基于对数应变的孔隙黏弹性有限变形理论

doi:10.19596/j.cnki.1001-246x.8723

摘要/Abstract

摘要：

在有限变形理论框架下, 基于对数应变和Kelvin流变模型提出一个孔隙黏弹性理论模型。该模型通过假设Kirchhoff应力和孔隙压力与对数应变和Lagrangian孔隙度的变化量存在线性关系, 将线性孔隙黏弹性模型中的无穷小应变直接替换为对数应变而得到。作为验证, 将该理论模型用于研究经典的Terzaghi一维固结问题, 通过与孔隙弹性有限变形模型的数值结果对比, 结果显示: 在固结早期, 孔隙固体骨架的黏弹性响应和弹性响应曲线几乎是一致的, 但随着时间的推移, 孔隙固体骨架的黏性逐渐主导骨架的变形, 并影响固结的最终结果; 并且, 骨架的黏性会延迟孔隙压力的扩散。此外, 通过设置黏性贡献系数ζ=0.001, 从数值角度实现了孔隙黏弹性响应"退化"为孔隙弹性响应, 这在一定程度上验证了模型的正确性。

关键词: 孔隙黏弹性, 对数应变, 有限变形, 有限元, 流变模型

Abstract:

In the framework of finite deformation theory, a theoretical model of poroviscoelasticity is proposed which is based on logarithmic strain and Kelvin rheological model. The model is obtained by assuming a linear relationship between Kirchhoff stress and pore pressure and logarithmic strain and the variation of Lagrangian porosity, and then directly replacing the infinitesimal strain in the linear pore viscoelastic model with the logarithmic strain. As a verification, the theoretical model is used to study the classic Terzaghi's one-dimensional consolidation problem. By comparing with the numerical results of the poroelastic finite deformation model, the results show that the viscoelastic response and elastic response curves of the pore solid skeleton are almost identical in the early stage of consolidation, but with the passage of time, the viscous response of the pore solid skeleton gradually dominates the deformation of the skeleton and affects the final result of consolidation. In addition, the viscous response of the skeleton delays the diffusion of pore pressure. In addition, by setting the viscosity contribution coefficient ζ=0.001, the poroviscoelastic response is numerically "degraded" to the poroelastic response, which verifies the correctness of the model to a certain extent.

Key words: poroviscoelasticity, logarithmic strain, finite deformation, finite elements, rheological model

中图分类号:

O345

唐熊, 郑佩. 基于对数应变的孔隙黏弹性有限变形理论[J]. 计算物理, 2024, 41(3): 287-297.

Xiong TANG, Pei ZHENG. Finite Deformation Theory of Poroviscoelasticity Based on Logarithmic Strain[J]. Chinese Journal of Computational Physics, 2024, 41(3): 287-297.

图/表 5

图1 一维广义Kelvin流变模型

Fig.1 One-dimension generalized Kelvin rheological model

图2 孔隙弹性与黏弹性材料的响应曲线(a) 表面沉降曲线; (b) 孔隙压力剖面图

Fig.2 Response curves of poroelastic material and poroviscoelastic material (a) surface settlement curves; (b) pore pressure profiles

图3 不同ζ值对应的孔隙黏弹性响应曲线(a) 表面沉降曲线；(b) 孔隙压力剖面图

Fig.3 Poroviscoelastic response curves with different ratios of ζ (a) surface settlement curves; (b) pore pressure profiles

图4 不同ζ值对应的孔隙弹性和黏性对数应变的演化曲线(a) 黏性对数应变主分量θv；(b) 黏性对数应变偏分量的主伸长$\lambda_1^{\overline{\boldsymbol{h}}_{\mathrm{v}}}$; (c) 弹性对数应变主分量θe；(d) 弹性对数应变偏分量的主伸长$\lambda_1^{\overline{\boldsymbol{h}}_{\mathrm{e}}}$

Fig.4 Evolution curves of logarithmic strain of viscous and elastic part with different ratios ofζ(a) viscous volumetric component of logarithmic strainθv; (b) stretch of viscous deviatoric component of logarithmic strain $\lambda_1^{\overline{\boldsymbol{h}}_{\mathrm{v}}}$; (c) elastic volumetric component of logarithmic strainθe; (d) stretch of elastic deviatoric component of logarithmic strain $\lambda_1^{\overline{\boldsymbol{h}}_{\mathrm{e}}}$

图5 孔隙弹性与黏弹性材料的响应曲线(a) 表面沉降曲线; (b) 孔隙压力剖面图

Fig.5 Response curves of poroelastic material and poroviscoelastic material (a) surface settlement curves; (b) pore pressure profiles

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