# Element 3d V2.2 Crack VERIFIED 44

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## element 3d v2.2 crack 44

In the last few decades, the material properties, as well as the structural performance of plain and fiber-reinforced concrete have been extensively investigated in laboratory environments. However, experiments are in general expensive and are limited to specific test configurations. Therefore, a variety of numerical models for concrete cracking, aiming at reliable prognoses of the fracture processes of concrete structures with or without reinforcement, have been proposed (see, e.g., [3,4,5,6,7,8] for an overview). The majority of models for structural analyses of FRC are conceptually based on cracking models for plain concrete, modifying the post-peak regime of the constitutive law in terms of an increase of the residual stress and the fracture energy, so as to represent the enhanced ductility of FRC at a phenomenological level [9,10,11,12]. To enable the analysis of the influence of specific fiber cocktails on the macroscopic material behavior of FRC, computational meso-scale models for FRC have been proposed, which include the explicit description of individual fibers within representative elementary volumes of FRC samples [13,14,15]. However, for computational analyses on a structural level, a multiscale-oriented approach allowing one to formulate the behavior of fiber and matrix and their mutual interactions at different length scales as proposed in [16] is required. Recently, the authors proposed a multilevel modeling framework, in which, at the lowest scale, the pullout behavior of different fiber types interacting with the concrete matrix at different inclination angles is considered by appropriate sub-models with the model information being appropriately transmitted across the scales depending on the specific fiber cocktail [17].

As an essential component of the modeling framework presented in [17], interface solid elements (ISE), i.e., degenerated solid finite elements with almost zero thickness proposed by Manzoli et al. [18,19,20,21], have been adopted and successfully applied to the failure analysis of plain and fiber-reinforced concrete structures. As compared with classical zero-thickness interface elements, ISE can be easily implemented based on standard finite element codes by using solid finite elements for the bulk material and for the interfaces. Employing a continuum damage model to approximate the interface degradation, it allows one to describe the interface behavior completely in the continuum framework. Consequently, those specific variational formulations, discrete constitutive relations and integration rules to obtain the internal forces associated with classical interface elements are not required. The artificial initial stiffness that is normally required in zero-thickness interface elements is automatically included in the elastic stiffness of ISE [18,20,21]. It is recognized that the interface solid elements share similar features with zero-thickness interface elements. The most notable advantage of this class of models is the fact that no special procedure for the tracking of evolving cracks is necessary. This contributes to its robustness and allows for 3D fracture simulations characterized by complex fracture patterns (see, e.g., [22]). The crack pattern obtained via discrete representations along prescribed element edges evidently suffers from a certain dependence on the mesh topology. However, the influence on the overall macroscopic material response is tolerable if unstructured meshes with reasonable resolution are used [20,23]. Furthermore, for analyses of heterogeneous materials on the mesoscale level, it was shown that the mesh-dependence of interface elements becomes less of a concern once the mesoscale heterogeneity is modeled [24,25,26]. This drawback can be alleviated, e.g., by continuously modifying the local finite element topology at the crack tip to enforce the alignment between the element edges and the crack propagation direction [27,28,29]. Alternatively, mesh refinement, at the cost of increased computational expense, can be applied to resolve the large elements along the crack path [30,31,32]. The increased computational demand resulting from the duplication of finite element nodes can be controlled by pre-defining the interface elements only in vulnerable regions or applying an adaptive algorithm for the mesh processing during computation [33,34,35,36,37].

Crack bridging model: (a) position and inclination of a fiber with respect to the crack; (b) unit area of an opening crack in FRC intercepted by fibers with length Lf; (c) sketch of the obtained traction-separation relations for different FRC composites.

