S. Tarasovs et alii, Frattura ed Integrità Strutturale, 35 (2016) 271-277; DOI: 10.3221/IGF-ESIS.35.31 272 mapped to the neighbor nodes of the background mesh. In this approach the mesh refinement around fibers is not necessary and the total number of degrees of freedom in the system remains unchanged. In present work two-scale numerical approach within finite element framework is proposed. The concrete matrix fracture is simulated using cohesive zone model. The cohesive elements in current model are embedded between all solid elements, as a result the crack growth direction is chosen automatically during the analysis, minimizing the potential energy of the model. The effect of reinforcing fibers is modeled using non-linear spring elements, connecting nodes of neighboring solid elements at the location of the fiber. The properties of the spring elements are defined using experimentally obtained pull-out characteristics of single fibers embedded in concrete matrix at different angles. This model allows to simulate the initiation and propagation of the crack, taking into account the non-uniform spatial distribution of reinforcing fibers. F INITE ELEMENT MODEL omplex nature of the reinforced concrete failure, involving the cracking of the concrete matrix and steel fiber debonding from concrete matrix and pull-out, does not allow to model this process explicitly. Therefore simplified two-scale approach within the framework of finite element method was used. Two major mechanisms of fracture, matrix cracking and fiber pull-out were modeled independently. The approach suggested in [5] was used to model the fracture of the heterogeneous concrete material without prior knowledge of the crack path. In this model the cohesive elements are embedded between all solid elements, thus allowing automatic formation of the fracture surface during the simulations. In-house Perl script was written for the cohesive elements embedding procedure. Input file generated by the finite element code ABAQUS and containing solid 3D model with defined surfaces, sets and boundary conditions is automatically transformed into new modified model with cohesive elements embedded between solid elements in the middle part of the specimen, where the crack is expected to grow. The procedure of cohesive elements embedding is shown in Fig. 1: at first solid elements are separated by creating duplicate nodes with the same coordinates, then these elements are connected by zero- thickness cohesive elements. The procedure takes care of converting all predefined sets, surfaces and boundary conditions, using new nodes, and generates new input file, which can be directly solved by ABAQUS explicit solver. Figure 1 : Procedure of embedding cohesive elements: original FE mesh; separated elements with duplicate nodes; cohesive elements embedded between all solid elements. Due to small size and large number of steel fibers, the fibers are not modeled explicitly. The fibers bridging action is approximately modeled by a number of non-linear springs, embedded into finite element mesh at the fibers location. The same Perl script randomly distributes steel fibers in the specimen’s volume, taking into account experimentally measured orientation of fibers [6], finds the elements faces, crossing the fiber, and connects these faces with non-linear spring elements, as shown in Fig. 2. The spring’s stiffness takes into account the fiber pull-out forces, obtained experimentally or using some simplified analytical model [7]. In many situations the general crack growth direction is known a priori and the angle between the fiber and pulling force and, therefore, the spring’s stiffness, can be estimated before the analysis. However, if the crack growth direction is not known, the spring’s stiffness has to be determined during the analysis, taking into account the local crack opening direction and fiber orientation, as shown in Fig. 3. This can be done using user defined element or similar approach, depending on finite element software used. C

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