A comparative study on three approaches to investigate the size independent fracture energy of concrete

Engineering Fracture Mechanics 138 (2015) 49–62

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

A comparative study on three approaches to investigate the size independent fracture energy of concrete N. Trivedi ⇑, R.K. Singh, J. Chattopadhyay Reactor Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India

a r t i c l e

i n f o

Article history: Received 20 December 2014 Received in revised form 11 March 2015 Accepted 14 March 2015 Available online 20 March 2015 Keywords: Concrete RILEM fracture energy Size independent fracture energy

a b s t r a c t This paper deals with investigation of size-independent fracture energy (GF) of concrete. The study involves numerical modeling of three point bend concrete beams that are geometrically similar having constant length to depth ratio with varying notch to depth ratios. RILEM fracture energy (Gf) values evaluated numerically and experimentally are found to be in reasonable agreement. GF is estimated from developed relationship of fracture energy release rate and through bilinear model by Gf values. Gf values have been utilized to develop a simple methodology for estimation of GF. Comparative analysis of GF from three different methodologies has been carried out. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Concrete, due to its excellent shielding capability, fire rating, long service life under normal and accidental conditions and ease in construction with relatively lower cost, is used extensively in building most of the civil engineering structures. In-spite of such salient features, the concrete structures generally consist of numerous micro-cracks that might result in fracture of the concrete structures under service loads, accidental load and/or exposure to regular environmental conditions. Thus a micro-crack in concrete may become a potential source of crack propagation leading to a probable catastrophic failure. In order to prevent such accidents, it is necessary to predict the failure mechanisms of structures, so that the safety of concrete structures throughout the service life can be assured. The failure mechanism can be studied by quantifying the energy consumed in crack propagation and formation of new crack surfaces. In a concrete structure, the crack growth requires a certain amount of energy that can only be studied through an energy based propagation criterion, which provides a fundamental basis for understanding the phenomenon of concrete fracture mechanism. Concrete despite predominantly elastic material response, exhibits a stable non-linear fracture response in tension loading, when tested under displacement control. The reason for the non-linearity is the development of a fracture process zone (FPZ) ahead of the crack tip. In a quasi brittle material like concrete the energy dissipated for the formation of FPZ ahead of the crack tip, is termed as fracture energy. The concrete fracture energy characterizing the failure process is still under extensive research. The various finite element studies [1–3], incorporating the concrete strain based softening model, showed the dependency on mesh size. The fracture energy based concrete softening model yields the consistent finite element results independent of mesh size. The size-independent fracture energy is the most useful parameter in the analysis of cracked concrete structure. Hu and Wittmann [4] proposed the bilinear function so that size-independent fracture energy GF can be estimated from the fracture energy data measured on laboratory-size specimens. The study by Elices and Planas [5] compared the theoretical ⇑ Corresponding author. Tel.: +91 22 25591548. E-mail address: [email protected] (N. Trivedi). http://dx.doi.org/10.1016/j.engfracmech.2015.03.021 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

50

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

Nomenclature gf Gf GF w wc a = a/W

r

local fracture energy at crack tip RILEM fracture energy size independent fracture energy crack opening displacement maximum widening of the crack separation displacement (terminal point where cohesive stress equals to zero) notch to depth ratio of three point bend beam cohesive stress

background and experimental aspect of various concrete fracture models like cohesive crack model, the equivalent elastic crack model, or models based on two parameters such as Bazant’s and Jenq-Shah’s [1]. The fictitious crack method is applied to determine the load–deflection diagrams of notched plain concrete beams under three-point bending using various forms of strain softening in the stress–deformation relationship and the results indicated that there is a need to determine a more realistic relationship [6]. The study by Luigi and Gianluca [7] deals with the identification of concrete fracture parameters through the size effect curves, associated with certain specimen geometry, to identify the tensile strength and the initial fracture energy, which are typically used to characterize the peak and the initial post-peak slope of the cohesive crack. Acoustic emission technique [8] on three point bend specimen is used to estimate the FPZ size. In the work of Karihaloo et al. [9], 26 test data sets from literature have been re-evaluated to assess the validity of obtaining the size-independent fracture energy of concrete by testing three point bend specimen. The fracture test on geometrically similar (constant length to depth ratio) three point bend (TPB) plain concrete beam specimens, made of aggregates, sand, cement and water, were performed by Raghu Prasad [10]. Concrete structures due to their unique material characteristics often need the application of quasi-brittle fracture mechanics. The investigation undertaken so far involved limited concrete fracture tests on laboratory size concrete specimens. The precise definition and test method for estimation of the fracture energy has been a subject of debate among researchers. Based on the fracture test results, a basic understanding of the fracture processes can be developed, but the exact quantification of size-independent fracture energy remains still elusive. Although some progress has been achieved but altogether detailed study on size independent fracture energy is not readily available. The present study investigates this aspect. The present work involves the numerical modeling of fracture tests of geometrically similar TPB beams having constant length to depth ratio [10]. The displacement controlled test on the universal TPB specimens was conducted for varying a/W ratio = 0.05, 0.25 and 0.33 to estimate the fracture energy [10]. The simulation of TPB specimen performed by the finite element analysis incorporating the concrete softening behavior, predicts the load–load line displacement curves. The numerically and experimentally observed maximum load, vertical displacement at maximum load is found to be in excellent agreement. The RILEM fracture energy (Gf) of concrete is estimated from the numerically predicted and experimentally observed load–load line displacement curves. The RILEM fracture energy values have been averaged out at various a/W for geometrically similar beams and the coefficient of variance is estimated. This is how a methodology is developed to investigate the size-independent fracture energy (GF) of concrete utilizing the values of Gf. Also GF values are estimated from the other two methodologies based on Hu and Wittmann bi-linear model and fracture energy release rate. The size-independent fracture energy of concrete is the most useful parameter in the analysis of cracked concrete structure. The present study investigates an easy and robust technique for the determination of the size independent fracture energy of concrete. To accurately analyze the concrete fracture phenomenon, the study involves the comparative analysis of size independent fracture energy by the three methodologies based on Hu and Wittmann bi-linear model, fracture energy release rate and presently proposed model based on Gf averaging for geometrically similar beams. The comparison of fracture energy values evaluated by these different methods has not been carried out before in the open literature. Thus the present study invokes a primary approach based on concrete fracture mechanics that helps to understand the failure mechanisms and load bearing capacity of concrete structures. 2. Theoretical background The fracture energy of concrete is the most important parameter in the fracture behavior of concrete that describes the mechanism of cracking. The commonly used method for measuring the fracture energy is the work-of-fracture method recommended by RILEM [11]. The local fracture energy model [4] and the energy release rate are the other methods for measuring the size-independent fracture energy of concrete. The method developed in this paper and other two popular methods to estimate the size independent fracture energy are described below. 2.1. Proposed methodology using RILEM fracture test RILEM technical committee recommended the guidelines for determination of fracture energy of cementitious materials by conducting TPB test on notched beam [11,12] as shown in Fig. 1. In order to obtain a complete load and load point

