fracture of rock: effect of loading rate - Semantic Scholar
0013-7944/93 $6.00+0.00 $6.00 + 0.00 0013-7944/93
Engineering 45, No.3, No. 3, pp. pp. 393-398, 393-398, 1993 1993 l?$tgineering Fracture Fracture Mechanics Mechanics Vol. Vol. 45,
© 0 1993 1993 Pergamon PergsmonPress PressLtd. Ltd.
Printed Printed in in Great Great Britain. Britain.
FRACTURE FRACTURE OF OF ROCK: EFFECT OF OF LOADING LOADING RATE ZDEN£K VINDRA GETTU ZDENEK P. P. BA~ANT, BAZANT, SHANG-PING SHANG-PING BAI BAI and and RA RAVINDRA GETTU Center Center for for Advanced Advanced Cement-Based Cement-Based Materials, Materials, Northwestern Northwestern University, University, Evanston, Evanston, IL IL 60208, 60208, U.S.A. U.S.A. Abstract-Fracture parameters Abstract-Fracture parameters of of limestone limestone at at loading loading rates rates ranging ranging over over four four orders orders of of magnitude magnitude in in the the static static regime regime are are determined determined using using the the size size effect effect method. method. Three Three sizes sizes of of three-point three-point bend bend notched notched specimens specimens were were tested tested under under crack-mouth crack-mouth opening opening displacement displacement control. control. The The fracture fracture toughness toughness and and nominal nominal strength strength decrease decrease slightly slightly with with aa decrease decrease in in rate, rate, but but the the fracture fracture process process zone zone length length and and the the brittleness brittfeness of of failure failure are are practically practically unaffected. unaffected. The The effect effect of of material material creep creep on on the the fracture fracture of of limestone limestone is negligible rate negligible in the time range studied studied here. However, However, the methodology methodology developed developed for characterizing ch~acte~ng effects in static materials. static fracture fracture can be easily applied applied to other other brittle-heterogeneous b~ttl~hetero~neous materials. The decrease decrease of fracture fracture toughness toughness as a function function of the crack propagation propagation velocity is described described with a power power law. A formula formula for the size- and rate-dependence rate-dependence of the nominal nominal strength strength is also presented. presented.
INTRODUCTION INTRODUCTION BoND I&MDRUPTURE RUPTURE is a rate rate process process governed governed by Maxwell Maxwell distribution distribution of of molecular molecular thermal thermal energies energies and and characterized characterized by activation activation energy. energy. Therefore, Therefore, fracture fracture in aU all materials materials is rate-sensitive. rate-sensitive. This has been been experimentally experimentally demonstrated demonstrated for rock rock in the dynamic dynamic range, range, but but not not in the static static range. range. However, However, knowledge knowledge of of this rate rate effect effect is very very important important for may may practical practical applications applications in mining, mining, geotechnical geotechnical engineering enginee~ng and and geology. geology. The The present present paper paper reports reports new experimental experimental results results on the static static fracture fracture of of limestone limestone at loading loading rates rates ranging ranging over over four four orders orders of of magnitude. magnitude. The The corresponding corresponding times to failure failure range range from from about about 2 sec set to almost almost 1 day. day.
EXPERIMENTAL EXPERIMENTAL DETAILS DETAILS specimens were were cut cut from from the same same block of Indiana Indiana (Bedford) (Bedford) limestone. limestone. Three Three sizes of of All specimens block of three-point bend (single-edge-notched) three-point bend (single-edge-notched) fracture fracture specimens specimens (Fig. I) 1) were tested. tested. The The depths, depths, d, of of the beams were 25, 25,5511 and and 102mm 102 mm (I, (it22 and the thickness,b, thickness, b, of 13 mm (0.5 in.). the beams were and 4 in.), in.), and and the of each each was 13 mm(0.5 in.). The The specimens specimens were were cut cut such such that that the the ending bending plane plane of of the the rock rock was normal normal to to the the load. load. Notches Notches of length equal of 1.3 mm mm (0.05 in.) in.) width width were were cut cut with with a steel steel saw blade. blade. Alumina Aluminum bearing bearing plates plates of oflength equal to to half half the the beam beam depth depth were were epoxied epoxied at at the the ends ends to to provide provide support. support. The The fracture fracture tests tests were were conducted under under constant constant crack-mouth crack-mouth opening opening displacement displacement (CMOD) (CMOD) rates rates in a 89 kN kN (20 kip) kip) conducted closed-loop controlled controlled machine machine with with a load load cell operating operating in the the 890 N (200 lb) Ib) range. range. The The CMOD CMOD closed-loop was monitored monitored with with a transducer transducer (LVDT (LVDT of of 0.127 0.127 mm mm range) range) mounted mounted across across the the notch. notch. Four Four series series was of of tests tests were were performed; performed; each each series series consisted consisted of of six specimens, specimens, two two in each each size (see Table Table 1). I). The The CMOD CMOD rates rates were were chosen chosen so that that all specimens specimens in a series series reached reached their their peak peak load load in about about the the same same time, time, tp. The The average average tp tp values values were were 2.3, 213, 21,420 21,420 and and 82,500 82,500 set sec for for the the different different series. series. The load-CMOD curves The typical typicalload-CMOD curves for for each each size are are shown shown in Fig. Fig. 2. From From the the initial initial slopes slopes of of these these curves, curves, the the initial initial elastic elastic modulus modulus _?& Eo of of the the rock rock was was calculated, calculated, for for each each test, test, using using linear linear elastic elastic Table 1. I. fracture mechanics mechanics (LEFM) (LEFM) formulas formulas [I]; [I]; see Table fracture
IDENTIFICATION OF OF FRACTURE FRACTURE PARAMETERS PARAMETERS IDENTIFICATION The used to to determine determine the the material material fracture fracture parameters parameters from from the the test test The size size effect effect method method [2] is used data. of limestone limestone [3], as as well well as as other other data. The The method method has has previously previously been been verified verified for for the the fracture fracture of rocks of the the effect effect of of loading loading rate rate rocks and and concrete concrete [4,5]. [4,5]. Recently, Recently, itit has has also also been been used used in in aa study study of on on the the fracture fracture of of concrete concrete [6]. The The method method is based based on on the the size size effect effect law law [7], which which is: (IN=
Bfu
..j(l
+ fJ) , 393 393
fJ
= ~, uo
(1)
Z. P. BA2ANT BAZANT et 01. al.
394
p
T d
I
GAd
~------~------~~
I. ---4-:--- .1 4.5d
d-thlckneu Fig. 1. 1. Fracture Fracture specimen specimen geometry. geometry.
