Damage Plasticity Model

In this section, basic components of a damage-plasticity model for the concrete material are shortly discussed and detailed investigation of it is provided in Appendix A. A damage-plasticity model can be formulated in terms of either effective or nominal stress. However, according to the local uniqueness condition in which a specific strain history results in a unique stress and internal hardening variables response (refer to Grassl and Jirasek [20]), damage-plasticity models formulated in terms of effective stress space are preferred in this study. Furthermore, these types of damage-plasticity models result in a simpler stress-updating algorithm. The material model used in this study is according to the Grassl et al. [27] work in which a modified version of Leon model(Etse and Willam [38]) is adopted.

4.1 Plasticity formulation

Following equations are used

Now, the introduced damage-plasticity model of concrete can be incorporated into the framework of large deformation plasticity. According to the algorithm of Box 1, if the trial stress state doesn’t lie within the elastic domain, the return mapping equations must be solved. To do so, these equations should be more simplified. By replacing from equation (20) into equation (7), it is concluded that: (29)

As the flow potential function is an isotropic function of T, besides the assumption of elastic and plastic isotropy and considering the fact that the potential function doesn’t depend on the lode angle, the tensorial equation (29) can be reduced to the following two nonlinear equations: (30) (31) in which K and G are bulk and shear modulus, respectively. So equations (23), (30),(31) together with the equation Q=0 form the final nonlinear system of equations which must be solved for unknowns P, Q, T and G. With converged values of P and Q the principal stress components can be found with the following