Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)


 The spectral ratio between horizontal and vertical components (H/V ratio) of microtremors measured at the ground surface has been used to estimate fundamental periods and amplification factors of a site, although this technique lacks theoretical background. The aim of this article is to formulate the H/V technique in terms of the characteristics of Rayleigh and Love waves, and to contribute to improve the technique. The improvement includes use of not only peaks but also troughs in the H/V ratio for reliable estimation of the period and use of a newly proposed smoothing function for better estimation of the amplification factor. The formulation leads to a simple formula for the amplification factor expressed with the H/V ratio. With microtremor data measured at 546 junior high schools in 23 wards of Tokyo, the improved technique is applied to mapping site periods and amplification factors in the area.

Ground Motion Characteristics Estimated from Spectral Ratio between Horizontal and Verticcl Components of Mietremors.

The English-language dictionary defines wrinkles as "small furrows, ridges, or creases on a normally smooth surface, caused by crumpling, folding, or shrinking". In this paper we review the scientific aspects of wrinkling and the related phenomenon of buckling. Specifically, we discuss how and why wrinkles/buckles form in various materials. We also describe several examples from everyday life, which demonstrate that wrinkling or buckling is indeed a commonplace phenomenon that spans a multitude of length scales. We will emphasize that wrinkling is not always a frustrating feature (e.g., wrinkles in human skin), as it can help to assemble new structures, understand important physical phenomena, and even assist in characterizing chief material properties.

Soft matter with hard skin : From skin wrinkles to templating and material characterization

A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity μ, Ā, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.

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On the third- and fourth-order constants of incompressible isotropic elasticity.

The present paper deals with the problem on Lamb waves propagation in periodically layered, compressible elastic media with initial deformations, in the case of long wavelength approximation (i.e. E = k.h < < 1, where k is the wave number , h is the thickness of one periodicity cell) . With the assumption that E < < 1, the dispersion equation is written as: 2 = ~2 = [h + c!\"h + c D3 + ... = L cmDm+l· m=O The main aim of this paper is to find formulae for determining the coefficients Di, ( i 2'. 1). In particular, we prove that D2n = 0 (n 2'. 1), derive formulae for D1 , D3, and construct recurrent formulae for D2n+i (n 2'. 2). Based on these formulae , the solution with any order of accuracy can be obtained. This research is an extension of the investigation by Norris and Santosa [Norris A. and Santosa F. , Wave Motion 16 (1992) , 33-55] from SH waves (one-component waves) to Lamb waves (two-component waves). Keyword. Lamb waves, Wave propagation, Periodically layered media, Asymptotic expansion.

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Asymptotic expansion of the dispersion equation of lamb waves in periodically layered elastic media

In this paper, we assess the reliability of explicit formulas introduced by Pham et al. [Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of...

Reliability assessment and data inversion using a surrogate model of wave propagation in functionally graded materials

The main purpose of this paper is to derive the explicit secular equation and formula of H/V ratio of Rayleigh waves propagating in an orthotropic micropolar elastic half-space. The polarization vector method is used so that the characteristic equation of Rayleigh waves need not to be solved. The obtained secular equation recovers the secular equations of Rayleigh waves propagating in isotropic micropolar elastic and orthotropic (purely) elastic half-spaces. Furthermore, for isotropic micropolar elastic materials of which the micropolar couple modulus is much smaller than the shear elastic modulus, explicit formulas for the velocity and the H/V ratio of Rayleigh waves are derived by utilizing the complex function method. It is shown numerically that the micropolar parameters affect Rayleigh waves weakly in the range of low frequencies (long wavelengths) but considerably in the range of high frequencies. The numerical analysis also indicates that the obtained formulas for the velocity and the H/V ratio are good tools for nondestructively evaluating the micropolar couple modulus using Rayleigh waves of short wavelengths, and if the data of phase velocity and H/V ratio are measured with the same accuracy in practice the H/V ratio formula could be better than phase velocity in inverse problem. 

Explicit secular equation and H/V ratio formula of Rayleigh waves in orthotropic micropolar elastic half-spaces

This paper is concerned with the propagation of Rayleigh waves in an incompressible orthotropic elastic half-space coated with a thin incompressible orthotropic elastic layer. The main purpose of the paper is to establish an approximate formula for the Rayleigh wave H/V ratio (the ratio between the amplitudes of the horizontal and vertical displacements of Rayleigh waves at the traction-free surface of the layer). First, the relations between the traction amplitude vector and the displacement amplitude vector of Rayleigh waves at two sides of the interface between the layer and the half-space are created using the Stroh formalism and the effective boundary condition method. Then, an approximate formula for the Rayleigh wave H/V ratio of third-order in terms of dimensionless thickness of the layer has been derived by using these relations along with the Taylor expansion of the displacement amplitude vector of the thin layer at its traction-free surface. It is shown numerically that the obtained formula is a good approximate one. It can be used for extracting mechanical properties of thin films from measured values of the  Rayleigh wave H/V ratio.

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Approximate formula for the H/V ratio of Rayleigh waves in incompressible orthotropic half-spaces coated by a thin elastic layer

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.

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Pham Chi Vinh

Papers

Asymptotic expansion of the dispersion equation of lamb waves in periodically layered elastic media

Reliability assessment and data inversion using a surrogate model of wave propagation in functionally graded materials

Explicit secular equation and H/V ratio formula of Rayleigh waves in orthotropic micropolar elastic half-spaces

Approximate formula for the H/V ratio of Rayleigh waves in incompressible orthotropic half-spaces coated by a thin elastic layer

An approximate secular equation of rayleigh waves in an elastic half-space coated by a thin weakly inhomogeneous elastic layer