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

51

Fig. 1. TPB geometry.

displacement curve, a closed loop servo hydraulic testing machine is recommended. The fracture behavior of concrete characterized by the RILEM measures the averaged fracture energy over the entire projected ligament area. According to RILEM recommendation, the fracture energy is evaluated by dividing the total applied energy with the projected ligament area. Therefore, for a specimen with a depth W and an initial crack length a as shown in Fig. 1, the Gf is given as:

Gf ða; WÞ ¼

1 ðW  aÞB

Z

Pdd

ð1Þ

where B is the specimen thickness, a = a/W, P is the applied load and d is the displacement at the loading point. Let us consider geometrically similar TPB specimens with depth W1, W2. . .Wn. Each specimen having different notch to depth ratios as: a = a1, a2. . .am. The a values are to be chosen in such a way that the crack tip is sufficiently away from the boundary. In other words, a values should not be too small or high [4,9]. The size independent average fracture energy GF is approximated as:

GF ðaverageÞ ¼

" # m X n 1 X Gf ðai ; W j Þ mn i¼1 j¼1

ð2Þ

Using the proposed Eq. (2) the size independent fracture energy for around 40 TPB specimens with a = 0.25 and 0.33 is estimated and % coefficient of variance is shown in Section 5.

2.2. Bi-linear approximation In bi-linear approximation, the fracture energy at crack tip known as local fracture energy (gf) is assumed to vary with ligament in a bilinear manner. Local fracture energy method addressed the effect of the free boundary of the specimen on the fracture process zone ahead of a real crack in a concrete structure [4,9,13]. The energy required to create a fresh crack decreases as the crack approaches the free boundary [4]. Initially, when the crack grows from a pre-existing notch, the rate of decrease is moderate, almost a constant, but it accelerates as the crack approaches the end of the un-cracked ligament [9,13]. The local fracture energy and the FPZ size were found to decrease rapidly as the crack approaches the back surface of the specimen. The maximum FPZ size is attained when the crack is far away from the boundary. In a large specimen, a region ahead of crack tip exists where the FPZ size is relatively constant resulting in a constant gf (or GF). A bilinear approximation is represented in Fig. 2 to simplify the boundary-induced reduction in the fracture energy. This bilinear function consists of a horizontal line with the value of the size-independent fracture energy and a descending branch that reduces to zero at the back surface of the specimen. The transition from GF to the rapid decrease occurs at the transition ligament length a⁄l , which is given by the intersection of these two lines. On the basis of the boundary effect method of Hu and Wittmann [4], the measured RILEM fracture energy, Gf, represents the average of the local fracture energy function over the ligament area (dotted line in Fig. 2). The relationship between all the involved variables is given by:

R Wa Gf ða=WÞ ¼

0

  g f ðxÞdx al =W ¼ GF 1  W a 2ð1  a=WÞ

1  a=W > al =W

ð3Þ

R Wa Gf ða=WÞ ¼

0

g f ðxÞdx 1  a=W ¼ GF 2al =W W a

1  a=W 6 al =W

ð4Þ

where W is the total depth of the specimens and a is the initial notch depth. To obtain the values of GF of a concrete, the RILEM fracture energy Gf of the specimens of different sizes for a range of the notch to depth (a/W) ratios is first determined by the work-of-fracture method. Then Eqs. (3) and (4) is applied for each specimen depth and a/W ratios. The number of the measured Gf(a/W) values is therefore much larger than the two unknowns GF and a⁄l in Eqs. (3) and (4). This gives an overdetermined system of equations which is solved by a least square method to obtain the best estimates of GF and is shown later in Section 5.

52

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

Fig. 2. Local fracture energy model of Hu and Wittmann [4].

2.3. Energy release rate Although the size independent fracture energy can be estimated by extrapolating the results of research investigations on laboratory size specimens but the range of specimen sizes is still being debated among various researchers. The concrete material ahead of crack tip develops relatively large FPZ which undergoes progressive softening due to micro-cracking. The assumption made in this approach is that the fracture energy is dissipated in the fracture process zone which exists in the un-cracked ligament of TPB specimens [14]. Based on the RILEM fracture energy values at various a/W ratios, a relationship between fracture energy release rate and the un-cracked ligament length is developed to estimate the size independent fracture energy GF. From the developed relationship, the fracture energy is found to reach almost a constant value implying that the fracture energy remains almost constant at larger ligament lengths. In other words, when ligament length becomes sufficiently large to accommodate the fully developed FPZ, fracture energy becomes constant being independent of specimen size and a/W. The fracture energy is proportional to the length of the FPZ size (lFPZ) for a given strength of concrete assuming constant width of FPZ. The relationship is given as:

Gf a lFPZ Similarly un-cracked ligament length: (W–a) a W From above relations;

Gf lFPZ ¼k W a W G

G

f f where k is a constant, and Wa is proportional to lFPZ . A curve of Wa vs. (W–a) is plotted (Fig. 3) from the RILEM fracture energy W values and plot is observed to follow a power law and almost asymptotic with the axis representing (W–a) for larger values of (W–a). The equation of the curve is:

Gf ¼ CðW  aÞb W a

ð5Þ

In which C and b are constants. As the un-cracked ligament length increases, Gf/(W–a) almost attains a constant value as is clear from Fig. 3. The size independent fracture energy is estimated by the following expression [14]:



 Gf ðW  aÞLarge ðW  aÞ ðWaÞ¼large

ð6Þ

Gf Gf /(W-a)

GF ¼

W −a

= C (W − a) − β

Uncracked ligament (W-a) Fig. 3. Typical curve following the power law.

53

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

3. Specimen material property and experimental setup The RILEM fracture test on geometrically similar (constant length to depth ratio) TPB plain concrete beams were conducted by Raghu Prasad [10] and Muralidhara [14]. The present work involves the numerical modeling of those tested TPB beams for investigation of size independent fracture energy. The mix proportion of M45 concrete grade prepared at BARC, Tarapur site is given in Table 1. The cement used was 43 grades Portland cement conforming to Indian Standard IS: 8112-1689. The dimensions of geometrically similar beams designated as D1, D2 and D3 are as shown in Table 2. Two different concrete mixes were prepared for both D1 and D2 using the maximum aggregate size of 12.5 mm and 20 mm. Further, three identical beams for each a/W value (Table 2) were cast for both D1 and D2. For D3, three identical specimens were prepared for each a/W, namely, 0.25 and 0.33 using the concrete mix prepared with maximum aggregate size of 20 mm only. Each beam was tested under crack mouth opening displacement (CMOD) control. To measure the central deflection, Linear Variable Differential Transducer (LVDT) was used. The D1 and D2 beams were tested in servo hydraulic Dartec machine of 500 kN capacity under three point bend condition. The servo controlled MTS machine of 1200 kN capacity was employed for D3 beam testing. Each beam was carefully placed on the roller supports. After everything was set, load, CMOD and time data were acquired at certain interval of time. The acquisition of load and displacement parameters was simultaneous. 4. Concrete material model and finite element analysis (FEA) The material behavior is incorporated by smeared/fictitious crack approach that has the capability for modeling concrete in all types of structures, including beams, trusses, shells and solids. The concrete smeared model does not track individual cracks; it rather performs the constitutive calculations independently at each integration point of the finite element model [15]. The post-failure behavior associated with the strain-softening behavior is incorporated by applying a fracture energy cracking criterion in the analysis [15,16]. The compressive and cracking behavior of concrete are incorporated by the uni-axial stress–strain response as shown in Fig. 4a–b. The details of characteristic of concrete mixtures are derived from uni-axial test on cylindrical specimens [17]. The fictitious crack method is applied in the present work to determine the response of the notched plain concrete beams under three-point bending using the linear strain softening stress-crack separation displacement (r–w) relationship. In the fictitious crack model, the tensile strength, fracture energy and stress-crack separation displacement relationship are incorporated to study the fracture behavior of plain concrete. The stress-crack separation displacement is one of the fundamental properties required to introduce the non-linearity in the fictitious crack model. The constitutive law that relates the cohesive stress (r) across the crack faces and the corresponding crack opening displacement w, i.e., r = f(w) is also known as softening function. The cohesive pressure r(w) is a monotonically decreasing function of crack separation displacement w and it tends to close the crack. The value of r(w) is equal to material tensile strength ft for w = 0 at the crack tip. A typical sketch of r–w curve is shown in Fig. 5a where the terminal point wc is the maximum widening of the crack separation displacement. The fictitious crack approach assumes that energy (fracture energy  Gf) produced by the applied load is completely balanced by the cohesive pressure as [11,12]:

Gf ¼

Z

w

rðwÞdw

ð7Þ

0

Table 1 Quantity of materials per cubic meter of concrete [14]. Property

Mix-with 20 mm and down size coarse aggregates

Mix-with 12.5 mm and down size coarse aggregates

Cement (kg) (c) Coarse aggregate (kg) (CA) 20 mm Coarse aggregate (kg) (CA) 12.5 mm Fine aggregate (kg) (FA) Water (kg) Superplasticizer (% weight of cement content) Water/cement ratio Mix proportion (c:FA:CA)

400 492 492 902 152 1.4 0.38 1:2.26:2.46

435 – 946 870 165 1.4 0.38 1:2:2.18

Table 2 Geometrical dimensions of beam. Type

Length (mm)

Depth (mm)

Width (mm)

Span (mm)

a/W

D1 D2 D3

375 750 3000

95 190 750

47.5 95 375

282 564 2250

0.05, 0.25, 0.33 0.05, 0.25, 0.33 0.25, 0.33

54

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

50

4

Stress (MPa)

Stress (MPa)

40 30 20 10 0

0

0.002

0.004

0.006

3 2 1 0 0

0.001

Strain

Strain

(a)

(b)

0.002

Fig. 4. Typical uni-axial (a) compressive and (b) tensile stress–strain curve for concrete.

Fig. 5. (a) Typical sketch of concrete softening (b) linear r–w relationship.

