Guided Waves in Structures for SHM: The Time - domain Spectral Element Method

Preface

This book is aimed at professionals whose scientific interests are directly associated with propagation of elastic waves in structural elements. This book may be useful not only for students of technical universities but also for researchers and engineers who solve practical problems involving propagation of elastic waves in structural elements made of isotropic materials or laminated composites.

Waves propagating in elastic media have been known for many centuries and have been the subject of scientific research of many scholars. Elastic waves result from stresses acting within the media and are associated with volume (compression and tension) and shape (shear) deformations. Better recognition and understanding of the complex phenomena behind the propagation of elastic waves in structural elements have promoted various novel and practical applications in many fields of technology. One such field is diagnostics of structural elements, where the use of elastic waves increases rapidly each year. Local methods employing elastic waves have been employed successfully for many years, but attempts to apply elastic waves in a global sense for diagnosing structural elements are still at an early stage of development. The measure of success in these attempts comes from various achievements made in parallel in several different fields. The first of them is the development of numerical simulation methods and tools aimed at modelling and analysing the phenomena associated with propagation of elastic waves in structural elements. The second, independent, one is the development of appropriate experimental methods and techniques allowing verification and validation results of numerical simulations. Recently these goals have become achievable in practice thanks to employing the most advanced measuring techniques based on three-dimensional (3D) laser scanning vibrometry.

This book is intended to report on the challenges associated with numerical simulation methods, analyses and experimental investigations related to the propagation of elastic waves in structural elements made of isotropic materials or composite laminates. For the first time the full spectrum of theoretical and practical issues associated with the propagation of elastic waves are presented and discussed in one study.

The first part of the book, devoted to various modelling and analysis issues associated with propagation of elastic waves, is focused on the Spectral Finite Element Method, which in the authors’ opinion is the most suitable modelling technique out of a variety of numerical methods used nowadays to solve wave propagation-related problems. This part of the book gives a broad overview of the existing state of the art and knowledge concerning modelling of elastic wave propagation in structural elements, while emphasising the problems associated with developing efficient numerical methods and tools and verifying them. Original solutions developed by the authors, suitable for constructing appropriate numerical models for simulating propagation of elastic waves in 1D, 2D and 3D structural elements made of isotropic and laminated composites are presented and discussed. Based on the developed spectral finite elements, a range of numerical tests has been carried out in order to verify the accuracy of the models, beginning from wave propagation in simple rods, beams, membranes and plates, and ending with shells or 3D structures.

The second part of the book, devoted to experimental measurements, presents the application of 3D laser scanning vibrometry for measuring, investigating and visualising the propagation of elastic waves in real-life structural elements. This part of the book naturally complements the theoretical and numerical investigations of the earlier part. Numerous scenarios and results of experimental measurements carried out on 1D, 2D and 3D structures are presented and discussed.

The last part of the book is concerned with various practical applications associated with wave propagation phenomena in structural elements. Problems of damage detection and location are discussed and investigated here. These problems are a part of a wide multidisciplinary research subject known as Structural Health Monitoring. Several damage detection methods developed or/and implemented by the authors and their practical applications in the context of Structural Health Monitoring are described in great detail, based on the results of either numerical or experimental investigations. The results of experimental studies included in this book make use of excitation and registration of elastic waves within structural elements using piezoelectric transducers. Additionally, and in parallel, independent registration of propagating elastic waves employs advanced 3D laser scanning vibrometry. These two techniques have been applied and investigated in order to qualitatively and quantitatively characterise the wave propagation phenomena.

The authors would like to underline the unique character of this book resulting from its complex and multidisciplinary character. Various acclaimed books dedicated to wave propagation phenomena in elastic media are usually theoretical in nature, while the question of appropriate verification of the developed numerical methods is addressed in a very limited manner. The authors of the studies mentioned often use analytical models of the wave propagation phenomena and/or apply different numerical methods based on either the finite element method or spectral methods in the frequency domain. The intention of the authors of this book is to present for the first time in one place new models of spectral finite elements defined in the time domain developed to facilitate analysis of propagation of elastic waves in structural elements. Originality of the material presented in this book comes from the attempt to connect together the results of both numerical and experimental investigations, as well as to indicate their practical implications. Until now, the original numerical models discussed in the book as well as the results of experimental studies using 3D laser scanning vibrometry have had no equivalents in published books dedicated to this field. Therefore the level of scientific research of this book, in the opinion of the authors, closely follows the latest trends in this area. It is worth noting that this book is accompanied by a demonstration version of software employing methods of analysing elastic wave propagation in structural elements using spectral finite elements. It should be emphasised that this software has been developed by the authors of this book.

