13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Based on Eq.(2), define the second-order tensor N , P and the forth-order tensor Qrespectively as ( ) 0 2 0 2 2 0 1 1 1 n ij i j ij T t ij t ip p j i j ij t t t ij t ik j l i j k l kl ij ij kl N P Q ξξ ε ε ξ ξ ξξ ε δ ξ ξ ξξ ξ ξ ε ε ε ε ∂Δ = ⋅ = ∂ ∂Δ = ⋅Δ ⋅ = − ∂ ⎛ ⎞ ∂Δ ∂Δ ∂ Δ = ⋅ ⋅ +Δ = − ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ξ εξ l l l (3) where ijδ being the Kronecker delta. According to this micro structure, the strain tensor of material[3] can be derived as 1 n t ij ij n ij t ij U U V σ ε ε ε ∂Δ ∂Δ ∂Φ ∂ ∂ = = ⋅ + ⋅ ∂ ∂Δ ∂ ∂Δ ∂ (4) and the tangent modulus of material as 2 2 2 2 2 2 2 2 2 1 n n n ijkl ij kl n kl ij n ij kl t t t t kl ij t ij kl t n n t n t kl ij t n kl ij U U C V U U U U ε ε ε ε ε ε ε ε ε ε ε ε ε ε ∂Δ ∂Δ ∂ Δ ∂ Φ ∂ ∂ = = ⋅ ⋅ + ⋅ ∂ ∂ ∂Δ ∂ ∂ ∂Δ ∂ ∂ ∂Δ ∂Δ ∂ Δ ∂ ∂ + ⋅ ⋅ + ⋅ ∂Δ ∂ ∂ ∂Δ ∂ ∂ ∂Δ ∂Δ ∂Δ ∂Δ ∂ ∂ + ⋅ ⋅ + ⋅ ⋅ ∂Δ∂Δ ∂ ∂ ∂Δ∂Δ ∂ ∂ (5) in which Φ denotes the strain energy density of a micro element;εthe strain tensor; V the volume of the micro element and ( ) 2 0 0 ( ) , sin D d d π π θ φ θ θ φ =∫ ∫ L L in spherical coordinates for 3D case and ( ) 2 0 ( )D d π θ θ =∫ L L for 2D case. Substituting Eqs.(1,2,3) into Eqs.(4,5), the stress tensor and tangent modulus tensor can be rewritten as 1 ij n ij t ij f N f P V σ = ⋅ + ⋅ (6) and ( ) 1 ijkl A ij kl B ijkl C ij kl D ij kl ij kl C f N N f Q f PP f PN N P V = ⋅ + ⋅ − ⋅ − ⋅ + (7) in which the coefficients are respectively
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