# A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials
## 1 Introduction
Based on Koiter, [PERSON] [23] presented a non-smooth yield surface plastic equation integration approach that is only useful when two yield surfaces cross. [PERSON] [24] proposed a convex programming-based technique for multi-surface plastic calculation that, in order to calculate the stress at an integral point, needs the Hessian matrices for each effective yield function. [PERSON] et al. and [PERSON] et al. [8] proposed the Gauss-Seidel projection-contraction (GSPC) algorithm, based on the Gauss-Seidel idea and projection-contraction algorithm. The GSPC algorithm converges for any strain increment size, the computational efficiency is higher than the return mapping algorithm, and it does not need to calculate the potential function's Hessian matrix. ABAQUS has been extensively utilized in civil engineering computation because it is a powerful generic nonlinear finite element analysis program that can be used to analyze and compute several difficult linear and nonlinear engineering issues. Through the user subroutine UMAT, [PERSON] et al. [25] integrated the extended Drucker Prager model into ABAQUS and applied it to tunnel construction. Based on the return mapping method, [PERSON] et al. [26] incorporated the Tresca yield criterion, plastic flow, and hardening rules into a UMAT in the finite element analysis program ABAQUS. [PERSON] and [PERSON] [27] proposed a novel precise stress integration strategy based on the combined linear hardening von Mises elastoplastic model, which is implemented in ABAQUS through its UMAT material interface. Some researchers [28; 29; 30; 31; 32] have also used the UMAT material interface in ABAQUS to numerically implement the existing model. The aforementioned research shows that it is challenging for the current failure criteria to strike a balance between a straightforward mathematical form and a distinct physical meaning. In this paper, a generalized M-C (GMC) failure criterion that considers the intermediate principal stress's impact is established. The suggested criterion simply regards its parameters of cohesion \(c\) and friction factor \(f\) as functions of the coefficient \(b\) of the intermediate principal stress, written as \(c(b)\) and \(f(b)\), respectively. In this way, the effect of \(\sigma_{2}\) can be naturally reflected in the failure criterion. A true triaxial failure criterion can be easily established simply using a set of traditional triaxial compression tests and a set of triaxial extension tests. The GMC failure criterion's predictions for several geomaterials were found to be congruent with experimental findings when they were compared to data from real triaxial testing. Based on the GSPC algorithm in the principal stress space, the UMAT subroutine of the three-dimensional GMC model is developed in ABAQUS using the Fortran programming language. The numerical simulation of a uniaxial compression test and the simulation of an underground chamber under hydrostatic pressure were performed to assess the established elastoplastic model's viability and the program's correctness. Finally, the GMC model was more accurate in determining the extent of the plastic zone and the surrounding rock stress by simulating tunnel excavation, which has a certain practical significance.