13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- For plane strain problem, stress fields and displacement fields may be expressed in terms of Muskhelishvili’s complex potentials [7] ( )z Φ and ( )z Ψ ( ) ( ) 2 yy xx z z σ σ ⎡ ⎤ + = Φ +Φ ⎣ ⎦ (4) ( ) ( ) ' 2 2 yy xx xy i z z z σ σ σ ⎡ ⎤ − + = ⎣ Φ −Ψ ⎦ (5) ( ) ( ) ( ) ( ) ( ) ' ' ' 2 x y z u u iz z z z z z z μ κ ⎡ ⎤ + =⎢Φ−Φ+Φ +Ψ⎥ ⎣ ⎦ (6) where ' x x u u θ =∂ ∂ , ' y y u u θ =∂ ∂ , ( ) ( ) ' z d z dz Φ = ⎡Φ ⎤ ⎣ ⎦ , the overbar represents the complex conjugate, μis the shear modulus of the bulk solid, υis Poisson’s ratio of the bulk solid , 3 4 κ υ = − for plane strain state. Under the assumption that the interface adheres to the bulk without slipping, and in the absence of body forces, according to Sharma et al. [6] based on Gurtin and Murdoch surface/interface model, the constitutive equation in the surface region is given as ( ) 0 0 0 0 0 0 2 θθ θθ σ τ μ λ τ ε = + + − (7) where 0 θθ σ and 0 θθ ε denote surface stress and strain, 0μand 0λare surface Lame constants, 0τ is the residual surface tension. According to Gao [8], the uniform eigenstrains could be represented to express the displacements produced by the mismatch strains εof the matrix and the effective medium. 0 0 2 ru iu R θ ε + = 2R ζ = (8) where εis dilatational or shrunk eigenstrain of neighboring voids in the matrix phase. The effective medium plasticity mismatches εmay be produced due to the yield stress and the strain hardening exponent on the nanovoid growth and coalescence. It is possible, of course, that to be attributed to mismatchs of the thermal expansion coefficient between the constituents. According to Muskhelishvili [7], two complex potentials ( ) 1 z Φ and ( ) 1 z Ψ in the matrix can be taken the following forms ( ) ( ) 0 1 1 10 0 1 z z z z z z γ γ Φ = + +Φ − − (9) ( ) ( ) ( ) ( ) 0 0 0 1 1 1 1 10 2 2 0 1 0 1 z z z z z z z z z z z z γ γ γ γ Ψ = + + + +Ψ − − − − (10) Two complex potentials ( ) 2 z Φ and ( ) 2 z Ψ in the effective medium can be taken the following forms ( ) ( ) ' ' 0 1 2 1 20 z z z γ γ + Φ = +Γ +Φ (11)
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