# A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials
## 1 Introduction  
Including soils' behaviors into a mathematical framework is required to calculate the bearing capacity. A failure criterion is typically used to define a material's strength capabilities under particular circumstances [1]. Here, the simple Mohr-Coulomb material model is by far the most frequently used [2]. Geotechnical materials belong to the collection of scattered particles, which are widely distributed and are closely related to engineering construction. A crucial mechanical characteristic of construction materials is their shear strength [3]. Hence, the shear strength must be easy to obtain from the failure criteria. Additionally, the M-C criterion directly provides the shear strength. Since the limit surface of the M-C criterion is composed of piecewise linear equations, the corner points are formed at the intersection of the two equations [4]. If the associated flow rule is utilized in deformation analysis, the presence of corner points will result in the plastic flow direction not being unique [5]. To overcome this problem, many scholars have proposed a variety of smooth shape functions (ridge functions) to replace the M-C failure criterion [6; 7]. Although ridge functions eliminate corner points, their mathematical expressions are cumbersome and lack clear physical meaning, making them not easy to apply. According to [PERSON] et al. [8; 9], however, the corner problem has been solved. The geotechnical materials' failure criteria are generally divided into two categories, from the perspective of whether they have physical meaning. One category is hypothesis criteria, based on physical concepts, and the other is statistical abstract criteria, based onthe findings of countless experiments. The former criteria include the M-C, the SMP [10], and the twin shear criteria [11]. The latter criteria include the Lade-Duncan [12], Hoek-Brown [13], and Yoshida criteria [14]. Due to its strong applicability, obvious physical meaning, and ease in calculating strength parameters using direct shear or traditional triaxial testing, the M-C failure criterion is frequently employed in geotechnical engineering. However, it ignores the intermediate principal stress's influence. Some researchers use elliptic, hyperbolic, and spatially moving plane deviatoric functions to extend the M-C criterion to three dimensions while taking the effect of \(\sigma_{2}\) into account [15; 16; 17]. However, the existing 3D M-C failure criterion has no obvious advantage in either the mechanical mechanism or mathematical form. For example, the Mogi-Coulomb criterion [18] has significantly improved the true triaxial data's fitting accuracy but does not meet the convexity condition in the deviatoric plane. [PERSON] et al. [19] proposed the nonlinear M-C criterion formula but found that this criterion's applicability was limited. A popular calculation technique for solving a variety of engineering problems is the finite element method. It can be combined with various numerical approaches [20], which is just one of its numerous benefits. The key part of numerically calculating geotechnical material plasticity is the stress update. Since the stress update is performed multiple times in each loading step, the process must be fast and accurate to ensure an effective numerical solution. In general, schemes based on explicit and implicit procedures constitute the two categories of constitutive equation-integration algorithms [21]. The explicit integration algorithm is simple and easy to implement, but there may be cases where the predicted stress deviates from the current yield surface, thus resulting in inaccurate calculation results. Implicit algorithms have attracted increasing attention because of the progress. The return mapping algorithm was first put forth by [PERSON] and [PERSON] [22], and it has since been rapidly refined and used. Most implicit algorithms fall within the return mapping scheme category. However, the implicit return mapping algorithm is difficult to develop and is actually locally convergent.