S U M M A R YFor a fluid saturated porous cylinder, in which the saturating fluid is allowed to move freely across the traction-free cylindrical boundary surface, the dispersion relation for extensional deformation, established over four decades ago by Gardner within the framework of Biot's theory, yield a trivial and two non-trivial roots. One of the non-trivial roots corresponds to the coupled Biot fast-compressional and shear waves and it is the poroelastic analogue of the extensional wave of the elasticity theory. The nature of the second non-trivial root is not fully understood yet. Also there is the suggestion that the scope of applicability of Gardner's first non-trivial root might be limited. Apart from these matters, since the Biot theory is also incapable of accounting for the viscous bulk-and shear-relaxations that occur within the saturating Newtonian fluid because its fluid stress tensor is devoid of the viscous stress tensor part, therefore, we re-examined this poroelastic extensional problem with viscosity-extended Biot constitutive relations. The erstwhile trivial third root of the Biot-Gardner framework is not non-vanishing in this viscosity-extended Biot framework. In the frequency regime below the Biot relaxation frequency, the other two non-trivial roots in this framework are the same as those obtained by Gardner within the classical Biot framework. We find there is nothing dubious about the formula of Gardner's first root. The scenario in which it is not expected to hold true, in fact the Biot theory itself breaks down. We find the Gardner's second root to be essentially the Biot slow-compressional wave, the out-of-phase/differential compressional motion of the two phases. The new non-vanishing third root is akin to the slow-shear process, the out-of-phase/differential shear motion of the two phases, and a process analogue to the Biot slow-compressional wave.The dispersion relation for the extensional wave propagation for fully saturated, homogeneous, isotropic, porous, circular cylinders, subjected to stress-free open-pore boundary condition was first studied by Gardner (1962). He utilized the framework of Biot's theory, and showed that there is one trivial and two non-trivial roots for the dispersion relation.Under the consideration that the associated axial wavelengths are larger than the radius of the cylinder, which is known as the slender rod approximation, and restricted to the frequency regime below the Biot relaxation frequency, Gardner developed the expressions for the lowest order modes. He showed that in this limit the phase velocity (squared) associated with one of the roots, which may be regarded as a fast extensional wave, is precisely given by the Young's modulus of the saturated-frame over the total mass density. White (1986) showed that this result is valid only up to a critical frequency. Below that frequency the half-wavelength of the Biot slow P wave is more than the radius of the cylinder. Since in this regime, Biot slow P wave is a diffusion process that dies off within...
In this paper, algorithms invariant to position, rotation, noise and non-homogeneous illumination are presented. Here, several manners are studied to generate binary rings mask filters and the corresponding signatures associated to each image. Also, in this work it is shown that digital systems, which are based on the k-law non-linear correlation, are k-invariant for 0 < k < 1. The methodologies are tested using greyscale fossil diatoms digital images (real images), and considering the great similarity between those images the results obtained are excellent. The box plot statistical analysis and the computational cost times yield that the Bessel rings masks are the best option when the images contain a homogeneous illumination and the Fourier masks digital system is the right selection when the non-homogeneous illumination and noise is presented in the images.
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