13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- , being i c i = + = σ σ σ σ AσA (16) and considering the adopted expressions of the free energy parts (Eq. 11-13), the elastic stress-strain equations can be derived, thus ( ) ( ) ( ) { } 1 tr tr 2 i i c cp i σ ω λ μ = − + − + ε ε ε A ε (17) ( ) ( ) ( ) ( ) ( ) 1- tr tr 2 c i c cp c cp σ λ ω μ = + − − + − ε ε ε I A ε ε (18) p p p h χ ξ = (19) 1 d d d d h ξ χ ξ = − (20) ( ) ( ) ( ) 2 1 tr 2 : tr tr 2 i i i i c cp ζ λ μ λ = + + − ε ε ε ε ε ε (21) where χp and χd are the static variables conjugate of the internal variables ξp and ξd respectively, and ζ the thermodynamic force conjugate of the damage variable ω. Making use of the elastic strain-stress equation and of the previous positions, the final expression of the instantaneous dissipation is obtained: : 0 c cp p p d d D χ ξ χ ξ ζω = − − + ≥ σ ε ɺ ɺ ɺ ɺ . (22) In order to regulate the activation of each dissipative mechanism, two different yield functions are defined in the space of the proper static variables, namely: ( ) ( ) , 0, , 0 c p p d d χ ζ χ Φ ≤ Φ ≤ σ (23) where Φp is the classical plastic yield function specifying the elastic contact domain assumed convex and Φd is the damage activation function also assumed convex. The activation of each dissipation mechanism can be effectively described by a variational formulation which is represented by the generalized principle of maximum intrinsic dissipation: ( ) , , , max : c p d c cp p p d d χ χ ζ χ ξ χ ξ ζω − − + σ σ ε ɺ ɺ ɺ ɺ (24) subject to the constraints (Eq. 23). The Kuhn-Tucker conditions of the maximum constrained problem provide the plastic and damage evolution laws of the interphase: ; p cp d p d c λ ω λ ζ ∂Φ ∂Φ = = ∂ ∂ ε σ ɺ ɺ ɺ ɺ (25) ; p d p p d d p d ξ λ ξ λ χ χ ∂Φ ∂Φ − = − = ∂ ∂ ɺ ɺ ɺ ɺ (26) ( ) ( ) , 0, 0, , 0 c c p p p p p p χ λ λ χ Φ ≤ ≥ Φ = σ σ ɺ ɺ (27) ( ) ( ) , 0, 0, , 0 d d d d d d ζ χ λ λ ζ χ Φ ≤ ≥ Φ = ɺ ɺ (28) being pλɺ and dλɺ the plastic and damage multiplier, respectively.
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