Issue 33

M. Cova et alii, Frattura ed Integrità Strutturale, 33 (2015) 390-396; DOI: 10.3221/IGF-ESIS.33.43 392         ,max ,min , 1 1 1 max , 2 2 2 2 n n T T T n a V V V n n n n n n            (4) Where the max{} function identifies the maximum value among the values within the bracket. At this stage, it is clear that the simple amplitude of the normal stress is dependent only from the time-variable stress tensor. Hence, the maximum value of normal stress amplitude is the half of the maximum eigenvalue of the [σ] V tensor; consequently, the direction “n” loaded by the maximum normal stress amplitude is the related eigenvector. The modulus is conveniently used in order to properly take into account even compressive loading.       , 1 max max 2 a n n a V eigvl      (5) Where eigvl[ ] function specifies the eigenvalues of the considered matrix. Anyway, the problem turns out much more complex if, as usual, the consider material has any kind of sensibility to mean or hydrostatic stress components. Let us consider a material showing a linear sensibility to mean value, i.e. the fatigue strength decreases linearly by increasing the mean stress value. This assumption is usually called “simplified Goodman” relationship. In this case, it is possible to consider an effective value of the stress amplitude by combining the actual stress amplitude with a portion of the applied mean stress value. , a eq a m b      (6) The constant “b” is dependent from the material, it is usually positive and lower than 1. The critical direction is no longer the direction previously identified, but the direction undergoing the maximum value of the equivalent amplitude. For a general direction n, the equivalent amplitude value turns out to be:                         , , , , 1 1 2 2 1 1 1 max , 2 2 2 1 1 max , 2 2 n a eq n a n m T T T V S V T T T T V V S V T T V S V S b n n b n n n n n n n n b n n n n b b n b n n b n                                                              (7) Finally, it is necessary to find the direction with the maximum value of equivalent amplitude. The linear combination of tensors is itself a tensor; hence, the maximum value, by changing the direction n, is again the maximum eigenvalue of the considered tensors.           , , , 1 1 max max , 2 2 a eq n n a eq V S V S b b eigvl b eigvl b                               (8) Applicative example Let us consider a time variable tensile loading, ranging from 0 to 2S, so that its amplitude is S. The equivalent amplitude is simply (b+1)S. Then, if the variable tensile loading is combined with a static torsional loading, the related tensors are:   0 0 0 0 0 0 0 S T T             ;   2 0 0 0 0 0 0 0 0 V S             (9) If we assume that the tensile loading is actually positive, the terms “b+1” are predominant and the resulting equivalent stress amplitude is: