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Dec 15, 2009 - R. Vidya Sagar · B. K. Raghu Prasad ·. B. L. Karihaloo. Received: 4 April 2009 / Accepted: 11 November 2009 / Published online: 15 December ...
Int J Fract (2010) 161:121–129 DOI 10.1007/s10704-009-9431-7

ORIGINAL PAPER

Verification of the applicability of lattice model to concrete fracture by AE study R. Vidya Sagar · B. K. Raghu Prasad · B. L. Karihaloo

Received: 4 April 2009 / Accepted: 11 November 2009 / Published online: 15 December 2009 © Springer Science+Business Media B.V. 2009

Abstract Notched three-point bend specimens (TPB) were tested under crack mouth opening displacement (CMOD) control at a rate of 0.0004 mm/s and the entire fracture process was simulated using a regular triangular two-dimensional lattice network only over the expected fracture process zone width. The rest of the beam specimen was discretised by a coarse triangular finite element mesh. The discrete grain structure of the concrete was generated assuming the grains to be spherical. The load versus CMOD plots thus simulated agreed reasonably well with the experimental results. Moreover, acoustic emission (AE) hits were recorded during the test and compared with the number of fractured lattice elements. It was found that the cumulative AE hits correlated well with the cumulative fractured lattice elements at all load levels thus providing a useful means for predicting when the micro-cracks form during the fracturing process, both in the pre-peak and in the post-peak regimes.

R. Vidya Sagar (B) · B. K. Raghu Prasad Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India e-mail: [email protected] B. K. Raghu Prasad e-mail: [email protected] B. L. Karihaloo School of Engineering, Cardiff University, Cardiff CF24 3 AA, UK e-mail: [email protected]

Keywords Concrete fracture · Lattice model · Fuller distribution · Acoustic emission · Beam element · Acoustic emission energy · Load-CMOD

1 Introduction A quasi-brittle material such as concrete is characterized by a gradually decreasing stress transfer capability after its tensile strength has been reached (Bazant and Planas 1998; Karihaloo 1995; Shah et al. 1995). One of the main characteristics of quasi-brittle materials is their heterogeneity as a result of which the damage in the form of micro-cracks always forms in tortuous bands. The phenomenon of micro-cracking is the most important feature of the concrete fracture behavior both before and after the attainment of its tensile strength. It is therefore useful to know when the micro-cracks form during the entire fracturing process. This knowledge can be gained by simulating the process on say, notched three-point bend beams using the lattice network model (Schlangen and van Mier 1991, 1992a,b; Schlangen and Garboczi 1996, 1997; Chiaia et al. 1997) and by comparing the simulated results with acoustic emission (AE) measurements (Maji et al. 1990; Ohtsu 1996; Landis and Baillon 2002) made during the testing of the same beams. A good correlation between the cumulative AE hits and the cumulative fractured lattice elements at all load levels would not only provide a useful means for predicting when the micro-cracks form during the entire fracturing process, but would also

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increase our confidence in the lattice modeling technique. This paper will show that such a correlation does indeed exist.

Table 1 Lattice element material properties (Alexander et al. 1968; Schlangen and van Mier 1991; Ince et al. 2003) Young’s Tensile Cross sectional modulus strength properties (N/ mm2 ) (N/ mm2 ) of the beam element

2 The lattice model

Depth (mm) Width (mm)

The lattice modeling technique for concrete fracture has been extensively described in the literature (Schlangen and van Mier 1991, 1992a,b; Schlangen and Garboczi 1996, 1997; Chiaia et al. 1997), but for the sake of completeness a brief review of the relevant material will be given here. A lattice model discretises a continuum by line elements capable of transferring forces and moments. In general, an isotropic continuum can only be discretised by a lattice network of truss elements. A lattice with bar or beam elements corresponds to a micro-polar continuum (Karihaloo et al. 2003). To account for the complex microstructure of concrete, it is customary to consider it at three scales (VanMier 1997): micro, meso and macro. The lattice model operates at the meso scale and distinguishes between the hardened cement paste, the fine and coarse aggregates, and a bond layer between the cement paste and aggregate constituents. All the three phases can be elastic and brittle although Arslan et al. (2002) and Ince et al. (2003) have simulated concrete beams by regarding the matrix phase as elastic and tension softening. The inherent disorder in the microstructure can be included either by using random lattices (Herrmann 1991) or by assigning random strength and/or stiffness properties to the elements in a regular lattice (Ince 2005). Fig. 1 A typical notched three-point bend specimen showing the locations of sensors

