Genesis of solitons arising from individual Flows of the Camassa‐Holm hierarchy

The present work offers a detailed account of the large‐time development of the velocity profile run by a single “individual” Hamiltonian flow of the Camassa‐Holm (CH) hierarchy, the Hamiltonian employed being the reciprocal of any eigenvalue of the underlying spectral problem. In this simpler scenario, I prove some of the conjectures raised by McKean [27]. Notably, I confirm the ultimate shaping into solitons of the cusps that appear, near blowup sites, of any velocity profile emanating from an initial disposition for which breakdown of the wave in finite time is sure to happen. The careful large‐time asymptotic analysis is carried from exact expressions describing the velocity in terms of initial data, the integration involving a “Lagrangian” scale and three “theta functions,” the rates at which the latter reach their common values at each end of the line characterizing the region where soliton genesis is expected. In fact, the present method also suggests how solitons may arise from initial conditions not leading to breakdown. The full CH flow is nothing but a superposition of such commuting “individual” actions. Therein lies the hope that the present account will pave the way to elucidate soliton formation for more complex flows, in particular for the CH flow itself. © 2005 Wiley Periodicals, Inc.     

Relaxation experiments of polymeric systems confined in very narrow gaps

A new way of investigating the relaxation processes of polymeric systems confined between solid surfaces is presented here. The test is based on draining experiments by means of a surface force apparatus and consists in monitoring the hydrodynamic force release following a drainage motion. Experiments carried out with solutions of high polymers exhibit two distinct relaxation processes. The longest one is connected to the relaxation of the chains adsorbed onto the solid surfaces and which are pinned in the area of closest distance between the solid surfaces. Its variations as a function of spacing are consistent with the bridging of some macromolecules. The fastest process is connected to the flow of the solvent molecules through the pseudonetwork formed by the adsorbed layers carried by the solid surfaces. These results have been compared favorably with those obtained by oscillatory measurements as far as relaxation time and viscosity are concerned. The accuracy of the experimental relaxation function is not sufficient for describing reasonably the viscoelastic behavior of the confined fluid. © 1996 John Wiley & Sons, Inc.     

An approximate solution to the problem of cone or wedge indentation of elastoplastic solids

The model developed in this Note makes it possible to determine the value of the mean indentation pressure usually named hardness from the elastoplastic properties of materials and also the shape of the cone or that of the wedge. The approximation rests upon the definition of a linear elastic solid which has the same indentation pressure as the material actually indented. Cases of cone and wedge indentation are studied. A method to determine the uniaxial stress–strain curve of materials from indentation tests is given. The results are validated using finite element simulations. To cite this article: G. Kermouche et al., C. R. Mecanique 333 (2005).     

Response of thin polymer layers confined between solid surfaces and submitted to normal oscillatory deformations

Summary The viscoelastic properties of liquids close to solid surfaces differ from the bulk. Nanorheology has been performed by using a surface force apparatus adapted to operate as a rheometer in a sphere-plane geometry. Axial oscillatory measurements have been carried out with high polymer solutions filling the gap. The deformations were kept sufficiently small not to perturb the film structure and were applied in a large range of frequency (10⁻³ to 10² s⁻¹). It is shown that the complex modulus characterizing the confined medium can be split into two components: a shear modulus (it accounts for the viscous dissipation due to the flow of solvent molecules through the mesh created by the long polymer chains which connect the two solid surfaces) and a compression modulus which is related to the normal stress response of the chains confined between the solid surfaces. The hydrodynamic screening lengthξ _h and the correlation length ξ deduced from the two moduli are compared and are found to scale in the same way as a function of the distance between the two surfaces. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Measuring the Soret coefficient of binary hydrocarbon mixtures in packed thermogravitational columns (contribution of Toulouse University to the benchmark test)

Within the framework of an international benchmark test, our group which researches packed thermogravitational columns has measured the porous medium thermodiffusion coefficient of two hydrocarbon binary mixtures: tetrahydronaphthalene-dodecane and tetrahydronaphthalene-isobutylbenzene. The weight fraction of each component was near 50 wt%.     

Extraction of Mechanical Properties with Second Harmonic Detection for Dynamic Nanoindentation Testing

In this article, a new method based on the detection of the second harmonic of the displacement signal to determine mechanical properties of materials from dynamic nanoindentation testing, is presented. With this technique, the Young’s modulus and hardness of homogeneous materials can be obtained at small penetration depths from the measurement of the second harmonic amplitude. With this innovative method, the measurement of the normal displacement is indirectly used, avoiding the need for very precise contact detection. Moreover, the influence of the tip defect and thermal drift on the measurements are reduced. This method was used for dynamic nanoindentation tests performed on fused silica and on an amorphous polymer (PMMA) because these materials are supposed not to exhibit an indentation size effect at small penetration depths. The amplitude of the second harmonic of the displacement signal was correctly measured at small depths, allowing to calculate the Young’s modulus and the hardness of the tested materials. The mechanical properties calculated with this method are in good agreement with values obtained from classical nanoindentation tests.     

Strain-rate effects on hardness of glassy polymers in the nanoscale range. Comparison between quasi-static and continuous stiffness measurements

Strain rate effects on Hardness and Young's modulus of two glassy polymers, poly(diethylene glycol bis allyl carbonate) (CR39) and bisphenol-A polycarbonate (PC), were studied in the nanoscale range. Before analyzing material behaviors, we focused on a particular phenomenon prevailing at the first stage of the contact between the surface of these polymers and the Berkovitch diamond tip used in the experiments, leading to an apparent increase of the tip defect (i.e., the missing tip of the diamond from having a shape equivalent to a perfect cone). The common methods based on calibration functions of the tip appear to be inaccurate to calculate correctly the contact area at the nanoscale range for these polymers. A new method based on Loubet et al.'s model to calculate the contact area by taking account of the apparent tip defect is proposed. The hardness values obtained this way were compared to the compressive yield stress using Tabor's relationship. A hardness-yield stress ratio close to 2.0, as expected on such polymers, was found. A strain-rate effect on the load-depth curve for these two polymers is interpreted as an increase of the hardness with the strain rate. The results from quasi-static and dynamic (the continuous stiffness method) measurements are compared. The strain-rate effect on Young's modulus in dynamic conditions should be taken into account in the hardness calculation.