Surface waves propagation in heterogeneous media

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Theoretical basis of considered models

In this chapter the models of ultrasonic waves propagation phenomena used for the developed concrete diagnostic methods are presented. Analysis of surface waves (SW) propagation in inhomogeneous medium and description of a reflection phenomena are mainly considered.
The first part of the chapter presents the theoretical basis of two models of macro-inhomogeneous media: the Haskell-Thomson model, in which a layered structure of medium is assumed (each layer is homogeneous, and heterogeneity is caused by differences between the layers), and the Gibson’s model where shear modulus change linearly with depth. For each model a sensitivity analysis is made. The approach of taking into account the attenuation as an independent source of dispersion (exception material heterogeneity) is presented.
In the second part of the chapter the theoretical aspects of a reflection coefficient are presented. Waves propagated in the air and reflected from a porous media are considered and modified Biot’s model without skeleton deformation is used. For that model also a sensitivity analysis is proceeded.

Surface waves propagation in heterogeneous media

Surface waves are the disturbances propagating along the surface of en elastic medium, when the surface bordered on another elastic medium or a vacuum. The disturbance consists in elliptical movement of the particles in a plane perpendicular to the surface of the material and parallel to the direction of propagation (Fig. II.1). The amplitude of the movement quickly decreases with depth. These types of waves propagating in a homogeneous elastic media are free form dispersion and are named Rayleigh waves (form the name of the author’s of its mathematical description) [Rayleigh 1885]. Rayleigh wave propagation velocity in homogeneous media cR is the root of the equation
The wave propagating along the surface of material being in contact with another medium generates so-called leaky waves [Victorov 1967] where direction of the emission is inclined also with angleR from the normal. Observation and acquisition of leaky waves is possible in non-contact manner and what is, it important gives us indirect information about the surface waves.
Each wave could be characterized by a phase V and group VGR velocities. The phase velocity of a monochromatic plane wave might be defined with as an angular frequency and k as a wave number. Considering not monochromatic wave as a superposition of monochromatic waves with angular frequencies close to the central frequency the group velocity can by introduced as [Mavko et al 2003]
In homogeneous half space without attenuation the Rayleigh waves of different frequencies (different wavelengths) propagate with the same velocity (V(f)=const.) In that case the phase and group velocities are equal (V=VGR=c).
Modeling of the wave propagation phenomena in heterogeneous medium is developed since the 50’ of XX century, mainly in geophysics. Usually, it is used to describe a propagation of seismic waves in rocks and soils for natural resources survey. The most common approach is the assumption of layered structure (eg Haskell model), where the heterogeneity arises from different mechanical parameters of the layers.
The model based on the concept of Gibson’s elastic half-space (in short Gibson model) is an another model used for generating the theoretical dispersion characteristics of surface waves in a heterogeneous media (degraded material). The linear relation of Shear modulus versus depth is assumed.

SW in layered inhomogeneous medium (Haskell model)

