F.M.F. Simões

Instability and ill-posedness in some friction problems

The steady frictional sliding of an elastic half-space in contact with a rigid flat surface is considered. Perturbed solutions in the elastic body are sought which correspond to surface (flutter) instabilities of the steady sliding state. The behavior of these surface solutions for vanishing small values of their arbitrary wavelengths is made clear. The effect of introducing intrinsic length scales in the problem by means of the earlier proposed non-local friction law and normal compliance law or by means of viscous dissipation in the deformable body is discussed.

Articular cartilage with intra- and extrafibrillar waters: a chemo-mechanical model

Abstract A three-phase multi-species electro-chemo-mechanical model of articular cartilage is developed that accounts for the effect of two water compartments, namely intrafibrillar water stored in between collagen fibrils and extrafibrillar water covering proteoglycans. The collagen fibers constitute the solid phase while intrafibrillar water and dissolved NaCl on one hand and extrafibrillar water, ions Na+ and Cl− and proteoglycans on the other hand form the two fluid phases. Chemical equilibrium between the fluid phases is assumed and only the mechanical aspects of the behaviour are considered. Although mass transfer occurs here, the theory is time-independent. The complete picture that includes time-dependent mass transfer between the two fluid phases, diffusion of water and ions and electrical field is postponed for a future study. Constitutive functions are motivated and material parameters calibrated. Simulations of confined compression and oedometric free swelling are compared with available data. The evolution of internal entities, e.g. mass and molar fractions of intra- and extrafibrillar waters and ions, is also documented during chemical and mechanical loading processes.

Flutter Instability and Ill-Posedness in Solids and Fluid-Saturated Porous Media

For elastoplasticity with locally smooth yield surface and plastic potential, the nature, real or complex, of the squares of the acceleration wave-speeds is analyzed. Emphasis is laid on the effect that some features of the constitutive equations have on that nature. Specifically, our reference problem contemplates an infinite elastic-plastic body endowed with elastic isotropy and deviatoric associativity. In a second step, individual or simultaneous deviations with respect to these reference properties are analyzed. The performed linearized analyses are essentially intended to detect, in the course of a loading process, the onset of wave-speeds whose squares are complex, a phenomenon called flutter. Finite element simulations show how perturbations evolve in such a situation, and, as expected, the plastic loading condition plays a crucial role.

Instabilities in elastic–plastic fluid-saturated porous media: harmonic wave versus acceleration wave analyses

Abstract A linearized analysis of the stability of the flow in elastic–plastic fluid-saturated porous media with incompressible constituents is performed. A relationship is established between the results of this analysis and the results of analyses based on acceleration waves performed by Loret and Harireche [Journal of the Mechanics and Physics of Solids 39, 569–606 (1991) ] and by Loret et al. [International Journal of Solids and Structures 34, 1583–1608 (1997) ]; a justification is found for the growth of the acceleration waves in the non-associative case when their speeds are real and a clarification is provided relative to the growth or decay of waves in the interior of flutter regions.

Dissipative graph solutions for a 2 degree-of-freedom quasistatic frictional contact problem

Abstract In this paper we discuss the nature of the quasitatic purely elastic limit to the dynamic viscoelastic solutions to a 2 degree-of-freedom (d.f.) frictional contact problem. In a significant situation for which a continuous equilibrium path cannot exist in the limit, we show that, when the mass and viscosity coefficients are decreased to zero in an appropriate manner, a connected graph is approached in the 3-dimensional space of the displacement components plus the time (load parameter) variable. This graph solution contains an instantaneous portion, a displacement discontinuity with respect to time, along which the total energy dissipated into some external sink is non-negative. These (possibly oscillatory and non-equilibrated) instantaneous paths and some of their qualitative features are discussed and compared with those resulting from quasistatic viscoelastic approximation procedures. For a particular class of graph solutions which may be approached by sequences of either dynamic or quasistatic viscoelastic solutions, we show that a near future evolution of the system does always exist and that direction of that evolution for the initiation or the continuation of an instantaneous path is uniquely defined.

