PSI - Issue 79

D. Marhabi et al. / Procedia Structural Integrity 79 (2026) 34–52

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Uniax Te Rb W W W   (1.e) The threshold stress σ * is considered as a stress of no damage crack at the microscale. By reference to a homogenous fully reversed load (tensile or rotating bending), the energy analyses is suggested by Banvilet and all (Eqs 1.a and 1.c). The authors deduce the threshold stress value σ *, on area initiating small crack. * 2 2 , 1 , 1 2 2 * * *2 , 1 , 1 Te Rb * 2( ) ( ) (1. .1) 2( ) ( ) W W Te Rb Te Rb f                     To identify the threshold stress near small crack initiation, many tests were performed for the steel Iron cast GS under constant amplitude tensile by [3]. Fatigue scanning electronic microscope observations were carried out at different fractions of life for constant stress above and below the endurance limit. These observations have confirmed that fatigue small crack and growth in middle-cycle regime (# 10 5 Cycles). Tensile mean stress and dynamic amplitude The study in this article is based on average and dynamic stress applied in straight section of specimen:   , sin Te mTe Te t t        A required quantity is the over energy on area elsto-plastic S* initiating small crack:     2 * 2 2 * 2 *2 , , 1 1 Eq Te Te mTe mTe W and W E E E           (1.g.1)     * Te Te * 2 * 2 , * W -W 1 1 A Te Eq Te Eq ds E A        (1.g.2) The postulate (1.d) for mean stress and dynamic loading is :   2 * 2 , 1 Uniax Te Rb Eq Te Eq E          (1.g.3) 3. ANALYTICAL OVER-ENERGY CONCEPT AND NUMERICAL INVESTIGATION We analyze an over-energy under rotating bending and expressed in the elliptical axes ( * a , * b ). An Asymptotic approach transformed the over-energy in biquadratic function predict the unknown threshold stress σ *. These threshold stresses σ * are lower than the endurance limit σ -1 . O VER -E NERGY UNDER ROTATING BENDING The service stress (Fig. 1 (5a) (b)) on a straight section by a cylindrical specimen: sin , y r t Rb mRb Rb R R        (2.a) The energy density given to each elementary under dissymmetrical rotating bending is:     2 2 2 2 , W m Rb Rb Rb y r E R E R     (2.b) The Iso-energy (Eq. (2.b)) is an ellipse on the cross-section of the specimen (Fig. 1 (1.b)). 2 2 , 1  , 1  (1. .2) f ( ) (  ) (1. .3) f unia   Te   Rb       Rb Te unia    W E E E    