13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where iΨ and cΨ represent the free energies related to the internal and contact parts of the strain state respectively and ic Ψ is the mixed term of the free energy which takes into account the co-presence of the contact and internal strains. ξd and ξp are the damage and plastic internal variables, respectively. The principle considered for developing the constitutive laws is that damage occurs in the bulk material, therefore the damage tensor ω appears in the two terms of the total free energy that are functions of the internal strains also. In this way the constitutive model takes into account the onset of microvoids and fractures along the thickness of the joint. On the other hand, debonding of the joint from the adherents, sliding and fractures developing on surfaces parallel to the middle plane of the interphase are modelled using elastoplasticity and the inelastic contact strains cp ε are the related internal variables. In this work a single scalar damage variable ω governs the loss of stiffness of the bulk material. It ranges from 0 to 1, with the inferior and superior limits having the meaning of a pristine and a fully damaged bulk material, respectively. The explicit expression of the components of the free energy is given below: ( ) ( ) ( ) 2 1 1 tr 2 : ln 1 2 i i i i d d d h ω λ μ ξ ξ Ψ = − + − + − ε ε ε (11) ( ) ( ) ( ) 2 2 1 1 tr 2 : 2 2 c c cp c cp c cp p p h λ μ ξ Ψ = − + − − + ε ε ε ε ε ε (12) ( ) ( ) ( ) , i c cp 1 tr tr i c ω λ Ψ = − − ε ε ε (13) where λ and μ are the Lamè’s constants, hd is a material parameter which governs the softening response associated to the damage onset and growth, and hp is a material parameter specifying isotropic hardening/softening interface response. 3. State equations and flow rules. In order to derive the interphase constitutive equations, the second principle of thermodynamics, taking into account also the balance equation (first principle) can be applied in the form of the Clausius-Duhem inequality. This inequality for isothermal purely mechanical evolutive process reads as : 0 D= −Ψ≥ σ εɺ ɺ (14) where D is the interphase dissipation (for unit surface) or net entropy production. From the assumed form of the Helmholtz free energy (Eq. 10) its general rate has the following expression: , , , , : : : : i i c c i c c i c i c cp i i c c cp cp i i c i c d p d p ω ξ ξ ω ω ξ ξ ∂Ψ ∂Ψ ∂Ψ ∂Ψ ∂Ψ ∂Ψ Ψ= + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂Ψ ∂Ψ ∂Ψ ∂Ψ + + + + ∂ ∂ ∂ ∂ ε ε ε ε ε ε ε ε ε ɺ ɺ ɺ ɺ ɺ ɺ ɺ (15) Particularizing Eq. (14) for a purely elastic incremental deformation process ( , 0 cp d p ω ξ ξ = = = = ε 0 ɺ ɺ ɺ ɺ ), assuming the decomposition of the stress state similar to that used for the strain state
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