# The Generalized Mohr-Coulomb Failure Criterion
## 1 Introduction  
Currently, strength evaluation and deformation calculation are still the two most important problems in geotechnical engineering, as in [PERSON]'s age [1]. For most geotechnical engineers, the phrase \"strength of soils\" conjures up images of Mohr-Coulomb (M-C) failure criteria [2]. Consequently, whenever new geotechnical materials are encountered, such as vegetated soil [3], lime-cement-improved clay [4], and steel slag aggregate [5], the models for evaluation of their shear strength are consistently established as close as possible to the M-C criterion. The requirements for failure criteria in strength problems differ significantly from those in deformation problems. In the solution of strength problems, such as the stability analysis of slopes [6], the earth pressure evaluation of retaining walls [7], and the loading capacity calculation of ground foundations [8], the shear strength is the most important quantity. Hence, it is better for the shear strength to be easily derived from the failure criterion, and the M-C criterion is such a criterion that directly gives the shear strength. This might explain why the M-C criterion has been dominant in the solution of strength problems. In the solution of deformation problems, on the other hand, a failure criterion is deemed better if the failure surface in the stress space is smooth, because the derivatives of the yield function are needed if plastic deformation is involved. From this perspective, the Lade-Duncan (L-D) criterion [9] and the SMP criterion [10] are more favorable than the M-C criterion because the M-C failure surface is not smooth in the stress space. According to the algorithms recently developed in [PERSON] et al. [11; 12], however, non-smooth yield surfaces bring no troubles to the update of stresses in the analysis of elastic-plastic deformation. Over the past decades, pursuing smooth yield surfaces and reflecting the influence of the intermediate stress \(\sigma_{2}\) on failure might be two agents in developing failure criteria. TheZienkiewicz-Pande criterion [13] and the Menetrey-Willam criterion [14] were derived by smoothing the M-C criterion. The Hoek-Brown criterion [15] does not pursue agreement with the M-C criterion, and has a smooth failure surface but does not reflect the effect of \(\sigma_{2}\) on shear strength. Moreover, the Hoek-Brown criterion has no explicit expression of shear strength. Hence, [PERSON] and [PERSON] developed a \"Generalized Tangential\" technique in the upper bound solution for ultimate bearing capacity with the modified Hoek-Brown failure criterion [16]. To reflect the influence of \(\sigma_{2}\) on strength, the concept of the spatially mobilized plane (SMP) was introduced by [PERSON] and [PERSON] [10], and they set up the SMP criterion that is expressed in terms of the three stress invariants to model the three-dimensional strength of geomaterials under different loading conditions. Meanwhile, the Lade-Duncan criterion was developed in terms of the first and the third stress invariants to model the three-dimensional strength of geomaterials based on the true triaxial compression tests of cohesionless soils [9]. The original versions of both the L-D and SMP criteria assume no cohesion, making them only applicable to sands and normally consolidated clays. The Drucker-Prager criterion, albeit commonly used, is a rather coarse approximation to the M-C criterion and inconsistent with actual cases of soils [17]. Several other failure criteria, for example, [PERSON] [18], [PERSON] et al. [19], [PERSON] and [PERSON] [20], [PERSON] et al. [21], [PERSON] et al. [22], [PERSON] et al. [23], and so on, have been proposed in terms of principal stresses to reflect the effect of \(\sigma_{2}\). However, according to comparative studies in [PERSON] et al. [24], [PERSON] [25], [PERSON] and [PERSON] [26], and [PERSON] [27], none of the existing failure criteria has a significant advantage over the others in both mechanical mechanism and mathematical form.