PSI - Issue 42

Michal Vyhlídal et al. / Procedia Structural Integrity 42 (2022) 1000–1007 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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its higher porosity compared to the bulk matrix which is related to the higher local water to cement ratio (Scrivener et al., 2004). The local increase in porosity leads to poorer bonds between the components, and thus to lower values of the mechanical fracture parameters of the ITZ, see e. g. Zacharda et al. (2018) or Vyhlídal and Klusák (2020). The properties of ITZ influence the overall fracture behavior of concrete. 2.2. Linear elastic fracture mechanics and its generalized form Fracture mechanics is a widely used tool for an assessment of crack behavior in materials. Linear elastic fracture mechanics supposes cracks in homogenous, isotropic, and linearly elastic material, see Anderson (2005). Williams (1957) presented a universal solution for the stress components around a crack tip in the form of infinite series. Near crack tip ( → 0), only the first (singular) term is used to describe the stress field, while the higher order terms of the infinite series can sometimes be neglected. Components of the stress tensor are given by the superposition of three basic failure modes defined by Irwin (1957), where the opening mode I is predominant failure mode. Generalized linear elastic fracture mechanics deals with an assessment of general singular stress concentrators, such as sharp notches, bi-material notches, bi-material inclusions, interface cracks etc. – see Klusák et al. (2013), De Corte et al. (2017), or Klusák et al. (2016). In most cases, the loading modes cannot be assigned to the series terms, so the terms are labeled by Arabic subscripts. The dependence of stress on polar coordinate r is expressed by a general stress singularity exponent p 1 : ij = 1 √2 − 1 1 (  ) ሺͳሻ where σ ij are the stress tensor components for i, j = r,  . Generalized stress intensity factor H 1 have the units [MPa  1 ], and F ij are the shape functions. For cracks in homogeneous media, the stress intensity factors can be determined directly in the FEM software, e.g. Ansys Inc. software (2021), while for the general singular stress concentrators, various direct or integration methods are used, see Ping et al. (2008), Klusák et al. (2008) or Profant et al. (2008). The values H 1 are ascertained from a numerical solution of the studied geometry, materials, and boundary conditions. 2.3. Criterion of stability based on average stress ahead of the crack tip In the paper, the crack propagation condition is determined from the generalized maximum tangential stress (GMTS) criterion. This stability condition is based on two well-known fracture mechanics stability conditions – (a) crack initiation occurs if the stress intensity factor K I reaches its critical value K Ic (also known as fracture toughness), see e.g. Anderson (2005), and (b) crack will propagate in the direction where the tangential stress σ θθ is maximal, see Erdogan and Sih (1963). The GMTS stability criterion employs the average stress σ̅ θθ ( ) calculated across a distance d ahead of the crack tip. The distance d is usually chosen in dependence on the mechanism of a rupture (dimension of a plastic zone of metals, dimension of process zone of quasi-brittle materials, material grain size, or finite supposed crack initiation/propagation increment). The average stress σ̅ θθ ( ) ahead of the crack tip is given by the following equation. ̅ = 1 ∫ ( , ) 0 (2) The crack propagation direction is supposed in the direction of maximum of the average tangential stress resulting from numerical analysis. The critical value of tangential stress σ̅ θθ ( ) corresponding to the crack initiation is determined from the knowledge of crack propagation conditions for a crack in homogeneous material under mode I, where fracture toughness K Ic is employed: ̅ , = 2 √2 (3)