The stress integration of critical soil model is usually based on implicit Euler algorithm, where the stress predictor is corrected by employing a return mapping algorithm. In the case of a large load step, the solution of the local nonlinear system to compute the plastic multiplier may not be attained. To overcome this problem, a sub-stepping scheme is generally used to improve the convergence of the local nonlinear system solution strategy. Nevertheless, the complexity of the tangent operator of the sub-stepping scheme is high. This complicates the use of Newton–Raphson algorithm to obtain global quadratic convergence. In this paper, a formulation for consistent tangent operator is developed for implicit sub-stepping integration for the modified Cam-Clay model and unified Clay and Sand model. This formulation is highly efficient and can be used with problem involving arbitrary large load step, such as tunnel simulation.
Bui et al. (Wed,) studied this question.