In this way, we are indirectly relating the sharpness of the needle tip to the cohesive law of the material. As explained in Sect. 2.3, the use of a displacement-based fracture criterion enables us to consider the relevant influence of the tip radius and sharpness on the condition of crack propagation during penetration.
a Illustration of the critical condition for crack propagation: the needle tip is pushed forward until the critical CTOD at a distance L from the equivalent crack tip is reached. b Sketch of the crack propagation path, defined as the locus of the equivalent crack-tip positions during the iterations. In a generic point, t and n locate, respectively, the tangent and the normal directions to the propagation path. c Schematic showing the MSED kinking criterion, with the radial mesh around the equivalent crack tip in the deformed configuration. The coloured contours are related to the strain energy density, in the case of a needle with a symmetric tip c=0 (left) and for the PBN (right)
Matlab R2013a Crack 983
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where the second condition states that the point should be a point of minimum of the function US(θ). In practice, from the ring of elements surrounding the current equivalent crack tip (Fig. 6b), strain energy densities are extracted at Gauss points and interpolated on an admissible angular range for propagation, based on the limit in pure shear loading (Sih 1974). The angle, within the admissible range, which corresponds to the local minimum of SED defines the crack propagation direction. Figure 6c illustrates the schematic contours of the SED in the case of a symmetric needle and for the PBN: in the latter, the SED distribution shows a minimum which is oriented at a certain angle with respect to the vertical direction of insertion.
View of the deformed mesh during the penetration. The enlarged insets show the moment when the critical condition is reached, at a distance Ladd from the current crack tip. At the same time, the kinking angle θc is computed, according to the MSED criterion. The equivalent crack tip is propagated, and a new cohesive element is added (highlighted in red) 2ff7e9595c
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