The inhomogeneous half-space in which material properties abruptly change with depth in z direction is considered. The surface waves propagating such medium are called pseudo-Rayleigh waves and they are subject to dispersion, ie wave velocity depends on frequency. This can be described as a structural dispersion which originates from waves of different lengths penetrate to different depths and related to that spread at different velocities (Fig. II.2a). In spectral domain it could be represented by a dispersion characteristic (Fig. II.2b), ie. the phase velocity characteristic in a function of frequency (or some times versus wavelength). In abrupt heterogeneity medium few modes can be generated which means existing of waves of different velocities at the same frequency. For a frequency tends to zero the first mode velocity (also referred to by the terms basic or fundamental mode) tends to the SW velocity in the half-space.
For frequency tending to infinity, the velocities of all modes tend to the velocity of surface waves in half-space with properties of layer near the surface (z=0).
There are few possibilities of mathematical description for SW propagation in layered medium. Their comparison is presented inter alia in the paper [Lowe 1995]. Most of models were developed for geophysical applications and for the structural and mechanical parameters of soil. In that case thick layers and the waves of low velocities and low frequencies are considered. The conditions need for modeling of wave’s propagation in depredated concrete diagnostic are considerably different and the thin layers and high frequencies are used. After bibliographic analysis the numerical implementation of Thomson-Haskell model (named Haskell model) was chosen as a model of pseudo-Rayleigh wave’s propagation in a layered heterogeneous concrete. This model is also named shortly Haskell model or Transfer matrix model.
The model firstly developed by Thomson in 1950 and corrected by Haskell in 1953 [Haskell 1953] concern the SW’s propagation in stratified heterogeneous material and the following assumptions must be fulfilled:
• the pseudo-Rayleigh waves (RW) are considered (shortly Rayleigh waves) with pulsation p and phase velocity V, and with the amplitude decreasing exponentially with depth z,
• there are n layers in considered system, and the last one is a half-space (layer of infinity depth),
• the stress and displacements are continues on the interfaces,
• there are no stresses across the free surface and there are no sources at infinity,
• each layer is isotropic and homogeneous solid,
• the particles displacements are in x-z plain.
Schema of considered structure is presented on the Figure II.3 and the properties of each layer are submitted in the Table II-1.
Numerical solution of the dispersion equation
Obtaining the dispersion characteristics from Haskell model requires development of an algorithm which searches the roots of the dispersion equation in a numerical way. For this purpose, the Equation II
Finding a solution of the Equation II.30 is realized on the velocity-frequency plane with fixed other models parameters. Elements of J matrix are depended on the system parameters and their complexity grows with the number of layers taken in the model, what could be well illustrated by the Equation II.27. Each layer is characterized by four parameters m,m,m, dm (see Table II-1) and the last layer is the infinity depth (half space). Thus,having n layers in the model 4n-1 parameters must be placed. That is why the number of layers has significant influence on the problem of complexity and on computation time. The solution of dispersion equation is to find a set of such pairs of variables V and f, for which the relation II.30 is satisfied. The roots of the Equation II.30 can be found by many different ways. One of it is presented in the paper [Lowe 1995], where the author proposes V(f) plane scanning with simultaneously changing both velocity and frequency. In this study a little bit different approach of solving dispersion equation is taken. The details of this are presented below.
In the first stage of the discussed problem the course of JJ function versus frequency and velocity was analysed. For the ultrasonic concrete diagnostic the SW frequencies from several dozen to several hundred kHz and the velocities in range of 1500m/s to 3000m/s are used. These facts are taken into account for the future consideration. The real part JJRe of the function JJ versus V and f is shown in Figure II.4a. The analysis of imaginary part JJIm of the function JJ showed that JJIm takes zero value in considered domain (JJIm=0, Fig II.4.b) and therefore its consideration may be omitted [Lowe 1995].

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Table of contents :

Chapter I Characteristics of the chosen problem, objectives and the range of work
Chapter II Theoretical basis of considered models
II.1 Surface waves propagation in heterogeneous media
II.1.1 SW in layered inhomogeneous medium (Haskell model)
II.1.2 Numerical solution of the dispersion equation
II.1.3 Study of the sensitivity of the Haskell model
II.1.4 Surface waves in the medium of a linear changes of a shear modulus (Gibson model) 41
II.1.5 Study of the sensitivity of the Gibson model
II.1.6 Comparison of the Gibson’s model with the Haskell’s model
II.1.7 Attenuation modelling
II.2 Modeling of the ultrasonic waves reflection from an interface of porous solid in the air
II.2.1 Description of the model with viscosity inclusion
II.2.2 Study of the model’s sensitivity
II.3 Conclusions
Chapter III Description and tests of the measurement devices
III.1 Surface waves propagation measurement system
III.1.1 Surface waves measurement device SWMD
III.1.2 Measured values and signal processing
III.1.3 Test with the reference material
III.2 Reflectometric measurement system
III.2.1 Reflectometric Measuring Device (RMD)
III.2.2 Measured values and signal processing
III.1.3 Test with the reference material
III.1 Conclusions
Chapter IV Identification of mechanical and structural parameters of concrete
IV.1 Brief review of the optimization methods
IV.2 Comparison of selected methods of optimization
IV.3 Identification of the medium parameters using the Haskell’s model – tests with synthetic data
IV.4 Identification of the medium parameters using the Gibson’s model – tests with synthetic data
IV.4.1 Studies of the error function sensitivity
IV.4.2 Tests of the program
IV.5 Procedure of the depth of degradation estimations based on the Gibson’s model
IV.5.1 Concept of the algorithm
IV.5.2 DDS program verification for the synthetic data
IV.6 Point Cloud program
IV.6.1 Concept of the algorithm
IV.6.2 Program verification for the synthetic data
IV.7 Identification of the structural parameters on the basis of reflection coefficient characteristic
IV.7.1 Concept of the algorithm
IV.7.2 Program verification for the synthetic data
IV.8 Conclusions
Chapter V Results and Discussion
III.1 Description of the tested materials
V.2 Test results for the concrete beams
V.3 Results of identification based on the surface wave propagation measurements
V.1 Results of identification based on the characteristics of reflection coefficient
V.5 Discussion about the experiences in use of the measurement systems
V.6 Conclusions
Chapter VI Summary and final conclusions


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