基于Biot形式波动方程的一维非饱和土柱瞬态响应研究

刘砚山; 吉姿洁; 赵云

doi:10.21656/1000-0887.460155

基于Biot形式波动方程的一维非饱和土柱瞬态响应研究

doi: 10.21656/1000-0887.460155

刘砚山^1,,
吉姿洁²,
赵云^2, ,

1.
河南省城乡建筑设计院有限公司，郑州 451452
2.
河南工业大学土木工程学院，郑州 450001

基金项目:

国家自然科学基金 52578399

河南省科技攻关计划 252102320021

河南工业大学拓新团队培育项目 2024TXTD15

河南省高等学校重点科研项目 26B410001

详细信息

作者简介:
刘砚山(1989—)，男，高级工程师(E-mail: yanshan_8903@163.com)

通讯作者:
赵云(1989—)，男，博士(通信作者. E-mail: zhaoyun1106@163.com)

中图分类号: TU435
计量
- 文章访问数: 34
- HTML全文浏览量: 11
- PDF下载量: 8
- 被引次数: 0
出版历程
- 收稿日期: 2025-08-28
- 修回日期: 2026-06-23
- 刊出日期: 2026-07-01

Transient Responses of 1D Unsaturated Soil Columns Based on the Biot-Type Wave Equations

Liu Yanshan^1
,,
Ji Zijie²,
Zhao Yun^{2
, ,}

1.
Urban & Rural Construction and Designing Institute of Henan Province Co., Ltd., Zhengzhou 451452, P.R. China
2.
College of Civil Engineering and Architecture, Henan University of Technology, Zhengzhou 450001, P.R. China

摘要

摘要: 基于Biot形式的非饱和土波动方程，本文研究了一维非饱和土柱在动力荷载作用下的瞬态响应问题.目的在于分析非饱和土中的波动特性，并推导该问题的解析解，为相关工程问题提供理论依据.在方法上，考虑饱和度对渗透系数和动剪切模量的影响，采用分离变量法和状态空间法相结合的方式系统求解了一维非饱和土柱在动载下的瞬态响应问题解析解，并通过与Comsol PDE模块的有限元数值结果对比，验证了解答的正确性.结果显示：在非饱和土中，由于渗透系数较低，仅能观测到一种压缩波；随着固有渗透系数的增大，孔隙水压力的瞬态响应峰值逐渐减小；同时，饱和度增加会导致孔隙水压力响应幅值明显上升.结论认为，饱和度与渗透系数对非饱和土中的波动响应具有显著影响，解析解能够有效预测瞬态响应特征，并为非饱和土动力学行为分析提供理论支持.
- 瞬态响应 /
- 非饱和土 /
- 解析解 /
- 状态空间法
Abstract: Based on the Biot-type wave equations for unsaturated soils, the transient responses of 1D unsaturated soil columns subjected to dynamic loading were investigated, to analyze wave propagation characteristics in unsaturated soils and derive analytical solutions to provide a theoretical basis for relevant engineering problems. The effects of saturation on both the permeability coefficient and the dynamic shear modulus were incorporated into the governing equations. With the separation of variables method plus the state space method, a systematic analytical solution of the transient response was obtained. The validity of the solution was verified through comparison with the finite element results from COMSOL's PDE module. The results indicate that, only 1 compression wave occurs in unsaturated soil due to its relatively low permeability. As the intrinsic permeability increases, the peak transient pore water pressure response will gradually decreases. Furthermore, an increase in the saturation leads to a significant rise in the amplitude of the pore water pressure response. Both the saturation and the permeability considerably influence the wave-induced responses in unsaturated soils. The proposed analytical method effectively predicts transient response characteristics and provides a theoretical support for analyzing the dynamic behaviors of unsaturated soils.
- transient response /
- unsaturated soil /
- analytical solution /
- state-space method

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图 1 一维非饱和土柱计算模型

Figure 1. The analytical model of the 1D unsaturated soil

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图 2 与有限元方法对比

Figure 2. Comparison with the finite element method

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图 3 与退化解对比

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 3. Comparison with the saturated soil solution

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图 4 不同渗透系数下孔压时程曲线图

Figure 4. Time histories of water pore pressures with different permeabilities

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图 5 放大渗透系数后孔压时程曲线图

Figure 5. Time histories of water pore pressures after magnifying the permeability coefficients

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图 6 不同饱和度下孔压时程曲线图

Figure 6. Time histories of water pore pressures with different saturations

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表 1 材料参数

Table 1. Values of parameters

phase	parameter	unit	value
air	K_n	kPa	145
	η_n	N·s/m²	1.8×10^-5
	ρ_n	kg/m³	1.28
water	K_w	GPa	2.25
	η_w	N·s/m²	0.001
	ρ_w	kg/m³	1 000
solid	K_s	GPa	35
solid	ρ_s	kg/m³	2 650
skeleton	K_b	MPa	8.33
	μ_s	MPa	3.85
	n	-	0.45
	S_r	-	0.2~1.0
	S_w0	-	0.05
	κ	m²	5.3×10^-1⁰
	ϕ′	(°)	10
	a	Pa^-1	1.0×10^-4
	m	-	0.5
	d	-	2

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