Revisiting the thermodynamics of hardening plasticity for unsaturated soils

A thermodynamically consistent extension of the constitutive equations of saturated soils to unsaturated conditions is often worked out through the use a unique ’effective’ interstitial pressure, accounting equivalently for the pressures of the saturating fluids acting separately on the internal solid walls of the pore network. The natural candidate for this effective interstitial pressure is the space averaged interstitial pressure. In contrast experimental observations have revealed that, at least, a pair of stress state variables was needed for a suitable framework to describe stress-strain-strength behaviour of unsaturated soils. The thermodynamics analysis presented here shows that the most general approach to the behaviour of unsaturated soils actually requires three stress state variables: the suction, which is required to describe the invasion of the soil by the liquid water phase through the retention curve; two effective stresses, which are required to describe the soil deformation at water saturation held constant. However a simple assumption related to the plastic flow rule leads to the final need of only a Bishop-like effective stress to formulate the stress-strain constitutive equation describing the soil deformation, while the retention properties still involve the suction and possibly the deformation. Commonly accepted models for unsaturated soils, that is the Barcelona Basic Model and any approach based on the use of an effective averaged interstitial pressure, appear as special extreme cases of the thermodynamic formulation proposed here.

💡 Research Summary

The paper presents a thermodynamically consistent framework for extending the constitutive equations of saturated soils to unsaturated conditions. Traditional extensions rely on a single “effective interstitial pressure,” usually taken as the space‑averaged pressure of the pore fluids, and they attempt to use a Bishop‑type effective stress in the same way as for saturated soils. Experimental evidence, however, shows that a single stress variable cannot capture the full stress‑strain‑strength response of unsaturated soils; at least two independent stress measures are required.

Starting from the first and second laws of thermodynamics, the authors formulate the free‑energy density and the dissipation potential with two internal variables: the degree of saturation (S_r) and the suction (u_a – u_w). By differentiating the free energy, they identify three independent stress work‑conjugate variables: (1) a Bishop‑type effective stress σ′ = σ − χ u_a − (1 − χ) u_w, (2) the suction itself, and (3) an additional effective stress that becomes relevant when the water saturation is held constant during deformation. The suction governs the invasion of the liquid phase and thus the soil’s retention behavior (the water retention curve), while the two effective stresses govern mechanical deformation.

A key insight is that, if the plastic flow rule is assumed to be associated with a single scalar plastic potential that depends only on the Bishop‑type stress, the additional effective stress can be eliminated from the plastic flow description. Consequently, the deformation part of the constitutive model can be expressed with a single Bishop‑like effective stress, whereas suction remains as an independent variable for the retention part. This simplification does not violate thermodynamic consistency because the dissipation inequality is still satisfied.

The authors then show that widely used models are special cases of their general formulation. The Barcelona Basic Model (BBM) corresponds to the choice χ = χ(S_r) and a suction‑only coupling in the retention law; models that employ a simple space‑averaged interstitial pressure correspond to the extreme limits χ = 1 (fully saturated) or χ = 0 (dry). Both are therefore subsets of the more general three‑stress‑variable framework.

In summary, the thermodynamic analysis demonstrates that a complete description of unsaturated soil behavior requires three stress state variables: suction for the hydraulic (retention) response, and two effective stresses for the mechanical response at constant saturation. By adopting a simple associated plastic flow rule, the mechanical response can be reduced to a single Bishop‑type effective stress, preserving the essential physics while simplifying implementation. This unified framework clarifies the physical meaning of model parameters, resolves inconsistencies in earlier approaches, and provides a solid basis for developing more accurate and versatile constitutive models for unsaturated soils in geotechnical engineering and related numerical simulations.