Non-local numerical treatment of non-linear behavior by means of Helmholtz equation, with variable coefficients. Application to reinforced concrete structures.

== Abstract ==

Numerous work has been done with the aim of modeling the cracking of reinforced concrete (RC) structures. Among the recent methods proposed in the literature, the combination of reinforcement-concrete equilibrium combined with the linear behavior of the interface leads to a Helmholtz equation which takes account of the slip between the homogenized reinforcements and the concrete in presence of localized cracks [1][2]. In the case of large cracks openings, it is necessary to consider the non-linear behaviors of material and interfaces, such as the plasticity of reinforcements or the damage of the matrix-reinforcement interface. These phenomena induce variations of the coefficients in the Helmholtz equation, which leads to two levels of iterative procedures: one at a global level considering equilibrium of homogenized RC, and another one at a non-local level taking account of equilibrium between reinforcement and concrete. The implementation of a convergence criterion is then needed at each level. The goal of this paper is to describe the developments implemented in the Finite Element code Cast3m to perform non-local Helmholtz type calculations with non-constant coefficients. This method, using an acceleration method [3] is illustrated by the cases of reinforced concrete tie and beam, with homogenized reinforcements. References : [1] A. Sellier and A. Millard, “A homogenized formulation to account for sliding of non-meshed reinforcements during the cracking of brittle matrix composites: Application to reinforced concrete,” Eng. Fract. Mech., vol. 213, pp. 182–196, May 2019, doi: 10.1016/j.engfracmech.2019.04.008. [2] A. Sellier and A. Millard, “Traitement numérique non local de phénomènes physiques par l’équation d’Helmholtz : les effets d’échelle et le glissement renfort-matrice,” in Club Cast3M 2018, Paris, 2018, vol. 1, no. 1, pp. 12–18. Available: http://www-cast3m.cea.fr/html/ClubCast3m/club2018/Presentation_Sellier.pdf. [3] A. C. Aitken, “On the iterative solution of a system of linear equations.,” Proc. Roy. Sot. Edinburgh, pp. 52–60, 1950.
                                                                                                
== Video ==
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== Document ==
<pdf>Media:Draft_Content_936662302506_paper.pdf</pdf>