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Post-failure instability analysis of a three-point bend round beam specimen of rock
Engbeerfng Fraewe hfechaniicsVol. 41, No. 1, pp. 7-11, 1992 Printed in Great Britain.
0013-7944/92
55.00 + 0.00
Pergamon RESSpk.
POST-FAILURE INSTABILITY THREE-POI~ BEND ROUND OF ROCK
ANALYSIS OF A BEAM SPECIMEN
WANG QIZHI and XIAN XUEFU Department
of Resources and Environmentat Engineering, Chongqing University, Chongqing 630044. P.R.C.
Almtract-T%is paper analyses the post-failure behavior of a three-point bend round beam specimen based on the idea of Carpinteri [J. Engng Me&. 115, 13754392 (1989)]. It is pointed out that the behavior of this specimen is different from that of Carpinteri’s specimen with a rectangular cross-section in the snap-back instability. A new instability condition is proposed for the round beam specimen.
INTRODUCTION RECENTLY, Carpinteri[l] proposed a method to analyse the snap-back fracture instability in the post-failure behavior of materials such as rocks, concrete etc. One application of this method is in fracture toughness testing of these materials, and a proposed brittleness number S, is found to have great influence on the load-displacement curve[l]. The bend specimen configuration in Carpinteri’s analysis is a three-point bend (3PB) specimen with a rectangular cross-section (hereafter called rectangular beam). It is noted by the present authors that specimens with a circular cross-section are more practical and popular in the rock mechanics community, since rock cores are often obtained in the shape of round bars. Based on Carpinteri’s idea[l], the 3PB round beam specimen (hereafter called round beam) is analysed for its post-failure instability. Meaningful results are obtained and compared with the rectangular beam and it is pointed out that the behavior of a round beam is different from that of a rectangular beam in some respects. Hence the instability condition proposed in ref. [l] may not be suitable for a round beam. A new instability condition is proposed for a round beam; it is shown that instability takes place more easily for a round beam than for a rectangular beam.
POST-FAILURE
INSTABILITY
ANALYSIS
Figure 1 shows a round beam with no initial crack. If a material’s behavior is linear elastic, its central deflection 6 is
where E is the modulus of elasticity, P is load, I is beam span and D is beam diameter. Equation (1) can be expressed in dimensionless form as p =_*-. 3% 1 li 4 A3
(2)
where 1 = Z/D is a measure of the specimen’s slenderness, and the dimensionless load P and central deflection 8 are respectively given by
F =z
61
GD
where a, and 6, are the ultimate tensile stress and strain respectively. 7
(4)
WANG QIZHI and XIAN XUEFU
Fig. 1. Case I: 3PB round beam with no crack.
Once 0, is achieved at the lowest point of the central cross-section, the fracture process is supposed to initiate; this is considered as the limit situation of a round beam with no initial crack. The limit load can be obtained easily as P = n/8. Thus for Case 1 in Fig. 1 we must have
Using eqs (2) and (5) it can be found that
Equations (5) and (6) are the conditions for the application of eq. (2), which illustrates the linear elastic relation between P and s’ for a round beam. After a crack initiates from the lowest point of the central cross-section of a round beam, the crack begins to propagate. For materials such as rocks and concrete, which exhibit softening in the ultimate behavior under loading and have a long fracture process zone (FPZ), a linear stres~ispla~ment cohesive law can be assumed to be appli~ble in this F’PZ. This law states that the force acting on the two interacting surfaces is proportional to the distance w which separates them, This force is effective when w is less than a critical value w,. Figure 2 is an illustration for Case 2, where FPZ extends to the highest point of the central cross-section. In Fig. 2, let x be the extension of a linearly distributed cohesive force or the length of FPZ. From geometrical considerations, it is easy to obtain that 6 -=-. I z
%I2 x
Thus
I
c_rcs__:____ 8
~--------
+
Fig. 2. Case 2: 3PB round beam in complete fracture.
