PSI - Issue 5

M. Madia et al. / Procedia Structural Integrity 5 (2017) 875–882 M.Madia/ Structural Integrity Procedia 00 (2017) 000 – 000

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( , ) = √ 2 ∙ ∙ [1 + 1 ∙ ( ) 1 2 + 2 ∙ ( ) + 3 ∙ ( ) 3 2 ] . (5) The coefficients have been calculated according to Wang et al. (1995) and an efficient integration scheme is adopted to avoid singularity problems (Moftakhar et al., 1992). There exist various options for the definition of the plasticity correction function. If the stress-strain behavior of the material is known (approximated by a Ramberg-Osgood in the present work), ( ) can be determined by the highest analytical analysis level of R6 (2013) ( ) = ( ∙ + 1 2 ∙ ∙ ∙ 2 ) − 1 2 , (6) where and are the reference strain and reference stress respectively, and is the elastic modulus. The meaning of and is depicted in Fig. 1(a). The application of the reference stress method to cyclic loading has been proposed by Zerbst et al. (2012), who applied the method to the determination of the stress-life curve of specimens made of Al5380-H321, containing large second phase particles. The method has been developed further and the cyclic -integral is determined by ∆ = ∆ ∙ [ (∆ )] −2 , (7) (∆ ) = ( ∙∆ ∆ + 1 2 ∙ ∆ ∙∆ ∙ ∆ 2 ) − 1 2 , (8) and ∆ = ∆ 2∙ 0 . (9) It is easy to recognize that the expressions of Eqs. (7) to (9) are formally equivalent to the expressions derived in case of monotonic loading, in that the loading variables are replaced by their variation (range). Note that it has been assumed that the Masing’s hypoth esis is verified for the materials under investigation, so that either branch of the cyclically-stabilized stress-strain material response is geometrically similar to the Ramberg-Osgood material law by a scale factor of two. The meaning of this statement is depicted in Fig. 1(b).

Fig. 1. Determination of the reference stress and strain in case of (a) monotonic and (b) cyclic loading.