Guided Waves in Structures for SHM: The Time-Domain Spectral Element Method is accompanied by a website (www.wiley.com/go/ostachowicz). The website contains and describes the EWavePro (Elastic Wave Propagation) software, which can be used for analysing phenomena of propagation of longitudinal, shear and flexural waves in two and three–dimensional thin–walled structures composed of isotropic materials or composite laminates. The abbreviation EWavePro is used here to distinguish the software developed by the authors. The software is developed in order to facilitate better understanding of elastic wave propagation phenomena and to be used as a tool in designing structural health monitoring systems based on changes in the elastic wave propagation patterns.

The authors want to thank their colleagues from the Department of Mechanics of Intelligent Structures: Dr P. Malinowski, Dr M. Radzienski and Dr T. Wandowski for assistance with writing Chapters 3 and 7, as well as Dr L. Murawski for involvement in writing the Appendix. Their efforts contributed to the development of the mentioned parts of this book cannot be overstated.

1

Introduction to the Theory of Elastic Waves

1.1 Elastic waves

Elastic waves are mechanical waves propagating in an elastic medium as an effect of forces associated with volume deformation (compression and extension) and shape deformation (shear) of medium elements. External bodies causing these deformations are called wave sources. Elastic wave propagation involves exciting the movement of medium particles increasingly distant from the wave source. The main factor differentiating elastic waves from any other ordered motion of medium particles is that for small disturbances (linear approximation) elastic wave propagation does not result in matter transport.

Depending on restrictions imposed on the elastic medium, wave propagation may vary in character. Bulk waves propagate in infinite media. Within the class of bulk waves one can distinguish longitudinal waves (compressional waves) and shear waves. A three-dimensional medium bounded by one surface allows for propagation of surface waves (Rayleigh waves and Love waves). Propagation of bulk waves and surface waves is used for describing seismic wave phenomena. Bounding the elastic medium with two equidistant surfaces causes compressional waves and shear waves to interact, which results in the generation of Lamb waves. One can say that a free boundary restricting an elastic body guides and drives waves; therefore the term guided waves is also used. Lamb waves and guided waves are used in broadly considered diagnostics and nondestructive testing. There are also waves that propagate on media boundary (interface waves) with names derived from their discoverers: in the interface between two solids Stoneley waves propagate, while in the one between a solid and a liquid Scholte waves propagate.

Figure 1.1 Distribution of displacements for the horizontal shear wave

1.1.1 Longitudinal Waves (Compressional/Pressure/ Primary/P Waves)

Longitudinal waves are characterised by particle motion alternately of compression and stretching character. The direction of medium point motion is parallel to the direction of wave propagation (i.e. longitudinal).

1.1.2 Shear Waves (Transverse/Secondary/S Waves)

Shear waves are characterised by transverse particle movements in alternating direction. The direction of medium particle motion is perpendicular to the propagation direction (transverse). The transverse particle movement can occur horizontally (horizontal shear wave, SH; see Figure 1.1) or vertically (vertical shear wave, SV; see Figure 1.2).

Figure 1.2 Distribution of displacements for the vertical shear wave

1.1.3 Rayleigh Waves

Rayleigh waves (Figure 1.3) are characterised by particle motion composed of elliptical movements in the xy vertical plane and of motion parallel to the direction of propagation (along the x axis). Wave amplitude decreases with depth y, starting from the wave crest. Rayleigh waves propagate along surfaces of elastic bodies of thickness many times exceeding the wave height. Sea waves are a natural example of Rayleigh waves.

Figure 1.3 Distribution of displacements for the Rayleigh wave

1.1.4 Love Waves

Love waves (Figure 1.4) are characterised by particle oscillations involving alternating transverse movements. The direction of medium particle oscillations is horizontal (in the xz plane) and perpendicular to the direction of propagation. As in the case of Rayleigh waves, wave amplitude decreases with depth.

Figure 1.4 Distribution of displacements for the Love wave

1.1.5 Lamb Waves

These waves were named after their discoverer, Horace Lamb, who developed the theory of their propagation in 1917 [1]. Curiously, Lamb was not able to physically generate the waves he discovered. This was achieved by Worlton [2], who also noticed their potential usefulness for damage detection. Lamb waves propagate in infinite media bounded by two surfaces and arise as a result of superposition of multiple reflections of longitudinal P waves and shear SV waves from the bounding surfaces. In the case of these waves medium particle oscillations are very complex in character. Depending on the distribution of displacements on the top and bottom bounding surface, two forms of Lamb waves appear: symmetric, denoted as S0, S1, S2, … , and antisymmetric, denoted as A0, A1, A2, … . It should be noted that numbers of these forms are infinite. Displacement fields of medium points for the fundamental symmetric mode S0 and fundamental antisymmetric mode A0 of Lamb waves are illustrated in Figures 1.5 and 1.6, respectively.

