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Two-Dimensional Poroelastic Problem for Saturated Soil Under Fractional Order Theory of Thermoelasticity

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Abstract

A two-dimensional (2D) half-space elastic subgrade with multiple physical field coupling is examined in this study with normal mode analysis (NMA) under Ezzat’s fractional order generalized thermoelasticity. The coupled multiple physical field is investigated considering the temperature, stress, and seepage fields. The influence of fractional derivatives and frequency on the thermo-hydro-mechanical (THM) coupling dynamic response of saturated porous elastic subgrade is analyzed when the upper surface of the saturated porous elastic subgrade is subjected to the action of a time-harmonic load including normal and thermal loads. The distributions of non-dimension temperature increment, excess pore water pressure, vertical stress, and vertical displacement are obtained accordingly. The NMA method is derived via weighted residuals, which eliminates the discrete error and truncation error otherwise caused by the integral transformation. Regardless of load type, the load frequency significantly influences all physical variables. When considering the thermal source on the upper surface, the fractional derivatives have obvious influence on all physical variables. While, during considering a mechanical load, the fractional derivatives only have a certain influence on the non-dimension temperature increment without markedly affecting other variables.

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Abbreviations

\(d\) :

\(d = \frac{{\rho_{w} g}}{{\kappa_{d} }}\)

\(c_{w}\) :

Pore water heat capacity

\(c_{s}\) :

Solid particles heat capacity

\(e\) :

Soil volumetric strain

\(g\) :

Gravity

\(K\) :

Thermal conductivity coefficient

\(m\) :

Heat capacity per unit volume

\(n_{0}\) :

Porosity

\(p\) :

Excess pore water pressure

\(Q\) :

Temperature load magnitude

\(q\) :

Magnitude of mechanical load

\(q_{i}\) :

Vector components of heat flow

\(T\) :

Absolute temperature

\(T_{0}\) :

Initial temperature

\(u_{i}\) :

Displacement vector component

\(\alpha_{1}\) :

Biot’s effective stress coefficient

\(\alpha_{s}\) :

Solid particles Thermal expansion coefficient

\(\alpha_{w}\) :

Pore water thermal expansion coefficient

\(\beta_{1}\) :

\(\beta_{1} = \left( {3\lambda + 2\mu } \right)\alpha_{s}\)

\(\delta_{ij}\) :

Kronecker delta

\(\varepsilon_{ij}\) :

Strain tensor component

\(\theta\) :

\(\theta = T - T_{0}\)

\(\kappa_{d}\) :

Permeability coefficient

\(\lambda ,\mu\) :

Lame’s constants

\(\upsilon\) :

Constant parameter (\(0 < \upsilon \le 1\))

\(\rho\) :

Density

\(\rho_{w}\) :

Pore water density

\(\rho_{s}\) :

Solid particles density

\(\sigma_{ij}\) :

Stress tensor component

\(\tau\) :

Thermal relaxation time factor

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Acknowledgements

This work was supported in part by Key Scientific Research Project of Henan Province (No. 22A130004).

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  1. School of Civil Engineering, Henan University of Science and Technology, Luoyang 471023, Henan, China

    Ying Guo, Jianjun Ma, Da Li & Chaosheng Wang

  2. School of Civil Engineering, Tianjin University, Tianjin, 300072, People’s Republic of China

    Chunbao Xiong

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  1. Ying Guo
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Guo, Y., Xiong, C., Ma, J. et al. Two-Dimensional Poroelastic Problem for Saturated Soil Under Fractional Order Theory of Thermoelasticity. Transp Porous Med 141, 695–712 (2022). https://doi.org/10.1007/s11242-021-01742-8

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