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

2066

2

or multilayered depending to fulfil various requirements frequently used as containers for fluid consumer products. The single foil itself may be a semi-finished products to be laminate foils and folded to become the container or may be the final product such as common aluminium household foils. The typical examples are foils with thicknesses of around 5 to 15 µ m. During handling or materials testing with a focus on fracture, buckling occurs as an additional event that compli cates understanding and prediction of the critical load leading to crack growth. The ability to predict failure loads is essential both for the production of the foil and the subsequent processing by manufacturers towards the final product or directly for consumers. The risk of buckling is imminent wherever there is compressive stress, typically appearing around notches, cracks, grips or material inhomogeneities. The buckling is spontaneously initiated when it can happen without any loss of potential energy. The buckling reduces the compressive stress at the cost of increased of elastic bending energy. As we all know the buckling occurs the moment any increase of load would lead to larger release of energy than what is required for the bending.

Nomenclature

a

half crack length

σ B , υ ∗ as dimensionless functions of ν and σ B /σ D

f , g

h

foil thickness

length of the cohesive zone S S Y , S S Y length of the cohesive zone at SSY, resp. LSY LS Y length of the cohesive zone at LSY r distance from the crack tip s stress singularity exponent u , υ, w displacements in, x , y , z -directions υ ∗ crack tip opening displacement, CTOD x , y , z Cartesian coordinates E elastic modulus G , G c crack growth energy release rate, critical ditto K I stress intensity factor K I c fracture toughness ν Poisson’s ratio σ x , σ y , τ xy stress tensor components σ D cohesive stress σ B buckling stress σ Y yield stress σ ∞ critical remote stress Ψ potential energy density CTOD crack tip opening displacement SSY, LSY small respectively large scales of yielding

For central cracks in homogeneously loaded strips the stress along the crack edges is compressive. Also sectors ahead with large shear stress provides compression that is reduced at buckling. Fig. 1 shows a typical experiment with a thin aluminium foil with a central crack. It was observed by Li et al. (2006) that the reduced stresses that follow from the buckling reduces the crack tip stress singularity ∼ r − s . The pre-buckling exponent s = 1 / 2 becomes reduced. According to elastic theory, the fracture energy release rate vanishes if the crack tip is modelled as perfectly sharp while the non-linear region is represented by a singular point. As of the observations the reduction of s would at most be to 0.025, meaning that the ratio of stress in the asymptotic field before respectively after buckling would then be, r 0 . 025 . This is a very weak function of r and for most applications the changes become insignificant, meaning that observed di ff erences between experiments with