The Experts below are selected from a list of 204 Experts worldwide ranked by ideXlab platform
T.c.t. Ting - One of the best experts on this subject based on the ideXlab platform.
-
A new Secular Equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact
Wave Motion, 2013Co-Authors: T.c.t. TingAbstract:Abstract We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new Secular Equation is obtained that covers all special cases in one Equation. One special case is when a Rayleigh wave (called the R -wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S -wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.
-
The polarization vectors at the interface and the Secular Equation for Stoneley waves in monoclinic bimaterials
Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2005Co-Authors: T.c.t. TingAbstract:Stoneley waves propagating in the direction of the x 1 –axis in a bimaterial consisting of two half–spaces x 2 ⩾ 0 and x 2 ⩽ 0 of dissimilar anisotropic elastic materials are considered. Several invariants, independent of x 2 , that relate the displacement and the stress are obtained for a general anisotropic elastic bimaterial. We then study the case when both materials have the symmetry plane at x 2 = 0 (or x 1 = 0). The displacement and the traction at the interface x 2 = 0 describe two separate elliptic paths. They are polarized on planes that contain the x 2 –axis (or x 1 –axis). Moreover, the x 2 –axis (or x 1 –axis) is one of the principal axes of the ellipse. In the rest of the paper we consider special monoclinic bimaterials with the symmetry plane at x 2 = 0, x 1 = 0 or x 3 = 0. The elastic constants for the two monoclinic materials are identical except that those elastic constants that would vanish if the material were orthotropic have the opposite sign for the two materials. It is shown that one of the ellipses degenerates into a line along a coordinate axis while the other ellipse is on a coordinate plane normal to this coordinate axis. When the symmetry plane is at x 3 = 0, both ellipses degenerate into lines along the x 1 – and x 2 –axes. Explicit expressions of the polarization vectors and the Secular Equation are presented for all three monoclinic bimaterials.
-
the polarization vector and Secular Equation for surface waves in an anisotropic elastic half space
International Journal of Solids and Structures, 2004Co-Authors: T.c.t. TingAbstract:The displacement at the free surface of an anisotropic elastic half-space x2>0 generated by a surface wave propagating in the direction of the x1-axis traces an elliptic path. It is represented by the polarization vector aR=e1+ie2, where e1, e2 are the conjugate radii of the ellipse on the polarization plane. The displacement traces the ellipse in the direction from e1 to e2. We present explicit expressions of e1, e2 and the Secular Equation without computing the Stroh eigenvalues p and the associated eigenvectors a and b. After presenting the expressions for a general anisotropic elastic material, the special cases are studied separately. For monoclinic materials with the symmetry plane at x3=0, the Secular Equation and the conjugate radii e1, e2 are identical to that for orthotropic materials when s′16=0 but s′26 need not vanish. For monoclinic materials with the symmetry plane at x1=0, e1 is along the x1-axis while e2 is on the plane x1=0. If the symmetry plane is at x2=0, e1 is on the plane x2=0 while e2 is along the negative x2-axis. In both cases, e1, e2 are the principal radii of the ellipse. We also present the derivative of aR with respect to x2, the depth from the free surface, that provides information on (i) whether the conjugate radii of the ellipse increase as x2 increases and (ii) whether the polarization plane rotates as x2 increases. New Secular Equations are obtained for monoclinic materials with the symmetry plane at x1=0 or x2=0.
-
explicit Secular Equations for surface waves in monoclinic materials with the symmetry plane x1 0 x2 0 or x3 0
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: T.c.t. TingAbstract:The traditional way of deriving the Secular Equation for surface waves propagating in the direction of the x1axis in an anisotropic elastic halfspace x20 is to find a general steadystate solution f...
-
Explicit Secular Equations for surface waves in monoclinic materials with the symmetry plane x1 = 0, x2 = 0 or x3 = 0
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: T.c.t. TingAbstract:The traditional way of deriving the Secular Equation for surface waves propagating in the direction of the x1axis in an anisotropic elastic halfspace x20 is to find a general steadystate solution f...