Aggregate 75000

10.0

1.443

80

Matrix

25000

5.0

1.443

80

Bond

25000

2.5

1.443

80

In the application of the lattice model to the study of the fracture process in notched three-point bend plain concrete specimens (Fig. 1), the heterogeneous microstructure of the plain concrete was modeled explicitly by a triangular beam lattice network only in the central third of the beam span where the fracture process was expected to be concentrated. The outside thirds of the beam span were treated as an isotropic continuum with E = 30, 000 N/ mm2 ( MPa) and ν = 0.2. Within the central lattice region, the heterogeneity was modeled explicitly by assigning different material properties to different beam elements within the lattice. All three phases were assumed to be linear elastic and brittle. The elastic properties of each phase are listed in Table 1. The Poisson ratio of the phases is the same and follows from the energy equivalence of the lattice model with its continuum counterpart. In fact, the Poisson ratio is related directly to the ratio of the height (h) to the length (l) of the beam lattice element (Schlangen and Garboczi 1997)

6

8

t

top view

4

2

2

4

1r

3

4

8

s ao

7

3

d

2

6

1

5

t S L

right side view

front view 1 to 8 represents sensors

123

d

t left side view

Verification of the applicability of lattice model to concrete fracture

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 p = 100

Fig. 2 Triangular lattice projected on to the particle structure of concrete

ν=

1− 3+

 h 2 l

 h 2

(1)

l

In the present study, a physically realistic Poisson’s ratio equal to 0.2 is assumed which gives h/l = 0.5772. The width of the lattice beam element in the perpendicular direction equals the width of the TPB specimen tested (80 mm). The properties of a lattice beam element depend on whether it represents an aggregate particle, the matrix, or the bond zone between the particle and matrix. The phase that a lattice element represents is established by applying an overlay technique in which a randomly generated particle structure (Fig. 2) is mapped on to the lattice network. An algorithm determines the position of the lattice element in relation to the overlay. If both nodes of a lattice element are within the same aggregate particle, it is assigned the material properties of the aggregate particle. If both nodes of a lattice element are located in the matrix, it is assigned the matrix properties. If one node is located in a particle and the other in the matrix then the element is assigned the bond zone properties. The overlay particle structure is generated according to the Fuller distribution. The Fuller curve is given by the percentage (by weight) of a mix with a maximum aggregate diameter Dmax passing a sieve with aperture diameter D (Walraven 1980),

D Dmax

(2)

Using the Fuller curve distribution, Walraven (1980) derived an expression to represent the circular aggregate distribution in two dimensions. This expression represents the most probable distribution of diameters of the intersection circles, which are located in a plane and are crossed by a line. For a given concrete mix, the distribution of circle diameters (representing spherical aggregates in three dimensions) in a certain cross section can be generated from which a cumulative distribution function representing the probability Pc that an arbitrary point in the concrete mix lying in an intersection plane is located in an intersection circle with a diameter D < D0    D0 0.5 Pc (D < D0 ) = Pk 1.065 Dmax     D0 4 D0 6 −0.053 − 0.012 Dmax Dmax 8  D0 −0.0045 Dmax    D0 10 −0.0025 (3) Dmax D0 , sieve size; Pc (D), Probability of the existence of an aggregate particle of size D; Dmax , maximum aggregate size; Pk , aggregate volume fraction (usually the value of pk is 0.75). For a given area A and aggregate fraction Pk , the number of aggregate particles ni with diameter Di can be obtained (Van Vliet 2000) from Eqs. (4a) and (4b). [ pc (D < Di + 0.5) − pc (D < Di − 0.5)] A Pk 0.25π Di2 Di < Dmax (4a) [ pc (D < Di ) − pc (D < Di − 0.5)] A Pk ni = 0.25π Di2 Di = Dmax (4b)

ni =

The particle structure in the central third region, 80 mm × 80 mm (Fig. 3) are obtained from Table 2, and placed according to the method proposed in literature (Hsu and Slate 1963; Schlangen and van Mier 1992a,b). The particles are randomly placed by calculating the x and y coordinates of the centre of a circle diameter D, beginning with the largest diameter particles. According to Hsu and Slate (1963) the condition for placement is

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R. Vidya Sagar et al. Table 3 Concrete mix proportions (kg/m3 ) and compressive strength

