PSI - Issue 47

Domenico Ammendolea et al. / Procedia Structural Integrity 47 (2023) 488–502 Author name / Structural Integrity Procedia 00 (2019) 000–000

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As its main advantage, the present combined methodology allows the failure response of nano-filled UHPFRC structures under general loading histories (in terms of both loading curves and related crack paths) to be accurately simulated with limited computational cost. The reliability of this methodology has been assessed by a series of numerical experiments involving crack propagation in simple laboratory-scale specimens. In particular, a novel calibration procedure has been proposed for obtaining very accurate R -Curves starting from the measured load-displacement curve related to a Mode-I fracture test. Furthermore, the role of nano-filler content on the Mixed-Mode fracture properties has been investigated, with reference to an asymmetric four-point bending test. 2. Theoretical Background This section reports the key theoretical aspects at the base of the proposed modeling strategy. At first, an overview of the Arbitrary Lagrangian-Eulerian (ALE) method is outlined, in which the fundamental mapping equations used to link the mesh and material systems of coordinates are provided. Next, the governing equations of solid mechanics and the moving mesh problem are discussed. Then, the ALE formulation of the Interaction Integral method ( i.e. , the M -integral) is presented. Finally, the R -curve approach is briefly recalled. It is worth noting that the proposed model aims to reproduce through-thickness mixed-mode (I/II) crack propagation phenomena in nano-filled Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) structures. Therefore, all the basic equations and formulations introduced in the following sections are developed for two-dimensional problems. 2.1. Arbitrary Lagrangian-Eulerian (ALE) formulation The proposed modeling strategy adopts a moving mesh strategy consistent with the Arbitrary Lagrangian Eulerian (ALE) formulation to reproduce the geometry variations of the computational domain because of the growing cracks. Specifically, the ALE changes the positions of the nodes of the computational mesh during the numerical simulation consistently with conditions dictated by Fracture Mechanics criteria, which define necessary conditions to establish the crack onset and the direction of crack propagation. In particular, the ALE formulation ensures the consistency of the mesh nodes' motion, varying the computational mesh with regularity, thus avoiding significant distortions for the finite elements. The use of the ALE formulation implies that the governing equations of the problem under investigation must be formulated with reference to the Mesh (or Referential) system of coordinates ( R  ). In particular, the χ coordinate of R  identifies each node inside the computational mesh (see (Ponthot and Belytschko, 1998; Funari and Lonetti, 2017; Greco et al., 2020b)). Figure 1 shows a schematic drawing of how the proposed model uses the moving mesh technique to simulate the crack propagation process of a pre-existing notch inside the material. In particular, the figure depicts the movement of the mesh nodes because of the growth of the pre-existing crack from the initial to a varied configuration of the material. Note that the detailed scheme reported on the right part of the figure highlights that the mesh nodes move differently from the material particles during the numerical simulation. Because the governing equations of solid mechanics are stated in Material coordinates, such an aspect denotes that a specific mapping function serves to link the Material and Referential system of coordinates, thus formulating the governing equations in Referential coordinates. To this end, the following bijective mapping function ( ) Ψ χ is introduced: ( )   χ Ψ χ X (1) where X is the material coordinate that identifies the i -th material particle of the continuum with reference to the material frame.