13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- non-homogeneous media may be not involved. In the following, we will present a numerical implementation of the dual boundary integral equations based on Yue’s solution [9]. 3.2. Yue’s solution based Dual BIEs Fig. 1 shows a three-dimensional crack in a multilayered solid. By collocating the source point on the uncracked boundary, the conventional displacement BIE can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), , , u P Q t QdS Q t P Qu QdS Q c P u P S j S Yij S j S Y ij j S ij S ∫ ∫ + − + − +Γ +Γ +Γ +Γ = + i j x y z , , , = (1) where Ps and Q are the source and field points, respectively; ( ) t P Q S Y ij , and ( ) u P Q S Y ij , are the tractions and displacements of Yue’s solution, respectively; ( ) t Q j and ( ) u Q j are the tractions and displacements of the field point Q on the boundaries; S is the uncracked boundary of the cracked body; +Γ and −Γ are two crack surfaces; ( ) ij S c P is a coefficient dependent on the local boundary geometry at the source point Ps. Before loading, the points −ΓQ and +ΓQ on two crack surfaces are completely coincident and there are opposite outward normal directions on the two points. Thus, there exist the following relationships of kernel functions of the points on two crack surfaces ( ) ( ), , , − + Γ Γ =− t P Q t P Q S Y ij S Y ij ( ) ( ) − + Γ Γ = u P Q u P Q S Y ij S Y ij , , (2) Assume that there is a balanced relationship of tractions: ( ) ( ) − + Γ Γ =− t Q t Q j j . The relative crack opening displacement (COD) can be described as ( ) ( ) ( ) j x y z u u u j j j , , Q Q Q , = − = Δ − + + Γ Γ Γ (3) Using the above relationships, the two integrals in equation (1) can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t PQuQdSQ t PQuQdSQ t PQ uQdSQ j S Y ij S j S Y ij S j S Y ij ∫ ∫ ∫ + + − Γ +Γ +Γ Δ + = , , , (4a) ( ) ( ) ( ) ( ) ( ) ( ) u P Qt QdSQ u P Qt QdSQ S j S Y ij S j S Y ij ∫ ∫ = + − +Γ +Γ , , (4b) Thus, equation (1) can be rewritten as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cPuP t PQuQdSQ t PQ uQdSQ u PQtQdSQ S j S Y ij j S Yij S j S Y ij j S ij S ∫ ∫ ∫ = Δ + + + Γ , , , (5) The integral equation (5) is a general form of the displacement boundary integral equation based on Yue’s solution. Fig. 1 A three-dimensional crack in a multilayered solid x+ xΓ + x Γ - i u it n S y z
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