Issue34

Y. Hos et alii, Frattura ed Integrità Strutturale, 34 (2015) 133-141; DOI: 10.3221/IGF-ESIS.34.14 138 F ATIGUE CRACK GROWTH SIMULATION n the experiments presented in the previous chapter the applied load amplitudes have been so large and the crack growth rates have been so fast that a fatigue crack growth simulation is expected to fail when based on the stress intensity factor range of the linear elastic fracture mechanics. From the variety of crack driving force parameters of the elastic-plastic fracture mechanics [8] the cyclic  J -integral was selected for describing fatigue crack growth, i i ( ) d d u J W y T s x                 (3) with ij ij ij 0 d( ) W          . (4) The symbol  preceding the stress tensor, strain tensor, traction vector and displacement vector components designates the changes of these quantities. These changes must be evaluated referring to a reference state. It serves as the new origin for defining the increments of the field variables, the latter preceded by the symbol  . The stress and displacement state at the moment of load reversal is a natural reference state. Then the symbol  designates the increments from the respective reference values. At the moment of the next reversal, these increments become the conventional cyclic ranges. However, the symbol  in  J and  W does not represent changes of J and W ; instead  J and  W are functions of their arguments as defined by Eqs. (3) and (4). For non-proportional loading the theoretical background is widely lost. Nevertheless, Hertel et al. [9] proposed an algorithm which delivers practically path-independent values when solving Eq. (3). Due to this path independence the thereby obtained value should be valid crack driving force. In case of non-proportional loading the load reversal points, in terms of  J , are detected by the following algorithm. If the change in  J after the integration of Eq. (3) turns out to be negative, it is presumed that the last equilibrium state was a load reversal point. In consequence, the reference variables are redefined with stress, strain and displacement values of the last equilibrium state. Thereupon, the integration is continued. This procedure provides the monotonic increase of  J for every load step. Hertel’s algorithm was extended here to take crack closure into account. Only the portions of a cycle with open crack surfaces have been considered. Therefore, the values obtained here can be labelled effective cyclic J -integral,  J eff . The loading step associated with the onset of crack closure was determined by applying a finite element based crack growth simulation. In the present paper only results for the tension-compression can be reported. The simulation started with three cycles applied to the uncracked structure. At maximum load of the next cycle, a crack growth step for 1 mm crack advance was executed. For this purpose the boundary condition of the relevant nodes on the crack growth plane was changed from fixed to unrestrained. Additionally, a contact plane was inserted at the new crack surface to prevent negative displacements of the crack surface nodes during subsequent cyclic loading. After the new equilibrium was found three cycles without crack growth were calculated. The node release procedure – 1 mm crack growth followed by three cycles – was repeated until reaching 5 mm crack length. Further repetitions of this scheme followed until reaching a crack length of 6 mm. However, the crack growth per repetition was continuously reduced. Growth steps of 0.5 mm, 0.25 mm and two times 0.125 mm were used. It is admitted that sound recommendations, see e.g. Herz et al. [10,11] and Zerres and Vormwald [12], for achieving a converged result are violated. However, with the plasticity model at hand converged results cannot be achieved either due to its ratcheting properties. A first estimate is expected to provide inside anyway. In Fig. 9 an example is shown for the dependence of  J eff from the stress range  . The calculations were stopped on the occurrence of the first crack surface contact. Recently, Hos et al. [6,7] showed that due to severe ratcheting the crack opening displacements increase from cycle to cycle without any trend of stabilization. This behaviour is typical for the Chaboche model. As a consequence of the large ratcheting the crack opening and closure loads are too low when compared to measured values using the digital image correlation technique. In contrast to the simulations presented in [7] the node release scheme was modified to reduce ratcheting. In especially, no intermediate cycles without a crack growth I