All bulk elements are considered to be linear elastic. The constitutive behavior of the degenerated solid elements is cast in a continuum form equipped with a damage law, which allows one to approximate the behavior of interfacial degradation mechanisms involved during the cracking in FRC materials:

It is noticed that the post-cracking behavior of the FRC material is highly nonlinear; such nonlinearity frequently results in numerical difficulties while performing structural simulations. In the present work, the IMPL-EXintegration scheme [45] is implemented in the context of the interface solid element for FRC. Consequently, due to the explicit nature of damage models, the computation does not require any iteration, neither on the structural level, nor on the constitutive level. This ensures the robustness and efficiency of the computational model in failure analyses of FRC structures even in the case of complex crack configurations.

Analysis of a three-point bending test on a notched FRC beam: (a) photo of the failure state of the specimen and the contour plot of the crack-opening magnitude in the deformed configuration; (b) crack patterns represented by the activated interface solid elements at different loading states; (c) comparison between the force-displacement relations predicted by the proposed model and from the experiments [41] for three different fiber cocktails.

Pre-processing (insertion of interface solid elements in the complete domain): (a) original finite element mesh; (b) phantom mesh obtained by duplicating the edges and shrinking the bulk elements; (c) actual mesh for computation, obtained after insertion of solid elements into all interfacial gaps.

Insertion of solid elements at a specific interface in 2D: (a) original mesh with two bulk elements; (b) Node-1 split, the first interface solid element inserted (ISE-I, in gray); (c) Node-3 split, the second interface solid element inserted (ISE-II, in gray).

Insertion of solid elements at one interface in 3D: (a) original mesh with two bulk elements; (b) Node-1 split; ISE-I inserted (gray colored); (c) Node-2 split; ISE-II inserted (gray); (d) Node-3 split; ISE-III inserted (gray).

In the following, the efficiency of the adaptive algorithm is discussed based on the results shown in Figure 14. The simulation starts with the original mesh, which contains approximately 19% of the number of nodes and 26% of the number of elements as compared to the case of full fragmentation; in other words, at the start of the analysis at increment i = 1,

Here, Ï‰ denotes the relative problem size while using the adaptive algorithm as compared with the case of full fragmentation. With the growth of the macroscopic crack, ISEs are gradually generated and located along the potential crack path. When the applied displacement reaches 0.6 mm, the propagating macroscopic crack almost penetrates the sample. Afterwards, the major crack continues to open, and the structure fails rapidly; the relative problem sizes change marginally, approaching approximately 34% for the number of nodes and 40% for the number of elements, respectively, at the end of simulation when u = 1 mm at increment i = 1000:

Results of the 2D simulation of uniaxial tension on a square specimen: (a) computed structural responses and crack pattern; (b) evolution of the relative problem sizes regarding the system degree of freedom and number of elements, as well as all of the created interface solid elements (blue lines in the mesh).

Results from a 3D simulation of the notched FRC beam made of plain concrete: (a) load-displacement diagram; (b) crack pattern (side view and bottom-side view with the contours representing the crack opening magnitude in the deformed configuration).

The original 3D FE-discretization contains 6743 nodes and 34,537 linear tetrahedral elements. For comparison, the simulation is first performed with full insertion of ISEs, which requires the completely fragmented mesh filled with three solid elements at every interfacial gap; consequently, the preprocessed mesh includes 138,148 nodes (approximately 20-times that of the original mesh) and 235,939 elements (approximately seven-times that compared to the original mesh), among which, 201,402 are ISEs.

In a first analysis, the beam is assumed to be made of plain concrete, and in a second analysis, a fiber-reinforced concrete beam is re-analyzed and compared with experimental results. This has been analyzed by the authors as a 2D problem without using an adaptive strategy for the ISE insertion in [17]. The simulation results for the plain concrete beam, including the load-displacement curve and the crack pattern, are shown in Figure 18.

Results from a 3D simulation of the notched FRC beam with adaptive insertion of ISEs: (a) evolution of the relative problem sizes in terms of the number of degrees of freedom and the number of elements, respectively; (b) generated ISEs at the end of the simulation (u = 0.5 mm).