13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- K f aπ σ = (1) x ar x a m a x f a d ( / ) ( , ) 0 π σ σ ⋅ = ∫ (2) where m(a, x) is the weight function for the given crack geometry, which can be derived from the crack opening displacements u(a, x) for a reference load case. a u a x K a E m a x ∂ ∂ = ( , ) ( ) ( , ) ' (3) where ' E =E for plane stress and ) /(1 2 ' η = − E E for plane strain. For the corresponding crack surface displacements [10]: = ( , ) u a x ∫ ⋅ a a f s s m s x s E 0 [ ( ) ] ( , )d ' π σ (4) From the above equations, it is seen that the central issue in the weight function method is the determination of weight function m(a, x) for the crack geometry in consideration. Various approaches have been used for determining m(a, x). One effective way is to derive m(a, x) through a reference load case. With this procedure, systematic derivations for the weight functions m(a, x) for a large number of two-dimensional crack configurations have been given by Wu and Carlsson [10], which can be readily utilized for the present analysis. For the edge crack configuration, which is applicable to the present case, i.e. a radial crack emanating from a semi-circular notch in a semi-infinite plate, the stress intensity factor and the corresponding crack face loading for the reference load case are expressed in polynomials of the type: i I i i r r a r a f ( ) ( ) 0 ∑ = = α (5) m M m m r r x S r x ( ) ( ) 0 ∑ = = σ σ (6) where r is the notch radius, and the coefficients αi and Sm for the present crack configuration are given in Section 3. The weight function was expressed in closed-form by Wu and Carlsson [10], as 2 3 4 1 ( ) (1 ) 2 ( , ) − = ∑ ⋅ − = i i i a x r a a r m a x β π (7)
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