PSI - Issue 28

Per Stahle et al. / Procedia Structural Integrity 28 (2020) 2065–2071

2067

P. Ståhle et al. / Structural Integrity Procedia 00 (2020) 000–000

3

Fig. 2. Cross sections after rupture. Necking regions are marked X.

Fig. 1. Buckling regions in the crack tip vicinity.

buckling allowed vis-a`-vis prevented by supporting side plates cannot be explained by the altered stress singularity alone. Therefore, this pilot study considers the finite extent of the line shaped necking region that propagates ahead of the tip of a crack. The following section describes the fracture process. The process does not include the fracture toughness since it is entirely plastic deformation without any visible trace of fracture surfaces, see Fig. 2. For that reason it is in the following denoted failure. This puts restrictions on the foil thickness, that has to be su ffi ciently small. The exact meaning of su ffi cient is described in the next section.

2. The model

A Cartesian coordinate system, x , y and z , is chosen so that the foil in its natural state is placed in the plane z = 0 . The crack occupies the region | x | < a , y = 0 and | z | < h / 2. The material is elastic plastic, homogeneous and isotropic material and plane stress conditions apply. The material model is based on a split of the strain components into elastic and plastic counterparts, i.e., x = e x + p x , y = e y ... , etc. for all strain components. Elastic strains are given by Hooke’s law, using the elastic modulus E and Poisson’s ratio ν . Plastic strains are incrementally accumulated, while von Mises’ e ff ective stress exceeds the yield stress, σ Y . The plastic is associated with the von Mises yield condition. The plastic deformation in thin metal foils is observed to be plastic slip across the foil thickness as was observed by Dugdale (1960) and many others after that. Fig. 2 from Andreasson (2019) shows the cross section slip as the 1:2 slopes adjacent to the line of final failure, i.e., the tips of the left and right beveled parts marked ”X”. At the tip there is nothing indicating a final fracture. The slip that occurs along 45 o diagonals in the square shaped active plastic region, leaves a surface slope with an angle to the initial foil surface of 26.6 o , a 1:2 slope, to preserve volume. The width of the necking region, measured along the y -axis, cf. Fig. 2, is initially one thickness h wide and at completed rupture the neck width increases an additional h . Since the plastic straining is confined to the narrow necking region and the material is elastic directly above and below it, to maintain continuity across the neck boundary, of strains along the necking region, the plastic strains p x in the necking region have to be practically zero. Therefore, the stress in the neck, in the x -direction, according to von Mises’ associated flow rule, have to be σ D / 2 and with the yield condition fulfilled, σ D is readily obtained as 2.1. The failure process

2 √ 3

σ Y ≈ 1 . 15 σ Y .

(1)

σ D =