The softening model requires a unique r(w) curve to quantify the value of energy dissipation. The linear softening law by Hillerborg [11,12] as shown in Fig. 5b, is used in the present analyses considering the tensile strength (ft) and Gf as the material properties. From Fig. 5b, the maximum crack opening (wc) before the crack ceases to transfer stresses is estimated using values of ft and Gf. The linear softening curve utilized in the present study is:



r ¼ ft 1 

 w ; for 0 6 w 6 wc wc

r ¼ 0; for w P wc and wc ¼

2Gf ft

ð8Þ

Finite element method with constant strain triangle (CST) elements is adopted for analyses of the concrete beams. Finite element in house code has been used. FEA performed based on the strain softening model yields the mesh-sensitive results. Mesh insensitive solutions could be easily achieved by incorporating the fracture energy based softening model of concrete as proved in literature [1–3]. The FE formulation of the TPB, D1 type notch beam (Table 2) having size of 95 mm (depth)  47.5 mm (thickness)  375 mm (length) with span of 282 mm for a/W = 0.25 is performed by incorporating the fracture energy based softening model. The material property of plain concrete beam samples is as shown in Table 3. In Fig. 6 the response of coarse mesh (20 mm element size), intermediate mesh (15 mm element size) and fine mesh (8 mm element size) is found to be almost mesh in-sensitive. The maximum load predicted numerically for D1 type beam is in range 3.3 to 3.6 kN which is very close to the experimentally observed value in range 3.5 to 3.8 kN. Similarly the numerical simulation of all the beams mentioned in Table 2 is performed to address the mesh sensitivity. 5. Results and discussion The finite element modeling of the TPB geometrical specimens (Table 2) with the a/W = 0.05, 0.25 and 0.33 has been carried out. The concrete behavior is incorporated in the numerical simulation of TPB specimen by assigning the concrete

55

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62 Table 3 Material parameters utilized in FE numerical simulation. Material parameters

Value

fck (MPa) E (N/m2) Poisson ratio (m) ft (MPa) Gf (N/m)

45 2.5e10 0.2 3.5 100

4

Load (kN)

3

2

Intermediate Fine Coarse

1

0

0

0.5

1

Displacement (mm) Fig. 6. Mesh in-sensitive result of D1 type beam with a/W = 0.25.

Table 4 Experimental and numerical results for D1 type beam. Exp. max load (kN)

FEM. max load (kN)

Exp. disp. at max load (mm)

FEM disp. at max load (mm)

Exp Gf (N/m)

FEM Gf (N/m)

D1P12.5A1 D1P12.5A2 D1P12.5A3

6.89 6.31 6.7

6.85

0.15 0.12 0.14

0.12

212 179 –

213

D1P12.5B2 D1P12.5B3 D1T12.5B2 D1T12.5B3

3.89 3.6 3.77 3.58

3.1

0.115 0.09

0.092

122 116

136

D1P12.5C1 D1P12.5C2 D1P12.5C3 D1T12.5C1 D1T12.5C2 D1T12.5C3

2.4 2.5 2.6 3.08 3.48 3.15

2.53

0.10 0.05 – 0.08 0.113 0.108

0.082

107 121 – 142 150 144

130

D1P20A2 D1T20A2

7.5 6.49

7.24

0.051

0.1

– –

240

D1P20B1 D1P20B3 D1T20B1 D1T20B2 D1T20B3

3.2 3.24 3.2 3.48 3.24

3.4

0.05 0.043 – – 0.09

0.09

212 185 – – 146

154

D1P20C1 D1P20C2 D1P20C3 D1T20C2 D1T20C3

2.7 2.69 2.73 2.63 3.1

2.8

0.11 0.053 – 0.05 0.12

0.083

153 159 – – 137

152

0.13

162

fracture properties as shown in Table 3. The nomenclature employed for various types of beam in Tables 4–6 is as follows: for example, in D1P12.5A, the first two letters (e.g. D1 here) represent the type, P and T indicate pour mix and trial mix respectively, the number 12.5 indicate the maximum size of the aggregate, beams with a/W = 0.05, 0.25 and 0.33 are designated by A, B and C respectively and at last ‘‘1’’ ‘‘2’’ and ‘‘3’’ designates the serial number of identical samples. The detailed results are described below.

56

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

Table 5 Experimental and numerical results for D2 type beam. Exp. max load (kN)

FEM. max load (kN)

Exp. disp. at max load (mm)

FEM disp. at max load (mm)

Exp Gf (N/m)

FEM Gf (N/m)

D2P12.5A1 D2T12.5A2 D2T12.5A3

21 25.2 26

25

0.2 – –

0.14

– – –

200

D2P12.5B1 D2P12.5B2 D2P12.5B3 D2T12.5B1 D2T12.5B2 D2T12.5B3

10.33 10.48 10.57 9.3 9.8 10.65

10.4

0.13 0.12 0.108 0.11 – –

0.105

130 114 141 122 – –

121

D2P12.5C1 D2P12.5C2 D2P12.5C3 D2T12.5C1 D2T12.5C3

9.35 8.7 9.7 7.5 7

8.3

0.16 0.18 0.13 0.158 –

0.11

132 125 135 140 –

113

D2P20A1

23.7

27

–

0.17

–

226

D2P20B2 D2T20B2 D2T20B3

7.4 9.5 9.8

11.28

0.19 0.11 –

0.13

144 151 –

123

D2P20C2 D2P20C3 D2T20C1

8 8 5.54

9.16

0.2 0.17 0.2

0.14

140 158 85

118

Table 6 Experimental and Numerical results for D3 type beam. Exp. max load (kN)

FEM. max load (kN)

Exp. disp. at max load (mm)

FEM disp. at max load (mm)

Exp Gf (N/m)

FEM Gf (N/m)

