## 5 Conclusions  
This paper introduces a viscoelastic finite element algorithm that simultaneously considers the relaxation behavior of elastic modulus and the creep Poisson's ratio in viscoelastic materials. The relaxation behavior of the elastic modulus is modeled using the generalized Maxwell model, while the creep Poisson's ratio is represented using the generalized Kelvin model. Starting from the integral constitutive relationship and incorporating the principle of virtual displacement, a convolution-based variable stiffness method for solving equations is proposed. The paper also presents recursive calculations for convolution-based numerical integration. Based on this, the calculation steps of the proposed algorithm are outlined. The main conclusions of this paper can be summarized as follows:  
1. In contrast to the initial strain method derived from the differential constitutive relationship, this algorithm can simultaneously account for the variation in the elastic modulus and Poisson's ratio over time, thus establishing a more precise simulation pathway and a novel perspective for modeling the mechanical properties of viscoelastic materials.
2. Compared to the initial strain method for linear viscoelastic materials, the proposed variable stiffness method considering the time-dependent Poisson's ratio effectively avoids the assumption of constant stress over small time intervals, thereby significantly enhancing the accuracy of the finite element calculations for linear viscoelastic materials.
3. Compared to the finite element method commonly used in commercial software, the proposed variable stiffness method considering the time-dependent Poisson's ratio, with the elastic modulus and Poisson's ratio as material parameters, establishes a simulation algorithm that aligns with experiments on viscoelastic materials.
4. The proposed variable stiffness method considering the time-dependent Poisson's ratio and its calculation process effectively fits the coefficients of the Prony series for the elastic modulus and Poisson's ratio directly based on experimental data, thereby reducing errors introduced during the transformation of material parameters when using commercial software. Future research will include additional experimental testing, convergence analysis, and investigation into computational efficiency.  
Conceptualization, X.W.; methodology, Y.W., J.G., J.B., X.W. and Z.Z.; software, J.B., C.L., Z.Z. and Y.W.; writing--original draft preparation, X.W., J.G., Y.W., J.B., Z.Z. and C.L.; writing--review and editing, X.W., J.G., Y.W., J.B., Z.Z. and C.L.; visualization, Z.Z. and Y.W.; supervision, Z.Z.; funding acquisition, X.W. and Y.W. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Stable Support Project of National Key Laboratory of Solid Rocket Propulsion (grant number SY41 YYF202309043), National Natural Science Foundation of China (grant number 52108387); China Postdoctoral Science Foundation (grant number 2022M713851), and Project of Central Guiding Local Science and Technology Development Funds (Fundamental Research Project: 216Z5402G). All support is gratefully acknowledged.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.