where Pu/bd == maximum where UN bN = PJbd maximum nominal nominal stresses stresses of of geometrically geometrically similar similar fracture fracture specimens, specimens, P beam depth), Puu = = maximum maximum load, load, d = = characteristic characteristic dimension dimension (chosen (chosen here here as the beam depth), b = = specimen specimen Bf. and parameters, and thickness thickness (constant, (constant, for for two-dimensional two-dimensional similarity), similarity), BS, and do do = = empirical empirical parameters, and P= brittleness brittleness number. P is very P~G 0.1), is almost independent of size, /I number. When When /3 very small (e.g. fl O.l), UN a,,, almost independent of as in plastic plastic limit ~ 10), limit analysis. analysis. When When P /I is large large (e.g. P b B lo), the size-dependence size-dependence follows follows LEFM LEFM In the transition (i.e. UNIX a,cc l/.}d). l/G). transition zone, zone, nonlinear nonlinear fracture fracture mechanics mechanics needs needs to be applied. applied. For parameters from data, For determining determining the parameters from UN oN data, eq. (I) (1) can can be transformed transformed to Y Y = AX + + C, where l/u~. Then, and where X = = d and and Y = = l/of,. Then, BJ.. BS, = = l/ft; l/fi and do 4 = = CIA C/A [4]. [4]. By linear linear regression regression analysis analysis of of the parameters and the data data for for the four four series of of tests, the parameters and coefficients coefficients of of variation variation of of errors, errors, (OYlx, oVX, have have been computed been computed and and are are listed listed in Table Table 2. The The data data and and the fits [eq. (I)] (l)] are shown shown in Fig. 3. It It can can be seen that that the size effect effect law represents represents the trend trend reasonably reasonably well, at all the loading loading rates. rates. It It is clear clear that that the data data cannot cannot be represented represented by either either LEFM LEFM (a straight straight line with a slope slope of of -1/2) - l/2) or Bf.). or strength strength criteria criteria (horizontal (horizontal line UN aN = Bf”). Using Bf. and parameters can 7]: Using the values values of of Bf, and do, do, fracture fracture parameters can be calculated calculated as follows follows [4,5, [4,5,7]: dog(r/..o)
Kic
KIc = Bf.J(dog(rJ..o», cf= g'(rJ..o) , Gf=p'
(2)
Table Table I. 1. Test data data Series Series
Dimensionst Dimensionst (mm x mm x mm) (mmxmmxmm) 457 x 102 102 x 13 13
Fast Fast
229 x 51 51 x 13 13 114 x 25 x 13 114x25x 13 457 x 102 102 x 13 13
Usual Usual
229 x 51 51 x 13 13 114 114 x 25 x 13 13 457 x 102 x 13 457X102X13
Slow
229 x 51 51 x 13 13 114 x 25 x 13 114x25~13 457 x 102 102 x 13 13
Very slow
229 x 51 51 x 13 13 114 x 25 x 13 114x25x 13
CMOD CMOD rate rate 6 mm/sec) (10(10-6mm/sec) 15,900 15,900 15,900 15,900 10,600 10,600 10,600 10,600 5770 5770 5770 5770
Peak load load (N) (N) 445 472 281 281 291 291 178 178 165 165
Time to peak peak (sec) (set) 2.1 2.1 2.2 2.0 2.4 2.4 2.2
Eot (GPa) 40 32 32 35 35 24 24 35 35 35 35
436 414 269 271 271 153 153 165 165
176 176 194 194 237 210 248 215
33 33 30 30 30 30 30 30 29 29 30 30
1.42 1.42 1.42 1.42 0.978 0.978 0.978 0.706 0.508
394 383 383 245 240 147 147 153 153
23,175 23,175 16,875 16,875 26,000 26,000 20,475 20,475 15,750 15,750 26,250 26,250
0.353 0.318 0.318 0.236 0.236 0.160 0.160
385 385 387 387 262 265 140 140 136 136
81,900 81,900 79,000 79,000 87,800 87,800 82,350 82,350 72,000 72,000 92,000 92,000
30 ,30 32 32 28 28 25 25 32 32 34 34 27 27 34 34 32 32 27 27 26 26 25 25
159 159 141 141 106 106 106 106 57.7 63.5 63.5
tLength tLength x depth depth x thickness. thickness. tInitia1 $Initial modulus modulus from from load-CMOD Ioad-CMOD compliance. compliance.
395 395
Fracture of of rock rock Fracture ~~-------------------------------.
~~-------------------------------.
,,= 51
(a)
=
100
2.1 *s 2.1 176 Is = 176 23200 Is = 23200 81900 ss = 81900
100
O.-----.---_,----~----_.----._--~
0.00
mm
0.01 0.01
c0.06 LO6
0.02 0.03 0.04 0.05 0.03 0.04 0.05 0.02 Crack Mouth t.loulh Opening Opening (mm) (mm) Crack
z
0.01 0.02 0.03 Crack Crack Mouth t.loulh Oprning Opening (mm) (mm)
O. 44
200.--------------------------------.
d = 25 mm
(c) 150
..-.
~ .., 100'
c
.3 2.4 •
215 •
26300 •
92000 • 0~------_.------~------_,------_4 0.000 0.005 0.010 0.015 0.020 o.obs o.oio O.dlS 0.
Crack Crack t.loulh Mouth Opening Opening (mm) (mm)
Fig. 2. Typical Typical load-CMOD IoadXMOD curves curves for each each specimen specimen size.
where fracture toughness, toughness, cf= c, = effective effective length length of of the fracture fracture process process zone, zone, and and G G,f == fracture fracture where KIc K,, = - fracture energy. energy. Function Function g(IX) g(a) is the non-dimensionalized non-dimensionalized energy energy release release rate rate defined defined by the LEFM LEFM relation relation G == P2 g (IX)IE'b 2d, where P2g(a)/E’b2d, where G == energy energy release release rate rate of of the specimen, specimen, P = load, load, IXa == aid a/d = relative relative crack = crack length, length, g'(IX) g’(a) == dg(IX)/dIX, dg(a)/da, a == crack crack length, length, IXo LY,, = aold, a,,/d, ao a, = notch notch length length of of traction-free traction-free crack crack 2 ) for plane length, length, E' E’ = E for plane plane stress, E' E’ = EI(1 E/(1 - vv2) plane strain, strain, E = Young's Young’s modulus, modulus, and and v = Poisson's Poisson’s ratio. ratio. Function Function g(IX) g(a) can be obtained obtained from from handbooks handbooks (e.g. [1]) [l]) or from from LEFM LEFM analysis. analysis. Fracture Fracture parameters parameters are defined defined here for the limiting limiting case of an infinitely infinitely large specimen specimen at failure. failure. Then, Then, an infinite-size infinite-size extrapolation extrapolation of of eq. (I) (1) provides provides material material parameters parameters [eq. (2)] (2)] that that are practically practically size- and shape-independent shape-independent [5]. [5]. Using the values g(IXo) g(a,) = = 62.84 and g'(IXo) g’(G) = = 347.7 (from (from [1]), [l]), and assuming assuming plane plane stress conditions, conditions, the fracture fracture parameters parameters for the four four series can be computed; computed; see Table Table 2, in which the average average values of K K,,1c and cf c/ as well as their their coefficients coefficients of of variation variation are listed. The E-value E-value for each series is is taken taken as the average average initial initial modulus modulus Eo, E,,, and is used in eq. (2) for computing computing G G,f (see Table Table 2). 2). VARIATION VARIATION OF FRACTURE FRACTURE PARAMETERS PARAMETERS
The The test test results show show that that as as the the time time to to peak load, load, ttp, increases, the fracture fracture toughness toughness Klc K,, p ' increases, decreases. decreases. Since Since the fracture fracture energy energy Gfis G, is proportional proportional to to Kin Ki,, its its decrease decrease with slower slower loading loading rates is is even even stronger. stronger. The The same same trends trends have also also been observed observed in in similar similar materials, materials, such such as as hardened hardened cement cement paste paste [8], [8], concrete concrete [6], [6], and and ceramics ceramics at at high temperatures temperatures [9]. [9]. To To describe describe the influence influence of of loading loading rate, we we follow follow several several other other investigators investigators by adopting adopting aa power power function function of of crack crack velocity velocity v: v: (3)
