Application of generalized logistic equation for b-value analysis in fracture of plain concrete beams under flexure

Accepted Manuscript Application of Generalized Logistic equation for b-value analysis in fracture of plain concrete beams under flexure Nitin B. Burud, J.M. Chandra Kishen PII: DOI: Reference:

S0013-7944(18)30673-8 https://doi.org/10.1016/j.engfracmech.2018.09.011 EFM 6146

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Engineering Fracture Mechanics

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4 July 2018 6 September 2018 6 September 2018

Please cite this article as: Burud, N.B., Kishen, J.M.C., Application of Generalized Logistic equation for b-value analysis in fracture of plain concrete beams under flexure, Engineering Fracture Mechanics (2018), doi: https:// doi.org/10.1016/j.engfracmech.2018.09.011

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Application of Generalized Logistic equation for b-value analysis in fracture of plain concrete beams under exure ∗ Nitin B. Burud, J. M. Chandra Kishen Dept. of Civil Engineering, Indian Institute of Science, Bangalore, India

Abstract This work investigates the eect of nite size and evolution of fracture process zone on bvalue of acoustic emission in concrete beams. It is shown that the AE b-value determined using the Gutenberg-Richter (GR) law varies with beam size and invalidates universality. Since GR law ts the frequency-magnitude curve partially, a recently proposed Maslov's generalized logistic equation (GLE) is used for b-value analysis.

The existence of cut-o

magnitude and nonlinearity of frequency-magnitude distribution is explained through crack interaction mechanisms occurring within the FPZ. The growth of micro and macro cracking in concrete is observed to follow logistic or sigmoid growth law rather than the power law. The b-values obtained from GR and GLE are compared and it is found that the b-value of GLE correlates well with the eective crack length during damage process of plain concrete thereby exhibiting damage compliant behavior which could be used in health monitoring of structures.

Keywords:

Acoustic emission, plain concrete, damage, b-value, generalized logistic

equation, structural health monitoring

∗

Corresponding author Email addresses:

Kishen)

[email protected] (Nitin B. Burud), [email protected] (J. M. Chandra

Preprint submitted to Engineering Fracture Mechanics

September 5, 2018

1

Nomenclature

a0

notch depth;

ae

eective crack length;

bGR bGLE

constant in GLE;

D

overall beam depth;

beam span ;

m

magnitude of events;

mu mN max

Nmax s

upper cut-o magnitude limit; cut-o magnitude at N-th event; number of events; maximum number of events at failure of beam; rate parameter of GLE;

t0

time of 100th event (considered as initial time);

ti

time of i'th event;

α

shape parameter of GLE;

∆m ∆t σ

4

mode-I stress intensity factor;

L

N

3

b-value using GLE;

C

KI

2

b-value using GR law;

magnitude binning interval; time binning interval; stress;

1. Introduction

Gutenberg-Richter

(GR) law [1] has ruled the earthquake statistics for decades after its

5

inception and still continues to do so, not only in seismology but in many other elds too.

6

With the ease in application, it represents a nonlinear chaotic system to appear somewhat

7

linear and systematic. In fact, at any given time instant, it provides a snapshot of a dynamical

8

system in the form of an

9

frequency-magnitude relation of earthquake events and the exponent of this power law is

10

known as

b-value.

ogive.

The GR law is regarded as a power law, characterizing the

The existence of power law entails 2

self similarity

and

scaling

in the

11

emerging processes like earthquake. Furthermore, b-value tends to a constant value (b

12

near criticality exhibiting

13

generic for applications due to its manifestation of universality and self similarity.

14

universality.

≈ 1)

Consequently, GR law becomes more profound and

Universality establishes a link between dierent complex processes participating in an

15

emerging system to an universal scenario (yet unknown) at critical transformations.

Self

16

similarity allows an extrapolation to predict the crucial large magnitude events from smaller

17

ones. Although, universality and self similarity are relevant properties, many researchers have

18

observed deviation in the cumulative frequency distribution (CFD) contradicting pure GR

19

law. The oversimplication of CFD by GR law does not account for small magnitude events

20

and it also disregards large magnitude cut-o. The cut-o magnitude and log-concavity are

21

important features of CFD and should not be neglected but rather accounted for in the b-

22

value analysis. In the present work, the log-concavity along with cut-o magnitude of CFD

23

are addressed together as nonlinearity of CFD in a general sense signifying nonlinear shape

24

on both semi-log and linear scales.

25

As GR law was rst introduced for statistical analysis of earthquakes and applied without

26

any modication to acoustic emission due to its reasonable interpretation of this phenomena

27

[2], it is necessary to review its development in seismological studies. Therefore, the present

28

work mostly reviews seismological and geophysical literature. Mainly, three dierent types of

29

approaches are followed in literature to improve b-value estimation: 1) employing statistics

30

based formulation by accounting nonlinearity while preserving pure GR law, 2) modifying

31

GR law at large magnitudes by associating it with a polynomial or exponential or logarithmic

32

cut-o and 3) suggesting an alternative model or generalizing GR law to consider the shape

33

of distribution. By idealizing the occurrence of earthquakes as a Poisson process, the rst

34

approach with validity of GR law has yielded various statistical b-value estimates such as

35

maximum likelihood estimate or Aki's formula [3], improved b-value [4], Weichert's formula

36

[5], generalized Aki-Utsu

37

robust tting [7] has been proposed which is shown to be more ecient than maximum

38

likelihood estimate. Cut-o on maximum magnitude approach has resulted in many of upper

39

bound models such as tapered GR, truncated Gamma law, truncated GR, etc. The third

40

approach has not received signicant contribution except for a few which are noted further.

β -estimator

[6] and few others. An alternate approach based on

3

41

The recent development in Tsallis nonextensive statistical mechanics has oered a generalized

42

form of GR law [8, 9, 10] based on Tsallis entropy. A Baysian approach [11] is also notable

43

in the category of generalized GR law.

44

an approach based on logistic equation proposed by Maslov et al.

45

its fractional power-law exponent. This two parameter model regards nonlinear CFD over

46

whole magnitude range with soft cut-o for larger magnitudes. The overall shape of CFD

47

suggest that the growth of microcracks is logistic or sigmoid rather power law growth. The

48

b-value obtained from logistic equation exhibits damage compliant behavior as concluded

49

in the present work. Therefore, the present work details application of generalized logistic

50

equation for b-value analysis of acoustic emission in single edge notched (SEN) plain concrete

51

beams.

52

Acoustic emission

Apart from the above mentioned contributions, [12] is distinct due to

phenomena has a close resemblance to earthquakes [13] despite volu-

53

minous contrast on the temporal, spectral, spatial and energetic scales [14]. The acoustic

54

emission (AE) is a non-destructive technique and dierent features of it have been used for

55

structural health monitoring (SHM) [15, 16, 17, 18, 19, 20]. It is an acoustic based passive

56

technique relying on stress waves generated by cracking of material.

57

has been greatly inuenced by seismological studies and nonetheless, AE has contributed

58

in understanding the complex earthquake process through laboratory experiments on rocks.

59

The ubiquitous GR law has been applied to AE and determination of b-value has became

60

an essential part in most of the AE studies.

61

Development of AE

Application of GR law for b-value analysis is usually carried out to assess the progress

62

of damage in a structural element.

Convergence of b-value approximately towards unity

63

is considered as a sign of near failure. Qualitative drop of b-value with increasing stress is

64

consistent throughout the damage progress but quantitatively it is still uncertain to associate

65

b-value to a specic damage level. Furthermore, according to fracture mechanics, the crack

66

size depends not only on stress but also on stress intensity at crack tip as the crack propagates.

