13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- and conventional elements. The stiffness matrix for the super wedge tip element of bi-material wedge under mechanical and thermal loading is obtained by considering the total potential expressed in the following form[15] { } 2 ( ) ( ) ( ) ( ) ˆ ˆ 1 2 ( ) ( ) * ( ) ( ) 0 ˆ ˆ 1 1 2 ˆ ˆ k k k k m k T T m k m k T T t k c k m k T T k k T k k d d d d λ λ λ λ Γ Γ λ Γ Γ Π Γ Γ Γ Γ Π + = = ⎧ ⎫ = − − ⎨ ⎬ ⎩ ⎭ + − + ∑ ∫ ∫ ∑ ∫ ∫ σ n u σ n u σ n u t u (1) The terms ( ) m k p λσ , ( ) m k p λu and ( ) t k p λσ , ( ) t k p λu associated with the homogeneous solutions of mechanical and thermal loadings, respectively, and in the case of thermal loading, ( ) t k c pσ and ( ) t k c pu represent the known complementary solutions for non-singular stress and displacement fields. These vectors can be defined as ( ) ( ) m k k m p λσ Σ K = , ( ) ( ) m k k m p λ = u U K (2, 3) ( ) ( ) t k k t p λσ Σ K = , ( ) ( ) t k k t p λ = u U K (4, 5) ( ) ( ) ( ) ( , ) t k k k c p c c r θ = σ F q , ( ) ( ) ( ) ( , ) t k k k c p c c k r r r T θ α = + Δ u G q e (6, 7) The details of ( )k Σ and ( )k U refer to Chen and Sze [14], and the definition of ( )k cF , ( )k cG , ( )k cq , T re and t K are listed in Barut et al. [15]. TΔ denotes the uniform temperature change. The unknown components of displacement vector along the common boundary segments, ˆ kΓ (shown in Fig. 1), are denoted by ( ) ˆ k u . * ( )k t is the known applied traction components along the common boundary segments. n contains the components of the unite normal along ˆ kΓ . 0Π represents the total potential associated with the known initial strain and stress components arising from thermal loading only. The vector of displacement components, ( ) ˆ k u , along the common boundary between the global element and the conventional elements can be expressed in terms of the nodal displacement of the conventional elements as ( ) ˆ k = u Lv (8) in which the matrix L contains the linear interpolation function compatible with those of the conventional elements. The vector v contains the nodal degrees of freedom associated with the conventional elements located on the boundary segment ˆ kΓ of the super notch tip element. Substituting for ( ) m k p λσ , ( ) m k p λu and ( ) ˆ k u from Eqs. (2)-(8) into the expression for the total potential leads to * 0 1 ˆ 2 m T m m T t m T T Π Π =− − + − + K H K K f K Gv v f (9) in which 2 ( ) ( ) T ( ) ( ) ( )T ( ) ( ) ( ) ˆ 1 1 d 2 k k T k T k k k k T k k u u k S σ σ Γ = ⎡ ⎤ = + ⎣ ⎦ ∑ ∫ H Σ Z n Z U U Z nZ Σ , 2 ( ) ( ) 1 d k T k T k S σ Γ = =∑∫ G Σ Z nL w 2 ( ) ( ) ( ) T ( ) ( ) ˆ 1 k t k k T k T k t k u c p k d σ λ Γ Γ + = =∑∫ f Σ Z n Z u , 2 * ( ) ( ) * ( ) ˆ 1 k k k k k d Γ Γ = =∑∫ f L t , ( ) ( ) ( ) t k t k t k c p p c p λ λ + = + u u u ( )k σZ and ( )k uZ are the coordinate systerm transformation matrices. In order to express the total potential in terms of one unknown vector, v , the first variation of the total potential with respect to
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