Multi-Time-Scale Analysis of Fatigue Crack Growth via Cohesive-Zone Models

The extreme difficulty in simulating fatigue-crack propagation in structures has led to design procedures mainly based on very expensive and time-consuming experimental fatigue tests, whose results are correlated using semi-empirical models derived from Paris’ law. Some of the proposals which have been put forward in the last decade to develop more predictive numerical models have been based on enhancing conventional cohesive-zone models (CZMs) by accounting for sub-critical damage propagation and hysteretic loading-unloading response [1-3]. One main advantage of these formulations is that no restrictions are needed for the cyclic loading function so that, for example, variable load amplitudes and ratios or variable frequencies can be analysed. On the other hand, their main disadvantage, when they are implemented within conventional incremental non-linear finiteelement analyses, is that the number of increments required for each analysis is quite higher than the number of cycles. For high-cycle fatigue this can lead to millions of increments required, which is often unfeasible for industrial applications. In this paper, a method to conduct an analysis using a CZM accounting for subcritical damage and hysteretic response using a number of increments significantly lower than the number of cycles is proposed. It builds on some earlier work by Violeau [4] and its underlying idea is that fatigue is a multi-scale process in time, as first observed by Fish [5], although the multi-time-scale procedure proposed here is different. Therefore, a large time scale and a small time scale are introduced. At the large time scale two minimum and maximum envelope functions replace the real oscillatory values, not only for loads or prescribed displacements but also for displacements, strains and stresses, both in the continuum and on the interface. At each (iteration of each) increment n, the maximum and minimum nodal displacements at the beginning and at the end of the increment are un,max , un,min , un+1,max and un+1,min , respectively. Correspondingly, the relative displacements δn,max , δn,min , δn+1,max and δn+1,min , are obtained at each point of the interface through usual interface-element shape functions. In the structure surrounding the damaging interface this results in solving two uncoupled problems for the minimum and maximum values if the behaviour is elastic. Instead, on the interface the two problems are indeed coupled because of the non-linear CZM, and the small time-scale analysis is used to obtain the increment of the interface stress and of any other state variable, like the damage parameter D, and the material tangent stiffness. This is done by assuming a suitable cyclic variation of the relative displacement in the increment from the initial limits ( δn,min , δn,max ) to the final limits (δn+1,min , δn+1,max ), and by determining the constitutive response, cycle-by-cycle, using the proposed CZM. The proposed strategy is described in detail in Ref. [6] and is summarised in the scheme of Figure 2.