PSI - Issue 28

Per Stahle et al. / Procedia Structural Integrity 28 (2020) 2065–2071 P. Ståhle et al. / Structural Integrity Procedia 00 (2020) 000–000

2071

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present result in the range of ± 7%. Interestingly enough it means that the fracture toughness at buckling is more or less independent of both Poisson’s ratio and the scale of yielding. This is said with the disclaimer that the foil thickness have to be small enough to allow for a Dugdale type of necking region to develop before failure load is reached. From (11) and using the results displayed in Figs. 5 and 6 we define the functions f = σ B (1 − ν 2 ) a 2 Eh 2 and g = 2 π . (14)

υ ∗ E σ D σ 2 B a

Elimination of the crack length leave us with the following ration of CTOD versus foil thickness, υ ∗ h 2 = π 2 4 f g 2 σ D (1 − ν 2 ) E σ B σ D 3

(15)

Thus, buckling will occur before failure if υ ∗ h < 1 ⇒ E σ Y > π 2 2 √ 3 f g 2 (1 − ν 2 ) σ B σ D 3 .

(16)

The consequences for an aluminium foil, with the modulus of elasticity E = 70GPa, ν = 0 . 3, leading to f ≈ 1 . 2 and g ≈ 1, at the ASTM limit σ B /σ D ≈ 0 . 3 is that buckling occurs before failure if E /σ Y > 0 . 1. It would be di ffi cult to find an aluminium alloy that does not fulfil the inequality. Due to the quite limited changes for large scale yielding case and reasonable changes of ν , f and g failure is not at all probable that there is a construction material that does not fulfil the inequality (16). On the other hand fracture may happen according to the conditions (6) if the foil is thick enough.

5. Conclusions

A pilot investigation of the relation between buckling loads and the presence of necking type of strain localisation in the form of a Dugdale cohesive zone is completed. The elastic and two plastic cases are studied. Of the plastic cases one load is at the ASTM-limit for linear fracture mechanics and the other is a twice as large load. A wide range of Poisson’s ratios from ν = − 0 . 9 in the elastic and -0.6 in the plastic cases are included. For all cases the upper limit ν = 0 . 5 is reached. The buckling load σ B obtain very large values for small values of ν . Obviously σ B ≈ 10 . 5 σ D for ν = − 0 . 9 as compared with 1 . 3 σ D for ν = 0 . 5. Fig. 5 shows the scaled version of buckling load σ B (1 − ν 2 ) giving a factor of 2 over the same range. A comparison of the remote load at plastic failure and the same at buckling show that failure happens for almost all reasonable construction materials. There may be exceptions for materials with extremely high elastic strain limits. K I controlled fracture for thicker plates is not covered by the present study.

Acknowledgements

Support from Blekinge Institute of Technology and Tetra Pak is gratefully acknowledged.

References

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