13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- Table 2. Characteristics of the spalling test Parameter L t0 v E ρ lc/a0 εD0 αt At Bt αc Unit cm s cms−1 MPa kgm-3 cm Value 20 4 1,5 1 1 4 1 1 1 2 0 4. Concluding remarks A new interaction-based non-local formulation has been proposed. In this formulation, 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. A new interaction-based weight function is then built from these interactions. This new interaction-based non-local model has been first validated on simple 1D problems and its performances have been compared with the classical integral-type non-local model. Different results have been presented in the paper: (i) In the course of damage, the crack opening displacement has been estimated and the comparisons show that the failure process is better described with the new formulation. Indeed the crack opening profile is very close to an ideal opening profile obtained for a strong discontinuity. (ii) At complete failure, the crack opening should be independent of the element size. Therefore, after complete failure, the strain in the cracked element should evolve in inverse proportion of the element size, assuming that the crack opening is smeared over the finite element that contains the discontinuous displacement. It has been shown that for the new formulation, the strain versus element size curve follows a linear trend in a logarithmic plot. Moreover the slope is coherent with the CMOD estimated at complete failure. (iii) Close to a boundary, it has been shown that the spalling failure is better described with the interaction-based model since the spall location is predicted inside the bar whereas the damage is maximum on the boundary with the original model. Finally, it has been shown that this new interaction-based formulation fulfill several deficiencies of the classical integral-type non-local model and the formulation has to be implemented in 2D in order to test its performance on more challenging issues of quasi-brittle materials failure such as reproducing the peak loads and even the whole softening load-displacement responses of notched and un-notched beams in three-point experimental bending tests [16]. Acknowledgements Financial support from ERC advanced grant Failflow (27769) is gratefully acknowledged. References [1] G. Pijaudier-Cabot, Z.P. Bažant, Nonlocal Damage Theory. J Eng Mech, 113 (1987) 1512–1533. [2] R. Peerlings, R. De Borst, W. Brekelmans, J. De Vree, I. Spee, Some observations on localisation in non-local and gradient damage models. Eur J Mech A-Solid, 15 (1996) 937–953. [3] G. Wells, L. Sluys, A new method for modelling cohesive cracks using finite elements. Int J for Numer Meth Eng, 50 (2001) 2667–2682. [4] N. Moës, T. Belytschko, Extended finite element method for cohesive crack growth. Eng Fract Mech, 69 (2002) 813–833. [5] J. Oliver, A. Huespe, E. Samaniego, E. Chaves, Continuum approach to the numerical simulation of material failure in concrete. Int J Numer Anal Met, 28 (2004) 609–632. [6] V. Tvergaard, A. Needleman, Effects of nonlocal damage in porous plastic solids. Int J Solids Struct, 32 (1995) 1063–1077.
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