Insights into the links between microstructure and Bishop’s χ parameter for unsaturated soils

The choice of stress state variables for constitutive modelling of unsaturated soils is a long standing debate. It is now acknowledged that two stress state variables are needed to describe in a comprehensive manner the mechanical behaviour of these soils. In recent years, Bishop’s proposal for an extended stress used in combination with suction as a second independent stress variable has gained a large audience. In his original paper (Bishop, 1959), no explicit choice of the χ parameter which weights the contributions of the pressures of the air and water phases is given. A simple allusion to the fact that χ depends on the degree of saturation of the water phase is made. Based upon energetic approaches or upscaling methods, many contributions now use this Bishop expression (whose name varies from Bishop’s stress to effective, constitutive, skeleton or, even, generalized stress...) where the χ factor is assumed to be equal to the degree of saturation Sr. It may be shown that this choice may lead to gross overestimations of the real contribution of suction to the effective stress. This is particularly true for clays where high suction values mean high effective stress values which would induce unrealistic volume compressions. These approaches often assume the fluid phases (generally air and water) to be continuous. This point implicitly confers to the water phase a unique role whatever its location within the porous structure of the material. It will be shown later on that two distinct parts of the porous water may be identified on the basis of microstructural considerations. This partition of the interstitial water permits a proper definition of the effective stress by correcting the contribution of suction to it. This proposal and its microstructural interpretation will be evaluated on the basis of experimental data on various types of soils, from fairly granular materials to high plasticity clays. The work presented in this paper focuses on the choice of the effective stress. In particular, attention will be given to a recurring assumption in recent works which consists in using an average constitutive stress, strictly equivalent to Bishop’s stress when χ = Sr. The consequences of such choice on particular features of unsaturated soils behaviour will be analyzed. It is worth saying that the term effective stress employed here refers to the conjugate variable of the strain variable. It does not hold the general definition given by Terzaghi in the case of saturated soils. It is acknowledged that partial saturation effects induce further couplings than those included in such stress definition thus requiring the use of a second independent stress state variable. ABSTRACT: In recent years, Bishop’s proposal for a generalized stress extending Terzaghi’s stress to unsaturated states gained a large audience. Despite the complexity introduced in some aspects of modelling or stress representation, such an approach allows to model in a simple manner particular features of unsaturated soil behaviour. In this paper, an alternative choice for this parameter is proposed and validated using either shear strength or elastic stiffness experimental data from a set of soils ranging from almost granular materials to high plasticity clays. A physical interpretation in terms of microstructurally trapped water is provided. It is shown that the amount of trapped water is closely linked to the definition of χ. When the amount of non-free water increases (increasing plasticity indices), a significant deviation of χ values from the degree of saturation is observed. It is thus concluded that the hypothesis stating that Bishop’s χ parameter equals the degree of saturation of water is only approximately valid for a particular class of soils, namely those having a marked granular nature