67

Therefore, if b-value is only correlated to stress then the conjecture of micro and macro crack

68

distribution associated with b-value is also ambiguous (same b-value does not imply same

69

crack size distribution). Most importantly, GR distribution ts only to the tail portion of

70

CFD neglecting small magnitude events, thereby making the b-value as a biased indicator. 4

71

On the other hand, the prominent size eect in concrete has been attributed to varying

72

fracture process zone size.

73

indicates widely distributed microcracks around the main crack. Therefore, large size beams

74

should result in higher b-value due to relatively higher microcracking compared to small size

75

beams.

76

In this work, the nonlinearity of CFD is dealt by employing Maslov's generalized logistic

77

equation (GLE) with two parameters.

78

Thus, the increasing width of fracture process zone with size

Such eect of nite size on b-value has not yet been discussed in the literature.

The present work is organized as follows.

Section 2 introduces Gutenberg-Richter law

79

followed by a brief discussion on the universality of b-value.

Maslov's generalized logistic

80

equation is introduced in Section 3. The b-value is a parameter which represents FPZ in the

81

statistical domain and hence fracture process in concrete and evolution of FPZ is described in

82

Section 4. Experimental details with analysis techniques used in the present work are given

83

in Section 5. The test results of geometrically similar beams that are tested are described in

84

Section 6. The details of fracture process zone as an interactive phenomenon of micro and

85

macro crack is thoroughly discussed with experimental observation in Section 7.1 and the

86

nonlinearity of CFD and existence of cut-o magnitude is justied. Application of GLE to

87

AE data acquired from tested beams is demonstrated in Section 7.3. The damage compliance

88

behavior of b-value obtained from GLE is shown in Section 7.4. The conclusions arising from

89

this study are briefed in Section 8.

90

2. Gutenberg-Richter Law and the universality of b-value

91

The ubiquity of Gutenberg-Richter (GR) law in many natural phenomena has been stud-

92

ied and conrmed. Although it was fundamentally proposed for earthquake occurrences in

93

seismology, it has been adopted and applied in many other areas too. Despite ubiquity, GR

94

law has received criticism due to its broad generality which makes it dicult to validate in

95

many other situations. In fact, in seismology itself it holds true only for the nite range of

96

magnitudes [21, 22]. The GR law can be expressed as follows,

log10 (N ≥ m) = a − bm

5

(1)

where N is the number of events greater than magnitude

97 98

and

b

m, a

is referred as productivity

is the negative slope of CFD over magnitude plotted on log-linear scale.

99

The GR law has been applied extensively for b-value analysis of AE events. However,

100

GR law partially represents the CFD of AE. There are possible reasons for it being partially

101

applicable to AE data, one of which is the incompleteness of small magnitude data as a con-

102

sequence of thresholding used during its acquisition. Further, saturation of large magnitude

103

events results in maximum amplitude cut-o, which the GR law fails to consider. Neverthe-

104

less, the CFD of AE is not linear and requires another model which can eectively describe

105

it.

106

seismology. In seismology, the b-value is used to predict the large (largest) size earthquake

107

by extrapolating smaller earthquakes for which existence of universality can be considered

108

useful. On the other hand, AE in structural health monitoring (SHM) is required to estimate

109

damage progress and not to extrapolate large size events and therefore small variations in

110

shape of CFD should also be regarded. This basic dierence in application of b-value in both

111

seismology and SHM is important and needs to be emphasized. The universality of b-value

112

is a topic of debate for several years and many studies have supported its existence [23, 24].

113

The present work provides evidence for non-existence of universality by an experimental

114

study using acoustic emission.

The universality of b-value is a conviction followed by AE community borrowed from

115

The universality of b-value has been a conundrum for decades and its observed variabil-

116

ity has added further dilemma among researchers. As a measure of variation in power law

117

distribution, the b-value obviously depends on the underlying physics of the system. There-

118

fore, the universality of b-value indicates that dierent emerging systems have the same

119

universal physical phenomenon behind them.

120

factors aecting b-value have been recorded and studied by many seismologists [25]. Such

121

studies are found to be scarce in the literature dealing with AE of concrete. Factors aecting

122

b-value in concrete fracture include heterogeneity[26, 27], stress level [4, 28, 29, 30], shape

123

of the tail distribution aected by cut-o magnitude [31, 32], dierent methods of b-value

124

evaluation [3, 4], sample size [33], magnitude binning [34, 3, 35, 36] and most important of

125

all is the specimen size. These factors can be broadly distinguished as numerical, material

126

and geometry dependent. The b-value dependence on the numerical factor can be reduced 6

Whether universality exists or not, several

127

by selecting appropriate sample size and binning magnitude.

128

dependence can not be avoided but can be quantied. Although, the major problem with

129

b-value analysis is more elemental, it needs to be addressed based on the physics of the

130

problem. This elemental problem is nothing but the validity of GR law itself. Therefore, the

131

present work advocates use of generalized logistic equation for b-value evaluation instead of

132

GR law. Besides application of GLE, the eect of nite size is also investigated by testing

133

geometrically similar concrete beams of three dierent sizes under exure.

134

3. Generalized Logistic Equation

135

The material and geometry

Recently, Maslov et al. [12] proposed a new equation to study the statistics of earthquake

the generalized logistic equation ".

136

distribution by calling it as "

137

generalized logistic equation can be written in the following form, −s

P (x) = 138

Ce 1−α x

A special case of Maslov's

1−α

−s

1 + Ce 1−α x

f or

1−α

0 ≤ x < ∞ and 0 ≤ α < 1

(2)

Equation 2 is not a perfect cumulative distribution function but can be considered ap-

C, where C

proximately as CFD for suciently large

140

of function

141

than

142

magnitude of earthquakes, the amplitude of acoustic emission events, etc.).

143

C/(1+C) for x=0

144

tor of Equation 2 is a normalizing function (similar to partition function used in statistical

145

mechanics). Therefore, in order to arrive at an analogous GR law expression, removing the

146

denominator and replacing variable

x

and

P(x). P(x) x

is a constant and

α and s

139

is an approximation to the number of all elements with size greater

is a representative variable of the size of an element in a structure (e.g, the

to

0

for

x→∞.

For suciently large

x

by magnitude

−s

N (m) = Ce 1−α m

147 148

are parameters

m

P(x) varies from

C (C/(1 + C) ≈ 1), the denomina-

results in,

1−α

(3)

Equation 3 can also be obtained as a rst-order approximation of Equation 2 by power expansion. Equation 3 can be written in a logarithmic form as,

log10 N (m) ≈ log10 C − 7

s m1−α log10 e 1−α

(4)

149

Equation 4 is analogous to the Gutenberg Richter law shown by Equation 1 for

150

and

151

cumulative event number

152

α

153

and

154

equation and equivalent to the GR law can be written as,

s=b.

The above equation considers the nonlinear relation between magnitude

N(m)

m

can also be thought of as a penalty on GR law due to nonlinear relation between

m.

and

contrary to linear relation assumed in GR law. The role of

N(m)

b

Therefore, the new b-value ( GLE ) based on approximation of generalized logistic

bGLE = 155

α=0, a=log10 C

s log10 e 1−α

(5)

As Equation 4 is simply a rst order approximation of Equation 2, for b-value analysis,

s

and

α

156

the estimation of parameters

should be performed using Equation 2 itself. Figure 1

157

shows the behavior of Equation 2 with respect to varying parameters

158

log-linear scales for comparison. The variation of

159

value of s=3.5. The head and tail portion of the curves are dominated by

160

smooth transition from small to large magnitudes. It is evident that larger

161

fewer small magnitude events while small

162

magnitude events. Figures 1b and 1c are plotted for two dierent values of

163

against varying s-values.