Z. P. BAlANT BtiAlUT et aI. al.
396
Table parameters Table 2. Fracture Fracture parameters
Series Series
lp Avg. tp (sec) (se4
Fast Fast Usual Usual Slow Very slow
2.3 213 21,400 82,500
Bf. (MPa) (2a)
(mm) (nil,
cony (QI1X
4Klc Avg. (MPaJiiiiii) (MPa@)
36.2 36.3 31.9 36.5
0.07 0.07 0.04 O.ll 0.11
33.1 33.1 30.8 27.5 28.2
do
0.693
0.645 0.614
0.589
cf
Avg. (mm)
o
(Q
0.13 0.12 0.08 0.19
6.5 6.6 5.8 6.6
o
E, Avg. Eo (GPa) @Pa)
0.19 0.19 0.12 0.28
33.5 30.3 30.2 28.5
(Q
Gf
(N/m) 0%) 32.7 31.3 25.0 27.9
0 = coefficient coefficient of of variation. variation.
(Q
where reference velocity, where ~ & is the the fracture fracture toughness toughness corresponding corresponding to to a reference velocity, vo, v,,, chosen chosen here here as Vo = 0.01 mm/sec. Since the effective (LEFM) crack tip is roughly at a distance cf from the notch v. mm/set. effective (LEFM) crack tip roughly distance c, from notch we use the approximation tip at the peak peak load, load, the approximation (4) (4)
v = q/t,.
Then, Then, by fitting fitting the the test results results with with eq. (3) (see Fig. 4), we obtain obtain n = 0.0173 and and ~ = 30.0 MPay'niffi. Note that, alternatively, beam deflection or crack & MPa@. Note that, alternatively, beam deflection or crack opening opening rates rates have have been used used instead been instead of of v in other other studies. studies. In similar similar tests of of concrete concrete [6], [6], it was found found that, that, with with an increase increase in time time to to failure, failure, the the group group of data for the three sizes of specimens shifts to the right, i.e. toward the LEFM asymptote, of data for three of specimens shifts to the right, toward the LEFM asymptote, when when Ip, the process process zone plotted as in Fig. 3. This plotted This implies implies that, that, for for higher higher tp, zone length length cf c, decreases decreases and and the the p [eq. (I)], brittleness of brittleness of failure, failure, characterized characterized by fl (l)], increases. increases. present results Rather Rather interestingly, interestingly, no no such such trend trend is observed observed from from the the present results of of limestone. limestone. For For all Ip, the data part of tp, data remain remain within within the the same same part of the the size effect effect curve. curve. This This is reflected reflected by the the fact fact that that cf practically constant brittleness of c, is practically constant (cf (c, z~ 6 mm; Table Table 2), implying implying that that the the brittleness of fracture fracture in limestone limestone is rate-independent rate-independent within within the time time range range studied studied here. here. This This difference difference in the behavior behavior (for (for the present load present load durations) durations) from from concrete concrete may may be explained explained by the lack lack of of significant significant creep creep [10]. [lo]. Concrete bulk of Concrete exhibits exhibits marked marked viscoelastic viscoelastic creep creep in the the bulk of the test test specimen, specimen, as well as high nonlinear process zone. nonlinear creep creep in and and near near the the fracture fracture process zone. 4.0 0 . 0 , - - - - - - - - - . . . , - - - - - - - - - - - ,
t, = 2.3 sec
(a)
-0.' ‘:~
iii-O.2 ;f, -0.z
........ \ ~ t?
'-' x 01
.2-0.3 P 4.2
I
0s
,
Bf.
0.
-oh
do
mm
b
log ;
0.
\
a
-0.00 . 0 , - - - - - - - - - . . : - - - - - - - - - - . ,
= 0.645 MPa = 36.3 mm
\ -o.4+---r------.--~--_._-~ 0.5 a s -O.SO -0.25 Oh 0. -0:2!, 0.60
109 b
-0.0...-------..--------__
tp = 21400 21400 sec sac +, =
(c) (c>
(d)
=
82500 sec SIC tp = 82500
-0.'
-0.'
-0.1
.
.
,....
,....
iii -0.2
iii -0.2
........
........
~ .......
~
'-'
~-0.3
= 213 213 sec sac t, ‘P =
-0.'
do = 36.2
4.1
~o,-----____..:__------~
-o.’(b) (b)
01
0.614 MPa = 31.9 mm =
.2 -o.J
‘5
Fig. 3. Size effect curves peak load. curves at different different times times to peak load.
397
Fracture of of rock rock Fracture 34.,.----------------,
0.60,-----------------,
33
0.55
32
0.50
........
~
-E 31 E
00.45
•
limestone
0...
~0.40
030
0...
:::E
6
'""'29 .!!
0.35
K.c=Ko(v/voY·
::.::
n=O.OI13
28
0.30 0.30
for vo=O.Ol mm/s. /2 • Ko=30.0MPamm' 27 +m---.--rrrrrrrr-..--,-,rrrn.,,-..,....,.-m-.,.,,--,-rrrn-nr--.--1 0.0001
0.001 V
0.01
= c,/t
p
0.25
0 I
(mm/s)
I 1
I1111111 10 IO
I I1111111
100(
tp
y&c>
Fig. 4. 4. Variation Variation of of fracture fracture toughness toughness with with crack crack velocity. velocity. Fig.
