Stiffness Model

The wide use of the concrete as structural material necessitates a deep understanding of its behavior. From experiments of concrete specimens under tension or low confined compression, it is observed that after reaching a specific threshold of failure, the stress decreases with increasing the strain which is known as softening. In the softening behavior of concrete, permanent deformations and stiffness reduction happen together. By increasing the confining pressure, the degradation of the stiffness is reduced and material behaves more ductile. Until now, several constitutive models have been proposed for characterizing this complicated behavior. Models which are only based on the plasticity cannot represent the stiffness reduction. For example

[17] and Feng [18] are in this category. Consequently, the most appropriate approach is to combine the plasticity and damage models to capture both irreversible deformations and stiffness reduction. Accordingly, several damage-plasticity models have been developed. For example, the works of Lee and Fenves [19], Grassl and Jirasek [20], Lubliner et al. [21], Luccioni and Rougier [22], Cicekli et al. [23], Voyiadjis et al. [24], Nguyen and Houlsby [25], Yu et al. [26], Grassl et al. [27] and Tysmans et al. [28] have combined the plasticity and damage for modeling the concrete material. In this study, the damage-plasticity model proposed by Grassl et al. [27] is utilized for modeling the concrete material. In this model, two separate damage variables are used for tension and compression which is suitable for loading conditions like impact in which several transitions from compression to tension

In RC structures in which steel bars make the structure behave more ductile, large rotations and deformations may occur as a result of the high-velocity impact. So, it is necessary to formulate the damage-plasticity model of the concrete within the large deformation framework. Accordingly, the general framework of large deformation plasticity which is based on the multiplicative decomposition of the deformation gradient tensor (refer to J.C. Simo [29]) is provided in this study. In high-velocity impact problems, the stress and strain values increase in a short period of time. So, using an appropriate method for updating the stress values plays a crucial rule in the numerical simulation. In plasticity frameworks, the stress values are usually updated using a two-steps elastic predictor/plastic corrector procedure. The plastic corrector algorithm usually forms a nonlinear system of equations which is known as return mapping equations. These equations are often strongly nonlinear and the ordinary Newton-Raphson algorithm for solving them has a small radius of convergence. Until now, several solutions have been proposed to overcome this nonlinearity. Some of them like de Souza Neto et al. [30], try to find appropriate initial guesses for these equations. As another solution, line-search procedures can be incorporated into the