郭颖^,*,2), 李文杰^*, 马建军^*, 梁斌^,*,3), 熊春宝^†*河南科技大学土木工程学院, 河南洛阳 471023
^†天津大学建筑工程学院, 天津 300072

DYNAMIC COUPLED THERMO-HYDRO-MECHANICAL PROBLEM FOR SATURATED POROUS VISCOELASTIC FOUNDATION¹⁾

Guo Ying^,*,2), Li Wenjie^*, Ma Jianjun^*, Liang Bin^,*,3), Xiong Chunbao^{† *}School of Civil Engineering, Henan University of Science and Technology, Luoyang 471023, Henan, China
^†School of Civil Engineering, Tianjin University, Tianjin 300072, China

通讯作者: 2)郭颖, 讲师, 主要研究方向: 介质的多场耦合动力响应. E-mail:gytha_ying@163.com;3)梁斌, 教授, 主要研究方向: 板壳理论. E-mail:liangbin4231@163.com

收稿日期:2020-11-12接受日期:2021-03-21网络出版日期:2021-04-08

基金资助:

1)国家自然科学基金资助项目.11502072

Received:2020-11-12Accepted:2021-03-21Online:2021-04-08
作者简介 About authors


摘要
天然土体由于沉积条件和应力状态不同, 往往会表现出一定的流变性. 本文研究地基上表面受外载荷作用时, 渗透系数和孔隙率变化对饱和多孔黏弹性地基热-水-力耦合动力响应问题的影响. 基于Biot波动方程、达西定律和Lord-Shulman广义热弹性理论, 并引入了考虑黏弹性松弛时间因子的Kelvin-Voigt黏弹性模型研究地基上表面受热/力源作用时, 孔隙率和渗透系数变化对均质各向同性饱和多孔黏弹性地基中所考虑的各无量纲量的影响. 根据不同的边界条件采用正则模态法推导出无量纲竖向位移、超孔隙水压力、竖向应力和温度的解析表达式, 结合算例分析了不同变量对各物理量的影响. 正则模态法是一种加权残差法, 可不经正、反积分变换将方程快速解耦并消除数值反变换的局限性. 结果表明: 无论何种载荷作用时, 载荷频率变化对所有考虑的物理量均有明显的影响; 孔隙率和渗透系数均对无量纲超孔隙水压力有明显的影响, 当仅考虑热载荷作用时, 孔隙率和渗透系数变化对无量纲温度均无影响. 正则模态法可广泛应用于岩土工程领域, 尤其适用于商业建筑、高速铁路和公路能源基础的热、力学特性研究中. 该研究结果可为工程施工奠定一定的理论基础, 具有一定的指导性意义.
关键词： Lord-Shulman广义热弹性理论;黏弹性松弛时间;热-水-力耦合黏弹性模型;正则模态法

Abstract
Natural soil often has the characteristics of rheology due to different depositional conditions and stress states. The present paper focuses on investigated the effects of different porosity and permeability coefficient in saturated porous foundation which considered the viscoelastic relaxation times with coupled thermo-hydro-mechanical fields under external load. A two-dimensional coupled thermo-hydro-mechanical dynamics problem for a half-space on an isotropic, uniform, fully saturated, and poroviscoelastic soil (THMVD) whose surface is subjected to either mechanical force or thermal load based on the Biot's wave theory of porous media, Darcy's law, and Lord-Shulman (L-S) generalized thermoelastic theory with Kelvin-Voigt viscoelastic model is investigated. The general relationships among the non-dimensional vertical displacement, excess pore water pressure, vertical stress, and temperature distribution are then deduced via normal mode analysis and depicted graphically. Normal mode analysis is a method using weighted residuals to derive analytical solutions. Via this method, the equation can be divided into two parts without integral transformation and inverse transformation, thereby increasing the speed of decoupling and eliminating the limitation of numerical inverse transformation. The effects of the porosity and the permeability coefficient on the four different physical variables have been investigated. It can be shown that: whatever load is being considered, the variation of load frequencies have obvious effect on all the considered physical variables; the porosity and permeability have the most obvious influence on non-dimension excess pore water pressure. When thermal loads were considered only, the variation of porosity and permeability coefficient had barly effect on non-dimension temperature. This proposed derivation method can be widely applied in the geotechnical engineering field, especially with regard to the mechanical and thermal behaviors of commercial buildings, high-speed railways, and highway energy foundations. The research results of this problem can lay a certain theoretical foundation for engineering construction and have a certain guiding significance.
Keywords：Lord-Shulman generalized thermoelastic theory;viscoelastic relaxation time;coupled thermo-hydro-mechanical viscoelastic dynamic;normal mode analysis


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本文引用格式
郭颖, 李文杰, 马建军, 梁斌, 熊春宝. 饱和多孔黏弹地基热-水-力耦合动力响应分析¹⁾. 力学学报[J], 2021, 53(4): 1081-1092 DOI:10.6052/0459-1879-20-385
Guo Ying, Li Wenjie, Ma Jianjun, Liang Bin, Xiong Chunbao. DYNAMIC COUPLED THERMO-HYDRO-MECHANICAL PROBLEM FOR SATURATED POROUS VISCOELASTIC FOUNDATION¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2021, 53(4): 1081-1092 DOI:10.6052/0459-1879-20-385

引言

1956年, Biot^[1]首次提出了耦合的热弹性理论, 该理论依旧认为热在介质中的传输速度是无限大的, 这与事实不符. 为了消除这一悖论****们纷纷提出了可解释热的波动效应的热传导理论以及相应的数学模型. 目前, ****们广泛应用的理论主要有: Lord-Shulman (L-S)^[2]广义热弹性理论、Green-Lindsay (G-L)^[3]广义热弹性理论以及Green-Naghdi (G-N)^[4-6]广义热弹性理论. 上述几种广义热弹性理论均能很好地可描述热的"次声效应". 除上述常用的广义热弹性理论外, 还有多种不同的热弹性理论, 适用于不同的环境、初始条件、边界条件等^[7-9]. Hetnarski和Ignaczak^[10]基于多种广义热弹性理论研究和探讨了热弹性问题解的唯一性.