I
I1111111
1000
sec) loo0
I 1111111, 10000 1OPOO
1111111 100000 100000
Fig. 5. Influence Influence of of specimen specimen size and and time time to to failure failure on on Fig. nominal strength. strength. nominal
EFFECT OF OF RATE RATE ON ON STRENGTH STRENGTH AND AND YOUNG’S YOUNG'S MOD~US MODULUS EFFECT Several investigators investigators have have demonstrated demonstrated that that the the strength strength of of rock rock generally generally increases increases with with an an Several [11, 12]). This is also observed here from Table 1. When the loading increase in the loading rate (e.g. increase the loading rate [ 11, 121).This also observed here from Table I. When the loading of magnitude, magnitude, the the maximum maximum nominal nominal stress stress decreases decreases by more more than than 16%. rate slows by by four four orders orders of rate 16%. is similar to the change in KIn has also been observed other This phenomenon, which has also been observed in other This phenomenon, which similar to the change K,,, may be attributed attributed to to the the statistical statistical nature nature of of the the failure failure of of molecular molecular bonds bonds materials [13]. It materials It may of thermal energies). (particularly the activation energy theory and the Maxwell distribution (particularly the activation energy theory and Maxwell distribution of thermal energies). of a quasi-brittle quasi-brittle heterogeneous heterogeneous material material is generally generally difficult difficult to to measure measure The strength strength of The of its dependence on specimen size and shape, and because failure does not objectively because objectively because of dependence on specimen and shape, and because failure does not at all points but is progressive. However, strength (or failure stress) occur simultaneously occur simultaneously at points but progressive. However, strength (or failure stress) is correlated to the fracture toughness since failure propagation; higher correlated to the fracture toughness failure occurs occurs by unstable unstable crack crack propagation; higher toughness implies higher resistance against failure. toughness implies higher resistance against failure. Equations Equations (I) (1) and and (2) can can be combined combined to to give the size effect effect on on the nominal nominal strength strength of the material fracture parameters [5]: (maximum nominal stress) in terms (maximum nominal stress) terms of material fracture parameters [5]: aN =
Kic -,-------=-=--.j(g'(r:J.O)Cf+ g(r:J.o)d)
(5)
Substituting Klc from Substituting for for -lu,, from eq. (3), and and cf cr from from eq. (4), one one obtains obtains a relation relation for for the the dependence dependence of of the the nominal nominal strength strength on on the the failure failure time:
Ku aN = .j(g'(r:J.O)Cf+ g(r:J.o)d)
(c)n vo~p .
(6) (6)
Since cf is not c/is not systematically systematically affected affected by the loading loading rate, rate, the average average value value of of 6.4 mm is considered. considered. Equation Equation (6) may may then then be plotted, plotted, along along with the test data, data, for the different different sizes tested tested (Fig. 5). The The agreement agreement is acceptable. acceptable. The The test test results results also indicate indicate that that the average average initial initial elastic elastic modulus modulus decreases decreases slightly slightly with an increase increase in the time time to peak peak load load (Table (Table 2). Such Such an effect effect has been been observed observed for several several rocks rocks in the the dynamic dynamic range range [14]. [14].
CONCLUSIONS CONCLUSIONS (1) For For times to peak peak load load ranging ranging from from 2 to 80,000 sec, set, the measured measured nominal nominal strengths strengths offracture of fracture of limestone agree with the size effect law. specimens specimens of limestone agree (2) The The fracture fracture toughness toughness and failure failure stress decrease decrease with increasing increasing failure failure time. However, However, the fracture fracture process process zone zone size and and the brittleness brittleness of of failure failure appear appear to be unaffected unaffected by the loading loading rate. rate. (3) Since there there is insignificant insi~ificant creep creep outside outside the process process zone zone of of limestone limestone in the time range range studied, studied, the effective effective process process zone zone size does does not not change change as the loading loading rate rate is varied. varied.
398 398
Z. P. BAUNT BAZANT et al.
Acknowledgements-This work partially supported Northwestern University, Acknowledgements-This work was partially supported by AFOSR AFOSR contract contract 91-0140 with Northwestern University, and and Center for Advanced Advanced Cement-Based Cement-Based Materials Materials at Northwestern University (NSF (NSF Grant Grant DMR-8808432). DMR-8808432). S. P. Bai is the Center Northwestern University grateful grateful for support support from from ISTIS, Taiyuen, Taiyuen, P.R.C., P.R.C., during during the course course of of this study. study.
REFERENCES REFERENCES [I] Analysis o/Cracks Handbook, 2nd Edn. Tada, P. C. Paris Paris and and G. R. Irwin, Irwin, The Stress Stress Analysis of Cracks Handbook, Edn. Paris Paris Productions, Productions, St. Louis, 111 H. Tada, MO (1985). (1985). [2] process zone Mater. Structures RILEM TC89, Size-effect Size-effect method method for determining determining fracture fracture energy energy and and process zone size of of concrete. concrete. Mater. Structures PI RILEM 23, 461-465 461-465 (1990). (1990). [3] properties from Baiant, R. Gettu Gettu and and M. T. Kazemi, Kazemi, Identification Identification of of nonlinear nonlinear fracture fracture properties from size effect tests tests and and 131Z. P. Bazant, Int. J. Rock Me&. Mech. Min. Min. Sci. structural based on geometry-dependent structural analysis analysis based geometry-dependent R-curves. R-curves. Inr. Sci. 28.43-51 28,43351 (1991); (1991); Corrigenda. Corrigenda. 