# Toughness Calculation of Postfire Normal Concrete

Authors: Qingyang Chen; Anjing Tang; Zhoudao Lu
Journal: Advances in Materials Science and Engineering
 (2014)

---

## Abstract

Fracture tests of postfire normal concrete with ten temperatures up to 600°C are implemented. Residual fracture toughness using analytical method is determined. Two situations are divided at critical load when calculating the cohesive fracture toughness. The initial and critical fracture toughness could be calculated from the complete load-crack opening displacement curves. Finally, the validation of double-K fracture model to the postfire concrete specimens is proved.


---

## Body

## 1. Introduction

The fracture process of concrete structures underwent three main stages: (i) crack initiation, (ii) stable crack propagation, and (iii) unstable fracture. Accordingly, the double-K fracture criterion initially introduced by Xu and Reinhardt [1] showed the crack initiation, crack propagation, and failure during a fracture process until the maximum load is reached. And the two size-independent parameters, initial cracking toughness, K
I
ini, and unstable fracture toughness, K
I
un, can be used to study the crack propagation of concrete.In order to determine the double-K fracture parameters analytically [2, 3] the value of cohesion toughness K
I
c due to cohesive stress distribution in the fictitious fracture zone was computed using method proposed by Jenq and Shah [4]. The influences of geometrical parameter [5–7] and size effect [2, 3, 8] on fracture toughness were studied by various researchers. It was found that the influence of a
o
/
D ratio and shape of test specimen on the values of fracture parameters were relatively less than the one of size effect.The influence of temperature on the fracture parameters was also considered by several researchers, but mainly on the fracture energy and material brittleness [9–13] and relatively fewer discussions on the fracture toughness [14, 15]. In the present paper, the calculation of residual fracture toughness of concrete is carried out. Wedge-splitting experiments of totally ten temperatures varying from 20°C to 600°C are implemented. The specimen sizes are of 230 × 200 × 200 mm with initial-notch depth ratios of 0.4. The validation of double-K fracture model to the postfire normal concrete is proved.


## 2. Analytical Determination of Cohesive Fracture Toughness

### 2.1. Effective Crack Extension Length and Residual Young’s Modulus

The linear asymptotic superposition assumption was considered in the analytical method presented by Reinhardt et al. [2, 3] to introduce the concept of linear elastic fracture mechanics for calculating the double-K fracture parameters. Detailed explanation of the above assumption can be found elsewhere [2].Based on this assumption, the value of the equivalent-elastic crack length for WS specimen is expressed as(1)
a
=
(
h
+
h
0
)
{
1
-
(
13.18
E
·
b
·
c
+
9.16
)
1
/
2
}
-
h
0
,
where c
=
CMOD
/
P is the compliance of specimens and b is specimens thickness; h is specimens height and h
0 is the thickness of the clip gauge holder. For calculation of critical value of equivalent-elastic crack length a
c, the value of crack mouth opening displacement (
CMOD
) and P is taken as CMOD
c and P
u, respectively.The residual Young’s modulusE is calculated using the P-CMOD curve as
(2)
E
=
1
b
c
i
[
13.18
×
(
1
-
α
)
2
-
9.16
]
,
where c
i
=
CMOD
ini
/
P
ini is the initial compliance before cracking, α
=
(
a
0
+
h
0
)
/
(
h
+
h
0
). The values of critical equivalent-elastic crack length a
c and residual Young’s modulus E are listed in Table 1.Table 1
The experimental results of fracture parameters.
Specimen
Temperature
P
ini/kN
P
max
⁡/kN
CMOD
c/mm
CTOD
c/mm
E GPa
G
F N/m
K
I
ini MPa·m1/2
K
I
C
-
A MPa·m1/2
K
I
un
-
E MPa·m1/2
K
I
un
-
A MPa·m1/2
WS6
65°C
6.98
11.31
0.195
0.078
21.73
425.91
0.550
1.007
1.594
1.557
WS7
3.88
8.23
0.163
0.100
24.79
482.62
0.303
1.63
1.664
1.933
WS8
6.88
10.41
0.212
0.087
19.43
487.75
0.557
1.006
1.518
1.563
WS9
7.94
10.71
0.164
0.087
23.25
480.51
0.511
1.174
1.685
1.685
WS10
6.32
11.67
0.229
0.086
16.60
522.36
0.562
0.903
1.507
1.465
Average
6.42
10.47
0.193
0.088
21.16
479.83
0.480
1.161
1.594
1.641
WS16
200°C
—
—
—
—
—
—
—
—
—
—
WS17
4.22
6.60
0.317
0.143
11.58
282.51
0.211
0.803
1.206
1.014
WS18
3.97
6.15
0.213
0.138
6.98
215.14
0.353
0.485
0.863
0.838
WS19
2.41
5.33
0.213
0.138
9.17
235.91
0.215
0.72
0.924
0.935
WS20
3.32
5.24
0.458
0.211
7.00
368.94
0.245
0.796
1.022
1.041
Average
3.50
5.72
0.352
0.157
8.68
376.95
0.256
0.688
1.044
0.957
WS31
400°C
1.55
3.61
1.032
0.492
2.56
532.09
0.138
0.572
0.795
0.710
WS32
1.78
3.19
1.218
0.548
1.42
396.93
0.158
0.333
0.574
0.491
WS33
2.37
3.27
0.768
0.299
2.12
434.89
0.174
0.236
0.621
0.410
WS34
2.36
4.53
0.736
0.302
1.71
598.93
0.125
0.356
0.468
0.494
WS35
—
—
—
—
—
—
—
—
—
—
Average
2.01
3.78
0.901
0.410
1.95
490.71
0.152
0.374
0.615
0.526
WS46
600°C
0.76
1.13
1.482
0.684
0.47
228.23
0.067
0.174
0.221
0.231
WS47
0.53
1.48
2.082
0.684
0.48
395.06
0.063
0.209
0.277
0.284
WS48
0.81
1.65
1.908
0.813
1.14
539.22
0.072
0.478
0.550
0.550
WS49
0.58
1.14
1.687
0.973
0.38
331.99
0.052
0.188
0.225
0.225
WS50
0.62
1.48
2.082
0.727
0.38
273.07
0.068
0.155
0.213
0.213
Average
0.62
1.38
1.848
0.799
0.57
353.51
0.064
0.241
0.297
0.301


