## 4 Conclusions  
This paper delves into the variations in CPS and CDS in permafrost under long-term cyclic loading, with respect to temperature and moisture content, through a series of dynamic triaxial tests. Cumulative permanent deformation under the influence of different temperatures and water contents, the calculation criterion of plasticity-stabilized creep limit as well as the calculation and influence analysis of CDS are presented in this paper. The following conclusions can be drawn from the results of this paper:  
1. The cumulative axial strain of frozen soil under cyclic loading is notably influenced by alterations in temperature and moisture content. Notably, under varying temperatures and water contents, the cumulative strain rate in Region A undergoes a swift decline with an increasing number of loading cycles, ultimately stabilizing at zero. Similarly, Region B experiences a rapid decrease in cumulative strain rate with more cycles. In Region C, while a general decrease is observed, certain curves exhibit a rebound trend, highlighting the complex interplay of temperature, moisture, and cyclic loading on the strain behavior of frozen soil.
2. Drawing upon the vertical CPS associated with the 100 th and 1000 th loading cycles, a definitive plastic stability and creep limit criterion is introduced for frozen silty clay. This criterion clearly establishes the plastic stability limit as the boundary between areas A and B, marked by a 0.8% difference, and the plastic creep limit as the threshold separating areas B and C, characterized by a 5.2% discrepancy. Employing the plastic limit criterion formula, we conclusively determine the CDS \(\sigma_{\text{d}}\) at both limits under varying stress levels, enabling a thorough analysis of how different temperatures and moisture contents impact this critical stress.
3. As temperature decreases, the CDS \(\sigma_{\text{d,s}}\) under the plasticity-stabilized creep limit exhibits a gradual upward trend in general. Assuming all other factors remain constant, lower temperatures lead to an enhancement in the strength and deformation resistance of frozen powdery clay, subsequently elevating the CDS.
4. As water content rises, the CDS \(\sigma_{\text{d,s}}\) under the plasticity-stabilized creep limit experiences an overall slow decline, with a slight recovery observed at a water content of 18.5%. At lower water contents, the decrease in CDS is gradual. Conversely, at higher water contents, the significant increase in the proportion of ice particles and the enhancement of their cementation result in markedly improved deformation resistance, thereby leading to an increase in the CDS.  
In the future, further research would consider the influence of soil type, particle size, loading frequency, and reinforced materials on the critical dynamic stress and plasticity calculation criteria.  
Conceptualization, J.W.; Methodology, J.D.; Project administration, J.W.; Validation, J.W. and J.D.; Investigation, J.D.; Writing--original draft preparation, J.D. and J.W.; Writing--review and editing, J.W. and J.D.; Funding acquisition, J.W. and Y.Z.; Supervision, Y.Z.;Data curation, Y.Z.; Formal analysis, Y.Z. and J.D. All authors have read and agreed to the published version of the manuscript.  
The research was partly supported by the National Natural Science Foundation of China (42101127) and the China Postdoctoral Science Foundation (2020M681621), the Fund Project for Central Government Guided Local Science and Technology Development of China (2022 ZY0052).  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare that there are no conflicts of interest regarding the publication of this paper.