13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- We are going to illustrate the influence of the inclusion size a on the weight function and we shall look at the evolution of the non-local contributions nearby the boundary of a damaged zone. For this purpose, we look at a one-dimensional problem of a clamped bar subjected to tension. The bar has a damaged zone in the middle, the rest being undamaged. The distribution of damage is fixed a priori. The strain distribution inside the bar corresponds to the onset of evolution of damage from this initial state. The strains εu and εt are obtained from a bilinear softening constitutive law (see Fig. 2). Although the bar is one-dimensional, the interactions are computed following a 2D, plane stress description. The weight functions are computed from a discrete set of circular inclusions located on the neutral axis of the bar (see Fig. 2). Their size is much smaller than the bar depth in order to avoid interactions with upper and lower boundaries. The finite element meshes consist in triangular elements with 1 integration point and the meshes are built in such a way that there are always 4 elements in the inclusion diameter. Each inclusion is dilated successively in order to reconstruct the weight functions. The weight functions are normalized afterwards so that their integral over the bar is equal to 1 (through the functional Ωr in Eq. 5). Fig. 3 presents the influence of the size of the inclusion on the weight function in the case where damage is equal to zero. Figure 3. Influence of the inclusion size on the weight function (Reproduced from [25]): (a) computed far from the boundaries; (b) normalized and computed far from the boundaries; (c) normalized and computed near the boundary. Far away from the center of the inclusion (Fig. 3.a), the weight function does not depend on the inclusion size and decreases as 1/X2 following the Eshelby’s theory (see e.g. [22]). Fig. 3.b presents the same influence of the size of the inclusion but on the normalized weight function. If the inclusion size tends to zero, the computation of the interactions reduces to the construction of Green functions in which it is well known that no internal length is involved. One can demonstrate from the construction of the normalized weight function that it becomes a Dirac delta function and the constitutive model becomes local. Fig. 3.c shows the same calculation nearby the boundary of the solid. The weight function is centered in the inclusion, which sits right next to the boundary. Again, upon decreasing of the radius of the inclusions, the weight function converges toward a Dirac Delta function. According to the results due to [12], [20] and [23], it is expected that at the boundary of
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