王颖泽等^[11]借助于Laplace积分变换及柱函数的渐近性质, 推导了循环热冲击作用时温度场、位移场和应力场的渐近表达式. 许新和李世荣^[12]基于Euler-Bernoulli梁理论和单向耦合的热传导理论, 定量分析了材料梯度指数、频率阶数、几何尺寸以及边界条件对热弹性阻尼的影响, 随后, 李世荣团队^[13]又对Mindlin矩形微板的热弹性阻尼问题进行了研究. 李妍等^[14]基于L-S型广义热弹扩散理论, 建立了考虑材料记忆依赖效应和空间非局部效应非局部广义热弹扩散耦合理论. 胡克强等^[15]利用Hankel积分变换法得到了表面受到轴对称热载荷作用的半无限大介质力-电-磁-热耦合问题. 王现辉等^[16]采用一种改进的勒让德正交多项式和分数阶积分相结合的方法, 详细分析了热弹性波在板中传播的问题. 上述研究没有考虑孔隙水的作用, 国内****白冰^[17]基于饱和多孔介质理论推导了热-水-力全耦合方程, 对不同耦合项的物理意义进行了解释. 随后, ****们研究了外载荷作用下空腔球壳、地基和隧道等不同介质的热-水-力耦合问题^[18-20]. 熊春宝等^[21]研究地基上表面受温度载荷和机械载荷时, 孔隙率各向异性参数变化对热-水-力耦合下饱和多孔弹性地基的影响.

上述研究考虑了介质的热-力耦合和热-水-力耦合问题, 但均是在弹性介质中进行研究的, 不能反映土体的流变特性. 黏弹性是材料的一种本构属性, 一般情况下, 往往认为其不仅具有弹性固体的性质, 还具有某些黏性流体的性质^[22]. 黏弹性介质的固有属性使其不仅会像固体一样对突发外载荷有明显的瞬态响应, 还也会像黏性流体那样受到动载荷作用后出现滞后现象, 这种现象所引起的滞后时间是介于弹性固体和黏性流体之间的. 基于上述几种广义热弹性理论, 结合常用的黏弹性模型, ****们研究了一些关于黏弹性材料的热弹动力响应问题. 采用Laplace-Fourier双重变换的方法, Ezzat等^[23]推导了材料的二维热黏弹基本方程, 在已有方程的基础上Ezzat和El-Bary^[24]又结合Kirchhoff理论研究了热导率变换对无限长黏弹性中空圆柱的影响. 随后, Ezzat和El-Karamany^[25]采用Laplace正、反变换的方法推导出了考虑两个热松弛时间的广义热黏弹理论, 并证明了方程的唯一性. 何天虎和井绪明^[26-27]基于L-S理论和G-L理论研究了两端固定有限长杆和半无限长杆的热黏弹问题. 李吉伟和何天虎^[28]结合热松弛和应变松弛现象, 分析了黏弹性介质的压电热弹问题. Kar和Kanoria^[29]在广义热黏弹理论下, 研究了边界受到温度载荷作用的均质各向同性黏弹性球壳动力响应问题, 对比和分析了两种不同理论中考虑黏弹性与不考虑黏弹性时各物理量之间的差异. Andreea^[30]选择了适当的时间加权值解决了与空间线弹性材料理论相关的时空问题, 最终建立了圣维南类型的空间估计方程. 基于双温度理论, Othman和Abouelregal^[31]采用Laplace正、反变换法描述了边界受到非高斯激光脉冲作用的均质各向同性黏弹半无限大体的动力响应问题. Sherief和Allam^[32]研究了外表面受到轴对称温度载荷作用时黏弹性实心球体的热黏弹问题. Zenkour和Abouelregal^[33]采用正则模态法研究了考虑四种不同黏弹松弛系数, 受温度载荷作用时的三维黏弹性板的热黏弹问题. 康建宏和谭文长^[34]的通过线性稳定性理论, 分析计算了多孔介质几何形状、热边界条件等因素对黏弹性流体热对流失稳的临界Rayleigh数的影响. Iesan^[35]基于Kelvin-Voigt黏弹性模型结合广义Darcy定律推导了可描述二元混合黏弹性介质的理论方程, 并证明了其唯一性. Fernández和Masid^[36]从数值分析角度出发, 研究了一个多孔混合黏弹性材料的热黏弹问题, 最终得到了该问题的一维和二维通解. Elhagary^[37]采用边界积分法和Laplace积分变换法研究了瞬态广义热黏弹问题, 并证明了其互易性. 上述研究虽然分析了黏弹性介质的一系列性质, 但是同样没有考虑孔隙水的作用, 徐长节和马晓华^[38]利用Laplace正、反变换的方法研究了考虑土骨架的黏性及流体与固体之间的耦合作用的黏弹性准饱和土中空腔球壳的动力响应问题. 祝彦知等^[39]借助Fourier展开、Laplace和Hankel积分变换方法推导出了考虑土骨架黏弹性的横观各向同性饱和土体的动力解析解, 结果表明在进行横观各向同性饱和土体动力分析时, 考虑土骨架的黏弹性十分有必要.

通过上述已有研究发现, 目前考虑孔隙水的研究主要在弹性介质中进行, 而考虑黏弹性介质的动态响应时一般都是忽略孔隙水的影响, 这样的研究仅适合在弹性介质中, 而对于具有流变性的天然地基来说并不适用, 为了能够更好研究外载荷作用下孔隙水对饱和多孔黏弹性半无限大地基的影响, 本文在Lord-Shulman广义热弹性理论的基础上引入考虑了黏弹性松弛时间因子的Kelvin-Voigt黏弹性地基模型, 从而建立了可以描述黏弹性地基渗流场、温度场和应力场多物理场耦合的数学模型, 采用更适合描述波的传播特性的正则模态法分析了热源和力源作用于半无限大地基上表面时无量纲超孔隙水压力、竖向位移、竖向应力以及温度的变化规律, 着重分析了渗透系数和孔隙率变化对饱和多孔黏弹性地基中各物理的影响, 地基模型详见图1.

