## 1. Introduction

[2] Many geotechnical problems concern coupling between the mechanical behavior of a soil and the transfer of water, in particular in the unsaturated zone. The behavior of soils in this zone remains one of the important subjects of soil mechanics which still presents some unknown and poorly understood aspects. Swelling and shrinkage problems of clayey unsaturated soils under water transfer are concerned by this study: after some drying/wetting cycles, unsaturated soils behave elastically. In France, for example, the disorders of constructions caused by dryness, or lashing rains, concern 5000 communes in 75 departments. The total damage caused by these disasters has been estimated to 3.3 billion euros from 1989 to 2002. The origin of these pathologies is a hydrous imbalance between the center and outsidewalls of constructions that produces differential movements. These movements can be large enough to create differential deformations which are at the origin of many damages. The objective of this article is to model the volume changes for an unsaturated soil along a hydromechanical loading path which does not cross the yield curve. The studies of overconsolidated soils, or swelling clays after some drying/wetting cycles, are directly concerned by this work.

[3] In geotechnical engineering, gravimetric water content *w* and void ratio *e* are commonly used to describe the volume changes of soils. In the case of a saturated soil, these two variables are linked (under the usual assumption of incompressibility of solid grains) by the relation *e* = *G*_{s}*w*, where *G*_{s} is the specific gravity of the grains of soil. Whatever is the hydromechanical loading applied to a saturated soil element, the variations of the void ratio are proportional to the variations of the water content. The description of the volume changes of a saturated soil element requires only one variable: *e* or *w*. The corresponding stress variable is the mean effective stress (*σ* − *p*_{w}*), where *σ* is the mean total stress (1/3 of the first invariant of stress tensor) and *p*_{w}* is the pore water pressure.

[4] In the case of an unsaturated soil, the variables water content and void ratio are linked by the relation *S*_{r}*e* = *G*_{s}*w*, where *S*_{r} is the degree of saturation. Along any loading path, the variations of void ratio and water content are independent [*Salager*, 2007; *Salager et al.*, 2007]. Variations in void ratio and water content are often represented according to the two variables: mean net stress (*σ* − *p*_{g}*) and suction *s* = (*p*_{g}* − *p*_{w}*) [*Coleman*, 1962], with *p*_{g}* as the pore gas pressure. For that reason, many authors [*Bishop and Blight*, 1963; *Matyas and Radhakrishna*, 1968; *Delage and Cui*, 2001] suggested to represent every experimental test with a loading or suction change path in a space (*e*,(*σ* − *p*_{g}*), *s*). When all the loading paths belong to the same surface, this surface is called state surface. These representations were resumed more recently by *Ho et al.* [1992], *Bolzon and Schrefler* [1995], *Gatmiri and Delage* [1995], *Gatmiri et al.* [1998], and *Fleureau et al.* [2002], who proposed a representation from the model of *Alonso et al.* [1990]. *Matyas and Radhakrishna* [1968], *Fredlund and Morgenstern* [1976], *Gatmiri and Delage* [1995], and *Gatmiri et al.* [1998] proposed the formulation of another state surface as a representation of the water volume variation.

[5] In this paper, a rigorous method of representation of these state surfaces is proposed. This method, which owes its development to Biot's works [*Biot*, 1972] and to those of *Coussy* [1995], was applied to the study of unsaturated soils in the framework of the thermoporoelasticity [*Lassabatère*, 1994; *Devillers*, 1998; *Laloui and Nuth*, 2005]. At first, the volume changes of an unsaturated soil element are modeled, along a loading path inside the elastic domain. The approach followed allows two relations to express the variations of void ratio and water content according to the variations of the mean net stress and the suction.

[6] Then, these two relations are represented in two three-dimensional spaces. Experimental results allow to propose a practical construction of the *e* = *f*((*σ* − *p*_{g}*), *s*) state surface on the basis of a reference state. The method of construction is then applied to the *G*_{s}*w* = *f*((*σ* − *p*_{g}*), *s*) state surface to propose a preliminary construction.