13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- undergoing damage in the course of failure [18, 19]. There are several mechanisms, which should be considered when looking at the non-locality due to the interaction between damaged points: (i) an interaction exists if there is damage, which produces this interaction. Assuming that damage corresponds to the growth of micro-cracks, this interaction grows with the size of the defect; (ii) shielding effects are also expected: the interaction between two points located apart from a crack should not exist; (iii) on free existing or evolving boundaries, and along the normal to these boundaries, non-local interactions should vanish as demonstrated in [20]. The internal length in the non-local model is the parameter inside the weight function that encompasses the non-locality and there is a consensus that this quantity may not be constant, but should depend on the geometry of the specimen or on the state of damage. Therefore enhanced non-local models accounting for a variation of the internal length have been proposed recently [13, 20, 21]. Proposals discussed in [13, 20] are considered on academic one-dimensional problems. Their implementation and extension to 2D or 3D problems are really not trivial as they involve the computation of path integrals, which are tedious in a finite element setting. The stress-based model in [21] is more tractable in 2D/3D computations but the evolution of non-locality is rather empirical. The purpose of this paper is to discuss a new approach to non-local interactions during failure in quasi-brittle materials and to upscale the relevant information present from the meso-scale to the macro-scale. Therefore, the paper focuses on the estimation of the non-local weight function directly from interactions. The material is modeled as an assembly of inclusions and the elastic interactions upon dilation of each inclusion are computed in a similar ways to a classical Eshelby’s problem [22]. A new interaction-based weight function is then built from these interactions. This new interaction-based non-local model is validated on simple 1D problems and its performances are compared with the classical integral-type nonlocal model. 2. A new interaction-based non-local model 2.1. Non-locality in integral-type macro-scale models In classical non-local models, such as the integral-type [1], the internal length is the parameter inside the weight function that encompasses the non-locality. Associated with a classical Gaussian weight function, it set how and how far the interactions produce inside the materials. However, the main drawback of the formulation is that this parameter is constant whatever the geometry and the failure process. For instance, close to a boundary, the part of the nonlocal averaging domain that protrudes outside the boundary is classically chopped off [1]. Improved models can be found in the literature, with a different averaging process close to the boundary of the solid [12,23] or with a varying internal length in the course of damage [13, 20, 21]. However, even if the internal length variations are based on micro-mechanical concepts, such as the crack growth interaction effect or the transfer of information through a damaged area, the final choice of the weigh function and thus the evolution of non-locality are rather empirical. 2.2. Non-locality in meso-scale models In meso-scale models, the non-locality is intrinsically included by representing the meso-structure of the materials (e.g. granular, matrix and interfaces in concrete). Therefore, the non-locality does not behave the same close to a boundary, close to a damaged area, at initiation or during the failure process. It has been shown recently that such models are able to capture challenging issues of quasi-brittle materials failure such as predicting the peak loads and even the whole softening load-displacement responses of notched and unnotched beams in three-point bending [15, 17]. In the following, we aim at building a new interaction-based non-local weight function, which will evolve intrinsically when damage occurs inside the materials.
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