图1

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图1饱和多孔黏弹性地基热-水-力耦合问题模型示意图

Fig.1Schematic diagram of the coupled thermo-hydro-mechanical problem of saturated porous viscoelastic foundation

1 基本方程

本文引入Kelvin-Voigt黏弹性模型同时考虑了温度和渗流作用从而建立了饱和多孔黏弹性地基热-水-力耦合动力模型, 分析了饱和多孔黏弹性地基的热-水-力耦合问题. 该模型中所有物理量均可以表达为坐标$x$和$z$以及时间$t$的函数形式. $x$轴方向为波的传播方向, $z$轴为地基深度方向, 假设外载荷在$z\to \infty $完全消失(详见图1). 其基本控制方程如下:

热传导方程

(1)$\begin{eqnarray} q_{i} +\tau \dot{q}_{i} =-K_{ij} \theta_{,i} \end{eqnarray} $
本构方程

(2)$\begin{eqnarray} \sigma _{ij} =2G_{* } \varepsilon _{ij} +\left( {\lambda_{* } e-\beta_{1* } \theta -p} \right)\delta_{ij} \end{eqnarray} $
不计体力的运动方程

(3)$\begin{eqnarray} \sigma_{ij,j} =\rho\ddot{u}_{i} \end{eqnarray} $
结合Darcy定律的质量方程

(4)$\begin{eqnarray} b\left( {\alpha_{{u}} \dfrac{\partial \theta }{\partial t}-\dfrac{\partial e}{\partial t}} \right)+\rho_{{w}} \dfrac{\partial^{2}e}{\partial t^{2}}+p_{,ii} =0 \end{eqnarray} $
几何方程

(5)$\begin{eqnarray} \varepsilon_{ij} =\dfrac{1}{2}\left(u_{i,j} +u_{j,i} \right) \end{eqnarray} $
根据Biot^[40]三相固结理论, 假设地基中的固体颗粒不可压缩. 并假设地基的固相和流相达到了热平衡从而产生小变形. 根据方程(2)和(3), 可以得到关于该黏弹性地基热-水-力耦合问题的动力平衡方程为

(6)$\begin{eqnarray} G_{* } u_{i,jj} +\left( {\lambda_{* } +G_{* } } \right)u_{j,ij} -\beta_{1* } \theta_{,i} -p_{,i} =\rho \ddot{{u}}_{i} \end{eqnarray} $
能量方程为

(7)$\begin{eqnarray} m\left( {\dfrac{\partial }{\partial t}+\tau \dfrac{\partial^{2}}{\partial t^{2}}} \right)\theta +\beta_{1* } T_{0} \left( {\dfrac{\partial }{\partial t}+\tau \dfrac{\partial^{2}}{\partial t^{2}}} \right)e=K\theta_{,ii} \end{eqnarray} $
式中, $q_{i} $为热流矢量分量, $T$为绝对温度, $T_{0} $为初始温度, $\theta =T-T_{0} $为温度增量, $p$为超孔隙水压力, $\varepsilon_{ij} $为应变分量, $\delta_{ij} $为Kronecker记号, $m=n_{0} \rho_{{w}} c_{{w}} +\left( {1-n_{0} } \right)\rho_{{s}} c_{{s}} $, $n_{0} $为孔隙率, $\rho_{{w}} $为孔隙水密度, $c_{{w}} $为孔隙水的比热容, $\rho_{{s}} $为固体颗粒的密度, $c_{{s}} $为固体颗粒的比热容, $\tau $为热驰豫时间因子, $K$为热传导系数, ${\sigma }_{ij} $为应力分量, $\rho $为地基的密度, $u_{i} $为位移分量, $b={\rho_{{w}} g}/{k_{{d}} }$, $\lambda_{* } =\lambda _{{e}} \left( {1+\alpha_{0} {\partial }/{\partial t}} \right)$, $G_{* } =G_{{e}} \left( {1+\alpha_{1} {\partial }/{\partial t}} \right)$, $\lambda_{{e}} $和$G_{{e}} $为拉梅常数, $\alpha_{0} $和$\alpha_{1} $为黏弹松弛时间因子, $\beta_{1* } =\beta_{{1e}} \left( {1+\beta_{1} {\partial }/{\partial t}} \right)$, $\beta _{1{e}} =\left( {3\lambda_{{e}} +2G_{{e}} } \right)\alpha _{{s}} $, $\alpha_{{s}} $为固体颗粒的线性热膨胀系数, $\beta_{1} =\left( {3\lambda_{{e}} \alpha_{0} +2G_{{e}} \alpha_{1} } \right){\alpha_{{s}} }/{\beta_{{1e}} }$, $\alpha_{{u}} =n_{0} \alpha_{{w}} +\left( {1-n_{0} } \right)\alpha_{{s}} $, $\alpha_{{w}} $为孔隙水的线性热膨胀系数.

位移分量可以展开为

(8)$\begin{eqnarray} u_{x} =u\left( {x,z,t} \right),\ \ u_{y} =0,\ \ u_{z} =w\left( {x,z,t} \right) \end{eqnarray} $
应变分量可以展开为

(9)$\begin{eqnarray} \left.\begin{array}{l} e_{xx} =\dfrac{\partial u}{\partial x},\ \ e_{zz} =\dfrac{\partial w}{\partial z},\ \ e_{xz} =\dfrac{1}{2}\left( {\dfrac{\partial u}{\partial z}+\dfrac{\partial w}{\partial x}} \right) \\ e_{xy} =e_{yy} =e_{yz} =0 \\ \end{array}\right\} \end{eqnarray} $
式中, $e$为体应力, 可以展开为

(10)$\begin{eqnarray} e=\dfrac{\partial u}{\partial x}+\dfrac{\partial w}{\partial z} \end{eqnarray} $
将动力平衡方程(6)和本构方程(2)展开, 可以得到

