Elastoplastic analysis of rock and soil masses based on smooth finite element method
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摘要:
随着大型工程项目的日益增多，工程中的岩土极限问题也频繁出现，致使采用数值方法处理时，经常面对极端的模型变形问题。传统有限元法在处理极端的模型变形，特别是采用低阶单元分析时，往往会产生收敛问题、体积锁定问题以及网格严重畸变而导致的应力失准问题。寻找一种新的方法进行数值分析具有重要意义。光滑有限元法是一种有效改进传统有限元法固有缺陷，提高求解精度和收敛速度的算法。基于光滑有限元法，结合一种修正的Mohr-Coulomb屈服准则和线性搜索优化算法，建立了岩土体弹塑性计算模型。最后，针对经典的条形基础承载力模型和边坡模型进行验算，数值计算结果与参考解均吻合较好。结果表明光滑有限元法的计算精度明显优于传统有限元法，验证了本研究算法的可行性和实用性。本研究采用光滑有限元法建立的计算模型进一步提高了岩土体弹塑性问题的计算精度，降低了采用传统有限元法分析时存在的计算误差及网格畸变引起的应力失准问题。
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关键词:
- 光滑有限元法 /
- 弹塑性 /
- 岩土体 /
- 修正Mohr-Coulomb屈服准则 /
- 线性搜索
Abstract:Objective With the increasing number of large engineering projects, geotechnical limit problems are becoming more common, often leading to extreme model deformation when numerical methods are employed. The traditional finite element method frequently encounters convergence issues, volume locking, and stress misalignment due to severe mesh distortion during the analysis of extreme model deformation, especially when low-order elements are used. Therefore, developing a new method for numerical analysis is of great importance.
Methods The smooth finite element method is an effective approach to address the inherent defects of the traditional finite element method, enhancing both solution accuracy and convergence speed. Thus, based on the smooth finite element method combined with a modified Mohr-Coulomb yield criterion and a linear search optimization algorithm, an elastoplastic calculation model for rock and soil masses is developed in this study.
Results The classical bearing capacity model for the strip foundation and slope model was tested, and the numerical results align well with the reference solutions. The findings indicate that the calculation accuracy of the smooth finite element method is clearly superior to that of the traditional finite element method, confirming the feasibility and practicality of the proposed algorithm.
Conclusion In this work, the calculation model developed using the smooth finite element method significantly improves the calculation accuracy for rock and soil elastoplastic problems, while reducing the calculation error and stress misalignment caused by mesh distortion in traditional finite element methods.
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图 1 边光滑有限元法光滑域划分示意图
Figure 1. Schematic diagram of smooth domain division for the ES-FEM
图 2 修正Mohr-Coulomb屈服准则在子午面内的拟合形式
$ \bar \sigma $为等效应力；$\sigma_{{\mathrm{m}}} $为平均应力；m为参数
Figure 2. Fitting form of the modified Mohr-Coulomb yield criterion in the meridional plane
图 3 修正M-C准则在$ \pi $平面上的屈服曲线
$ \theta $为洛德角；$ \sigma_1 $，$\sigma_2 $，$\sigma_3 $均为主应力；m为参数
Figure 3. Yield curve of the modified M-C criterion in the $ \pi $-plane
图 5 不同数值方法下压力系数与沉降中心的关系
Figure 5. Relationship between the pressure coefficient and center settlement under different numerical methods
图 6 不同数值方法下的塑性应变云图
Figure 6. Plastic strain contour maps under different numerical methods
图 7 不同数值方法下承载力误差与自由度的关系
Figure 7. Relationship between bearing capacity errors and degrees of freedom under different numerical methods
图 9 边坡最大位移与强度折减系数的关系
Figure 9. Relationship between the maximum displacement of the slope and the strength reduction coefficient
图 10 强度折减系数$ S R F = 1.6 $时的等效塑性应变分布
Figure 10. Distribution of the equivalent plastic strain with $S R F = 1.6$
表 1 ES-FEM和FEM在不同自由度下的压力系数及误差
Table 1. Pressure coefficients and errors of ES-FEM and FEM under different degrees of freedom
自由度
DOFFEM ES-FEM 压力系数${L_F}$ 误差/% 压力系数${L_F}$ 误差/% 578 17.74 19.55 15.10 1.81 882 17.25 16.28 15.00 1.08 1250 16.89 13.84 14.94 0.72 1682 16.61 11.97 14.91 0.51 2178 16.39 10.50 14.89 0.40 2738 16.22 9.31 14.88 0.31 3362 16.07 8.33 14.87 0.24 
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