164

scale is zero indicating an uniform distribution of events of all magnitude. Increasing s-value

165

reduces the contribution of large magnitude events in CFD therefore the slope of the curve

166

is dominated by s-value.. It follows that the s-value is a rate parameter and

167

parameter of GLE model. Consequently,

168

and cut-o magnitude of CFD. The GLE model sets a soft limit over large magnitude events,

169

analogous to Gamma law advocated by Kagan [37, 38]. Soft magnitude cut-o considers the

170

possibility of large magnitude events with progressively smaller probabilities compared to

171

GR law and therefore it allows smooth transition of the dissipative dynamic system.

α

α

and

s

on linear and

is plotted in Figure 1a for a constant

α

which ensures

α

accounts for

α accounts for the relatively large number of small α (α=0, α=0.5 )

For s=1 in Figure 1b, the slope of the curve on the log-linear

α

α

is a shape

is the parameter which accounts for nonlinearity

172

As discussed above, GLE appears to be an appropriate candidate for b-value analysis.

173

However, before moving to b-value evaluation using GLE, it is necessary to briey introduce

174

fracturing process and fracture process zone of quasi-brittle material such as concrete.

8

α= 0 . 0 α= 0 . 4

F o r s = 3 .5 a n d C = 1 0 0 0 α= 0 . 1 α= 0 . 2 α= 0 . 5 α= 0 . 6

α= 0 . 3 α= 0 . 7

3

P (m -m

0

)

(C *P (m -m

0

))

1 .0

0

lo g

1 0

0 .5

F o r C = 1 0 0 0 a n d α=0

0 .0 0

F o r C = 1 0 0 0 a n d α=0

2 M a g n itu d e

4

0

2 M a g n itu d e

4

(a)

s = 1

F o r α= 0 a n d C = 1 0 0 0 s = 2 s = 5 s = 7

1 .0

s = 1 0

0

))

3

(C *P (m -m

) 0

-3

1 0

0 .5

lo g

P (m -m

0

F o r C = 1 0 0 0 a n d α=0.5

-9

-1 2

0 .0 0

-6

2 M a g n itu d e

4

F o r C = 1 0 0 0 a n d α=0.5

0

2 M a g n itu d e

4

(b)

s = 1

F o r α= 0 . 5 a n d C = 1 0 0 0 s = 2 s = 5 s = 7

(C *P (m ))

3

0 .5

lo g

1 0

P (m )

1 .0

s = 1 0

0 -3 -6 -9

-1 2

0 .0 0

2

M a g n itu d e

4

0

2 M a g n itu d e

4

(c)

Figure 1: Plot of Equation 2, a) for C=1000, s=3.5, varying α=0 to 0.7, b) for C=1000, α=0, varying s=1 to 10, c) for C=1000, α=0.5, varying s=1 to 10 9

175

4. Evolution of fracture process zone in quasi-brittle materials

176

Although resemblance between occurrence of earthquakes and AE events does exists, the

177

mechanics behind both of these phenomena have a vast dierence. The earthquakes originate

178

from fault lines due to sliding movement of tectonic plates. On the other hand, the devel-

179

opment of FPZ in quasi-brittle material, like concrete, originates from cracking of material

180

due to local tensile stresses. The area of fault plane is considered as an invariable, in seis-

181

mology. Whereas, the volume of fracture evolves in size during fracture process of concrete.

182

In fact, volume of the FPZ depends on structure size, geometry and loading conditions to

183

which the material is subjected. Therefore, it is necessary to understand the evolution of

184

fracture process in concrete to appreciate b-value variation in AE. Fracture process zone is

185

a cluster of micro and macroscopic cracks around the crack tip produced by stress redis-

186

tribution and energy dissipation mechanism due to underlying heterogeneity as shown in

187

Figure 2. In quasi-brittle materials like concrete, the heterogeneity primarily arises due to

188

its constituents of varying particle sizes from ne to coarse aggregates and it is further ele-

189

vated by discontinuities and pores developed during the hydration of cement. The randomly

190

dispersed discontinuities and pores play the role of probable sites for crack initiation due

191

to stress concentration. At the crack initiation stage, microcracks develop and coalesce due

192

to applied stress and form a macrocrack near the crack tip oriented perpendicular to the

193

principal tensile stress direction. After crack initiation, the fracture mechanism becomes an

194

interactive process between various crack sizes [39]. Microcracks usually occur around the

195

main crack and their density reduces as the distance from the crack tip increases. Otsuka

196

et al. [40] studied the development of FPZ in concrete under tensile loading using AE with

197

X-ray and proposed the existence of a sub-zone within the FPZ and named

198

(FCZ) as shown in Figure 2. Similar behavior by considering AE energy is recorded in [41].

199

FCZ is an area of densely distributed microcracks which further develops into a main crack.

200

Such dispersed microcracking around a macrocrack oers a bi-fold contribution to fracture

201

process by relieving the stress singularity as a consequence of stress redistribution which is

202

accompanied by energy dissipation.

203

main crack tip often documented as toughening mechanism by microcracks. Not all, but the

204

majority of microcracks participate in shielding while others amplify stress intensity at the

fracture core zone

This bi-fold contribution adds shielding eect at the

10

Aggregates Fracture core zone Fracture process zone

Notch

Figure 2: Schematic diagram of FPZ and FCZ. FCZ is shown in red shaded region 205

main crack tip. Microcracks which are far away from the FCZ do not interact signicantly

206

with the main crack and remain idle after dissipation of energy.

207

collective mechanism of interaction between micro-micro and macro-micro cracks. The in-

208

teractive process of dierent crack sizes is further elaborated with experimental observations

209

in Section 7.

210

5. Experimental program

211

5.1. Experimental setup

Therefore, shielding is a

212

An experimental program is designed to study the b-value behavior of geometrically

213

similar plain concrete beams of three dierent sizes. The geometrically similar notched plain

214

concrete beams are casted from the same concrete mix. The mix design of concrete is done

215

using the ACI method and the mix proportion of the cement, ne aggregate and coarse

216

aggregate obtained is 1:1.86:2.61 by weight.

217

mm and the ne aggregates (river sand) pass through 4.75 mm sieve. A water to cement

The maximum size of coarse aggregate is 12

11

Table 1: Details of beam dimensions Designation Small

Depth

Width

Span

Notch

(mm)

(mm)

(mm)

depth(mm)

75

50

337.5

15

Medium

150

50

675

30

Large

300

50

1350

60

Load

D

200 mm a0=0.2D L=4.5D - AE sensors on front face - AE sensors on rear face Figure 3: Beam dimensions with AE sensor location for large beam

218

ratio of 0.5 is used for preparing the concrete mix. Table 1 gives the geometrical details of the

219

beams. A computer controlled servo hydraulic machine is employed for testing the beams in

220

exure under three-point loading using crack mouth opening displacement (CMOD) control.

221

Monotonic loading rate is set to 1µm/sec for all specimens. Midpoint deection of beams is

222

measured using a linear variable dierential transformer (LVDT), while the load is recorded

223

using a load cell of 35 kN capacity. Three specimens are tested for each size of beam.

