Issue 75

R. Ince et alii, Fracture and Structural Integrity, 75 (20YY) 435-462; DOI: 10.3221/IGF-ESIS.75.30

The double- K fracture model, based on equivalent elastic fracture models, simulates failure in quasi-brittle structures by utilizing two quantities: initiation and unstable fracture toughness ( K un Ic and K ini Ic ). Therefore, contrary to the TPM, both crack initiation and unstable crack growth in a quasi-brittle structure can be determined using Eqn. (1) through this model, where the values of a ₀ and the first cracking load ( P i ) are utilized to determine K un Ic . As pointed out above, the only unknown points in Eqns. (1) and (2) are the critical (or effective) crack depth ( a c in the procedure for determining fracture quantities in the aforementioned fracture approaches. According to concrete fracture codes recommended by RILEM (International Union of Laboratories and Experts in Construction Materials, Systems and Structures), as illustrated in Fig. 2b, this critical quantity can be evaluated from the load–crack mouth opening displacement (P-CMOD) response of the test specimen (preferably a beam). However, to determine the P-CMOD response of the specimen, the use of a feedback-testing machine with CMOD control is necessary for the TPM, whereas no such requirement exists for the double- K model. In approaches based on RILEM, Young’s modulus ( E ) of the tested specimen is initially determined from the initial compliance value ( C i = CMOD i /P i ), where P i is commonly assumed to be P c /2 due to difficulties in determining P i (Fig. 2b). Subsequently, assuming that E remains constant for any pair of P and CMOD , the a c value of the notched specimen is computed using the secant compliance value ( C s ) in the double- K model or the unloading compliance value ( C u ) in the TPM, as indicated in Fig. 2b. Although both models employ different test equipment and compliance techniques, experimental studies on beams and wedge-splitting specimens by Xu et al. revealed that the fracture toughness and CTOD c values obtained using the TPM were in good statistical agreement with those of the double- K model. An alternative method, which utilizes only the ultimate loads of at least three specimens with different notch depths, was developed for the TPM by Tang et al. [18]. In this so-called peak-load method in concrete fracture, both fracture parameters ( K s Ic and CTOD c ) are treated as unknowns instead of a values for the specimens. Consequently, this system, which consists of multiple equations but only two unknowns, can only be solved using an optimization-based approach. To achieve this, Tang et al. proposed that K s Ic and CTOD c curves corresponding to specific notch depths should first be determined for each specimen with a distinct a 0 . The values of the sample standard deviation ( s ) corresponding to the same K s Ic value are then calculated using Eqn. (6):     2 1, 1,2,..., 1 n s K s CTOD CTOD n i n c ci Ic i       (6) Here, CTODc is the mean value of CTOD ci values. Consequently, the K s Ic and CTOD c values, where   s K s Ic is at its minimum, are accepted as the fracture quantities according to the peak-load method. As stated above, determining the initial cracking load ( P i ) of the specimens is difficult. Therefore, Xu and Reinhardt [4] recommended a simplified approach, known as the inverse method in concrete fracture, as follows: ini un c K Ic K Ic K Ic   (7) Here, K c Ic refers to the cohesive fracture toughness. As depicted in Fig. 3a, K c Ic corresponds to the toughness value consisting of cohesive stresses assumed to occur along the critical crack extension line (  a c ). According to the inverse method, this cohesive stress distribution is trapezoidal, such that the maximum value of this distribution is supposed to correspond to the tensile strength of the material ( f t ) while its lower value,  s ( CTOD c ), is the stress value determined according to the chosen stress–crack tip opening displacement function (  - w ), as indicated in Fig. 3a. In many applications up to the peak load in concrete fracture, using a linear function, as shown in Fig. 3b, provides an adequate approach for  - w [19]. The area under this function is equal to the fracture energy ( G f ), which is converted to fracture toughness using Irwin’s G-K relation:   2 un G f K Ic E  [1,3]. On the other hand, the toughness value created by such a trapezoidal stress distribution at the crack tip can be calculated by integrating the toughness values created by the individual forces ( P =  dx ) shown in Fig. 3c:

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