Alessandro Gajo

A kinematic hardening constitutive model for sands : The multiaxial formulation

This paper explores the possibility of using well-accepted concepts—Mohr-Coulomb-like strength criterion, critical state, existence of a small strain elastic region, hyperbolic relationship for representing global plastic stress–strain behaviour, dependence of strength on state parameter and flow rules derived from the Cam-Clay Model—to represent the general multiaxial stress–strain behaviour of granular materials over the full range of void ratios and stress level (neglecting grain crushing). The result is a simple model based on bounding surface and kinematic hardening plasticity, which is based on a single set of constitutive parameters, namely two for the elastic behaviour plus eight for the plastic behaviour, which all have a clear and easily understandable physical meaning. In order to assist the convenience of the numerical implementation, the model is defined in a ‘normalized’ stress space in which the stress–strain behaviour does not undergo any strain softening and so certain potential numerical difficulties are avoided. In the first part the multiaxial formulation of the model is described in detail, using appropriate mixed invariants, which rationally combine stress history and stress. The model simulations are compared with some experimental results for tests on granular soils along stress paths lying outside the triaxial plane over a wide range of densities and mean stresses, using constitutive parameters calibrated using triaxial tests. Furthermore, the study is extended to the analysis of the effects induced by the different shapes of the yield and bounding surfaces, revealing the different role played by the size and the curvature of the bounding surface on the simulated behaviour of completely stress- and partly strain-driven tests. Copyright

Chemo-mechanical coupling in saturated porous media: elastic-plastic behaviour of homoionic expansive clays

Chemically active saturated homoionic clays are considered in a two-phase framework. The solid phase contains the clay particles, absorbed water and salt. The fluid phase, or pore water, contains free water and salt. Water, and possibly salt, can transfer between the two phases. In addition free water may diffuse through the porous medium. A global understanding of the phenomena involved, namely deformation, transfer and diffusion, is proposed. Emphasis is laid on the chemo-mechanical constitutive equations in an elastic–plastic setting. Elastic chemo-mechanical coupling is introduced through a potential, in such a way that the tangent poro-elasticity matrix remains symmetric. Material parameters needed to quantify the coupling are calibrated from specific experiments available in the recent literature. The elasto-plastic behaviour aims at reproducing qualitatively and quantitatively the typical experimental responses observed on almost pure Na-Montmorillonite clays during chemical and mixed chemo-mechanical loadings. Increase of the salinity of pore water at a constant confinement stress leads to a volume decrease, so-called chemical consolidation. Subsequent exposure to a distilled water solution produces swelling: however, the latter is smaller than the chemical consolidation so that the chemical loading cycle results in a net contractancy, the amount of which increases with the confinement. In fact, plastic yielding takes place at low salinities of pore water, and when it stops, chemical preconsolidation is generated. Natural clays which contain cations of different species are considered in a companion paper, Gajo et al. [Int. J. Solids Struct., this issue], as they require to account for electro-chemo-mechanical couplings. � 2002 Published by Elsevier Science Ltd.

Electro-chemo-mechanical couplings in saturated porous media: elastic-plastic behaviour of heteroionic expansive clays

Chemically active saturated clays containing several cations are considered in a two-phase framework. The solid phase contains the negatively charged clay particles, absorbed water and ions. The fluid phase, or pore water, contains free water and ions. Electroneutrality is ensured in both phases, which gives rise to electrical fields. Water and ions can transfer between the two phases. In addition, a part of free water diffuses through the porous medium. A global understanding of all phenomena, deformation, transfer, diffusion and electroneutrality, is provided. Emphasis is laid on the electro-chemo-mechanical constitutive equations in an elastic–plastic setting. Elastic chemo-mechanical coupling is introduced through a potential, in such a way that the tangent elastic stiffness is symmetric. Material parameters needed to estimate the coupling are calibrated from specific experiments available in the recent literature. The elastic–plastic behaviour aims at reproducing qualitatively and quantitatively typical experimental phenomena observed on natural clays during chemical and mixed chemo-mechanical loadings, including chemical consolidation and swelling already described in Int. J. Solids Structures (39 (10), 2773–2806) in the simpler context of Na-Montmorillonite clays. Crucially, the successive exposure of a clay to pore solutions with chemical content dominated by a cation already present in the clay or quasi-absent leads to dramatically different volume changes, in agreement with experimental data. � 2002 Elsevier Science Ltd. All rights reserved.

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part I – small strain formulation

Mechanical densification of granular bodies is a process in which a loose material becomes increasingly cohesive as the applied pressure increases. A constitutive description of this process faces the formidable problem that granular and dense materials have completely different mechanical behaviours (nonlinear elastic properties, yield limit, plastic flow and hardening laws), which must both be, in a sense, included in the formulation. A treatment of this problem is provided here, so that a new phenomenological, elastoplastic constitutive model is formulated, calibrated by experimental data, implemented and tested, that is capable of describing the transition between granular and fully dense states of a given material. The formulation involves a novel use of elastoplastic coupling to describe the dependence of cohesion and elastic properties on the plastic strain. The treatment falls within small strain theory, which is thought to be appropriate in several situations; however, a generalization of the model to large strain is provided in Part II of this paper.