224

A Physical Acoustic Corporation (PAC) system is used to monitor the acoustic emission

225

throughout the test. Six resonant type R6D AE sensors are mounted on beams as shown

226

in Figures 3 and 4. R6D sensors having sensitivity and frequency response over the range

227

of 35 - 100 kHz with resonant frequency around 55 kHz are used. AE sensors are attached

228

to concrete surface using vacuum grease as a couplant and adhesive tape is used to ensure

229

xity of sensor to the surface.

230

dB gain was set for signal amplication. A threshold limit of 40 dB was set for background

Due to weak strength of AE signals, preamplier with 40

12

Figure 4: Beam with AE sensors, LVDT and CMOD gauge 231

noise reduction. Sampling rate of 1 MHz was used to ensure good time and signal frequency

232

resolution.

233

length as a function of loading, digital image correlation (DIC) technique is adopted. The

234

surface of the beam specimen is sprayed with speckles of a black paint and images of the

235

beam are captured continuously during the loading stages by a digital camera mounted on

236

a tripod.

237

5.2. Analysis of AE data

238

Signals below the threshold level are neglected.

In order to obtain the crack

Continuous monitoring of the beams up to failure by using AE and DIC techniques re-

239

sulted in a large data generation which are analyzed.

AE can be thought of as a passive

240

acoustical microscope which provides a glimpse of the internal cracking mechanism of the

241

material. On the other hand, a whole-eld based DIC technique provides surface information

242

regarding deformation/strain development and cracking due to the applied loading. Accord-

243

ing to the AE terminology, AE signal acquired by a single sensor is called as a hit and if

244

the same hit is acquired by multiple sensors then it is counted as an event. Events can be

245

localized in space by using triangulation method based on dierences in arrival time of the

246

stress waves.

247

source, though the possibility of multiple crack occurrence can not be denied. The amplitude

An event is considered to be originated from a single micro or macro crack

13

248

of AE signal is considered to be correlated to the microcrack volume and therefore often used

249

as an alternative to AE energy. In the present work, the amplitude of hit acquired by the

250

rst sensor, in a group of sensors which acquired the event, is considered as the source am-

251

plitude. The magnitude scale is derived by subtracting threshold amplitude of 40 dB from

252

the acquired raw source amplitudes of stress waves ranging from 40 dB to 99 dB and then

253

dividing it by 20 makes it useful for b-value analysis. The factor 20 accounts for conversion

254

from voltage to decibel scale with reference voltage of 1µV referred to the preamplier input.

255

For example, 65 dB amplitude will be denoted as 1.25 ((65-40)/20) on magnitude scale. The

256

recorded magnitude for each event are further used for the b-value analysis. The referred

257

meaning of magnitude in AE is straight forward and should not be confused with dierent

258

magnitude scales used in seismology.

259

There are two methods for windowing on time series for the b-value analysis usually fol-

260

lowed in the literature: i) instantaneous b-value by considering a xed length of time/event

261

sliding window and, ii) long-term b-value by considering all events occurred before a particu-

262

lar time instant. The long-term b-value is preferred as it represents overall b-value variation

263

throughout the test duration. If

264

events occurred during interval

265

time

266

any time instance i , the number of events is always greater than 100 due to the cumulative

267

eect.

268

Magnitude binning also plays an important role in the accuracy of b-value and therefore the

269

magnitude interval is kept as low as possible i.e.

270

1 dB with respect to source amplitudes). Equation 2 is then tted to CFD using particle

271

swarm optimization and parameters

272

ticle swarm optimization is the authors personal preference although any other optimization

273

technique can be used as an alternative for parameter estimation. The obtained parameters

274

are then substituted in to Equation 5 to determine

275

log-linear scale is not required for evaluation of

276

CFD nonlinearity through exponent

277

which is often emphasized on log-linear scale.

ti .

The time

t0

t0

t0

is the time of test initiation, then at any time

to

ti

ti

all the

should be used to evaluate the long-term b-value at

is considered as the time of occurrence of 100th event and therefore at

t

Increment of 100 events is considered such that

ti = 100.i + 100, f or

∆m = 0.05

(magnitude binning interval is

si and αi are determined for incremental ti .

α.

bGLE

bGLE .

i = 0, 1....

Use of par-

Transformation from linear to

as Equation 2 implicitly incorporates

Therefore, GLE discards the CFD linearity pretext

14

278

DIC analysis is performed on images captured by high resolution camera using GOM

279

correlate software package [42]. Successive images are captured at random intervals during

280

load application. The crack tip determination is achieved manually by varying strain and

281

displacement threshold on experience basis.

282

6. Results of mechanical testing and acoustic emission

283

The load versus crack mouth opening displacement (CMOD) response for all the three

284

sizes of beams tested is shown in Figure 5a. The shaded region depicts the scatter observed

285

in the response for three specimens of each size.

286

the area in between curves of each specimen of the same size.

287

obtained by averaging the variable over x axis.

288

crack propagation resulting in the development of FPZ which is evident from the softening

289

behavior exhibited by the beams in the post-peak regime. Similar scatter is also seen in the

290

acquired cumulative AE events as shown in Figure 5b. Table 2 lists the observed statistics

291

of AE hits and events during progressive cracking for each specimen.

292

of cumulated event over magnitudes is shown in Figure 6a at various stages of damage

293

progress. Figure 6b represents the same CFD on log-linear scale. The GR b-value is usually

294

determined by least square tting of CFD on log-linear scale. The slope of the tted line is

295

called as the b-value. Along with nonlinearity, another noticeable feature of CFD is the cut-

296

o magnitude denoted by vertical dotted lines in Figure 6b. The cut-o magnitude is dened

297

as the highest AE magnitude observed at any instance of damage. The log-linear plot clearly

298

exhibits the incremental cut-o magnitude (mmax ) with the increasing number of events and

299

as an example shown in Figure 6b for

300

upper cut-o limit, irrespective of the beam size at 99dB as seen in Figure 6b denoted by

301

mu .

302

this upper limit is crossed, the CFD curve starts shifting upward indicating accumulation

303

of increasing number of events with magnitude

304

related to material properties at extreme stress conditions. The localized events are shown

305

in Figures 7 and 8 with data density and color mapped bubble plot showing magnitude

306

of events in decibels. The images analyzed through DIC are shown in Figure 9 at various

N=100

The scatter plot is obtained by lling The average dark line is

The use of CMOD control allows stable

by

=100 mN max .

The typical CFD

There is also an evidence of

No event has been observed throughout the experiment above magnitude

15

mu .

mu

and once

Such upper limit on magnitude can be

7

M e d iu m

6

L a rg e

3 0 k

5 4 3 2 1 0

0 .0

0 .2

0 .4 C M O D

0 .6 (m m )

0 .8

S m a ll M e d iu m

C u m u la tiv e E v e n ts

S m a ll

L o a d (k N )

8

L a rg e

2 0 k

1 0 k

0

1 .0

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

T im e ( s e c s ) (b)

(a)

Figure 5: a) Load vs. CMOD and b) Cumulative AE events for three dierent size specimen (shaded area shows scatter while dark line shows average of the variable plotted) Table 2: Statistics of AE Events and Hits Size

Small

Medium

Large

Specimen

Events

Hits S1

S2

S3

S4

S5

S6

B1

9008

28938

23766

18254

26175

-

-

B2

1388

21304

18358

7491

4446

-

-

B3

10651

23793

26666

28069

25416

-

-

B1

7956

12386

9876

12902

8220

14809

9744

B2

19142

23797

25605

28247

26045

27555

39383

B3

16446

26226

24235

25772

18276

26526

12706

B1

26664

52979

40254

35464

25037

46583

45987

B2

27169

57772

44570

23043

39355

52371

50187

B3

34166

54582

46150

40312

48433

58827

52138

307

stages of loading history. The crack length and crack tortuosity is evident from the analyzed

308

images.

