Toughness Calculation of Postfire Normal Concrete

Fracture tests of postfire normal concrete with ten temperatures up to 600C 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.


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,  ini  , and unstable fracture toughness,  un  , can be used to study the crack propagation of concrete.
In order to determine the double- fracture parameters analytically [2,3] the value of cohesion toughness    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][6][7] and size effect [2,3,8] on fracture toughness were studied by various researchers.It was found that the influence of   / 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][10][11][12][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- fracture model to the postfire normal concrete is proved.

Analytical Determination of Cohesive
Fracture Toughness

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- fracture parameters.Detailed explanation of the above assumption can be found elsewhere [2].
Based on this assumption, the value of the equivalentelastic crack length for WS specimen is expressed as  = (ℎ + ℎ 0 ) {1 − ( 13.18  ⋅  ⋅  + 9. 16 ) where  = CMOD/ is the compliance of specimens and  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   , the value of crack mouth opening displacement (CMOD) and  is taken as CMOD  and   , respectively.The residual Young's modulus  is calculated using the P-CMOD curve as where   = CMOD ini / ini is the initial compliance before cracking,  = ( 0 + ℎ 0 )/(ℎ + ℎ 0 ).The values of critical equivalent-elastic crack length   and residual Young's modulus  are listed in Table 1.

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 of COD along the fracture process zone; for practical purposes the value of COD () at the crack length  is computed using the following expression [3]: For calculation of critical value of crack tip opening displacement CTOD  , the value of  and  (see Figure 3) in ( 3) is taken to be   and   , respectively.The value of cohesive stress along the fictitious fracture zone to the corresponding crack opening displacement is evaluated using bilinear stressdisplacement softening law as given in (6).

Softening Traction-Separation Law of Postfire Concrete.
The softening traction-separation law is a prerequisite determine the double- fracture parameters, at room temperature; many expressions have been proposed based on direct tensile tests [16][17][18][19][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   .Therefore, one could get the following equation: where   is tensile strength of postfire specimens,   is the crack width at break point of softening curve,   is the cohesive stress at the break point of softening curve, and  0 is crack width at stress-free point.
As a consequence, a general form of the simplified bilinear expression of the softening traction-separation law is given as follows: Different values of the break point (  ,   ) and the crack width  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:

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: where and  () is the cohesive force at crack length , see Figure 3, and its expression is shown in (9) or (11).At critical condition the value of  is taken to be   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 Figure 2, two conditions at critical load, that is, CTOD  ≤   and   ≤ CTOD  ≤   , may arise at the notch tip while using bilinear softening function.For specimens subjected to temperatures less than 120 ∘ C, the critical CTOD  is less than   , whereas, for temperatures higher than 120 ∘ C, the critical CTOD  is wider than   .
(A) When the critical CTOD  corresponding to maximum load   is less than   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,   ≤  ≤   or 0 ≤ CTOD ≤ CTOD  , is written as where,  (CTOD  ) is the critical values of cohesive stress being at the tip of initialnotch.The value of  (CTOD  ) is determined by using bilinear softening function: (B) When the critical CTOD  corresponding to maximum load   is wider than   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,   ≤  ≤   or 0 ≤ CTOD ≤ CTOD  , is written as The value of  (CTOD  ) is determined by using bilinear softening function: The limits of integration of ( 7) should be taken in two steps:   ≤  ≤   for cohesive stress  1 () and   ≤  ≤   for cohesive stress  2 (), respectively.The same Green's function  (/, /ℎ) for a given effective crack extension will be determined using (8).The calculated formula is listed as follows: The effective crack length at break point   (shown in Figure 3(b)) is computed from the following nonlinear expression [4] by substituting COD (  ), CMOD,   and ℎ: where COD (  ) is the crack opening displacement at   ,   is the effective crack length (according to (1)), and ℎ is the specimen height.

Calculation of Double-𝐾 Fracture Parameters
The two parameters ( ini  and  un  ) of double- fracture criterion for wedge-splitting test are determined using linear elastic fracture mechanics formula given in XU [8]: () = 3.675 × [1 − 0.12 ( − 0.45)] The empirical expression ( 15) is valid within 2% accuracy for 0.2 ≤  ≤ 0.8.Equations ( 15) and ( 16) can be used in calculation of unstable fracture toughness,  un  at the tip of effective crack length   , in which  =   and  = maximum load,   for TPBT and CT test specimen geometries, respectively.The initiation toughness,  ini  , is calculated using ( 15) and ( 16) when the initial cracking load,  ini , at initial crack tip is known.In the present paper, the  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 toughness  ini  is far less than the value of critical fracture toughness,  un  , especially for higher temperatures.So much more consideration is put to the critical fracture toughness  un  .In double- fracture model, the following relation can be employed: Here we donate the experimental value, analytical value of critical fracture toughness as  un-  ,  un-  , respectively, and from which we would judge the validation of double- fracture model to the postfire concrete.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   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].

Experimental Results.
Figure 5 shows typical complete load-displacement curves for different heating temperatures up to 600 ∘ C. The figure shows that the ultimate load   decreases significantly with increasing temperatures   , whereas the crack-mouth opening displacement (CMOD) increases with   .The initial slope of ascending branches decreases with heating temperatures, and the curves become gradually shorter and more extended.
The recorded maximum load   , the recorded crack mouth opening displacement CMOD  at   , the calculated crack tip opening displacement CTOD  based on ( 14), the initial cracking load  ini determined by graphical method, the calculated residual Young's modulus  based on (2), the double- fracture parameters, that is,  ini  and  un-  , and the residual fracture energy   are summarized in Table 1.Here we only list part of the statistics.

Discussion.
In order to express the influence on the residual fracture toughness in detail, Figure 6 plots the tendency of initial fracture toughness  ini  and the unstable fracture toughness  un  with heating temperatures   .It is concluded that the toughness of the two fractures decreases monotonously with   because of the thermal damage induced by the heating temperatures.
Comparing the results shown in Table 1, it can be known that the value of  un-  evaluated by (17) has good coincidence with the one calculated by inserting the values of  max and   = 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  un-  and  un-  of 22 specimens is below 5% and of 40 specimens is below 15%, which accounts for 89% of total specimens.

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 critical CTOD  is less than   , whereas, for temperatures higher than 120 ∘ C, the critical CTOD  corresponding to maximum load   is wider than   .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- fracture model to the postfire concrete specimens is proved.In totally 45 effective specimens, the deviation between analytical value  un-  and experimental  un-  of 22 specimens is below 5% and of 40 specimens is below 15%, which accounts for 89% of total specimens.

Figure 2 :
Figure 2: Two different situations for CTOD  and   .

Figure 3 :
Figure 3: Cohesive force distribution along the crack length at critical load.

Table 1 :
The experimental results of fracture parameters.