PSI - Issue 26

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

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γ supplementary V-notch opening angle and crack surface intersection angle 2 α V-notch opening angle u displacement field, subscripts r , θ , z denote directions λ stress singularity, subscript indicating the mode of loading G shear modulus ν Poisson’s ratio f i j angular functions for stress field K i mode i (notch) stress intensity factor (N)SIF; superscript ∞ for far-field (N)SIF

1. Introduction

In a previous recent our paper Maurizi and Berto (2020), we reviewed the state of the art on three dimensional e ff ects on Fracture Mechanics. Specifically, the main numerical results regarding the in-plane and out-of-plane shear coupled modes were described and contextualized in the various analytical frameworks so far developed. Moreover, 3D corner point singularities were briefly introduced, without an in-depth-analysis. To the best of our knowledge, Benthem (1977, 1980) was the first author to analytically prove the existence of a particular state of stress at the vertex, i.e. at the intersection of the crack front with a surface (from here the term surface singularity ), of a quarter infinite crack in a half-space. The author solved an eigenvalue problem in the linear elastic fracture mechanics (LEFM) framework, assuming the existence of a state of stress as σ i j = r λ − 1 f i j ( λ, θ, Φ ) around the vertex of a crack in spherical coordinates ( r , θ , Φ ), whose infinite and enumerable eigenvalues and eigenfunctions were the singularity exponents λ − 1 and stress functions, respectively; obviously, the displacements will behave like r λ . Analogously to Benthem (1977, 1980), Bažant and Estenssoro (1979) based their analysis on the assumption of separation of variables on the stress and displacement functions, meaning that at di ff erent angles, identified by the couple ( θ, Φ ), the behavior is invariant with respect to r . From the strain energy, they derived a variational principle, proving how this latter, together with the eigenvalue problem, must be nonsymmetric, leading to complex eigenvalues. Their main finding was that to establish a numerical relation between the stress corner singularity (1 − λ ), the Poisson’s ratio ν , the crack surface angle γ and the crack front angle β (Fig. 1).

Fig. 1: (a) Crack front intersection angle. (b) Crack surface intersection angle.

Expressing the energy flux which flows into the crack front edge by means of the Rice’s J-integral, they deduced that the condition of propagating crack is Re ( λ ) = 1 / 2, which is the classical 2D stress singularity. However, from the numerical relationship between the angles in Fig. 1 and λ , these authors have also shown that the crack front edge angle ( β ) of a propagating crack must be oblique, whose amplitude depends on the crack surface angle ( γ ). Therefore, the shape of a propagating crack front is adjusted such that the classical square root singularity is held, intersecting the surface at a critical angle β c . This latter was empirically quantified by Pook (1994), linking it to the Poisson’s ratio for every mode of fracture, assuming λ = 1 / 2. Numerical and experimental studies, such as that of Heyder et al. (2005) confirmed this continuous process of adjusting β to keep constant the stress singularity during a crack propagation. Nonetheless, for the fracture assessment of brittle materials knowing the behavior of the stress singularity and (notch)