# Application of Dimension Extending Technique to Unified Hardening Model
## 1 Introduction  
The modified Cam-Clay model (MCC) [1] is a widely used elasto-plastic constitutive model [2], which can reflect the friction, compression and shear shrinkage of soil [3]. However, it can only reflect the characteristics of normal consolidated or light over-consolidated clay, and cannot reasonably reflect the dilatancy of heavy over-consolidated clay [4]. Based on critical state theory, various constitutive models have been proposed [5; 6; 7]. The unified hardening (UH) model is one of them. The UH model is proposed by [PERSON] et al. in the framework of the MCC model and combined with the concept of the sub-loading surface model [8], which can reasonably reflect the hardening, softening, dilatancy, and shrinkage characteristics of soil. The constitutive equation of the elasto-plastic model is generally expressed in the rate form [9]. The numerical algorithm for updating the stress of the elasto-plastic constitutive model for geotechnical materials requires integration of the rate form constitutive equation along the strain increment path within a finite time step. This is also one of the core issues in elasto-plastic numerical calculations, which directly affects the accuracy of the calculation results [10]. The common numerical algorithms of stress updating can be roughly divided into two kinds: the explicit algorithm (the forward Euler method)and the implicit algorithm (the backward Euler method). Using an explicit algorithm, if the entire strain increment includes both elastic and elastic-plastic behavior, it is necessary to find the intersection point of the stress increment vector and the yield surface [11]. The gradients of the yield surface and plastic potential surface are calculated based on the stress state at the starting point of the incremental step. The procedure is easy to implement, but the computational accuracy is low, and the solution is prone to drift from the yield surface. Implicit algorithms generally include two parts: elastic prediction and plastic correction, which require obtaining the gradients of yield and plastic potential surfaces based on unknown stresses. Therefore, an iterative method is needed. Although iterative calculation improves computational accuracy, it also increases the difficulty of derivation. The closest point projection method (CPPM) [12] is currently a widely used implicit algorithm with good computational stability. However, the Hessian matrix is needed in the algorithm. For some complex constitutive models or those that introduce stress Lode angle, the theoretical derivation process is relatively cumbersome, especially for highly non-linear constitutive models, which are prone to the singularity of the Jacobean matrix and non-convergence during calculation. [PERSON] et al. [13] improved the initial iteration values of the algorithm using the auxiliary projection surface method, but how to effectively construct the auxiliary projection surface has not yet been well addressed. To avoid calculating the Hessian matrix, [PERSON] et al. [14; 15] proposed the cutting-plane algorithm (CPA). Although the simplicity of this algorithm is highly attractive for large-scale calculation, the precise linearization cannot be obtained in a closed form [16]. Regarding the comparison results of these two algorithms from theory to numerical computation, [PERSON] et al. [17] have conducted a more detailed analysis. The rate form elasto-plastic constitutive model defines an initial value problem of an ordinary differential equations (ODEs) about the stress tensor \(\sigma\) changing in the elastic domain \(E_{\alpha}\). For the perfect-plastic model, \(E_{\alpha}\) is invariant in the stress space. The Kuhn-Tucker complementarity condition and the consistency condition provide constraints for solving the ODEs. Based on the above analysis, the constitutive integration of the perfect-plastic model can be reduced to a mixed complementarity problem (MiCP), which is also a finite-dimensional variational inequality (_VI_).