13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- fully accounted for in the empirical design approaches that are in practice today. More specifically, an enriched space-time finite element method (XTFEM) based on the time discontinuous Galerkin formulation is developed to handle the multiple temporal scales in fatigue problems. XTFEM is coupled with the two-scale continuum damage mechanics model for evaluating fatigue damage accumulation, with a damage model governing the fatigue crack-initiation and propagation as the simulation progresses. High Cycle Fatigue (HCF) simulations are performed using the developed methodology on a notched specimen of AISI 304L steel to predict total fatigue life under cyclic conditions. 2. Technical Approach 2.1 A Computational Framework of Enriched Space-time FEM As the name suggests, space-time method introduces discretization both in the spatial and time domains. The main advantage of the space-time method over the semi-discrete scheme based methods is reflected in the ability to introduce approximations in the temporal domain. Space-time methods are also known for their capability of reducing artificially oscillations and handling loads with sharp gradients. These features of space-time method make it an attractive platform for fast and accurate simulation of the fatigue loading. In the semi-discrete scheme, approximations are established in space x for each time instance t. In the space-time formulation, the approximations are built simultaneously with both space x and time t. If finite element method is used, we have the following approximation in the space/time description for a general three-dimensional case ( ) ( ) , , I t N t =∑ I I u x x d (1) in which ( ) ,x IN t is the finite element shape functions at nodes indexed by I and dI is the corresponding nodal displacement vector. Note that although I N and dI look similar to the ones in the semi-discrete scheme, they are defined on a space/time grid as opposed to space only. To describe our approach what is also known as the time discontinuous Galerkin method (TDG) [5], the 2D space/time grid plotted in Figure 1 will be used as a reference. We consider a space/time domain Q, which is the Cartesian product of space domain Ω and time duration ( )Τ,0 , i.e., ( ) Ω Τ,0 = × Q . As can be seen from Figure 1, the time domain is subdivided into time slabs with the n-th time slab given as ( ) n n n t t Q ,1− =Ω× . In the current method, we will restrict the presence of discontinuities to the mechanical field only. We define a jump operator [ ], given as ( ) [ ] ( ) ( ) − + = − w t w t w t and ( ) ( ) ( ) x x x w w w + − = − (2) with ( ) ( )ε = ± ± w t w t , ( ) ( ) x x n w w ε ± = ± , ε represents infinitesimal perturbation, n is the
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