Issue 41

M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 41 (2017) 98-105; DOI: 10.3221/IGF-ESIS.41.14 99 I NTRODUCTION lthough practical fatigue analyses usually involve variable amplitude loads (VAL), material data (if available) normally is measured in constant amplitude load (CAL) tests. So, fatigue design routines must use some damage accumulation rule to deal with VAL. The classic linear damage accumulation rule, known as the Palmgren-Miner rule (or as Miner’s rule) [1-2], predicts fatigue failures when the sum of the damage induced by each load event  (D i ) equals the critical damage D C the piece can sustain. D C is usually arbitrarily defined as D C  1   (n i /N i ) = 1 , where n i is the number of cycles of the i -th load event, and N i is the number of cycles the piece would last if only that event loaded it. This heuristic hypothesis implicitly assumes that the various load events are independent. Partial loss of available life is a simple way to quantify fatigue damage, but it is not the only one. As terminal fatigue failures occur at a critical crack size a C , accumulated damage can be defined e.g. by the ratio D  a/a C in crack growth problems, where a is the current crack size. Too many other semi-empirical rules are available to model fatigue damage accumulation under VAL. Some try to consider sequence effects not predictable by Palmgren-Miner (if it is used non-sequentially like a statistics), while others try to use less heuristic damage concepts. Fatemi and Yang [3] list 56 such rules, subdividing them into 6 groups: (i) linear damage evolution rules; (ii) bilinear damage evolution rules; (iii) rules that use modified fatigue life curves ( SN or  N ) to consider some sequence effects; (iv) Fracture Mechanics-based rules; (v) Continuum Damage Mechanics-based rules; and (vi) rules based on energy methods. Although damage accumulation modeling still is a fascinating research subject, none of such other (too many) rules got consensus to forever retire the old Palmgren-Miner rule, which has been used by structural engineers since 1924. Indeed, despite countless criticisms, it keeps on being by far the most used damage accumulation tool for fatigue design applications, bravely resisting to its poor performance with ordered loads, not to mention its lack of academic sophistication. Continuum Damage Mechanics, in particular, is a modern technique that has many interesting features, among them be based on much more sound mechanical fundamentals that most of the concurrent heuristic rules. It associates different damage mechanisms to the apparent Young modulus’ variation they cause in standard specimens, or to other similarly distributed parameters [4-5]. Therefore, it is no surprise it has been more useful to describe damage accumulation when it is caused by distributed damage mechanisms. However, its localized damage models, needed to describe most fatigue failures, still are at best controversial. The value D C  1 usually chosen to quantify the critical damage is certainly arbitrary, since it is based on heuristic arguments, not on suitable mechanical foundations. Hence, it is no surprise to find  n i /N i  1 in practical applications. Miner himself quoted 0.7 <  n i /N i < 2.2 as typical values obtained in fatigue tests performed under random loads or load blocks [2]. Juvinall [6] affirms that usually  n i /N i >> 1 in two step loads when all small load events are applied before the large ones, whereas  n i /N i  << 1 when all large load events act before the small ones, and that such ordered load blocks may have a huge dispersion, namely 0.18 <  n i /N i < 23 . However, as such well-ordered loading is rare in practice, for engineering purposes the linear damage accumulation rule produces reasonable and frequently conservative predictions in many fatigue design problems under real service loads, the main cause for its perennial popularity. However, such a popularity does not mean that SN models applied with Miner’s rule can safely solve all practical fatigue crack initiation assessment problems, even if applied to a rainflow-counted representative sample of the load history. In fact, fatigue lives are intrinsically sensitive to the order of VAL events, which may e.g. induce residual stresses and change the critical point stress state, much affecting its subsequent residual life. All statistics ignore sequence effects, but the Palmgren-Miner rule does not need to be used as so. Indeed, fatigue damage can be sequentially accumulated considering at least residual stress variations associated with cyclic yielding at every load event [7], using linear or fancier damage accumulation models. To do so, ordered damage calculations by  N techniques may recognize such plasticity-induced effects, following the critical point stress/strain history in a cycle-by-cycle basis, and they can even be included in simpler SN calculation routines to quantify the effect of residual stress changes caused by high-load events that induce local yielding. This strategy can be numerically efficient when such macroscopic plasticity events are rare. However, the order of the individual load events is also important even in the absence of macroscopic plasticity, probably due to its effect on localized microplasticity around the crack initiation point. This is an important issue when measuring Gassner curves obtained by applying sequences of load blocks composed by a series of CAL steps containing many cycles each. Indeed, even under the high cycle fatigue regime, it cannot be expected that the damage caused by a single ordered load block, with all its steps applied in a low  high  low sequence, is identical to the damage caused by the same steps, A