Fig. 3 Representation of a beam with lattice beam elements (middle third) and standard finite element mesh (outer thirds)

Property

Mix-F

Water/binder ratio

0.34

Cement

450

Micro silica

56

Fine aggregate

707

Coarse aggregate

1058

Water

151

Superplasticiser, (% of weight of cement)

1.2

Compressive strength, MPa (28-days)

78

Table 2 Distribution of circular aggregates in a plane Distribution of circles Diameter (mm)

Number

Diameter (mm)

Number

1

434

11

21

2

173

12

15

3

127

13

9

4

100

14

4

5

82

15

0

6

67

16

0

7

55

17

0

8

45

18

0

9

36

19

0

10

28

20

0

Fig. 4 The complete test setup

that the distance X0 between two aggregates A and B has to be equal, or larger than  (D A + D B ) (5) X 0 = 1.1 2

trolled testing. At the end of an iteration, the lattice beam element in which the maximum tensile stress



(|Mi | , M j )max f (6) σt < + A W

Figure 2 shows a regular triangular lattice that has been projected on to this particle structure. A TPB notched specimen meshed with lattice beam elements in a regular triangular network in the central region is shown in Fig. 3. The fracture process zone is expected to be concentrated in this region. The remaining parts of the specimen are modeled with a coarse triangular finite element mesh. This hybrid discretisation of a continuum should reduce substantially the simulation time. The fracture process was simulated by performing a linear elastic analysis of the TPB notched beam under displacement control. The mid-span top node of the lattice mesh was allowed to move vertically downward 0.065 mm per iteration to simulate displacement con-

exceeded the corresponding tensile strength (Table 1) was removed and the analysis continued until no more elements reached the critical state in this iteration. In Eq. (6), σt is the tensile strength, f is the axial force in the element, A is the cross sectional area, W the section modulus of the lattice member, and Mi and M j are the bending moments at the end nodes i and j. The failure criterion (6) resembles the Rankine criterion and was introduced by Schorn and Rode (1991). The lattice member whose stress has reached the limiting value was ascribed a negligible value of modulus of elasticity in the next iteration. This process was continued until the complete failure of the TPB notched beam specimen.

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Table 4 Details of the specimen tested Specimen name Notch details

Dimensions of the specimen (mm)

(a0 ) (mm) (a0 /d) L FSN1 a

12

0.15

S

t

Sensors (Resonant type)

Threshold (dB)

Depth

290 240 80

80

Sensor position from centre of the notch (mm) r

R6D

40

s

32 32

Table 5 Record of AE events and fractured lattice elements at different times (and load levels) Time (s) Pre-peak zone

Post-peak zone

75

154

93.7

299

475

601

912

1192

1219

1433

Load (kN)

3.88

5.56

6.74

4.63

2.51

1.88

1.0

0.478

0.45

0.281

AE hits

18

48

183

908

1823

2197

2783

3000

3024

3160

Fractured elements

167

879

984

1054

1136

1137

1145

1148

***

***

Fig. 5 Variation of AE energy released with load and time

3 Experimental program 3.1 Materials and specimens A schematic diagram of the three-point bend test specimen is shown in Fig. 1. The specimen was loaded at mid-span and was simply supported over a span S. A notch was cut into the cured beam with a diamond saw. The span-to-depth ratio of the specimen was 3.0

and the notch to depth ratio was 0.15. The 28-day compressive strength of the concrete mix was 78 MPa and the maximum size of coarse aggregate was 20 mm.The mix proportions are given in Table 3. 3.2 Testing arrangement The experimental setup consisted of a servo-hydraulic loading frame with a data acquisition system and

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Load (kN)

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R. Vidya Sagar et al. 7.50 6.75 6.00 5.25 4.50

4 Results and discussion 1 - Experiment 2 - Simulation 2 1

3.75 3.00 2.25 1.50 0.75 0.00 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.20 0.23 0.25 0.28

CMOD (mm)