(11)$G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)\nabla^{2}u+\left[\lambda_{{e}} \left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)+\right. \left. G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right) \right]\dfrac{\partial e}{\partial x}- \beta_{{1e}} \left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)\dfrac{\partial \theta }{\partial x}- \dfrac{\partial p}{\partial x}=\rho \dfrac{\partial^{2}u}{\partial t^{2}} $
(12)$G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)\nabla^{2}w+\left[ \lambda_{{e}} \left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)+\right. \left. G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)\right]\dfrac{\partial e}{\partial z}- \beta_{{1e}} \left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)\dfrac{\partial \theta }{\partial z}-\dfrac{\partial p}{\partial z}=\rho \dfrac{\partial^{2}w}{\partial t^{2}} $
(13)$\sigma_{xx} =2G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)\dfrac{\partial u}{\partial x}+\lambda_{{e}} \left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)e- \beta_{{1e}} \left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)\theta -p $
(14)$\sigma_{zz} =2G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)\dfrac{\partial w}{\partial z}+\lambda_{{e}} \left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)e- \beta_{{1e}} \left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)\theta -p $
(15)$\sigma_{xz} =G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)\left( {\dfrac{\partial u}{\partial z}+\dfrac{\partial w}{\partial x}} \right) $
式中, $\nabla^{2}={\partial^{2}}/{\partial x^{2}}+{\partial ^{2}}/{\partial z^{2}}$为一个二维的Laplace算子.

为了后续计算方便, 引入如下无量纲量^[41-42]

(16)$\begin{eqnarray} \left.\begin{array}{l} \left( {{x},{z}} \right)=\dfrac{\bar{{\omega }}}{c_{1} }\left( {x,z} \right),\ \ \left( {{u},{w}} \right)=\dfrac{\rho c_{1} \bar{{\omega }}}{\beta_{{1e}} T_{0} }\left( {u,w} \right)\\ {\theta }=\dfrac{\theta }{T_{0} },\ \ {p}=\dfrac{p}{\beta_{{1e}} T_{0} }\quad {\sigma }_{ij} =\dfrac{\sigma_{ij} }{\beta_{{1e}} T_{0} }\\ \left( {{t},{\tau },{\alpha }_{0} ,{\alpha }_{1} ,{\beta }_{1} } \right)=\bar{{\omega }}\left( {t,\tau ,\alpha_{0} ,\alpha_{1} ,\beta_{1} } \right) \\ \end{array}\right\} \end{eqnarray} $
式中, $\bar{{\omega }}={m\left( {\lambda_{{e}} +2G_{{e}} } \right)}/{\rho K}$, $c_{1}^{2} =({\lambda_{{e}} +2G_{{e}} })/{\rho }$.

根据方程(16), 方程(4), (7), (11) $\sim$ (15)可以写成如下形式

(17)$\left[ {\dfrac{\lambda_{{e}} \left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)}{G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)}+1} \right]\dfrac{\partial e}{\partial x}-\dfrac{\beta^{2}}{\left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)}\dfrac{\partial p}{\partial x} - \dfrac{\beta^{2}\left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)}{ {1+\alpha_{1} \dfrac{\partial }{\partial t}} }\dfrac{\partial \theta }{\partial x}+\nabla^{2}u=\dfrac{\beta^{2}}{ {1+\alpha_{1} \dfrac{\partial }{\partial t}} }\dfrac{\partial^{2}u}{\partial t^{2}} $
(18)$\left[ {\dfrac{\lambda_{{e}} \left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)}{G_{{e}} \left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)}+1} \right]\dfrac{\partial e}{\partial z}-\dfrac{\beta^{2}}{ {1+\alpha_{1} \dfrac{\partial }{\partial t}}}\dfrac{\partial p}{\partial z} - \dfrac{\beta^{2}\left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)}{ {1+\alpha_{1} \dfrac{\partial }{\partial t}} }\dfrac{\partial \theta }{\partial z}+\nabla^{2}w=\dfrac{\beta^{2}}{ {1+\alpha_{1} \dfrac{\partial }{\partial t}} }\dfrac{\partial^{2}w}{\partial t^{2}} $
(19)$\nabla^{2}\theta =\left( {\dfrac{\partial }{\partial t}+\tau \dfrac{\partial^{2}}{\partial t^{2}}} \right)\left[ {\theta +\varphi_{0} \left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)e} \right] $
(20)$\nabla^{2}p=\varphi_{1} \dfrac{\partial e}{\partial t}+\varphi_{2} \dfrac{\partial \theta }{\partial t}+\varphi_{3} \dfrac{\partial^{2}e}{\partial t^{2}} $
(21)$\sigma_{xx} =\dfrac{2\left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)}{\beta^{2}}\dfrac{\partial u}{\partial x}-\left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)\theta -p + {\left( {1-\dfrac{2}{\beta^{2}}} \right)\left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)}e $
(22)$\sigma_{zz} =\dfrac{2\left( {1+\alpha_{1} \dfrac{\partial }{\partial t}} \right)}{\beta^{2}}\dfrac{\partial w}{\partial z}-\left( {1+\beta_{1} \dfrac{\partial }{\partial t}} \right)\theta -p + {\left( {1-\dfrac{2}{\beta^{2}}} \right)\left( {1+\alpha_{0} \dfrac{\partial }{\partial t}} \right)} e$
(23)$\sigma_{xz} ={\dfrac{1+\alpha_{1} \dfrac{\partial }{\partial t}}{\beta^{2}}}\left( {\dfrac{\partial u}{\partial z}+\dfrac{\partial w}{\partial x}} \right) $
式中

$\begin{eqnarray*} &&\varphi_{0} =\dfrac{T_{0} \beta_{{1e}}^{2} }{m\left( {\lambda _{{e}} +2G_{{e}} } \right)},\ \ \varphi_{1} =\dfrac{b}{\eta \left( {\lambda_{{e}} +2G_{{e}} } \right)}\\ &&\varphi_{2} =-\dfrac{b\alpha_{{u}} }{\eta \beta_{{1e}} },\ \ \varphi_{3} =-\dfrac{\rho_{{w}} }{\rho },\ \ \beta^{2}=\dfrac{\lambda_{{e}} +2G_{{e}} }{G_{{e}} },\ \ \eta =\dfrac{m}{K} \end{eqnarray*}$

2 正则模态分析

所考虑的物理变量的解均可以按照以下正则模态的形式进行分解^[43]

(24)$\begin{eqnarray} && \left[ {u,w,e,\sigma_{ij} ,p} \right]\left( {x,z,t} \right)= \\&&\qquad \left[ u^{* }\left( z \right),w^{* }\left( z \right),e^{* }\left( z \right), \sigma_{ij}^{* }\left( z \right),p^{* }\left( z \right) \right]\cdot\\&&\qquad\exp \left( {\omega t+\mbox{i}ax} \right) \end{eqnarray} $
式中, 为频率(复时间常数), $\mbox{i}=\sqrt {-1} $为虚数单位, $a$为$x$方向的波数, $\;u^{* }\left( z \right)$, $w^{* }\left( z \right)$, $e^{* }\left( z \right)$, $\theta^{* }\left( z \right)$, $\sigma_{ij}^{* }\left( z \right)$和$p^{* }\left( z \right)$分别为所考虑的物理场的振幅.

