13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- By collocating the stress boundary integral equation on the source point +ΓP on the crack surface +Γ shown in Fig. 1, the conventional stress BIE can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). , , 2 1 2 1 T P Q t QdS Q T P Qu QdS Q P P S k Y ijk S k Y ijk ij ij ∫ ∫ + − + + − + − + +Γ +Γ Γ +Γ +Γ Γ Γ Γ = + + σ σ (6) where Y ijk T and Y ijk U are the new kernel functions obtained by using the numerical difference of the derivatives of Yue’s tractions and displacements. Multiplying equation (6) by the outward unit normal ( ) +Γ n P i and noticing that ( ) ( ) − + Γ Γ =− n P n P i i , the traction boundary integral equation on the crack surface results in ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) U P Q t QdS Q T P Qu QdSQ nP t P t P n P S k Yijk i S k Y ijk i j j ∫ ∫ + − + + + − + + − + +Γ +Γ Γ Γ +Γ +Γ Γ Γ Γ Γ = + − , , 2 1 2 1 (7) 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 Γ Γ = =− TPQ TPQ UPQ UPQ S Y ijk S Yijk S Y ijk S Y ijk (8) Applying expressions (8), the two integrals in equation (7) can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T P QuQdSQ T P QuQdSQ T P QuQdSQ k Y ijk S k Y ijk S k Y ijk ∫ ∫ ∫ + + + + − + Γ Γ Γ +Γ +Γ Γ Δ + = , , , (9a) ( ) ( ) ( ) ( ) ( ) ( ) U P Qt QdSQ U P Qt QdSQ S k Yijk S k Y ijk ∫ ∫ + + − + Γ +Γ +Γ Γ = , , (9b) Using expressions (8) and (9), equation (7) can be further written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , nP U P Qt QdSQ tP nP T P QuQdSQ nP T P QuQdSQ S k Y ijk i k Y ijk i S k Y ijk i j ∫ ∫ ∫ + + + + + + + + Γ Γ Γ Γ Γ Γ Γ Γ = Δ + + (10) The integral equation (10) is a general form of the traction boundary integral equation based on Yue’s solution. Equations (5) and (10) give explicit expressions of the dual boundary integral equations based on Yue’s solution. These two integral equations do not contain the integrations on the interfaces of multilayered media because Yue’s solution strictly satisfies the interface conditions. Collocating equation (5) on the uncracked boundary S and equation (10) on +Γ constitutes the dual boundary integral equations for crack problems. In these equations, the unknown quantities are the tractions and displacements on the uncracked boundary and the discontinuous displacements on the crack surfaces. When the dual BIEs are applied for the study of the crack problems in a multilayered medium of infinite extent, the displacement boundary integral equation is not required and then dual BEM degenerates into the discontinuous displacement method (DDM) [10,11].
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