Fig. 6 Variation of load-crack mouth opening displacement (CMOD) with load

7

B A

6

load, kN

5 C

4 3

D

2

E F

1

G

H

0 0

150

300

450

600

750

900 1050 1200 1350 1500 1650 1800 1950 2100

time, sec

Fig. 7 Typical load—time plot. Letters indicate the moments of capture of AE events (Fig. 8)

an acoustic emission monitoring system. The specimen was tested under crack mouth opening displacement (CMOD) control at the rate of 0.0004 mm/s. The mid-span displacement was measured using a linearly varying displacement transducer (LVDT), placed at the center on the underside of the beam. The AE monitoring system had 8 channels, one for each of the eight resonant type sensors (Fig. 1, Table 5). As the crack propagation will start from the pre-existing notch tip, the AE sensors were placed close to the notch. High vacuum silicon grease was used to improve the acoustic coupling between the sensors and concrete. The complete experimental setup is shown in Fig. 4. The AE signals were amplified with a gain of 40 dB in a preamplifier. The threshold value of 40 dB was selected to ensure a high signal to noise ratio. The total AE energy released was calculated by summing up the AE energy release values of all 8 channels. The sensor location data are presented in Table 4.

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A typical plot of load versus crack mouth opening displacement (CMOD) is shown in Fig. 6. It is interesting to note that the CMOD obtained from the lattice model analysis agrees fairly well with the experimental results. It was observed from the analysis that the bond elements were the first to fail under increased loading and that the number of fractured elements increased near, and after, the peak load, i.e. once the crack started to propagate. Figure 5 shows the variation of load and cumulative AE energy released (in relative units) with time. The units of the cumulative AE energy released shown in the Fig. 5 are volts squared times microseconds and not in terms of the absolute units, Joules (Landis and Baillon 2002). From Fig. 5 it can be observed that until the peak load is reached the AE energy accumulated is quite small in relative terms. This is quite consistent with the fact that pre-peak work of fracture is much less compared to that in the post-peak region. There is a jump in the AE energy release at or near the peak load, possibly as a result of the onset of macrocrack growth. The specific fracture energy of the concrete used in this study determined according to the RILEM Committee 50-FMC recommendations (RILEM 1985) is 180.6 N/m. After the peak-load, both the AE energy released and the number of fractured lattice elements increase. Table 5 shows the load, time, number of AE hits registered during the fracture of the TPB specimen, and the number of fractured lattice elements. Figure 7 also shows the times on the load-time plot when the AE measurements were recorded. Figure 8 shows the plots of the recorded AE hits at these times. Most of the AE activity takes place on either side of the pre-existing notch tip, as seen in Fig. 8, thus justifying the use of the lattice network model only in the middle third of the beam. The typical recorded load-time curve and the locations of the AE event sources are compared with lattice simulation results in Fig. 9, whereas Fig. 11 compares the locations of the AE events and of the number of fractured lattice elements at load levels identified in Fig. 10. The trend in the cumulative number of recorded AE hits with load, as shown in the Table 5 and Fig. 11 matches closely the trend in the cumulative number of fractured elements at the same load. In other words, the pattern in which the AE hits are distributed around the notch has the same trend as that of

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Fig. 8 The AE source locations in TPB specimen at times noted in Fig. 7: a 139 s b 174 s c 300 s d 475 s e 600 s f 912 s g 1192 s h 1219 s. (X-axis represents distance from left along the span and Y-axis represents the depth of the beam) 340

lattice model approach

320

AE hits

AE experiment

No.of failed lattice members

300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0

Distance from left along the span (mm)

0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240

Distance from left along the span (mm)

Fig. 9 Comparison of AE hits and fractured lattice members 7 II 6

III

5

load (kN)

the fractured elements around the notch (Fig. 11 and Table 5), which is in support of lattice model. The differences are likely due to the attenuation and scattering of the acoustic waves and also due to fact that whereas an AE event takes place at the micro scale, the lattice element size used in this study is at the meso scale (2.5 mm).

4 I

3 2

IV

1

5 Conclusions Based on the above results the following two major conclusions can be drawn. 1.

The lattice network modeling technique predicts a response diagram that agrees reasonably well

0

0

250

500

750

1000

1250

1500

1750

2000

time (sec)

Fig. 10 Load-time plot showing the load levels at which the locations of the AE sources and of the fractured lattice members are compared

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R. Vidya Sagar et al. 90 80

I I Experiment

I

Depth (mm)

70

Simulation

60 50 40 30 20 10 0 0

20

40

60

80

100 120

140

160 180

200 220

240

Distance from left along the span of the beam (mm) 90

II

80

II II Experiment

Depth (mm)

70

Simulation

60 50 40 30 20 10 0 0

20

40

60

80

100

120

140

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180

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240

Distance from the leftside of the beam (mm) 90

III

80

III

70

Depth (mm)