### 2.2. Crack Opening Displacement along the Fracture Process Zone

Since the cohesive stress distribution along the fracture process zone depends on the crack opening displacement and the specified softening law, it is important to know the value of crack opening displacement along the fracture line. It is difficult to measure directly the value ofCOD along the fracture process zone; for practical purposes the value of COD
(
x
) at the crack length x is computed using the following expression [3]:
(3)
COD
(
x
)
=
CMOD
{
(
1
-
x
a
)
2
+
(
1.018
-
1.149
a
h
)
[
x
a
-
(
x
a
)
2
]
}
1
/
2
.For calculation of critical value of crack tip opening displacementCTOD
c, the value of x and a (see Figure 3) in (3) is taken to be a
o and a
c, respectively. The value of cohesive stress along the fictitious fracture zone to the corresponding crack opening displacement is evaluated using bilinear stress-displacement softening law as given in (6).


### 2.3. Softening Traction-Separation Law of Postfire Concrete

The softening traction-separation law is a prerequisite determine the double-K fracture parameters, at room temperature; many expressions have been proposed based on direct tensile tests [16–20]. Based on numerical studies, simplified bilinear expressions for the softening traction-separation law (illustrated in Figure 1) were suggested by Petersson in 1981 [16], Hilsdorf and Brameshuber in 1991 [19], and Phillips and Zhang in 1993 [20]. The area under the softening curve was defined as the fracture energy G
F. Therefore, one could get the following equation:
(4)
G
F
=
1
2
(
f
t
w
s
+
σ
s
w
0
)
,
where f
t is tensile strength of postfire specimens, w
s is the crack width at break point of softening curve, σ
s is the cohesive stress at the break point of softening curve, and w
0 is crack width at stress-free point.Figure 1
The bilinear softening traction-separation law.As a consequence, a general form of the simplified bilinear expression of the softening traction-separation law is given as follows:(5)
σ
=
f
t
-
(
f
t
-
σ
s
)
w
w
0
,
0
≤
w
≤
w
s
σ
=
σ
s
(
w
0
-
w
)
(
w
0
-
w
s
)
,
w
s
≤
w
≤
w
0
σ
=
0
,
w
≥
w
0
.Different values of the break point (σ
s, w
s) and the crack width w
0 at stress-free point were used for the expression proposed by different researchers. In the present work, the bilinear softening function of concrete proposed by Petersson is used for postfire specimens:
(6)
σ
s
=
f
t
3
,
w
s
=
0.8
G
F
f
t
,
w
0
=
3.6
G
F
f
t
.


