13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The displacement fundamental solution in equation (4) is given in Ref. [11] by Wang. He derived two-dimensional elastostatic fundamental solutions for general anisotropic solids by the use of Stroh's formalism [12]. The displacement fundamental solution in equation (4) can be given as an explicit expression: ( ) ( ) ( ) ( ) * 1 1 , Im log = ⎡ ⎤ = ⋅ − ⎣ ⎦ ∂∑ M ij m ij m m A u Dη η π η m x y d y x (6) with ( ) ( ) 1, ⎡ ⎤ = ⎣ Γ ⎦ ij m ij m A adj η η , (7) ( ) ( ) det 1, ⎡ ⎤ = ⎣ Γ ⎦ m ij m D η η . (8) ( ) 1, = m mη d . (9) In equations (6)-(9) Γij(1,ηm) is defined by ( ) ( ) 2 2 2 2 1 2 1 1 1 1, Γ = ⋅ + + ⋅ + ij m ij m ij ji m ij C C C C η η η , (10) where Cαijβ is the elasticity tensor. D(ηm) in (6) and (8) is a polynomial function of order four and has M=2 complex roots ηm and two complex conjugates of ηm, which satisfy the following characteristic equation: ( ) 0= m D η . (11) The traction fundamental solution in equation (4) is defined by the following closed expression [10]: ( ) ( ) ( ) ( ) ( ) * 1 1 , Im = ⋅ = ∂ ⋅ − ∑ M ij m ij m m B p Dη η π η m m d n y x y d y x (12) with ( ) ( ) ( ) 2 2 2 1 = ⋅ + ⋅ ij m ip m ip pj m B C C A η η η , (13) and the outward unit normal vector n. The stress and higher-order stress fundamental solutions in equation (5) are provided by: ( ) ( ) ( ) ( ) * 1 1 , Im = =− ∂ ⋅ − ∑ M ij m ij m m B d D γ γ η η π η m m d x y d y x , (14) ( ) ( ) ( ) ( ) ( ) * 2 1 1 , Im = ⋅ ⋅ =− ∂ ⎡ ⎤ ⋅ − ⎣ ⎦ ∑ M ij m ij m m C s D γ γ η η π η m m m d d n y x y d y x , (15) with ( ) ( ) ( ) 2 1 = ⋅ + ⋅ ij m ip m ip pj m B C C A γ γ γ η η η , (16) ( ) ( ) ( ) ( ) 2 1 2 2 2 1 = ⋅ + ⋅ ⋅ ⋅ + ij m ip m ip pt m jt m jt C C C A C C γ γ γ η η η η . (17) The integration of the fundamental solutions (6), (12), (14) and (15) are performed fully analytically. It should be noted that the solutions have a weak logarithmic singularity log[y-x] (6), a strong singularity 1/[y-x] (12,14), and a hypersingularity 1/[y-x]2 in the higher-order stress fundamental solution (15). For spatial discretization of equation (4) and (5) a collocation method is utilized. In Ref. [9] and [13] a substructure technique is presented, which enables coupling of individual homogeneous substructures by use of continuity condition. The implementation of this technique in the present method allows to represent each grain by a homogeneous, elastic anisotropic substructure which is combined with other grains to a continuous microstructure.
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