Simulation

Simulation

60 50 40 30 20 10 0 0

20

40

60

80

100 120 140 160 180 200 220 240

Distance from left along the span (mm) 90

IV Simulation

80 70

Simulation

60

Depth (mm)

IV

50 40 30 20 10 0 0

20

40

60

80

100

120

140

160

180

200

220

240

Distance from the left along the span (mm)

Fig. 11 Comparison of the locations of the AE sources and of the fractured lattice members (X-axis represents the distance from left along the span and Y-axis represents the depth of the beam)

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Verification of the applicability of lattice model to concrete fracture

2.

with the experimental results. Importantly, it has been shown that the fine lattice network need only within the expected width of the fracture process zone, whereas the rest of the specimen can be discretised by a coarse finite element mesh thus leading to a substantial reduction in the simulation time. The cumulative number of AE hits registered during the test correlates well with the cumulative number of fractured lattice elements practically at all load levels. This provides confirmation that the lattice modeling technique is eminently suited to the study of fracture process in concrete which has hitherto only been presumed.

References Alexander KM, Wardlaw J, Gilbert DJ (1968) Aggregate-cement bond, cement paste strength and strength of concrete. Proceedings of international conference on the structure of concrete, cement and concrete association, pp 59–81 Arslan A, Ince R, Karihaloo BL (2002) Improved lattice model for concrete fracture. J Eng Mech ASCE 128(1):57–65 Bazant ZP, Planas J (1998) Fracture and size effect in concrete and other quasi brittle materials. CRC press, Boca Raton, FL Chiaia B, Vervuurt AHJM, van Mier JGM (1997) Lattice model evaluation of progressive failure in disordered particle composites. Eng Frac Mech 57(2/3):301–318 Herrmann HJ (1991) Patterns and scaling in fracture, in fracture process in concrete, rock and ceramics. In: Van Mier JGM, Rots JG, Bakker A (eds) Chapman & Hall, New York, p 195 Hsu TTC, Slate FO (1963) Tensile bond strength between aggregate and cement paste or mortar. J Am Conc Inst 4(4):465– 485 Ince R (2005) A novel meso-mechanical model for concrete fracture. Str Eng Mech 18(1):28–36 Ince R, Arslan A, Karihaloo BL (2003) Lattice modeling of size effect in concrete strength. Eng Fract Mech 70:2307–2320

129 Karihaloo BL (1995) Fracture mechanics and structural concrete. Addison Wesley, Longman, UK Karihaloo BL, Shao PF, Xiao QZ (2003) Lattice modeling of the failure of particle composites. Eng Fract Mech 70:2385– 2400 Landis EN, Baillon L (2002) Experiments to relate acoustic emission energy to fracture energy of concrete. J Eng Mech 128(6):698–702 Maji AK, Ouyang C, Shah SP (1990) Fracture mechanism of quasi-brittle materials based on acoustic emission. J Math Res 5(1):206–217 Ohtsu M (1996) The history and development of acoustic emission in concrete engineering. Mag Conc Res 48(177):321– 330 Schlangen E, Garboczi EJ (1996) New method for simulating fracture using an elastically uniform random geometry lattice. Int J Eng Sci 34(10):1131–1144 Schlangen E, Garboczi EJ (1997) Fracture simulations of concrete using lattice models: computational aspects. Eng Fract Mech 57(2/3):319–332 Schlangen E, Van Mier JGM (1991) Experimental and numerical analysis of micro mechanisms of fracture of cement based composites, report no. 25. 5.-91-1/VFC., TU., Delft Schlangen E, van Mier JGM (1992a) Experimental and numerical analysis of micromechanicsm of fracture of cement based composites. Cem Conc Comp 14:105–118 Schlangen E, van Mier JGM (1992b) Simple lattice model for numerical simulation of fracture of concrete materials. Mater Struct 25:534–542 Schorn H, Rode U (1991) Numerical simulation of crack propagation from micro cracking to fracture. Cem Conc Comp 13:87–94 Shah SP, Swartz SE, Ouyang C (1995) Fracture mechanics of concrete: applications of fracture mechanics to concrete, rock and other quasi-brittle materials. Wiley, New York VanMier JGM (1997) Fracture process of concrete: assessment of material parameters for fracture models. CRC press, Boca Raton, FL Van Vliet MRA (2000) Size effect in tensile fracture of concrete and rock. PhD thesis, Delft University Press, Delft Walraven JC (1980) Aggregate interlock: a theoretical and experimental analysis. PhD thesis, Delft University of Technology

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