Figure 1.5 Distribution of displacements for the fundamental symmetric mode of Lamb waves

Figure 1.6 Distribution of displacements for the fundamental antisymmetric mode of Lamb waves

1.2 Basic Definitions

A specific case of waves as harmonic initial perturbations is considered here:

(1.1) Numbered Display Equation

The notions of wavenumber inline and wavelength inline are common for waves of every type. Wavenumber inline refers to the spatial frequency of perturbations. Wavelength inline refers to the spatial period of perturbations (Figure 1.7) and is expressed by the following formula:

(1.2) Numbered Display Equation

Figure 1.7 Harmonic wave of length inline

Solution of Equation (1.1) can be expressed in a general form as:

(1.3) Numbered Display Equation

where inline is wave amplitude and inline is angular velocity. The first term in square brackets is associated with wave propagation to the right (or forwards), while the second term is associated with wave propagation to the left (or backwards). Considering a wave propagating to the right, this can be written as:

(1.4) Numbered Display Equation

The phase of this wave is inline . For the constant phase inline it is inline . Thus, a point of constant phase moves with velocity:

(1.5) Numbered Display Equation

The harmonic wave propagating to the right with velocity inline is presented in Figure 1.8.

Figure 1.8 Harmonic wave propagating with velocity inline

The phase velocity of a wave describes the relationship between spatial frequency inline and temporal frequency inline of the propagating waves. The dependency inline is called the dispersion relationship. If this relation is linear, that is inline , the wave is nondispersive. In a nondispersive medium, the phase velocity is constant for all velocities.

Besides phase velocity, the term of group velocity is also associated with wave propagation. Group velocity refers to propagation of a group of waves called a wave packet. In order to understand the term of wave group velocity two waves propagating to the right, having the same amplitudes, but different frequencies and wavenumbers are considered:

(1.6) Numbered Display Equation

Application of universally known trigonometric identities for the sum of sinus functions leads to:

(1.7)

In formula (1.7) one can distinguish a term associated with modulation and one associated with a carrier wave:

(1.8) Numbered Display Equation

The wave packet is a superposition of a carrier wave and a modulating wave in the form of a window, as presented in Figure 1.9.

Figure 1.9 Wave packet as the superposition of a carrier wave and a modulating wave

The propagation velocity of a modulating wave defines the propagation velocity of a wave packet. For a constant phase inline , this is inline . Thus, group velocity at the limit transition inline , inline is defined as:

(1.9) Numbered Display Equation

One should note that for the nondispersive media the group velocity is equal to the phase velocity. In the dispersive media these velocities differ, which manifests directly as wave packet deformation during propagation. First of all, the wave packet amplitude decreases and the packet stretches.

1.3 Bulk Waves in Three-Dimensional Media

1.3.1 Isotropic Media

In infinite elastic medium waves propagate freely in every direction and are called bulk waves. The basis for discussing bulk waves is the three-dimensional theory of elasticity. The full set of equations is as follows:

(1.10) Numbered Display Equation

(1.11) Numbered Display Equation

(1.12) Numbered Display Equation

where inline (Einstein summation convention) and inline is the Kronecker delta. Equation (1.10) covers three motion equations, Equation (1.11) describes linear relationships between deformations and displacements (six independent equations) and Equation (1.12) covers six independent constitutive equations for the isotropic case. In Equation (1.12) Lame constants have been used; these are defined as:

(1.13) Numbered Display Equation

Equations (1.10) to (1.12) may be expanded using Cartesian notation. Thus, the motion equation can be written as:

(1.14) Numbered Display Equation

where inline is the mass density. The relationships between stress components are governed by symmetry, that is inline . The deformation–displacement equations take the following form:

(1.15) Numbered Display Equation

They are also subject to symmetry, that is inline . The constitutive Equation (1.10) in Cartesian notation is as follows:

(1.16)

Equations (1.14) and (1.15) remain valid for any continuous medium; the specific type of the discussed medium is introduced by Equations (1.16) – in this case it is isotropic. Elimination of stresses and deformations from Equations (1.14) to (1.16) leads to:

(1.17) Numbered Display Equation

Motion Equations (1.17) containing only particle displacements are displacement-type partial differential equations. These equations are also known as Navier equations and in Cartesian notation take the following form [3]:

(1.18)

If the area where the solution is sought is infinite, these equations are sufficient for describing elastic wave propagation. If the area is finite, on the other hand, boundary conditions are necessary for the problem to be well-posed. These boundary conditions take the form of imposed stresses and/or displacements at area boundaries.