309

7. Discussion of results

310

7.1. Eect of nite size and crack interaction on fracture process zone (FPZ) evolution

311

The overall topology of the FPZ as shown in Figure 2 and described in the previous section

312

is well studied and understood in literature. There are two aspects of crack distribution in

313

FPZ which can be observed from AE. The rst aspect is the concentration or density of

314

the micro and macro cracks with respect to the main crack path. 16

Main crack in single

C u m u la tiv e F r e q u e n c y d itr ib u tio n

N = 1 N = 5 N = 1 N = 2 N = 4 N = 6 N = 8 N = 1 N = 1 N = 2 N = 2

2 5 0 0 0

2 0 0 0 0

1 5 0 0 0

1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 7 1 6 0 0 0 9

5 0 0 0

0 0

1

2

3

M a g n itu d e (a)

1 0 0 0 0

m

N = 1 0 0 m a x

m u

1 0 0

lo g

1 0

(C F D )

1 0 0 0

1 0

1 0

1

2

3

M a g n itu d e (b)

Figure 6: Cumulative frequency distribution at diernt damage instances, a)linear scale, b)log-linear scale, vertical dash lines shows cut-o magnitude at dierent damage instances (refer legend of gure (a) for gure (b) also) 17

L o a d

H e ig h t

L o c a liz e d A E e v e n t

L e n g th

D e n s ity 1 .4 0 5 E -0 4

3 0 0

1 .2 2 9 E -0 4 2 5 0

B e a m

h e ig h t ( m m )

1 .0 5 4 E -0 4 2 0 0

8 .7 8 1 E -0 5

1 5 0

7 .0 2 5 E -0 5 5 .2 6 9 E -0 5

1 0 0 3 .5 1 3 E -0 5 5 0

n o tc h

1 .7 5 6 E -0 5 0 .0 0 0

0 5 0 0

5 5 0

6 0 0

6 5 0

B e a m

7 0 0

7 5 0

8 0 0

8 5 0

le n g th ( m m )

Figure 7: Qualitative data density of localized events in a large size beam

18

3 0 0

B u b b le -

B e a m

h e ig h t ( m m )

2 5 0 -

5 5 d B - 6 2 d B -

6 3 d B - 8 4 d B -

8 5 d B - 9 1 d B -

9 2 d B - 9 9 d B

2 0 0

1 5 0

- M a g n itu d e 4 0 d B - 5 4 d B

1 0 0

5 0

0 0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

B e a m

7 0 0

8 0 0

9 0 0

1 0 0 0

1 1 0 0

1 2 0 0

1 3 0 0

le n g th ( m m )

Figure 8: Localized events in large size beams 315

edge notched beams under three point bending initiates at notch tip and progresses towards

316

the loading point.

317

events.

318

from Figure 7. Cracking is highly dense along main crack and reduces farther away from it.

319

The second aspect is the location and number of dierent sized cracks with respect to main

320

crack location. The bubble plot of AE localized events as shown in Figure 8 depicts events

321

of dierent magnitude by color mapped bubbles. It can be clearly seen that large magnitude

322

events are rare and are centrally located. Small magnitude events are dispersed abundantly

323

around large magnitude events.

324

order as the large magnitude events are scattered over small magnitude events to bring out

325

the contrast between various magnitude events. The events between 92-99 dB do not occur

326

frequently but instead occurs in relatively large intervals. Most of these large events have

327

occurred between 100-250 mm depth range in large size beam. There are few large events

328

visible at notch tip around 60mm depth due to sudden release of strain energy. In the top

Figure 7 shows color mapped kernel density contours of localized AE

The overall shape of FPZ with variation of its width along main crack is evident

The event locations in Figure 8 are not in chronological

19

Strain %

Main crack

Figure 9: DIC images at various load level in a large beam

20

329

range of 250-300mm, 92-99dB events are again rare. One can argue that at this range where

330

nal failure becomes catastrophic, large size events must be present.

331

events are less probable near failure due to compressive stresses caused by exure above the

332

neutral axis and reduced applied load. At this stage, the material is softened and does not

333

have sucient strength as well as volume to generate large size events.

334

However, large size

Although dispersed microcracks around centrally located macrocrack is an overall sim-

335

ple depiction of FPZ, a complex fracture process lies underneath.

A macrocrack is more

336

susceptible to growth due to its relatively large size and higher tensile stresses in the mid-

337

span location than the microcracks surrounding it.

338

has emerged, it will grow further and generate a chain of macrocracks of equivalent or big-

339

ger size up to failure.

340

macrocracks leads to unstable crack growth heading to catastrophic failure.

341

brittle materials, quasi-brittle materials like concrete exhibit stable crack growth due to the

342

toughening mechanism of microcracks as mentioned in Section 4. Interestingly, macrocracks

343

grow in more restricted environment than the microcracks around it. The rst restriction

344

on macrocracks is its interaction with neighboring microcracks during which the energy sup-

345

plied through loading is consumed, thereby limiting further growth of macrocrack. Secondly,

346

the macrocracks have dimensional restrictions as they grow centrally at the awakening of

347

two new surfaces and therefore their growth is limited in two dimensional space.

348

contrary, microcracks are dispersed in volume around the main crack which can grow in

349

three dimensional space. The third restriction arises due to the type of loading and stresses

350

to which the material is subjected.

351

compression stabilizes it. The energy dissipation during tensile loading is slow and stable

352

for quasi-brittle material compared to compression where it occurs in large bursts close to

353

failure when tested under CMOD or displacement control. During exural deformation of

354

the beams, the crack growth is stable with the combination of compression and tension.

355

The stresses ahead of crack tip in compression zone restricts further growth of macrocrack.

356

This restriction in the compression zone forces wider dispersion of microcracks which may

357

result in widening of FPZ. Small amounts of cracking might actuate in compression zone

358

too, but these do not lead to any failure in plain concrete beams. Compression zone crack

This implies that, once a macrocrack

This is a typical behavior of brittle materials wherein the chain of Contrary to

On the

Direct tensile stresses tend to accelerate cracks and

21

359

may not help tensile crack propagation directly as both of them are oriented normal to each

360

other. The compression zone cracks soften the material and help tensile cracks indirectly.

361

The softened compression zone might be the reason for observed scarcity of large magnitude

362

events (92-99 dB) within the top 50 mm range of the large size beams as shown in Figure 8.

363

Consequently, restricted macrocracks produce incremental magnitude cut-o during pro-

364

gressive damage. Similarly, the size and density of dispersed microcracks diminishes away

365

from the main crack as shown in Figures 7 and 8 ensuing a at plateau which is evident in

366

the CFD curve at small magnitudes (Figure 6). Inevitable thresholding and attenuation also

367

play an important role in acquiring less number of small magnitude events during acoustic

368

emission. In summary, large events experience cut-o while small events are undersized re-

369

sulting in nonlinear CFD dominated by intermediate size cracks and deviating from the GR

370

law. Gutenberg and Richter also [1] noted such deviation for large earthquake magnitudes

371

which further led to the upper bound power laws [38, 43].

