Complete band gaps in two-dimensional phononic crystal slabs.

The propagation of acoustic waves in a phononic crystal slab consisting of piezoelectric inclusions placed periodically in an isotropic host material is analyzed. Numerical examples are obtained for a square lattice of quartz cylinders embedded in an epoxy matrix. It is found that several complete band gaps with a variable bandwidth exist for elastic waves of any polarization and incidence. In addition to the filling fraction, it is found that a key parameter for the existence and the width of these complete band gaps is the ratio of the slab thickness, d, to the lattice period, a. Especially, we have explored how these absolute band gaps close up as the parameter d/a increases. Significantly, it is observed that the band gaps of a phononic crystal slab are distinct from those of bulk acoustic waves propagating in the plane of an infinite two-dimensional phononic crystal with the same composition. The band gaps of the slab are strongly affected by the presence of cutoff frequency modes that cannot be excited in infinite media.

Determining the Zeta Potential of Porous Membranes Using Electrolyte Conductivity inside Pores.

The zeta potential is an important and reliable indicator of the surface charge of membranes, and knowledge of it is essential for the design and operation of membrane processes. The zeta potential cannot be measured directly, but must be deduced from experiments by means of a model. The possibility of determining the zeta potential of porous membranes from measurements of the electrolyte conductivity inside pores (lambda(pore)) is investigated in the case of a ceramic microfiltration membrane. To this end, experimental measurements of the electrical resistance in pores are performed with the membrane filled with KCl solutions of various pHs and concentrations. lambda(pore) is deduced from these experiments. The farther the pH is from the isoelectric point and/or the lower the salt concentration is, the higher the ratio of the electrolyte conductivity inside pores to the bulk conductivity is, due to a more important contribution of the surface conduction. Zeta potentials are calculated from lambda(pore) values by means of a space charge model and compared to those calculated from streaming potential measurements. It is found that the isoelectric points are very close and that zeta potential values for both methods are in quite good agreement. The differences observed in zeta potentials could be due to the fact that the space charge model does not consider the surface conductivity in the inner part of the double layer. Measurements of the electrolyte conductivity within the membrane pores are proved to be a well-adapted procedure for the determination of the zeta potential in situations where the contribution of the surface conduction is significant, i.e., for small and charged pores. Copyright 2001 Academic Press.

Electrokinetic Phenomena in Homogeneous Cylindrical Pores.

The electrokinetic phenomena occurring in homogeneous cylindrical pores containing symmetric electrolytes are studied. The local relations for flow in the pores (Nernst-Planck and Navier-Stokes equations) are developed. The analysis includes a numerical solution of the nonlinear Poisson-Boltzmann equation. The integral expressions of the phenomenological coefficients coupling the solvent flow and the electrical current with the hydrostatic pressure and the electrical potential gradients are established and calculated numerically. The mobilities of anions and cations are individually specified and the electroviscous effects as well as the surface conductance are taken into account. Streaming potentials obtained from numerical calculations are compared with results given by classical relations in a range of zeta potentials and electrokinetic radii that may commonly occur in experimental investigation of micro- and ultrafiltration membranes. In this work, it is shown that classical approximated relations can give rise to very misleading conclusions and that the determination of the true zeta potential requires a full analysis (including numerical calculations) of the basic relations for flow and potential distribution in charged pores. Copyright 1999 Academic Press.

On the mechanical characterization of compact bone structure using the homogenization theory.

In a previous paper (Crolet et al., 1993, J. Biomechanics 26, 677-687), a modelling of the mechanical behavior of compact bone was presented, in which the homogenization theory was the basic tool of computation. In this simulation, approximations were used for the modelling of the lamellae and the osteons: the lamella and the osteon were divided into cylindrical sectors, each sector being approximated as a parallelepiped having a periodic structure (fibrous composite for the lamella, superimposition of plates for the osteon). The present study deals with a new model without these approximations. First, it can be proved that the homogenized elasticity tensor for a lamella, which has non-periodic structure, is obtained at each geometrical point as a homogenized tensor of a periodic problem. Similarly, for the osteonal structure, the components of the homogenized tensor are determined at each point as the result of a periodic homogenization. The software OSTEON, which is the computational method associated with this model, allows one to obtain a better understanding of the effects of many bony parameters. The obtained results are in accordance with experimental data.

Compact bone: numerical simulation of mechanical characteristics.

One of the main difficulties encountered in the numerical simulation of the anisotropic elastic characteristics of compact bone is to account for the Haversian microstructure when determining the overall macroscopic behavior. Engineering analyses of such problems are usually based on 'homogenized approximations'. Compact bone is not exactly a composite material, but rather a heterogeneous medium which exhibits a multiscale composite structure. If the homogenized approximation is precise enough (and this is true for the mathematical theory of homogenization), it is then possible to simulate the macroscopic behavior from the microscopic mechanical characteristics. The present paper is devoted to such mathematical developments. Moreover, the 'inverse simulation' allows the computation of the microscopic stress fields in the haversian structure from the macroscopic stress fields, taking into account bone microstructure.