13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Δuis a compatible displacement increment on e ∂Ω and Δf is a traction increment on the traction boundary t e ∂Ω . An element complementary energy function can be expressed as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 1 , 2 ( ) e e mc e e b m m c c b b te b m m m m m c c c c c m c m m m m m m m m m m d d d d d d d Ω ∂Ω ∂Ω ∂Ω ∂Ω ∂Ω ∂Ω Π Δ Δ = +Δ +Δ Ω− ⋅ +Δ ⋅ +Δ ∂Ω ∫ ∫ + +Δ ⋅ +Δ ∂Ω+ ⋅ +Δ − −Δ ⋅ +Δ ∂Ω ∫ ∫ − ⋅ +Δ ⋅ +Δ ∂Ω− ⋅ +Δ ⋅ +Δ ∂Ω ∫ ∫ − ⋅ +Δ ⋅ +Δ ∂Ω ∫ − ⋅ +Δ ⋅ +Δ u S n u u f f u u n u u n u u n u u n u u n u σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ ( ) 3 3 m m d ∂Ω ∂Ω ∫ u (1) where, b Δu , c Δu , 1m Δu , 2m Δu and 3m Δu are the displacement increments of the crack edges in an element. The total energy for an entire heterogeneous structure that contains N inclusions is obtained by adding the element energy functions for N elements: 1 N mc mc total element e= Π = Π∑ (2) By setting the first variation of mc element Π in Eq.(1) with respect to the stress incrementsΔσto zero, the element displacement relations in each of the element eΩ can be obtained. Setting the first variation of mc element Π with respect to boundary displacement increments Δu to zero, obtains the traction reciprocity conditions on the inter-element boundaries and traction boundaries as shown in Fig. 1, and setting the first variation of mc element Π with respect to boundary displacement increments b Δu , c Δu , 1m Δu , 2m Δu and 3m Δu to zero, obtains the traction reciprocity conditions on the interfaces of inclusion-matrix as shown as follows: ( ) ( ) b c c b m m ⋅ +Δ = ⋅ +Δ n n σ σ σ σ on bonded interface b ∂Ω ( ) c c c ⋅ +Δ = n 0 σ σ on debonded interface c ∂Ω ( ) 1 1 1 m m m ⋅ +Δ = n 0 σ σ on debonded interface 1m ∂Ω (3) ( ) 2 2 2 m m m ⋅ +Δ = n 0 σ σ on the first crack edgee 2m ∂Ω ( ) 3 3 3 m m m ⋅ +Δ = n 0 σ σ on the second crack edge 3m ∂Ω 2.2. Method of Solution The stresses in the matrix and inclusion phases can be individually described to accommodate stress jumps across the interface. The expressions may be assumed for stress functions ( ) ,x y Φ in the matrix and inclusion phases in the form as m m m Δ = Δ P σ β (in mΩ ) (4) c c c Δ = Δ P σ β (in cΩ ) (5) where m Δβ and c Δβ correspond to a set of undetermined stress coefficients; mP and cP are matrixes of interpolation functions. Compatible displacement increments on the element boundary e ∂Ω as well as on the crack edges b ∂Ω , c ∂Ω and m ∂Ω are generated by interpolating in terms of generalized nodal values as: { }e Δ = Δ u L q , on e ∂Ω ;
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