# Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils
## 1 Introduction  
The mechanism and prediction of critical state is an important research topic in soil mechanics [1; 2; 3; 4], especially the non-unique critical state of gap-graded soils that attracts many researchers' attention [5; 6]. The non-unique critical state is related to the distance between the critical state line (CSL) and the isotropic consolidation line (ICL), and the distance varies with stress paths or particle size distributions. So, predicting the non-unique critical state of soils is important for solving soil mechanics problems.  
Over the past decades, considerable efforts have been devoted to the constitutive modeling of soils. A unified hardening model was proposed and improved to reflect the overconsolidation, anisotropy, and thermo-elasto-plastic behavior of clays [7; 8; 9]. A double-yield surface model was proposed and modified to investigate dilatancy, fabric anisotropy, and consolidation conditions of sands [10; 11; 12; 13]. Several other types of constitutive models were formulated for predicting the effect of soil structure [14; 15; 16] and unsaturated characteristics [17; 18; 19]. Recently, the framework of the bounding surface plasticity model proposed by [PERSON] et al. [20] has been widely developed, and it shows a robust capacity in predicting the overconsolidated soils under monotonic and cyclic loading, such as saturated clays [21; 22; 23], unsaturated clays [24; 25], sands [26], and frozen soils [27]. The bounding surface plasticity model was also applied to solve multi-axial wave propagation problems [28] and boundary value problems [29].  
However, predicting the non-unique critical state of soils is still a challenge, because the predicted distance between CSL and ICL using classical constitutive models does not vary with soil types. Lists of experimental results indicate that the distance varies with soil types, especially for gap-graded soils [5; 6]. To solve the problem, a material parameter \"spacing ratio \(r\)\" was improved to reflect the distance between CSL and ACL (see Figure 1) [30], and a structural bounding surface plasticity model with spacing ratio \(r\) (SBSP-R model) was formulated [31]. The SBSP-R model well captures the non-unique critical state of soils because of the spacing ratio \(r\) (see Figure 1).  
Furthermore, the algorithm of the SBSP-R model is essential in capturing the non-unique critical state of soils. The SBSP-R model was once implemented using an explicit algorithm to predict the non-unique critical state and strain-softening of clays, and the limitations of the explicit algorithm are obvious: (a) it even does not capture the non-unique critical state of soils with large spacing ratio \(r\); (b) it consumes much computation time in predicting critical state at the intersection of the distorted bounding surface and CSL (see Figure 1b). So, it is necessary to formulate an algorithm for the SBSP-R model to predict the non-unique critical state of soils. Fortunately, the implicit algorithm has been proven to be an efficient method to eliminate the limitations [32]. The implicit algorithm of the SBSP-R model may be an alternative approach to the previous explicit algorithm.  
This paper aims to formulate the implicit algorithm of the SBSP-R model to efficiently predict the non-unique critical state of soils with larger spacing ratios. The implicit algorithm mainly undergoes elastic prediction and plastic correction. The plastic correction is conducted using a Newton-Simpson scheme with a list of controlling equations related to consistency condition, plastic flow, hardening parameter, bounding surface equation, plastic modulus, and mapping rule. The implicit algorithm is analyzed via case studies and is further verified by comparisons between calculated results and experimental results of triaxial compression tests on soils including Sandy clay, Kaolin clay, and Intact clayey loess.