372

The clear distinction between sizes of micro and macro crack has not yet been dened

373

rmly, and it is also experimentally infeasible to associate crack size with AE magnitudes

374

in concrete. Indeed, crack size itself depends on applied stress, stress intensity factor (SIF)

375

and geometry of the specimen. Therefore, a specic crack size at dierent stages of damage

376

progress will produce dierent magnitude events depending on SIF and stress condition

377

at that instant.

378

Consequently, the AE magnitude distribution progresses as a collective eect of applied

379

stress, SIF, boundary conditions, crack density and their orientation.

380

7.2. b-value analysis using GR law

381

The orientation of the crack also inuences the stress wave parameters.

The conventional b-value analysis using GR law is carried out on the AE magnitudes

382

obtained from beams of dierent sizes.

As beam size increases, eventually to dissipate

383

proportionate amount of applied energy, the volume of FPZ increases. The maximum width

384

of FPZ in a beam depends on maximum aggregate size and uncracked ligament length [40]

385

and therefore for the same maximum aggregate size, the increase in the width of FPZ solely

386

depends on the beam size as shown in Figure 10.

387

in Figure 10 is determined as the width of localized AE events in length direction at one

22

The maximum width of FPZ shown

µ ± σ,

389

if a data distribution is approximately normal then about 68 % of the data values are

390

within one standard deviation of the mean). Large volume of the FPZ accommodates large

391

number of microcracks due to its larger width and length. The b-value should reect such

392

relative change in the proportion of micro and macro cracks with size. Consequently, dierent

393

beam sizes should exhibit dierent b-values instead of converging to unity.

394

universality of the b-value (

395

eect on b-value has not been yet studied and discussed.

396

determined by least square tting (bLS ) and Aki's formula (bM L ) at nal failure for the three

397

beam sizes, where size dependence of b-value is evident. The

and

bM L

are obtained from

398

GR law and therefore these b-values will be collectively referred as

bGR

henceforth in the

399

present work. According to Carpinteri et al. [24, 44], the energy dissipation in disordered

400

materials takes place in fractal domain with dimension lower than 3, intermediate between

401

surface and volume.

402

relation

403

cumulative number of AE events and characteristic linear dimension has been proposed by

404

Carpinteri et al. [24]. The observed average variation of

405

1.29 and 1.33 resulting in fractal dimension of 2.3, 2.58 and 2.66 for small, medium and large

406

size beams respectively. Therefore, an alternate explanation for size eect on

407

fractal theory could be that the damage localization in exure of concrete beams is shifting

408

from two dimensional to three dimensional fractal space with increase in the size of specimen.

409

Such increment in fractal dimension is due to the widening of FPZ. Size eect on b-value may

410

appear insignicant, if absolute b-values are considered, due to small dierence between b-

411

values and the reason behind such insignicance is the low variability range of b-value itself.

412

The range of b-value for concrete has been reported in the range 0.5-1.5. Though b-value

413

variation for dierent sizes appears insignicant, the trend is considerable and needs further

414

detailed investigation.

D=3bGR /c

is mean and

σ

standard deviation from the mean (

b≈1 )

where

µ

388

c=1.5.

Accordingly,

should not hold true for dierent beam sizes. Such size

Aki [45] related the fractal dimension with

is standard deviation,

Similar relation for

23

bGR

Figure 11 shows the b-value

bLS

D

to seismic

bGR

through the

and fractal dimension based on

bGR

for the present study is 1.15,

bGR

based on

7 0

F P Z w id th ( m m )

6 0 5 0 4 0 3 0 2 0 1 0 5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

s iz e ( m m ) Figure 10: Width of FPZ vs. size of the specimen

2 .0

b b

L S

b - v a lu e

1 .5

M L

1 .0

0 .5 5 0

1 0 0

1 5 0

2 0 0

2 5 0

s iz e ( m m ) Figure 11: Eect of nite size on b-value near failure.

24

N o r m a liz e d

C F D

G R 1 0

3

1 0

2

1 0

1

f it

G L E f it

(N ) 1 0

1 .0

lo g

N o r m a liz e d N

2 .0

0 .0 0 .0

1 .0

2 .0

0 .0

M a g n itu d e

1 .0

2 .0

M a g n itu d e

(a )

(b )

Figure 12: GR vs GLE t for 1000 AE events on a) Linear scale and b) Log-linear scale 415

7.3. Application of GLE for b-value analysis

416

The applicability of GLE for b-value analysis is demonstrated in two steps in this section.

417

First, its superiority of tting the CFD compared to GR law is demonstrated by tting 1000

418

randomly sampled AE events from a tested beam. Then it is applied to the whole population

419

of events up to failure in chronological order to observe time evolution of damage.

420

parameter s-value obtained from GLE t is compared with GR b-value and their equivalent

421

behavior is illustrated. The b-values obtained from GR and GLE are not the same, in fact

422

it is argued that the

423

7.3.1. Application to randomly sampled 1000 events

bGLE

is more generalized and informative than

The

bGR .

424

Before proceeding to the application of GLE to tested beams, this section demonstrates

425

the comparative quality of tting by GLE and GR. To testify, 1000 randomly sampled

426

events are taken from the population of events acquired from a tested beam up to failure.

427

The samples are not consecutive in time rather sampled randomly to eliminate any bias. To

428

determine

429

For

430

obtain

431

the GR law does not t for the small magnitudes on a linear scale although it shows the

bGLE , Equation 2 is tted to the CFD (shown in Figure 6a) of these sampled events.

C=1000, the parameters s-value and α are obtained and substituted into Equation 5 to bGLE .

Figure 12 shows a comparative t of GR and GLE. One can observe that

25

8

L o a d (k N )

(a ) 6

M a x . L o a d 1 0 0 th E v e n t 4 2 0

(b )

1 .5

b - v a lu e

1 .4 1 .3 1 .2 1 .1

L S b v a lu e

1 .0

(c )

b - v a lu e

0 .9 6 0 .9 4 0 .9 2

A k i's b - v a lu e 0 .9 0 4 .4 4 .2 4 .0 3 .8 3 .6 3 .4

s - v a lu e

(d )

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

T im e ( s e c s )

Figure 13: a) Load vs. time, Comparison between b)Least square b-value, c) Aki's b-value and d)s-value of GLE

26

432

apparent t only on the log-linear scale.

The residual errors for large magnitudes appear

433

large while for small magnitudes appear small on the log-linear scale. Therefore, the least

434

square regression on the log-linear scale provides apparent t for the GR law and biased to

435

large magnitudes. On the other hand, GLE does not need the use of log-linear scale and

436

ts CFD on the linear scale itself without showing any magnitude bias. The superiority of

437

GLE t is clearly seen on both linear and log-linear scale. It is important to stress that the

438

GLE uses two parameters for tting CFD and perhaps it is an obvious reason for a better

439

t whereas GR law uses a single rate parameter.

440

Despite the simplicity of GR law, if it does not t small magnitude events with consid-

441

erable accuracy then the use of GR law should be avoided at least for AE in quasi-brittle

442

materials as far as damage assessment is concerned. Use of GLE is justiable due to its supe-

443

rior quality of tting CFD for small as well as for large magnitude events. Accounting small

444

magnitude events is necessary as they represent microcracking. Microcracking is a benet

445

oered by quasi-brittle material for reducing crack severity and it should not be overlooked.

