13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 3. Definition of the stress intensity factor (SIF) The singular stress field around the inclusion corner apex under thermo-mechanical loads can usually be expressed in a form of singular terms as: t c 1 ( ) n N M n n n K r f λ θθ θθ θθ σ θ σ + = = + ∑ (6) t c 1 ( ) n N M n r n r r n K r f λ θ θ θ σ θ σ + = = + ∑ (7) where ( , ) r θ is a local polar coordinate system centered at the inclusion corner apex, and the axis of 0 θ= is the bisector of the two wedge apexes shown in Fig.1; N represents the total number of complex singularity orders nλ between -1 and 0, and M is the number of real singularity orders nλ; ( ) n f θθ θ and ( ) n rf θ θ are the notch angular variation of normal stress fields and shear angular variations associated with nλ, respectively; t c θθ σ and t c r θ σ are regular stresses caused by a thermal load. According to Chen’s work [17], the singularity orders nλ have only two roots, i.e., 1λ and 2λ; When Dundur’s composite parameters α and β [18] meet the condition ( ) 0 β α β − > , 1λ and 2λ are always real within the range of 1 2 1 Re( , ) 0 λ λ − < < . On the moment, expressions (6) and (7) can be rewritten as: 1 2 1 2 t 1 2 c ( ) ( ) K r f K r f λ λ θθ θθ θθ θθ σ θ θ σ = + + (8) 1 2 1 2 t 1 2 c ( ) ( ) r r r r K r f K r f λ λ θ θ θ θ σ θ θ σ = + + (9) Defining 1 ( ) f θθ θ and 2 ( ) rf θ θ in such a way that 1 0 ( ) 1 f θθ θ = and 2 0 ( ) 1 rf θ θ = , where 0θ can be chosen arbitrarily, then we have 1 0 t 2 t 1 c 0 c 1 2 0 0 0 1 lim [ ( ) ( )] [1 ( ) ( )] r r r r K r f f f λ θθ θθ θθ θ θ θ θ θ θθ σ σ θ σ σ θ θ − = → = − − ⋅ − − ⋅ (10) 2 0 t 1 t 2 c 0 c 1 2 0 0 0 1 lim [ ( ) ( )] [1 ( ) ( )] r r r r r K r f f f λ θ θ θ θθ θθ θ θ θ θθ σ σ θ σ σ θ θ − = → = − − ⋅ − − ⋅ (11) Once the values of 1K and 2K are obtained, the singular stress fields at every θ can be solved from Eqs.(8) and (9).
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