13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Fig. 1. Schematic illustration of the contour integrals and domain integrals. The J-integral can be deduced to an equivalent domain integral (EDI) by using the divergence theorem, that is ,1 1 , ,1 1 , ( ) ( ) ij i j j ij i j j A A J u qdA u q dA σ δ σ δ = − + − ∫ ∫ W W (3) In the interaction energy integral method, the auxiliary fields, including displacements ( aux u ), strains ( aux ε ), and stresses ( aux σ ) are used. These auxiliary fields need to be suitably defined in order to evaluate T-stress and TSIFs in nonhomogeneous materials. Here we adopt analytical fields originally developed for homogeneous materials and the “incompatible formulation” is chosen in this paper [14]. As shown in Fig.1, there is a point force F applied to the crack tip in an infinite plane. The auxiliary displacement fields and the auxiliary stress fields for T-stress are chosen as follows [11, 12], 2 1 2 ( 1) ln sin 8 4 ( 1) sin cos 8 4 tip aux tip tip tip aux tip tip F r F u d F F u κ ω πμ πμ κ ω ω ω πμ πμ + =− − − =− + (4) 3 11 2 22 2 12 cos cos sin cos sin aux aux aux F r F r F r σ ω π σ ω ω π σ ω ω π =− =− =− (5) where d is the coordinate of a fixed point on the 1x axis, tipμ is the shear modulus evaluated at the crack-tip, and 3- (plane stress) 1 3-4 (plane strain) tip tip tip tip ν ν κ ν ⎧ ⎪ = ⎨ + ⎪ ⎩ (6) And the auxiliary strain fields ( ) aux aux ij ijkl kl S ε σ = x , which is incompatible with the auxiliary displacement fields [15]. Since ijkl S is the compliance tensor of the actual materials, not the compliance tensor of the crack tip, it can be got that: , , ( ) / 2 aux ij i j j i u u ε ≠ + . Superposition of the actual and auxiliary fields leads to a new equilibrium state.
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