Equations of State in Soil Compression Based on Statistical Mechanics

Abstract

The authors have published a very interesting contribu­tion in which they present a new and general relationship between void ratio and overburden pressure in the com­pression process of soils. The study is based on the law of interparticle energy distribution used in statistical mechanics and, in particular, the compression process of a soil is described according to its initial and final void ra­tios, and to a parameter /? which is related to the potential energy of a soil element. Such potential energy depends not only on the mass and elevation of soil but also on the interactions between them. To account for such interactions, the authors introduce the concept of imaginary particles, which allows them to use em­pirically-calibrated P values in the formulation without the need to consider interparticle interactions explicitly. In fact, the results presented by the authors show that such approach reproduces successfully the compression behaviour of a wide range of situations and soils. The purpose of this discussion is to extend and clarify some theoretical aspects that might be helpful for a better understanding of Fukue and Mulligan's work. To that end, we build on the work of Edwards and Oakeshott (1989), who presented a study of how statistical mechan­ics applies to powders. We believe that there are some fundamental differences between both approaches that deserve some discussion, with the important conceptual conclusion that P should not depend on the product kT. For an ideal gas, and pressure are directly related to the kinetical energy of the gas molecules, and that makes the distribution of energy levels to depend on temperature. In a soil at equilibrium, however, particles are not in motion, pressure is related to forces transmit­ted between particles rather than to their movements, and is related to the vibration of the atoms in the lattice of their structure rather than to their motion (which happens on a different scale). Such differences in­dicate that, for soil at equilibrium, the gravitational energy is much higher than thermal energy, which sug­gests that there are other variables (different to tempera­ture) that have a much stronger influence on p. This idea is explained in detail below. The total physical energy of a soil at equilibrium con­sists of both intraparticle energies (e.g., temperature, lat­tice energy, deformation energy, etc.) and of external energies, related to the arrangement of particles. At equilibrium, the external energies are only of potential type (e.g., gravitational, electrical, . . .) and, if particles are not crushing during compression of the soil, the ex­pected energy change is only due to changes of the arran­gement energy (since neither heat transfer nor change in the internal energy are expected). Therefore, although the 1st and the 2nd laws of thermodynamics are (of course) still valid, the amount of heat energy is negligible and the system is not governed by the typical equation state of a pVT system (such as an ideal gas). In this context, when Fukue and Mulligan describe the physical meaning of fi, they use the thermodynamical equation which expresses the change in internal energy, Eq. (28), but since in soil compression internal energy is hardly varied, this equa­tion is probably not very adequate from a theoretical per­spective. This fundamental idea was considered by the statistical mechanics theory of powders developed by Edwards and Oekshott (1989). They argued that, in powders, it is their density (or, equivalently, their volume) which plays the role of energy in conventional statistical mechanics (i.e., as applied to ideal gases). Of course, both approaches should coincide, because potential and interaction energy are related to volume. They also defined the partition function (analogous to Eq. (25) in Fukue and Mulligan's work) in terms of volume and defined a sort of en­tropy and a sort of temperature that are mathemati­cally similar to those used in thermodynamics but which have a different physical meaning. Table 1 in their paper elegantly summarizes the formal analogy between classi­cal statistical mechanics and their formulation of statisti­cal mechanics for powders. It is important to note {see (35)) that, for ideal gases, the Boltzmann parameter is e~