Flexural vibrations of finite composite poroelastic cylinders

Abstract

This paper deals with the flexural vibrations of composite poroelastic solid cylinder consisting of two cylinders that are bonded end to end. Poroelastic materials of the two cylinders are different. The frequency equations for pervious and impervious surfaces are obtained in the framework of Biot’s theory of wave propagation in poroelastic solids. The gauge invariance property is used to eliminate one arbitrary constant in the solution of the problem. This would lower the number of boundary conditions actually required. If the wavelength is infinite, frequency equations are degenerated as product of two determinants pertaining to extensional vibrations and shear vibrations. In this case, it is seen that the nature of the surface does not have any influence over shear vibrations unlike in the case of extensional vibrations. For illustration purpose, three composite cylinders are considered and then discussed. Of the three, two are sandstone cylinders and the third one is resulted when a cylindrical bone is implanted with Titanium. In either case, phase velocity is computed against aspect ratios.

List of symbols:

(r, 𝜃, z)

cylindrical coordinate system

A, N, Q, R, A ^∗, N ^∗, Q ^∗, R ^∗

poroelastic constants

b

dissipation

d

phase velocity

e

solid dilatation

k, k ^∗

wave numbers

L+c

length of the composite cylinder

R ₀

ratio of length of cylinder-I to radius

R ₁

ratio of lengths of cylinders

r ₁

radius of the composite cylinder

s, s ^∗

fluid pressure

$\vec {u}$

solid displacement

$\vec {u}$

fluid displacement

$V_{1} ,V_{2} ,V_{3} ,V_{1}^{\ast },V_{2}^{\ast },V_{3}^{\ast }$

dilatational wave velocities of first and second type, shear wave velocity

ε

fluid dilatation

m

non-dimensional phase velocity

$\rho _{11} ,\rho _{12} ,\rho _{22} ,\rho _{11}^{\ast },\rho _{12}^{\ast },\rho _{22}^{\ast }$

mass coefficients

$\sigma _{ij} ,\sigma _{ij}^{\ast }$

stresses

$\phi {\kern 1pt}_{1} ,\phi {\kern 1pt}_{2} ,\vec {\psi }_{1} ,,\vec {\psi }_{2}$

potential functions

ω

frequency

∇²

Laplacian operator