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.     

Mechanics of granular materials: The discrete and the continuum descriptions juxtaposed

In recent years a discussion could be followed where the pros and cons of the applicability of the Cosserat continuum model to granular materials were debated [Bardet, J.P., Vardoulakis, I., 2001. The asymmetry of stress in granular media. Int. J. Solids Struct. 38, 353–367; Kruyt, N.P., 2003. Static and kinematics of discrete Cosserat-type granular materials. Int. J. Solids Struct. 40, 511–534; Bagi, K., 2003. Discussion on “The asymmetry of stress in granular media”. Int. J. Solids Struct. 40, 1329–1331; Bardet, J.P., Vardoulakis, I. 2003a. Reply to discussion by Dr. Katalin Bagi. Int. J. Solids Struct. 40, 1035; Kuhn, M., 2003. Discussion on “The asymmetry of stress in granular media”. Int. J. Solids Struct. 40, 1805–1807; Bardet, J.P., Vardoulakis, I., 2003b. Reply to Dr. Kuhn’s discussion. Int. J. Solids Struct. 40, 1809; Ehlers, W., Ramm, E., Diebels, S., D’Addetta, G.A., 2003. From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702; Chang, C.S., Kuhn, M.R., 2005. On virtual work and stress in granular media. Int. J. Solids Struct. 42, 3773–3793]. The authors follow closely this debate and try, with this paper, to provide a platform where the various viewpoints could find their position. We consider an ensemble of rigid, arbitrarily shaped grains as a set with structure. We establish a basic mathematical framework which allows to express the balance laws and the action–reaction laws for the discrete system in a “global” form, through the concepts of “part”, “granular surface”, “separately additive function” and “flux”. The independent variable in the balance laws is then the arbitrary part of the assembly rather than the single grain. A parallel framework is constructed for Cosserat continua, by applying the axiomatics established by [Noll, W., 1959. The foundation of classical mechanics in the light of recent advances in continuum mechanics. In: The axiomatic method, with special reference to Geometry and Physics, North-Holland Publishing Co., Amsterdam pp. 266–281, Gurtin, M.E., Williams, W.O., 1967. An axiomatic foundation of continuum thermodynamics. Arch. Rat. Mech. Anal. 26, 83–117, Gurtin, M.E., Martins, L.C., 1976. Cauchy’s theorem in classical physics. Arch. Rat. Mech. Anal. 60, 305–324]. The comparison between the two realisations suggests the microscopic interpretation for some features of Cosserat Mechanics, among which the asymmetry of the Cauchy-stress tensor and the couple-stress.     

Effects of the pH on the mechanical behavior of articular cartilage and corneal stroma

Electro-chemo-mechanical couplings in articular cartilages and corneal stroma are due to the presence of electric charges on proteoglycans. In addition, at non-physiological pH, collagen molecules become charged as well. Variation of the pH of the electrolyte has strong implications on the electric charge of these tissues, and by the same token, on their transport and mechanical properties. Indeed, articular cartilages and corneal stroma swell and shrink depending on the composition of the electrolyte, they are in contact with.Emphasis is laid here on the combined effects of pH, ionic strength, calcium and chloride binding on mechanical properties.The tissues are viewed as three-phase multi-species porous media. The constitutive framework is phrased in the theory of thermodynamics of deformable porous media. Acid–base reactions, as well as ion binding, are embedded in this framework. Although, macroscopic in nature, the approach accounts for a number of biochemical details defining collagen and proteoglycans.The model is used to simulate laboratory experiments where specimens of articular cartilages and corneal stroma are put in contact with a bath of controlled chemical composition. Chemical loadings, where the ionic composition and pH of the bath are varied, are intermingled with mechanical loadings. The variations of the stress and strain are observed to depend strongly on the ionic strength and ion type present in the bath: sodium chloride leads to a stiffer response than calcium chloride and hydrochloric acid. Moreover, when the bath changes from basic to acidic, the change of sign of the fixed charge across the isoelectric point has definite mechanical implications, and it gives rise to non-monotonous evolutions of the stress, strain and chemical content.While the chemo-mechanical effect is a key phenomenon that governs the behavior of tissues with fixed charges, the converse mechano-chemical effect is significant in corneal stroma due to its low stiffness.     

