# A Modified Plastic Burgers Model for the Non-Decaying Creep of Frozen Soils
## 2 Modified Plastic Burgers Model  
In the plastic Burgers model [29], total strain is decomposed into three components, i.e., Maxwell (\(\varepsilon_{ij}^{\text{M}}\)), Kelvin (\(\varepsilon_{ij}^{\text{K}}\)) and plastic (\(\varepsilon_{ij}^{p}\)) strains, as shown with Formula (1). \[\varepsilon_{ij}=\varepsilon_{ij}^{M}+\varepsilon_{ij}^{K}+\varepsilon_{ij}^{p} \tag{1}\]  
In Formula (1), the Maxwell strain (Formula (2)) is used to describe the instantaneous strain and stable creep stage with a constant strain rate on the typical frozen soil creep curve (Figure 1); the Kelvin strain (Formula (3)) is used for describing the creep strain decrease process; and the plastic strain (Formula (4)) is employed when plastic flow is involved. \[\varepsilon_{ij}^{M}=\frac{\sigma_{ij}}{2G_{M}}+\frac{\sigma_{ij}}{2H_{M}}t \tag{2}\]  
\[\varepsilon_{ij}^{K}=\frac{\sigma_{ij}}{2G_{\text{K}}}\bigg{[}1-\exp\biggl{(} -\frac{G_{\text{K}}}{H_{\text{K}}}t\biggr{)}\bigg{]} \tag{3}\]  
\[\varepsilon_{ij}^{p}=\int_{0}^{t}\varepsilon_{ij}^{p}dt=\int_{0}^{t}(\lambda \frac{\partial g}{\partial\sigma_{ij}}-\frac{1}{3}\dot{\varepsilon_{p}^{p}} \delta_{ij})dt \tag{4}\]  
\[\dot{\varepsilon_{v}^{p}}=\lambda(\frac{\partial g}{\partial\sigma_{1}}+ \frac{\partial g}{\partial\sigma_{2}}+\frac{\partial g}{\partial\sigma_{3}}) \tag{5}\]  
In the above equations, \(G_{\text{M}}\) and \(G_{\text{K}}\) are the elastic shear modulus for the Maxwell and Kelvin strains, respectively; \(H_{\text{M}}\) and \(H_{\text{K}}\) are the viscosity coefficients of the Maxwell and Kelvin strains, respectively; and \(g\) is the plastic potential function. It is expressed as  
\[g=\sigma_{1}-\sigma_{3}(1+\sin\psi)(1-\sin\psi) \tag{6}\]  
where \(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\) are the minimum, intermediate and maximum principal stresses, respectively; \(\psi\) is the dilation angle; \(\lambda\) is equal to zero when the material is under an elastic stress state; and when material plastic flow is taking place, \(\lambda\) is a non-zero parameter, which is determined by the application of the plastic yield condition \(f=0\)[29]. Here, the Mohr-Coulomb yield criteria are employed, and the plastic yield function \(f\) is expressed as  
\[f=\sigma_{1}-\sigma_{3}N_{\phi}+2C\sqrt{N}_{\phi} \tag{7}\]  
where \(N_{\phi}=(1+\sin\phi)/(1-\sin\phi)\), \(\phi\) is the internal friction angle and \(C\) is the cohesion. With Formulas (1) to (7), the framework of the plastic Burgers model is introduced in brief. Generally, according to the previous creep model-evaluating works [30], the conventional plastic theory-based creep models are capable of describing a decaying or  
Figure 1: Typical creep curve of the frozen soils [1]. non-decaying creep process separately. However, such creep models cannot capture the developing process from the decaying to non-decaying creep stages, as shown in Figure 1. This is due to the fixed plastic yield envelope being employed in the conventional plastic models, where the shear strength parameters are constant for a certain kind of soil. For frozen soil, it has been proven by a large amount of experimental work that the cohesion is decaying with time under constant overloading, while the internal friction angle is independent of time [1; 22]. Subsequently, the plastic yield envelope moves downward with time in the principal stress coordinates (Figure 2). For an initial stress point lower than the plastic yield envelope, the material is under the elastic stress state, and only the decaying creep process happens; then, with ongoing time, the plastic yield envelope approaches the stress point, gets lower than it (Figure 2), plastic flow gets involved, the plastic strain rate increases sharply and the non-decaying creep process begins (Figure 1). This implies that when the decay of the shear strength parameter of frozen soil is embedded into the plastic Burgers model, the successive developing process from the decaying to non-decaying creep stages can be reasonably described.