13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 2.3. A new interaction-based non-local weight function The purpose of the 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, we aim at estimating the non-local weight function directly from interactions. Before we get to the weight function to be inserted in a non-local integral model, let us first consider elastic interactions. In order to compute the effect of point ξ on point x, we look at the strain induced at point x due to the dilation of ε∗ of a circular inclusion of radius a centered at point ξ (see Fig. 1). Figure 1. Non-local contribution seen by a point x when a perturbation is produced in ξ. Assuming now that the induced strain at point x has been computed, numerically for instance. The growth of damage is often defined from energy considerations and we shall look at a norm of this strain, denoted as A, instead of the strain tensor itself: A(x, ξ, ε*,a) = ε i (x) 2 i=1 3 ∑ (1) where εi(x) is the ith principal strain. Note that we could have chosen the true elastic energy instead of a norm of the strain tensor. It would not have changed much the following development. Then, the interaction is represented by this norm transmitted from the dilation in the inclusion centered at ξ to x. It depends on the geometry of the solid, on the inclusion size a, and on the material elastic properties inside and outside the inclusion. Formally, the norm A transmitted to x by the dilation ε* writes also: A(x, ξ, ε*,a) = ε* A*(x, ξ,a) (2) where A* represents the interaction produced at x due to ξ for a unit dilation. 2.4. Final formulation 2.4.1. Non-local averaging We assume now that it is this interaction A*, which governs the weight function involved in non-local averaging. This non-local averaging writes: εeq(x) = 1 Ωr ψ(x, ξ) εeq( ξ)d ξ Ω∫ with Ωr = ψ(x, ξ)d ξ Ω∫ (3) where εeq is the non-local strain and εeq is the effective strain defined by Mazars [24] as: εeq = εi + 2 i=1 3 ∑ (4)
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