13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where s ∂Ω is the boundary of the local subdomain which consists of three parts s s st su L ∂Ω = ∪Γ ∪Γ [16]. Here, sL is the local boundary that is totally inside the global domain, st Γ is the part of the local boundary which coincides with the global traction boundary, i.e., st s t Γ =∂Ω ∩Γ , and similarly su Γ is the part of the local boundary that coincides with the global displacement boundary, i.e., su s u Γ =∂Ω ∩Γ . By choosing a Heaviside step function as the test function * ( ) ik u x in each subdomain the local weak-forms (6) and (7) are converted into the following local integral equations ( , ) ( , ) ( , ) ( , ) s su s st s i i i i L t d u d t d X d τ ρ τ τ τ +Γ Ω Γ Ω Γ− Ω=− Γ− Ω ∫ ∫ ∫ ∫ x x x x % & , (10) ( , ) ( , ) ( , ) ( , ) s su s st s i i i i L h d w d h d g d τ ρ τ τ τ +Γ Ω Γ Ω Γ− Ω=− Γ− Ω ∫ ∫ ∫ ∫ x x x x % & . (11) The expressions for the traction and the generalized traction vectors result from the constitutive equations and are given as , , ( , ) ( ) ( , ) ( ) ( , ) ( ) i ijkl k l ijkl k l j t c u R w n τ τ τ ⎡ ⎤ = + ⎣ ⎦ x x x x x x , (12) , , ( , ) ( ) ( , ) ( ) ( , ) ( ) i ijkl k l ijkl k l j h K w R u n τ τ τ ⎡ ⎤ = + ⎣ ⎦ x x x x x x , (13) where ( ) j n x is the unit outward normal vector to the boundary s ∂Ω . The trial functions are chosen to be the MLS approximations by using a number of nodes spreading over the domain of influence. The approximated functions for the phonon and phason displacements can be written as [16] 1 垐 ( , ) ( ) ( ) ( ) n h T a a a τ φ τ = = ⋅ =∑ u x Φ x u x u , (14) 1 ˆ ( , ) ( ) ( ) n h a a a τ φ τ = =∑ w x x w , (15) where the nodal values ( ) 1 2 垐 ( ) ( ), ( ) T a a a u u τ τ τ = u and ( ) 1 2 垐 ( ) ( ), ( ) T a a a w w τ τ τ = w are fictitious parameters for the phonon and phason displacements, respectively, and ( ) aφ x is the shape function associated with the node a. The number of nodes n used for the approximation is determined by the weight function ( ) am x . A 4th order spline-type weight function [16] is applied in the present work.
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