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.

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