### 2.4. Determination of Stress Intensity Factor Caused by Cohesive Force

The standard Green’s function [21] for the edge cracks with finite width of plate subjected to a pair of normal forces is used to evaluate the value of cohesive toughness. The general expression for the crack extension resistance for complete fracture associated with cohesive stress distribution in the fictitious fracture zone for Mode I fracture is given as follows:(7)
K
I
c
=
∫
a
0
a
2
σ
(
x
)
F
(
x
/
a
,
a
/
h
)
π
a
d
x
,
where
(8)
F
(
x
a
,
a
h
)
=
3.52
(
1
-
x
/
a
)
(
1
-
a
/
h
)
3
/
2
-
4.35
-
5.28
x
/
a
(
1
-
a
/
h
)
+
{
1.30
-
0.30
(
x
/
a
)
3
/
2
1
-
(
x
/
a
)
2
+
0.83
-
1.76
x
a
}
×
{
1
-
(
1
-
x
a
)
a
h
}
and σ
(
x
) is the cohesive force at crack length x, see Figure 3, and its expression is shown in (9) or (11).At critical condition the value ofa is taken to be a
c in (7) and (8). The integration of (8) is done by using Gauss-Chebyshev quadrature method because of existence of singularity at the integral boundary.As shown in Figure2, two conditions at critical load, that is, CTOD
c
≤
w
s and w
s
≤
CTOD
c
≤
w
c, may arise at the notch tip while using bilinear softening function. For specimens subjected to temperatures less than 120°C, the critical CTOD
c is less than w
s, whereas, for temperatures higher than 120°C, the critical CTOD
c is wider than w
s.(A)
When the criticalCTOD
c corresponding to maximum load P
u is less than w
s as shown in Figure 2(a), the distribution of cohesive stress along the fictitious fracture zone is approximated to be linear as shown in Figure 3(a). The variation of cohesive stress along the fictitious fracture zone for this loading condition, that is, a
o
≤
a
≤
a
c or 0
≤
CTOD
≤
CTOD
c, is written as
(9)
σ
(
x
)
=
σ
(
CTOD
c
)
+
(
f
t
-
σ
(
CTOD
c
)
)
(
x
-
a
0
)
(
a
c
-
a
0
)
,
where,σ
(
CTOD
c
) is the critical values of cohesive stress being at the tip of initialnotch. The value of σ
(
CTOD
c
) is determined by using bilinear softening function:
(10)
σ
(
CTOD
c
)
=
σ
s
(
w
s
)
+
w
s
-
CTOD
c
w
s
(
f
t
-
σ
s
(
w
s
)
)
.
(B)
When the criticalCTOD
c corresponding to maximum load P
u is wider than w
s as shown in Figure 2(b), the distribution of cohesive stress along the fictitious fracture zone is approximated to be bilinear as shown in Figure 3(b). The variation of cohesive stress along the fictitious fracture zone for this loading condition, also, a
o
≤
a
≤
a
c or 0
≤
CTOD
≤
CTOD
c, is written as
(11)
σ
1
(
x
)
=
σ
(
CTOD
c
)
+
(
σ
s
(
w
s
)
-
σ
(
w
)
)
(
x
-
a
0
)
(
a
s
-
a
0
)
,
a
s
≤
x
≤
a
0
,
σ
2
(
x
)
=
σ
s
(
w
s
)
+
(
f
t
-
σ
s
(
w
s
)
)
(
x
-
a
s
)
(
a
c
-
a
s
)
,
a
s
≤
x
≤
a
c
. The value of σ
(
CTOD
c
) is determined by using bilinear softening function:
(12)
σ
(
CTOD
c
)
=
w
0
-
CTOD
c
w
0
-
w
s
σ
s
(
w
s
)
.
The limits of integration of (7) should be taken in two steps: a
o
≤
x
≤
a
s for cohesive stress σ
1
(
x
) and a
s
≤
x
≤
a
c for cohesive stress σ
2
(
x
), respectively. The same Green’s function F
(
x
/
a
,
a
/
h
) for a given effective crack extension will be determined using (8). The calculated formula is listed as follows:
(13)
K
I
c
=
∫
a
0
a
s
2
σ
1
(
x
)
F
(
x
/
a
c
,
a
c
/
h
)
π
a
c
d
x
+
∫
a
s
a
c
2
σ
2
(
x
)
F
(
x
/
a
c
,
a
c
/
h
)
π
a
c
d
x
.
The effective crack length at break point a
s (shown in Figure 3(b)) is computed from the following nonlinear expression [4] by substituting COD
(
a
s
), CMOD, a
c and h:
(14)
COD
(
a
s
)
=
CMOD
{
(
1
-
a
s
a
c
)
2
+
(
1.018
-
1.149
a
c
h
)
[
a
s
a
c
-
(
a
s
a
c
)
2
]
}
1
/
2
,
where COD
(
a
s
) is the crack opening displacement at a
s, a
c is the effective crack length (according to (1)), and h is the specimen height.Two different situations forCTOD
c and w
s.
(a)
WhenCTOD
c
≤
w
s
(b)
WhenCTOD
c
>
w
sCohesive force distribution along the crack length at critical load.
(a)
The linear distribution of cohesive force
(b)
The bilinear distribution of cohesive force