根据方程(24), 方程(17), (18)以及(21) $\sim$ (23)可以变成如下形式

(25)$\nabla^{2}u+\left( {\dfrac{\lambda_{{e}} \left( {1+\alpha_{0} \omega } \right)}{G_{{e}} \left( {1+\alpha_{1} \omega } \right)}+1} \right)\dfrac{\partial e}{\partial x}-\dfrac{\beta^{2}}{1+\alpha_{1} \omega}\dfrac{\partial p}{\partial x} - \dfrac{\beta^{2}\left( {1+\beta_{1} \omega } \right)}{1+\alpha_{1} \omega}\dfrac{\partial \theta }{\partial x}=\dfrac{\beta^{2}}{1+\alpha_{1} \omega }\dfrac{\partial^{2}u}{\partial t^{2}} $
(26)$\nabla^{2}w+\left[ {\dfrac{\lambda_{{e}} \left( {1+\alpha_{0} \omega } \right)}{G_{{e}} \left( {1+\alpha_{1} \omega } \right)}+1} \right]\dfrac{\partial e}{\partial z}-\dfrac{\beta^{2}}{1+\alpha_{1} \omega}\dfrac{\partial p}{\partial z} - \dfrac{\beta^{2}\left( {1+\beta_{1} \omega } \right)}{1+\alpha_{1} \omega}\dfrac{\partial \theta }{\partial z}=\dfrac{\beta^{2}}{ 1+\alpha_{1} \omega}\dfrac{\partial^{2}w}{\partial t^{2}} $
(27)$\sigma_{xx} =\dfrac{2\left( {1+\alpha_{1} \omega } \right)}{\beta^{2}}\dfrac{\partial u}{\partial x}-\left( {1+\beta_{1} \omega } \right)\theta -p+ {\left( {1-\dfrac{2}{\beta^{2}}} \right)\left( {1+\alpha_{0} \omega } \right)}e $
(28)$\sigma_{zz} =\dfrac{2\left( {1+\alpha_{1} \omega } \right)}{\beta^{2}}\dfrac{\partial w}{\partial z}-\left( {1+\beta_{1} \omega } \right)\theta -p+ {\left( {1-\dfrac{2}{\beta^{2}}} \right)\left( {1+\alpha_{0} \omega } \right)}e $
(29)$\sigma_{xz} =\dfrac{1+\alpha_{1} \omega }{\beta^{2}}\left( {\dfrac{\partial u}{\partial z}+\dfrac{\partial w}{\partial x}} \right) $
在方程(25)和(26)中分别对$x$和$z$求微分得

(30)$\begin{eqnarray} &&\left[ {\left( {\dfrac{\lambda_{{e}} \left( {1+\alpha_{0} \omega } \right)}{G_{{e}} \left( {1+\alpha_{1} \omega } \right)}+2} \right)\nabla^{2}-\dfrac{\beta^{2}}{1+\alpha_{1} \omega}\dfrac{\partial^{2}}{\partial t^{2}}} \right]e -\\&&\dfrac{\beta^{2}\left( {1+\beta_{1} \omega } \right)}{1+\alpha_{1} \omega}\nabla^{2}\theta -\dfrac{\beta^{2}}{1+\alpha_{1} \omega}\nabla^{2}p=0 \end{eqnarray} $
在后面的计算中, 为了简便, 设以下关系式

(31)$\begin{eqnarray} \left.\begin{array}{l} \omega_{1} =\left( {1+\alpha_{0} \omega } \right)\quad \omega_{2} =\left( {1+\alpha_{1} \omega } \right) \\ \omega_{3} =\left( {1+\beta_{1} \omega } \right)\quad \omega_{4} =\omega \left( {1+\tau \omega } \right) \\ \end{array}\right\} \end{eqnarray} $
根据方程(31), 方程(19)、(20)和(30)可简化并得到关于$\theta^{* }\left( z \right)$、$p^{* }\left( z \right)$和$e^{* }\left( z \right)$的关系式

(32)$\left[ {\left( {\dfrac{\lambda_{{e}} \omega_{1} }{G_{{e}} \omega_{2} }+2} \right)\left( {{D}^{2}-a^{2}} \right)-\dfrac{\beta^{2}\omega^{2}}{\omega_{2} }} \right]e^{* }\left( z \right)= \dfrac{\beta^{2}\left( {{D}^{2}-a^{2}} \right)}{\omega_{2} }\left[ {\omega_{3} \theta^{* }\left( z \right)+p^{* }\left( z \right)} \right] $
(33)$\left[ {{D}^{2}-a^{2}-\omega_{4} } \right]\theta^{* }\left( z \right)=\varphi_{0} \omega_{3} \omega_{4} e^{* }\left( z \right) $
(34)$\left( {{D}^{2}-a^{2}} \right)p^{* }\left( z \right)=\left( {\varphi_{1} \omega +\varphi_{3} \omega^{2}} \right)e^{* }\left( z \right) + \varphi_{2} \omega \theta^{* }\left( z \right) $
式中, ${D}={{\mbox{d}}/{\mbox{d}z}}$.