28, Zs, 233 (1991). (1991). [4] brittleness number. ACI Marer. Mater. Baiant and and P. A. Pfeiffer, Pfeiffer, Determination Determination of of fracture fracture energy energy from from size effect and and brittleness number. ACI [41 Z. P. Baiant JI84, JI 84, 463-480 463-480 (1987). (1987). [5] process zone Baiant and and M. T. Kazemi, Kazemi, Determination Determination of of fracture fracture energy, energy, process zone length length and and brittleness brittleness number number from from PI Z. P. Baiant size effect, with application Int. J. Fracture Fracture 44. application to rock rock and and concrete. concrete. Inr. 44, 111-131 111-131 (1990). (1990). [6] Baiant and and R. Gettu, Gettu, Rate Rate effects and and load relaxation relaxation in static static fracture fracture of of concrete. concrete. Rep. No. 9O-3/498r, 90-3/498r, (61 Z. P. Bazant Center Northwestern Univ., ACI Mater. Mater. JI89(5), Center of of Advanced Advanced Cement-Based Cement-Based Materials, Materials, Northwestern Univ., Evanston, Evanston, IL (1990). (1990). Also Also ACI JI 89(S), 456-468 456-468 (1992). (1992). [7] blunt fracture: Engng Mech. Mech. 110. Baiant, Size effect in blunt fracture: concrete, concrete, rock, rock, metal. J. Engng 110, 5128-5135 (1984). (1984). [71 Z. P. Bazant, [8] Application 0/ Fracture Mechanics Mechanics to Mindess, Rate Rate of of loading loading effects on the fracture fracture of of cementitious cementitious materials, materials, in Application of Fracture PI S. Mindess, Nijhoff, Dordrecht Cementitious Cementitious Composites Composites (Edited (Edited by S. P. Shah), Shah), pp. 617-638. 617-638. Martinus Martinus Nijhoff, Dordrecht (1985). (1985). [9] Knickerbocker, A. Zangvil Zangvil and and S. D. Brown, Brown, Displacement Displacement rate rate and and temperature temperature effects in fracture fracture of of a 191S. H. Knickerbocker, hot-pressed Am. Ceram. hot-pressed silicon silicon nitride nitride at 1100° 1loo” to 1325°C. 1325°C. J. Am. Ceram. Soc. Sot. 67, 67, 365-368 365-368 (1984). (1984). [10] Griggs, Creep Creep of of rocks. rocks. J. Geology Geology 47. 47, 225-251 (1939). (1939). WI D. Griggs, [II] basalt and Kumar, The effect of of stress stress rate and and temperature temperature on the strength strength of of basalt and granite. granite. Geophysics Geophysics 33. 33, 501-510 Ull A. Kumar, (1968). (1968). [12] propagation and behavior of uniaxial tension. Int. J. Rock Rock Peng, A note note on the fracture fracture propagation and time-dependent time-dependent behavior of rocks rocks in uniaxial tension. Inf. iI21 S. S. Peng, Mech. Min. Min. Sci. Mech. Sci. 12, If, 125-127 (1975). (1975). [13] Advances in Fracture Fracture Research Research (Edited Hsiao, Kinetic Kinetic strength strength of of solids, solids, in Advances (Edited by K. Salama, Salama, K. Ravi-Chandar, Ravi-Chandar, P31 C. C. Hsiao, D. M. R. Taplin Taplin and and P. Rama Rama Rao), Rao), Volume Volume 4, pp. 2913-2919. Pergamon Pergamon Press, Press, Oxford Oxford (1989). (1989). [14] properties of Chong, J. S. Harkins, Harkins, M. D. Kuruppu Kuruppu and and A. I. Leskinen, Leskinen, Strain Strain rate rate dependent dependent mechanical mechanical properties of Western Western 1141K. P. Chong, Oil Shale, in 28th Rock Mechanics Mechanics (Edited 28rh U.S. U.S. Symp. Symp. on Rock (Edited by I. W. Farmer, Farmer, 1. J. J. K. Daemen, Daemen, C. S. Desai, Desai, C. E. Glass Glass Neuman), pp. 157-164. A. A. Balkema, and and S. P. Neuman), Balkema, Rotterdam Rotterdam (1987). (1987). (Received April 1992) (Received 8 April
Engineering 45, No.3, No. 3, pp. pp. 393-398, 393-398, 1993 1993 l?$tgineering Fracture Fracture Mechanics Mechanics Vol. Vol. 45,
© 0 1993 1993 Pergamon PergsmonPress PressLtd. Ltd.
Printed Printed in in Great Great Britain. Britain.
FRACTURE FRACTURE OF OF ROCK: EFFECT OF OF LOADING LOADING RATE ZDEN£K VINDRA GETTU ZDENEK P. P. BA~ANT, BAZANT, SHANG-PING SHANG-PING BAI BAI and and RA RAVINDRA GETTU Center Center for for Advanced Advanced Cement-Based Cement-Based Materials, Materials, Northwestern Northwestern University, University, Evanston, Evanston, IL IL 60208, 60208, U.S.A. U.S.A. Abstract-Fracture parameters Abstract-Fracture parameters of of limestone limestone at at loading loading rates rates ranging ranging over over four four orders orders of of magnitude magnitude in in the the static static regime regime are are determined determined using using the the size size effect effect method. method. Three Three sizes sizes of of three-point three-point bend bend notched notched specimens specimens were were tested tested under under crack-mouth crack-mouth opening opening displacement displacement control. control. The The fracture fracture toughness toughness and and nominal nominal strength strength decrease decrease slightly slightly with with aa decrease decrease in in rate, rate, but but the the fracture fracture process process zone zone length length and and the the brittleness brittfeness of of failure failure are are practically practically unaffected. unaffected. The The effect effect of of material material creep creep on on the the fracture fracture of of limestone limestone is negligible rate negligible in the time range studied studied here. However, However, the methodology methodology developed developed for characterizing ch~acte~ng effects in static materials. static fracture fracture can be easily applied applied to other other brittle-heterogeneous b~ttl~hetero~neous materials. The decrease decrease of fracture fracture toughness toughness as a function function of the crack propagation propagation velocity is described described with a power power law. A formula formula for the size- and rate-dependence rate-dependence of the nominal nominal strength strength is also presented. presented.
INTRODUCTION INTRODUCTION BoND I&MDRUPTURE RUPTURE is a rate rate process process governed governed by Maxwell Maxwell distribution distribution of of molecular molecular thermal thermal energies energies and and characterized characterized by activation activation energy. energy. Therefore, Therefore, fracture fracture in aU all materials materials is rate-sensitive. rate-sensitive. This has been been experimentally experimentally demonstrated demonstrated for rock rock in the dynamic dynamic range, range, but but not not in the static static range. range. However, However, knowledge knowledge of of this rate rate effect effect is very very important important for may may practical practical applications applications in mining, mining, geotechnical geotechnical engineering enginee~ng and and geology. geology. The The present present paper paper reports reports new experimental experimental results results on the static static fracture fracture of of limestone limestone at loading loading rates rates ranging ranging over over four four orders orders of of magnitude. magnitude. The The corresponding corresponding times to failure failure range range from from about about 2 sec set to almost almost 1 day. day.
