PSI - Issue 25

Marco Maurizi et al. / Procedia Structural Integrity 25 (2020) 268–281

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M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

u displacement field, subscripts x , y , z denote direction w displacement along z -direction dependent only on x and y λ Lame’s modulus; eigenvalue with subscripts 1, 2, 3s, 3a and 0; strength of corner point singularity G shear modulus k shear factor ν Poisson’s ratio N i j i , j − th stress resultant, subscripts x , y , z denote directions; without subscript denotes the mean in-plane g i j angular functions for plane elastic crack problems Φ Airy stress function K N i mode i notch stress intensity factor (NSIF); without superscript stress intensity factor (SIF); with superscript C coupled mode; with ∞ far-field (N)SIF Φ Airy stress function F i dimensionless scale e ff ect function for coupled mode i β crack front intersection angle The history of fracture mechanics starts with the work of Inglis (1913), who provided the analytical exact 2D solution of the stress field around an elliptical hole (notch) in a plate, allowing to extend this result to cracks. Plane elasticity, under the hypothesis of plane stress or plane strain, constitutes the analytical framework under which Linear Elastic Fracture Mechanics (LEFM) has been studied and applied to assess the structural integrity of materials since its emergence. Based on this, Gri ffi th (1995) proposed an energetic failure criterion, Westergaard found out the closed form solution of a center through-the-thickness cracked infinite plate under biaxial tension, Irwin modified the Gri ffi th theory introducing a correction for small scale plasticity at the crack tip and defined the concept of stress intensity factor ( K ) related to that of stress singularities (stress tend to infinite as the crack / corner is approached). These latter have been investigated for angular corners (notches) of plates by the well-known work of Williams (1952), where the 2D stress field σ i j ∝ r λ − 1 has been found to show a singularity for r →− 0, with i , j = 1 , 2 , r the radial distance from the notch tip, and λ the eigenvalue obtained by imposing the boundary conditions and representing the strength of the singularity. These results paved the way for the theoretical generalization of the stress intensity factor (often indicated as K N ) to V-shaped notches (crack is a particular case of it), formally defined by Gross and Mendelson (1972), giving physical meaning to the constants of proportionality of the William’s formulae; subsequently, Lazzarin and Zappalorto (2012) extended the analytical formulation also to blunt notches, for which the singularity disappears. The analytical and numerical 2D results in the field of LEFM have been used as bases for standards, such as ASTM E1820-18 for measurement of fracture toughness and ASTM E647-00 for measurement of fatigue crack growth, which are hence based on the assumption of 2D elasticity. Experimental evidence Pook (2013), Doquet et al. (2009, 2010), Bertolino and Doquet (2009) seems to contradict some 2D analytical results, which for instance do not well predict the site of crack initiation in notched thick members, shifting the failure location from the middle plane to the corner (near the free surface) as the thickness increases. In fact, stress intensity factors and energy release rate have a variable profile along the crack front Berto et al. (2013a,b), and coupled modes (see paragraph 4) exist for shear loading, meaning a breaking down of classical 2D solutions. This could explain the bowed fatigue crack fronts Doquet et al. (2010) as well as the inconsistency of the predicted strength size e ff ect by using the 2D formulation with experimental results Sinclair and Chambers (1987). Crack / notch tip displacements are classically described by three modes of relative motion between the crack / notch surfaces, if considered as points in the 2D frame. However, a 3D description of the surfaces gives rise to regarding them as composed of infinitesimal elements, which can also rotate relative to each other; six distinct movements are hence possible for one element on a crack / notch tip surface, correspondent to Volterra distorsioni (distortions) of dislocation and disclination, as reported by Pook (2013). The former represent the three well-known modes of fracture Mode I (along y axis), II (in-plane shear, along x axis) and III (out-of-plane shear, along z axis), whereas the latter are rotations about the same three axes used to identify the dislocation distortions (Fig.1). 1. Introduction