## 2.1. Effective Crack Extension Length and Residual Young’s Modulus

The linear asymptotic superposition assumption was considered in the analytical method presented by Reinhardt et al. [2, 3] to introduce the concept of linear elastic fracture mechanics for calculating the double-K fracture parameters. Detailed explanation of the above assumption can be found elsewhere [2].Based on this assumption, the value of the equivalent-elastic crack length for WS specimen is expressed as(1)
a
=
(
h
+
h
0
)
{
1
-
(
13.18
E
·
b
·
c
+
9.16
)
1
/
2
}
-
h
0
,
where c
=
CMOD
/
P is the compliance of specimens and b is specimens thickness; h is specimens height and h
0 is the thickness of the clip gauge holder. For calculation of critical value of equivalent-elastic crack length a
c, the value of crack mouth opening displacement (
CMOD
) and P is taken as CMOD
c and P
u, respectively.The residual Young’s modulusE is calculated using the P-CMOD curve as
(2)
E
=
1
b
c
i
[
13.18
×
(
1
-
α
)
2
-
9.16
]
,
where c
i
=
CMOD
ini
/
P
ini is the initial compliance before cracking, α
=
(
a
0
+
h
0
)
/
(
h
+
h
0
). The values of critical equivalent-elastic crack length a
c and residual Young’s modulus E are listed in Table 1.Table 1
The experimental results of fracture parameters.
Specimen
Temperature
P
ini/kN
P
max
⁡/kN
CMOD
c/mm
CTOD
c/mm
E GPa
G
F N/m
K
I
ini MPa·m1/2
K
I
C
-
A MPa·m1/2
K
I
un
-
E MPa·m1/2
K
I
un
-
A MPa·m1/2
WS6
65°C
6.98
11.31
0.195
0.078
21.73
425.91
0.550
1.007
1.594
1.557
WS7
3.88
8.23
0.163
0.100
24.79
482.62
0.303
1.63
1.664
1.933
WS8
6.88
10.41
0.212
0.087
19.43
487.75
0.557
1.006
1.518
1.563
WS9
7.94
10.71
0.164
0.087
23.25
480.51
0.511
1.174
1.685
1.685
WS10
6.32
11.67
0.229
0.086
16.60
522.36
0.562
0.903
1.507
1.465
Average
6.42
10.47
0.193
0.088
21.16
479.83
0.480
1.161
1.594
1.641
WS16
200°C
—
—
—
—
—
—
—
—
—
—
WS17
4.22
6.60
0.317
0.143
11.58
282.51
0.211
0.803
1.206
1.014
WS18
3.97
6.15
0.213
0.138
6.98
215.14
0.353
0.485
0.863
0.838
WS19
2.41
5.33
0.213
0.138
9.17
235.91
0.215
0.72
0.924
0.935
WS20
3.32
5.24
0.458
0.211
7.00
368.94
0.245
0.796
1.022
1.041
Average
3.50
5.72
0.352
0.157
8.68
376.95
0.256
0.688
1.044
0.957
WS31
400°C
1.55
3.61
1.032
0.492
2.56
532.09
0.138
0.572
0.795
0.710
WS32
1.78
3.19
1.218
0.548
1.42
396.93
0.158
0.333
0.574
0.491
WS33
2.37
3.27
0.768
0.299
2.12
434.89
0.174
0.236
0.621
0.410
WS34
2.36
4.53
0.736
0.302
1.71
598.93
0.125
0.356
0.468
0.494
WS35
—
—
—
—
—
—
—
—
—
—
Average
2.01
3.78
0.901
0.410
1.95
490.71
0.152
0.374
0.615
0.526
WS46
600°C
0.76
1.13
1.482
0.684
0.47
228.23
0.067
0.174
0.221
0.231
WS47
0.53
1.48
2.082
0.684
0.48
395.06
0.063
0.209
0.277
0.284
WS48
0.81
1.65
1.908
0.813
1.14
539.22
0.072
0.478
0.550
0.550
WS49
0.58
1.14
1.687
0.973
0.38
331.99
0.052
0.188
0.225
0.225
WS50
0.62
1.48
2.082
0.727
0.38
273.07
0.068
0.155
0.213
0.213
Average
0.62
1.38
1.848
0.799
0.57
353.51
0.064
0.241
0.297
0.301