EXPERIMENTAL EXPERIMENTAL DETAILS DETAILS specimens were were cut cut from from the same same block of Indiana Indiana (Bedford) (Bedford) limestone. limestone. Three Three sizes of of All specimens block of three-point bend (single-edge-notched) three-point bend (single-edge-notched) fracture fracture specimens specimens (Fig. I) 1) were tested. tested. The The depths, depths, d, of of the beams were 25, 25,5511 and and 102mm 102 mm (I, (it22 and the thickness,b, thickness, b, of 13 mm (0.5 in.). the beams were and 4 in.), in.), and and the of each each was 13 mm(0.5 in.). The The specimens specimens were were cut cut such such that that the the ending bending plane plane of of the the rock rock was normal normal to to the the load. load. Notches Notches of length equal of 1.3 mm mm (0.05 in.) in.) width width were were cut cut with with a steel steel saw blade. blade. Alumina Aluminum bearing bearing plates plates of oflength equal to to half half the the beam beam depth depth were were epoxied epoxied at at the the ends ends to to provide provide support. support. The The fracture fracture tests tests were were conducted under under constant constant crack-mouth crack-mouth opening opening displacement displacement (CMOD) (CMOD) rates rates in a 89 kN kN (20 kip) kip) conducted closed-loop controlled controlled machine machine with with a load load cell operating operating in the the 890 N (200 lb) Ib) range. range. The The CMOD CMOD closed-loop was monitored monitored with with a transducer transducer (LVDT (LVDT of of 0.127 0.127 mm mm range) range) mounted mounted across across the the notch. notch. Four Four series series was of of tests tests were were performed; performed; each each series series consisted consisted of of six specimens, specimens, two two in each each size (see Table Table 1). I). The The CMOD CMOD rates rates were were chosen chosen so that that all specimens specimens in a series series reached reached their their peak peak load load in about about the the same same time, time, tp. The The average average tp tp values values were were 2.3, 213, 21,420 21,420 and and 82,500 82,500 set sec for for the the different different series. series. The load-CMOD curves The typical typicalload-CMOD curves for for each each size are are shown shown in Fig. Fig. 2. From From the the initial initial slopes slopes of of these these curves, curves, the the initial initial elastic elastic modulus modulus _?& Eo of of the the rock rock was was calculated, calculated, for for each each test, test, using using linear linear elastic elastic Table 1. I. fracture mechanics mechanics (LEFM) (LEFM) formulas formulas [I]; [I]; see Table fracture
IDENTIFICATION OF OF FRACTURE FRACTURE PARAMETERS PARAMETERS IDENTIFICATION The used to to determine determine the the material material fracture fracture parameters parameters from from the the test test The size size effect effect method method [2] is used data. of limestone limestone [3], as as well well as as other other data. The The method method has has previously previously been been verified verified for for the the fracture fracture of rocks of the the effect effect of of loading loading rate rate rocks and and concrete concrete [4,5]. [4,5]. Recently, Recently, itit has has also also been been used used in in aa study study of on on the the fracture fracture of of concrete concrete [6]. The The method method is based based on on the the size size effect effect law law [7], which which is: (IN=
Bfu
..j(l
+ fJ) , 393 393
fJ
= ~, uo
(1)
Z. P. BA2ANT BAZANT et 01. al.
394
p
T d
I
GAd
~------~------~~
I. ---4-:--- .1 4.5d
d-thlckneu Fig. 1. 1. Fracture Fracture specimen specimen geometry. geometry.
where Pu/bd == maximum where UN bN = PJbd maximum nominal nominal stresses stresses of of geometrically geometrically similar similar fracture fracture specimens, specimens, P beam depth), Puu = = maximum maximum load, load, d = = characteristic characteristic dimension dimension (chosen (chosen here here as the beam depth), b = = specimen specimen Bf. and parameters, and thickness thickness (constant, (constant, for for two-dimensional two-dimensional similarity), similarity), BS, and do do = = empirical empirical parameters, and P= brittleness brittleness number. P is very P~G 0.1), is almost independent of size, /I number. When When /3 very small (e.g. fl O.l), UN a,,, almost independent of as in plastic plastic limit ~ 10), limit analysis. analysis. When When P /I is large large (e.g. P b B lo), the size-dependence size-dependence follows follows LEFM LEFM In the transition (i.e. UNIX a,cc l/.}d). l/G). transition zone, zone, nonlinear nonlinear fracture fracture mechanics mechanics needs needs to be applied. applied. For parameters from data, For determining determining the parameters from UN oN data, eq. (I) (1) can can be transformed transformed to Y Y = AX + + C, where l/u~. Then, and where X = = d and and Y = = l/of,. Then, BJ.. BS, = = l/ft; l/fi and do 4 = = CIA C/A [4]. [4]. By linear linear regression regression analysis analysis of of the parameters and the data data for for the four four series of of tests, the parameters and coefficients coefficients of of variation variation of of errors, errors, (OYlx, oVX, have have been computed been computed and and are are listed listed in Table Table 2. The The data data and and the fits [eq. (I)] (l)] are shown shown in Fig. 3. It It can can be seen that that the size effect effect law represents represents the trend trend reasonably reasonably well, at all the loading loading rates. rates. It It is clear clear that that the data data cannot cannot be represented represented by either either LEFM LEFM (a straight straight line with a slope slope of of -1/2) - l/2) or Bf.). or strength strength criteria criteria (horizontal (horizontal line UN aN = Bf”). Using Bf. and parameters can 7]: Using the values values of of Bf, and do, do, fracture fracture parameters can be calculated calculated as follows follows [4,5, [4,5,7]: dog(r/..o)
Kic
KIc = Bf.J(dog(rJ..o», cf= g'(rJ..o) , Gf=p'
(2)
Table Table I. 1. Test data data Series Series
Dimensionst Dimensionst (mm x mm x mm) (mmxmmxmm) 457 x 102 102 x 13 13
Fast Fast
229 x 51 51 x 13 13 114 x 25 x 13 114x25x 13 457 x 102 102 x 13 13
Usual Usual
229 x 51 51 x 13 13 114 114 x 25 x 13 13 457 x 102 x 13 457X102X13
Slow
229 x 51 51 x 13 13 114 x 25 x 13 114x25~13 457 x 102 102 x 13 13
Very slow
229 x 51 51 x 13 13 114 x 25 x 13 114x25x 13
CMOD CMOD rate rate 6 mm/sec) (10(10-6mm/sec) 15,900 15,900 15,900 15,900 10,600 10,600 10,600 10,600 5770 5770 5770 5770
Peak load load (N) (N) 445 472 281 281 291 291 178 178 165 165
Time to peak peak (sec) (set) 2.1 2.1 2.2 2.0 2.4 2.4 2.2
Eot (GPa) 40 32 32 35 35 24 24 35 35 35 35
436 414 269 271 271 153 153 165 165
176 176 194 194 237 210 248 215
33 33 30 30 30 30 30 30 29 29 30 30
1.42 1.42 1.42 1.42 0.978 0.978 0.978 0.706 0.508
394 383 383 245 240 147 147 153 153
23,175 23,175 16,875 16,875 26,000 26,000 20,475 20,475 15,750 15,750 26,250 26,250
0.353 0.318 0.318 0.236 0.236 0.160 0.160
385 385 387 387 262 265 140 140 136 136
81,900 81,900 79,000 79,000 87,800 87,800 82,350 82,350 72,000 72,000 92,000 92,000
30 ,30 32 32 28 28 25 25 32 32 34 34 27 27 34 34 32 32 27 27 26 26 25 25
159 159 141 141 106 106 106 106 57.7 63.5 63.5
tLength tLength x depth depth x thickness. thickness. tInitia1 $Initial modulus modulus from from load-CMOD Ioad-CMOD compliance. compliance.
395 395
Fracture of of rock rock Fracture ~~-------------------------------.
