# Three-Dimensional Probabilistic Semi-Explicit Cracking Model for Concrete Structures
## 3 Estimation of the Coefficients of Variation of Fracture Energy
### A Brief Presentation of the Experimental Study Performed on the DCB Specim  
The geometry and loading conditions of the DCB test are depicted in Figure 2. The test is governed by the application of a constant notch displacement rate set at 25 \(\mu\)m/min. Throughout the test, crack propagation occurs from the bottom to the top of the structure. The load is applied 17.5 cm from the lower side of the beam. Crack opening is measured by averaging the results from two extensometers--one on each side. To guide and maintain the crack within the median plane, the section was thinned and a longitudinal prestressing (via post-tensioning) was introduced using multiple cables.  
During the test, the load/notch opening curve is recorded. Upon reaching the peak load, signifying the initiation of the macrocrack, the specimen undergoes several cycles of notch closing and reopening, all at the same frequency. These cycles enable the tracking of the specimen's stiffness evolution, a critical factor used to calculate \(K_{IC}\) and \(G_{IC}\) employing the classical compliance method, a well-established technique in linear fracture mechanics theory [28; 38; 39; 40]. A more detailed description of the entire test procedure can be found in [28; 38].  
The analysis of the Double Cantilever Beam (DCB) test using the compliance method results in the derivation of a curve depicting \(K_{IC}\) versus the equivalent crack length (ECL). This ECL represents an idealized crack length--essentially a straight and smooth crack within a linear elastic material--that results in the same compliance as experimentally obtained from the specimen. Conceptually, this ECL can be visualized as an average crack length, encompassing the real macrocrack along with its associated process zone--a  
Figure 2: Detail of the geometry and of the loading conditions related to the DCB specimen. (**a**) Geometry of the specimen, (**b**) Details about load and prestress force application.  
microcracked region at the macrocrack tip during propagation. Typically, the evolution of the specimen compliance with the idealized crack length (analytical relation) is obtained by performing linear elastic finite element analyses [28; 38].  
The experimental test campaign involved various types of concrete, including plain concrete with and without additional prestressing loads, concrete with fibers, and reinforced concrete. However, this numerical study focuses solely on the cases involving plain concrete. The mix design details for plain concrete are provided in Table 1 as reported in [28; 38]. The values for compressive strength (\(f_{c}\)), splitting tensile strength (\(f_{l}\)), the Young's modulus (obtained from average values during compressive tests), and the mean values of \(G_{IC}\) for plain concrete are provided in Table 2.  
During the tests conducted with the DCB specimen composed by plain concrete, three prestressing force (F) conditions were applied: case 1 with \(F_{1}=1230\) KN, case 2 with \(F_{2}=682\) KN, and case 3 with no prestressing force (\(F_{3}=0\) KN). To focus on using the most rectilinear crack propagation, only the test with the highest prestressing load (\(F_{1}\)) is utilized for the numerical simulations. The purpose of prestressing load was to guide the crack propagation to the central part of the specimen (in the thinnest zone) and prevent any premature bifurcation, as observed in the case with no prestressing load. Figure 3 displays the load-notch opening curve obtained from the considered experimental test. It is noteworthy that the curve exhibits a softening phase, characterized by a continuous decrease, which correlates with the very rectilinear propagation of the crack.