446

Hence, microcracking is an essential part of the FPZ development and neglecting microcrack

447

contribution to b-value can result in partial information regarding damage progress.

448

7.3.2. Application to AE events acquired up to failure of a beam

449

As superiority of GLE over GR law is demonstrated in the previous section, its application

450

for damage analysis of beams is explored further. Equation 2 is used to t CFD of AE events

451

at various time instances during the progress of damage in the beams. The constant

452

taken as

453

coecient of determination not less than 0.99. The time of 100th event is indicated after

454

which the b-value analysis is performed to avoid sample size bias.

455

progress of

456

of its parameter, the s-value with

457

bLS

458

s-value is negatively correlated to stress (as GR b-value is negatively correlated to stress).

459

Consequently, the s-value can be considered as a representative of the CFD slope although

460

not being its physical slope (slope of a curve varies at every point and thus the s-value

and

C = N (0)

bGLE

bM L .

C

is

at each time instant and other parameters of GLE are determined with

Before illustrating the

with damage, it is helpful to understand the analogues behavior of one

bGR .

Figure 13 shows the progress of s-value along with

The overall trend of s-value correlates with GR b-values indicating that the

27

8

L o a d (k N )

(a ) 6

M a x . L o a d 4

1 0 0 th E v e n t 2 0 5

(b ) G L E

4

b

3 2

(c )

s - v a lu e

4 .2 3 .9 3 .6 3 .3

(d )

0 .7

α

0 .6 0 .5 0 .4 0

2 0 0

4 0 0

6 0 0

8 0 0

0 .3 1 0 0 0

T im e ( s e c s ) Figure 14: a) Load vs time, b) bGLE , c) s-value, d) α

28

L o a d b - v a lu e

S m a ll S m a ll

M e d iu m M e d iu m

L a rg e L a rg e

5 .4

9

5 .0 8

4 .5 6

b

G L E

L o a d (k N )

4 .0 3 .5 4

3 .0 2 2 .5 2 .0 0 0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

N o r m a liz e d tim e Figure 15: Typical evolution of bGLE for three dierent size specimen 461

can be considered as a slope of CFD curve in an average sense).

462

value and GLE s-value are equivalent but not equal quantities. Figure 14 plotted over the

463

test duration shows evolution of

464

increases approximately from 0.35 to 0.7 indicating shift from small to large magnitude event

465

towards failure. Fluctuations in

466

changes in CFD and such uctuations are averaged out in

467

bGLE α

along with s-value and

α.

Accordingly, the GR b-

As damage progresses,

α

and s-value show the sensitivity of these parameters for

Figure 15 shows the evolution of

bGLE

bGLE .

along with applied load on a normalized time axis

for three dierent sizes of beam specimens.

469

70-80% of peak load in prepeak regime where 100th event is detected. The initial growth

470

rate of

471

to softening behavior, the damage growth stabilizes due to various toughening mechanism

472

taking place as a result of microcracking and

473

It is important to notice that the

474

to failure.

475

considered as a damage compliant b-value.

bGLE

The growth of

bGLE

468

starts approximately at

is steep due to accelerated damage growth. As crack grows and load drops due

This behavior of

bGLE

bGLE

bGLE

approaches a constant value near failure.

increases monotonically (with some uctuations) up

is consistent with damage growth and hence it can be

29

476

7.4. bGLE versus eective crack length

477

To determine a one to one relation between crack size and event magnitude is infeasible.

478

Therefore, the causal inference is inevitable for deriving some important conclusions in the

479

present work. The GR b-value and GLE s-value have been shown correlated negatively to

480

stress.

481

causes variation in s-value, then what is the source of incremental cut-o magnitude in

482

CFD? The obvious explanation comes from the mechanics of fracture of the problem. The

483

far-eld applied stress may cause crack propagation but the state of stress near the crack

484

tip is dened by the stress intensity factor (SIF). The crack interaction of dierent sizes (as

485

discussed in Section 7.1) relieves stress singularity and restricts crack growth which results in

486

cut-o magnitude. The parameter

487

and therefore it must be correlated to the SIF at the instance of the damage. Considering

488

the general expression of stress intensity factor for mode-I as,

Therefore, an important question that needs to be answered here is, if the stress

α accounts for nonlinearity and cut-o magnitude of CFD

√ KI = σ πaf (a) =⇒ a = 489

where

KI

1  KI 2 πf (a) σ

is the mode-I stress intensity factor due to stress

a = a/d, where d

σ

(6)

at crack length

a. f (a) is

490

geometry shape factor and

is size of the specimen. Equation 6 is applicable

491

to brittle materials and does not account for dierent crack sizes and their interaction but for

492

simplicity of argument this equation serves the purpose. According to linear elastic fracture

493

mechanics, Equation 6 implies that the eective crack length at any point during damage

494

progression can be expressed as a ratio of stress intensity factor and stress normalized by

495

geometry shape function as follows,

ae ≈ f (σ, KI , geometry) 496 497

The parameter

(7)

α increases with nonlinearity and cut-o magnitude in CFD and therefore

it can be considered as correlated to SIF as

KI ∝ α ∝

30

1 and the stress can be correlated (1−α)

Table 3: Range of dierent parameters

498

to s-value as

s ∝ 1/σ .

Parameter

Range

s-value

4.0 - 7.5

α bGLE

0.3 - 0.75

Therefore, for the same geometry,

bGLE = 499

2.5 - 5.5

Equation 8 signies that the

bGLE

s =⇒ f (σ, KI ) (1 − α)

(8)

is a function of stress and stress intensity factor in

500

distribution domain. Consequently, a change in

501

crack length during damage progress.

bGLE can be considered correlated to eective

∆bGLE ∆ae ∝ ∆t ∆t 502

Figure 16 shows

bGLE

(9)

on the left and eective crack length on the right axis plotted

503

over time to demonstrate the correlation expressed by Equation 9.

504

determined from digital image correlation technique as discussed in Sections 5.2 and 6. A

505

reasonably good correlation as seen between eective crack length and

506

damage compliant behavior of

507

7.5. bGLE versus scale eect

508

The eective crack is

bGLE

illustrates the

bGLE .

Figure 17 shows the eect of nite specimen size on

bGLE .

α

and s-value which eventually

509

results in scale eect on

Data scatter is noticeable in small size beams and it is

510

negligible for large beams.

511

obviously due to prominent heterogeneity eect (heterogeneity of material is unchanged but

512

its eect gets intense with decrease in dimensions of the beams). As beam size increases,

513

such heterogeneity eect averages out resulting in reduced scatter in parameters. Similar to

514

b-values determined by GR law as shown in Figure 11, the size eect on

515

negligible but the trend is again signicant.

516

shown in Table 3.