1.3.2 Christoffel Equations for Anisotropic Media

Wave propagation in infinite anisotropic elastic solids is governed by the full set of equations of the three-dimensional theory of elasticity. Compared to the isotropic case, the difference lies in a more general constitutive equation. The full set of equations of the theory of elasticity for homogeneous anisotropic media is as follows:

(1.19) Numbered Display Equation

(1.20) Numbered Display Equation

(1.21) Numbered Display Equation

By combining Equations (1.19), (1.20) and (1.21) and ignoring external forces the motion equations are obtained:

(1.22) Numbered Display Equation

The tensor of elasticity constants inline is symmetric with regard to inline and inline , and therefore:

(1.23) Numbered Display Equation

A flat harmonic plane wave propagating forwards is assumed:

(1.24) Numbered Display Equation

where inline is the imaginary unit, inline is the wavenumber, inline is a vector of wave amplitudes and inline is angular frequency. Substitution of Equation (1.24) into Equation (1.22) leads to:

(1.25) Numbered Display Equation

It can be seen that inline ; therefore:

(1.26) Numbered Display Equation

This is the Christoffel equation for an anisotropic medium. The Christoffel tensor can be defined as:

(1.27) Numbered Display Equation

where inline are direction cosines normal to the wavefront. Furthermore, taking into account the relationships:

(1.28) Numbered Display Equation

leads to:

(1.29) Numbered Display Equation

By recalling the definition of phase velocity:

(1.30) Numbered Display Equation

Equation (1.29) is brought to the following form:

(1.31) Numbered Display Equation

This is a uniform system of three equations. The system has a nontrivial solution if the determinant of the coefficient matrix is equal to zero. This is a classic eigenvalue problem. The solution is composed of three velocities (eigenvalues with regard to inline ) and the corresponding eigenvectors. Depending on the arrangement of eigenvectors in space, one can be dealing with: a P wave together with SH and SV waves, a quasi-P wave together with SH and SV waves, a P wave together with quasi-SH and quasi-SV waves or a quasi-P wave together with quasi-SH and quasi-SV waves [4]. One should note that phase velocities depend on the direction of propagation, which results from the definition of the Christoffel tensor (Equation (1.27)). In an isotropic medium there are always pure waves: a longitudinal one and two shear ones, the phase velocities of which do not depend on the direction of propagation.

1.3.3 Potential Method

Bulk waves connected with wave propagation in an isotropic infinite media are considered in this section. When no external forces inline are present, Equation (1.18) can be expressed in vector form as:

(1.32) Numbered Display Equation

The motion Equation (1.32) can be simplified further by applying Helmholtz decomposition and the potential method [4–6]. Such an operation is only possible for isotropic media. It is assumed that the displacement vector inline can be expressed through two potential functions: the scalar potential inline and the vector potential inline , that is:

(1.33) Numbered Display Equation

Equation (1.33) is known as the Helmholtz solution complemented by the condition:

(1.34) Numbered Display Equation

By applying Equation (1.33), components of Equation (1.32) can be expressed as:

(1.35)

(1.36)

(1.37) Numbered Display Equation

By substituting Equations (1.35), (1.36) and (1.37) into Equation (1.32) the following formula is obtained:

(1.38)

Noting that inline (commutativity of differentiation), Equation (1.38) after transformations yields:

(1.39)

Equation (1.39) is satisfied for any point in space at any time, if the terms in parentheses vanish, that is:

(1.40) Numbered Display Equation

(1.41) Numbered Display Equation

After dividing by inline and ordering, Equations (1.40) and (1.41) become wave equations for the scalar potential inline and the vector potential inline , that is:

(1.42) Numbered Display Equation

(1.43) Numbered Display Equation

where inline is the longitudinal wave velocity, defined as:

(1.44) Numbered Display Equation

and inline is the shear wave velocity, defined as:

(1.45) Numbered Display Equation

As a result, the motion Equation (1.32) was decomposed into two simplified wave Equations (1.42) and (1.43). Assuming that the rotational part inline of Equation (1.33) is equal to zero, the longitudinal wave equation is obtained:

(1.46) Numbered Display Equation

Assuming that displacements in Equation (1.33) contain the rotational part only, the shear wave equation is obtained as:

(1.47) Numbered Display Equation

1.4 Plane Waves

A specific case of three-dimensional waves are plane waves. These waves are invariant in one direction along the wave crest. Such a situation happens when the wave crest is parallel to the z axis (cf. Figures 1.1 to 1.6). Moreover, the normal vector of the wave crest is perpendicular to the z axis. Invariance in the direction of the z axis means that all wave functions are independent of z, and therefore their derivatives with respect to z are equal to zero, that is:

(1.48) Numbered Display Equation

After substituting Equation (1.48) into Equation (1.33) and expanding, the expression for displacement is obtained:

(1.49)

Although movement is invariant with respect to the z axis, Equation (1.49) indicates that displacement components appear in all three directions (x, y and z). It is noteworthy that the displacement component inline depends only on potentials inline and inline that are associated with the horizontally polarised shear wave (SH wave). Displacement components