D3T20B1 D3T20B2 D3T20B3

110.58 101 103.4

114

0.36 0.4 0.32

0.28

168 109 124

83

D3T20C1 D3T20C2 D3T20C3

98.4 80.38 84.56

93

0.28 0.3 0.33

0.25

145 95 141

64

5.1. Load vs. load point displacement response and RILEM fracture energy (Gf) The numerically and experimentally observed response of load vs. load point vertical displacement of D1 beam for a/W = 0.25 and a/W = 0.33, prepared with the concrete mix of 12.5 mm maximum aggregate size, are shown in Fig. 7. The numerically and experimentally observed plot of load vs. load point vertical displacement of D1 beam, prepared with the concrete mix of 20 mm maximum aggregate size, for a/W = 0.25 and a/W = 0.33 are shown in Fig. 8. In Figs. 7 and 8, the ‘‘experiment-1’’, ‘‘experiment-2’’, ‘‘experiment-3’’ designate the experimental results of identical specimens and ‘‘numerical’’ designates the results of finite element simulation. There exists some scatter in elastic range of the experiments as clear from Figs. 5 and 6 which is not uncommon and is also reported in literature [18,19]. Similarly the loads vs. load point vertical displacement curves of other TPB beams of Table 2 are obtained. The numerically predicted and experimentally observed maximum load is found to be in excellent agreement in most of the specimens for D1, D2 and D3 as is clear from Tables 4–6. From Tables 4 and 5, it is being noted that vertical displacement at maximum load is in excellent agreement in almost all data set of D1 and D2 with a/W = 0.25 and a/W = 0.33 except with a/W = 0.05. Too small a/W and too large a/W are not preferable for concrete fracture test due to boundary effect, which is reported in literature [9]. Since the a/W = 0.25 and a/W = 0.33 are away from the boundary, therefore best suited to estimate the size independent fracture energy. Based on the numerically predicted and experimentally observed load vs. load point displacement response of TPB specimens of Table 2, the RILEM fracture energy (Gf) values are evaluated as per Eq. (1). The fracture energy estimated through the load vs. load line displacement response obtained from the finite element simulation and set of experiments for D1, D2 and D3 type beam is also presented in Tables 4–6 where, the ‘‘Exp’’ and ‘‘FEM’’ designate the results of experiment and finite element method respectively, ‘‘Max’’ and ‘‘Disp’’ stand for the maximum and displacement respectively. 5.2. Size independent fracture energy from presently proposed methodology The limited number of D3 specimens were tested due to its heavy weight and handling issues. Hence, focus has been put mainly on D1 and D2 beams having plenty of data sets to estimate the size independent fracture energy. The RILEM fracture

57

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

4

3

D1-numerical D1-12.5 experiment-1 D1-12.5 experiment-2 D1-12.5 experiment-3

2

Load (kN)

Load (kN)

3

0

0.2

0.4

0.6

D1-12.5-numerical D1-12.5-experiment-1 D1-12.5-experiement-2 D1-12.5-experiement-3

1

1

0

2

0

0.8

0

Displacement (mm)

0.2

0.4

0.6

0.8

Displacement (mm)

(a)

(b)

Fig. 7. Numerical and experimental load vs. load line displacement response comparison of D1 beam cast with 12.5 mm aggregate size for (a) a/W = 0.25 and (b) a/W = 0.33.

3

2

2

Load (kN)

Load (kN)

3

D1-numerical D1-20 experiment-1 D1-20 experiment-2 D1-20 experiment-3

1

1

0

0

0.5

1

D1-20-numerical D1-20-experiment-1 D1-20-experiment-2 D1-20-experiment-3

1.5

Displacement (mm)

(a)

0

0

0.5

1

1.5

Displacement (mm)

(b)

Fig. 8. Numerical and experimental load vs. load line displacement response comparison of D1 beam cast with 20 mm aggregate size for (a) a/W = 0.25 and (b) a/W = 0.33.

energy (Gf) values evaluated from numerically predicted and experimentally observed load–deflection data are presented in Fig. 9. The averaging of RILEM fracture energy values as per Eq. (2) performed for various combinations such as: all numerical datasets, experimental datasets and experimental datasets excluding outlier are shown in Table 7. Averaging the numerical values of RILEM fracture energy as per Eq. (2) of all the specimens with a/W = 0.25 and a/W = 0.33, the size independent fracture energy (GF) is found to be 134 N/m and 128 N/m respectively shown in Table 7. Similarly the various band of GF along with coefficient of variation (COV) is estimated by averaging the RILEM fracture energy as per the Eq. (2) which is presented in Table 7. From Table 7, by excluding the size independent fracture energy with high COV value, the band of GF is found to be 126–145 N/m. Also from Table 7 for experimental and numerical results excluding the outliers, the band of GF with a/W = 0.25 and a/W = 0.33 is found to be 134136 N/m. Thus considering the sets of RILEM fracture energy (Gf) with a/W = 0.25 and a/W = 0.33, the unique value of size independent fracture energy in range of 134–136 N/m is estimated. 5.3. Size Independent fracture energy: bilinear approximation The RILEM fracture energy Gf for around 49 beam specimens having three different depths and three different a/W ratios are determined as illustrated in Section 5.1 using Eq. (1). In Eqs. (3) and (4) the number of unknowns are two, namely, GF and

58

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

Fracture Energy (N/m)

200 150

D1-12.5 D1-20 D2-12.5 D2-20 D3-20 D1-12.5N D1-12.5N D2-12.5N D2-20N D3-20N

100 50 0

0

0.1

0.2

0.3

0.4

a/W Fig. 9. Experimentally and numerically observed RILEM fracture energy considering all data sets.

