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. 8: Out-of-plane shear stress ( τ yz ) distributions in the proximity of sharp-V-notches with opening angle 2 α = 45 ◦ loaded under mode 3, as function of the distance from the notch front ( r ). (a) Distribution along di ff erent planes z / h = constant in the range 0 to 1. Plate thickness equal to 20 mm. (b) Distribution along a 1 MPa shear-loaded surface. Plate thickness equal to 2 mm.

As obtained by Pook et al. (2015); Berto et al. (2017) for cracks, for sharp V-notches an analogous e ff ect on the τ yz distribution close to the free surface was found. As the free surface is approached, the stress component distribution is deflected from the straight line, i.e. the single power-law (as Eq. (3)). Besides, the distance from the notch front at which the deviation occurs is smaller as the distance from the free surface is reduced. We postulate that this e ff ect is caused by the 3D vertex singularity, in agreement with the results of the previous studies. However, further confirmations are needed. The core of the apparent paradox lies on the boundary conditions at the model surface, which is usually traction-free. However, this latter hypothesis, adopted by Benthem (1977); Bažant and Estenssoro (1979); Benthem (1980) in the analytical framework of self-equilibrating states of stress (eigenvalue problem), is not necessary so that the surface singularity can a ff ect the out-of-plane shear stress distribution, as shown in Fig. 8b. It was indeed obtained as a result of a numerical simulation, in which a shear surface force field along the y − direction of intensity equal to 1 MPa was applied to the model surface. It is hence proved that the 3D vertex singularity still influences the anti-plane shear stress in a region close to the notch front, even if the surface is not traction-free. The present work was conducted to understand the generative mechanism of 3D vertex singularities and their e ff ects on sharply-V-notched plates loaded under mode 1 as well as to highlight the presence of the apparent paradox on the out-of-plane shear stress component under mode 3. What is already known for cracks, was also underlined for sharp V-notches subjected to in-plane tensile loading, highlighting that the Poisson’s ratio is the fundamental parameter which causes a non uniform transversal strain region around the corner point, therefore confuting the usual assumption of generalized plane strain in the current available 3D analytical frameworks. These latter approaches cannot in fact predict neither the stress intensity factor variation (only a linear change of K 3 ) along the crack / notch front nor the 3D vertex singularity, which have been proven in this work to be strictly related to each other for sharply V-notched plates under mode 1. Therefore, a new analytical breakthrough for 3D cracked / notched plates seems to be needed. A new e ff ect on auxetic materials was further observed: numerical simulations of sharply-V-notched plates with negative values of Poisson’s ratio loaded under mode 1 revealed an opposite e ff ect both on the stress singularity and on the notch stress intensity factor. This latter was seen to increase in the proximity of the free surface, predicting a possible change in failure location for brittle auxetic materials compared to the classic ones. Despite auxetic materials are not nowadays widespread, inconceivable future applications, where brittle fracture represents the main mechanism of failure, might take advantage of the understanding previously developed. 4. Conclusions