Issue 48

L. Reis et alii, Frattura ed Integrità Strutturale, 48 (2019) 318-331; DOI: 10.3221/IGF-ESIS.48.31 323 Once the equivalent strain history is obtained, it is identified the maximum peak of the equivalent strain history, ௜௝஺ , which will be the reference strain. All points prior to the time instant of ௜௝஺ , ஺ , are moved to the end of the history and the equivalent relative strain is then calculated, as shown in Eq. (17), resulting in a new strain history, where the first half-cycle goes from 0 to 2. ௘௤ ሺ ሻ ൌ ௘௤ ቀ ௜௝ ሺ ሻ െ ௜௝஺ ሺ ஺ ሻቁ (17) After obtaining the relative strain history, it is performed the extraction of reversals. The extraction of a reversal starts at t 0 , which once was t A , and stops whenever a descending path is found. When this occurs the extraction of that reversal is resumed after an ascending path, with a higher relative strain, is found. This procedure goes on until the end of the relative strain history. Once a reversal is extracted, the equivalent shear strain data points, from this reversal, are removed from the original strain history, and the relative strain is again computed, followed by the extraction method. This procedure is conducted until there are no more reversals to extract. Finally, the Wang-Brown’s damage parameter is calculated for each reversal, as formulated in Eq. (18). ̂ ≡ ଴.ହሺ∆ఊ ೘ೌೣ ሻାௌሺఋఌ ೙ ሻ ଵାఔ ᇲ ାሺଵିఔ ᇲ ሻௌ (18) where ∆ ௠௔௫ is the maximum shear strain range, found in all the planes of projection, and ௡ is the normal strain excursion calculated between the time interval of ∆ ௠௔௫ and in the same plane of ∆ ௠௔௫ . ᇱ is the effective Poisson coefficient and is a material constant. The number of reversals to failure, 2 ௙ , is then calculated for every reversal, as formulated in Eq. (19). ̂ ൌ ఙ ᇲ ೑ ିଶ∙ఙ ೙,೘೐ೌ೙ ா ∙ ൫2 ௙ ൯ ௕ ൅ ᇱ ௙ ∙ ൫2 ௙ ൯ ௖ (19) where ᇱ ௙ and ᇱ ௙ are the fatigue resistance coefficient and the fatigue ductility coefficient, respectively. and refer to the fatigue strength exponent and the fatigue ductility exponent. The damage of a loading block is calculated using a linear damage accumulation rule, as shown in Eq. (20). Fatigue life, in blocks, is estimated as formulated in Eq. (21). ௕௟௢௖௞ ൌ ∑ ଵ ଶே ೑ ೔ # ௢௙ ௥௘௩௘௥௦௔௟௦ ௜ୀଵ (20) ௕௟௢௖௞ ൌ ଵ ஽ ್೗೚೎ೖ (21) Damage accumulation rules In this work, two non-linear damage accumulation models were calculated for comparison, the Palmgren-Miner’s rule and the Morrow’s rule [25]. Palmgren proposed damage concept defined by the ratio between the number of cycles performed and the number of cycles to failure at a given load level. Using this concept, Miner proposed the so called Palmgren-Miner’s Rule. This damage accumulation rule computes fatigue damage block by block and adds the damage linearly, as shown in Eq. (22). ൌ ∑ ௡ ೔ ே ೔ ௞ ௜ୀଵ (22) where ௜ and ௜ are the number of cycles in a block and the number of cycles to failure of the i th block, respectively, at a given stress level. is the number of stress levels present in the loading history. The rule states that the damage at failure moment is equal to 1. Morrow proposed a nonlinear damage accumulation rule that accounts for load interaction effects in variable amplitude loadings. This rule updates the Miner’s damage by a parameter that is computed by the ratio between the maximum stress amplitude at the i th stress level and the maximum normal stress of the loading spectrum, as shown in Eq. (23).