PSI - Issue 28

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

2070

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

2 ) a 2 / ( Eh 2 ) vs. Poisson’s ratio ν

2 B a ) vs. Poisson’s ratio ν

Fig. 5. Buckling load σ B (1 − ν

Fig. 6. Crack tip displ. 2 υ ∗ σ

D E / ( πσ

constrained along the external edges at x = L and at y = L . The resulting buckling loads for the nonlinear response are obtained in an iterative process of repeated calculations of the pre-stress and buckling load until the di ff erence between the pre-stress and the buckling load is less than 0.5%.

4. Results and discussion

On dimensionless form the buckling stress may, according to e.g. Guz et al. (2004), Brighenti (2009), preferably be written σ B = (1 − ν 2 )( a / h ) 2 / E which removes, E , a , and h , from its only occurrence in the governing equations Eq. (7). Thus, the solution only depends on the mentioned quantities in the combination E ( h / a ) 2 . Only two parameters, are required for a complete solution, i.e., Poisson’s ratio and the yield stress. The latter is given as its de facto replacement, the cohesive stress. The non-dimensional ratio σ D /σ B is used. The following complete solution is sought, σ B = f (1 − ν 2 )( a / h ) 2 / E , (11) where the function f = f ( ν, σ D /σ B ). The physically allowed range for Poisson’s ratio is − 1 < ν < 1 / 2 , (12) where as the lower limit is approaching the unshearable limit ( E / (1 + ν ) → ∞ ) and at the upper, the incompressibility limit is approached. For the cohesive stress the lower limit, if the fully plastic limit should be avoided, is σ D /σ B > 1 . (13) The intension to cover the full region for Poisson’s ratio, ν , was not fully achieved. Due to reasons that are yet not understood by the authors, the buckling load could not be obtained for Poisson’s ratios below -0.9 in the elastic case and for below around -0.6 for the plastic cases. To some concolation, the important results are for natural reasons Poisson’s ratios in the fully covered range 0. to 0.5. As is observed in this important range the buckling loads drop from around 1.5 to 1 in the elastic case. Of the plastic cases σ B = 0 . 3 σ D represents the ASTM limit of linear fracture mechanics. The length of the cohesive zone is then around 15% of the half crack length, a , cf. Fig. 4. As is observed in Fig. 5 the drop of buckling load as compared with the elastic case is slightly below 10% lower for ν = 0 and around 20% lower for ν = 0. The case σ B = 0 . 6 σ D is taken as a representative of large scale of yielding. The length of the cohesive zone is then around 55% of the half crack length a . In spite of this pretty large scale of yielding the drop of buckling load from the elastic case is not more than around 20%. Figure 6 shows the crack tip opening displacement 2 υ ∗ (CTOD). The CTOD also represents the energy release rate in the necking region which allow us to include the elastic asymptotic limit. Comparing the two cases gives first that both cases have a CTOD’s that are rather independent of Poisson’s ratio. Second the the CTOD is close to be twice as large for the for the lower cohesive stress that is half of the higher cohesive stress. The accuracy is according to the