13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- effect of the internal stresses into the analysis. This enhancement of the ZTI is known as interphase model, for the first time proposed by Giambanco and Mròz [11]. The interphase element has been formulated by authors as a new contact element and introduced in a scientific oriented finite element code. Patch tests have been carried out in elasticity to investigate the numerical performance and convergence of the element. All the results are shown in the paper written by Giambanco el al. [12]. In particular, in that paper is shown how strategies such as the Reduced Selective Integration or the Enhanced Assumed Strain methods are necessary to avoid shear locking effects of the element. In this work the interphase element is implemented for nonlinear applications by introducing separate limit conditions for the joint bulk material and for the physical interfaces. In particular the damage mechanics theory is used to simulate the formation and propagation of fractures in the bulk material. The elastoplastic limit condition of Mohr-Coulomb type with a tensile cut-off is adopted to describe the decohesion process of the interfaces. The contact strains are subdivided in an elastic and a plastic part. The overall model is thermodynamically consistent and the flow rules are derived by applying the Lagrangian method. With the aim to show the effectiveness of the model the interphase constitutive laws have been implemented in an open-source research-oriented finite element analysis program for 2D applications and by using the Selective Reduced Integration. The paper is organized as follows. In Section 2 the general assumptions of the model are reported and the expression of the Helmholtz free energy is furnished. In Section 3 the state equations and the flow rules are determined on the base of the thermodynamically consistent theory. Section 4 is finally dedicated to numerical applications in order to show the effectiveness of the proposed model. 2. General assumptions and Thermodynamics. Let us consider, in the Euclidean space 3ℜ referred to the orthonormal frame ( ) , , , O 1 2 3 i i i , a structure formed by two adherents +Ω , −Ω connected by a third material Ω in contact with the two bodies by means of the two physical interfaces +Σ and −Σ respectively, as in Fig. 1. Ω+ Ω− Σ+ Σ− Ωj Γ+ u Γ− u h Γ+ t Γ− t i1 i2 i3 Ω+ Ω− Σ Γ+ u Γ− u Γ+ t Γ− t i1 i2 i3 e 1 e2 e3 (a) (b) Figure 1. (a) Mechanical scheme of a third body interposed between two adherents; (b) Interphase mechanical scheme. It is assumed that the thickness h of the joint is small if compared with the characteristic dimensions of the bonded assembly. The boundary of the two adherents is divided in the two parts u ±Γ and t ±Γ , where kinematic and loading conditions are specified respectively. The joint interacts with the two adherents through the following traction components: 1 1 2 2 3 3 t t t ± ± ± ± = + + t e e e (1) which can be considered as the external surface loads for the joint.
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