13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- decomposition in Eq.(6) can be further written in direct notation, i.e., ( ) u x, Nu Nd Na h t = = + where N is the FEM shape function matrix and N is the enrichment shape function matrix. N N,N = and [ ]T u d,a = with d the nodal displacement vector and a the vector for the enriched degree of freedom. Note that the way that the enrichment function ( ) ,x t φ being built in Eq.(6) is an equivalent statement of the “partition-of-unity” concept. It ensures the consistency of the approximation. An example of regular and enrichment space-time shape functions is shown in Figure 2. In this case, we have used a fine scale harmonic function for ( ) ,x t φ with a temporal scale that is smaller than the temporal element size of the regular space-time FEM. We have employed harmonic enrichment function considering the fine scale harmonic components of the loading and the solution (displacement) in the case of HCF. i.e., ( ) ( ) , sin x k k t t φ ω =∑ in which kω represents the k-th characteristic frequency. Complete spatial domain will be enriched with the time enrichment function as the whole structure is subjected to fatigue loads. With the addition of the enrichment, the final discretized equation is in the form of { } { } rr re r e er ee K K d F F K K a = (7) with subscripts “r” and “e” representing the contribution from the regular and enrichment components, respectively. 2.2 Modeling of Material Failure under Cyclic Loading Modeling damage under fatigue loading is challenging due to the strong coupling to the loading path. Many techniques have been developed, including fracture mechanics, porous plasticity and continuum damage mechanics (CDM). In HCF, we introduce the two-scale CDM approach proposed by Lemaitre et al. [9] and Desmorat et al. [10]. This approach assumes that material behavior at mesoscale is elastic and damage can be treated as quasi-brittle in HCF. This quasi-brittle damage is modeled by introducing damage variables that represent the microcracks and microvoid Figure 2: Examples of regular and enrichment shape functions.
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