The reason behind scatter of

α

and s-value in small beams is

bGLE

also appears

The observed ranges of GLE parameters are

31

0 .0 8 0 .0 6 4

b

G L E

0 .0 4 3 M e d iu m

0 .0 2 b

C r a c k le n g th ( m )

5

G L E

C r a c k le n g th

2

2 0 0

4 0 0

6 0 0

8 0 0

0 .0 0 1 0 0 0

T im e ( s e c s ) (a)

0 .1 5

0 .1 0 4

b

G L E

0 .0 5 3 b L a rg e G L E

C r a c k le n g th ( m )

5

0 .0 0

C r a c k le n g th

2 0

5 0 0

1 0 0 0

1 5 0 0

T im e ( s e c s ) (b)

0 .3

0 .2 4

b

G L E

0 .1 3 b

G L E

C r a c k le n g th

2

2 0 0

4 0 0

6 0 0

8 0 0

C r a c k le n g th ( m )

5

0 .0

1 0 0 0

T im e ( s e c s ) (c)

Figure 16: Correlation between crack length and b-value of GLE for a)Small, b)Medium and c)Large size specimen 32

6 .4

1 0 0

2 0 0

3 0 0

1 0 0

2 0 0

3 0 0

(a )

b

G L E

4 .8 3 .2 1 .6 0 .0 0 .9 2

(b )

α

0 .6 9 0 .4 6 0 .2 3

s - v a lu e

7 .8

(c )

6 .5 5 .2 3 .9

S iz e ( m m )

Figure 17: Eect of nite size on b-value obtained from GLE

33

517

7.6. Uncertainty of bGLE versus bGR

518

The b-values obtained from GLE and GR law show dierent scatter pattern with respect

519

to ligament depth of the beams as shown in Figures 18a and 18b enveloped by maximum and

520

minimum values with average nonlinear trend (dashed lines). The uncertainty in b-value is a

521

result of uncertain occurrence of cracks in concrete due to its heterogeneous microstructure

522

which produces uncertain events of varying magnitudes. To compare uncertain behavior of

523

the material at dierent scales, the AE information needs to be normalized by volume of FPZ

524

to obtain crack density. However the volume of FPZ is unknown and therefore the ligament

525

depth is used as a representative of beam size for normalization.

526

normalized by ligament depth measures mean events per unit ligament depth.

527

normalized standard deviation of events for a beam size can be determined.

528

shows the variation of normalized mean and standard deviation

529

statistics of AE events listed in Table 2.

530

with increase in beam size as shown in Figure 18c, the normalized dispersion of these events

531

reduces with the increase in beam size as shown in Figure 18d.

532

uncertainty of AE generation in concrete reduces with larger beam size. Here, uncertainty

533

of AE detectability is not considered as AE sensors are arranged closely around the crack

534

path to avoid any data loss by attenuation.

535

bGLE

The observed scatter of

bGLE

of

537

accounts for uncertainty.

Similarly, Figure 18d

derived from

This illustrates that the

reduces for large beam sizes.

538

a log-linear scale.

539

existence of relationship between

540

damage evaluation. Such type of relation is not certain for

541

8. Conclusions

On the contrary,

Figure 18e shows a linear relation between

Nmax

Nmax

Nmax

(Figure 18a ) reects this uncertainty caused by beam size

whereas dispersion of

542

(σ)

of

Though cumulative number of events increases

536

As

(µ)

The mean

Nmax

bGR and

does not

bGLE

on

can be predicted by relating it with ligament depth of beam,

Nmax

and

bGLE

should help to predict expected

bGR

bGLE

for

as shown in Figure 18f.

The GR distribution has been successfully used for eective prediction of nal failure in

bGR , its bias for certain magnitude range

543

structural elements. Despite such productivity of

544

and inadequacy to characterize CFD makes it uncertain for quantitative damage evaluation.

34

5 .5 1 .3

5 .0

1 .2 G R

4 .0

1 .1

b

b

G L E

4 .5

3 .5 1 .0 3 .0 0 .9

2 .5 6 0

9 0

1 2 0

1 5 0

1 8 0

2 1 0

2 4 0

6 0

L ig a m e n t d e p th d ( m m )

9 0

1 2 0

1 5 0

1 8 0

2 1 0

2 4 0

L ig a m e n t d e p th d ( m m )

(a)

(b)

3 8 k

1 4 0

=1 1 5 d −5 6 8 . 8 8 N

3 0 k

R

1 2 5

=0 .7 5 5 8 2

µ / d , σ /d

2 0 k

N

m a x

1 0 0

1 0 k

7 5 5 0

µ/d σ/d

2 5

0 6 0

9 0

1 2 0

1 5 0

1 8 0

2 1 0

6 0

2 4 0

L ig a m e n t d e p th d ( m m )

9 0

1 2 0

1 5 0

1 8 0

2 1 0

2 4 0

L ig a m e n t d e p th d ( m m )

(c)

(d)

4 0 k

4 0 k

L o g R

2

1 0

( N ) =1 . 8 5 + 0 . 4 8 5 b

R

G L E

=0 .9 5

1 0 k

2

=0 .5 8

N

N

m a x

m a x

1 0 k

1 k

1 k 2

3

4

b

5

6

0 .8

1 .0

1 .2

b

G L E

(e)

1 .4

G R

(f )

Figure 18: Envelope of a) bGLE and b) bGR for three ligament depths, c) Linear t between Nmax and ligament depth, d) Normalizes mean (µ) and standard deviation (σ) of Nmax , Log-linear relationship between Nmax and e) bGLE and f) bGR . 35

545

Currently, two critical stages, one near peak load and other near failure, are used to charac-

546

terize damage where b-value varies from 1.5 to 1.0 respectively. The present work emphasis

547

on monotonic increase in b-value with incremental damage such that the damage can be

548

characterized in between these two critical conditions quantitatively. If one looks closer to

549

CFD evolving over time, as an animated series of curves superposed in chronological order,

550

the subtle changes in its shape become visible with progressive damage. Quantication of

551

these shape changes by single parameter b-value oversimplies and approximates the non-

552

linearity in CFD. One to one correlation between damage level and

553

dierent damage levels sometimes show the same b-values. On the other hand, the logistic

554

(sigmoidal) growth of microcracks is more appealing, natural and evident in experimental

555

studies over the power law growth. The two parameters, s-value as a rate parameter and

556

α

557

proposed

558

range of magnitude but accounts for cut-o magnitude too. GLE approaches the problem

559

in more statistical way as a problem of parameter estimation without the need of log-linear

560

scale transformation whereas GR law is merely a linear least square tting technique on

561

log-linear scale.

562

age compliant as it correlates with crack length. Therefore, one to one correlation between

563

damage level and

564

AE generation uncertainty and size eect makes

565

practical applications.

566

bGR

is also absent as

as a shape parameter of GLE, eectively captures the subtle shape changes of CFD. The

bGLE

is rather a generalized version of

bGR

which not only ts well in the whole

Moreover, the monotonic incremental behaviour of

bGLE

bGLE

is possible to evaluate damage quantitatively.

bGLE

makes it dam-

Accommodation of

superior and predictable over

bGR for

In practice, damage initializes by dispersed micro cracking and then a dominant crack

bGLE can be eectively used to evaluate damage

567

takes over the damage process. In such cases

568

quantitatively. Small size geometrically similar specimens are often tested in laboratories to

569

understand the behavior of real world large structural elements.

570

based research on concrete revolves around quantifying the prominently observed size eect

571

through such laboratory experiments. Therefore, it is also necessary to evaluate size eect on

572

b-value for its consideration as an absolute damage indicator for structural health monitoring.

573

Therefore, the present work is motivated by the fact that instead of apparent universality,

574

an absolute b-value near failure for each size of beam can be determined in laboratory and 36

The fracture mechanics

575

can be extrapolated to account for size eect in real world application.

576

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577 578

579 580

581 582

583 584

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41

1. Effect of size and evolution of fracture on b-value of acoustic emission in concrete is studied. 2. Commonly used Gutenberg-Richter law is shown to invalidate universality when applied to fracture of concrete. 3. A generalized model is proposed for computing the b-value and is shown to be universal and damage compliant. 4. The growth of micro and macro cracking in concrete is shown to follow the proposed law.