13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- In Eq. (1) 1e , 2e and 3e are the unit vectors of the local reference system, with 3e oriented along the normal to the middle surface Σ and directed towards the adherent +Ω . The joint can be regarded as an interphase model. It is assumed that the fibers inside the interphase and directed along 3e are maintained rectilinear during the deformation process. In view of this hypothesis the interphase displacement field u can be easily obtained from the displacement +u and −u of the interfaces +Σ and −Σ , thus ( ) ( ) 3 3 1 2 3 1 2 1 2 1 1 ( , , ) , , 2 2 x x x x x x x x x h h + − = + + − u u u (2) with 1x , 2x and 3x the Cartesian coordinates in the orthonormal frame ( ) 1 2 3 , , , Oe e e . Since the thickness of the joint is generally small if compared to the characteristic dimensions of the adherents, we can assume the strain state ε uniform along the 3e direction and given by: ( ) 2 1 2 1 2 3 3 2 1 ( , ) , , h s h x x x x x dx h − = ∇ ∫ ε u (3) Substituting the Eq. (2) we have: [ ] [ ] ( ) ( ) 1 2 1 1 ( , ) 2 2 s x x h + − = ⊗ + ⊗ + ∇ + ε u n n u u u (4) where [ ] + − = − u u u , n is the unit normal vector to the interphase plane and s∇ is the symmetric gradient operator defined as ( ) 1 2 s T ∇ = ∇+∇ . Let us note that in the interphase model the joint curvatures generated by displacement field (2) and the related flexural effect are neglected. Equilibrium equations are derived by applying the principle of virtual displacements (PVD) that asserts that the external work produced by the contact tractions equals the internal work developed in the joint. According to the hypothesis of a constant strain state, by applying the divergence theorem and assuming that + − Σ=Σ =Σ , the PVD leads to the following local equilibrium relations of the interphase model: ; , 2 2 h h div div on + − − ⋅ + = + ⋅ + = Σ t σ n σ 0 t σ n σ 0 (5) on ⋅ = Γ m σ 0 . (6) The basic kinematical hypotheses are the additive decomposition of total strain in the internal (i) and contact (c) parts and, for the contact strain only, a further decomposition in elastic (e) and inelastic (p) parts: i c = + ε ε ε (7) c ce cp = + ε ε ε (8) with i = ε AεA (9) being = − ⊗ A I n n the unit second order tensor. In order to comply with thermodynamic requirements, the interphase Helmholtz free energy is introduced in the following form: ( ) ( ) ( ) ( ) , , , , , , , , , , , , , i c cp i i c c cp i c i c cp d p d p ξ ξ ξ ξ Ψ =Ψ +Ψ +Ψ ε ε ε ω ε ω ε ε ε ε ε ω ; (10)
RkJQdWJsaXNoZXIy MjM0NDE=