Finite element simulations of chemo-mechanical coupling in elastic–plastic homoionic expansive clays

Abstract Chemically active saturated clays are considered in a two-phase framework. The solid phase contains clay particles, absorbed water and a single salt. The fluid phase, or pore water, contains free water and salt. Water, and possibly salt, can transfer between the two phases. In addition, part of both species diffuse through the porous medium. A global understanding of all phenomena, mass transfer, diffusion/advection and deformation is provided. The coupled constitutive equations associated to these three phenomena are developed. Emphasis is laid on the chemo-mechanical constitutive equations in an elastic–plastic setting. A finite element formulation embodying all the above phenomena is proposed and simulations of oedometer tests are presented and commented.

SILENT BOUNDARY CONDITIONS FOR WAVE PROPAGATION IN SATURATED POROUS MEDIA

Wave propagation both in one- and in two-dimensional saturated elastic porous media is analysed by means of a two-field finite element model with silent boundaries. An extension of the elastic ‘multidirectional’ transmitting boundary to two-phase media is developed to simulate the silent boundary condition. The theoretical assessment and the numerical formulation of the first-order silent boundary technique is presented in detail. Some examples are used to demonstrate the reliability of the first-order method, especially for problems with plane and axisymmetric waves having various angles of incidence. Finally, the wave propagation along a pile shaft is presented, to simulate a common non-destructive dynamic pile test.

A general approach to isothermal hyperelastic modelling of saturated porous media at finite strains with compressible solid constituents

A general approach is proposed for defining the macroscopic free-energy function of a saturated porous medium subjected to finite deformations, under isothermal conditions, in the case of compressible fluid and solid constituents. Reference is made to an elementary volume treated as an ‘open system’, moving with the solid skeleton. The strain-like variables are characterized by a suitable strain measure and the variation of fluid-mass content. Although the free energy is obtained by assembling the contributions of the single constituents, the resulting free energy is not the simple sum of the free energies of the single constituents. It is shown that the simplified approaches previously proposed in the literature are recovered as particular cases of the more general framework proposed in the present article and, in addition, new simplified constitutive frameworks are investigated. The proposed approach paves the way to the consistent hyperelastic–plastic and thermo-elastic modelling of saturated porous media with compressible fluid and solid constituents.

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state. The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.  2005 Elsevier SAS. All rights reserved.

Deformable Porous Media Saturated by Three Immiscible Fluids: Constitutive Modelling and Simulations of Injection and Imbibition Tests

A number of environmental and petroleum engineering applications involve the coexistence of three non-miscible fluids. In this work, basic constitutive relations and computational schemes are developed in order to simulate fluid injection and imbibition processes in a deformable rock through the finite element method. For this purpose, the following ingredients are worked out: (i) simple, but general formulas for the effective saturations; (ii) constitutive expressions for the relative permeabilities of water, oil and gas in terms of effective saturations; and (iii) constitutive capillary pressure relationships. These ingredients are introduced in a domestic finite element code where the primary variables are the solid displacement vector and the three fluid pressures. Given the abundance of experimental data in the petroleum engineering field, the whole framework is firstly tested by simulating gas injection into a rock core sample initially saturated by water and oil. Sensitivity analyses are performed upon varying key constitutive, loading and numerical parameters, to assess the physical and computational outputs of the proposed framework. Particular attention is given to the influence on the model predictions of several expressions defining relative permeabilities. Simulations of water-alternated-gas injection and of counter-current water imbibition tests are also performed, to establish the reliability of the proposed constitutive and computational framework.

A non-linear analysis of non-isothermal wave propagation in linear-elastic fluid-saturated porous media

Abstract The non-isothermal dynamic behaviour of saturated porous media is analysed numerically employing the finite element method and taking energy convection due to large pore fluid displacements into account. A different pore fluid reference temperature is introduced in order to allow properly for heat convection: this concept is usually neglected in the literature and is discussed and analysed herein. The numerical procedure is validated in a simple problem of hot fluid injection in a steady seepage flow and by comparing the numerical results, neglecting energy convection, with those obtained with a novel solution of the linearised equations, presented herein, which is based on the transfer functions and Fourier transforms method. Finally, the effects of energy convection in wave propagation are analysed: in a pervious porous medium the flux of energy due to energy convection is much greater than the one due to heat conduction; in any case, wave propagation can be considered completely adiabatic even when energy convection is taken into account. Thus the validity of the results presented in the literature and based on the linearised theory is demonstrated.