Table 7 Assessment of size independent fracture energy (GF). Combinations

Mean fracture energy (N/m) = GF and coefficient of variation = COV a/W = 0.25

All numerical results Numerical results excluding D3 All experimental results Experimental results excluding outliers Experimental results excluding outliers and D3 Experimental and numerical results Experimental and numerical results excluding outliers and D3

a/W = 0.33

GF

% COV

GF

% COV

134 123 145 138 139 131 136

11.5 21 8.6 5.5 5.8 15 11

128 115 139 137 140 126 134

13.7 28 10 6.7 5.8 20 10

Table 8 Estimated GF from experimental Gf including all outlier. W (mm)

Gf (N/m) values for a/W = 0.25

Gf (N/m) values for a/W = 0.33

Gf (N/m) values for a/W = 0.25 and 0.33

94

122, 116, 162, 212, 185, 188, 89

188

112, 114, 141, 122, 174, 198, 144, 151, 190 168, 109, 124 137 18

107, 121, 170, 142, 150, 144, 153, 159, 164, 137 132, 125, 135, 140, 159, 140, 158, 85

122, 170, 112, 132, 168, 132 10.7

750 GF (N/m) % COV

145, 95, 141 124 11.4

116, 142, 114, 125, 109,

162, 150, 141, 135, 124,

212, 144, 122, 140, 145,

185, 188, 153, 159, 174, 198, 159, 140, 95, 141

89, 107, 121, 164, 137 144, 151, 190, 158, 85

Table 9 Estimated GF from experimental Gf excluding outlier. W (mm)

Gf (N/m) values for a/W = 0.25

Gf (N/m) values for a/W = 0.33

Gf (N/m) values for a/W = 0.05, 0.25 and 0.33

Gf (N/m) range for a/W = 0.25 and 0.33

94

122, 116, 162, 212, 185

188

114, 141, 122, 144, 151

750 GF (N/m) % COV

168, 109, 124 124 18

107, 144, 132, 158, 145, 124 11.4

212, 142, 114, 140, 168, 127 11

122, 150, 114, 140, 168, 126 11.2

121, 142, 150, 153, 159, 137 125, 135, 140, 85 95, 141

179, 150, 141, 158, 109,

122, 144, 122, 85 124,

116, 162, 212, 185, 107, 121, 153, 159, 137 144, 151, 132, 125, 135, 145, 95, 141

116, 144, 141, 158, 109,

162, 153, 122, 85 124,

212, 185, 107, 121, 142, 159, 137 144, 151, 132, 125, 135, 145, 95, 141

a⁄l , whereas the measured Gf(a/W) values are more. For this reason, the over determined system of equations is solved by a least square method to obtain the best estimates of GF. Tables 8–10 show the GF values with coefficient of variation (COV) for various combinations based on a/W values, and GF ranging from 110 to 137 N/m for a/W = 0.25 and 0.33 is observed. Too small a/W is not preferable for concrete fracture test as reported in literature [9] which is clear from Table 10 where the GF value for a/W = 0.05 is observed to be 215 N/m with high COV. From Tables 8–10 the exact average estimates of GF for a/W = 0.25 and 0.33 is found to be 123 N/m. On excluding the outlier the exact average estimate of GF for a/W = 0.25 and

59

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62 Table 10 Estimated GF from experimental Gf and FEA Gf excluding outlier. W (mm)

Gf (N/m) values for Gf (N/m) values for a/W = 0.05 a/W = 0.25

Gf (N/m) values for a/W = 0.33

Gf (N/m) values for a/W = 0.05, 0.25 and 0.33

94

213, 240, 212, 179 136, 154, 122, 116, 162, 130, 152, 107, 121, 142, 213, 240, 212, 179, 136, 154, 212, 185 150, 144, 153, 159, 137 122, 116, 162, 212, 185, 130, 152, 107, 121, 142, 150, 144, 153, 159, 137 188 200, 226 121, 123, 114, 141, 122, 113, 118, 132, 125, 135, 200, 226, 121, 123, 114, 141, 144, 151 140, 158, 85 122, 144, 151, 113, 118, 132, 125, 135, 140, 158, 85 750 – 83, 168, 109, 124 64, 145, 95, 141 83, 168, 109, 124, 64, 145, 95, 141 GF (N/m) 215 113 110 125 % COV 23 15 12.8 12.4

Gf (N/m) values for a/W = 0.25 and 0.33 136, 154, 122, 116, 162, 212, 185, 130, 152, 107, 121, 142, 150, 144, 153, 159, 137 121, 123, 114, 141, 122, 144, 151, 113, 118, 132, 125, 135, 140, 158, 85 83, 168, 109, 124, 64, 145, 95, 141 113 10

Table 11 Numerically and experimentally observed values of Gf and G/W–a. Beam type

a/W

W–a (mm)

Gf (N/m)

Gf/W–a (N/m2)

FEA D1P12.5A D1P12.5B D1P12.5C D1P20A D1T20B D1P20C D2T12.5A D2T12.5B D2P12.5C D2P20A D2P20B D2T20C D3T20B D3T20C

0.05 0.25 0.33 0.05 0.25 0.33 0.05 0.25 0.33 0.05 0.25 0.33 0.25 0.33

89.3 70.5 62.98 89.3 70.5 62.98 178.6 141 125.96 178.6 141 125.96 562.5 502.5

213 136 130 240 154 152 200 121 113 226 123 118 83 64

2.385218 1.929078 2.064147 2.68757 2.184397 2.413465 1.119821 0.858156 0.89711 1.265398 0.87234 0.936805 0.147556 0.127363

Experimental D1P12.5A1 D1P12.5A2 D1P12.5B2 D1P12.5B3 D1T12.5B3 D1P20B1 D1P20B3 D1T20B3 D1P12.5C1 D1P12.5C2 D1T12.5C1 D1T12.5C2 D1T12.5C3 D1P20C1 D1P20C2 D1T20C3 D2P12.5B2 D2P12.5B3 D2T12.5B1 D2P12.5B1 D2P20B2 D2T20B2 D2P12.5C1 D2P12.5C2 D2P12.5C3 D2T12.5C1 D2P20C2 D2P20C3 D2T20C1 D3T20B1 D3T20B2 D3T20B3 D3T20C1 D3T20C2 D3T20C3