(7)
Post-failure instability analysis of rock
9
But we must have
From eqs (8) and (9) it is obvious that
(10) where a dimensionless brittleness number SE is defined as SE = wJ26. Since only bending geometrical @ations, i.e. eqs (8) and (9), are used in the derivation of eq. (lo), it is natural for the result of 6 obtained in this paper and ref. [l] to be identical for this case. The limit load P of Fig. 2 is obtained from consideration of the rotation equilibrium around point A for each half of the specimen, which is acted on by cohesive force and support reaction. Although calculation of the moment equilibrium is simple for a rectangular beam in ref. [l], it is complex for the round beam analysed in this paper. The difficulty arises from the fact that the width of crack front is changing all the time and is not a constant like that of a rectangular beam. According to Fig. 2 the relation between the width of crack front b and crack front position x’ is as follows: b =2J(m
(11)
where x’ describes the position of the crack front. The moment of this linearly distributed cohesive force around point A is calculated as follows: Mw,=
f Through detailed integration
xx--’ --,.x’*2/mdx’. 0
(12)
x
we obtain (see Appendix A) that
The moment of supporting reaction around point A is simply Msup= ;
.
(14)
The moment equilibrium condition requires that
Not, = Mup.
(15)
From eqs (3), (13), (14) and (15) we can get the relation of P and x:
where the contents of the brace ( } are the same as in eq. (13). The P-x relation is more complicated than the corresponding one in ref. [l]. Using eqs (16) and (8) we can obtain the PJ relation, which is also complicated. The limit value of P can be obtained by substituting x = D into eq. (16), then P = 3x/16, which must be the maximum load of Case 2, then generally we have
p
(17)
It is noted that eq. (17) is not coincident with eq. (5). This behavior of a round beam is different from that of a rectangular beam. In ref. [l] it was proved that the same relation P < 2/3 was applicable in both Case 1 and Case 2 for a rectangular beam.
WANG QIZHI and XIAN XUEFU
10
lb)
0 Fig. 3. Load-deflection
diagrams of a rectangular beam[l]. (a) J2 z- 8,. (b) 8, < s,, with snap-back instability.
Fig. 4. Load-deflection
diagrams of a round beam. (a) & > 8, and determination of 8. (b) 5, < s;, with snap-back instability.
An instability criterion is proposed in ref. [l] that instability will occur when two cases are partially overlapped, i.e. gZ < $, as shown in Fig. 3b. In this figure the dashed line connecting two curves has a highly negative or even positive slope, which corresponds to snap-back instability. The behavior of a round beam is different and the instability condition 8, < 8, seems to be too harsh for a round beam, because the maximum load of Case 2 is greater than the maximum load of Case 1 here, as when s, is a little greater than s,, a normal curve cannot connect these two curves, as shown in Fig. 4a. A reasonable and comparable solution for a round beam is to extend the straight line of Case 1 and let it intersect the horizontal line P = 3x/16, and find s; = ,13/4. Instability will occur when gZ < 8, i.e.
i.e.
SE<1
(1%
c,I ‘2’
For comparison
the instability condition for a rectangular
CONCLUSIONS
beam[l] is listed as follows:
AND DISCUSSION
1. Under the same assumption and using the same analytical method as proposed by Carpinteri[l], the post-failure behavior of a 3PB round beam specimen is different from that of a 3PB rectangular beam specimen. The snap-back instability condition is proposed to be SE/c,2 < l/2 for the round beam.
Post-faihrre instability anatysis of rock
11
2, For a 3PB round beam specimen, the relationship between load and central deflection in Case 2 (Fig. 2) is more complicated than that for a 3PB rectangular beam. The limit load in Case 2 is greater than that in Case 1 (Fig. 1) for a round beam, and snap-back instability occurs more easily for a round beam than for a rectangular beam. 3. Ve~fication of the above conclusions should be done by n~e~~al calculation or experimental detection. Suggested methods have been proposed by Carpinteri[l] and Bocca[2,3]. REFERENCES [If A. Carpinteri, Size effect on strength, toughness, and ductility. J. Engng Me&. 115,1375-1392 (1989). [2] P. Bocea, Evaluation of the released energy during catastrophic failure. Exp. Meek 13, 25-28 (1989). [3] P. Bocca and A. Carpinteri, Snapback failure instability in rock specimen: experimental detection through a negative impulse. Engng Fracture Me&. 35, 241-250 (1990). APPENDIX A: CALCULATION OF THE MOMENT OF COHESIVE FORCE AROUND POINT A IN FIG. 2 Beginning from eq. (12) we have
Let y = x’ - D/2, then x’ = y + D/2, dx’ = dy and eq (12) &I be rewritten as
First, we calculate the following three integrals respectively:
+ WY
(D/2)4 R . x -D/2 -aarcsin-+---. 8 2 8 D/2 Substituting eqs (A2), (A3) and (A4) into eq. (Al) we obtain eq. (13), i.e.