Pham Chi Vinh - One of the best experts on this subject based on the ideXlab platform.
-
Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact
Meccanica, 2016Co-Authors: Pham Chi Vinh, Vu Thi Ngoc AnhAbstract:In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer and the half-space are both compressible and they are in spring contact with each other. The main aim of the paper is to derive explicit exact Secular Equation of the wave. This Equation has been derived by using the effective boundary condition method. From the obtained Secular Equation, the Secular Equations for the welded and sliding contacts can be derived immediately as special cases. For the welded contact, the obtained Secular Equation recovers the Secular Equations previously obtained for the isotropic and orthotropic materials. Since the obtained Secular Equation is totally explicit it is a good tool for nondestructively evaluating the adhesive bond between the layer and half-space as well as their mechanical properties.
-
on a technique for deriving the explicit Secular Equation of rayleigh waves in an orthotropic half space coated by an orthotropic layer
Waves in Random and Complex Media, 2016Co-Authors: Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh LinhAbstract:The Secular Equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this Secular Equation in both papers. This Secular Equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explic...
-
Rayleigh waves in orthotropic fluid-saturated porous media
Wave Motion, 2016Co-Authors: Pham Chi Vinh, Abdelkrim Aoudia, Pham Thi Ha GiangAbstract:Abstract In this paper, we are interested in the propagation of Rayleigh waves in orthotropic fluid-saturated porous media. This problem was investigated by Liu and Liu (2004). The authors have derived the Secular Equation of the wave but that Secular Equation is still in implicit form. The main aim of this paper is to derive explicit Secular Equation of the wave. By employing the method of polarization vector, the Secular Equations of Rayleigh waves in explicit form is obtained. This Equation recovers the dispersion Equation of Rayleigh waves propagating in pure orthotropic elastic half-spaces. Remarkably, the Secular Equation obtained is not a complex Equation as the one derived by Liu and Liu, it is a really real Equation.
-
AN APPROXIMATE Secular Equation OF RAYLEIGH WAVES IN AN ELASTIC HALF-SPACE COATED BY A THIN WEAKLY INHOMOGENEOUS ELASTIC LAYER
Vietnam Journal of Mechanics, 2015Co-Authors: Pham Chi Vinh, Vu Thi Ngoc AnhAbstract:In this paper, the propagation of Rayleigh waves in a homogeneous isotropic elastic half-space coated with a thin weakly inhomogeneous isotropic elastic layer is investigated. The material parameters of the layer is assumed to depend arbitrarily continuously on the thickness variable. The contact between the layer and the half space is perfectly bonded. The main purpose of the paper is to establish an approximate Secular Equation of the wave. By applying the effective boundary condition method an approximate Secular Equation of second order in terms of the dimensionless thickness of the layer is derived. It is shown that the obtained approximate Secular Equation has good accuracy.
-
Rayleigh waves in an incompressible orthotropic half-space coated by a thin elastic layer
Archives of Mechanics, 2014Co-Authors: Pham Chi Vinh, Nguyen Thi Khanh Linh, Vu Thi Ngoc AnhAbstract:The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer. The halfspace and the layer are both incompressible and they are in welded contact to each other. The main purpose of the paper is to establish an approximate Secular Equation of the wave. By using the effective boundary condition method an approximate Secular Equation of third-order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate Secular Equation has high accuracy. From it an approximate formula of third-order for the velocity of Rayleigh waves is obtained and it is a good approximation. The obtained approximate Secular Equation and formula for the velocity will be useful in practical applications.
Vu Thi Ngoc Anh - One of the best experts on this subject based on the ideXlab platform.
