Issue 48

L. Reis et alii, Frattura ed Integrità Strutturale, 48 (2019) 318-331; DOI: 10.3221/IGF-ESIS.48.31 322 Liu Liu’s proposal [21] consists in an energy-based model with two different formulations, depending on fatigue failure’s mode. For tensile failure mode the damage parameter, ∆ ூ is given by Eq. (13), and by Eq. (14) for shear failure mode, ∆ ூூ . ∆ ூ ൌ max ఏ ሺ∆ ௡ ∙ ∆ ௡ ሻ ൅ ሺ∆ ∙ ∆ ሻ (13) where ∆ ௡ is the normal stress range, ∆ ௡ is the normal strain range. ∆τ and ∆γ are the shear stress range and the shear strain range, respectively. ∆ ூூ ൌ ሺ∆ ௡ ∙ ∆ ௡ ሻ ൅ max ఏ ሺ∆ ∙ ∆ ሻ (14) This method uses a virtual strain energy (strain multiplied by stress) concept (VSE). The plane θ with the higher VSE will be the critical plane. Cycle counting As it was described, when a part or material sample is subjected with variable fatigue damage due to variable amplitude loading conditions, it is essential to design a model that can count the cycles in order to estimate the damage output in to the material. In this work three different counting methods were applied, the Rainflow, Virtual Cycle Counting and the Wang-Brown method. They are now briefly described. Rainflow counting method Matsuiski and Endo proposed cycle counting method based on the concept of a raindrop falling from a pagoda roof. The method identifies closed hysteresis loops, in a stress-strain plot, as cycles. A series of rules must be followed to extract cycles from a strain-time or stress-time history [22]. Virtual cycle counting (vcc) Anes et al. [23] proposed a cycle counting method for variable amplitude multiaxial loadings, based on the SSF equivalent shear stress time evolution. In SSF equivalent shear stress history, the number of virtual cycles is given by adding the maximum absolute value of the peaks and valleys found between two zero stress points. This sum is then divided by the double of the maximum equivalent shear stress in the considered loading block. This method is formulated in Eq. (15). ௕௟௢௖௞ ൌ ∑ ௔௕௦൫ఛ ೐೜ ൯ ೛೐ೌೖ ା ∑ ௔௕௦൫ఛ ೐೜ ൯ ೡೌ೗೗೐೤ ଶ∙൫ఛ ೐೜ ൯ ೘ೌೣ,್೗೚೎ೖ (15) The formulation in Eq. (15) assumes that the block extraction method was applied. Otherwise, the denominator would be the maximum equivalent shear stress in the entire loading spectrum. Wang-Brown Wang and Brown [24] proposed a method to estimate fatigue life in variable amplitude loading conditions. In this method, it was proposed a new procedure to count cycles. Given a multiaxial strain history, the von Mises equivalent strain is calculated, as shown in Eq. (16). ௘௤ ൌ ଵ √ଶሺଵାఔ ᇲ ሻ ⎩⎨ ⎧ ൣ ௫ െ ௬ ൧ ଶ ൅ ൣ ௬ െ ௭ ൧ ଶ ൅ ሾ ௭ െ ௫ ሿ ଶ ൅ ቀ ଷ ଶ ቁ ൣ ௫௬ଶ ൅ ௬௭ଶ ൅ ௭௫ଶ ൧ ⎭⎬ ⎫ భ మ (16)