0.05 0.05 0.25 0.25 0.25 0.25 0.25 0.25 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.25 0.25 0.25 0.25 0.25 0.25 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.25 0.25 0.25 0.33 0.33 0.33

89.3 89.3 70.5 70.5 70.5 70.5 70.5 70.5 62.98 62.98 62.98 62.98 62.98 62.98 62.98 62.98 141 141 141 141 141 141 125.96 125.96 125.96 125.96 125.96 125.96 125.96 562.5 562.5 562.5 502.5 502.5 502.5

212 179 122 116 162 212 185 146 107 121 142 150 144 153 159 137 114 141 122 130 144 151 132 125 135 140 140 158 85 168 109 124 145 95 141

2.37402 2.004479 1.730496 1.64539 2.297872 3.007092 2.624113 2.070922 1.698952 1.921245 2.254684 2.381708 2.28644 2.429343 2.524611 2.175294 0.808511 1 0.865248 0.921986 1.021277 1.070922 1.047952 0.992379 1.071769 1.111464 1.111464 1.254366 0.674817 0.298667 0.193778 0.220444 0.288557 0.189055 0.280597

60

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

3

Gf /W-a

Gf /W-a

3

2 y = 260.72x-1.13

1

0

0

200

2 y = 192.9x-1.06

1

400

600

0

0

200

400

Uncracked Ligament (W-a)

Uncracked Ligament (W-a)

(a)

(b)

600

Fig. 10. Plot of Gf/(W–a) vs. (W–a) consisting (a) experimental and FEA Gf data (b) purely experimental Gf data.

Table 12 Values of slope and fracture energy for different values of ligament length using the curve in Fig. 10(a). W–a (mm)

Gf (N/m)

Gf/W–a (N/m2)

d{Gf/W–a}/d(W–a) (N/m3)

40 50 70 90 125 140 200 300 400 500 600 700 800 900 1000 1300 1500

161.3887 156.7743 150.0646 145.2411 139.1691 137.1338 130.9203 124.1982 119.6391 116.2184 113.4962 111.2444 109.33 107.6687 106.204 102.6428 100.751

4.034719 3.135486 2.14378 1.61379 1.113353 0.979527 0.654602 0.413994 0.299098 0.232437 0.18916 0.158921 0.136662 0.119632 0.106204 0.078956 0.067167

0.08992 0.04959 0.0265 0.0143 0.00892 0.00542 0.00241 0.00115 0.00067 0.00043 0.0003 0.00022 0.00017 0.00013 9.1E05 5.9E05 4.2E05

0.33 is found to be 119 N/m from Tables 9 and 10. Also for a/W = 0.25 and a/W = 0.33 separately the exact average estimate of GF is found to be 119 N/m and 117 N/m respectively. 5.4. Size independent fracture energy: energy release rate G

f The estimated RILEM fracture energy Gf and Wa values of about 49 beams, having three different depths and three different a/W ratios (see Table 2), are shown in Table 11. The size independent fracture energy GF is evaluated based on the

G

G

f f vs. (W–a). Wa vs. (W–a) data points are plotted and relationship between them is derived. relationship between Wa Fig. 10a depicts the relationship between Gf/(W–a) and (W–a) from experimental and FEA data sets of Gf whereas Fig. 10b illustrates the same for experimental data sets of Gf only. Two power law equations generated by best fit of these data in Fig. 10a–b are as follows:

Gf ¼ 260:7  ðW  aÞ1:13 W a

ð9Þ

Gf ¼ 192:9  ðW  aÞ1:06 W a

ð10Þ

From the

Gf Wa

vs. (W–a) data, slope

dðGf =ðWaÞÞ dðWaÞ

is calculated by differentiating Eqs. (9) and (10), which are extrapolated

beyond test range till (W–a) = 1800. Tables 12 and 13 show the values of slope and fracture energy for different ligament lengths (W–a) for the datasets in Fig. 10a–b respectively. Data of Tables 12 and 13 are plotted in the form of variation of dðGf =ðWaÞÞ dðWaÞ

vs. (W–a) and shown in Fig. 11a–b. From Fig. 11a–b, the rate of decrease of slope of the curve approaches to almost

61

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62 Table 13 Values of slope and fracture energy for different values of ligament length using the curve in Fig. 10(b). W–a (mm)

Gf (N/m)

Gf/W–a (N/m2)

d{Gf/W–a}/d(W–a) (N/m3)

40 50 70 90 125 140 200 300 400 500 600 700 800 900 1000 1300 1500 1800

154.5997 152.5436 149.4949 147.2576 144.3835 143.4051 140.3687 136.9951 134.6507 132.8599 131.4145 130.2046 129.1656 128.256 127.4478 125.4572 124.3846 123.0314

3.864992 3.050872 2.135641 1.636195 1.155068 1.024322 0.701844 0.45665 0.336627 0.26572 0.219024 0.186007 0.161457 0.142507 0.127448 0.096506 0.082923 0.068351

0.08141 0.04576 0.02497 0.01375 0.00872 0.00537 0.00245 0.0012 0.00071 0.00047 0.00033 0.00025 0.00019 0.00015 0.0001 6.8E05 4.9E05 3.6E05

0

1000

2000

3000

0

1000

2000

3000

0

(W-a)-mm

-0.01

(W-a)-mm

-0.004

-0.012

-0.05 -0.07

-0.016

d{Gf/W-a}/d(W-a)

-0.008

d{Gf/W-a}/d(W-a)

-0.03

-0.09

(a)

(b)

Fig. 11. A plot of d{Gf/W–a}/d(W–a) vs. (W–a) from the results of (a) Table 12 (b) Table 13.

