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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reported in Section 2.4 and exploits the available experimental data in terms of the load-displacement curves for a bending fracture test under pure Mode-I conditions. Such an approach allows accurate expressions for the shape functions   ˆ v  and   ˆ k  , which are needed to derive the R -curve, to be obtained in the following form:

1 c c

                        4 3 2 1 2 3 4 4 3 2 c c c c c c c c

ˆ ˆ v

(14)

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k

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starting from the results of preliminary finite element computations. The entire numerical procedure, which is described in a more detailed way in (Ammendolea et al., 2023), consists of three distinct steps: 1. Finite Element analysis for evaluating discrete values of the dimensionless shape functions   ˆ v  and   ˆ k  . In this step, several independent linear computations are conducted for the same reference problem by varying the relative crack length in a discrete manner. Therefore, the outcomes of this step are two discrete point databases   , ˆ i i v  and   , ˆ i i k  , with 0 i n   , n being the number of crack increments (starting from the initial crack length a 0 ). 2. Curve fitting analysis for obtaining analytical expressions for   ˆ v  and   ˆ k  . In this step, closed-form expressions for these shape functions are derived starting from the two discrete point databases found in the first step. The outcomes of this step are the coefficients   1, ,10 i c i   appearing in Eq. (14) as computed from two independent polynomial regression analyses via a standard least squares approach. 3. R -curve calibration based on the experimental loading curve. In this step, the fracture toughness K I c as a function of the effective crack extension Δ a is computed starting from the available experimental load-displacement curves. Such a function is obtained by approximating all the points   I , i i c K  obtained by solving the system of equations described through Eqs. (13) for all the couples   , i i u P taken from the experimental results. The outcome of this step is a piecewise R -curve in the form   I c K   , Δ α being the equivalent crack extension. 3.2. Moving Mesh approach for mixed-mode crack propagation In this section, the proposed crack propagation algorithm for simulating mixed-mode fracture in nano-filled UHPFRC structures is briefly described. This algorithm requires the preliminary definition of the geometry, material, and boundary conditions of the specimen. In particular, a pre-crack must be inserted in the initial geometry, as the proposed approach is not able to handle crack onset starting from an uncracked configuration. The current crack, represented as a polyline in the present approach, is split into two parts: a fixed one, representing the real crack portion, responsible for tracking the previous crack path in the current configuration, and a moving one, referred to as stretching segment, which is forced to move consistently with the surrounding mesh, in order to fulfill the crack propagation criterion 0 F f  for each loading step. The same mesh is used during the analysis, unless one of the two following stop conditions is met: the first condition is the occurrence of an excessive distortion of the moving mesh (after which a remeshing operation is required), whereas the second one is the occurrence of an excessively large kinging angle (which requires an entire geometry update, aimed at inserting a new stretching segment to the current crack). The simulation is ended when either a total failure condition or a crack arrest state is reached. 4. Results The section is devoted to the numerical experiments aimed at validating the proposed numerical methodology. In detail, two examples have been considered, involving a single crack propagation problem for the same pre-notched beam under different loading configurations.