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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mixed-mode SIFs of the actual state, two interaction integrals must be used. Such interaction integrals are derived assuming the analytic solution associated with a pure Mode-I and a pure Mode-II as auxiliary fields. Then, using Eq. (11) the mixed-mode SIFs of the actual field are evaluated as follows:     , , 0 and 0 2 2 act aux I act aux II act aux I act aux II I II II I aux I aux II I II EM EM K K K K K K             (12) 2.4. The R-curve approach An R -curve approach has been adopted in the present work as an effective way to capture the fracture toughness behavior of nano-filled UHPFRCs, thus overcoming the limitations of classical LEFM concepts usually experienced for quasi-brittle materials. In particular, this approach assumes the critical stress intensity factor as a function of only the crack extension, in the spirit of the approach proposed by (Zhao et al., 2015). As the key novelty point of the present work, a rigorous calibration approach is presented to determine accurate R -curves for nano-filled reinforced concretes as a function of the type, size and content of embedded nanoparticles. This is an FE-based approach that exploits the experimental data related to complex bending fracture tests involving Mode-I crack propagation for which no analytical solutions are readily available. In detail, the proposed numerical strategy consists in computing the critical effective crack length, being required for obtaining the R -curve, from the deformability properties of the specimen, in the spirit of the secant compliance approach. The analytical equations governing the Mode-I fracture problem, appearing in (Ammendolea et al., 2023) but not reported here for conciseness, involve the two dimensionless shape functions   ˆ v  and   ˆ k  depending on the relative crack length α , the latter being defined as the ratio of the crack length a to the specimen height D . The function   ˆ v  is regarded as a normalized measure of the mid-span total deflection u , whereas the function   ˆ k  is the dimensionless counterpart of the SIF K I . These functions can be defined by inverting the following relations:

P

  

ˆ v

u



'

bE

(13)

P

ˆ

  

  

K

k



I

b D

where E ’ is the effective elastic modulus, b is the out-of-plane specimen thickness, and P is the applied load. It is useful to note that explicit closed-form expressions for the shape functions   ˆ v  and   ˆ k  are available only for common fracture problems, so that in the present work, they are numerically estimated by means of suitable finite element computations (performed in an “off-line” processing stage) used in combinations with an accurate curve fitting analysis, as it will be explained in Section 3.1. 3. Numerical implementation The weak forms of the governing equations briefly recalled in Section 2 are implemented in the commercial finite element environment COMSOL Multiphysics ((COMSOL AB, 2022)), chosen for its advanced scripting capabilities. Furthermore, the proposed numerical strategy for the R -curve determination as well as the adopted Moving Mesh approach for Mixed-Mode crack propagation, detailed in Sections 3.1 and 3.2, respectively, are

implemented via the development of suitable scripts in MATLAB language. 3.1. Numerical determination of the R-curve for nano-filled UHPFRC specimens

In this section, the numerical procedure proposed to obtain complete R -curves for nano-filled UHPFRC specimens is described. This is a hybrid numerical/analytical approach that uses the general theoretical results