### Poromaterials and Poromechanics

#### Introduction

Porous materials are ubiquitous in engineering applications, ranging from traditional materials (such as soils, rocks, concrete, etc.) to new materials (such as membranes, ceramics, biomaterials, energy materials, etc.). Mechanics of porous materials is crucial to address many emerging engineering problems, i.e., utilization of unconventional fossil energy (i.e. shale gas), thermo-chemo-mechanical damage of separator membrane in Lithium Ion battery, durability of engineering tissues, mechanical responses of micro-nano porous materials, etc. Solution to these problems requires further fortify the foundation of porous materials mechanics.

However, a problem arises when a porous medium is more complicated than a binary system, e.g., a ternary system, which includes a solid skeleton permeated by a wetting fluid (water) and a non-wetting fluid (air). That is, the effective stress concept, which describes the interactions between the solid and fluid phases in a binary system (e.g., saturated soils), cannot be directly extended to ternary (e.g., unsaturated soils) or more complicated systems. Various attempts have been made from engineering perspectives to intuitively extend the effective stress concept in saturated soils to unsaturated soils (Skempton, 1960). The corresponding theories, however, were not completely physically-based and failed to give a general description on porous media mechanics involving multiple fluid phases. There are other approaches developed via the fusion of mixture theory and the concept of volume fraction. These approaches are more rigorous. However, the effective stress formulation with volume fractions relies on some assumptions that are not well supported or validated.

Another problem arises due to the incapability of classic soil mechanics to describe the coexistence of multiple pore fluids and their interactions. In a ternary system such as an unsaturated soil, it has been proven that a relationship exists between the pressure difference (suction) and the volume ratio (saturation) between water and air (Barbour, 1998). This relationship is believed to stem from the morphology of internal structure and the physical chemistry of interfaces between phases (Fredlund, 1994). However, the apparent contact angle between water-air and air-solid interfaces has been conventionally assumed to be zero (Arya, 1999; Bachmann, 2002). This simplification leads to the incapability of the soil water characteristic curve (SWCC) to consider significant phenomena such as thermally-induced liquid flows and hystereses. Lacking the general science in interfacial interactions has also dwarfed attempts to extend the SWCC to a more complicated system involving multiple fluids. Such extended relationships are crucial for the mathematical closure of porous media systems.

#### Holistic Theories

The key concept in the soil mechanics based on the work of von Terzaghi is the formulation of effective stress. A general form of the effective stress can be formulated as Eq. 1 (Gary, 2007)

\[\sigma=\sigma '- p^s -\mathbf{I},\]

where \(\sigma\) is the total stress tensor, \(\sigma '\) is the effective stress, \(p^s\) is the solid phase pressure (pressure acting on the solid phase), and I is the unit tensor. Equation 1 was obtained for a binary system (e.g., saturated soil) by assuming both solid grains and pore fluid are incompressible. However, the grain compressibility is not always negligible to that of the skeleton. Formulation of effective stress (Eq. 2) considering the compressibility of solid grains was proposed by Skempton (1960) for soils and was later confirmed by Nur (1971) for rocks.

\[\sigma=\sigma ' -\alpha p^s \mathbf{I},\]

where \(\sigma\) is related to the compressibility of the grains, \(C_s\) and the drained bulk compressibility, \(C\), by Eq. 3

\[ \alpha=1-\frac{C_s}{C}.\]

But a dispute emerged when the classical effective concept is applied to a ternary system (e.g., unsaturated soil). The first form of effective stress for unsaturated soils was proposed by Bishop (1959) under the assumption of incompressible grains as Eq. 4,

\[ \sigma=\sigma ' -[\chi p^w+(1-\chi)p^n]\mathbf{I},\]

where \(p^w\) and \(p^n\) are the water (wetting) and gas (non-wetting) pressures, respectively; and \(\chi\) is the Bishop parameter which was believed to be related to the saturation. This type of formulation of effective stress is usually called single effective stress. Fredlund (1977) argued that \(\chi\), which is a state variable independent of soil properties. Alternatively, the use of any two of the three possible stress variables (solid, water and air) was suggested. A good example in favor of this argument is the poroelastic model proposed by Biot (1941) for compressible saturated soils (Eqs. 5-6):

\[ \varepsilon _v=\frac{1}{K} \sigma + \frac{1}{H} p^w,\]

\[ \zeta _v=\frac{1}{H} \sigma + \frac{1}{R} p^w.\]

