Issue 37

M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 139 cycles depending on the choice of the initial counting point of a periodic load history [2, 3]. Furthermore, multiaxial fatigue damage evaluation requires the semi-empirical calculation of path-equivalent stress or strain ranges from the rainflow-counted paths, increasing even more its computational burden [4]. On the other hand, a completely different fatigue calculation approach assumes damage as a continuous variable, whose increments can be computed as the loading proceeds. Most works based on this idea use Continuum Damage Mechanics concepts [5], which need to be supplemented by purely phenomenological damage evolution equations that are difficult to calibrate, to say the least. In fact, despite their academic appeal, such models remain controversial and have not found a wide acceptance in the fatigue design community. Other continuous damage approaches are based on an integration of elastoplastic work. However, the accumulated total work required to initiate a microcrack by fatigue certainly is not a material property. Moreover, the elastoplastic work still depends on the number of cycles, thus it is impossible to calculate without previous load cycle and/or reversal detection. Therefore, even if it could be assumed that fatigue damage can be quantified by this parameter, its calculation routine still would need to include a rainflow or other similar load event counter. Alternatively, instead of integrating dubious strain energy or energy-based damage parameters, a more reasonable path is to continuously quantify fatigue damage itself, using some well-proven model that can properly describe multiaxial fatigue damage in the material in question. The so-called Incremental Fatigue Damage (IFD) approach integrates the chosen parameter until reaching 1.0 or any other suitable critical-damage value using traditional accumulation concepts, as originally performed a long time ago for the uniaxial case by Wetzel under Topper's guidance in 1971 [6], and again by Chu in 2000 [7]. It is important to emphasize that in such calculations fatigue damage is continuously computed after each infinitesimal stress or strain increment, so its quantification does not require the prior identification of load cycles. In this work, the IFD approach is revisited and extended to general multiaxial fatigue problems, including no-proportional ones, based on a direct analogy with non-linear incremental plasticity concepts, however calculating damage instead of plastic strains at each load increment. T HE I NCREMENTAL F ATIGUE D AMAGE A PPROACH he Incremental Fatigue Damage approach was proposed for uniaxial load histories in Wetzel and Topper’s rheological model [6, 8]. It makes use of the derivative of the normal stress  with respect to damage D , called here generalized damage modulus D  , thus D d dD D dD (1 D ) d            (1) Consider, for instance, a uniaxial constant amplitude loading history with stress amplitude  a . During a loading half-cycle, the excursion of the stress  from  a to  a could be integrated according to Eq. (1) to find the associated fatigue damage D  1/2N , however without explicitly calculating the fatigue life N . Assuming the material is initially virgin, the damage D from the first half-cycle is initially zero in the initial valley when  a and thus  a )  0 , and continuously grows toward D = 1/2N until  reaches the peak  a , when  a ) = 2  a . For simplicity, Wöhler’s stress-based fatigue damage model is adopted below (but strain-based models will be considered later). A simplified relation between the current stress state  and the continuous damage D from the half-cycle excursion  a   a can then be obtained from Wöhler’s curve e.g. written in Basquin’s notation:   1/b b b a c a c a c ( 2N ) 2 [ ( )] 2 D D ( ) 2                     (2) The generalized damage modulus D  during this half-cycle is thus such that   1/b a c a 1 D dD d ( ) 2 [ b( )]              (3) from which the fatigue damage D  1/2N can be calculated using the integral T