-
Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact
Meccanica, 2016Co-Authors: Pham Chi Vinh, Vu Thi Ngoc AnhAbstract:In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer and the half-space are both compressible and they are in spring contact with each other. The main aim of the paper is to derive explicit exact Secular Equation of the wave. This Equation has been derived by using the effective boundary condition method. From the obtained Secular Equation, the Secular Equations for the welded and sliding contacts can be derived immediately as special cases. For the welded contact, the obtained Secular Equation recovers the Secular Equations previously obtained for the isotropic and orthotropic materials. Since the obtained Secular Equation is totally explicit it is a good tool for nondestructively evaluating the adhesive bond between the layer and half-space as well as their mechanical properties.
-
on a technique for deriving the explicit Secular Equation of rayleigh waves in an orthotropic half space coated by an orthotropic layer
Waves in Random and Complex Media, 2016Co-Authors: Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh LinhAbstract:The Secular Equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this Secular Equation in both papers. This Secular Equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explic...
-
AN APPROXIMATE Secular Equation OF RAYLEIGH WAVES IN AN ELASTIC HALF-SPACE COATED BY A THIN WEAKLY INHOMOGENEOUS ELASTIC LAYER
Vietnam Journal of Mechanics, 2015Co-Authors: Pham Chi Vinh, Vu Thi Ngoc AnhAbstract:In this paper, the propagation of Rayleigh waves in a homogeneous isotropic elastic half-space coated with a thin weakly inhomogeneous isotropic elastic layer is investigated. The material parameters of the layer is assumed to depend arbitrarily continuously on the thickness variable. The contact between the layer and the half space is perfectly bonded. The main purpose of the paper is to establish an approximate Secular Equation of the wave. By applying the effective boundary condition method an approximate Secular Equation of second order in terms of the dimensionless thickness of the layer is derived. It is shown that the obtained approximate Secular Equation has good accuracy.
-
Rayleigh waves in an incompressible orthotropic half-space coated by a thin elastic layer
Archives of Mechanics, 2014Co-Authors: Pham Chi Vinh, Nguyen Thi Khanh Linh, Vu Thi Ngoc AnhAbstract:The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer. The halfspace and the layer are both incompressible and they are in welded contact to each other. The main purpose of the paper is to establish an approximate Secular Equation of the wave. By using the effective boundary condition method an approximate Secular Equation of third-order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate Secular Equation has high accuracy. From it an approximate formula of third-order for the velocity of Rayleigh waves is obtained and it is a good approximation. The obtained approximate Secular Equation and formula for the velocity will be useful in practical applications.
-
Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact
Wave Motion, 2014Co-Authors: Pham Chi Vinh, Vu Thi Ngoc Anh, Vu Phuong ThanhAbstract:Abstract In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate Secular Equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate Secular Equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the Secular Equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate Secular Equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.
Nguyen Thi Khanh Linh - One of the best experts on this subject based on the ideXlab platform.
-
on a technique for deriving the explicit Secular Equation of rayleigh waves in an orthotropic half space coated by an orthotropic layer
Waves in Random and Complex Media, 2016Co-Authors: Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh LinhAbstract:The Secular Equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this Secular Equation in both papers. This Secular Equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explic...
-
Rayleigh waves in an incompressible orthotropic half-space coated by a thin elastic layer
Archives of Mechanics, 2014Co-Authors: Pham Chi Vinh, Nguyen Thi Khanh Linh, Vu Thi Ngoc AnhAbstract:The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer. The halfspace and the layer are both incompressible and they are in welded contact to each other. The main purpose of the paper is to establish an approximate Secular Equation of the wave. By using the effective boundary condition method an approximate Secular Equation of third-order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate Secular Equation has high accuracy. From it an approximate formula of third-order for the velocity of Rayleigh waves is obtained and it is a good approximation. The obtained approximate Secular Equation and formula for the velocity will be useful in practical applications.
-
an approximate Secular Equation of generalized rayleigh waves in pre stressed compressible elastic solids
International Journal of Non-linear Mechanics, 2013Co-Authors: Pham Chi Vinh, Nguyen Thi Khanh LinhAbstract: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
Meccanica, 2013Co-Authors: Pham Chi Vinh, Nguyen Thi Khanh LinhAbstract: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.