## 2.2. Crack Opening Displacement along the Fracture Process Zone

Since the cohesive stress distribution along the fracture process zone depends on the crack opening displacement and the specified softening law, it is important to know the value of crack opening displacement along the fracture line. It is difficult to measure directly the value ofCOD along the fracture process zone; for practical purposes the value of COD
(
x
) at the crack length x is computed using the following expression [3]:
(3)
COD
(
x
)
=
CMOD
{
(
1
-
x
a
)
2
+
(
1.018
-
1.149
a
h
)
[
x
a
-
(
x
a
)
2
]
}
1
/
2
.For calculation of critical value of crack tip opening displacementCTOD
c, the value of x and a (see Figure 3) in (3) is taken to be a
o and a
c, respectively. The value of cohesive stress along the fictitious fracture zone to the corresponding crack opening displacement is evaluated using bilinear stress-displacement softening law as given in (6).


## 2.3. Softening Traction-Separation Law of Postfire Concrete

The softening traction-separation law is a prerequisite determine the double-K fracture parameters, at room temperature; many expressions have been proposed based on direct tensile tests [16–20]. Based on numerical studies, simplified bilinear expressions for the softening traction-separation law (illustrated in Figure 1) were suggested by Petersson in 1981 [16], Hilsdorf and Brameshuber in 1991 [19], and Phillips and Zhang in 1993 [20]. The area under the softening curve was defined as the fracture energy G
F. Therefore, one could get the following equation:
(4)
G
F
=
1
2
(
f
t
w
s
+
σ
s
w
0
)
,
where f
t is tensile strength of postfire specimens, w
s is the crack width at break point of softening curve, σ
s is the cohesive stress at the break point of softening curve, and w
0 is crack width at stress-free point.Figure 1
The bilinear softening traction-separation law.As a consequence, a general form of the simplified bilinear expression of the softening traction-separation law is given as follows:(5)
σ
=
f
t
-
(
f
t
-
σ
s
)
w
w
0
,
0
≤
w
≤
w
s
σ
=
σ
s
(
w
0
-
w
)
(
w
0
-
w
s
)
,
w
s
≤
w
≤
w
0
σ
=
0
,
w
≥
w
0
.Different values of the break point (σ
s, w
s) and the crack width w
0 at stress-free point were used for the expression proposed by different researchers. In the present work, the bilinear softening function of concrete proposed by Petersson is used for postfire specimens:
(6)
σ
s
=
f
t
3
,
w
s
=
0.8
G
F
f
t
,
w
0
=
3.6
G
F
f
t
.