zero when compared with the previous values and hence the Gf/(W–a) could be considered as almost a constant at larger values of (W–a). This implies that the fracture energy variation over larger values of (W–a) is negligibly small. From Fig. 11a–b it is clear that the slope almost tends to zero for ligament (W–a) is around 1000 mm. Using Eqs. (5) and (6), the GF value corresponding to large un-cracked ligament length is estimated and found to be in range 106–125 N/m. 5.5. Comparison of size independent fracture energy values The range of size independent fracture energy (GF) from the presently proposed methodology based on averaging of RILEM fracture energy is found to be in range of 134–136 N/m. GF values from the bi-linear approximation and energy release rate are observed to be in range 119–117 N/m and 106–125 N/m respectively. It is seen that fracture energy values evaluated by these three different methods are reasonably close to each other. 6. Conclusion The size-independent fracture energy of concrete is the most useful parameter in the analysis of cracked concrete structure which is investigated in the present work. The finite element simulations of the TPB specimens of different sizes with a/ W = 0.05, 0.25 and 0.33 are performed by incorporating the concrete properties based on fracture energy strain softening model. Following conclusions are drawn from the above study:  The numerically predicted and experimentally observed maximum load and vertical displacement at maximum load are found to be in excellent agreement.  The estimated RILEM fracture energy (Gf) values of concrete from the numerically predicted and experimentally observed load–load line displacement curves are found to be in reasonably good agreement.

62

N. Trivedi et al. / Engineering Fracture Mechanics 138 (2015) 49–62

 The RILEM fracture energy values have been averaged out over various a/W values and beam sizes for geometrically similar (constant length to depth ratio) beams and a methodology is developed that investigates the size-independent fracture energy (GF) of concrete. The present work develops an easy and robust technique for the determination of the size independent fracture energy of concrete. The study also invokes a primary approach based on concrete fracture mechanics that describes the nonlinear aspects of concrete behavior through load–load line responses for a range of a/W ratios and help to understand the failure mechanisms and load bearing capacity of concrete structures.  The size-independent fracture energy of concrete is also investigated by the other methodologies based on Hu and Wittmann bi-linear model and fracture energy release rate.  A comparative analysis of the three methods used to measure the size-independent fracture energy of concrete has been carried out. The observed values of GF from the three methodologies are found to be in reasonable agreement. It is, therefore, concluded that either method can be used to obtain a unique value of the size-independent fracture energy of concrete due to consistent trend of fracture energy values evaluated by all methods. The numerical results in post peak zone may differ if the numerical analysis is performed by incorporating the different softening models such as exponential, non-linear, bi-linear (Petersson) and bi-linear (CEB-FIP). However the estimated size independent fracture energy should not differ much. References [1] Bazant ZP, Planas J. Fracture and size effect in concrete and other quasi-brittle materials. Florida: CRC Press; 1998. [2] Singh RK, Basha SM, Singh Rajesh K. Fracture energy and size effect studies for nuclear concrete structures. In: Proc SMiRT-20, Espoo, Espoo, Finland; 2009. [3] Trivedi N, Singh RK. Prediction of impact induced failure modes in reinforced concrete slabs through nonlinear transient dynamic finite element simulation. Ann Nucl Energy 2013;56:109–21. [4] Duan Kai, Xiaozhi Hu, Wittmann FH. Boundary effect on concrete fracture and non-constant fracture energy distribution. Engng Fract Mech 2003;70:2257–68. [5] Elices Manuel, Planas Jaime. Fracture mechanics parameters of concrete. Adv Cem Based Mater 1996;4:116–27. [6] Sundara Raja Iyengar KT, Raghu Prasad BK, Nagaraj TS, Patel Bharti. Parametric sensitivity of fracture behaviour of concrete fracture mechanics parameters of concrete. Nucl Engng Des 1996;163:397–403. [7] Luigi Cedolin, Gianluca Cusatis. Identification of concrete fracture parameters through size effect experiments. Cement Concr Compos 2008;30:788–97. [8] Muralidharan S, Raghu Prasad BK, Hamid Eskandari, Karihaloo BL. Fracture process zone size and true fracture energy of concrete using acoustic emission. Constr Build Mater 2010;24:479–86. [9] Karihaloo BL, Abdalla HM, Imjai T. A simple method for determining the true specific fracture energy of concrete. Mag Concr Res 2003;55:471–81. [10] Raghu Prasad BK. Experimental evaluation of fracture properties of concrete. Interim progress report under collaborative research project between BARC and Indian Institute of Science, Bangalore; 2009. [11] Shah SP, Swartz Stuart E, Chengsheng Ouyang. Application of fracture mechanics to concrete⁄rock and other quasi-brittle materials. Wiley Interscience; 1995. [12] Shailendra Kumar, Sudhirkumar Barai V. Concrete fracture models and applications. Berlin Heidelberg: Springer-Verlag; 2011. [13] Cifuentes H, Alcalde M, Medina F. Measuring the size-independent fracture energy of concrete. An Int J Exp Mech 2013;49:54–9. [14] Muralidhara S. Fracture energy and process zone in plain concrete beams. PhD dissertation. Banglore: Indian Institute of Science; 2010. [15] Bangash MYH, Thomas Telford. Manual of numerical methods in concrete. Thomas Telford Publishing Ltd; 2001. [16] Mehta PKumar, Monteiro Paulo JM. Concrete microstructure property and materials. 3rd ed. McGraw-Hill; 1997. [17] Trivedi N, Singh RK. Assessment of in-situ concrete creep: cylindrical specimen and prototype nuclear containment structure. Constr Build Mater 2014;71:16–25. [18] Hoover G, Christian C, Bazant ZP, Jan Vorel, Roman Wendner, Mija Hubler H. Comprehensive concrete fracture tests: description and results. Engng Fract Mech 2013;114:92–103. [19] Hoover Christian G, Bazant ZP. Comprehensive concrete fracture tests: size effects of Types 1 & 2, crack length effect and postpeak. Engng Fract Mech 2013;110:281–9.