PSI - Issue 75

Andrew Halfpenny et al. / Procedia Structural Integrity 75 (2025) 219–233

232 14

Author name / Structural Integrity Procedia (2025) The crack growth rate at the ℎ integration point ( ) is determined using the standard ‘Cycle -by- Cycle’ algorithm except that the crack length is not incremented between cycles. This reduces the computational effort in determining the SIF and enables block loads to be dealt with using linear accumulation. The number of cycles to propagate the crack between limits and are then calculated using a standard numerical interpolation routine integrating the reciprocal value for ( ) . The ‘Rapid Integration’ looping algorithm is summarised in Fig. 4. This algorithm is compatible with all the retardation models presented in this paper. According to Brussat, where retardation is required, it is recommended to initialise the retardation model before each integration point calculation to establish the overload plastic zone size for the highest tensile load in the spectrum. Unlike the ‘Cycle -by- Cycle’ algorithm, the ‘Rapid Integration’ algorithm is not able to implicitly determine yield or the critical crack size, , and requires an explicit calculation. In particular, a check is necessary before the calculation process is started to ensure that the user does not try to set an integration point beyond . Three types of discretisation and numerical integration methods can be used within this algorithm (see Press et al. (2007)): • Equal spacing : this is where the integration points are equally spaced at constant crack length intervals. The numerical integration can then use a simple Trapezoidal rule or Simpson’s rule. These methods are ideal for estimating the intermediate points and are therefore ideal for plotting crack growth curves and calculating the estimated initial flaw size. This method is based on a prescribed discretisation and yields no indication of convergence. • Non-equal spacing : where, as the name suggests, the integration points are non-equally spaced. This effectively excludes Simpson’s rule as it requires three equally spaced integration points. This method is also based on a prescribed discretisation and yields no indication of convergence. • Adaptive integration : this is an iterative approach that starts using a fairly coarse discretisation and refines this comparing convergence along the way. The advantage of using this approach is that a tolerance for convergence can be entered, and the crack is discretised appropriately. This can yield very efficient calculations as there is generally no need for over conservative discretisation. The downside of this approach is that it does not provide intermediate points and so proves unsuitable for plotting crack growth curves. 5. Conclusions In the second half of the 20 th century, a significant number of crack growth laws and crack retardation models were proposed. Unfortunately, this invaluable inheritance is scattered across numerous scientific publications making the review and comparison difficult and time consuming. This paper collects the most relevant fatigue crack growth laws, crack retardation models and looping algorithms with the aim of helping engineers, dealing with fatigue crack growth applications, to efficiently review these laws and models and make informed decisions. In this paper, several well-known fatigue crack growth laws were described, starting from the simpler but groundbreaking Paris law, suitable to describe only the Region II of a crack growth curve, to subsequent laws proposed to consider mean stress effects, crack closure, threshold behaviour (Region I) or the onset of fast fracture (Region III). Crack retardation was also described, and a wide range of retardation models were reviewed, from the Wheeler and Willenborg models, that neglect delayed retardation, overshoot and crack acceleration, to the Austen-Willenborg model that includes all three phenomena. Finally, two looping algorithms allowing the implementation of the crack growth analysis in commercial durability software were also described. References

Bannantine, J. A., Comer, J. J., Handrock, J. L., 1990. Fundamentals of Metal Fatigue Analysis. Prentice Hall.