13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where • + is the positive part function, Ω is the volume of the structure, Ωr is a characteristic volume introduced in such a way that the non-local operator does not affect an uniform distribution of equivalent strain far away from the boundary when no damage occurs in the structure. The analogy between the interactions defined above and the weight function ψ suggests: ψ(x, ξ) ≡A*(x, ξ,a) = A(x, ξ, ε*,a) ε* ≡ εi (x) 2 i=1 3 ∑ ε* with Ωr = A0 *(x, ξ,a)d ξ Ω∫ (5) where A0 *(x, ξ,a) is the interaction function reconstructed when no damage occurs in the structure (typically at the beginning of the computation). Practically, the computation of the interactions (function A∗) is performed using a finite element setup, which is identical to that of the mechanical problem to be solved, with the same mesh. The finite elements which belong to each inclusion centered on a given integration point are subjected to a thermal expansion (ε∗ = α∆TI) where α is the thermal expansion coefficient, I is the identity tensor, and T is the temperature. If the structure has n inclusions (integration points), n elastic computations are performed to build the weight function at each loading step. Since the construction process of the interaction-based weight function is cinematically driven by the successive thermal expansions, all boundary conditions are clamped during the reconstruction process in order to avoid the perturbation of the kinematics on the boundary. The single model parameter which remains to be determined is the inclusion size a. This inclusion size is the internal length involved in the formulation. It ought to be related to the average size of the heterogeneities in the underlying heterogeneous material to be modeled. 2.4.2. Constitutive model Damage is considered to be isotropic. Temperature and time-dependent effects are neglected. Damage is a function of the amount of extension in the material, defined locally by the equivalent strain (see Eq. 4). The evolution of damage is a function of the non-local equivalent strain and it is governed by the Kuhn-Tucker loading-unloading condition (see [15] or [16] for details). 3. Validation and performances 3.1. Clamped bar in tension (L =10 cm, σY =3 MPa, E =30 GPa, εD0 =10 -4, σ 0 =1.8 MPa, D =70 %) Figure 2. Simple problem of a clamped bar in tension.
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