13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- normal to the surface of discontinuity. Figure 1: Illustration of TDG discretization of space-time finite elements. By applying the variational principle to the governing equations of momentum within each time slab, we develop the so-called weak form that serves as the basis for computational implementation. For the the n-th time slab, we have the following bilinear form: ( ) ( ) ,h h h DG n DG n B L= w u w (3) with ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , h h h h h h h h h h DG n n n n n n n Q Q B dQ dQ t t d t t d ρ ρ + + + + − − − − Ω Ω = ⋅ + ∇ ⋅ ∇ + ⋅ Ω+ ∇ ⋅ ∇ Ω ∫ ∫ ∫ ∫ w u w u w σ u w u w σ u ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) 1 1 1 1 b t n h h h h h h h h DG n t n n n n i n n n Q L dQ d t t d t t d d γ γ ρ γ ρ γ + − + − − − − − Ω Ω = ⋅ + ⋅ + ⋅ Ω+ ∇ ⋅ ∇ Ω+ ⋅ ⋅ ∫ ∫ ∫ ∫ ∫ w w w w u w σ u w σ (4-5) in which hu and hw are respectively the trial and test function for displacement. In Eqs.(4) and (5), the combination of the first two terms represents the momentum balance and traction boundary conditions. The 3rd and 4th terms in Eqs.(4) and (5) combined gives the continuity conditions between the time slabs. The last term in Eq.(5) is due to internal discontinuity. In addressing the multiple temporal scales, we propose that the space-time approximation from Eq.(1) can be further improved by introducing enrichment [6-8]. In the enrichment formulation, the displacement field is approximated as ( ) ( ) ( ) ( ) ( ) ( ) , , , , , u x x d x x x a h I I J J J J I N J N t N t N t t t φ φ ∈ ∈ = + − ∑ ∑ (6) in which ( ) ,x IN t is the regular space/time finite element shape functions, ( ) ,x t φ is the called an enrichment function and is selected based on the physics of the problem. Correspondingly, dI is the nodal displacement, and aJ is the enriched degree of freedom. The multiscale
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