根据方程(31), 方程(25) $\sim$ (29)可简化为

(35)$\nabla^{2}u+\left( {\dfrac{\lambda_{e} \omega_{1} }{G_{e} \omega_{2} }+1} \right)\dfrac{\partial e}{\partial x}-\dfrac{\beta^{2}\omega_{3} }{\omega_{2} }\dfrac{\partial \theta }{\partial x} - \dfrac{\beta^{2}}{\omega_{2} }\dfrac{\partial p}{\partial x}=\dfrac{\beta^{2}}{\omega_{2} }\dfrac{\partial^{2}u}{\partial t^{2}} $
(36)$\nabla^{2}w+\left( {\dfrac{\lambda_{e} \omega_{1} }{G_{e} \omega_{2} }+1} \right)\dfrac{\partial e}{\partial z}-\dfrac{\beta^{2}\omega_{3} }{\omega_{2} }\dfrac{\partial \theta }{\partial z} - \dfrac{\beta^{2}}{\omega_{2} }\dfrac{\partial p}{\partial z}=\dfrac{\beta^{2}}{\omega_{2} }\dfrac{\partial^{2}w}{\partial t^{2}} $
(37)$\sigma_{xx} =\dfrac{2\omega_{2} }{\beta^{2}}\dfrac{\partial u}{\partial x}+\left( {1-\dfrac{2}{\beta^{2}}} \right)\omega_{1} e-\omega_{3} \theta -p $
(38)$\sigma_{zz} =\dfrac{2\omega_{2} }{\beta^{2}}\dfrac{\partial w}{\partial z}+\left( {1-\dfrac{2}{\beta^{2}}} \right)\omega_{1} e-\omega_{3} \theta -p $
(39)$\sigma_{xz} = {\dfrac{\omega_{2} }{\beta^{2}}} \left( {\dfrac{\partial u}{\partial z}+\dfrac{\partial w}{\partial x}} \right) $
方程(32) $\sim$ (34)间相互消元就可以得到满足$\theta^{* }\left( z\right)$、$p^{* }\left( z \right)$和$e^{* }\left( z \right)$的六阶偏微分方程

(40)$\begin{eqnarray} \left( {{D}^{6}-L_{1} {D}^{4}+L_{2} {D}^{2}-L_{3} } \right)\left( {e^{* },\theta^{* },p^{* }} \right)\left( z \right)=0 \end{eqnarray} $
所考虑的各物理量的解析表达式如下

(41)$\theta^{* }\left( z \right)=\sum\limits_{i=1}^3 {A_{1i} R_{i} \left( {a,\omega } \right)} {e}^{-k_{i} z} $
(42)$p^{* }\left( z \right)=\sum\limits_{i=1}^3 {A_{2i} } R_{i} \left( {a,\omega } \right){e}^{-k_{i} z} $
(43)$w^{* }\left( z \right)=F{e}^{-nz}+\sum\limits_{i=1}^3 {A_{3i} } R_{i} \left( {a,\omega } \right){e}^{-k_{i} z} $
(44)$\sigma_{zz}^{* } \left( z \right)=-\dfrac{2\omega_{2} n}{\beta^{2}}F{e}^{-nz}-\sum\limits_{i=1}^3 {A_{4i} R_{i} \left( {a,\omega } \right)} {e}^{-k_{i} z} $
(45)$\sigma_{xz}^{* } \left( z \right)= {\dfrac{\omega_{2} }{\beta^{2}}}\bigg\{ -\dfrac{\mbox{i}}{a}\bigg[ \sum\limits_{i=1}^3 {A_{5i} } R_{i} \left( {a,\omega } \right){e}^{-k_{i} z}+ \left( {n^{2}+a^{2}} \right)F{e}^{-nz} \bigg]\bigg\} $
式中, $F=F\left( {a,\omega } \right)$为关于$a$和$\omega $的相关参量,

$A_{1i}=\dfrac{\varphi_{0} \omega_{3} \omega_{4} }{k_{i}^{2} -a^{2}-\omega_{4}}$

$A_{21} =\dfrac{\varphi_{1} \omega +\varphi_{3} \omega ^{2}}{k_{i}^{2} -a^{2}}+\dfrac{\varphi_{0} \varphi_{2} \omega \omega_{3} \omega_{4} }{\left( {k_{i}^{2} -a^{2}} \right)\left( {k_{i}^{2} -a^{2}-\omega_{4} } \right)}$

$A_{3i} =-\dfrac{\beta^{2}\omega_{3} k_{i} g_{1} }{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}-\dfrac{\beta ^{2}k_{i} g_{2} }{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}+\dfrac{\left( {\dfrac{\lambda_{e} \omega_{1} }{G_{e} \omega_{2} }+1} \right)k_{i} }{{k_{i}^{2} -n^{2}}}$

$A_{4i} =-\dfrac{2\omega_{2} \omega_{3} k_{i}^{2} g_{1} }{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}+\dfrac{2\omega_{2} \left( {\dfrac{\lambda _{e} \omega_{1} }{G_{e} \omega_{2} }+1} \right)k_{i}^{2} }{k_{i}^{2} -n^{2}}-\dfrac{2\omega_{2} k_{i}^{2} g_{2} }{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}-\left( {1-\dfrac{2}{\beta^{2}}} \right)\omega_{1} +\omega_{3} g_{1} +g_{2}$

$A_{5i} =-\dfrac{\beta^{2}\omega_{3} k_{i} g_{1} \left( {k_{i}^{2} +a^{2}} \right)}{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}-\dfrac{\beta ^{2}k_{i} g_{2} \left( {k_{i}^{2} +a^{2}} \right)}{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}+ \dfrac{\left( {\dfrac{\lambda_{e} \omega_{1} }{G_{e} \omega_{2} }+1} \right)\left( {k_{i}^{2} +a^{2}} \right)k_{i} }{k_{i}^{2} -n^{2}}+k_{i}$

正则模态法不仅可直接对多场耦合方程进行快速解耦, 使得公式推导过程更简便; 还可以消除积分变换时数值反变换中离散误差和截断误差的存在而不能全面反映热的"次声效应"的局限^[44]. 为了确定$R_{i}$ $(i=1,2,3)$和$F$的表达式, 必须引入边界条件, 根据上述对问题的描述, 可以得到以下的边界条件.

