A Multi-scale Model of Overburden Pressure and Water Content in Montmorillonite-Bearing Clayey Soils

Abstract

Soils containing swelling clays, such as montmorillonite, can develop significant pressures due to their water content, or can absorb significant surrounding water, leading to potentially dramatic volume changes. This work presents a generalized Terzaghi’s stress principle that when rewritten in terms of measurable quantities, relates overburden pressure to volumetric change. In the limiting case of 100% clay it simplifies to the pressure-volumetric relationship developed by Phillip Low, and in the other extreme (no clay) Terzaghi’s stress principle is obtained. The relationship is derived using Hybrid Mixture Theory, a multi-scale mixture theoretical framework, which allows for developing a more generalized mathematical model for multi-physics problems. The equation relating overburden pressure to volume incorporates the clay and water content, bulk liquid pressure, and four empirical constants. The resulting constitutive equation is validated against existing experimental data over a wide range of pressures and clay content.

Appendix: Notation

The following symbols are used in this paper:

• $(\cdot )^A =$ property of the interacting (clay/adsorbed water phase);

• $(\cdot )^{A_\ell } =$ property of adsorbed water;

• $(\cdot )^{A_s} =$ property of clay solids;

• $(\cdot )^{B} =$ property of the non-interacting phase;

• $(\cdot )^{B_\ell } =$ property of bulk water;

• $(\cdot )^{B_s} =$ property of non-clay solids;

• $(\cdot )^{\ell } =$ property of liquid;

• $(\cdot )^{s} =$ property of solid;

• $(\cdot )_0=$ property of the undeformed system;

• $\underline{\underline{\overline{\varvec{C}}}}^\alpha =$ phase $\alpha$ modified Cauchy-Green Tensor $\textsf {1}$

• $f^c =$ clay mass fraction $\textsf {1}$;

• $F^\alpha =$ extensive Helmholtz potential ML$^2$ T$^{-2}$;

• $K=$ bulk modulus ML$^{-1}$ T$^{-1}$;

• $m^\alpha =$ phase $\alpha$ mass $\textsf {M}$;

• $p^\alpha =$ phase $\alpha$ mechanical pressure ML$^{-1}$ T$^{-1}$;

• $\bar{p}^{\alpha }_{\beta } =$ phase $\alpha$/$\beta$ thermodynamic pressure difference ML$^{-1}$ T$^{-1}$;

• $p^e =$ effective pressure ML$^{-1}$ T$^{-1}$;

• $P_{\textrm{atm}} =$ atmospheric (ambient) pressure ML$^{-1}$ T$^{-1}$;

• $T=$ temperature $\textsf {Q}$;

• $V^\alpha =$ phase $\alpha$ volume L$^3$;

• $w =$ gravimetric water content $\textsf {1}$;

• $w^\alpha =$ gravimetric adsorbed water content $\textsf {1}$;

• $\alpha =$ Low’s coefficient for clay pressure (not a superscript) $\textsf {1}$;

• $\varepsilon ^\alpha =$ phase $\alpha$ volume fraction $\textsf {1}$;

• $\lambda ^\alpha=$ phase alpha lagrange multiplier for mass conservation enforcement;

• $\pi ^{\alpha }_{\beta } =$ phase $\alpha$/$\beta$ structural pressure difference ML$^{-1}$ T$^{-1}$;

• $\rho ^\alpha =$ phase $\alpha$ intrinsic density ML$^{-3}$;

• $\varvec{\sigma }^\alpha =$ phase $\alpha$ Cauchy stress tensor ML$^{-1}$ T$^{-1}$;

• $\psi ^\alpha =$ phase $\alpha$ Helmholtz potential L$^2$ T$^{-2}$;