Showing papers by "Pham Chi Vinh published in 2013"

An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids

Abstract: The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic half-space coated with a thin pre-stressed elastic layer. The half-space and the layer are assumed to be compressible and in welded contact with each other. By using the effective boundary condition method, an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function. When the pre-strains are absent, the secular equation obtained coincides with the one for the isotropic case. Numerical investigation shows that the approximate secular equation obtained is a good approximation. Since explicit dispersion relations are employed as theoretical bases for extracting pre-stresses from experimental data, the secular equation obtained will be useful in practical applications.

Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity

Abstract: This paper is concerned with the propagation of Rayleigh waves in an incompressible isotropic elastic half-space overlaid with a layer of non-viscous incompressible water under the effect of gravity. The authors have derived the exact secular equation of the wave which did not appear in the literature. Based on it the existence of Rayleigh waves is considered. It is shown that a Rayleigh wave can be possible or not, and when a Rayleigh wave exists it is not necessary unique. From the exact secular equation the authors arrive immediately at the first-order approximate secular equation derived by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. When the layer is assumed to be thin, a fourth-order approximate secular equation is derived and of which the first-order approximate secular equation obtained by Bromwich is a special case. Some approximate formulas for the velocity of Rayleigh waves are established. In particular, when the layer being thin and the effect of gravity being small, a second-order approximate formula for the velocity is created which recovers the first-order approximate formula obtained by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. For the case of thin layer, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate second-order polynomials of the third- and fourth-powers.

Homogenization of very rough interfaces separating two piezoelectric solids

Abstract: The main aim of this paper is to derive homogenized equations in explicit form of the linear piezoelectricity in two-dimensional domains separated by an interface which highly oscillates between two parallel straight lines. First, the basic equations of the linear theory of piezoelectricity are written down in matrix form. Then, following the techniques presented recently by these authors, the explicit homogenized equation and the associate continuity condition, for generally anisotropic piezoelectric materials, are derived. They are then written down in component form for some specific cases. Since the obtained equations are totally explicit, they are significant in practical applications.

Explicit homogenized equations of the piezoelectricity theory in a two-dimensional domain with a very rough interface of comb-type

Abstract: The main purpose of this paper is to derive explicit homogenized equations of the linear piezoelectricity in two-dimensional domains separated by a very rough interface of comb-type. In order to do that, first, the basic equations of the theory of piezoelectricity are written down in matrix form. Then, following the techniques presented recently by these authors, the explicit homogenized equation in matrix form and the associate continuity condition, for generally anisotropic piezoelectric materials, are derived. They are then written down in component form for a special case when the solids are made of tetragonal crystals of class 4̄2m. Since the obtained equations are totally explicit, they are significant in use.