## 2.4. Determination of Stress Intensity Factor Caused by Cohesive Force

The standard Green’s function [21] for the edge cracks with finite width of plate subjected to a pair of normal forces is used to evaluate the value of cohesive toughness. The general expression for the crack extension resistance for complete fracture associated with cohesive stress distribution in the fictitious fracture zone for Mode I fracture is given as follows:(7)
K
I
c
=
∫
a
0
a
2
σ
(
x
)
F
(
x
/
a
,
a
/
h
)
π
a
d
x
,
where
(8)
F
(
x
a
,
a
h
)
=
3.52
(
1
-
x
/
a
)
(
1
-
a
/
h
)
3
/
2
-
4.35
-
5.28
x
/
a
(
1
-
a
/
h
)
+
{
1.30
-
0.30
(
x
/
a
)
3
/
2
1
-
(
x
/
a
)
2
+
0.83
-
1.76
x
a
}
×
{
1
-
(
1
-
x
a
)
a
h
}
and σ
(
x
) is the cohesive force at crack length x, see Figure 3, and its expression is shown in (9) or (11).At critical condition the value ofa is taken to be a
c in (7) and (8). The integration of (8) is done by using Gauss-Chebyshev quadrature method because of existence of singularity at the integral boundary.As shown in Figure2, two conditions at critical load, that is, CTOD
c
≤
w
s and w
s
≤
CTOD
c
≤
w
c, may arise at the notch tip while using bilinear softening function. For specimens subjected to temperatures less than 120°C, the critical CTOD
c is less than w
s, whereas, for temperatures higher than 120°C, the critical CTOD
c is wider than w
s.(A)
When the criticalCTOD
c corresponding to maximum load P
u is less than w
s as shown in Figure 2(a), the distribution of cohesive stress along the fictitious fracture zone is approximated to be linear as shown in Figure 3(a). The variation of cohesive stress along the fictitious fracture zone for this loading condition, that is, a
o
≤
a
≤
a
c or 0
≤
CTOD
≤
CTOD
c, is written as
(9)
σ
(
x
)
=
σ
(
CTOD
c
)
+
(
f
t
-
σ
(
CTOD
c
)
)
(
x
-
a
0
)
(
a
c
-
a
0
)
,
where,σ
(
CTOD
c
) is the critical values of cohesive stress being at the tip of initialnotch. The value of σ
(
CTOD
c
) is determined by using bilinear softening function:
(10)
σ
(
CTOD
c
)
=
σ
s
(
w
s
)
+
w
s
-
CTOD
c
w
s
(
f
t
-
σ
s
(
w
s
)
)
.
(B)
When the criticalCTOD
c corresponding to maximum load P
u is wider than w
s as shown in Figure 2(b), the distribution of cohesive stress along the fictitious fracture zone is approximated to be bilinear as shown in Figure 3(b). The variation of cohesive stress along the fictitious fracture zone for this loading condition, also, a
o
≤
a
≤
a
c or 0
≤
CTOD
≤
CTOD
c, is written as
(11)
σ
1
(
x
)
=
σ
(
CTOD
c
)
+
(
σ
s
(
w
s
)
-
σ
(
w
)
)
(
x
-
a
0
)
(
a
s
-
a
0
)
,
a
s
≤
x
≤
a
0
,
σ
2
(
x
)
=
σ
s
(
w
s
)
+
(
f
t
-
σ
s
(
w
s
)
)
(
x
-
a
s
)
(
a
c
-
a
s
)
,
a
s
≤
x
≤
a
c
. The value of σ
(
CTOD
c
) is determined by using bilinear softening function:
(12)
σ
(
CTOD
c
)
=
w
0
-
CTOD
c
w
0
-
w
s
σ
s
(
w
s
)
.
The limits of integration of (7) should be taken in two steps: a
o
≤
x
≤
a
s for cohesive stress σ
1
(
x
) and a
s
≤
x
≤
a
c for cohesive stress σ
2
(
x
), respectively. The same Green’s function F
(
x
/
a
,
a
/
h
) for a given effective crack extension will be determined using (8). The calculated formula is listed as follows:
(13)
K
I
c
=
∫
a
0
a
s
2
σ
1
(
x
)
F
(
x
/
a
c
,
a
c
/
h
)
π
a
c
d
x
+
∫
a
s
a
c
2
σ
2
(
x
)
F
(
x
/
a
c
,
a
c
/
h
)
π
a
c
d
x
.
The effective crack length at break point a
s (shown in Figure 3(b)) is computed from the following nonlinear expression [4] by substituting COD
(
a
s
), CMOD, a
c and h:
(14)
COD
(
a
s
)
=
CMOD
{
(
1
-
a
s
a
c
)
2
+
(
1.018
-
1.149
a
c
h
)
[
a
s
a
c
-
(
a
s
a
c
)
2
]
}
1
/
2
,
where COD
(
a
s
) is the crack opening displacement at a
s, a
c is the effective crack length (according to (1)), and h is the specimen height.Two different situations forCTOD
c and w
s.
(a)
WhenCTOD
c
≤
w
s
(b)
WhenCTOD
c
>
w
sCohesive force distribution along the crack length at critical load.
(a)
The linear distribution of cohesive force
(b)
The bilinear distribution of cohesive force


## 3. Calculation of Double-K Fracture Parameters

The two parameters (K
I
ini and K
I
un) of double-K fracture criterion for wedge-splitting test are determined using linear elastic fracture mechanics formula given in XU [8]:
(15)
K
(
P
,
a
)
=
P
×
1
0
-
3
th
1
/
2
f
(
α
)
,
(16)
f
(
α
)
=
3.675
×
[
1
-
0.12
(
α
-
0.45
)
]
(
1
-
α
)
3
/
2
,
α
=
a
h
.
The empirical expression (15) is valid within 2% accuracy for 0.2
≤
α
≤
0.8.Equations (15) and (16) can be used in calculation of unstable fracture toughness, K
I
un at the tip of effective crack length a
c, in which a
=
a
c and P
=
maximum load, P
u for TPBT and CT test specimen geometries, respectively. The initiation toughness, K
I
ini, is calculated using (15) and (16) when the initial cracking load, P
ini, at initial crack tip is known. In the present paper, the P
ini is determined by graphical method using the starting point of nonlinearity in P-CMOD curve described in the following section.Generally, for postfire concrete specimens the value of initial fracture toughnessK
I
ini is far less than the value of critical fracture toughness, K
I
un, especially for higher temperatures. So much more consideration is put to the critical fracture toughness K
I
un. In double-K fracture model, the following relation can be employed:
(17)
K
I
un
=
K
I
ini
+
K
I
c
.
Here we donate the experimental value, analytical value of critical fracture toughness as K
I
un-
e, K
I
un-
A, respectively, and from which we would judge the validation of double-K fracture model to the postfire concrete.