Mmh = 2a,
x2 5xD 13D= --b_+8-32+i5;i-y
(A4)
5 D3
(13) (Received 13 November 1990)
0013-7944/92
55.00 + 0.00
Pergamon RESSpk.
POST-FAILURE INSTABILITY THREE-POI~ BEND ROUND OF ROCK
ANALYSIS OF A BEAM SPECIMEN
WANG QIZHI and XIAN XUEFU Department
of Resources and Environmentat Engineering, Chongqing University, Chongqing 630044. P.R.C.
Almtract-T%is paper analyses the post-failure behavior of a three-point bend round beam specimen based on the idea of Carpinteri [J. Engng Me&. 115, 13754392 (1989)]. It is pointed out that the behavior of this specimen is different from that of Carpinteri’s specimen with a rectangular cross-section in the snap-back instability. A new instability condition is proposed for the round beam specimen.
INTRODUCTION RECENTLY, Carpinteri[l] proposed a method to analyse the snap-back fracture instability in the post-failure behavior of materials such as rocks, concrete etc. One application of this method is in fracture toughness testing of these materials, and a proposed brittleness number S, is found to have great influence on the load-displacement curve[l]. The bend specimen configuration in Carpinteri’s analysis is a three-point bend (3PB) specimen with a rectangular cross-section (hereafter called rectangular beam). It is noted by the present authors that specimens with a circular cross-section are more practical and popular in the rock mechanics community, since rock cores are often obtained in the shape of round bars. Based on Carpinteri’s idea[l], the 3PB round beam specimen (hereafter called round beam) is analysed for its post-failure instability. Meaningful results are obtained and compared with the rectangular beam and it is pointed out that the behavior of a round beam is different from that of a rectangular beam in some respects. Hence the instability condition proposed in ref. [l] may not be suitable for a round beam. A new instability condition is proposed for a round beam; it is shown that instability takes place more easily for a round beam than for a rectangular beam.
POST-FAILURE
INSTABILITY
ANALYSIS
Figure 1 shows a round beam with no initial crack. If a material’s behavior is linear elastic, its central deflection 6 is
where E is the modulus of elasticity, P is load, I is beam span and D is beam diameter. Equation (1) can be expressed in dimensionless form as p =_*-. 3% 1 li 4 A3
(2)
where 1 = Z/D is a measure of the specimen’s slenderness, and the dimensionless load P and central deflection 8 are respectively given by
F =z
61
GD
where a, and 6, are the ultimate tensile stress and strain respectively. 7
(4)
WANG QIZHI and XIAN XUEFU
Fig. 1. Case I: 3PB round beam with no crack.
Once 0, is achieved at the lowest point of the central cross-section, the fracture process is supposed to initiate; this is considered as the limit situation of a round beam with no initial crack. The limit load can be obtained easily as P = n/8. Thus for Case 1 in Fig. 1 we must have
Using eqs (2) and (5) it can be found that
Equations (5) and (6) are the conditions for the application of eq. (2), which illustrates the linear elastic relation between P and s’ for a round beam. After a crack initiates from the lowest point of the central cross-section of a round beam, the crack begins to propagate. For materials such as rocks and concrete, which exhibit softening in the ultimate behavior under loading and have a long fracture process zone (FPZ), a linear stres~ispla~ment cohesive law can be assumed to be appli~ble in this F’PZ. This law states that the force acting on the two interacting surfaces is proportional to the distance w which separates them, This force is effective when w is less than a critical value w,. Figure 2 is an illustration for Case 2, where FPZ extends to the highest point of the central cross-section. In Fig. 2, let x be the extension of a linearly distributed cohesive force or the length of FPZ. From geometrical considerations, it is easy to obtain that 6 -=-. I z
%I2 x
Thus
I
c_rcs__:____ 8
~--------
+
Fig. 2. Case 2: 3PB round beam in complete fracture.
(7)
Post-failure instability analysis of rock
9
But we must have
From eqs (8) and (9) it is obvious that
(10) where a dimensionless brittleness number SE is defined as SE = wJ26. Since only bending geometrical @ations, i.e. eqs (8) and (9), are used in the derivation of eq. (lo), it is natural for the result of 6 obtained in this paper and ref. [l] to be identical for this case. The limit load P of Fig. 2 is obtained from consideration of the rotation equilibrium around point A for each half of the specimen, which is acted on by cohesive force and support reaction. Although calculation of the moment equilibrium is simple for a rectangular beam in ref. [l], it is complex for the round beam analysed in this paper. The difficulty arises from the fact that the width of crack front is changing all the time and is not a constant like that of a rectangular beam. According to Fig. 2 the relation between the width of crack front b and crack front position x’ is as follows: b =2J(m
(11)
where x’ describes the position of the crack front. The moment of this linearly distributed cohesive force around point A is calculated as follows: Mw,=
f Through detailed integration
xx--’ --,.x’*2/mdx’. 0
(12)
x
we obtain (see Appendix A) that
The moment of supporting reaction around point A is simply Msup= ;
.