(1) 地基上表面处受到力源作用时的边界条件如下: ①应力边界条件, 地基上表面受到力源作用

(46)$\begin{eqnarray} \sigma_{xz} =0\quad \sigma_{zz} =-q\psi \left( {x,t} \right) \end{eqnarray} $
②温度边界条件, 地基上表面不考虑温度作用

(47)$\begin{eqnarray} \theta =0 \end{eqnarray} $
③超孔隙水压力边界条件, 地基上表面可透水

(48)$\begin{eqnarray} p=0 \end{eqnarray} $
式中, $q$为力源的大小, $\psi \left( {x,t} \right)$为$x$轴上力源的分布函数. 结合方程(24), $\psi \left( {x,t} \right)$可以改写成

(49)$\begin{eqnarray} \psi \left( {x,t} \right)=\psi^{* }\left( {a,\omega } \right)\exp \left( {\omega t+\mbox{i}ax} \right) \end{eqnarray} $
(2) 地基上表面处受到热源作用时的边界条件如下: ①应力边界条件, 地基上表面应力自由

(50)$\begin{eqnarray} \sigma_{xz} =0\quad \sigma_{zz} =0 \end{eqnarray} $
②温度边界条件, 地基上表面受到热源作用

(51)$\begin{eqnarray} \theta =Q\psi \left( {x,t} \right) \end{eqnarray} $
③超孔隙水压力边界条件, 地基上表面可透水

(52)$\begin{eqnarray} p=0 \end{eqnarray} $
式中, $Q$为热源的大小.

根据上述边界条件, 可以得到$R_{i}$ $(i=1,2,3)$和$F$的表达式, $R_{i}$ $(i=1,2,3)$的表达式比较繁杂, 需要借助Maple软件计算得到, 在此处不在列出, $F$的表达式如下

(53)$\begin{eqnarray} F=-\dfrac{1}{{n^{2}+a^{2}}}\sum\limits_{i=1}^3 {A_{6i} } R_{i} \end{eqnarray} $
其中

$\begin{eqnarray*} &&A_{6i} =-\dfrac{\beta^{2}\omega_{3} k_{i} g_{1} \left( {k_{i}^{2} +a^{2}} \right)}{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)}-\dfrac{\beta ^{2}k_{i} g_{2} \left( {k_{i}^{2} +a^{2}} \right)}{\omega_{2} \left( {k_{i}^{2} -n^{2}} \right)} +\dfrac{\left( {\dfrac{\lambda_{{e}} \omega_{1} }{G_{{e}} \omega _{2} }+1} \right)\left( {k_{i}^{2} +a^{2}} \right)k_{i} }{{k_{i}^{2} -n^{2}} }+k_{i} \end{eqnarray*}$

3 算例及结果讨论

本文以饱和多孔半无限大黏弹性地基为例, 分别在地基的上表面施加了力源和热源等外载荷作用. 结合边界条件最终得到了渗透系数和孔隙率变化等对该地基中无量纲竖向应力、超孔隙水压力、竖向位移以及温度的影响. 本算例中所需参数同参考文献[45,46,47].

Table 1
表1
表1考虑三场耦合效应的饱和多孔黏弹性地基计算参数
Table 1Calculation parameters of saturated porous viscoelastic foundation considering the coupled three-fields effect

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由于载荷频率满足$\omega =\omega_{0} +\mbox{i}\varsigma $, 式中$\mbox{i}$为虚数单位, 则${e}^{\omega t}={e}^{\omega_{0} t}\left( {\cos \varsigma t+\mbox{i}\sin \varsigma t} \right)$, 当作用时间$t$很小时, 即可取$\omega =\omega_{0} $. 其他参数$a=1.2$, $\psi^{* }=1$.

3.1 力源作用时地基响应特性分析

地基上表面考虑力源作用时, 取载荷频率非常小($\omega =0.002)$, 波数$a=1$时, THMVD模型和THMD模型(饱和多孔弹性地基的热-水-力耦合动力模型)均可以认为是准静态地基模型. 设地基上表面受到了与力源波长幅值大小相同的静力载荷作用, 将静力载荷沿$x$轴方向等分成若干小份, 每份视为均布载荷, 根据Boussinesq解推出的均布载荷作用时土中的应力计算公式求出每一份均布载荷作用下土的应力, 并将所有应力结果叠加, 用以验证本文中解析方法的适用性. 三条曲线都吻合得很好, 从而验证了正则模态法计算结果的可靠性.

图2

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图2不同方法计算水平方向上竖向应力分布情况的对比验证

Fig.2Comparison and verification of vertical stress distribution in horizontal direction calculated by different methods



从图3 $\sim\!$图6可以看出, 当载荷频率相同时, 文中饱和多孔黏弹性地基中超孔隙水压力、竖向应力、竖向位移和温度曲线与文献[48]的相应结果分布趋势一致, 从而证明了方法的合理性和结果的可靠性.

图3

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图3力源作用下THMVD模型中超孔隙水压力与文献^[48]对比

Fig.3Comparison of the excess pore water pressure in THMVD model under mechanical force with the work in Ref.[48]

图4

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图4力源作用下THMVD模型中竖向应力与文献[48]对比

Fig.4Comparison of the vertical stress in THMVD model under mechanical force with the work in Ref.[48]

图5

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图5力源作用下THMVD模型中竖向位移力与文献[48]对比

Fig.5Comparison of the vertical displacement in THMVD model under mechanical force with the work in Ref.[48]

图6

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图6力源作用下THMVD模型中温度力与文献[48]对比

Fig.6Comparison of the temperature in THMVD model under mechanical force with the work in Ref.[48]



图7$\sim\!$图10描述了饱和多孔黏弹地基上表面受到不同频率的力源作用时, 孔隙率和渗透系数变化对地基中各量纲量的影响.