~~-------------------------------.
,,= 51
(a)
=
100
2.1 *s 2.1 176 Is = 176 23200 Is = 23200 81900 ss = 81900
100
O.-----.---_,----~----_.----._--~
0.00
mm
0.01 0.01
c0.06 LO6
0.02 0.03 0.04 0.05 0.03 0.04 0.05 0.02 Crack Mouth t.loulh Opening Opening (mm) (mm) Crack
z
0.01 0.02 0.03 Crack Crack Mouth t.loulh Oprning Opening (mm) (mm)
O. 44
200.--------------------------------.
d = 25 mm
(c) 150
..-.
~ .., 100'
c
.3 2.4 •
215 •
26300 •
92000 • 0~------_.------~------_,------_4 0.000 0.005 0.010 0.015 0.020 o.obs o.oio O.dlS 0.
Crack Crack t.loulh Mouth Opening Opening (mm) (mm)
Fig. 2. Typical Typical load-CMOD IoadXMOD curves curves for each each specimen specimen size.
where fracture toughness, toughness, cf= c, = effective effective length length of of the fracture fracture process process zone, zone, and and G G,f == fracture fracture where KIc K,, = - fracture energy. energy. Function Function g(IX) g(a) is the non-dimensionalized non-dimensionalized energy energy release release rate rate defined defined by the LEFM LEFM relation relation G == P2 g (IX)IE'b 2d, where P2g(a)/E’b2d, where G == energy energy release release rate rate of of the specimen, specimen, P = load, load, IXa == aid a/d = relative relative crack = crack length, length, g'(IX) g’(a) == dg(IX)/dIX, dg(a)/da, a == crack crack length, length, IXo LY,, = aold, a,,/d, ao a, = notch notch length length of of traction-free traction-free crack crack 2 ) for plane length, length, E' E’ = E for plane plane stress, E' E’ = EI(1 E/(1 - vv2) plane strain, strain, E = Young's Young’s modulus, modulus, and and v = Poisson's Poisson’s ratio. ratio. Function Function g(IX) g(a) can be obtained obtained from from handbooks handbooks (e.g. [1]) [l]) or from from LEFM LEFM analysis. analysis. Fracture Fracture parameters parameters are defined defined here for the limiting limiting case of an infinitely infinitely large specimen specimen at failure. failure. Then, Then, an infinite-size infinite-size extrapolation extrapolation of of eq. (I) (1) provides provides material material parameters parameters [eq. (2)] (2)] that that are practically practically size- and shape-independent shape-independent [5]. [5]. Using the values g(IXo) g(a,) = = 62.84 and g'(IXo) g’(G) = = 347.7 (from (from [1]), [l]), and assuming assuming plane plane stress conditions, conditions, the fracture fracture parameters parameters for the four four series can be computed; computed; see Table Table 2, in which the average average values of K K,,1c and cf c/ as well as their their coefficients coefficients of of variation variation are listed. The E-value E-value for each series is is taken taken as the average average initial initial modulus modulus Eo, E,,, and is used in eq. (2) for computing computing G G,f (see Table Table 2). 2). VARIATION VARIATION OF FRACTURE FRACTURE PARAMETERS PARAMETERS
The The test test results show show that that as as the the time time to to peak load, load, ttp, increases, the fracture fracture toughness toughness Klc K,, p ' increases, decreases. decreases. Since Since the fracture fracture energy energy Gfis G, is proportional proportional to to Kin Ki,, its its decrease decrease with slower slower loading loading rates is is even even stronger. stronger. The The same same trends trends have also also been observed observed in in similar similar materials, materials, such such as as hardened hardened cement cement paste paste [8], [8], concrete concrete [6], [6], and and ceramics ceramics at at high temperatures temperatures [9]. [9]. To To describe describe the influence influence of of loading loading rate, we we follow follow several several other other investigators investigators by adopting adopting aa power power function function of of crack crack velocity velocity v: v: (3)
Z. P. BAlANT BtiAlUT et aI. al.
396
Table parameters Table 2. Fracture Fracture parameters
Series Series
lp Avg. tp (sec) (se4
Fast Fast Usual Usual Slow Very slow
2.3 213 21,400 82,500
Bf. (MPa) (2a)
(mm) (nil,
cony (QI1X
4Klc Avg. (MPaJiiiiii) (MPa@)
36.2 36.3 31.9 36.5
0.07 0.07 0.04 O.ll 0.11
33.1 33.1 30.8 27.5 28.2
do
0.693
0.645 0.614
0.589
cf
Avg. (mm)
o
(Q
0.13 0.12 0.08 0.19
6.5 6.6 5.8 6.6
o
E, Avg. Eo (GPa) @Pa)
0.19 0.19 0.12 0.28
33.5 30.3 30.2 28.5
(Q
Gf
(N/m) 0%) 32.7 31.3 25.0 27.9
0 = coefficient coefficient of of variation. variation.
(Q
where reference velocity, where ~ & is the the fracture fracture toughness toughness corresponding corresponding to to a reference velocity, vo, v,,, chosen chosen here here as Vo = 0.01 mm/sec. Since the effective (LEFM) crack tip is roughly at a distance cf from the notch v. mm/set. effective (LEFM) crack tip roughly distance c, from notch we use the approximation tip at the peak peak load, load, the approximation (4) (4)
v = q/t,.
Then, Then, by fitting fitting the the test results results with with eq. (3) (see Fig. 4), we obtain obtain n = 0.0173 and and ~ = 30.0 MPay'niffi. Note that, alternatively, beam deflection or crack & MPa@. Note that, alternatively, beam deflection or crack opening opening rates rates have have been used used instead been instead of of v in other other studies. studies. In similar similar tests of of concrete concrete [6], [6], it was found found that, that, with with an increase increase in time time to to failure, failure, the the group group of data for the three sizes of specimens shifts to the right, i.e. toward the LEFM asymptote, of data for three of specimens shifts to the right, toward the LEFM asymptote, when when Ip, the process process zone plotted as in Fig. 3. This plotted This implies implies that, that, for for higher higher tp, zone length length cf c, decreases decreases and and the the p [eq. (I)], brittleness of brittleness of failure, failure, characterized characterized by fl (l)], increases. increases. present results Rather Rather interestingly, interestingly, no no such such trend trend is observed observed from from the the present results of of limestone. limestone. For For all Ip, the data part of tp, data remain remain within within the the same same part of the the size effect effect curve. curve. This This is reflected reflected by the the fact fact that that cf practically constant brittleness of c, is practically constant (cf (c, z~ 6 mm; Table Table 2), implying implying that that the the brittleness of fracture fracture in limestone limestone is rate-independent rate-independent within within the time time range range studied studied here. here. This This difference difference in the behavior behavior (for (for the present load present load durations) durations) from from concrete concrete may may be explained explained by the lack lack of of significant significant creep creep [10]. [lo]. Concrete bulk of Concrete exhibits exhibits marked marked viscoelastic viscoelastic creep creep in the the bulk of the test test specimen, specimen, as well as high nonlinear process zone. nonlinear creep creep in and and near near the the fracture fracture process zone. 4.0 0 . 0 , - - - - - - - - - . . . , - - - - - - - - - - - ,
t, = 2.3 sec
(a)
-0.' ‘:~
iii-O.2 ;f, -0.z
........ \ ~ t?
