13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- the solid the response becomes local. In mesoscale models, there is a wall effect on the boundaries and large inclusions may not be fitted in a boundary layer smaller than their radius. Nearby a boundary, the inclusion size is constrained by the distance to the free surface, as it cannot protrude outside the solid. This feature can be easily introduced in the present model: at points located close to the boundary, the inclusion size is decreased so that it cannot protrude outside the solid. In a boundary layer of thickness l, inclusions of diameter l shall be considered only when l < a. Thus interactions tend to vanish as we consider points closer and closer to the boundary of the solid. Fig. 4.b presents the normalized non-local contributions when the non local strain is computed at the center of the inclusion located close to the damage band in the region which unloads on the left side. The damage band contains 7 inclusions (a = 1.25mm, h = 8.75mm). A comparison with the gauss-type weight function used in the classical non-local damage model is also provided (Fig. 4.a). With the Gauss weight function and because the strain inside the damage band is larger than outside the damage band, the non-local contribution from points lying inside the band is much larger than those of points lying outside the band. This will trigger the propagation of the damage band, which should expand in the course of the calculation eventually. Fig. 4.b shows that a shielding effect is observed with the new formulation. The non-local weights outside the damage band are the most important. As a matter of fact, the weight at points lying inside the band is decreasing with increasing damage. There is a shielding effect due to damage, which derives directly from the method used for the calculation of interactions. In the extreme case of a fully damaged band, the dilation of an inclusion sitting inside the band will not be transmitted to the stiffer zone outside the band. Figure 4. Response close to a damaged area (Reproduced from [25]): (a) original formulation; (b) interaction-based formulation. There is, however, a limitation to the shielding effect when the radius of the inclusions is larger than the width of the damage band. In this case, the interaction induced by the dilation of the inclusion will extend across the band. It is expected then that a point lying on one side of the band will feel the interaction from points lying on the other side. We recover here the case of an inclusion located near a boundary, a fully formed crack being two free boundaries facing each other. In order to avoid this problem, we impose that the radius of the circular inclusion reduces as damage grows and we adopt the following rule, which encompasses the situation where an inclusion is centered at a point nearby a damage zone or near a boundary of the solid:
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