Unit cell calculations keeping all stress ratios constant

Ductile fracture in metals and metal alloys occurs through nucleation, growth, and coalescence of micron-scale voids nucleated at second phase particles. A detailed experimental investigation of the stages of ductile fracture, especially that of void nucleation and coalescence, is rather challenging and requires guidance from numerical simulations. With the studies in the literature, finite element voided unit cell calculations performed on ideal materials containing periodically distributed voids have become a convenient tool for numerical simulations of ductile fracture. In the last century, the effect of stress triaxiality on ductile fracture was well known, but shear loads were disregarded. Therefore, voided unit cell calculations were performed by prescribing only the ratios of normal stresses, i.e. ρ_11=Σ_11/Σ_22 and ρ_33=Σ_33/Σ_22. Experimental studies in the last decades, however, revealed a pronounced effect of shear loads on ductile fracture at low stress triaxiality, incorporated into ductile fracture simulations through the Lode parameter. In order to account for the effect of shear loads on void growth and coalescence, Tekoğlu (2014) developed a finite element framework where the stress state on a cubic unit cell containing a spheroidal void at its center is represented by three non-dimensional stress ratios, i.e. ρ_11=Σ_11/Σ_22, ρ_33=Σ_33/Σ_22 and ρ_12=Σ_12/Σ_22. The present study attempts to extend the work of Tekoğlu (2014) by allowing a complete definition of the stress state on a unit cell with five non-dimensional stress ratios, i.e. ρ_11=Σ_11/Σ_22, ρ_33=Σ_33/Σ_22, ρ_12=Σ_12/Σ_22, ρ_13=Σ_13/Σ_22 and ρ_23=Σ_23/Σ_22. Aside from being numerically efficient, the developed framework allows keeping the errors in the prescribed stress ratios below a few percent. The developed framework is generic and can be applied to any unit cells containing voids or particles. In addition to this framework, an alternative method that lowers modeling and computation costs is proposed for unit cell calculations performed only under principal stress state.