Transient analysis of ionic replacements in elastic–plastic expansive clays

Chemically active saturated clays are considered in a two-phase framework. The solid phase contains clay particles, absorbed water and dissolved ions, Na⁺, K⁺ and Cl⁻. The fluid phase, or pore water, contains free water and the same ionic species. Water and ions can transfer between the two phases. In addition, they 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 phenomena are developed. Emphasis is laid on the electro-chemo-mechanical constitutive equations in an elastic–plastic setting.A finite element formulation embodying all the above aspects is proposed and simulations of oedometer tests are presented and commented. Of particular interest are the consolidation and swelling that occur during salinization and desalinization of an external reservoir in contact with the specimen, and the more subtle, but important effects of replacing an NaCl pore solution by a KCl pore solution, and conversely.     

A note on the critical points of equilibrium paths in imperfect structures

In a recent work (Int. J. Solids Struct. 37 (2000) 1561) by one of the authors, an extended system for calculating critical points of equilibrium paths in imperfect structures was presented. However, the extremum nature of these points was not analyzed explicitly in that paper. In this note, we will fill in the gap and establish a sufficient condition for determining the buckling strength of imperfect structures.     

On the importance of thermo-elastic cooling in the fracture of glassy polymers at high rates

In a previous thermo-mechanical analysis [Estevez, R., Basu, S., van der Giessen, E., 2005. Analysis of temperature effects near mode I cracks in glassy polymers. Int. J. Fract. 132, 249–273] in which shear yielding of the bulk and failure by crazing were accounted for, we examined which of these two viscoplastic processes contributed to heat in mode I fracture. The present study completes this work by investigating the conditions for thermo-elastic cooling prior to crack propagation as reported experimentally by Rittel [Rittel, D., 1998. Experimental investigation of transient thermo-elastic effects in dynamic fracture. Int. J. Solids Struct. 35, 2959–2973] and Bougaut and Rittel [Bougaut, O., Rittel, D., 2001. On crack tip cooling during dynamic crack propagation. Int. J. Solids Struct. 38, 2517–2532] on high strain rate loading of PMMA. To this end, coupled thermo-mechanical finite element simulations are carried out by accounting for the thermo-elastic source, in addition to the heat sources related to shear yielding and crazing. The bulk as well as cohesive zone parameters for crazing realistically describe PMMA as they are obtained from detailed calibration experiments. Our results show that if significant thermo-elastic cooling has to be observed in the vicinity of the crack tip of a polymeric material, suppression of shear yielding as well as suppression of crazing is necessary. It seems that at these high strain rates a brittle fracture mechanism activated at very high stresses takes over from crazing, or at least that craze initiation occurs for stress levels very different to those for quasi-static conditions.     

Non-uniform frictional sliding under cyclic loading with frictional characteristics changing in the process of sliding

Frictional sliding on a crack with non-uniform frictional characteristics is considered. The present work continues the investigation of Gorbatikh et al. [Int. J. Solids Struct., in press] and focuses on the cyclic loading. The evolution of the sliding process in loading–reloading–unloading cycles is analyzed. We also extend the analysis to the important case when the frictional resistance changes in the process of sliding (such changes may model “degradation” of the sliding surface during sliding, as well as other physical factors, not necessarily related to the sliding itself).     

Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction

Explicit formulas are derived for the van der Waals (vdW) interaction between any two layers of a multi-walled carbon nanotube (CNT). Based on the derived formulas, an efficient algorithm is established for the buckling analysis of multi-walled CNTs, in which individual tubes are modeled as a continuum cylindrical shell. The explicit expressions are also derived for the buckling of double-walled CNTs. In previous studies by Ru (J. Appl. Phys. 87 (2000b) 7227) and Wang et al. (Int. J. Solids Struct. 40 (2003) 3893), only the vdW interaction between adjacent two layers was considered and the vdW interaction between the other two layers was neglected. The vdW interaction coefficient was treated as a constant that was not dependent on the radii of the tubes. However, the formulas derived herein reveal that the vdW interaction coefficients are dependent on the change of interlayer spacing and the radii of the tubes. With the increase of radii, the coefficients approach constants, and the constants between two adjacent layers are about 10% higher than those reported by Wang et al. (Int. J. Solids. Struct. 40 (2003) 3893). In addition, the numerical results show that the vdW interaction will lead to a higher critical buckling load in multi-walled CNTs. The effect of the tube radius on the critical buckling load of a multi-walled CNT is also examined.