Issue 72

H. S. Vishwanatha et alii, Fracture and Structural Integrity, 72 (2025) 80-101; DOI: 10.3221/IGF-ESIS.72.07

model is an effective representation of the stress-COD relationship. The parameters of this model include the turning point stress σ 1 , w 0 , w 1 and f t .

Figure 17: The bilinear softening constitutive model of concrete.

௙ ௧

(µm) ௡

Average w 0

w 0

Average G f (N/m)

Beam ID

Iterations

Pc 51 53 54 69 74 76 73

P 7

P 8

P 9 54 51 47

P 10

1 2 3

50.35 47.00 53.15 50.17 5.01 83.34 90.12 88.51 87.32 3.31 92.33 88.02 90.67 90.34 1.96

13.99 13.05 14.76 13.93 5.02 23.15 25.03 24.59 24.26 3.31 25.64 24.45 25.19 25.09 1.96 40.56 33.93 32.11 35.53 10.22 41.39 34.81 35.59 37.26 7.88

58.2 59.8 59.8

52.1 52.15 50.75 3.83 48

45 49 48

B-SB75

52

3.73

Avg. CoV

52.67 2.37

59.27 1.27

50.67 5.66

47.33 3.59

1 2 3

66 74 70 70 95 97 85

68 77 70

74 73 71

80 78 72

B-MB150

72.8

3.00

Avg. CoV

71.67 5.38

72.67 1.72

76.67 4.43 99.8

4.03

4.67

1 2 3

95

98 94

100

100

95

88 80

B-LB250

98

90.6 94.20 3.21

85.4 93.47 6.48

93.30

3.72

Avg. CoV

97.67 2.10

92.33 5.69

89.27 9.11

1 2 3

146

128 145 138 137 5.09 171 175 165

136 131

137 132

101

129

122.15 115.62 127.92 10.21 125.32 128.11 134.14 149

98

125.8

B-VB500

120.6 129.20 4.96 187.8 185.5

113.6 127.53 7.89 195 188.9 193.5

96.1 98.37 2.05

124

123.68

3.48

Avg. CoV

126.27

1.64 240 235 220

1 2 3

222 212 208

B HB1000

176

198.30

5.32

Avg. CoV

170.3 2.41

183.10

192.47 214.00

231.67

7.88

2.79

1.35

2.75

3.67

Table 5: Evaluation of  n . The bilinear model is highly effective in simulating the softening properties of the concrete FPZ. Wittmann et al. [14] proposed a constitutive model for concrete softening based on this approach, observing that the ratio of σ 1 / f t for plain concrete is typically below 0.25. Asmaro [15], through inverse analysis and mode I fracture tests on plain concrete, determined that σ 1 / f t falls within the range of 0.4 to 0.53. Using the cohesive crack model, Huang [16] developed a numerical solution for the softening curve, concluding that σ 1 / f t is 0.25 and w 1 / w 0 is 0.4. Petersson [17] suggested that the stress-free crack opening displacement ( w 0 ) can be obtained by defining the cohesive stress and crack opening displacement at the kink point as f t /3 and 0.8 G f /f t , respectively, where f t represents tensile strength and G f denotes the fracture energy of concrete. The value of G f is calculated as the area under the load-displacement curve

96