Issue 37

D. Angelova et alii, Frattura ed Integrità Strutturale, 37 (2016) 258-264; DOI: 10.3221/IGF-ESIS.37.34 259 newly-proposed mathematical models of short fatigue crack growth, and making comparative analysis with Steel A. The microstructure, chemical compositions and slightly different hour-glass shape of specimens of Steel A and Steel B are presented in [6, 7]. Steel A. Twelve hourglass-shaped specimens were subjected to symmetric cyclic rotating-bending fatigue (RBF) at different stress ranges (Δσ = 800, 1000, 1200, 1400, 1500 MPa), frequency of f=11 Hz and air environment. Steel B. Fourteen hourglass-shaped specimens were exposed to fully reversed torsion fatigue (TF), and frequency f=5 Hz for two sets of stress ranges Δτ = (915, 1080, 1106 MPa in air), (404, 601, 815, 900 MPa in corrosion). For both steels a traditional replication technique was used; all details are given in [6]. Fatigue crack growth modeling Since the beginning of the last two decades it has been well established that the fatigue behaviour of metals can be described by three distinct regimes - microstructurally short cracks (MSC), physically small cracks (PSC) and long cracks (LC). Each type of crack requires different analytical approach characterizing its behaviour, i.e. microstructural fracture mechanics (MFM), elastic-plastic fracture mechanics (EPFM) and linear elastic fracture mechanics (LEFM), respectively. In terms of driving stresses there exist two basically different forms of crack, a Stage I crack developed by shear stresses, and a Stage II crack developed by tensile stresses. The limits of the transition zone between a propagating Stage I shear crack and a continuously growing Stage II tensile crack (the PSC zone) are emphasized by Miller [8] as two fundamentally different threshold conditions (or barriers d 1 and d 2 ) representing a "microstructural-dependant state" and "mechaniical stress-strain-dependant state". The barrier d 1 is defined by short fatigue crack experiment and d 2 - by long fatigue crack experiment using the same material. Based upon the experimental data for short fatigue crack behaviour there are a few basic models describing it, the most known of which are those of Hobson and Brown [9] - H-B and Akid and Murtaza[10] - A-M, which present crack growth rate as a combined parabolic (Stage I of MSC) and linear (Stage II of PSC and LC) functions of crack length. The H-B model (for symmetrical push-pull fatigue) consist of one parabola and one line, considering one barrier of type d 1 and one, d 2 . The A-M model (for TF) includes four barriers of type d 1 and one of type d 2 for in-air fatigue; and two barriers d 1 and one d 2 for corrosion fatigue. The H-B and A-M models are revised by Angelova [11] - becoming H-B-A and A-M-A - who introduces a special modelling of PSC zone between a Stage I shear crack and a Stage II tensile crack. In this way two important tasks were solved: a more precise description of PSC behaviour; and finding the barrier of type d 2 from the same short fatigue crack experiment. The complicated defining of the barriers d 1 and d 2 , and the parabolic functions in H-B-A and A-M-A models was replaced with simplified methodic by Yordanova [12] which is more convenient for practical use and a quick (express) evaluation of fatigue at keeping the basic concept for separated description of MSC (parabolic function), PSC (parabolic function) and LC (linear function) and conducting only short fatigue crack experiment. The new model is presented in Eq.1: M(a):                           III D l II ps I ms Nd a aD dN da LC N d d aDaD aD dN da PSC Nd a aDaD aD dN da MSC ; ; / : ; ; , ; / : ; ; , ; / : 2 7 2 1 6 5 2 4 1 0 3 2 2 1 8 (1) where:       n n n n n N N a a dN da       1 1 1 / / is the crack growth rate da/dN ; D i , i=1-8 – materials constants; III II I NNN , , – the corresponding number of cycles to crack regimes, MSC, PSC, LS; d 1 and d 2 – the microstructural barriers analytically determined after Yordanova's methodic, described in detail in[12]. This model is supported by the comparison of the fatigue lifetime predicted by Eq. (2), mf N , , and the actual fatigue lifetime, exp , f N : III II I f mf N N N N N     0 , or in more detail, [12]: 0 , f mf N N  da dN da ms d a ) / (/1 1 0      da dN da ps d d ) / /(1 2 1 da dN da l a d f ) / /(1 2  (2) where 0 f N is the number of cycles to crack initiation, a 0 – the initial crack length and a f – the final crack length. The following formulae   %, / 100 exp , exp , , f f mf N N N  is used as an evaluation of the adequacy of the proposed new model and if its value is in the error band  25 %, the mathematical description is accepted as adequate.