Growth and decay of acceleration waves in non-associative elastic-plastic fluid-saturated porous media

The modes under which acceleration waves can propagate in elastic-plastic fluid-saturated porous media have been obtained in Loret and Harireche [J. Mech. Phys. Solids39, 569–606 (1991)]. The implications of the existence of complex wave-speeds are analyzed here. First, like elastic mixtures, elastic-plastic mixtures with an associative flow rule may be characterized by a positive and finite decay coefficient: acceleration waves propagate with an amplitude that either strictly decreases or remains constant. Moreover, the range of variation of the decay coefficient is the same as for elastic mixtures. In contrast, when the squares of the wave-speeds are real, nonassociative flow rules may give rise to negative and/or unbounded decay coefficients. When the squares of the wave-speeds are complex, the so-called flutter phenomenon, the decay coefficient is found to be positive and finite. The analytical derivations require the material state on the wave front to be constant; on the other hand, the analysis is valid independently of the compressibilities of the solid and fluid constituents.

Some Notes on Friction and Instabilities

These lectures address some topics related to instability, bifurcation and non-smoothness in finite dimensional frictional contact problems, as well as instability, ill-posedness and regularization procedures in some infinite dimensional problems.

Surface boundary conditions trigger flutter instability in non-associative elastic-plastic solids

Abstract It is shown that surface flutter instability may be triggered by the simultaneous influence of non-associativity and boundary conditions even if, taken independently, neither the elastic-plastic constitutive law (satisfying deviatoric associativity) nor the boundary conditions (no applied traction rates) would lead to flutter. More specifically, it is shown that, for orthotropic elastic-plastic constitutive tensors with an orthotropy axis tangent to the rate-traction-free boundary, the onset of surface flutter instability coincides with the incipience of plasticity for any non-associative flow rule, whenever the normal to the boundary does not coincide with one of the other two orthotropy axes. In the case of deviatoric associativity and coaxiality between the directions of orthotropy and the normal and tangent to the rate-traction-free boundary, surface flutter instability may also occur, but only for unusual values of material parameters. Additional results and discussion are also presented for the onset of stationary surface waves in associative and non-associative elastic-plastic bodies. In contrast to the onset of surface flutter instabilities, the condition for the onset of stationary surface waves involves material properties only, that is, it does not discriminate among different orientations of the rate-traction-free boundary with respect to the material orthotropy directions.

Effects of pH on transport properties of articular cartilages

Articular cartilages swell and shrink depending on the ionic strength of the electrolyte they are in contact with. This electro-chemo-mechanical coupling is due to the presence of fixed electrical charges on proteoglycans (PGs). In addition, at nonphysiological pH, collagen fibers become charged. Therefore, variation of the pH of the electrolyte has strong implications on the electrical charge of cartilages and, by the same token, on their transport and mechanical properties. Articular cartilages are viewed as three-phase multi-species porous media. The constitutive framework is phrased in the theory of thermodynamics of porous media. Acid–base reactions, as well as calcium binding, are embedded in this framework. Although macroscopic in nature, the model accounts for a number of biochemical details defining collagen and PGs. The change of the electrical charge is due to the binding of hydrogen ions on specific sites of PGs and collagen. Simulations are performed mimicking laboratory experiments where either the ionic strength or the pH of the bath, the cartilage piece is in contact with, is varied. They provide the evolutions of the chemical compositions of mobile ions, of the sites of acid–base reactions and calcium binding, and of the charges of collagen and glycosaminoglycans, at constant volume fraction of water. Emphasis is laid on the effects of pH, ionic strength and calcium binding on the transport properties of cartilages, and, in particular, on the electrical conductivity and electro-osmotic coefficient.

A Chemo-Mechanical Model for Articular Cartilage

A three-phase multi-species electro-chemo-mechanical model of articular cartilage is developed that accounts for the effect of two water compartments, namely intrafibrillar water stored in between collagen fibrils and extrafibrillar water covering proteoglycans. The collagen fibers constitute the solid phase while intrafibrillar water and dissolved NaCl on one hand and extrafibrillar water, ions Na+ and Cl− and proteoglycans on the other hand form the two fluid phases. Chemical equilibrium between the fluid phases is assumed and only the mechanical aspects of the behaviour are considered.