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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Fig. 7: Normalized notch stress intensity factor under mode 1 along the notch front, varying the notch opening angle and the Poisson’s ratio. (a) Normalized notch stress intensity factor K I / K ∞ I vs. z / h , notch opening angle 2 α = 30 ◦ . (b) Normalized notch stress intensity factor K I / K ∞ I vs. z / h , notch opening angle 2 α = 60 ◦ .

correct, together with the previous observation, the estimate of the normalized NSIF can be considered su ffi ciently representative of the real trend. For ν = 0, the NSIF is constant along the notch front, as expected, assuming a value slightly smaller than K ∞ I due to the asymptotic character of the plane stress region, hypothesized to be completely developed at r = h (to save computational time); nonetheless, the behavior and the relative change of K I as a function of ν were una ff ected. The typical increase of the NSIF over the 2D value toward the mid-plane (Pook (2013)), i.e. for z / h < 0 . 5, is clear, especially for ν = − 0 . 3, 0 . 3 and 0 . 45. It can further be noted that for positive values of Poisson’s ratio, the intensity drops near the free surface, whereas for ν = − 0 . 1 and − 0 . 3 it rises. Such observations could explain for example why the crack initiation of sharply notched plates, with typical positive Poisson’s ratio, loaded in mode 1 is located around the mid-plane, where the hypothesis of plane elasticity represents a good approximation, neglecting the slight increment of the 3D solution. The opposite trend exhibited at negative values of ν might reveal a new phenomenon: auxetic notched materials (Greaves et al. (2011)) subjected to in-plane tensile loading might fail locally to the free surface, where the NSIF tends apparently to infinity or simply to a maximum. Future experimental confirmations are anyway needed. The thickness was considered a constant parameter. However, being the relation between 3D corner point singularities and plate thickness still unclear (Kotousov (2010)), a deeper research focused only on the e ff ect of the thickness may be necessary in the future. Additionally, it must be noted that the normalized NSIF in Fig. 7 is not formally adimensional close to the corner point. In fact, although K ∞ I has physical dimensions [ F ][ L ] − 2 [ L ] 1 − λ ∞ 1 , the corresponding ones for K I are [ F ][ L ] − 2 [ L ] 1 − λ 1 , where [ F ] and [ L ] represents the dimensions of force and length, respectively. The dimensional contribution is provided by the di ff erence between λ 1 , the 3D stress singularity at the coordinate z , and λ ∞ 1 , the asymptotic plane stress singularity. Recent works, such as Pook et al. (2015) and Berto et al. (2017), have dealt with an apparent paradox related to the 3D stress field around cracks loaded under mode 3. They observed that the stress intensity factor as well as the out-of-plane shear stress ( τ yz , Fig. 2a for coordinate system) at the free surface of cracked plates and discs is not zero, as the boundary conditions, i.e. external surface loads equal to zero, impose. In fact, evaluating the SIF along the crack front at fixed distance from it, the SIF seems tending to zero near the free surface; however, its distribution at the free surface as a function of r suggests an increase as the crack front is approached. This contradiction is in part corroborated by Bažant and Estenssoro (1979), who predicted the mode 3 SIF ( K 3 ) to tend to infinity in the proximity of the free surface. In Fig. 8a the out-of-plane shear stress component ( τ yz ) as a function of the distance from the notch front in a log-log plot is featured for a set of planes at z / h = constant (codified by scale of colors). 3.2. Out-of-plane Mode 3 loading