图7

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图7力源作用下渗透系数和孔隙率变化对超孔隙水压力的影响

Fig.7Variations of the excess pore water pressure with different porosity and permeability coefficient under mechanical force

图8

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图8力源作用下渗透系数和孔隙率变化对竖向应力的影响

Fig.8Variations of the vertical stress with different porosity and permeability coefficient under mechanical force

图9

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图9力源作用下渗透系数和孔隙率变化对竖向位移的影响

Fig.9Variations of the vertical displacement with different porosity and permeability coefficient under mechanical force

图10

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图10力源作用下渗透系数和孔隙率变化对温度的影响

Fig.10Variations of the temperature with different porosity and permeability coefficient under mechanical force



图7中无量纲超孔隙水压力从零开始随地基深度增大而逐渐增大, 达到峰值后再逐渐减小, 这是因为假设上边界处可透水. 在$\omega =5$时, 靠近地基上表面的一定区域内, 三条曲线基本重合, 主要是由于孔隙水可以快速排出而造成的. 载荷频率增大使超孔隙水压力快速变大, 且峰值向着地基更深处移动. 此外, 随着载荷频率增大, 两不同渗透系数和不同孔隙率引起的超孔隙水压力曲线间差异明显增大.

孔隙率和渗透系数变化对无量纲竖向应力和竖向位移(图8和图9)并没有明显的影响, 而载荷频率对这两个物理量有较为明显的影响, 载荷频率大的曲线衰减的速度更快. 在本研究中, 正负值与大小无关, 正值表示处于受拉状态, 负值表示处于受压状态. 图8中无量纲竖向应力处于压缩状态, 渗透系数和孔隙率变化对竖向应力的影响主要体现在深度达到$z=0.5$后到应力扰动完全消失之前的区域, 在这个区域中, 渗透系数和孔隙率较小的曲线衰减的速度更快, 随着载荷频率增大, 这个差异更明显一些.

图10中无量纲温度曲线趋势与超孔隙水压力类似, 曲线随载荷频率增大而逐渐变大, 但随孔隙率和渗透系数增大逐渐减小. 无论是载荷频率变化还是孔隙率亦或是渗透系数变化, 温度曲线的峰值都基本在地基同一深度出现. 载荷频率大的曲线衰减的速度更快. 随着载荷频率增大, 两不同孔隙率的温度曲线间的差异逐渐增大, 但随着载荷频率的增大, 两不同渗透系数的温度曲线之间的差异基本一致. 此外, 当载荷频率一定, 在接近地基上表面处, 三条曲线基本重合, 这主要是因为研究时假设地基上表面可透水, 在地基深度较浅时, 地基中的孔隙水更容易排出, 所以差异不太明显, 随着深度增大, 孔隙水不再容易排出, 曲线间差异就愈发明显. 虽然仅有单位力源作用所引起的无量纲温度的数值不大, 但是仍然能看出孔隙水的存在对温度的影响很明显.

3.2 热源作用时地基响应特性分析

图11$\sim\!$图14分析了上表面受到热源作用时, 不同载荷频率($\omega =1.6$和$\omega =5)$和不同孔隙率($n_{0} =0.3$和$n_{0} =0.4)$以及不同渗透系数($k_{{d}} =10^{-7}$和$k_{{d}} =10^{-8})$对无量纲超孔隙水压力、竖向应力、竖向位移和温度的影响.

图11

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图11热源作用下渗透系数和孔隙率变化对超孔隙水压力的影响

Fig.11Variations of the excess pore water pressure with different porosity and permeability coefficient under thermal load

图12

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图12热源作用下渗透系数和孔隙率变化对竖向应力的影响

Fig.12Variations of the vertical stress with different porosity and permeability coefficient under thermal load

图13

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图13热源作用下渗透系数和孔隙率变化对竖向位移的影响

Fig.13Variations of the vertical displacement with different porosity and permeability coefficient under thermal load

图14

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图14热源作用下渗透系数和孔隙率变化对温度的影响

Fig.14Variations of the temperature with different porosity and permeability coefficient under thermal load



图11和图12中, 无量纲超孔隙水压力、竖向应力曲线均是从零开始, 先逐渐增大, 达到峰值后再逐渐减小. 无论是随载荷频率增大还是随孔隙率亦或者渗透系数增大, 无量纲超孔隙水压力和竖向应力均随之增大. 随着载荷频率增大, 不同渗透系数和孔隙率所引起的无量纲超孔隙水压力曲线间的差异也越发明显. 在载荷频率一定时, 随着渗透系数和孔隙率的增大, 曲线峰值均向着地基上表面方向移动.

图13中, 无量纲位移一开始处于压缩状态, 随着地基深度逐渐增大, 慢慢进入膨胀状态. 渗透系数和孔隙率变化对无量纲竖向位移的影响不太明显, 主要在曲线进入膨胀区域的附近以及曲线峰值处影响较为明显. 在载荷频率一定时, 渗透系数小的曲线在地基上表面处和曲线峰值处均略大一些, 随着孔隙率增大, 无量纲位移逐渐增大.

图14中无论是孔隙率变化还是渗透系数变化均对无量纲温度没有明显影响, 随着载荷频率增大, 温度曲线在地基上表面处明显增大, 随地基深度增大, 其衰减速度明显增大.

4 结论

本文旨在基于研究饱和多孔黏弹性地基中渗透系数和孔隙率变化对地基中各无量纲量的影响. 建立了黏弹性地基的热-水-力耦合动力模型. 详细的分析和探讨了饱和多孔黏弹性地基上表面受到外载荷作用时无量纲超孔隙水压力、竖向位移、竖向应力以及温度等物理量的分布情况和变化规律. 主要结论如下:

(1) 正则模态法作为一种加权残差法, 能够快速对多场耦合方程进行解耦, 最终得到了所考虑的各物理量的变化规律, 该方法的引入为多孔黏弹性地基的热-水-力耦合问题的求解提供了有效的计算方法.

(2) 地基上表面受到力源作用时, 孔隙率变化仅对无量纲超孔隙水压力和温度有明显的影响; 地基上表面受到热源作用时, 孔隙率变化对无量纲超孔隙水压力和竖向应力的影响更为明显一些. 总体而言, 无论何种载荷作用, 孔隙率变化对无量纲超孔隙水压力影响比较明显, 而对无量纲竖向位移却没有明显的影响.

(3) 地基上表面受到力源作用时, 渗透系数变化同样仅对无量纲超孔隙水压力和温度有明显影响, 尤其是在峰值处. 当地基上表面受到热源作用时, 渗透系数变化对除无量纲温度外的各物理量均有一定的影响, 但对无量纲竖向应力和超孔隙水压力的影响更为明显.

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