-
an approximate Secular Equation of rayleigh waves propagating in an orthotropic elastic half space coated by a thin orthotropic elastic layer
Wave Motion, 2012Co-Authors: Pham Chi Vinh, Nguyen Thi Khanh LinhAbstract:Abstract The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer and the main purpose of the paper is to establish an approximate Secular Equation of the wave. By using the effective boundary condition method an approximate Secular Equation of third-order in terms of the dimensionless thickness of the layer has been derived. From this Equation two different third-order approximate Secular Equations are obtained for the case when the half-space and the layer are both isotropic, one of which recovers the Secular Equation of second-order derived by Bovik [P. Bovik, A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers, J. Appl. Mech. 63 (1996) 162–167]. An explicit second-order approximate formula for the Rayleigh wave velocity has been created based on the obtained approximate Secular Equation. Since explicit dispersion relations are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data, the obtained Secular Equation and formula for the velocity may be useful in practical applications.
Frank Szmulowicz - One of the best experts on this subject based on the ideXlab platform.
-
a novel Secular Equation for the coupled band envelope function approximation for superlattices and application to inas inxga1 xsb superlattices
Superlattices and Microstructures, 1997Co-Authors: Frank SzmulowiczAbstract:Abstract This paper derives and demonstrates a new Secular Equation for the coupled-band envelope-function approximation (EFA) formalism for superlattices in order to overcome the difficulty of handling rapidly growing or decaying wavefunction components, in particular, the ‘wing solutions’. In a second development, the generally nonHermitian Secular Equation is made Hermitian, making it easier to locate multiply degenerate roots. In the process, the simple Kronig–Penney model is recast into a form closely related to that for the underlying quantum well (QW) problem. The present method is applicable to Burt's EFA formalism. An InAs/InGaSb type-II superlattice is used to demonstrate the method.
-
A novel Secular Equation for the coupled-band envelope-function approximation for superlattices and application to InAs/InxGa1 − xSb superlattices
Superlattices and Microstructures, 1997Co-Authors: Frank SzmulowiczAbstract:Abstract This paper derives and demonstrates a new Secular Equation for the coupled-band envelope-function approximation (EFA) formalism for superlattices in order to overcome the difficulty of handling rapidly growing or decaying wavefunction components, in particular, the ‘wing solutions’. In a second development, the generally nonHermitian Secular Equation is made Hermitian, making it easier to locate multiply degenerate roots. In the process, the simple Kronig–Penney model is recast into a form closely related to that for the underlying quantum well (QW) problem. The present method is applicable to Burt's EFA formalism. An InAs/InGaSb type-II superlattice is used to demonstrate the method.
-
Numerically stable Hermitian Secular Equation for the envelope-function approximation for superlattices.
Physical review. B Condensed matter, 1996Co-Authors: Frank SzmulowiczAbstract:A method is developed for implementing the coupled-band envelope-function-approximation (EFA) formalism for the calculation of the electronic structure of superlattices. The approach overcomes the difficulties in handling exponentially growing and decaying wave-function components, in particular, the so-called wing solutions, as is the case with existing Secular Equations. As importantly, the Secular Equation, which, in general, is general complex, is recast into a Hermitian form, which makes it easy to separate degenerate eigensolutions of the superlattice problem. Therefore, it is not necessary to first find a unitary transformation to eliminate the Kramers degeneracy of the starting k\ensuremath{\cdot}p Hamiltonian. In the case of the simple Kronig-Penney model, the present formalism recasts the characteristic Equation into a form that directly exhibits its parentage to the underlying quantum-well problem. The present method can be used in conjunction with Burt's EFA formalism in the form of a coupled differential Equation with piecewise-constant coefficients. The method is demonstrated on the example of the technologically important semiconducting InAs/${\mathrm{In}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Sb type-II superlattice. \textcopyright{} 1996 The American Physical Society.
