note that the wedge is assumed frictionlesso This assumption can be removed, however, at the cost of added, algebraic complexity, 2 Other failure criteria are available, e.g. [5], including a more detailed. modification of Jaeger's criterion, i.e. [6]1 It is felt, however, that the ad.d.ed. complexities of introducing a more refined, failure criterion is not justified in view of the assumptions. See Appendix A for detailed algebra. A linear crushing law is assumed for convenience. Note that K, equation (10), is independent of crushing lawo 5Note that equations (16) cannot be used for finding S1, S2 for 1 = 0. The set of roots introduced by squaring, ioe., A2 + B2 0, are imaginaryo 19

TABLE I EXPERIMENTAL DATA FOR INDIANA LIMESTONE (29 = 90~) (Numbers in Parentheses Refers to Number of Tests Making up the Average) Compressive Crushing Slope of Specific _Y _ Strength(psi) Slope-k(psi) Envelope-K(psi) Energy (psi) 00 10,900(5) 172,900 (6) 8,700 (3),950(8) 90 10,000(5) 154,900(4 77,9OO 1) 7,220(6) 5~ ~ ~ ~~~790,2 6 20

TABLE II VALUES OF THE SHEAR CONSTANTS AS A FUNCTION OF > Shear Constants = 5=~ = 100~ = 20~ (psi) Si 5,320 4,490 3,680 S2 2,350 1,090 460 21

TABLE III NUMERICAL COMPUTATIONS Failure Angle - f Slope of Envelope - K (psi) 7Y X = 5o_ 0 = 10~oo = 20~ O = 5 = 10~o = 20~ 0~ 17.4 16.6 12.4 40,570 55,790 92,140 90~ 21.6 18.0 12.8 83,900 835,160 115,630 Specific Energy-Con. Rate - E (psi) Spec. Energy-Con. Load - E (psi) 7 -ab X = 5~ X = 10~ _ = 200~ = 5~ = 10~ _ = 20~ 00~ 5,560 4,990 6,910 5,650 7,280 8,420 90~ 11,360 9,236 10,550 13,750 11,10 10,995 22

Figure 1. Figure 2. Figure 3. Figure 4. FIGURE LEGENDS Bedding plane direction. Incipient chip formation. Constant load and other test conditions. Work done in constant rate test Z A.); and constant load j=l test E (Aj + Bj) j=l 23

WEDGE BEDDING PLANE Figure 1

P+l WEDGE R di* I LINE OF FAILURE & L Figure 2 25

K I P pa = Kd* \ Pj+ k [di+l-dT I -CrNqI TANT I An1 TF:T P I O, /s\ /-ACTUAL LOADING CONDITION Pi-l __\ //CONSTANT RATE TEST,Pi~ - di-2 di-I d Figure 3

P Pi pi-ar - I ro P -2 d di*-2 d i* Figure 4

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