'-' x 01
.2-0.3 P 4.2
I
0s
,
Bf.
0.
-oh
do
mm
b
log ;
0.
\
a
-0.00 . 0 , - - - - - - - - - . . : - - - - - - - - - - . ,
= 0.645 MPa = 36.3 mm
\ -o.4+---r------.--~--_._-~ 0.5 a s -O.SO -0.25 Oh 0. -0:2!, 0.60
109 b
-0.0...-------..--------__
tp = 21400 21400 sec sac +, =
(c) (c>
(d)
=
82500 sec SIC tp = 82500
-0.'
-0.'
-0.1
.
.
,....
,....
iii -0.2
iii -0.2
........
........
~ .......
~
'-'
~-0.3
= 213 213 sec sac t, ‘P =
-0.'
do = 36.2
4.1
~o,-----____..:__------~
-o.’(b) (b)
01
0.614 MPa = 31.9 mm =
.2 -o.J
‘5
Fig. 3. Size effect curves peak load. curves at different different times times to peak load.
397
Fracture of of rock rock Fracture 34.,.----------------,
0.60,-----------------,
33
0.55
32
0.50
........
~
-E 31 E
00.45
•
limestone
0...
~0.40
030
0...
:::E
6
'""'29 .!!
0.35
K.c=Ko(v/voY·
::.::
n=O.OI13
28
0.30 0.30
for vo=O.Ol mm/s. /2 • Ko=30.0MPamm' 27 +m---.--rrrrrrrr-..--,-,rrrn.,,-..,....,.-m-.,.,,--,-rrrn-nr--.--1 0.0001
0.001 V
0.01
= c,/t
p
0.25
0 I
(mm/s)
I 1
I1111111 10 IO
I I1111111
100(
tp
y&c>
Fig. 4. 4. Variation Variation of of fracture fracture toughness toughness with with crack crack velocity. velocity. Fig.
I
I1111111
1000
sec) loo0
I 1111111, 10000 1OPOO
1111111 100000 100000
Fig. 5. Influence Influence of of specimen specimen size and and time time to to failure failure on on Fig. nominal strength. strength. nominal
EFFECT OF OF RATE RATE ON ON STRENGTH STRENGTH AND AND YOUNG’S YOUNG'S MOD~US MODULUS EFFECT Several investigators investigators have have demonstrated demonstrated that that the the strength strength of of rock rock generally generally increases increases with with an an Several [11, 12]). This is also observed here from Table 1. When the loading increase in the loading rate (e.g. increase the loading rate [ 11, 121).This also observed here from Table I. When the loading of magnitude, magnitude, the the maximum maximum nominal nominal stress stress decreases decreases by more more than than 16%. rate slows by by four four orders orders of rate 16%. is similar to the change in KIn has also been observed other This phenomenon, which has also been observed in other This phenomenon, which similar to the change K,,, may be attributed attributed to to the the statistical statistical nature nature of of the the failure failure of of molecular molecular bonds bonds materials [13]. It materials It may of thermal energies). (particularly the activation energy theory and the Maxwell distribution (particularly the activation energy theory and Maxwell distribution of thermal energies). of a quasi-brittle quasi-brittle heterogeneous heterogeneous material material is generally generally difficult difficult to to measure measure The strength strength of The of its dependence on specimen size and shape, and because failure does not objectively because objectively because of dependence on specimen and shape, and because failure does not at all points but is progressive. However, strength (or failure stress) occur simultaneously occur simultaneously at points but progressive. However, strength (or failure stress) is correlated to the fracture toughness since failure propagation; higher correlated to the fracture toughness failure occurs occurs by unstable unstable crack crack propagation; higher toughness implies higher resistance against failure. toughness implies higher resistance against failure. Equations Equations (I) (1) and and (2) can can be combined combined to to give the size effect effect on on the nominal nominal strength strength of the material fracture parameters [5]: (maximum nominal stress) in terms (maximum nominal stress) terms of material fracture parameters [5]: aN =
Kic -,-------=-=--.j(g'(r:J.O)Cf+ g(r:J.o)d)
(5)
Substituting Klc from Substituting for for -lu,, from eq. (3), and and cf cr from from eq. (4), one one obtains obtains a relation relation for for the the dependence dependence of of the the nominal nominal strength strength on on the the failure failure time:
Ku aN = .j(g'(r:J.O)Cf+ g(r:J.o)d)
(c)n vo~p .
(6) (6)
Since cf is not c/is not systematically systematically affected affected by the loading loading rate, rate, the average average value value of of 6.4 mm is considered. considered. Equation Equation (6) may may then then be plotted, plotted, along along with the test data, data, for the different different sizes tested tested (Fig. 5). The The agreement agreement is acceptable. acceptable. The The test test results results also indicate indicate that that the average average initial initial elastic elastic modulus modulus decreases decreases slightly slightly with an increase increase in the time time to peak peak load load (Table (Table 2). Such Such an effect effect has been been observed observed for several several rocks rocks in the the dynamic dynamic range range [14]. [14].
CONCLUSIONS CONCLUSIONS (1) For For times to peak peak load load ranging ranging from from 2 to 80,000 sec, set, the measured measured nominal nominal strengths strengths offracture of fracture of limestone agree with the size effect law. specimens specimens of limestone agree (2) The The fracture fracture toughness toughness and failure failure stress decrease decrease with increasing increasing failure failure time. However, However, the fracture fracture process process zone zone size and and the brittleness brittleness of of failure failure appear appear to be unaffected unaffected by the loading loading rate. rate. (3) Since there there is insignificant insi~ificant creep creep outside outside the process process zone zone of of limestone limestone in the time range range studied, studied, the effective effective process process zone zone size does does not not change change as the loading loading rate rate is varied. varied.
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Z. P. BAUNT BAZANT et al.
Acknowledgements-This work partially supported Northwestern University, Acknowledgements-This work was partially supported by AFOSR AFOSR contract contract 91-0140 with Northwestern University, and and Center for Advanced Advanced Cement-Based Cement-Based Materials Materials at Northwestern University (NSF (NSF Grant Grant DMR-8808432). DMR-8808432). S. P. Bai is the Center Northwestern University grateful grateful for support support from from ISTIS, Taiyuen, Taiyuen, P.R.C., P.R.C., during during the course course of of this study. study.
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