## 4. Experimental Validation and Comparison of Results

### 4.1. Experimental Program and Experimental Phenomena

50 concrete specimens with the same dimensions 230 × 200 × 200 mm were prepared; the geometry of the specimens is shown in Figure4 (b
=
200 mm, d
=
65 mm, h
=
200 mm, f
=
30 mm, a
0
=
80 mm, θ
=
15°). The concrete mix ratios (by weight) were Cement: Sand: Coarse aggregate: Water = 1.00 : 3.44 : 4.39 : 0.80, with common Portland cement-mixed medium sand and 16 mm graded coarse aggregate. The compressive strength at 28 days is 34 MPa. Nine heating temperatures, ranging from 65°C to 600°C (T
m
=
65°C, 120°C, 200°C, 300°C, 350°C, 400°C, 450°C, 500°C, 600°C), were adopted with the ambient temperature as a reference. Each wedge splitting specimen was embedded with a thermal couple in the center of specimen for temperature control. An electric furnace with net dimensions 300 × 300 × 900 mm was used for heating. When the designated T
m was reached, the furnace was shut down, and the specimens were naturally cooled for 7 days prior to the test. It averagely took 50, 95, 135, 182, 218, 254, 294, 342, and 453 mins for the specimens to reach the final temperatures, respectively (from 65°C to 600°C). The detailed experimental information would be found elsewhere [22].Figure 4
The geometry of specimens.


### 4.2. Experimental Results

Figure5 shows typical complete load-displacement curves for different heating temperatures up to 600°C. The figure shows that the ultimate load P
u decreases significantly with increasing temperatures T
m, whereas the crack-mouth opening displacement (CMOD) increases with T
m
.The initial slope of ascending branches decreases with heating temperatures, and the curves become gradually shorter and more extended.Figure 5
P versus CMOD curves of specimens with temperatures.The recorded maximum loadP
u, the recorded crack mouth opening displacement CMOD
c at P
u, the calculated crack tip opening displacement CTOD
c based on (14), the initial cracking load P
ini determined by graphical method, the calculated residual Young’s modulus E based on (2), the double-K fracture parameters, that is, K
I
ini and K
I
un-
E, and the residual fracture energy G
F are summarized in Table 1. Here we only list part of the statistics.


### 4.3. Discussion

In order to express the influence on the residual fracture toughness in detail, Figure6 plots the tendency of initial fracture toughness K
I
ini and the unstable fracture toughness K
I
un with heating temperatures T
m. It is concluded that the toughness of the two fractures decreases monotonously with T
m because of the thermal damage induced by the heating temperatures.The tendency of residual fracture toughness with heating temperaturesT
m.
(a)
The tendency ofK
I
ini with T
m
(b)
The tendency ofK
I
un with T
mThe initial fracture toughness continuously decreases from 0.498 MPa·m1/2 at room temperature to 0.269 MPa·m1/2 at 200°C, 0.115 MPa·m1/2 at 450°C, and finally 0.064 MPa·m1/2 at 600°C, with a significant loss of 0.434 MPa·m1/2 or 96%. The unstable fracture toughness decreases from 1.186 MPa·m1/2 at room temperature to 0.297 MPa·m1/2at 600°C, with a significant loss of 0.889 MPa·m1/2 or 75%.Comparing the results shown in Table1, it can be known that the value of K
I
un-
A evaluated by (17) has good coincidence with the one calculated by inserting the values of P
max
⁡ and a
c
=
h into (15), that is, the critical fracture toughness from analytical and experimental method. Figure 7 shows the relationship between the two parameters. In totally 45 effective specimens, the deviation between K
I
un-
A and K
I
un-
E of 22 specimens is below 5% and of 40 specimens is below 15%, which accounts for 89% of total specimens.Figure 7
Comparison betweenK
I
un-
A and K
I
un-
E.


