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

Modelling plasticity of unsaturated soils in a thermodynamically consistent framework

Constitutive equations of unsaturated soils are often derived in a thermodynamically consistent framework through the use a unique 'effective' interstitial pressure. This later is naturally chosen as the space averaged interstitial pressure. However, experimental observations have revealed that two stress state variables were needed to describe the 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 retention properties of the soil and two effective stresses, which are required to describe the soil deformation at water saturation held constant. Actually, it is shown that a simple assumption related to internal deformation leads to the need of a unique effective stress to formulate the stress-strain constitutive equation describing the soil deformation. An elastoplastic framework is then presented and it is finally shown that the Barcelona Basic Model, a commonly accepted model for unsaturated soils, as well as all models deriving from it, appear as special extreme cases of the thermodynamic framework proposed here

Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.Comment: 41 pages, 24 figures; moderate revision after the first round of review, Appendix A re-written completel

Deformation Regimes for Sphere-Plane Contact: Revisiting Tabor’s Criteria for Differential Hardness

This chapter presents an update of theories involving the differential hardness problem, starting from the hypothesis made by Tabor for the contact between a sphere and a plane. In this way, the reader interested in problems affected directly by these formulations, such as contact area and contact fatigue, can take part of a fundamental theoretical basis to perform investigations in this field

Multiphase-field modelling of crack propagation in geological materials and porous media with Drucker-Prager plasticity

A multiphase-field approach for elasto-plastic and anisotropic brittle crack propagation in geological systems consisting of different regions of brittle and ductile materials is presented and employed to computationally study crack propagation. Plastic deformation in elasto-plastic materials such as frictional, granular or porous materials is modelled with the pressure-sensitive Drucker-Prager plasticity model. This plasticity model is combined with a multiphase-field model fulfilling the mechanical jump conditions in diffuse solid-solid interfaces. The validity of the plasticity model with phase-inherent stress and strain fields is shown, in comparison with sharp interface finite element solutions. The proposed model is capable of simulating crack formation in heterogeneous multiphase systems comprising both purely elastic and inelastic phases. We investigate the influence of different material parameters on the crack propagation with tensile tests in single- and two-phase materials. To show the applicability of the model, crack propagation in a multiphase domain with brittle and elasto-plastic components is performed

A small deformations effective stress model of gradient plasticity phase-field fracture

A variational formulation of small strain ductile fracture, based on a phase-field modeling of crack propagation, is proposed. The formulation is based on an effective stress description of gradient plasticity, combined with an AT1 phase-field model. Starting from established variational statements of finite-step elastoplasticity for generalized standard materials, a mixed variational statement is consistently derived, incorporating in a rigorous way a variational finite-step update for both the elastoplastic and the phase-field dissipations. The complex interaction between ductile and brittle dissipation mechanisms is modeled by assuming a plasticity driven crack propagation model. A non-variational function of the equivalent plastic strain is then introduced to modulate the phase-field dissipation based on the developed plastic strains. Particular care has been devoted to the formulation of a consistent Newton–Raphson scheme for the case of Mises plasticity, with a global return mapping and relative tangent matrix, supplemented by a line-search scheme, for the solution of the gradient elastoplasticity problem for fixed phase field. The resulting algorithm has proved to be very robust and computationally effective. Application to several benchmark tests show the robustness and accuracy of the proposed model

Rheology of granular flows across the transition from soft to rigid particles

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