For simplicity, Eqs. 5-6 which correspond to an isotropic loading condition are used here. In the equations, \(\varepsilon_{ij}\) is the volumetric strain, \(\zeta\) is the increment of water content, \(\sigma\) is the isotropic applied stress, \(p^w\) is the water pressure, \(K\) is the drained bulk modulus, \(1/K\) is the poroelastic expansion coefficient, \(1/R\) is the specific storage coefficient, and σ corresponds to the total stress \(\sigma\). The independent stress concept encouraged several preeminent contributions such as the elasto-plastic model developed by Alonso (1990). More recently, Lu (2006) proposed a micromechanical conceptualization of the single effective stress and suggested to use the relationship between the effective stress and water content to characterize soils. Therefore, the contradiction between the two types of effective stress models has not been settled yet and could confuse researchers in the area. The statement of Fredlund and Morgenstern (1977) holds water while the concept presented by Lu (2006) is supported by experiments and is convenient for applications. In fact, this paradox can be reconciled by revisiting Eq. 2 and Eq. 5. To do this, let us first reformat Eq. 5 into Eq. 7

\[ \sigma=K \varepsilon_v-\frac{K}{R} p^w=\sigma '-B p^w,\]

where \(\sigma ' \) is the effective stress under isotropic applied stress and $B$ is usually called the Biot-Willis coefficient. It is seen that Eq. 7 is equivalent to Eq. 2 with \(B=\alpha\). Therefore, we can draw a conclusion that the formulations of the single effective stress ever since Skempton (1960), which accounted for the properties of porous media, are not merely concepts of stresses, but also constitutive relations. By recognizing this fact, the validity of the single effective stress is proved without violating Fredlund’s statement. Therefore, the two types of stress formulations are essentially equivalent.

The above dispute occurred within the context of engineering approaches for porous media, namely, soil mechanics and unsaturated soil mechanics. However, a greater discrepancy has been preserved from the controversy between von Terzaghi and Fillunger. The outline laid down by Fillunger was resumed in the recent decades in the context of mixture theory. Other than the mechanical approaches following von Terzaghi, the models with the mixture theory were developed based on the conservation of mass, energy and momentum. Mechanical principles and thermodynamics are thus strictly ensured. Moreover, these models can be easily extended to porous media with more than three phases. Considering the difficulties of the engineering approaches in a ternary system, the mixture theory is superior in a theoretical perspective. This type of model poses a new direction for the porous materials mechanics. Instead of Eqs. 1-3, the stress in current mixture theories is formulated by Eq. 8,

\[ \sigma=\theta^s \sigma ^s + \theta^w\sigma ^w + \theta^n \sigma ^n,\]

where \(\theta J\) and \( \sigma J\) are the volume fraction and stress tensor of phase (\(J (J=s,w,n)\)), respectively. The way to relate the mixture theory to engineering approaches has been identified as Eqs. 9-11 (Hutter, 1999; Lewis, 1998; Ehlers, 2004),

\[ \sigma^s = \sigma '- [Sp^2 +(1-S) p^n] \mathbf{I}, \]

\[ \sigma^w=-p^w \mathbf{I},\]

\[ \sigma^n=-p^n \mathbf{I},\]

and an additional assumption (Eq. 12):

\[ S=\chi.\]

The relationships above rendered a consistent theory for porous media. However, the assumption (Eq. 12) is in contradiction with most experimental evidence (Jennings, 1960; Bishop, 1961). Furthermore, the unique relationship between \(\chi\) and \(S\) and even the validity of Bishop’s model were questioned because the model could not explain phenomena such as the collapse in unsaturated soils (Jennings, 1962). Nuth (2008) proposed to use Bishop’s stress concept with the volume fraction as the Bishop parameter for unsaturated soils.

#### LiuRG's Work

Attempts have been made to address the issues on the level of fundamental theories based on both experiments and simulations from micro- and meso- to macro-scale. The following showing the development of stress and strain in a typical porous materaisl element based on fundamental forces between atoms. This purpose is to cast light into historical disputes in the mechanics of porous mmaterials for a holistic and consistent porous material theory.