## 4.1. Experimental Program and Experimental Phenomena

50 concrete specimens with the same dimensions 230 × 200 × 200 mm were prepared; the geometry of the specimens is shown in Figure4 (b
=
200 mm, d
=
65 mm, h
=
200 mm, f
=
30 mm, a
0
=
80 mm, θ
=
15°). The concrete mix ratios (by weight) were Cement: Sand: Coarse aggregate: Water = 1.00 : 3.44 : 4.39 : 0.80, with common Portland cement-mixed medium sand and 16 mm graded coarse aggregate. The compressive strength at 28 days is 34 MPa. Nine heating temperatures, ranging from 65°C to 600°C (T
m
=
65°C, 120°C, 200°C, 300°C, 350°C, 400°C, 450°C, 500°C, 600°C), were adopted with the ambient temperature as a reference. Each wedge splitting specimen was embedded with a thermal couple in the center of specimen for temperature control. An electric furnace with net dimensions 300 × 300 × 900 mm was used for heating. When the designated T
m was reached, the furnace was shut down, and the specimens were naturally cooled for 7 days prior to the test. It averagely took 50, 95, 135, 182, 218, 254, 294, 342, and 453 mins for the specimens to reach the final temperatures, respectively (from 65°C to 600°C). The detailed experimental information would be found elsewhere [22].Figure 4
The geometry of specimens.


## 4.2. Experimental Results

Figure5 shows typical complete load-displacement curves for different heating temperatures up to 600°C. The figure shows that the ultimate load P
u decreases significantly with increasing temperatures T
m, whereas the crack-mouth opening displacement (CMOD) increases with T
m
.The initial slope of ascending branches decreases with heating temperatures, and the curves become gradually shorter and more extended.Figure 5
P versus CMOD curves of specimens with temperatures.The recorded maximum loadP
u, the recorded crack mouth opening displacement CMOD
c at P
u, the calculated crack tip opening displacement CTOD
c based on (14), the initial cracking load P
ini determined by graphical method, the calculated residual Young’s modulus E based on (2), the double-K fracture parameters, that is, K
I
ini and K
I
un-
E, and the residual fracture energy G
F are summarized in Table 1. Here we only list part of the statistics.


## 4.3. Discussion

In order to express the influence on the residual fracture toughness in detail, Figure6 plots the tendency of initial fracture toughness K
I
ini and the unstable fracture toughness K
I
un with heating temperatures T
m. It is concluded that the toughness of the two fractures decreases monotonously with T
m because of the thermal damage induced by the heating temperatures.The tendency of residual fracture toughness with heating temperaturesT
m.
(a)
The tendency ofK
I
ini with T
m
(b)
The tendency ofK
I
un with T
mThe initial fracture toughness continuously decreases from 0.498 MPa·m1/2 at room temperature to 0.269 MPa·m1/2 at 200°C, 0.115 MPa·m1/2 at 450°C, and finally 0.064 MPa·m1/2 at 600°C, with a significant loss of 0.434 MPa·m1/2 or 96%. The unstable fracture toughness decreases from 1.186 MPa·m1/2 at room temperature to 0.297 MPa·m1/2at 600°C, with a significant loss of 0.889 MPa·m1/2 or 75%.Comparing the results shown in Table1, it can be known that the value of K
I
un-
A evaluated by (17) has good coincidence with the one calculated by inserting the values of P
max
⁡ and a
c
=
h into (15), that is, the critical fracture toughness from analytical and experimental method. Figure 7 shows the relationship between the two parameters. In totally 45 effective specimens, the deviation between K
I
un-
A and K
I
un-
E of 22 specimens is below 5% and of 40 specimens is below 15%, which accounts for 89% of total specimens.Figure 7
Comparison betweenK
I
un-
A and K
I
un-
E.


## 5. Conclusion

The determination of residual fracture parameter using analytical method is carried out in present research. In calculating the cohesive fracture toughness, two conditions are divided at critical load: for specimens subjected to temperatures less than 120°C, the criticalCTOD
c is less than w
s, whereas, for temperatures higher than 120°C, the critical CTOD
c corresponding to maximum load P
u is wider than w
s. This part of work would be a useful supplement to the existed analysis.Wedge-splitting tests with ten temperatures varying from 20°C to 600°C are implemented. The complete load-crack opening displacement curves are obtained and the initial and critical fracture toughness could be calculated experimentally.The validation of double-K fracture model to the postfire concrete specimens is proved. In totally 45 effective specimens, the deviation between analytical value K
I
un-
A and experimental K
I
un-
E of 22 specimens is below 5% and of 40 specimens is below 15%, which accounts for 89% of total specimens.



---

*Source: 452763-2014-02-19.xml*