(14)
The moment equilibrium condition requires that
Not, = Mup.
(15)
From eqs (3), (13), (14) and (15) we can get the relation of P and x:
where the contents of the brace ( } are the same as in eq. (13). The P-x relation is more complicated than the corresponding one in ref. [l]. Using eqs (16) and (8) we can obtain the PJ relation, which is also complicated. The limit value of P can be obtained by substituting x = D into eq. (16), then P = 3x/16, which must be the maximum load of Case 2, then generally we have
p
(17)
It is noted that eq. (17) is not coincident with eq. (5). This behavior of a round beam is different from that of a rectangular beam. In ref. [l] it was proved that the same relation P < 2/3 was applicable in both Case 1 and Case 2 for a rectangular beam.
WANG QIZHI and XIAN XUEFU
10
lb)
0 Fig. 3. Load-deflection
diagrams of a rectangular beam[l]. (a) J2 z- 8,. (b) 8, < s,, with snap-back instability.
Fig. 4. Load-deflection
diagrams of a round beam. (a) & > 8, and determination of 8. (b) 5, < s;, with snap-back instability.
An instability criterion is proposed in ref. [l] that instability will occur when two cases are partially overlapped, i.e. gZ < $, as shown in Fig. 3b. In this figure the dashed line connecting two curves has a highly negative or even positive slope, which corresponds to snap-back instability. The behavior of a round beam is different and the instability condition 8, < 8, seems to be too harsh for a round beam, because the maximum load of Case 2 is greater than the maximum load of Case 1 here, as when s, is a little greater than s,, a normal curve cannot connect these two curves, as shown in Fig. 4a. A reasonable and comparable solution for a round beam is to extend the straight line of Case 1 and let it intersect the horizontal line P = 3x/16, and find s; = ,13/4. Instability will occur when gZ < 8, i.e.
i.e.
SE<1
(1%
c,I ‘2’
For comparison
the instability condition for a rectangular
CONCLUSIONS
beam[l] is listed as follows:
AND DISCUSSION
1. Under the same assumption and using the same analytical method as proposed by Carpinteri[l], the post-failure behavior of a 3PB round beam specimen is different from that of a 3PB rectangular beam specimen. The snap-back instability condition is proposed to be SE/c,2 < l/2 for the round beam.
Post-faihrre instability anatysis of rock
11
2, For a 3PB round beam specimen, the relationship between load and central deflection in Case 2 (Fig. 2) is more complicated than that for a 3PB rectangular beam. The limit load in Case 2 is greater than that in Case 1 (Fig. 1) for a round beam, and snap-back instability occurs more easily for a round beam than for a rectangular beam. 3. Ve~fication of the above conclusions should be done by n~e~~al calculation or experimental detection. Suggested methods have been proposed by Carpinteri[l] and Bocca[2,3]. REFERENCES [If A. Carpinteri, Size effect on strength, toughness, and ductility. J. Engng Me&. 115,1375-1392 (1989). [2] P. Bocea, Evaluation of the released energy during catastrophic failure. Exp. Meek 13, 25-28 (1989). [3] P. Bocca and A. Carpinteri, Snapback failure instability in rock specimen: experimental detection through a negative impulse. Engng Fracture Me&. 35, 241-250 (1990). APPENDIX A: CALCULATION OF THE MOMENT OF COHESIVE FORCE AROUND POINT A IN FIG. 2 Beginning from eq. (12) we have
Let y = x’ - D/2, then x’ = y + D/2, dx’ = dy and eq (12) &I be rewritten as
First, we calculate the following three integrals respectively:
+ WY
(D/2)4 R . x -D/2 -aarcsin-+---. 8 2 8 D/2 Substituting eqs (A2), (A3) and (A4) into eq. (Al) we obtain eq. (13), i.e.
Mmh = 2a,
x2 5xD 13D= --b_+8-32+i5;i-y
(A4)
5 D3
(13) (Received 13 November 1990)